detached eddy simulation of unsteady turbulent flows in the draft tube of a bulb turbine

DETACHED EDDY SIMULATION OF UNSTEADY TURBULENT FLOWS IN THE DRAFT TUBE OF A BULB TURBINE Thèse

ARASH TAHERI

Doctorat en génie mécanique Philosophiae Doctor (Ph.D.)

Québec, Canada © Arash Taheri, 2015

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Résumé Les aspirateurs de turbines hydrauliques jouent un rôle crucial dans l’extraction de l’énergie disponible. Dans ce projet, les écoulements dans l’aspirateur d’une turbine de basse chute ont été simulés à l'aide de différents modèles de turbulence dont le modèle DDES, un hybride LES/RANS, qui permet de résoudre une partie du spectre turbulent. Déterminer des conditions aux limites pour ce modèle à l’entrée de l’aspirateur est un défi. Des profils d’entrée 1D axisymétriques et 2D instationnaires tenant compte des sillages et vortex induits par les aubes de la roue ont notamment été testés. Une fluctuation artificielle a également été imposée, afin d’imiter la turbulence qui existe juste après la roue. Les simulations ont été effectuées pour deux configurations d’aspirateur du projet BulbT. Pour la deuxième, plusieurs comparaisons avec des données expérimentales ont été faites pour deux conditions d'opération, à charge partielle et dans la zone de baisse rapide du rendement après le point de meilleur rendement. Cela a permis d’évaluer l'efficacité et les lacunes de la modélisation turbulente et des conditions limites à travers leurs effets sur les quantités globales et locales. Les résultats ont montrés que les structures tourbillonnaires et sillages sortant de la roue sont adéquatement résolus par les simulations DDES de l’aspirateur, en appliquant les profils instationnaires bidimensionnels et un schéma de faible dissipation pour le terme convectif. En outre, les effets de la turbulence artificielle à l'entrée de l’aspirateur ont été explorés à l'aide de l’estimation de l’intermittence du décollement, de corrélations en deux points, du spectre d'énergie et du concept de structures cohérentes lagrangiennes. Ces analyses ont montré que les détails de la dynamique de l'écoulement et de la séparation sont modifiés, ainsi que les patrons des lignes de transport à divers endroits de l’aspirateur. Cependant, les quantités globales comme le coefficient de récupération de l’aspirateur ne sont pas influencées par ces spécificités locales.

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Abstract Draft tubes play a crucial role in elevating the available energy extraction of hydroturbines. In this project, turbulent flows in the draft tube of a low-head bulb turbine were simulated using, among others, an advance hybrid LES/RANS turbulent model, called DDES, which can resolve portions of the turbulent spectrum. Providing appropriate inflow boundary conditions for such models is a challenging issue. In this regard, different inflow boundary conditions were tested, including axisymmetric 1D profiles, and unsteady 2D inflow profiles that take runner blade wakes and vortices into account. Artificial fluctuation at the inlet section of the draft tube was also included to mimic the turbulence existing after the runner. Simulations were conducted for two draft tube configurations of the BulbT project. For one of them, intensive comparisons with experimental data were done for two operating conditions, one at part load and another in the sharp drop-off portion of the efficiency hill after the best efficiency point. This allowed to assess the effectiveness and shortcomings of the adopted turbulence modeling and boundary conditions through their effects on the global and local quantities. The results showed that the runner-related vortical structures and wakes are appropriately resolved using stand-alone DDES simulation of the draft tube flows. This is achieved by applying unsteady 2D inflow profiles along with adopting low dissipation scheme for the convective term. Furthermore, the effects of applying artificial turbulence at inlet were explored using separation intermittency, two-point correlation, energy spectrum and Lagrangian coherent structure concepts. These analyses revealed that the type of inflow boundary conditions modifies the details of the flow and separation dynamics as well as patterns of the transport barriers in different regions of the draft tube. However, the global quantities such as recovery coefficient are not influenced by these local features.

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Table of Contents Résumé ……………………………………………………………………………………..iii Abstract ……………………………………………………………………………………..v Table of contents ………………………………………………………………………..…vii List of tables ………………………………………………………………………………xiii List of figures……………………………………………………………………………....xv Nomenclature ……...…………………………………………………………………….xxxi Dedication……………………………………………………………………………….....xli Acknowledgments………………………………………………………………………..xliii

Chapter 1. Introduction ……………………………………………………1 1.1 Preface ………………………………………………………………………......1 1.2 Overview of the hydro-power plants.……………………………………………1 1.2.1 Conventional hydro-power plants..……………………………………3 1.2.2 Different types of the hydro-turbines………………………………….4 1.2.3 Hydro-turbine and draft tube performance ...………………………….7 1.3 BulbT geometry and the defined coordinate system …………………………..10 1.4 Motivation……………………………………………………………………...12 1.5 Statement of the object ………………………………………………………...14 1.6 Thesis outline ….……………………………………………………………….16

Chapter 2. Turbulence treatment, state of the art ……………………….17 2.1 Introduction …………………………………………………………………....17 2.2 Turbulence in wall-bounded flows ………………………………………….....18 2.3 Turbulence treatment methods ………………………………………………...24 2.3.1 Direct numerical simulation (DNS) approach ……………….…........25 2.3.2 Large eddy simulation (LES) approach ……………………………..26 2.3.3 Unsteady Reynolds Averaged Navier-Stokes (URANS) approach…..28

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2.3.4 Hybrid RANS/LES approach ……………………………………...31 2.4 Detached eddy simulation (DES) approach …………………………………...33 2.4.1 DES97 in the external and internal flow applications…………..…..35 2.4.2 Grid-induced separation problem………………………....………...35 2.4.3 DDES in the external flow applications………..…………………...38 2.4.4 DDES in the internal flow applications……………………..............38 2.5 Summary ………………………………………………………………………41

Chapter 3. Computational methodology …………………………………43 3.1 Introduction ……………………………………. …………………...................43 3.2 Governing equations of fluid flow motions …………………………………...43 3.3 Finite volume method (FVM) …………………………………………………46 3.3.1 Discretization of the Laplacian (diffusive) term ………….................47 3.3.2 Discretization of the convection term ………………….....................48 3.3.3 Discretization of the source term …………………………................50 3.3.4 Discretization of the unsteady term …………………….....................50 3.3.5 Pressure-velocity coupling …………...………………………….…..51 3.3.6 Uncertainty in hydro-turbine turbulent flow simulations …................53 3.4 OpenFOAM CFD platform …………………………………………................56 3.5 Parallel processing & HPC ………………………………………….................57 3.5.1 Speed-up test ………………………………………………………...59 3.6 Summary ……………………………………………………………................61

Chapter 4. Inflow condition for draft tube flow simulations….................63 4.1 Introduction ……………………………………………………………………63 4.2 RANS versus LES inflow conditions …………………………...……………..64 4.3 Inflow condition for draft tube-only flow simulation ………………................65 4.3.1 Normalization reference scales ……………………………...............68 4.3.2 Circumferential averaged-1D inlet profile ………………..................68 4.3.3 Unsteady 2D inflow profile………………………………..................69

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4.3.4 Unsteady 2D-rotating profile ………………………………………..70 4.3.5 Inflow turbulent viscosity treatment ………………………………...74 4.4 Synthetic inflow turbulence: Artificial Fluctuation Generation (AFG) techniques …………………………………………………………………77 4.4.1 Review of AFG methods.…………………………………………….79 4.4.1 (a) Physical space-based AFG methods …………..………....79 4.4.1 (b) Mapped space-based AFG methods ………..…................83 4.4.1 (c) Mixed AFG methods…………………………..................86 4.4.2 Anisotropic inflow turbulence ……………………………………….94 4.5 Summary ……………………………………………………………….........108

Chapter 5. Inflow condition considerations on the Clausen conical diffuser flow simulations……………………………..………111 5.1 Introduction ………………………………………………………………......111 5.2 Experimental set-up ………………………………………………………......111 5.3 Computational domains ………………………………………………………112 5.4 Numerical simulations of the extended-case …………………………………114 5.4.1 Computational grid specifications ………………………………….114 5.4.2 Numerical setups of the extended-case simulations ……………......114 5.4.3 Inflow and boundary conditions ……………………………………115 5.4.4 Extended-case simulation results ……………………………….......117 5.5 Numerical simulations of the base-case ………………………………….......120 5.5.1 Computational grid specifications ………………………………….121 5.5.2 Numerical setups of the base-case simulations …………………….121 5.5.3 Inflow and boundary conditions ……………………………………121 5.5.4 1D inflow profiles of mean velocity and turbulent quantities………124 5.5.5 Effect of extended-case turbulence model on the base-case DDES simulations ……………………………………………….....127 5.5.6 Effect of inflow radial velocity on the base-case simulations ……...132 5.5.7 Effect of inflow profile near-wall treatment on DDES simulations...135 5.5.8 Tuning of inflow near-wall turbulent viscosity……………………..138 5.6 Summary …………………………………………………………………......141 ix

Chapter 6. BulbT draft tube flow simulations: basic geometry …..........143 6.1 Introduction …………………………………………………………..............143 6.2 Basic draft tube geometry and selected operating point ………..…….............143 6.3 Computational grid specifications ……………………………………………145 6.4 Numerical setup for turbulent flow simulations …………….………………..146 6.5 Basic geometry RANS simulation …….……...……..………………………..147 6.6 Basic geometry DDES simulations …………………………………………..149 6.6.1 Effect of wall-zone turbulent viscosity at inflow section..………….150 6.6.2 Effect of the discretization of the convective term ….......................157 6.6.3 Grid independence test …………………………...………………...162 6.6.4 Unsteady 2D inflow profile ………………………...........................166 6.7 Summary ……………………………………………….. ……………...........175

Chapter 7. BulbT draft tube flow simulations: final geometry …..........177 7.1 Introduction …………………………………………………………..............177 7.2 Final draft tube geometry and selected operating points ……….……………177 7.3 Available experimental data ……………………………….…………………180 7.4 Computational grid specifications ……………………………………………182 7.5 Numerical setup for turbulent flow simulations ………………………...........184 7.6 Circumferential-averaged 1D inflow profile variants ………………………..184 7.6.1 Inflow profile near-wall treatment …………………………………185 7.6.2 Circumferential and radial velocity profile corrections ……............192 7.7 URANS simulations of the BulbT draft tube turbulent flows ……………….194 7.7.1 Simulation numerical setup …………………………………...........194 7.7.2 Inflow velocity profile correction effects …………………………..195 7.7.2 (a) Comparison to the LDV-measurements data……………195 7.7.2 (b) Separation topology …………………………………….199 7.7.2 (c) Recovery coefficient…………………………………….201 7.7.2 (d) Comparison to the PIV-measurements data…………….203 7.8 DDES simulations of the draft tube turbulent flows …………………………210 7.8.1 Simulation numerical setup……………………………………........210 x

7.8.2 Simulation results using 1D inflow profiles………………………...212 7.8.2 (a) Effect of inflow velocity corrections…………………....212 7.8.2 (b) Adjustment of inflow near wall turbulent viscosity using OP.1 ……………………………………………………220 7.8.3 DDES simulations at OP.4 ……………………………………........232 7.8.3 (a) 2D inflow velocity profile modification………………...239 7.8.3 (b) Simulation results using unsteady 2D inflow profiles ….241 7.8.3 (c) Effects of the synthetic inflow turbulence on separation topology………………………………………………..247 7.8.3 (d) Effects of the inflow synthetic turbulence on the turbulence energy spectrum at plane B………………….…………255 7.9 Effect of synthetic inflow turbulence: two-point correlation analysis ……...258 7.10 Flow skeleton: Lagrangian Coherent Structure (LCS) analysis …………..271 7.10.1 Introduction……………………………………………………….272 7.10.2 Mathematical formulation…………………………………………274 7.10.3 Effect of integration time on the LCS structures: Lid-driven cavity test case…………………………………………………………….275 7.10.3 (a) Numerical simulation of the lid-driven cavity flow…...275 7.10.3 (b) LCS structures of the cavity flow …………………….276 7.10.4 Definition of the LCS planes for draft tube flow analysis ………..278 7.10.5 Core separation bubble analysis at OP.1 using LCS………………280 7.10.6 Near-wall flow skeleton at OP.4…...……………………………...283 7.10.7 Core flow skeleton at OP.4…………………...…………………...287 7.11 Summary…………………………………………………………………….295

Chapter 8. Conclusion and future directions……………………………297 8.1 Summary of results …………………………………………………………..298 8.2 Future directions…………………...………………………………………....301

References…………………………………………………………….……305 Appendices…………………………………………………………………323 xi

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List of Tables Table 5.1 Statistics of the computational grids for the extended-case (including y  on the ERCOFTAC diffuser solid wall) ………………………………………………...114 Table 5.2 Different simulation cases within extended-case configuration ………………115 Table 5.3 Position of the hot-wire measurement lines …………………………………...118 Table 5.4 Statistics of the computational grids for base-case (including y  on the ERCOFTAC diffuser solid wall) ………………………………………………...120 Table 5.5 First series of DDES simulations with base-case configuration (effects of the extended-case turbulence model) ………………………………………………...128 Table 5.6 URANS simulations of the base-case configuration (effects of inflow radial velocity) …………………………………………………………………………..133 Table 5.7 Second series of DDES simulations with base-case configuration (effect of inflow near-wall treatment) ……………………………………………………....137 Table 5.8 Global performance quantities obtained from base-case simulations (effect of near-wall turbulent viscosity tuning) …………………………………………….139 Table 6.1 Details of the selected operating point for the numerical simulations................144 Table 6.2 Grid statistics for k   (mesh A) and S-A, DDES simulations (mesh B & C)….. ……..……………………………………………………………………………...145 Table 7.1 Details of the selected operating points for numerical simulations……………178 Table 7.2 Grid statistics for k   (mesh A) and S-A, DDES simulations (mesh B)…….183 Table 7.3 Average y  on the hub and shroud at inlet section of the draft tube……..……186 Table 7.4 Mean separation topology for different inflow profile variants using k   and S-A URANS simulations…………………………………………………………200 Table 7.5 Draft tube loss coefficient obtained from DDES simulations with amplification of vt in the WZ and applying inflow u -correction at OP.1………………………..226 Table 7.6 Draft tube mesh convergence test at OP.4……………………………………..232 Table 7.7 Probe locations to study reverse-flow intermittency quantity convergence…...238 Table 7.8 Probe locations on plane 1 to study u1† (t ) ……………………………………..261

Table 7.9 Probe locations to study the turbulent fluctuations…………………………….269 Table F.1 Discretization schemes in fvSchemes dictionary ……………………………..345 Table F.2 Details of fvSolution dictionary…………….…………………………………346 Table F.3 Boundary conditions on patches……………..………………………………..348

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List of Figures Fig 1.1 Electricity source in Canada in 2009 (adopted from library of parliament of Canada; http://www.parl.gc.ca).................................................................................................2 Fig 1.2 Schematic of a hydro-turbine site (modified from Hydro-Québec)………………...3 Fig 1.3 Classification of hydraulic turbines based on different energy generation mechanisms………………………………………………………………………….4 Fig 1.4 Graph of hydro-turbine selection (courtesy of Voith Hydro Inc.) ……………….…5 Fig 1.5 Schematic of the different types of the hydro-turbines [Ingram 2009] ….…………6 Fig 1.6 Schematic of a bulb turbine installation in a real site [Keck et al. 2008] ………......9 Fig 1.7 A cut view of BulbT assembly installed on the LAMH test-bench (top); a photo of test bench showing the BulbT draft tube (bottom) ………………………………...11 Fig 2.1 Different layers in the near-wall region (modified from [Lemay J. 2008]) ……….21 Fig 2.2 Boundary layer subjected to an adverse pressure gradient (APG) (modified from [Nakayama Y. 2000]) ……………………………………………………………...21 Fig 2.3 Schematic of a hairpin structure in the near wall-region ………………………….22 Fig 2.4 Schematic of a hairpin packet structure in the near wall-region (modified from [Adrian R.J. 2000]) ………………………………………………………………..23 Fig 2.5 Visualization of hairpin structures in the turbulent boundary layer on a flat plate at Rem  4300 obtained by isosurfaces of negative 2 from DNS results. The

structures have been colored by wall distance [Schlatter et al. 2011] …………….23 Fig 2.6 Typical comparison of turbulence treatment strategies [Buntic et al. 2005] ……..26 Fig 2.7 Resolved and subgrid scale eddies (fluctuations) on a given mesh in the LES ……………………………………………………………………………………...27 Fig 2.8 Turbulence treatment in LES vs. RANS …………………………………………..28 Fig 2.9 Turbulence treatment in VLES vs. LES method…………………………………..31 Fig 2.10 Ambiguous grid in the near wall zone inside the boundary layer (P: a point inside B.L.,  : B.L. thickness, d: distance to the wall) ………………………………….36 Fig 2.11 Turbulent flow structures ( Q  350 ) resolved by DDES turbulent treatment using unsteady 2D inflow profile in the BulbT draft tube at an overload condition ….....39 Fig 3.1 Control volumes in the finite volume method used for the discretization (modified from [OpenFOAM programmer’s guide]) ……………………...…………………47 Fig 3.2 Solution procedure for the utilized PIMPLE method……………………………..52 Fig 3.3 A typical OpenFOAM case structure for transient DDES simulations …………...57 Fig 3.4 Speed-up curve for the BulbT draft tube flow simulation on Colosse cluster …….60 xv

Fig 3.5 Speed-up curve for the BulbT draft tube flow simulation on Guillimin cluster …..60 Fig 4.1 Schematic of a typical turbulent velocity signal …………………………………..65 Fig 4.2 Sketch of the hydraulic profile sections in the full-machine simulation…………..65 Fig 4.3 Final version of the draft tube geometry ………………………………………….66 Fig 4.4 Circumferential averaged-1D inlet profile obtained from full machine k   simulation at OP.4 ………………………………………………………………...69 Fig 4.5 General Sketch on applying a transient 2D inflow profile for DDES simulation of the draft tube using OpenFOAM …………………………………………………..70 Fig 4.6 2D variation of the axial velocity uZ (left) and turbulent eddy viscosity  t (right) at inlet plane from k   RANS at OP.4 …………………………………………71 Fig 4.7 Delaunay triangulation at the draft tube inlet plane at one instant of time ………..71 Fig 4.8 Rotation of the 2D velocity profiles at the draft tube inlet plane for OP.4, top: radial velocity ( ur ) middle: circumferential velocity ( u ) bottom: axial velocity ( u z )...72 Fig 4.9 Power spectrum of the axial velocity signal generated by 2D rotating profile at one Eulerian probe placed at the normalized position ( x / RInlet , y / RInlet )  (0.3,  0.55) ……………………………………………………………….……………………..73 Fig 4.10 Schematic of the RANS (B.L.) & LES (core flow) zones at draft tube inlet plane ……………………………………………………………………………………...74 Fig 4.11 Function f d at the draft tube inlet plane for OP.4……………………………….75 Fig 4.12 Subgrid scale viscosity treatment using Smagorinsky model for OP.4 at the draft tube inlet plane……………………………………………………………………..76 Fig 4.13 Hybrid turbulent/subgrid-scale viscosity treatment for OP.4…………………….76 Fig 4.14 Energy spectrum of a real turbulent velocity signal (solid curve) & the white noise signal (horizontal dashed line) …………………………………………………….86 Fig 4.15 Definition of intermediate random angles, wave number vector and Fourier mode direction ……………………………………………………………………………87 Fig 4.16 Isotropic AFG fluctuation signals in axial direction with/without time-correlation before applying runner rotation and inflow anisotropy ……………………………90 Fig 4.17 Isotropic AFG turbulent velocity fluctuation signals with time-correlation and considering runner rotation (top: full signal, bottom: zoomed area) ……………...91 Fig 4.18 Inflow AFG fluctuation field of axial velocity component at t  1.2 s for OP.4 (top: isometric view, bottom: side view) …………………………………………..93 Fig 4.19 Experimental RMS values of the inflow velocity fluctuations at OP.4 (LDV) [top left to bottom right: circumferential, axial, radial velocities and sketch of flow pattern observed experimentally at inlet plane (plane A)] ………………………...95 Fig 4.20 Multi-layer perceptron architecture designed for each azimuthal angle …………96 xvi

Fig 4.21 Zonal training approach for the designed MLPR ………………………………..97 Fig 4.22 RMS values of the fluctuations OP.4 (LDV measurements) (left: circumferential velocity, right: radial velocity) …………………………………………………….98 Fig 4.23 Learning history of the designed ANN with l  0.005 at   25 …………….98 Fig 4.24 Learning history of the ANN in the unstable region with l  0.1 at   25 …...99 Fig 4.25 Radial distribution of ANN (MLPR) predictions for the parameter  radial at OP.4 (from top left to bottom right:   10 ,   40 ,   60 ,   90 ) …………….100 Fig 4.26 Two-dimensional variation of the  radial parameter obtained from ANN (MLPR) at the inlet plane for OP.4 ………………………………………………………..100 Fig 4.27 Quarter of the RMS field of radial velocity fluctuations using ANN for OP.4…101 Fig 4.28 Complete RMS field of the radial velocity fluctuations predicted using ANN for OP.4 ………………………………………………………………………………101 Fig 4.29 1D circumferential-average of 2D RMS field of radial velocity fluctuations at OP.4 ………………………………………………………………………………102 Fig 4.30 1D circumferential average of 2D RMS field of the experimental velocity fluctuations at OP.4 (top: circumferential velocity, bottom: axial velocity) ……..102 Fig 4.31 Scatter plot of the isotropic turbulent velocity fluctuations generated by the AFG method before applying rotation and scaling factors sampled at a point with the normalized position ( x / RInlet , y / RInlet )  (0.3,  0.55) on the inlet plane………...103 Fig 4.32 Scatter plot of u z  uy fluctuations generated by the AFG method after applying rotation and scaling factor sampled at a point with the normalized position ( x / RInlet , y / RInlet )  (0.3,  0.55) on the inlet plane …………………………….104 Fig 4.33 Scatter plot of u x  u z fluctuations generated by the AFG method after applying rotation and scaling factor sampled at a point with the normalized position ( x / RInlet , y / RInlet )  (0.3,  0.55) on the inlet plane …………………………….105 Fig 4.34 Scatter plot of ux  uy fluctuations generated by the AFG method after applying rotation and scaling factor sampled at a point with the normalized position ( x / RInlet , y / RInlet )  (0.3,  0.55) on the inlet plane …………………………….106 Fig 4.35 Power spectra of turbulent velocity signals with/without artificial fluctuations sampled at a probe positioned at ( x / RInlet , y / RInlet )  (0.3,  0.55) on the inlet plane …………………………………………………………………………………….107 Fig 5.1 Schematic sketch of the ERCOFTAC conical diffuser with swirling generator [Sketched base on the geometry, used by Clausen et al. 1993] ………………….112 Fig 5.2 Computational domain adopted for DDES turbulence treatment in the ERCOFTAC diffuser: a) base-case and b) extended-case ……………………………………...113 xvii

Fig 5.3 Variation of the C coefficient in DNS of channel flow [Pope S. 2004] ………..116 Fig 5.4 Convergence history of the k   simulation case (i.e. A-1 case in table 5.2)…...117 Fig 5.5 Normalized mean axial-velocity field on the mid-plane of the diffuser obtained from k   simulation case (i.e. A-1 case in table 5.2) …………………………..118 Fig 5.6 Comparison of the wall-parallel ( us ) and circumferential ( u ) component of mean velocity at different sections with the experimental data for extended-cases…….119 Fig 5.7 Inflow grid reconstructed by Delaunay triangulation at S1 section ……………..121 Fig 5.8 2D-velocity profile extracted at S1 section from extended-case simulations……122 Fig 5.9 Turbulent viscosity extracted at S1 section from extended-case simulations…….123 Fig 5.10 Normalized 1D-velocity profile extracted at S1 section extracted from extendedcase simulations (with k   and S-A turbulence models) …………………........124 Fig 5.11 Normalized 1D-velocity profile extracted at S1 section extracted from extendedcase simulations (with S-A turbulence model with/without wall-function)……...125 Fig 5.12 Comparison of the 1D turbulent quantity profiles at S1 section obtained from the extended-case k   simulation (A-1 case in table 5.2) with experimental data (left: turbulent dissipation rate, right: turbulent kinetic energy) …………….………...126 Fig 5.13 1D profile of turbulent viscosity at S1 section coming from the selected extendedcase simulations (table 5.2) ……………………………………………………....127 Fig 5.14 Wall-parallel ( us ) and circumferential ( u ) component of mean velocity obtained from DDES simulations of base-case using different inflow conditions (inflow profiles coming from extended-case k   , S-A simulations listed in table 5.5)...129 Fig 5.15 Axial-velocity field and separation-zone isosurface (with u z  0 ) obtained from the base-case DDES simulations listed in table 5.5 ……..…………………….....130 Fig 5.16 Typical f d function variation in the base-case DDES simulation with inflow profile coming from k   extended-case simulation (B-1 case in table 5.5) [ f d  0 : RANS , f d  1: LES ] ………………………………………………....130  Fig 5.17 Variation of utotal versus y  at the base-case inflow section ( S1 section) extracted from various extended-case simulations …………………………………………131

Fig 5.18 Wall-parallel us and circumferential u components of mean velocity obtained from URANS simulations of the base-case indicating effect of inflow radial velocity (the different inflow conditions for the cases are presented in table 5.6) …………………………………………………………………………………….134  versus y  of the constructed profile at the S1 section ……….136 Fig 5.19 Variation of utotal

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Fig 5.20 Variation of vt versus y  of the constructed profile at the S1 section ……….....137 Fig 5.21 Wall-parallel us and circumferential u component of the mean velocity obtained from DDES simulations of the base-case indicating effect of inflow near-wall treatment (cases are listed in table 5.7) ….……..………………………………...137 Fig 5.22 Wall-parallel us and circumferential u component of mean velocity obtained from DDES simulations of the base-case (effect of tuning of the inflow near-wall vt )..140 Fig 6.1 Basic geometry of the draft tube with its geometrical divergence angles………..144 Fig 6.2 Separation zone (blue iso-surface) in the basic geometry of the draft tube obtained from S-A (left) and k   (right) RANS simulations at BEP ……………………147 Fig 6.3 1D circumferential-averaged of mean velocity profiles in the conical part of the basic geometry extracted from S-A (red) and k   (black) RANS simulations at BEP……………………………………………………………………………….148 Fig 6.4 Draft tube recovery coefficient (left) and swirl number (right) evolution in the streamwise direction obtained from S-A and k   RANS simulations at BEP...149 Fig 6.5 f d function at the inlet of the draft tube with ‘basic geometry’ at BEP…………..150 Fig 6.6 Mean separation-zone (blue iso-surface) and coherent structures ( Q  500 ) obtained from DDES simulations: effect of inflow vt amplification in the WZ…151 Fig 6.7 Draft tube recovery coefficient (left) and swirl number (right) evolution in the streamwise direction obtained from DDES simulations at BEP: effect of vt amplification in WZ……………………….……………………………………...152 Fig 6.8 Mean axial velocity field on the mid-plane of the draft tube obtained from DDES simulations at BEP: effect of inflow vt amplification in WZ…………………….153 Fig 6.9 1D circumferential-averaged of mean velocity profiles in the conical part of the basic geometry extracted from DDES at BEP: effect of vt amplification in WZ..154 Fig 6.10 Energy spectrums of the turbulent axial velocity at different axial position of the axisymmetric center line of the draft tube: effect of inflow vt amplification in WZ………………………………………………………………………………...155  Fig 6.11 Sampling-lines utilized to obtain utotal versus y  graph in the transition part...…156  Fig 6.12 Variation of utotal versus y  on sampling-lines S1 , S2 and S3 extracted from DDES

simulations: effect of inflow vt amplification in the WZ…………………………156 Fig 6.13 Mean separation-zone (blue iso-surface) and coherent structures ( Q  500 ) in the DDES simulations: effect of the discretization scheme of the convective term …158 Fig 6.14 Instantaneous axial velocity field on the draft tube mid-plane obtained from DDES simulations at BEP: effect of the discretization scheme of the convective term…158

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Fig 6.15 Draft tube recovery coefficient (left) and swirl number (right) evolution in zdirection of DDES simulations at BEP: effect of the convective term discretization scheme….................................................................................................................159 Fig 6.16 1D circumferential-averaged of mean velocity profiles in the conical part in the DDES simulations at BEP: effect of the convective term discretization scheme...160 Fig 6.17 Energy spectrums of the turbulent axial velocity at different positions of the axisymmetric center-line of draft tube: effect of convective term discretization scheme…………………………………………………………………………….161  versus y  on sampling-lines S1 , S2 and S3 extracted from the Fig 6.18 Variation of utotal DDES simulations: effect of the convective term discretization scheme………...162

Fig 6.19 Mean separation-zone (blue iso-surface) and coherent structures ( Q  500 ) in the DDES simulations: grid independence test………………………………………163 Fig 6.20 Draft tube recovery coefficient (left) and swirl number (right) evolution in the streamwise direction of the DDES simulations at BEP: grid independence test…164 Fig 6.21 Energy spectra of the turbulent axial velocity at different axial position of the axisymmetric center-line of the draft tube: grid independence test………………165  Fig 6.22 Variation of utotal versus y  on the sampling-lines S1 , S2 and S3 extracted from the DDES simulations: grid independence test ………………………………………166

Fig 6.23 A snapshot of the generated artificial fluctuations in different spatial directions in DDES simulation of the ‘basic geometry’ using ‘2D rotating+AFG’ inflow profile……………………………………………………………………………..168 Fig 6.24 Time-evolution of the turbulent fluctuation signals in different spatial directions adopted for DDES simulation of the ‘basic geometry’ with ‘2D rotating+AFG’ profile……………………………………………………………………………..169 Fig 6.25 Coherent structures with Q  350 (left) and mean separation zone (right) obtained from DDES simulations with ‘2D rotating+AFG’ inflow profile………………..170 Fig 6.26 Instantaneous flow separation obtained from DDES simulations of the ‘basic geometry’ with ‘2D rotating+AFG’ inflow profile………………………………170 Fig 6.27 Instantaneous axial velocity field on the draft tube mid-plane ( x  0 ) extracted from DDES simulation using ‘2D rotating+AFG’ and ‘1D’ inflow profiles at BEP……………………………………………………………………………….171 Fig 6.28 Draft tube recovery coefficient (left) and swirl number (right) evolution in the streamwise direction of the DDES simulations at BEP: ‘2D’ vs. ‘1D’ profiles….172 Fig 6.29 1D circumferential-averaged of mean velocity profiles in the conical part obtained from DDES simulations at BEP: ‘2D’ vs. ‘1D’ profiles………………………….173 Fig 6.30 Energy spectrums of the turbulent axial velocity at different axial position of the axisymmetric center-line of the draft tube: ‘2D’ vs. ‘1D’ profiles……………….174 Fig 7.1 Final geometry of the draft tube with its geometrical divergence angles………...178 xx

Fig 7.2 Selected operating points on the turbine efficiency curve………………………..179 Fig 7.3 Pressure sensor positions and LDV measurement axis…………………………..180 Fig 7.4 PIV-measurement planes namely B1, B2, S3 and S4 [Duquesne et al. 2014-3]…181 Fig 7.5 Near-wall zone of inflow profile treatments for draft tube turbulent flow simulations using low-Re treatments e.g. S-A and DDES………………………..186 Fig 7.6 2D variation of y  quantity on the runner-shroud surface i.e. upstream component of the draft tube stemmed from full machine k   RANS simulations…………..187 Fig 7.7 2D variation of y  quantity on the runner-hub surface i.e. upstream component of the draft tube stemmed from full machine k   RANS simulations……………..187 Fig 7.8 Buffer-layer velocity profile variants ……………………………………………188 Fig 7.9 Turbulent viscosity in the near-wall zone of the draft tube using Reichardt model for two selected operating points i.e. OP.1 and OP.4…………………………….189 Fig 7.10 1D inflow profiles of u z and u velocities and turbulent viscosity vt at Plane A used for draft tube flow simulations resolving to the wall at OP.1 (left) and OP.4 (right) (solid black line: reconstructed profile, solid red line with circles: LDV measurements)…………………………………………………………………….190 Fig 7.11 1D inflow profiles of radial velocity at OP.1 (left) and OP.4 (right) at Plane A (solid black line: reconstructed profile, solid red line with circles: approximation profile)…………………………………………………………………………….191 Fig 7.12 Circumferential and radial velocity inflow profile correction for OP.1 (left: original profile, right: corrected profile based on experimental data i.e. red line)……………………………………………………………………………….192 Fig 7.13 Circumferential and radial velocity inflow profile correction for OP.4 (left: original profile, right: corrected profile based on the experimental data i.e. red line)……………………………………………………………………………….193 Fig 7.14 Effect of variant inflow profiles on the mean flow obtained from k   URANS simulations at OP.1 compared to the LDV-experimental data…………………...196 Fig 7.15 Effect of variant inflow profiles on the mean flow obtained from S-A URANS simulations at OP.1 compared to the LDV-experimental data…………………...197 Fig 7.16 Effect of variant inflow profiles on the mean flow obtained from k   URANS simulations at OP.4 compared to the LDV-experimental data…………………...198 Fig 7.17 Effect of variant inflow profiles on the mean flow obtained from S-A URANS simulations at OP.4 compared to the LDV-experimental data …………………..199 Fig 7.18 Mean separation zone (blue iso-surface) obtained from S-A (top) and k   (bottom) URANS simulations with inflow u -correction at OP.1……………….201 Fig 7.19 Draft tube recovery coefficient evolution in the streamwise direction for different variants of the URANS simulations………………………………………………202

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Fig 7.20 Error measure of the velocity field between k   URANS simulation results and the PIV experimental data at OP.1, plane B1 (upstream/downstream)…………...205 Fig 7.21 Error measure of the velocity field between k   URANS simulation results and the PIV experimental data at OP.4, plane B1 (upstream/downstream) …………..206 Fig 7.22 Mean streamlines for k   and S-A simulations at OP.1/OP.4 on the plane B1: downstream; (red: experimental measurements, Black: numerical simulation).......................................................................................................................207 Fig 7.23 Error measure of the velocity field between S-A URANS simulation results and the PIV experimental data at OP.1, plane B1 (upstream/downstream)……...……208 Fig 7.24 Error measure of the velocity field between S-A URANS simulation results and the PIV experimental data at OP.4, plane B1 (upstream and downstream)…..…..209 Fig 7.25 Mean streamline pattern for DDES simulations using different variants of the ‘original’ inflow profile, at OP.1 on plane B1, upstream/downstream (red: experimental measurements, black: numerical simulations)……………………..213 Fig 7.26 Mean flow separation zone topology obtained from DDES simulations with different inflow profiles at OP.1………………………………………………….215 Fig 7.27 f d function variation extracted from DDES simulation of the draft tube with u corrected inflow profile at OP.1…………………………………………………..216 Fig 7.28 Effect of “original” inflow profile variants on the mean flow obtained from DDES simulations at OP.1 compared to the LDV-experimental data…………………...217 Fig 7.29 Draft tube recovery coefficient evolution in streamwise direction obtained from DDES simulations for both operating points……………………………………..218 Fig 7.30 Effect of “original” inflow profile variants on the mean flow obtained from DDES simulations at OP.4 compared to the LDV-experimental data…………………...218 Fig 7.31 Turbulent kinetic energy ( k ) variation at the inlet plane at OP.1………………219 Fig 7.32 Effect of amplification of the turbulent viscosity in the WZ on the mean flow obtained from DDES simulations with u -corrected inflow profile at OP.1…….221 Fig 7.33 Mean streamline pattern for DDES simulations with amplification of vt ( WZ  10n ) at OP.1 on plane B1: downstream (red: measurements, Black: simulations)……………………………………………………………………….222 Fig 7.34 Error measure of the velocity field obtained from DDES simulation results with amplification of vt in the WZ ( WZ  10n ) at OP.1 at plane B1 (upstream/ downstream)………………………………………………………………………223 Fig 7.35 Mean flow separation topology obtained from DDES simulations with turbulent viscosity amplification in WZ applied on both hub and shroud zones at OP.1…..224 Fig 7.36 Effect of amplification of vt in the WZ ( WZ  10n ) on the draft tube recovery coefficient evolution obtained from DDES simulations at OP.1…………………225 xxii

Fig 7.37 Time evolution of the global draft tube coefficient (  ) for DDES simulation applying inflow u -correction and amplification of vt in the WZ ( n  3 ) at OP.1.........................................................................................................................227 Fig 7.38 Effect of amplification of vt in the WZ applied just for hub/shroud with/without u -correction at OP.1……………………………………………………………..227 Fig 7.39 Effect of amplification of vt in the WZ applied just for hub/shroud with/without u -correction on the mean flow separation topology at OP.1……………………229 7.40 Effect of amplification of inflow vt in the WZ applied just for hub/shroud on the error measure of the velocity field with/without u -correction at plane B1 at OP.1 a) u corrected, n  3for both hub & shroud ; b) u -corrected, n  3 just for hub , c) u corrected, n  3 just for shroud ; d) Original velocity profile, n3 for both hub & shroud …………………………………………………………….230 7.41 Coherent structures formed in the BulbT draft tube turbulent flow using DDES at OP.1 a) Reverse-flow bubble, b) Coherent structures visualized with Q  2000 ………………………………………………………………………….231 7.42 Topology of the mean separated region for intermediate/fine girds at OP.4 visualized by iso-surface of the negative velocity …………………………………………...233 Fig 7.43 Error measure obtained from DDES simulations at OP.4 for both the intermediate (mesh B) and fine (mesh C) grids on the planes B1, B2 (upstream/ downstream)………………………………………………………………………234 Fig 7.44 Error measure obtained from DDES simulations at OP.4 for both the intermediate (mesh B) and fine (mesh C) grids on the planes S3, S4 (upstream/downstream)...235 Fig 7.45 Reverse-flow intermittency on the different slices extracted from the DDES simulation using 1D-inflow profile at OP.4………………………………………237 Fig 7.46 Evolution of the axial velocity (top) and reverse-flow intermittency (bottom) for two probes extracted from the DDES simulation using 1D inflow profile at OP.4……………………………………………………………………………….238 Fig 7.47 Reconstruction of the u profile for DDES simulations with unsteady 2D-inflow profiles ……………………………………………………………………………240 Fig 7.48 Circumferential-averaging of the blended 2D u -profile at OP.4 (black: averaged curve of the 2D profile, red: experimental data)………………………………….241 Fig 7.49 Turbulent flow coherent structure anatomy resolved by Q  350 coming from DDES simulation using “2D rotating+AFG” inflow profile at OP.4…………….243 Fig 7.50 Effect of different inflow profiles on the resolved coherent structures at one instant of time obtained from DDES simulations at OP.4 (colorbar: u z uref . )…………..244

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Fig 7.51 Effect of type of the inflow profile on the mean flow obtained from DDES simulations at OP.4 compared to the LDV-experimental data…………………...245 Fig 7.52 Effect of inflow profile type on the draft tube recovery coefficient obtained from DDES simulations compared to the experiment at OP.4……………….………...246 Fig 7.53 Effect of inflow profile type on the time evolution of the global draft tube coefficient (  ) obtained from DDES simulations at OP.4……………………….246 Fig 7.54 Topology of the mean separated region for different unsteady 2D inflow profiles at OP.4 visualized by iso-surface of the negative velocity …………………………247 Fig 7.55 Error measure obtained from DDES simulations using different inflow profiles at OP.4 on planes B1, B2 (up/downstream) [ a) E (u x ) ,1D; b) E (u x ) ,2D rotating; c) E (u  ) , 2D rotating+AFG; d) E (u  ) ,1D; e) E (u  ) , 2D rotating; f) E (u  ) ,2D x

z

z

z

rotating +AFG]……………………………………………………………………248 Fig 7.56 Error measure obtained from DDES simulations using different inflow profiles at OP.4 on planes S3 x+/x- (up/downstream) [ a) E (u y ) ,1D; b) E (u y ) ,2D rotating; c) E (u  ) , 2D rotating+AFG; d) E (u  ) ,1D; e) E (u  ) , 2D rotating; f) E (u  ) ,2D z

y

z

z

rotating+AFG]…………………………………………………………………….249 Fig 7.57 Error measure obtained from DDES simulations using different inflow profiles at OP.4 on the planes S4 x+/x- (upstream/downstream) [ a) E (u y ) ,1D; b) E (u y ) ,2D rotating; c) E (u  ) ,2D rotating+AFG; d) E (u  ) ,1D; e) E (u  ) , 2D rotating; f) y

z

z

E (u z ) ,2D rotating+AFG]………………………………………………………...251 Fig 7.58 Reverse-flow intermittency on the different slices extracted from the DDES simulations using 2D inflow profiles at OP.4…………………………………….252 Fig 7.59 Evolution of the axial velocity (left) and reverse-flow intermittency (right) at probe A; extracted from DDES simulations using 2D inflow profiles at OP.4.........................................................................................................................253 Fig 7.60 Evolution of the axial velocity (left) and reverse-flow intermittency (right) at probe B; extracted from DDES simulations using 2D inflow profiles at OP.4……………………………………………………………………………….254 Fig 7.61 LES-content via LES  IQv obtained from DDES simulations with different unsteady 2D inflow profiles at OP.4……………………………………………...255 Fig 7.62 Energy spectra of the turbulent velocity signals obtained from DDES simulations with different inflow profiles at different x-positions of plane B………………...256 Fig 7.63 Triple view of the probe positions adopted to calculate two-point correlation (top: isometric view, bottom-left: side-view, bottom-right: front-view)……………….259 Fig 7.64 Normalized two-point correlation field for the radial velocity unsteady term i.e. †, norm 11 extracted from DDES simulations at OP.4 with/without AFG on planes 1, 2, 3…………………………………………………………………………………...260 xxiv

Fig 7.65 Variation of the unsteady portion of the turbulent radial velocity signal u1† (t ) in terms of time obtained from DDES simulations with “2D rotating +AFG” inflow profile at OP.4 at two probe positions presented in table 7.8 (red: probe C, blue: probe D) ………………………………………………………………………….261 Fig 7.66 Normalized two-point correlation field for circumferential velocity unsteady term,  †,22norm , extracted from DDES simulations at OP.4 with/without AFG on planes 1, 2, 3………………………………………………………………………………...262 Fig 7.67 Normalized two-point correlation field for axial velocity unsteady term i.e.  †,33norm extracted from DDES simulations at OP.4 with/without AFG on planes 1, 2, 3…………………………………………………………………………………...263  norm Fig 7.68 Normalized two-point correlation field of radial velocity turbulent term i.e. 11 extracted from DDES simulations at OP.4 with/without AFG on planes 1, 2, 3…………………………………………………………………………………...264

Fig 7.69 Normalized two-point correlation field of circumferential velocity turbulence term i.e. 22norm extracted from DDES simulations at OP.4 with/without AFG on planes 1, 2, 3………………………………...……………………………………………265 Fig 7.70 Normalized two-point correlation field of axial velocity turbulence term i.e. 33norm extracted from DDES simulations at OP.4 with/without AFG on planes 1, 2, 3…………………………………………………………………………………...266 Fig 7.71 1D plot of iinorm , i  1, 2,3 extracted from the DDES simulation with inflow AFG at OP.4 plotted at different rows (ic=1,..,10) of the probe locations on planes 1, 2, 3…………………………………………………………………………………...267 Fig 7.72 Sketch of plane 1 and the hub-vortex at OP.4 indicating positions of the probes S1 to S8 utilized for the scatter plots ……..………………………………………….268 Fig 7.73 Scatter plot of the turbulent fluctuation clouds along the axial path corresponding to the probes (S1: blue; S2: green; S3: magenta, S4: red; S5: black)…………….270 Fig 7.74 Scatter plot of the turbulent fluctuation clouds along the radial path corresponding to the probes (S5: blue; S6: green; S7: red, S8: black)…………………………...271 Fig 7.75 Flow trajectory in the proximity to the two types of the dynamical manifolds (solid red circle: initial position; hollow red circle: final position of fluid particles) …..273 Fig 7.76 Streamlines and velocity vector field of the cavity fluid flow stemmed from k   URANS simulation at Re  1.5 106 ………………………………………276 Fig 7.77 FTLE field of the velocity field in the cavity flow case obtained from forward time integration ( T  nCTC , TC  2 s )…………………………………………………...277 Fig 7.78 FTLE field of the velocity field in the cavity flow case obtained from backward time integration ( T   nCTC , TC  2 s )…………………………………………….278 Fig 7.79 LCS planes defined to study flow structures in the draft tube at OP.4…………279 xxv

Fig 7.80 LCS planes defined to study reverse–flow bubble using LCS at OP.1 ………...279 Fig 7.81 Axial velocity field on the LCS planes with C = 0 o ,45o ,90 o and 135o from top to bottom, at one instant of time, obtained from DDES simulation applying ‘1D’ inflow profile with u -correction and amplification of vt in the WZ ( n  3 ) at OP.1………………………………………………………………………………280 Fig 7.82 FTLE field with forward-time integration (repelling LCS) on the LCS planes with C = 0 o ,45o ,90 o and 135 o from top to bottom, at one instant of time, at OP.1……………………………………………………………………………….281 Fig 7.83 FTLE field with backward-time integration (attracting LCS) on the LCS planes with C = 0 o ,45o ,90 o and 135 o from top to bottom, at one instant of time, at OP.1……………………………………………………………………………….282 Fig 7.84 Axial velocity field on the yz1 LCS plane obtained from DDES simulation using ‘2D rotating+AFG’ inflow condition at t  0.5s , at OP.4………………………..283 Fig 7.85 FTLE field with forward-time integration (repelling LCS) on the yz1 LCS plane calculated from DDES simulations using different inflow conditions at OP.4…..284 Fig 7.86 FTLE field with backward-time integration (attracting LCS) on the yz1 LCS plane calculated from DDES simulations using different inflow conditions at OP.4…..285 Fig 7.87 Axial velocity field on the xz1 LCS plane obtained from DDES simulation using ‘2D rotating+AFG’ inflow (top) and ‘1D’ inflow (bottom), at t  0.5s , at OP.4.........................................................................................................................285 Fig 7.88 FTLE field with forward-time integration (repelling LCS) on the xz1 LCS plane calculated from DDES simulations using different inflow conditions at OP.4 (top: ‘2D rotating+AFG’, middle: ‘2D rotating’, bottom: ‘1D’) ……………………...286 Fig 7.89 FTLE field with backward-time integration (attracting LCS) on the xz1 LCS plane calculated from DDES simulations using different inflow conditions at OP.4 (top: ‘2D rotating+AFG’, middle: ‘2D rotating’, bottom: ‘1D’) ……………………...287 Fig 7.90 Axial velocity field on the xz 2 LCS plane obtained from DDES simulation using ‘2D rotating+AFG’ inflow profile, at t  0.5s , at OP.4 …………………………..288 Fig 7.91 FTLE field with forward and backward-time integrations on the xz 2 LCS plane calculated from DDES simulations using different inflow conditions at OP.4 (top: ‘2D rotating+AFG’, middle: ‘2D rotating’, bottom: ‘1D’)………………………289 Fig 7.92 Axial velocity field on the xy1 LCS plane obtained from DDES simulation using ‘2D rotating+AFG’ inflow profile, at t  0.5s , at OP.4 …………………………..290 Fig 7.93 FTLE field with forward-time integration (repelling LCS) on the xy1 LCS plane calculated from DDES simulations using different inflow conditions at OP.4.….291 Fig 7.94 FTLE field with backward-time integration (attracting LCS) on the xy1 LCS plane calculated from DDES simulations using different inflow conditions at OP.4…..291

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Fig 7.95 Axial velocity field on the xy 2 LCS plane superimposed with repelling LCS (black structure) and attracting LCS (cyan structure) obtained from DDES simulation using ‘2D rotating+AFG’ inflow profile, at t  0.5s , at OP.4………..292 Fig 7.96 FTLE field with forward-time integration (repelling LCS) on the xy 2 LCS plane calculated from DDES simulations using different inflow conditions at OP.4.….293 Fig 7.97 FTLE field with backward-time integration (attracting LCS) on the xy 2 LCS plane calculated from DDES simulations using different inflow conditions at OP.4……………………………………………………………………………….293 Fig 7.98 Axial velocity field on the xy3 LCS plane obtained from DDES simulation using ‘2D rotating+AFG’ inflow profile, at t  0.5s , at OP.4…………………………..294 Fig 7.99 FTLE field with forward-time integration (repelling LCS) on the xy3 LCS plane calculated from DDES simulations using different inflow conditions at OP.4.….294 Fig 7.100 FTLE field with backward-time integration (attracting LCS) on the xy3 LCS plane calculated from DDES simulations using different inflow conditions at OP.4……………………………………………………………………………….294 Fig B.1 Nonlinear model of a neuron……………………………………………………..331 Fig D.1 Computational grid for draft tube k   RANS/URANS simulations (top: Full domain draft tube mesh, bottom left: Top view of middle plane mesh intersection, bottom right: zoomed area near the hub of middle plane mesh intersection)……338 Fig D.2 Computational grid for draft tube DDES simulations. (top: Full domain draft tube mesh, bottom left: Top view of middle plane mesh intersection, bottom right: zoomed area near the hub of middle plane mesh intersection) …………………..339 Fig G.1 Comparison of the simulation mean velocity field to the PIV experimental data k   URANS simulations, OP.1, plane B1: upstream …………………………..350 Fig G.2 Comparison of the simulation mean velocity field to the PIV experimental data k   URANS simulations, OP.1, plane B1: downstream………………………..351 Fig G.3 Comparison of the simulation mean velocity field to the PIV experimental data k   URANS simulations, OP.4, plane B1: upstream …………………………..351 Fig G.4 Comparison of the simulation mean velocity field to the PIV experimental data k   URANS simulations, OP.4, plane B1: downstream ……………………….352 Fig G.5 Comparison of the simulation mean velocity field to the PIV experimental data k   URANS simulations, OP.4, plane B2: upstream …………………………..352 Fig G.6 Comparison of the simulation mean velocity field to the PIV experimental data k   URANS simulations, OP.4, plane B2: downstream ……………………….353 Fig G.7 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.1, plane B1: upstream …………………………………353 Fig G.8 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.1, plane B1: downstream………………………………354 xxvii

Fig G.9 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.4, plane B1: upstream …………………………………355 Fig G.10 Simulation mean velocity field in the case of u , ur inflow corrected profile S-A URANS simulations, OP.4, plane B1: upstream …………………………………355 Fig G.11 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.4, plane B1: downstream ………………………...355 Fig G.12 Simulation mean velocity field in the case of u , ur inflow corrected profile S-A URANS simulations, OP.4, plane B1: downstream………………………………356 Fig G.13 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.4, plane B2: upstream ……………………………356 Fig G.14 Simulation mean velocity field in the case of u , ur inflow corrected profile S-A URANS simulations, OP.4, plane B2: upstream………………………………….357 Fig G.15 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.4, plane B2: downstream ………………………...357 Fig G.16 Simulation mean velocity field in the case of u , ur inflow corrected profile S-A URANS simulations, OP.4, plane B2: downstream ……………………………...357 Fig G.17 Effects of vt amplification on the simulation mean velocity fields DDES simulations, OP.1, plane B1: upstream …………………………………………..358 Fig G.18 Effects of vt amplification on the simulation mean velocity fields DDES simulations, OP.1, plane B1: downstream ……………………………………….358 Fig G.19 Effects of vt amplification with inclusion/exclusion hub/shroud on the simulation mean velocity fields obtained from DDES simulations, OP.1, plane B1: upstream …………………………………………………………………………………….359 Fig G.20 Effects of vt amplification with inclusion/exclusion hub/shroud on the simulation mean velocity fields obtained from DDES simulations, OP.1, plane B1: downstream …………………………………………………………………………………….360 Fig G.21 Effects of mesh resolution on the simulation mean velocity fields obtained from DDES simulations using 1D profile, OP.4, plane B1…………………………….360 Fig G.22 Effects of mesh resolution on the simulation mean velocity fields obtained from DDES simulations using 1D profile, OP.4, plane B2…………………………….361 Fig G.23 Effects of mesh resolution on the simulation mean velocity fields obtained from DDES simulations using 1D profile, OP.4, plane S3 x+………………………….362 Fig G.24 Effects of mesh resolution on the simulation mean velocity fields obtained from DDES simulations using 1D profile, OP.4, plane S4 x+………………………….363 Fig G.25 Effects of type of the inflow profile on the DDES simulation results, OP.4, plane B1: upstream ……………………………………………………………………..363

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Fig G.26 Effects of type of the inflow profile on the DDES simulation results, OP.4, plane B1: downstream ………………………………………………………………….364 Fig G.27 Effects of type of the inflow profile on the DDES simulation results, OP.4, plane B2: upstream ……………………………………………………………………..365 Fig G.28 Effects of type of the inflow profile on the DDES simulation results, OP.4, plane B2: downstream ………………………………………………………………….365 Fig G.29 Effects of type of the inflow profile on the DDES simulation results, OP.4, plane S3 x+: upstream …………………………………………………………………..366 Fig G.30 Effects of type of the inflow profile on the DDES simulation results, OP.4, plane S3 x-: upstream …………………………………………………………………...366 Fig G.31 Effects of type of the inflow profile on the DDES simulation results, OP.4, plane S3 x+: downstream ……………………………………………………………….366 Fig G.32 Effects of type of the inflow profile on the DDES simulation results, OP.4, plane S3 x+: downstream ……………………………………………………………….367 Fig G.33 Effects of type of the inflow profile on the DDES simulation results, OP.4, plane S4 x+: upstream ……………………………………………………………..……367 Fig G.34 Effects of type of the inflow profile on the DDES simulation results, OP.4, plane S4 x-: upstream …………………………………………………………………...368 Fig G.35 Effects of type of the inflow profile on the DDES simulation results, OP.4, plane S4 x+: downstream ……………………………………………………………….368 Fig G.36 Effects of type of the inflow profile on the DDES simulation results, OP.4, plane S4 x-: downstream ………………………………………………………………..369

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Nomenclature

Latin Symbols

Az

Cross-section area of each section perpendicular to z direction [m2]

AInlet

Cross-section area of the inlet [m2]

aN

Off-diagonal coefficients in the discretized transport equation [-]

aP

Diagonal coefficient in the discretized transport equation [-]

aij

Amplitude tensor in Cholesky decomposition [m.s-1]

B

Constant in the non-dimensional relation of log-layer [-]

bij

Coefficients in linear combination of AFG signal [m.s-1]

bip

Bias added for each neuron [-]

CDES

Detached eddy simulation model constant (  0.65 ) [-]

CS

Smagorinsky constant (  0.1  0.2 ) [-]

C

Empirical Coefficient in eddy-viscosity model [-]

cv

Constant in the LES-index formulation [kg.m-3]

D

Diameter of hydro-turbines [m]

D Dt

Material (substantial) derivative (   t  u j  x j ) [s-1]

d

Wall distance in Spalart-Allmaras model [m]

d

DES length scale [m]

E (ks )

Energy in the modified von Karman spectrum [m3.s-2]

E ()

Difference (error) measure between PIV-2D field and numerical counterpart [-]

F

Mass flux [kg.s-1]

fd

An intermediate function defined in DDES [-]

xxxi

fij

Uncorrelated random fields [-]

fc

Cut-off frequency [s-1]

g

Acceleration of gravity [m.s-2]

H

Water head [m]

kV

Von-Karman constant (  0.41 ) [-]

k

Turbulent kinetic energy per unit mass [m2.s-2]

k nj

Wave number vector in Davidson-Billson AFG method [m-1]

k1

Smallest wave number in the turbulence spectrum [m-1]

ke

Most energetic eddy wave number in the turbulence spectrum [m-1]

kmax

Highest wave number in the turbulence spectrum [m-1]

ks

Wave number in the modified von Karman spectrum [m-1]

k

Intermediate wave number in the modified von Karman spectrum [m-1]

Lt

Turbulent length scale [m]

LES  IQv

LES index indicating LES content of the simulation [-]

l

Largest length scale in the flow field [m]

lm

Mixing length scale [m]

m

Time step number in time digital filtering [-]

m

Mass flow rate [kg.s-1]

N

Runner rotation speed [rev.s-1]

ns

Specific speed [-]

nv

Constant in LES index (=0.53) [-]

 n

Unit vector pointing outward of a surface [-]

n

Iteration number in ANN learning [-]

nC

Time factor in the cavity test case [-]

N11

Speed factor [-]

xxxii

Ne

Number of eddies at inlet plane in SEM [-]

NK

Number of modes in Kraichnan AFG method [-]

Nf

Number of Fourier modes in Davidson-Billson AFG method [-]

NC

Number of cores hired for parallel computing [-]

p

Static pressure [N.m-2]

P

Total pressure [N.m-2]

pD

Scaling factor of most energetic eddy to the largest eddies [-]

p z

Average pressure of the four azimuthal mean pressures (sensors) [N.m-2]

p Inlet

Average pressure of the four azimuthal mean pressures at inlet [N.m-2]

p,ij

Hessian of pressure [N.m-4]

q

Fluid kinetic energy [N.m-2]

Q

Second invariant of velocity gradient tensor

Qf

Flow rate [m3/s]

r

Radial component in cylindrical coordinate system [m]

RP

RHS of the discretized transport equation [-]

rd

An intermediate parameter defined in DDES [-]

ri

Random white-noise signal [-]

Re

Reynolds number [-]

Re

Reynolds number based on momentum thickness of B.L. [-]

RInlet

Radius of the draft tube cone at the inlet plane [m]

RB

Radius of the section B after the hub [m]

Sw

Swirl number [-]

S

Source term in the general transport equation [-]

S , S

Strain rate tensor (symmetric part of velocity gradient tensor) [s-1]

Tt

Turbulent time scale [s]

xxxiii

T

Time interval FTLE [s]

u

Friction velocity [m.s-1]

 U

Velocity vector [m.s-1]

U i, j

Velocity gradient [s-1]

uref .

Reference velocity [m.s-1]

ui

Mean velocity component (i=1, 2, 3 corresponding to x,y,z directions) [m.s-1]

u

Non-dimensionalized velocity in the wall units [-]

ui

Turbulent velocity signal [m.s-1]

uim

Mean velocity in RANS and URANS [m.s-1]

ui

Coherent part of turbulent velocity signal [m.s-1]

uil

Low frequency fluctuating part of the turbulent velocity signal [m.s-1]

ui† (t )

Unsteady part of the velocity signal in the Reynolds triple decomposition [m.s-1]

ui

High frequency fluctuating part of the turbulent velocity signal [m.s-1]

us

Sub-grid scale (residual) fluctuations in LES [m.s-1]

us

Velocity component parallel to the wall in ERCOFTAC diffuser [m.s-1]

u

Resolved velocity in LES [m.s-1]

u x

Lateral component of velocity defined in PIV-planes [m.s-1]

u y

Vertical component of velocity defined in PIV-planes [m.s-1]

uz

Streamwise component of velocity defined in PIV-planes [m.s-1]

uˆ n

Modified von Karman spectrum amplitude [m.s-1]

u

Circumferential velocity fluctuation [m.s-1]

u z

Axial velocity fluctuation [m.s-1]

ur

Radial velocity fluctuation [m.s-1]

VP

Volume of the control volume [m3]

xxxiv

vip

Activation potential of neuron i in ANN [-]

wip, j

Synaptic weights in an ANN inter-connection [-]

 x

Position vector in flow computational domain [m]

( x, y )

Position at the draft tube inlet plane

x jp

Neuron input in the neuron model [-]

y

Non-dimensionalized distance from the wall in the wall units [-]

y

Distance from the wall [m]

y jp

Neuron output in the neuron model [-]

zˆI

Separation distance in the definition of two-point correlation [m]

Greek Symbols



Fluid density [kg.m-3]



Dynamic viscosity [Pa.s]



Turbulent eddy dissipation [m2.s-3]

 ji

Sign on vortex j in i dir. (independent random steps of values +1 or -1) [-]



Vorticity [s-1]



Dimensionless power coefficient [-]

D

Diffusion Coefficient [m2.s-1]

 FTLE

FTLE (or equivalently DLE) field [s-1]



Hydro-turbine efficiency [-]

l

Learning rate in ANN learning process [-]



Recovery coefficient [-]



Experimental recovery coefficient [-]

C G max

Eigenvalues of the finite–time Cauchy-Green tensor [-]

xxxv

 ANN

ANN activation function [-]

 is+1

Local gradient at neuron i in layer (s +1) [-]

 pattern

2D pattern of the inflow radial velocity field [-]

L

Loss coefficient [-]

 j ,k

Discrete wavelets [-]

n

Phase shift in Davidson-Billson AFG method [-]

 Kol .

Kolmogorov length scale [m]

ij

Random amplitude in the Kraichnan AFG method [m.s-1]

WZ

Amplification factor of the turbulent viscosity in the near-wall zone [-]

 i , j ,k

Wavelet coefficients [m.s-1]

 Runner

Runner opening angle [-]

 radial

Scaling parameter in ANN-based radial velocity prediction [-]

 s1

Relative amplitude of the radial to axial fluctuations [-]

s2

Relative amplitude of the circumferential to axial fluctuations [-]

r  scaling

Scaling factor of the fluctuations in radial direction [-]

  scaling

Scaling factor of the fluctuations in circumferential direction [-]

z  scaling

Scaling factor of the fluctuations in axial direction [-]

i j

Random amplitude in the Kraichnan AFG method [m.s-1]

 kj

Wave number vector component in the Kraichnan AFG method [m-1]

 kj

Phase vector component in the Kraichnan AFG method [s-1]



Boundary layer thickness [m]

v

Viscous length scale [m]

 ij

Kronecker delta [-]

 Intermittency

Reverse-flow intermittency [-]

xxxvi



Distance between stretched points in time interval T in the FTLE [m]

 in

Fourier mode direction in Davidson-Billson AFG method [-]

W

Wall shear stress [N.m-2]

 ij

Reynolds stress tensor (   uiuj ) [kg.m-1.s-2]

 ijS

Sub-grid scale (SGS) stress tensor [kg.m-1.s-2]

v

Kinematic viscosity [s-1.m2] ( v    )

vt

Turbulent eddy viscosity [s-1.m2]

v

Modified turbulent viscosity in SA model [s-1.m2]

vSGS

Sub-grid scale turbulent viscosity [s-1.m2]

vDES

DES turbulent viscosity [s-1.m2]

veff .

Effective viscosity in the LES-index formulation [s-1.m2]

vnum

Numerical viscosity in the LES-index formulation [s-1.m2]



Azimuthal angle component in cylindrical coordinate system [-]

m

Momentum thickness [m]

C

Cutting-angle of the planes to study separation bubble [-]

2

Negative Eigen value of the corrected Hessian of pressure tensor [-]



Passive scalar quantity [-]



Diffusion coefficient [-]

t

Time step [s]

 C G

Finite-time Cauchy-Green deformation tensor [-]



Grid spacing measure [m-1]

f

Filter width in LES method [m-1]



Coefficient in the time digital filtering [-]



Coefficient in the time digital filtering [-]



Constant in Reichardt turbulent viscosity model (  11 ) [-]

xxxvii



Scaling factor for Reichardt turbulent viscosity model [-]

V

Cell volume [m3]

 †,norm ii

Normalized two-point correlation for unsteady term ui† (t ) [-]

iinorm

Normalized two-point correlation for random turbulence term ui(t ) [-]



Rotation rate tensor (anti-symmetric part of velocity gradient tensor) [s-1]

Subscripts/ Superscripts

Norm.

Normalized quantity

CS

Control surface

CV

Control volume

DB

Database

net

Overall (net)

V

Von-Karman

List of abbreviations

ACEnet

Atlantic Computational Excellence Network

AFG

Artificial Fluctuation Generation

ANN

Artificial Neural Network

APG

Adverse Pressure Gradient

B.C.

Boundary condition

B.L.

Boundary Layer

BulbT

Bulb turbine project at LAMH

CFD

Computational Fluid Dynamics

CFL

Courant-Friedrichs-Lewy number

CLUMEQ

Consortium Laval, Université du Québec, McGill and Eastern Québec

CPU

Central Processing Unit

xxxviii

DDES

Delayed Detached Eddy Simulation

DES

Detached Eddy Simulation

DLE

Direct Lyapunov Exponent

DNS

Direct Numerical Simulation

ERCOFTAC European Research Community on Flow, Turbulence and Combustion Exp.

Experimental

FDM

Finite Difference Method

FEM

Finite Element Method

FSI

Fluid Structure Interaction

FTLE

Finite Time Lyapunov Exponent

FVM

Finite Volume Method

GCC

GNU Compiler Collection

GIS

Grid-Induced Separation

HPC

High Performance Computing

HPCVL

High Performance Computing Virtual Laboratory

HSV

Horseshoe Vortex System

IBVP

Initial Boundary Value Problem

ICC

Intel C++ Compiler

ILES

Implicit Large Eddy Simulation

LAMH

Hydraulic Machinery Laboratory at Laval University

LCS

Lagrangian Coherent Structures

LDV

Laser Doppler Velocimetry

LES

Large Eddy Simulation

LHS

Left-hand side

LLM

Log-Layer Mismatch

LMFN

Numerical Fluid Mechanic Laboratory- Laval University

LUD

Linear Upwind Differencing Scheme

MLPR

Multi-Layer Neural Network

MSD

Modeled Stress Depletion xxxix

ODE

Ordinary Differential Equation

OP

Operating Points

OpenFOAM Open Field Operation and Manipulation PDE

Partial Differential Equation

PISO

Pressure Implicit with Splitting of Operators

PIV

Particle Image Velocimetry

POD

Proper Orthogonal Decomposition

QUICK

Quadratic Upwind Differencing Scheme for Convective Kinetics

RANS

Reynolds Average Navier-Stokes Simulation

RHS

Right-hand side

RMS

Root Mean Square

RQCHP

Réseau Québécois de Calcul de Haute Performance

RSM

Reynolds Stress Model

S-A

Spalart-Allmaras RANS turbulence model

SEM

Synthetic Eddy Method

SGS

Sub Grid Scale

SHARCNET Shared Hierarchical Academic Research Computing Network SIMPLE

Semi-Implicit Method for Pressure-Linked Equations

URANS

Unsteady Reynolds Averaged Navier-Stokes Simulation

VLES

Very Large Eddy Simulation

WestGrid

Western Canada Research Grid

WZ

Wall-Zone (region close to the draft tube wall at the inflow plane)

ZPG

Zero Pressure Gradient

xl

Dedicated to The messenger of kindness

Prophet Muhammad

(peace be upon him)

& To all kindhearted & caring people all over the world

In the past, present & in the future

“Seek knowledge from the cradle to the grave” Prophet Muhammad

(570-632)

“Conduct yourself in this world as if you are here to stay forever, and yet prepare for eternity as if you are to die tomorrow” Prophet Muhammad

(570-632)

xli

xlii

Acknowledgments “Only a life lived for others is a life worthwhile.” Albert Einstein (1879-1955)

First of all, I would like to prostrately thank our creator, to whom all praise belongs, for donating me the joy of existence and giving me the ability to seek his signs in the extremely stupendous nature. Hope to live in the rest of my life in such a way that God likes the most by helping others. This work has been conducted in the Hydraulic Machines Laboratory (LAMH) at Laval University under supervision of Prof. Claire Deschênes, undoubtedly among few top labs in the field of hydro-turbines worldwide. I would like to express my deepest gratitude to my thesis advisor, Prof. Claire Deschênes, for her invaluable advice, vital encouragements and endless support during my Ph.D. study period. I have really learned from her not only science but also how to be a professional researcher and a kind and caring person at the same time. She taught me how to become a memorable teacher and a symbol of moral values for the students. I am also indebted to my co-advisor, Prof. Guy Dumas for his proper guidance and meticulous comments, which played a crucial role in this accomplishment. Indeed, I have learned a lot from him. I would also like to thank Mr. Sébastien Houde for his helpful discussions on different aspects of the project. I am also grateful to all my thesis committee members including Dr. Maryse Page, Prof. Håkan Nilsson and Prof. Alain De Champlain for their valuable comments and suggestions. Indeed, their presence has made this work more valuable. I also had a unique chance to work with some talented and friendly colleagues in LAMH, all of them are sincerely appreciated for their help and interesting discussions on different aspects of fluid mechanics and more importantly due to their friendship, including Vincent Guénette, Edwin Roman Ortiz , Dr. Gabriel Ciocan, Jean-David Buron, JeanPhilippe Taraud, Dr. Jean-Mathieu Gagnon, Julien Vuillemard, Maxime Coulaud, Mélissa Fortin, Dr. Monica Iliescu, Philippe Gouin, Dr. Pierre Duquesne and Quentin Longchamp, Richard Fraser, Sébastien Beaulieu, Sébastien Lemay, Dr. Vincent Aeschlimann and also LMFN (Numerical Fluid Mechanics Laboratory) research group members, especially Jeanxliii

Christophe Veilleux, Dr. Mathieu Olivier, Philippe B. Vincent and Simon Lapointe are truly appreciated. You all will be in my heart and memory forever. Mr. Normand Rioux from computer technical support center of Laval University and Mr. Maxime Boissonneault from Colosse cluster team are sincerely acknowledged for their excellent technical-computer support in the whole of my Ph.D. study at Laval University. I also wish to thank my dear friends Ahmad, Afshar, Ali, Hossein, MohammadHossein, Nami, Nima, Reza, Saeed, Salman, Shahab and Yousef for their friendship, help and support during my residing in the beautiful Québec City. I really owe to the numerous people in my life related to the place that I am standing now including all of my teachers in the past including elementary, intermediate and high schools and my professors at university level in my lovely motherland, Iran. Unfortunately, it is not possible for these few pages to name all. Among them, I would like to emphasize on the exceptional role of my deceased father, Shahmir, a wonderful teacher who taught me first to be kind and caring to others and behaved as a role model for me in the whole of my life and also love and care I have unconditionally received from my mother, Mahnaz, and my only sister, Farnaz, during the years since childhood. Last but an important one to me, I would like to express my hearty and deepest gratitude to my dear wife, Ameneh, undoubtedly without her love, patience, support, understanding and care this accomplishment was not possible. She kindly accompanied me all the time through thick and thin, encouraged me and endured me during my timeconsuming Ph.D. study, which was annoying sometimes, with exceptional understanding. As a colleague, she also provided me lots of new ideas and suggestions via her fruitful discussions. My success is shared with you, Ameneh. I would like to also acknowledge the participants on the Consortium on Hydraulic Machines for their financial support and contribution in this project including: Alstom Renewable Power Inc., Andritz Hydro Canada Inc., Hydro-Québec, Laval University, NRCan, Voith Hydro Inc. and Natural Sciences and Engineering Research Council of Canada (NSERC). My sincere appreciation also goes to Compute-Canada parallel processing facility team, specially Colosse and Guillimin cluster teams for providing adequate computing-time and enough computational resource allocation to this project, which made this study possible.

xliv

Chapter 1

Introduction

1.1 Preface Nowadays, computational fluid dynamics (CFD) is extensively adopted to study fluid flows encountered in engineering applications. In the field of hydraulic turbine engineering, the capability to estimate the behaviour of the draft tube is crucial for estimation of the whole hydro-machine performance. In general, CFD prediction of the draft tube flow is a challenging problem due to the complexity of flow originated from different factors such as high Reynolds number, inlet swirl, adverse pressure gradient and presence of vortex rope, separation zones, horseshoe vortices and reversed flow recirculation regions. In this research, with the aid of parallel processing facilities, high Reynolds number flow within the draft tube of low head turbines is investigated in details using advanced hybrid turbulence models. This chapter describes motivation, objectives, methodology and organization of this thesis. Furthermore, it states general overview and related topics to the investigated problem.

1.2 Overview of the hydro-power plants The demand for energy has been increasing in the world due to population growth, industrial needs and economic development. Water power is the most reliable and widely used source of renewable and green energy nowadays. In general, the water power is obtained by conversion of energy of the flowing water into more easily used electrical energy without any carbon dioxide emission. Hydropower produces about 60% of the total 1

electricity production of Canada and accounts for 6% of the total U.S. electricity generation. Figure (1.1) indicates the contribution of different sources of electricity production in Canada. According to the data of the Natural Resources Canada, the province of Québec has the largest installed hydroelectric capacity among all provinces of Canada, feeding 98% of its needs. The water power production methods can be classified in two most important categories: conventional hydropower and marine or bio-kinetic hydropower. The concept of conventional hydropower is relatively easy. Dammed water flowing through a turbine makes it to rotate. The rotation of the runner turns the shaft of the electrical generator, which produces electricity. Marine and hydro-kinetic technologies transform the energy of ocean currents, waves and ocean thermal gradient to the electricity. The bulb group of turbines, which belongs to the conventional hydropower group, is a low-head reaction turbine presenting a high flow rate and high range of specific speed. In this type of turbines, the diffuser (draft tube) becomes an important part of the energy extraction process by increasing significantly the available head of the turbine. In these turbines, the draft tube is often shortened to reduce the civil work, thus the diffuser angle increases to such a level that the adverse pressure gradient might lead to a flow separation near the walls or in areas of transitions between circular and rectangular sections. In addition, the vortical flow which comes from the runner into the draft tube may contain a flow separation in the runner section on the hub and shroud walls. Therefore, the flow in the draft tube becomes really complex to analyze numerically. On the other hand, in these days, with emerging of fast computers, parallel processing facilities and development of appropriate hybrid turbulence models, it is quite feasible to simulate complex flows encountered in engineering applications mostly at high Reynolds numbers with resolving methods like hybrid RANS/LES, LES and in future even DNS.

Fig 1.1 Electricity source in Canada in 2009 (adopted from library of parliament of Canada; http://www.parl.gc.ca)

2

In general, computational fluid dynamics (CFD) is a promising tool for parametric studies of the engineering systems like hydroturbines. It is much less costly in comparison to experimental tests and more flexible. In fact, experimental results provide a limited number of quantities for a pre-defined fraction of the domain. In contrast, CFD simulations provide detail studies of the flow field on the whole domain. Ideally, the ultimate goal of the CFD is to serve as a “numerical test rig” but errors that originate from different sources such as physical modelling, discretization, numerical algorithms, etc. always limit the accuracy, thus effectiveness of all numerical simulations should be proved via experimental results. This is especially true far from the best efficiency point (BEP), where the flows are really complex. This thesis concerns the numerical study of unsteady flow inside the draft tube low-head bulb turbines. The details are presented in the upcoming sections. 1.2.1

Conventional hydro-power plants

To understand the function of conventional hydro-turbines, the water cycle is an essential concept that should be understood. Water of the sea and ocean surfaces evaporates by the solar energy and then the evaporated water is condensed in the form of clouds and falls back to the earth surface in the form of rain or snow. Ultimately, by the effect of gravitational force, the water comes back to the ocean and sea through rivers and the cycle repeats itself. In the conventional methods, the river water energy can be stored in dams. Figure (1.2) presents the schematic of a hydro-turbine site with a reaction turbine. In an arrangement of a hydro-turbine site, typically, the released dammed water flows through the penstock or intake, guide vanes and runner blade passages and pushes the runner blades, which ultimately makes them to rotate. Finally, the water flows through the draft tube before entering to the tail-water.

Generator

Guide vanes

Head water

Tail water Penstock Runner

Draft tube

Fig 1.2 Schematic of a hydro-turbine site (modified from Hydro-Québec) 3

The penstock or intake length varies according to the required arrangement of the site. Function of the spiral casing and guide-vanes is to transform the flowing flow into a pure vortex and to distribute the flow around the runner as symmetric as possible. The runner absorbs the flow swirl and the degraded flow enters in the draft tube at the final stage before entering in the tail-water. At the best efficiency point, the runner absorbs most of the swirl of the flowing water, but in off-design points a remained swirl passes through the draft tube and affects considerably the fluid dynamic involved. The draft tube is the most important part of a hydraulic turbine after the runner that plays a key role in the machine performance behaviour. 1.2.2

Different types of the hydro-turbines

As mentioned, hydraulic turbines are used to convert the energy of flowing water to the more useful electrical energy. These green energy convertors have specific advantages compared to the thermal power stations using gas-turbines and steam turbines as well as nuclear power-plants. High efficiency, renewable energy generation characteristics and low side effects on the environment are some of their advantages. Regarding mechanisms of the energy generation involved, the hydraulic turbines can be classified in different types as shown in Figure (1.3).

Fig 1.3 Classification of hydraulic turbines based on different energy generation mechanisms The selection of a special type of hydraulic turbine suitable for use in a particular site depends on the available head and flow rate. The specific speed, ns, of a turbine characterizes the turbine's shape regardless of the turbine size and permits the comparison of turbines with different sizes and types. In practice, ‘specific speed’ is defined in nondimensional form, as below: ns 

4

1 1 P 2 Q f 2 1

 2H

5

4

(1.1)

where P , Q f ,  and H stands for shaft power, flow rate, fluid density and head, respectively. For practical purposes, a ‘speed factor’ is also defined as follows:

N11 

ND H

(1.2)

where D , N represents the runner diameter and runner rotation speed, respectively. Figure (1.4) indicates the usage of different hydraulic turbines depending on different site conditions. For impulse turbines e.g. Pelton, the reaction of impingement of flow stream exiting from nozzles with high velocity on the buckets makes the runner to rotate (figure 1.5). This type of turbines is appropriate for high head and low specific speed flow conditions. Reaction turbines do not need a high head, and work within intermediate to high specific speeds. In this type of turbines, the pressure of flowing water through the hydraulic passage makes the runner, which is immersed completely in water, to turn. As mentioned in Figure (1.3), there are two major types of reaction turbines including radial and axial flow turbines. Radial flow turbines e.g. Francis turbines, operate within 30 m to 500 m total head; in these turbines, the flow comes in the radial direction. After the runner, the flow turns about 90 degrees within the draft tube and enters in the tailrace. For vertical axial flow turbine, there is also a similar bend in the draft tube to change the flow direction from vertical to horizontal. In general, the presence of this 90 degree bend increases the risk of draft tube loss and of course the turbine as a whole in comparison to horizontally installed hydro-turbines. In the axial turbines, the flow direction through the runner blade is parallel to the axis of runner rotation.

Fig 1.4 Graph of hydro-turbine selection (courtesy of Voith Hydro Inc.) As illustrated in figure (1.4), the axial flow turbines are used for a high flow rate range, so they should be strong enough to withstand against a high loading created by fluidsolid interaction in this range of flow rates. This characteristic dictates the usage of long 5

blades with large chords in practice, which results in the pitch/chord ratio in the range of 11.5 and implies 4-6 bladed runner [Sayers 1990]. As shown in figure (1.5), bulb turbines are used in very low head (2-30m) and high specific speed conditions. Topology of this kind of axial turbines involves a relatively large bulb containing an electrical generator in the hydraulic passage ahead of the runner. Presence of the bulb in the water passage has advantages and disadvantages. The configuration of the turbine will be more compact than other types of hydro-turbines and provide better cooling of the electrical generator. On the other hand, in the bulb configuration, probable maintenance of the generator will be more problematic, and the risk of water pollution increases, for example by oil leakage. The shape of the bulb component is designed to provide a minimum loss and disturbances on the flow entering the distributor (guide vanes) and the runner.

Fig 1.5 Schematic of the different types of hydro-turbines [Ingram 2009] Draft tubes are always shortened in bulb turbines to reduce the construction cost and to compact the system as much as possible, as mentioned. This configuration results in high values of draft tube divergence angle, which may lead to a flow separation. In general, the draft tubes of bulb turbines are designed as a straight diffuser, which reduces loss of the component considerably in comparison to the other types of hydro-turbines. However, the design of this component is more crucial for low head turbines, because a relatively small additional loss in the draft tube creates a large degradation of the turbine efficiency. Furthermore, to have a less civil engineering and excavation work and to lower the cost of

6

hydro turbine site establishment, there is usually a transformation from a circular to a rectangular shape in the draft tube topology. From a hydraulic point of view, this feature adds a more complexity in the flow field, which increases the possibility of flow separation and corner vortex formation; as a result, unsteadiness increases in the flow field. To reinforce the structure of draft tube and lower harmful vibrations, especially for large draft tubes, piers are sometimes used as supports. In a hydraulic point of view, these obstacles create more loss and complexity into the flow and specially induce horseshoe vortices near the top and low surfaces of the draft tube [Shingai et al. 2008]. In fact, the boundary layers developed on the top and bottom induce lateral vortices, which are deflected in the vicinity of the piers and forms horseshoe vortices. Presence of these vortices creates even more unsteadiness in the flow field, although piers are not present in the BulbT draft tube. In the next subsection, a theoretical background for the estimation of draft tube performance and corresponding parameters are presented. 1.2.3 Hydro-turbine and draft tube performance

In this subsection, the theoretical background of hydro-turbine and draft tube performance computation, and important corresponding parameters, such as swirl number, recovery and loss coefficients are presented. For engineering systems with large number of describing variables, performing dimensional analysis can reduce the number of variables for experimental and numerical analyses and also allow to compare different turbines with different parameters, such as speed, size and fluid conditions. For hydro-turbines, i.e. turbomachines working with incompressible fluid, the shaft power ( P ) can be expressed as a general function of fluid and geometry parameters as below [Sayers 1990]: P  f   , N ,  , D, Q f , ( gH net ) 

(1.3)

where  , N ,  , D and Q represents fluid density, runner rotation speed, dynamic viscosity, turbine diameter and flow rate, respectively. The last term in the above equation indicates the net energy across the water turbine. By performing the dimensional analysis, important dimensionless parameters can be obtained, which are called power (  ), flow (  ) and head ( ) coefficients:



P ,  N 3 D5



Qf ND

3

,  

gH N 2 D2

(1.4)

Another similarity parameter is the Reynolds number ( Re =  ND 2  ). Based on dimensionless hydro-turbine coefficients, the overall efficiency of the hydraulic turbine ( ) is defined as follow [Sayers 1990]:

7



Power delivered to runner P     gQ f H net  Available power

(1.5)

As mentioned, the draft tube behaviour is crucial in the operation of whole turbine especially for low head turbines. The role of the draft tube is to convert the remaining kinetic energy entering into the draft tube to static pressure as much as possible. This can be achieved simply by increasing its cross-section from inlet to outlet (A7 < A8, as shown in figure 1.6). The created adverse pressure gradient resists flow motion in the low velocity boundary layer region and may result in a separation, which degrades the turbine efficiency. In a hydraulic turbine, the flow leaving the runner has a swirl. As proved by experiments [Clausen et al. 1987 & 1993] and numerical simulations [Page et al. 1996, Duprat et al. 2008] in the case of the ERCOFTAC conical diffuser, the presence of swirl stabilizes the boundary layer and prevents flow separation in the diffusers. On the other hand, presence of the inlet swirl leads to a vortex rope formation, which increases the unsteadiness and thus complexity of the flow field phenomena involved. The swirl number is an important characteristic parameter of the draft tube flow and is defined as the ratio of angular momentum flux of the entering flow into the draft tube to its axial momentum flux [Cervantes et al. 2007]. If z-direction is considered as the streamwise direction, then the parameter is defined as below in the cylindrical coordinates: R

Sw 

r

2

u z u dr

0

(1.6)

R

R  ru z dr 2

0

Figure (1.6) shows different sections of a bulb turbine installation in a real site. As mentioned, the draft tube increases efficiency of a hydraulic turbine by converting the remained kinetic energy from the runner to the static pressure. Writing a simple Bernoulli’s energy equation between the headwater and tail-water levels gives: H net  H1  h fp 

V8 2 2g

(1.7)

where H1 is the total head at the headwater level and h fp indicates the hydraulic loss of the flow passage from the headwater to the distributor (guide vanes) at the station 6. As it is clear in this equation, the net head available to the runner, and thus the outcome of hydroturbine site, is developed by the reduction of the outlet velocity of draft tube. That is why the draft tubes in the hydro-turbine topology are designed to reduce the outlet speed.

8

1 2 3 4

Head water Inlet trash racks Stoplog slots Generator bulb

5 Upper and bottom pillars 6 Distributor 7 Runner 8 Tail-water

Fig 1.6 Schematic of a bulb turbine installation in a real site [Keck et al. 2008] There are two major coefficients to characterize the draft tube performance called recovery coefficient,  , and loss coefficient , L .To achieve a well-designed draft tube, there should always be a trade-off between these two factors. The recovery factor indicates how efficient a draft tube is to increase the static pressure within the flow passage. For a diffuser with a pure axial inlet velocity, the ‘recovery coefficient’ is defined, as follows:



O 

 poutlet  pinlet m Soutlet  qaxial

     pU  n dS  pU  n dS m S  inlet

1 2 U axial 2

(1.8)

where p , qaxial and m indicates the static pressure, kinetic energy only involving the axial component of velocity and mass flow rate, respectively. This definition over-estimates the recovery factor coefficient for draft tube flows because there are two other velocity components in the circumferential and radial directions, which contribute to the total inlet kinetic energy beside the axial velocity component. An appropriate generalized definition of the recovery coefficient, applicable to the swirling flows is defined as follows [McDonald 1971]:





 poutlet  pinlet m Soutlet  qtotal

     pU  n dS  pU   n dS m Sinlet   1   U  U dS 2Sinlet Sinlet

(1.9)

9

In fact, the presence of circumferential and radial components of the velocity increases the friction loss of the draft tube. A loss coefficient can be defined based on the total pressure difference between the inlet and outlet of the component. For the draft tube, the ‘loss coefficient’  L is defined as follows:      PU  n dS  PU  n dS  m Sinlet m Soutlet    1   U  U dS 2Sinlet Sinlet



L 

Pinlet  Poutlet qtotal



(1.10)

where P stands for the total pressure. In general, fluid flow phenomena like the flow separation and vortices are the major loss sources in the draft tube performance. The guide vanes and specially the runner are usually replaced by more efficient or powerful ones in the rehabilitation process of old hydro-turbine sites, however the old draft tube is still utilized in the new configuration to avoid an expensive replacement of parts enclosed in the concrete infrastructure of the hydro-turbine site. In some circumstances, this new configuration shows an efficiency drop near the best efficiency operating point of the hydro-machine [Tridon et al. 2008]. Therefore, the ability to predict the draft tube performance is essential before starting the rehabilitation of the old site. To do so, CFD simulations can be performed with a reasonable cost in comparison to experimental tests. On the other hand, to establish a reliable virtual test-rig based on computational fluid dynamics for industrial applications, it is crucial to validate numerical simulation strategies in different flow conditions within the operating point zone of the hydro-turbines. In this project, the fluid flow within the draft tube of a very low head turbine i.e. bulb turbine is investigated numerically in details.

1.3 BulbT geometry and the defined coordinate system After successful experience with the AxialT project, the BulbT project was initiated in 2011 within the framework of an international research partnership in the Hydraulic Machines Laboratory (LAMH) of Laval University [Deschenes et al. 2010]. The goal of the BulbT project was to experimentally and numerically characterize the flow field in a low head model-scale bulb turbine. To perform measurements, a model-scale of the bulb turbine was constructed, which is called ‘BulbT’ hereafter, and was installed on the LAMH test bench. Figure (1.7) shows a cut view of the BulbT assembly. As one can see in this figure, there is an intake part, which brings the water flow from an upstream reservoir into the bulb section. For installation of the bulb in the middle of the hydraulic passage, two pillars are installed on the top and bottom of the bulb unit. There are also 16 guide vane blades before

10

the runner section forming the distributor, which regulates the flow rate and induce an angular momentum. In addition, there is a four-blade runner in the turbine assembly. The blade angles are adjustable to absorb the flow energy as much as possible in the different flow conditions; the blades are adjustable between 10 to 38 degrees. After the runner section, the flow enters into the draft tube component, which consists of two parts: a conical cone and a transition from a circular to rectangular section. The cone part is made of acrylic to permit the flow visualization, LDV and PIV measurements via the optical accesses [Vuillemard 2015]. In the non-transparent transition part, there are also some access windows, which are adopted for the PIV measurements [Duquesne 2015].

Top pilar

y x

Guide vanes

o

Runner blade

Intake

z

Bulb Conical part Transition part (trumpet) Transition (trumpet)

Acrylic cone

Pressure sensors

Fig 1.7 A cut view of BulbT assembly installed on the LAMH test-bench (top); a photo of test bench showing the BulbT draft tube (bottom) 11

Figure (1.7) also illustrates the global coordinate system adopted in the BulbT project. The origin of the coordinate system is located on the axisymmetric axis of the runner. As seen, Z is the axial direction, Y is the upward direction and X represents the lateral direction.

1.4 Motivation To understand the flow physics within the BulbT project framework, the flow field in the different parts of the BulbT turbine were studied both experimentally and numerically. For instance, the flow field in the intake channel of the turbine was studied using the LDV measurement data obtained on a pre-defined plane positioned in the intake along with the numerical k   simulation results [Longchamp 2014]. Furthermore to understand the flow behaviour in the inter-blade channel of the runner, measurements were conducted by LDV and stereoscopic PIV techniques [Lemay 2014]. The interesting results of this thesis show the formation of runner-linked vortices that are convected into the draft tube. Since the start of BulbT project, a special emphasis was placed on the understanding of the fluid flow phenomena in the draft tube component; in this regard, both experimental and numerical methods are adopted to study the flow field in the draft tube. It is important to mention that at the beginning of the project, the draft tube geometry- called ‘basic geometry’ hereafter- involved a relatively mild divergence angle i.e. 6.6 in the conical part. After a while, by progressing of the project, the draft tube geometry was modified in a way to have a more aggressive divergence angle i.e. 10.25 , in order to better reproduce more challenging fluid flow phenomena, like a chaotic separation on the draft tube walls; the new geometry is called ‘final geometry’ hereafter. It is also important to emphasize that all experimental measurements like those explained above were performed on the BulbT assembly involving ‘final geometry’ of the draft tube, whereas the ‘basic geometry’ configuration suffers from the lack of experimental data. The turbine hill-chart was initially obtained in the case of the ‘basic geometry’ of the draft tube with the aid of numerical simulations. The flow field in the full-machine assembly involving the ‘basic geometry’ was simulated using the k   RANS turbulence model [Guénette 2012]. Then the full-machine flow simulations with the ‘final geometry’ of the draft tube were also considered for five selected operating points [Houde et al. 2014]. The hill-chart was also measured on the LAMH test bench [Duquesne 2014-1]. Due to the importance of the draft tube component in the behaviour and performance of the low-head turbines like the BulbT turbine, a special attention was made 12

both experimentally and numerically to perform the detail investigations of draft tube flow. In this regard, 1D circumferential-averaged and 2D velocity profiles were measured at the inlet section of the draft tube. These data can be used for both comparison and correction purposes as seen in the following chapters of this thesis. In addition, the velocity profile downstream of the runner-hub was also measured [Vuillemard 2014]. The wall-separation and other related phenomena in the draft tube was studied in details experimentally using PIV measurements on certain planes defined in the draft tube [Duquesne 2014-3, 2015]. These data will be used for the validation of numerical simulations. In pursuing the goals of BulbT project, the present study is an attempt to simulate the unsteady turbulent flows in the draft tube with the aid of a hybrid turbulence treatment, namely delayed detached eddy simulations (DDES). The DDES adopted in this project is formulated based on Spalart-Allmaras (S-A) turbulence model. In the literature, URANS simulations with different turbulence models like k   and SST are widely used to simulate the draft tube flow; examples are reviewed in chapter 2. In contrast to RANS and URANS techniques, a portion of the turbulence spectrum can be resolved by hybrid methods, whereas the computational cost of the technique is lower than pure LES techniques. The comparison of the turbulence treatment techniques are presented in chapter 2. To mention briefly here, in a hybrid approach adopted for the simulations of wall-bounded flows, the wall-zone is left to URANS model whereas the LES-content is directly resolved in the core flow. The literature shows that DDES turbulence treatment has been recently adopted to simulate swirling flows. For example, Paik et al. studied strongly swirl flows through an abrupt expansion using DDES simulations via an in-house code [Paik et al. 2010]. In this study, a 1D inflow profile containing a flow swirl was imposed at the inflow section. In the case of stand–alone simulation of the draft tube for example, the draft tube of a Francis turbine was studied using DDES turbulent treatment in OpenFOAM [Vincent 2010]. In the study, the circumferential-averaged 1D inflow profile was applied at the draft tube inlet section just after the runner. The results showed the superiority of DDES over URANS simulations in resolving turbulent flow structures. In another attempt, a flow field in a swirl generator was simulated with the aid of DDES in the OpenFOAM platform using a fixed velocity value at the inlet section [Gramlich M. 2012]. More recently, the flow field in the draft tube of a Francis turbine was simulated using DDES in ANSYS software considering 2D variation of the draft tube inflow profile. The results showed more complete representation of the turbulent flow structures especially the structures under the runner of the Francis turbine [Beaubien 2013]. In this project, the OpenFOAM code is adopted to perform DDES simulations of the turbulent flow in the BulbT draft tube at high-Re number. Details about the 13

OpenFOAM are presented in chapter 3, just to mention briefly here, the OpenFOAM is a C++ object oriented platform capable of almost any modification. OpenFOAM is a free-touse code with no license cost, in contrast to the relatively expensive commercial software packages. These features make it attractive for LES-type simulations, which typically need large number of cores in parallel computations. Moreover, the literature shows that there is a lack of stand-alone DDES simulation cases for the draft tube flow considering 2D variations of the inflow condition in OpenFOAM. Applying 2D profiles at the inlet is essential to resolve the runner-related vortical structure and wakes in the case of stand-alone draft tube simulations as seen in the upcoming chapters. In addition, the effect of applying synthetic inflow turbulence over the 2D profile has not been studied in the case of draft tube flow simulation using a hybrid turbulence approach, like DDES. As an attempt, but for LES simulations, Duprat studied the effect of synthetic inflow turbulence on a wall-modeled LES simulation. In the study, the effect of inflow turbulence for a swirling flow in ERCOFTAC diffuser was considered and good results were obtained compared to the experiment [Duprat 2010]. More research in this field is needed especially in the case of hybrid turbulence treatment techniques applied for the draft tube flow simulations. A comprehensive review on the different methods capable to generate an artificial turbulence is presented in chapter 4. The goal of the present project, in short, is to improve numerical simulation strategies using DDES hybrid turbulence treatments, in order to enhance the current knowledge of the turbulent unsteady flow through the draft tube of a low-head bulb turbine near BEP and at overload operating conditions.

1.5 Statement of the object This project aims to study a draft tube turbulent flow in the low head BulbT model turbine numerically by applying DDES turbulence treatment along with providing appropriate boundary conditions. Especially an accurate and appropriate inflow boundary condition becomes a key factor when moving towards unsteady CFD methods like DES or LES, resolving portions of the turbulent fluctuation spectrum. In this regard, DDES simulations for some selected operating points of the machine are considered. These are for a fixed runner blade angle and thereby along a propeller curve of the model. The simulations are performed in the OpenFOAM platform considering two selected operating points, namely OP.1 and OP.4, in the case of the ‘final geometry’ of the BulbT draft tube and the best efficiency point for the ‘basic geometry’. As these simulations require a large numerical effort, use of the CLUMEQ parallel processing

14

facilities including Colosse and Guillimin supercomputers, installed at Laval and McGill Universities respectively, was necessary. The strategy for this project is to start with the well-proven RANS simulations of the full machine including an intake, guide vanes, a runner and a draft tube. The inlet profiles are then extracted from these full-machine RANS simulations and along with their variants, are applied for the DDES simulations of the draft tube flow. By reducing the domain size to the draft tube only, or to the runner and draft tube, it becomes feasible nowadays to apply LES-type turbulence models, i.e. DES, to study the high-Re turbulent flows. In the present study, the computational domain is limited to the BulbT draft tube. In order to achieve this goal, an appropriate inflow boundary condition needs to be defined including a velocity field and turbulent viscosity or subgrid scale viscosity, turbulence length and time scales. Here various options were evaluated, namely the use of the results from the baseline calculation or usage of experimentally modified flow profiles obtained at the runner exit. Additionally, LES-type methods may require a prescription of velocity fluctuation levels and also time-dependent inflow conditions. The impact of the selection of inflow conditions on the CFD predictions is quantified in this project. The effects of applying circumferential-averaged 1D profile, 2D profiles with/without synthetic inflow turbulence are especially considered. This Ph.D. research was done in a collaboration with the BulbT experimental research to define appropriate measurement positions and required measurement quantities in order to validate the unsteady numerical results, but also to determine appropriate inflow conditions to the reduced-domain simulations. Also, the identification of flow features e.g. vortex breakdown, flow separation and their validation against the experimental findings, provided post-processing methodologies and procedures appropriate for the bulb turbine applications. Considering the above discussion, the main objectives of the thesis can be summarized, as follow:

1) To develop a numerical methodology to simulate unsteady high-Re turbulent flows within draft tubes of low head turbines using DDES hybrid turbulence modelling in the OpenFOAM platform. 2) To resolve the vortical structures and wakes ejected by runner into the draft tube using unsteady 2D inflow profiles in the OpenFOAM platform. 3) To study the impacts of applying different ‘1D’ and ‘2D’ inflow boundary conditions on the transient simulation of the BulbT draft tube flows using DDES turbulence treatment.

15

4) To assess the effects of synthetic inflow turbulence on the behaviour of the DDES, as a hybrid turbulence treatment, in the draft tube flow simulations. 5) To investigate physical fluid phenomena in the BulbT draft tube using numerical simulation data. 6) To compare simulation results to the experimental data. 7) To clarify fluid flow transport barriers inside the BulbT draft tube using Lagrangian coherent structures (LCS) concept.

1.6 Thesis outline This thesis has been prepared in eight chapters. First chapter involves general introduction to the topic, motivation and objectives. Chapters 2, 3 and 4 concerns methodologies utilized in the thesis. In chapter 2, the turbulence-modeling concept is introduced and its challenging aspects are discussed. The adopted turbulence model in the present research is described in details as well. Chapter 3 reveals the details of the numerical strategy adopted to simulate the flow field, such as the finite volume method, especially with an application to the hydro-turbine flows. In the chapter, different aspects of the flow solver and parallel computing are also discussed. Chapter 4 states the methodology developed to generate the unsteady inlet boundary conditions for the draft tube flow simulation in details. Chapters 5 to 7 present studies for three different draft tube geometries. In chapter 5, results of the flow field simulation within the ERCOFTAC conical diffuser are presented as an initial test case by comparison to the experimental data. The flow field within the draft tube of the BulbT turbine with the ‘basic geometry’ is assessed in chapter 6. No experimental data are available in this case, but different comparative simulations are performed to understand the effects of different simulation factors on the DDES simulation of BulbT draft tube. In chapter 7, the turbulent flow fields in the BulbT draft tube with the ‘final geometry’ are simulated using URANS and DDES methods at two selected operating points, namely OP.1 and OP.4. The results are also validated by comparison to the experimental data measured in the BulbT project (LAMH- Laboratoire de Machines Hydrauliques - Laval University), available in January 2015. Effects of the inlet boundary condition including synthetic turbulence on the flow field predictions are studied in details in this chapter, along with statistical analysis of the flow field, such as two-point correlation, energy spectrum and LCS analyses. The conclusions and listed contributions of the present research, as well as recommendations for the future directions are stated in the final chapter.

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Chapter 2

Turbulence treatment, state of the art

2.1

Introduction

Fluid flows encountered in the nature and industry above a critical Reynolds number ( Re ) are unstable. In the low Re number range, flows are stable and laminar; it means that any disturbance is damped due to the dominancy of the viscous effects. In relatively large Re number, some kind of random and chaotic motions of fluid particles are observed within the flow field, and the velocity and pressure field vary on and on in terms of time. NavierStokes equations govern the dynamics of the Newtonian fluid flow in all regimes including laminar, transitional and turbulent. With the aid of some simplifying assumptions, these equations can be reduced and an analytical solution is found for very simple problems; in general, the closed form solution of the Navier-Stokes equations is not on hand and remains an open problem to the mathematicians1. In fact, the probable closed form solution should reproduce the complex motion of the turbulent flows; this explains why it is difficult to obtain the analytical solution in general. To study the physical phenomena encountered in complex fluid motions in scientific and engineering applications, numerical solution of the Navier-Stokes equations can be used to gain appropriate insight into the fluid flow details. To do this, one should consider properly the effects of turbulence for most flows. In large enough Re number, the 1

Millennium Prize Problems, Clay Mathematics Institute (CMI), Cambridge, Massachusetts, USA. 17

resulting flow is always unsteady and time-dependent even under steady boundary conditions. In this situation, the flow characteristics change in terms of time non-stop and the flow is called ‘turbulent flow’. One of the most important characteristics of the turbulent flows is the chaotic nature of the phenomena and the presence of a wide range of temporal and spatial scales in motions of the fluid particles, which makes the turbulent flows so sophisticated to be understood and tackled. Undoubtedly, the turbulent flow prediction is a challenging problem. The chaotic and non-linear nature of the turbulent flows in general, along with the presence of solid walls in confined flows make the problem so intricate. The problem will be severe for flows with an adverse pressure gradient, separated flows and flows on the verge of separation. In the following subsection, important features of the wall bounded turbulent flows are described.

2.2

Turbulence in wall-bounded flows

In high Re flows, the effect of inertia is dominant to the viscosity, in other words, the viscous forces are not strong enough in comparison to the inertia forces to damp unsteadiness arising in the flow field. As a result, the unsteadiness in the different parts of the flow is amplified and interacts with each other to produce a chaotic and somehowrandom motion which is called ‘turbulence’. It should be emphasized that turbulence is not a completely random phenomenon; in fact it involves some hidden structures called ‘coherent structures’ that govern the turbulence dynamics. In the classical picture of the phenomenon, turbulent flows consist of eddies with different sizes from the largest (integral) scale comparable to the dimension of the flow field ( l ) to the smallest scale motion in the size of Kolmogorov scale ( Kol . ), which itself is much larger than the fluid molecular distance. This concept was introduced for the first time by Richardson [Richardson 1922] and quantified in details via some simplifying hypothesis by Kolmogorov [Kolmogorov 1941]. According to the energy cascade picture for an isotropic turbulence, the energy of large scale eddies is transferred to the smaller scales and finally is dissipated (converted to heat) by the effect of viscosity at the Kolmogorov scale level. At this level, viscosity smoothes velocity fluctuations and damps all the unsteadiness. At a large enough Re , the largest scale motion of the turbulence is independent of Re but the smallest scales decrease as Re increases. From Kolmogorov theory, by increasing the Reynolds number, the difference between the large and small scales widens, in mathematical form it can be written as below: l

 Kol . 18

 Re

3 4

(2.1)

where Re is defined based on the integral scale of the flow. As can be expected, for industrial flows which possess high Re number, according to equation (2.1) a wide range of scales is present in the turbulent flows. For instance, for hydro-turbine flows, the Reynolds number even for the model scale can be easily in the order of 106 or 108, thus the ratio between the largest to smallest scales of motion can be in the order of 106. It is worth mentioning that shear stress in the mean flow is the main source to maintain fluctuations in a turbulent flow field [Tennekes et al. 1972]. In the case of the confined (internal) flow, undoubtedly presence of the wall in the flow field adds more complexity to the dynamics of the turbulence. The ‘no-slip’ boundary condition at the wall provides a non-stop source of shear stress in the flow field. In fact, the classical theory of turbulence, essentially based on the local isotropy hypothesis (i.e. no directional preference), is not applicable in the anisotropic wall-turbulence region. For all external and internal wall flows, a boundary layer develops on the wall, where the viscous effects cannot be neglected. By increasing the Reynolds number, the boundary layer gets thinner and thinner but always stays as a source of turbulence generation in the flow field. The boundary layer region is traditionally composed of two different zones including the inner and outer regions. There is also an overlap layer which links these two regions. Inner region comprises the layers titled viscous sub-layer and buffer layer. To express the behavior of the near wall region, it is preferable to state all the quantities in wall units. To do so, the velocities and distances are scaled by a friction velocity ( u ) and viscous length scale (  v ) which can be defined as below [Pope 2004]:

W 

u 

v u

v 

(2.2) (2.3)

The mean velocity and distances from the wall ( y ) are expressed in wall units in the near wall region, as below [Pope 2004]:

y 

u 

y

v

u u

(2.4) (2.5)

In the inner region near the wall (typically y  v  0.1 , where  v is the boundary layer thickness), the behavior of the flow is approximately universal and can be expressed in generalized form based on y  and u  . The viscous sub-layer region is a very thin region 19

near the wall, inside the inner region ( y   5 ) where the effect of viscosity is dominant to the inertia and the contribution of the Reynolds stresses to the total shear stress is negligible. In this region the linear relation governs the behavior of the normalized mean velocity as below:

u  y

(2.6)

Figure (2.1) indicates the behavior of boundary layer flow in the near-wall region with the local equilibrium assumption. This assumption is completely valid in the case of zero pressure gradient (ZPG) flow [Lemay 2008]. In the logarithmic region ( y   30 and y   0.3 ), the relation below can be derived for the velocity distribution:

u 

1 ln y   B kV

(2.7)

where kV is the von-Karman constant and B is also a constant. There are some variations for the k and B values in the literature, typically it is assumed kV  0.41 and B  5.2 (Fig 2.1). The region which connects viscous sub-layer region to the logarithmic region ( 5  y   30 ) is called buffer layer. In fact, this region behaves as a transitional region which stitches viscosity-dominated and turbulence-dominated parts of the flow [Pope 2004]. Beyond the logarithmic region, which is called the wake region, the behavior of the flow depends on type of the flow and Re number and it is no longer universal. As mentioned, this classical picture of the near wall flow is valid under the assumption of local equilibrium between the production and dissipation of turbulence. It is worth mentioning that zero pressure gradient flows satisfy this assumption properly. Industrial flows like flows inside the hydraulic turbines usually experience adverse and favorable pressure gradient due to the cross-sectional area changes happening through the fluid flow passages. In a favorable pressure gradient region ( p s  0 ), the pressure helps the fluid particles to flow over the wall surface, thus it stabilizes the boundary layer and prevents the flow separation. In contrast, in the zones where the fluid particles are opposed to the adverse pressure gradient ( p s  0 ), pressure provides resistance to the fluid particle motion; this is especially critical inside the low speed zones of the flow as boundary layers. Eventually, fluid particles near the wall reach to a point where their energy is not enough to follow the wall curvature and they separate from the wall, this point is called separation point. After this point, a recirculation zone forms which involves a flow with reverse direction in comparison to the bulk flow (Fig 2.2). Under these circumstances, the classical picture of the boundary layer is not valid anymore.

20

Fig 2.1 Different layers in the near-wall region (modified from [Lemay 2008])

Fig 2.2 Boundary layer subjected to an adverse pressure gradient (APG) (modified from [Nakayama 2000]) The mechanism behind the generation of the turbulence in the boundary layer is not completely understood and remains as an active field of research nowadays. In the classical picture, it is postulated that the coherent structures i.e. hairpins or hairpin packets govern the dynamics of the turbulence and momentum transfer in the near wall region [Robinson 1991]. Theoretically, the hairpin formation mechanism can be explained by the concept of perturbation in the lateral vorticity line in the transional regime of a boundary layer. Above a critical Reynolds number, all disturbances created by noises are amplified, developed and 21

roll up to form the hairpin structures. In fact, creation of streamwise vorticity component due to the disturbances produces an upwash velocity component which leads to a hairpin nose up phenomenon and forms the head or arch of the hairpin. The velocity profile distribution in the boundary layer, from zero at the wall location to the free stream velocity at the boundary layer edge, causes different streamwise convection velocities for different portions of a lifted vorticity line. This process generates a stretched and (according to the Helmhotz rule) intensified vorticity line with two counter-rotating streamwise vorticity legs, this entity is called ‘hairpin’. Figure (2.3) shows a schematic view of a hairpin structure.

Fig 2.3 Schematic of a hairpin structure in the near wall-region It is postulated that hairpin structures produces high (HSS) and low (LSS) speed streaks in the buffer layer. The hairpin structures, which move with approximately the same convection velocity, form a hairpin packet in the near wall region (Fig 2.4). The hairpin and hairpin packets have a temporal coherency which means that they exist for a relatively long time in terms of time scale of the turbulent events [Adrian 2007]. This characteristic allows studying them as physical entities in the flow. Study of these structures provides a deep insight about the building blocks of the turbulence in the wall-bounded flows. Nowadays, with the aid of accurate numerical methods and non-destructive experimental measurement techniques, e.g. particle image velocimetry (PIV), it is feasible to study the coherent structures in more details. In the case of zero pressure gradient conical flows, the presence of hairpins and hairpin packets has been proven by the study of signature of the hairpin eddies in different cross-section planes [Adrian 2007].

22

Fig 2.4 Schematic of a hairpin packet structure in the near wall-region (modified from [Adrian 2000]) In the case of flows with an adverse pressure gradient (APG), the presence of hairpins and hairpin packet structures have been also studied by the aid of PIV measurements based on their signature in different cross-sections with different orientations [Shafiei 2009, Rahgozar 2012].

Fig 2.5 Visualization of hairpin structures in the turbulent boundary layer on a flat plate at Rem  4300 obtained by isosurfaces of negative 2 from DNS results. The structures have been colored by wall distance [Schlatter et al. 2011] According to direct numerical simulation (DNS) data, the hairpin structures are clearly formed at low Re [Wu et al. 2009] but at high Re , the coherency is lost and the classical hairpins are not visible anymore neither near the wall nor in the outer region [Schlatter et al. 2011]. At moderate Reynolds number, isolated instances of arches belonging to the hairpin vortices can be found on the top of outer structures but at high Re number, there is a forest of worm-like coherent structures but the classical hairpins are not seen in the flow [Schlatter et al. 2011]. As it can be seen in figure (2.5), at high Reynolds 23

number, Rem  4300 defined base on the boundary layer momentum thickness i.e.  m , the transional vortices completely disappear and the worm-like coherent structures do span the whole boundary-layer height. If we turn to the problem of interest in this project, fluid flows within the draft tube of hydro-turbines are high- Re bounded turbulent flows exposed to an adverse pressure gradient and flow swirl which typically experiences a flow separation on the curved bounding surfaces. After this brief discussion, it can be imagined that under these circumstances the near wall turbulence dynamics is very sophisticated and also computationally too costly to be captured appropriately. On the other hand, the use of the classical wall function such as equations (2.6) and (2.7) implies that the local equilibrium between the production and dissipation in the boundary layer; the picture is not really appropriate for this kind of flows that involve a separation or develop at the verge of the separation. As expressed before, the classical definition of the wall function only considers the effect of wall friction. Improvements are found in the literature; for instance, considering the pressure gradient and convective and unsteady terms can improve the accuracy of wall function predictions and widen its applicability range to the nonequilibrium separated regions [Duprat 2010]. In the present study, the turbulence structures in the boundary layer are modeled (not directly resolved) up to the viscous sub-layer ( y   1 ) using the detached eddy simulation (DES) method. In fact, DDES is capable to mimic the near wall and core flow turbulence appropriately by a combination of modeling and resolving techniques, and no wall function is utilized. In the next subsections, details of the DDES approach and other models to treat turbulent flows are discussed.

2.3

Turbulence treatment methods

There is a wide range of turbulence treatment approaches to simulate real fluid flows encountered in engineering applications and fundamental studies, from Reynolds Averaged Navier-Stokes Equations (RANS) towards Large Eddy Simulation (LES) and ultimately to Direct Numerical Simulations (DNS). As is discussed in the following, selection of an appropriate method between the different turbulence treatment approaches for a given case is performed based on a compromise between required accuracy and acceptable amount of the computational effort. In the following subsections, underlying concepts of each strategy are discussed shortly to understand its capability and limitations for hydro-turbine applications. In the literature, different turbulence models have been applied to study different aspects of fluid flow through the hydro-turbine systems and reported in a vast number of articles. In the 24

following, only some relevant studies related to the draft tube flow simulation and bulb turbines are reviewed. Furthermore, the utilized hybrid turbulence treatment approach applied in this thesis to study the draft tube flow, i.e. DES method, is explained in details in section (2.4). 2.3.1

Direct numerical simulation (DNS) approach

In DNS approach, all scales of the turbulent motion from smallest scale (Kolmogorov scale) to the integral scale of the flow are directly resolved without using any ad-hoc empirical input. The accuracy of the method is the highest. In fact, in DNS approach the Navier-Stokes equations are solved without any averaging or filtering techniques and the only error entering into the calculation originates from the solution procedure of the governing PDEs including the discretization method and the cut-off error of computing system. To resolve all of the wide range of the spatial and temporal spectrum of the turbulent flow motions, the DNS grid size and time step interval used in a simulation should be prohibitively small, to resolve the smallest dissipative (Kolmogorov) scale and smallest lifetime physical fluid phenomena involved. With increasing the Reynolds number, the smallest spatial and temporal scales decrease considerably. It can be concluded that DNS approach is incredibly expensive for high Re number flows encountered in industrial applications. Reynolds estimated the required resolution for DNS simulation about Re2.25 [Reynolds 1990]. It means for instance that for Re of 106, a DNS simulation would typically need a mesh in the order of 1013.5, grid points which is prohibitive for current computing resources. This fact along with the incredible small time step prevent to resolve the smallest lifetime turbulent motions which makes DNS a powerful research technique to study low to moderate Reynolds number flows and for fundamental investigation of the underlying physical mechanisms present in the fluid flow [Malm et al. 2012]. As explained in the previous section, the near wall turbulence contains entities like hairpins, hairpin packets and low and high speed streak regions. Under these circumstances, DNS can help to uncover the underlying physical mechanisms involved in the wall turbulence by capturing all the scales of the turbulent motions (Fig 2.5). An appropriate DNS numerical simulation of the flow field can provide an accurate data-base less noisy than the measurement results obtained in the case of state-of-the-art PIV and LDV measurements [Rahgozar et al. 2012]. Performing a flow simulation with the DNS method is unfeasible in industrial applications like hydro-turbine flows, which includes complex geometries and high Re numbers; therefore, other turbulence treatment strategies like LES should be

25

utilized. Figure (2.6) indicates a comparison between the degree of modelling and the computational effort of different turbulence modelling strategies. As it is clear, the DNS method is the most accurate approach without modelling, but also provides maximum computational cost. This makes the method almost infeasible for hydro-turbine applications.

Fig 2.6 Typical comparison of turbulence treatment strategies [Buntic et al. 2005] 2.3.2

Large eddy simulation (LES) approach

Large eddy simulation (LES) is a promising approach which is less costly in comparison to the DNS method, thus it can be expected to be more feasible to study the industrial fluid flows. In the LES approach, the flow field is viewed as containing large and small scale eddy motions. In practice with the aid of spatial filtering techniques, the flow motions are categorized in large and small scale eddy motions. It is postulated that the energetic large scale eddy motions (anisotropic turbulent structures) govern the major portion of the flow field dynamics. In contrast, small scale motions (isotropic turbulent structures) of the flow field are less critical and behave approximately universally. As a conclusion, it is reasonable to resolve the large scale eddies directly and to model the small scale motions; this is the original idea of the LES method. Nowadays, even by increasing the computational power of the processors and using the parallel processing techniques, performing a well-resolved LES simulation for industrial flows is still costly but quite feasible. In fact, LES is an appropriate method for wake flows which include turbulent fluctuations with large spatial scales. For this type of flows, large fluctuations can be resolved on a fairly coarse grid by LES. For wall bounded flows, the presence of wall breaks the Kolomogrov picture which is only valid for sufficiently high Re number flows. In the near wall region, the boundary layer cannot be considered as a high Re number flow. Small scale eddies in this zone are extremely important in turbulence dynamics. To resolve these small structures in the wall 26

bounded flows, adopted spatial and temporal resolution should be prohibitively small. As showed by Reynolds, the required resolution of LES for wall-bounded flows does not differ strongly with DNS and is in the order of Re1.75 [Reynolds 1990]. To perform a reliable LES simulation without wall-modeling that would resolve the important small structures in the boundary layer, the streamwise, spanwise and wall-normal resolutions of the utilized grid in wall units should be 100, 30 and 1, respectively [Davidson 2011]. This requirement results in a mesh with large number of cells which makes pure LES costly although still possible, in the industrial flow applications; like computational aerosciences [Slotnick et al. 2013] and similarly in the hydro-turbine flow applications. It is also worth to mention that by a wall-modeling technique, one can considerably attenuate the computational cost of LES. For example, Duprat et al. performed draft tube flow simulations in LES by developing a wall-function to treat the near wall zone and obtained some promising results [Duprat et al. 2008, 2009 and 2010]. In LES method, the mean flow, unsteady large scale (energy-containing sub-range) and intermediate scale (inertial sub-range) of the turbulent motions are resolved directly and the effect of small scale motions (dissipation sub-range) is modelled. To do so, a spatial filtering operation is applied to the Navier-Stokes equations; on the other hand, to utilize the maximum capacity of the mesh in resolving the unsteady turbulent structures, the width of the filter is typically considered as the mesh size. In fact, eddies smaller than the mesh size cannot be captured by the approach (Figure 2.7) and their effects should be considered via the ‘subgrid scale model (SGS)’. Larger eddies can be directly resolved by the filtered equations in LES approach (for more details refer to appendix A1).

Fig 2.7 Resolved and subgrid scale eddies (fluctuations) on a given mesh in the LES Figure (2.8) indicates the difference between LES and RANS modelling in terms of resolved and unresolved scales of the turbulent motions [Gode et al. 2006]. As explained in the energy cascade of turbulent flows, the energy is transferred to the small scales from 27

energy containing eddies and ultimately is dissipated and converted to the heat by the effect of viscosity at Kolmogorov scales. As shown in figure (2.8), the role of the subgrid scale model is to behave as a sink of energy for high frequency eddies. In Implicit LES (ILES) approach, no subgrid scale model is added in the formulation and the turbulent eddies are resolved by the mesh directly (like DNS) and the dissipation of energy in the spectrum is achieved via dissipative discretization schemes. In other words, the truncation error of the utilized numerical discretization acts as an implicit subgrid scale model [Grinstein et al. 2007, Hickel and Adams 2008, Uranga et al. 2009]. As being explained in the following subsection in details, all scales in the turbulent motions are modelled in RANS method, from lowest to highest frequencies; whereas in LES method about 80% of the turbulent kinetic energy is resolved directly by the method and the rest of the spectrum is modeled [Pope 2004]. As can be expected, for an appropriate LES simulation, the mesh should be fine enough to resolve the major portion of the unsteady turbulent structures. In fact, to obtain reliable results, the mesh size should be set in the inertial sub-range exhibiting the famous ( 5 3 ) decay. Therefore, one can obtain the power spectral density of the velocity signals in the LES region and check the decay rate to examine justness of a LES simulation. Due to the high computational cost of simulations, LES method is not adopted in this study.



5 3

Fig 2.8 Turbulence treatment in the LES vs. RANS 2.3.3 Unsteady Reynolds Averaged Navier-Stokes (URANS) approach

Nowadays, the Reynolds Averaged Navier-Stokes (RANS) strategy is a common approach for industrial fluid flow simulations owing to a relatively low computational cost. In RANS method, the complete behavior of turbulence is considered through turbulence models which covers all turbulence (eddies) scales. In other words, no portion of the turbulence spectrum is directly resolved. The turbulence model in this case should reproduce all effects of the wide range of the turbulence motions and that’s why the level of physical

28

representativity of the technique is limited. For the flows where steady mean flow is an acceptable approximation of reality, RANS can be applied. The RANS treatment involves the Reynolds decomposition, where an instantaneous flow variable is decomposed into a mean value and a fluctuating part. Then this decomposition is substituted in the Navier-Stokes equations and a time-averaging operation is applied on the equations. The result is equations for mean quantities that include an extra term called ‘Reynolds stress tensor’ (  uiuj ), where the over-bar indicates the timeaveraging. In fact, the contribution of turbulent eddy motions to the mean flow is introduced through this term. The above-mentioned procedure results in a ‘closure problem’ which means that the resulting set of equations is not ‘closed’ and the number of unknowns is larger than the number of equations. This is due to the creation of the Reynolds stress tensor in the derivation process. To remedy the problem and balance the number of equations and unknowns, it is necessary to utilize some ad-hoc expressions and semi-experimental assumptions to connect Reynolds stress tensor to the mean flow quantities. Different RANS closure models have been developed for this purpose. Many of them are established on the Boussinesq assumption based on an analogy between the molecular and turbulent diffusions (see appendix A2). Generally, RANS models are considered too dissipative (diffusive), in fact they damp instabilities and spread concentrated vorticities in the flow field. As a conclusion, RANS approach is not able to resolve properly the unsteady characteristics of the flow field, leads to an improper description of unsteady phenomena such as vortex shedding, wake flows behind blunt bodies, or formation of the vortex rope in the draft tube of hydraulic turbines and separated flow regions. In RANS equations, there is no unsteady term, and the governing equations are steady state. If there is a low frequency variation of the mean flow or in other words a clear separation of time scales between the mean flow (with relatively large time-scale) and turbulent flow motions (with relatively small time-scale), then the time-variant mean flow can be somehow resolved using a special technique called URANS. It is obtained by applying a ‘triple decomposition’ on the velocity signal. In this regard, the velocity can be rewritten using ‘triple decomposition’, as below: ui (t )  uim  uil (t )  ui(t )

(2.8)

where uim denotes the mean velocity and uil (t ) is the low frequency fluctuating part produced by low frequency changes of flow field e.g. in the rotating machinery due to the

29

runner rotation. The ui(t ) is the high frequency fluctuating part introduced by turbulent fluctuations. It is worth to mention that the uil (t ) is a deterministic part of the velocity signal and is not created by the turbulence. It is generated by the predictable motions in the flow field like the runner rotation in hydro-turbines applications. In contrast, ui(t ) is the random highfrequency part of the velocity signal related to the random motion of the turbulence. By this decomposition, the low frequency variations of the flow field can be resolved by the method, but the effect of turbulence is still modeled for all scales of the turbulence spectrum (for more details, refer to appendix A2). In the case of draft tube flow simulation, performing URANS simulation can improve the prediction of the fluid flow including more accurate estimation of vortex rope dynamics [Vincent et al. 2009, Vincent P.B. 2010, Petit et al. 2011, 2012] and fluid flow separation in comparison to RANS simulations [Gehrer et al. 2004]. In fact, some distinguishable frequencies of the flow phenomena which are much smaller than the turbulent structure’s frequency can be resolved by URANS method. Therefore, it does not provide a complete picture of the turbulent flow phenomena, but provides a better insight about the low frequency variation of flow field, like what is observed in hydro-turbine flows. Furthermore, URANS models are still too dissipative and are not able to appropriately reproduce unsteady phenomena like vortex shedding, flows including strong non-equilibrium and high anisotropy features like recirculating flows happening in the separation zones. In the literature, RANS/ URANS simulations are vastly adopted to investigate turbulent flows in the hydro-turbine applications for decades, due to its simplicity to apply and its low computational cost. To keep it concise, some studies only related to the draft tube or bulb turbine are mentioned here. For example, the following studies can be highlighted: k   RANS of a bulb turbine with a conical draft tube [Qian et al. 1996] and [Skoda et al. 2000], the k   and k   RANS of a bulb turbine with a draft tube consisting a conical part and a trumpet [Gehrer et al. 2002]. The k   , k   and SST unsteady RANS (i.e. URANS) of a bulb turbine [Gehrer et al. 2004] and [Benigni et al. 2006], k   RANS of a bulb turbine during a transient shut-down process [Kolsek et al. 2006], SST RANS of a bulb turbine with a draft tube consisting a conical part and a trumpet [Coehlo 2006] and [Santos et al. 2008], SST RANS/URANS of a bulb turbine [Hubner et al. 2008], RNG k   RANS of a bulb turbine draft tube [Shingai et al. 2008]. Almost in the all above-mentioned studies, a full-machine configuration is adopted in the simulations. The results show the dissipative nature of RANS/URANS techniques in capturing unsteady flow features. The results of URANS simulations also highlight that the

30

quality of the CFD predictions increases by taking into account the flow unsteadiness [Gehrer et al. 2004]. 2.3.4 Hybrid RANS/LES approach

The hybrid RANS/LES approach is a relatively new generation of strategies for turbulent flow simulation which combines the strength of both LES and RANS methods in certain regions in a unified method. The general idea of the hybrid method is to use LES in small localized sub-domains where an accurate description of the flow is intended, while computing the rest of the domain with a lower-accuracy URANS method. In fact, different hybrid RANS/LES models can be classified under the very large eddy simulation ‘VLES’ category [Helmrich et al .2002]. VLES is similar to LES but only a smaller part of the turbulence spectrum is resolved in this approach. In this sense, ‘VLES’ needs less computational resources in comparison to LES, therefore it is more feasible for fluid flow simulations in industrial applications. Figure (2.9) presents turbulence treatment in VLES compared to LES. As one can see in the figure, a larger portion of the spectrum is modeled in this approach than with LES.



5 3

Fig 2.9 Turbulence treatment in the VLES vs. LES method As explained in section (2.2), streak process (including generation of the low-speed and high-speed streaks) is responsible for the major portion of the turbulence production. For an accurate and reliable LES simulation, these small structures should be resolved using a very fine mesh. For high Re wall-bounded flows, this results in a prohibitive computational cost. In general, hybrid RANS/LES modelling is inherently a LES model designed to asymptote to a RANS model in vicinity of solid walls. In this strategy, the near wall region is modeled by a low- Re URANS approach and in the outer region LES mode of the method is activated to resolve the important turbulence structures. With the aid of this concept, computational demands decreases and simulations of industrial flows can be performed with a reasonable computational cost. 31

Hybrid methods can be classified based on the type of the interface between RANS and LES regions and its time-variant behavior during the computation. Although there is no generally accepted classification in literature, due to the frequent usage, the hybrid methods can be classified into two groups:  

Zonal approach Non-zonal approach

In the first approach i.e. zonal method, the zones of RANS and LES are declared explicitly, in other words, they are pre-defined by the user. The computational domain dealt with is divided into two separate zones, and two transport equations are solved separately for RANS and LES regions [Quemere and Sagaut 2002]. One of the disadvantages of this approach is the necessity for the user to have insight into the solution prior to performing simulations to divide the domain into the LES and RANS zones. This fact limits its application to simple geometries, while its usage for more complex geometries and unsteady flows is not straightforward. Another issue of the method is related to the transferring of data between RANS and LES zones and coupling between two zones through the interface. In contrast, in non-zonal methods (like Detached Eddy Simulation), the same eddy viscosity transport equation is solved over the entire computational domain. These methods are based on the fact that the shape of the time averaged Navire-Stokes equations in RANS and the spatially filtered equations in LES is exactly the same even if they involve fundamentally different characteristic length scales in the definition of Reynolds stress and subgrid scale models, respectively. In this method, there is no explicit declaration of the RANS and LES zones and the method based on predefined criteria switches between the zones. In another classification, the hybrid methods are categorized based on its interface dynamics in terms of time. If the interface is constant in time, it is called a hybrid method with ‘hard interface’. It is named hybrid method with ‘soft interface’ if the interface changes in time depending on the computed solution [Frohlich and Von Terzi 2008]. Just for completeness of this subsection, there is also another class of the turbulence treatment strategies called ‘Scale-Adaptive Simulation’, or in short ‘SAS’, which resolves turbulent unsteadiness. SAS as a second generation of the URANS technique can be classified in the Hybrid RANS/LES methods. In this technique, LES-content of the flow are resolved to some extent by introducing the von-Karman length-scale L k into the turbulence scale transport equations [Menter et al. 2003, 2005, 2010]. The von-Karman length scale is defined as L k  kS  2U ; where k and S stand for the turbulent kinetic 32

energy and strain rate tensor, respectively. The method is capable to exhibit a LES-like behavior in the regions involving flow unsteadiness, whereas it recovers URANS in the stable regions. In the literature, SAS method has been also used for hydro-turbine simulations to resolve larger portion of the turbulence spectrum. For example: a k   SST-SAS simulation of the draft tube of a Francis turbine [Jost et al. 2009] and a k   SST-SAS simulation of an abrupt expansion [Javadi and Nilsson 2014]. Some other studies also used techniques developed based on the SAS concept to simulate turbulent flows in the draft tube, like: [Helmrich et al. 2002], [Ruprecht et al. 2002], [Buntic et al. 2005], [Gyllenram and Nilsson 2006], [Gode et al. 2006] and [Nilsson and Gyllenram 2007]. In the present research project, a special version of the detached eddy simulation (DES) approach, which is called delayed detached eddy simulation (DDES), is utilized to simulate unsteady fluid flow through the bulb turbine. This method of turbulence treatment belongs to non-zonal hybrid RANS/LES methods. In terms of the interface dynamic, DES is a hybrid method with ‘hard interface’, whereas DDES is a hybrid method having ‘soft interface’. More details about DES and DDES turbulence treatment are presented in the following section.

2.4

Detached eddy simulation (DES) approach

DES was originally developed for external aerodynamic applications to address the challenging task of numerical simulation of high-Reynolds number flows with thin boundary layers and massive separations. The method (hereafter ‘DES97’) was established by Spalart et al. in 1997, based on one-equation Spalart-Allmaras model [Spalart et al. 1997, Spalart 2009]. As explained in details in appendix A3, Spalart-Allmaras (S-A) eddy viscosity model solves one transport equation for a dummy variable that is ultimately related to the turbulent viscosity [Spalart and Allmaras 1994]. The aim of the DES method, like any typical hybrid method for simulation of highRe number flows, is to model the boundary layer with URANS and resolve the outer detached eddies with LES; the method switches automatically to RANS and LES based on predefined criteria. In contrast to the zonal approach, the computational domain in DES approach does not contain any explicitly defined interface between the RANS and LES zones, so prior insight to the computed solution is not necessary. This makes the method attractive for industrial applications. In fact, the governing equations of the method are reduced to LES equations along with appropriate subgrid scale model when the grid has enough resolution. In the near wall region, where the resolution of the mesh is not enough for the LES type simulation, the 33

equations are reduced to URANS turbulence model equations. Considering one-equation SA model, this can be easily achieved by introducing a new parameter to replace the wall distance. In this regard, wall distance d in the S-A model is replaced by a new defined variable, called ‘DES length scale’, d , as below: d  min  d , CDES  

(2.9)

where  is typically defined using a local grid cell size as the following:   max  x, y, z 

(2.10)

Also CDES is a constant in the model which is calibrated typically based on the decay of isotropic turbulence and its recommended value is 0.65 [Shur et al. 1999]. In fact, the DES length scale can switch between the RANS length scale in the near wall region ( d ) and a measure of LES length scale ( CDES  ) in the LES zone. In other words, in a close proximity to the confined flow walls, where d  CDES  , the DES length scale d is reduced to d , which means that the method reduces to the Spalart-Allmaras model. Far from the walls, where d  CDES  , the DES length scale is reduced to CDES  , which implies that the turbulent length scale would be proportional to the local grid cell size (   ) and behaves like the LES approach. It is trivial that the interface is located where d  CDES  . As can be observed, the DES length scale in DES97 (equation 2.9) depends only on the geometrical characteristics of the computational domain and the generated mesh. Therefore the position of the interface between the RANS and LES regions does not change in time (as the convention, called ‘hard interface’) and is fixed for a given mesh and computational domain. In the LES zone, DES behaves as a one-equation subgrid scale model. As explained in appendix A1, for any typical LES method the turbulence length scale is considered as the local mesh size (  ). For example, in the Smagorinski sub-grid scale model, the sub-grid  2 (see appendix A1 for scale turbulent viscosity is related to the local mesh size as vSGS  S the mathematical definition of v and S ). In the S-A model (appendix A3), the destruction term is proportional to the square of modified kinematic turbulent viscosity ( v ) over the wall distance i.e. (v d ) 2 . In the case of DES, this term is modified to (v d ) 2 . By applying a balance between the destruction and production terms, one can deduce that the DES eddy   2 . Therefore, far from the solid walls, it becomes v  S  2 , viscosity becomes v  Sd DES

DES

which proves that the formulation is reduced to LES formulation and behaves similar to the one-equation Smagorinsgy subgrid scale model.

34

As can be expected, the grid resolution of DES is different from pure LES only in the boundary layer zone, where the RANS method is used. This removes the need to have a high spatial resolution in the three directions of the boundary layer; as a result, the required resolution of DES gets much lower than LES [Bunge et al. 2007]. 2.4.1 DES97 in the external and internal flow applications

In the literature, DES97 has been used to study the external flows in the different aerodynamic applications like: flow on airfoils experiencing massive separations [Shur et al. 1999] and [Schmidt and Thiele 2003], flow over delta wings and fighter configurations [Forsythe et al. 2000, 2004], [Morton et al. 2002, 2003] and [Mitchell et al. 2002], flow over a circular cylinder and a sphere [Travin et al. 1999], [Constantinescu and Squires 2003, 2004] and [Mockett et al. 2007], flow over a generic airplane landing gear [Hedges et al. 2002], a train in a cross flow wind [Wu 2004], flow past a corner-mounted block [Paik et al. 2004], flow over typical military aircrafts and business jets [Chalot et al. 2007], flow over a wing-body junction [Paik et al. 2007], flow over turbine blades [Magagnato et al. 2008] and a cavity flow [Ashworth 2008]. In general, the abovementioned studies showed the superiority of DES97 to the URANS simulations, for instance, in capturing unsteady flow features, vortical structures, frequency spectrum and vortex breakdown phenomena. In the study of Schmidt and Thiele, by performing 2D and 3D DES simulations of flow over airfoils it was shown that only 3D DES is capable to capture unsteady features of the flow. This is expected a-priori due to the 3D nature of turbulent flow structures [Schmidt and Thiele 2003]. The beauty of the DES method in capturing a richer turbulent content at a modest computational cost motivates researchers to adopt the simulation strategy for internal turbulent flow simulations as well. For example, Nikitin et al. performed the DES97 simulation for a standard channel flow test case with a friction Re number of 180 to 8 104 [Nikitin et al. 2000]. They could obtain stable and fairly accurate results, although a loglayer mismatch (LLM) was also observed in the predicted velocity profiles. 2.4.2 Grid-induced separation problem

In 2003, Caruelle and Ducros studied the attached and detached boundary layers on a flat plate in details with the aid of DES97 as classical examples to construct a guideline for flow simulation in complex geometries. In the first test case, they studied an over-resolved temporally evolving turbulent boundary layer and in the second one, they investigated a compressible laminar boundary layer suddenly exposed to an adverse pressure gradient. Moreover, they investigated the effect of CDES constant on the obtained results. The results 35

showed the major problem of the DES97 called ‘grid induced separation (GIS)’ leads to an under-estimated skin-friction coefficient about 25%. They concluded that DES97 simulation is very sensitive to the mesh quality; to get good results the mesh should be carefully chosen [Caruelle and Ducros 2003]. The ‘modeled stress depletion (MSD)’ which in extreme cases leads to the GIS problem, was noticed at the beginning of the formulation of the DES97 by its creators [Spalart et al. 1997]. As depicted in figure (2.10), if the mesh becomes over-resolved, or in other words when wall parallel grid spacing is smaller than the boundary layer thickness, then the model switches to LES mode. For example, for point P ( d  CDES  ), according to equation (2.9), DES length scale is reduced to d  CDES  and LES mode of the method is activated inside the boundary layer zone, however, the spatial resolution of the grid is not fine enough to resolve the required LES content or coherent structures. As a result, the model (subgrid) eddy viscosity becomes lower than the expected value, which directly leads to lowered modeled Reynolds stresses. In this situation, the predicted wall skin friction is reduced which can ultimately lead to a premature separation. These problematic situations involving GIS can be easily generated via a grid refinement process in grid independency tests or in the condensate grid zones in the proximity of regions with high gradients during a mesh adaptation. This situation can also be generated for flow simulations involving shallow separation regions and boundary layer thickening [Spalart et al. 2006]. The latter case is typically observed for flows passing through the draft tube of hydro-turbines experiencing an adverse pressure gradient via increasing the cross sectional area.

Fig 2.10 Ambiguous grid in the near wall zone inside the boundary layer (P: a point inside B.L.,  : B.L. thickness, d: distance to the wall) To remedy the GIS problem in the original version of DES, Spalart provided some instructions for an appropriate mesh generation for DES simulations, namely ‘youngperson’s guide’ [Spalart et al. 2001]. This improved the applicability of the method, however to generalize the method and bring it to a powerful tool for external and internal 36

flow, Spalart et al. proposed a new version of DES called ‘delayed-detached eddy simulation (DDES)’ [Spalart et al. 2006]. In this version, the DES length scale is not only a function of geometrical features of the grid and distance to the nearest wall, but also depends on the physical solution of the fluid flow. Therefore, the interface in this version is time-dependent with ‘soft interface’. The aim of the new version is to ensure that the boundary layer is treated by RANS, which switches to LES in the outer region of the boundary layer. In this regard, a new parameter rd is defined as below: rd 

  t U i , jU i , j kV 2 d 2

(2.11)

where  and  t are the kinematic molecular and eddy viscosity, respectively. The U i , j is the velocity gradient, kV is the von-Karman constant and d indicates the distance to the wall. The value of rd is equal to 1 in the logarithmic region and gradually falls to 0 towards the edge of the boundary layer [Spalart et al. 2006]. Afterward, an intermediate function f d is defined as below:





0 RANS 3 f d  1  tanh 8rd    1 LES

(2.12)

This function is designed to be 1 where rd  1 in the LES region and 0 elsewhere [Spalart et al. 2006]. The values 3 and 8 were picked out based on the required shape of f d using the DDES simulation of the boundary layer flow on a flat plate. As mentioned by Spalart et al., these values ensure that the method remains in RANS mode even if  is much smaller than the boundary layer thickness [Spalart et al. 2006]. In the final step, the DES97 length scale d in equation (2.9) is modified and forms the ‘DDES length scale’ as below: d  d  f d max  0, d  CDES  

(2.13)

If the value of f d is equal to 0, then the length scale reduces to the distance to the wall meaning that RANS mode recovers ( d  d ), and in the case of f d equals to 1, away from the walls, the model behaves in LES mode and DES97 is recovered in LES region as intended. It is worth to mention that DDES switches from RANS to LES more abruptly than DES97. In other words, the gray zone between the RANS and LES regions is narrower in the DDES approach which enhances the growth of turbulent content leading to attenuation 37

of LLM discrepancy [Spalart et al. 2006]. It is also important to notice that although DES97 and DDES were originally developed based on the S-A model; it is quite possible to modify any RANS turbulent model to construct a DES counterpart formulation. This can be simply done by comparing the RANS length scale (which is a function of the turbulent model variable or the wall distance in SA) to the LES length scale involving a grid cell size. As an example, in 2001, Stretlet proposed a DES formulation [Strelets 2001] based on the SSTRANS turbulence model [Menter 1994]. 2.4.3 DDES in the external flow applications

In the literature, the DDES approach is also widely adopted to study different types of external flow applications, like: flow over a flat plate [Spalart et al. 2006], flow over a circular cylinder [Spalart et al. 2006] and [Squires et al. 2008], flow on an airfoil experiencing a separation [Spalart et al. 2006], [Martinat et al. 2008], [Frederich et al. 2009] and [Im and Zha 2014], flow over a delta wing [Ludeke and Leicher 2008] and [Cummings and Schütte 2012, 2013], flow under wing flutter condition [Chen et al. 2010] and [Wang and Zha 2011], flow over a car [Martinat et al. 2008], a wind gust on a vehicle [Favre and Efraimsson 2010], flow over two wall-mounted cubes in tandem [Paik et al. 2009], flow over a submarine [Vaz et al. 2010], flow over an aircraft landing gear [Langtry et al. 2013], flow over the fuselage of a helicopter [Fuchs et al. 2015]. In short, in all cases DDES formulation shows its superiority to the URANS simulations in capturing the LES content of the flow in the detached eddy regions. In fact, the formulation not only provides a rich dynamics similar to the DES97 but also avoids the grid induced separation (GIS) intendedly in all cases in contrast to the basic formulation i.e. DES97. 2.4.4 DDES in the internal flow applications

The ability of DDES approach, to resolve turbulent coherent structures at a moderate computational cost motivated researchers to apply it for internal flow simulations although this was not the original intention of its developers. The application domain of detached eddy simulation covers a wide range from subsonic to the hypersonic flow regimes. To name a few, the DDES was successfully used to study a flow in a swirl generator [Benim et al. 2008], flow in a highly-curved open channel [Constantinescu et al. 2010], flow in a supersonic inlet [Trapier et al. 2008] and supersonic flow in a diverging nozzle [Deck 2009]. In all above-mentioned cases, the DDES was successfully applied for the internal flow simulations to resolve very large eddies in the flow field and simultaneously to avoid the GIS.

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Interesting features of the DDES turbulence treatment to resolve turbulent structures in the LES zone gives a motivation to apply it in the unsteady simulations of draft tube flows. Furthermore, its well-proven character in capturing separated flow dynamics in the massive separation cases is another key factor in selection of the DDES for simulation of the BulbT draft tube flows. Figure (2.11) shows the turbulent structures resolved by DDES simulation of the BulbT draft tube at an overload operating point, namely OP.4. uz uref .

y x z

Fig 2.11 Turbulent flow structures ( Q  350 ) resolved by DDES turbulent treatment using unsteady 2D inflow profile in the BulbT draft tube at an overload condition (OP.4) There are some studies in the literature trying to apply DDES to study draft tube flows in the hydro-turbine applications. In 2005, with the aid of an in-house finite volume code Paik et al. simulated the incompressible swirling flow inside a typical hydro-turbine draft tube with DES97 and URANS methods. The draft tube involved a 90 curved elbow part with two piers. An axisymmetric steady velocity profile was applied at the inlet section of the draft tube without any explicit unsteady forcing. In the DES simulation, they studied the vortex breakdown phenomena and observed interesting interaction phenomena of the swirling flow with the low-speed near-wall flow resulting in a complicated instability mode. They found that both URANS and DES97 are capable of capturing the large scale unsteadiness in the flow, although the DES97 provides richer flow dynamics. Both methods also yield a similar frequency band from the low to moderate frequencies, but the magnitude of the power in the DES97 is higher indicating a more intense unsteadiness [Paik et al. 2005]. In the configuration including two piers, it was also observed that the distribution of the flow in the three different channels of the draft tube is extremely sensitive to flow asymmetries at an inlet section.

39

In another attempt, Vincent et al. from LMFN group at Laval University studied the flow through a Francis turbine draft tube using DES97 and DDES in OpenFOAM [Vincent et al. 2009, Vincent 2010]. Their motivation was to investigate the difference sometimes observed between on-site efficiency measurements and URANS simulations at IREQ1 in the performance analysis of the Francis turbines. Like in the Paik et al. study, they utilized the steady 1D circumferential-averaged velocity profile at the inlet section, with no inflow synthetic turbulence. Their simulation results were very promising in terms of capturing the rich turbulence dynamics involved. In addition, they observed that DES can capture precessing vortex rope unsteadiness more completely; although it did not solve the discrepancy observed between the experimental and numerical efficiency curves [Vincent 2010]. In another study, Paik and Sotiropoulos investigated a strongly swirling flow through an abrupt expansion over a range of swirl numbers (equation 1.5) from 0.17 to 1.23. Their results reproduced well both the axial and circumferential mean velocities and turbulence intensity profiles in comparison to experimental data. They also emphasized that an appropriate specification of the inflow conditions at a sufficiently upstream position of the expansion section as well as a fine enough grid generation, are crucial factor to simulate the flow properly [Paik and Sotiropoulos 2010]. In another attempt, Javadi and Nilsson studied the same geometry with a swirl number in the range of 0.6 to 1.23 using the DDES and LES simulations in OpenFOAM. The results showed that LES and DES are both capable to capture mean and unsteady features of the flow including the intensity and turbulence anisotropy [Javadi and Nilsson 2015]. As mentioned in the ‘motivation’ section of the first chapter, Gramlich simulated the flow field in a swirl generator using the DDES in the OpenFOAM platform [Gramlich 2012]. In the study, a fixed velocity value was imposed at the inlet section of the swirl generator. In another recent study, the flow field in the draft tube of a Francis turbine was simulated using the DDES considering 2D variation of the draft tube inflow profile in ANSYS software. The results showed the more complete representation of the turbulent flow structures especially the structures under the runner of the Francis turbine [Beaubien 2013]. The study showed the effect of the grid and time resolutions on the diffusion of the structures generated by the runner and convected into the draft tube. The work also highlighted the importance of the radial velocity component of the inflow profile on the DDES simulation of the draft tube flow. In general, the quality of the simulation of draft tube flow strongly depends on the capability of the utilized turbulence model in capturing the turbulent flow structures as well as accuracy of the applied boundary conditions. The inlet boundary condition has a crucial 1

Hydro-Québec Research Institute (IREQ)

40

impact on the justness of a flow simulation in the hydro-turbine draft tube. In this research work, the effects of the steady and unsteady inflow boundary conditions on the predictions of the DDES internal flow simulation are studied in details. In chapter 4, the strategy utilized to generate unsteady inlet data for the simulation of a draft tube flows is reviewed in details. In the literature, there are few studies dealing with the response of the DES turbulence treatment in the LES regions under applied inflow turbulence. As an example, but for the external aerodynamic application, Gilling et al. simulated flows with and without an inflow turbulent content over NACA 0015 airfoil at high Re number 1.6 106 and at various angles of attack before and after the stall. Their results showed that including a resolved turbulent inflow could improve the predicted results compared to the experimental data. Especially, the turbulent inflow content leads to a better prediction of the separation occurrence near the stall. To generate the inflow turbulence, they used a hybrid approach including an artificial fluctuation generation and a precursor simulation. In this approach, the random phase of the generated artificial signal was corrected by performing a precursor simulation [Gilling et al. 2009]. Without applying the inflow turbulence, the stall happens at a too high angle of attack, however considering inflow turbulence triggers the separation in a way that provides better results in comparison to the experiment. In fact, providing the inlet turbulence is mandatory for an accurate simulation of DNS and LES, where the inlet boundary condition is not in the laminar zone. For the RANS simulations, only applying the mean component of velocity is accurate enough to perform simulations. Hybrid methods as explained before are a combination of RANS and LES methods. The idea is that applying the turbulent fluctuations at the inlet plane better mimics the reality and improves the predicted results. In chapter 4, details of the adopted strategy to generate flow unsteadiness at the inlet are presented. As a conclusion of this chapter, the simulation of draft tube flows is still a challenging and active field of research far from a reliable global simulation strategy and many open questions remained to be answered. As discussed, due to the promising capabilities of LES-like simulations, DDES method is utilized for numerical simulations of the draft tube flow fields in this thesis. In this regard, the impact of different inflow conditions on the DDES simulations is also considered. The DDES method resolves upto the wall based on the S-A RANS method in the boundary layer ( y   1 ), therefore it eliminates the necessity for wall functions. The details of numerical strategy adopted to simulate the fluid flow along with the strategies to generate the inlet flow conditions are presented in the two following chapters.

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2.5 Summary

The major points of this chapter can be summarized, as follows: 

Flow field in the BulbT draft tube at all selected operating points is turbulent with Re in the order of 106 , defined based on the draft tube inlet diameter and the draft tube mean inflow velocity.



For wall-bounded flows as in the BulbT draft tube, real representation of the turbulent flow is a challenging task, due to the complexity of the near wall turbulence dynamics.



DNS approach is not feasible and LES method is still too costly to be applied for BulbT draft tube flow simulations at multiple operating points.



RANS/URANS techniques can be applied for draft tube flow simulations, although the dissipative nature of the techniques leads to suppressing flow unsteadiness intensively.



As a compromise, hybrid techniques such as DES and SAS are very promising in capturing the LES content of the flow with a moderate cost. This is done by excluding the near-wall zone in the LES calculation; in this sense, they resemble the wall-modeled LES.



DES97 was originally developed based on the S-A model for external aerodynamic applications, although its applicability was soon extended to the internal flow domain.



DDES was developed as a modified version of DES97 to solve the GIS (or MSD) problem, essentially by protecting the boundary layer.



DES and DDES are classified as non-zonal hybrid methods with hard and soft interfaces, respectively.



In this project, DDES is applied for the simulation of turbulent flows in the BulbT draft tube. The emphasis is placed on the effect of inflow boundary conditions.

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Chapter 3

Computational methodology

3.1

Introduction

Analytical solutions of the governing equations of fluid flow motions are not possible for most applications due to their non-linear and complex nature. Therefore, the only feasible choice is to solve the equations by numerical methods, which is the main objective of the computational fluid dynamics (CFD). In this chapter, the fundamental basis utilized for numerical simulations of the hydro-turbine fluid flow field are reviewed. First, governing equations of the fluid flow motions called ‘Navier-Stokes equations’ are presented and discussed briefly. In the next subsection, the finite volume method (FVM) used to solve the Navier-Stokes equations in the present research is presented. In addition, OpenFOAM adopted as the flow solver in this research project is introduced. The parallel processing technique adopted to speed-up the numerical computations is discussed, ultimately.

3.2

Governing equations of fluid flow motions

Fluid flows can be considered as a continuum media in general, except in the case of the rarefied fluid flows. For high Re number flows, the occurrence of turbulence in the fluid flows do not change the situation and fluid flow media keeps its continuous nature as before. In fact, the smallest Kolmogorov scales in the high Re number fluid flows are still much larger than molecular length scales, so the fluid flow can be treated as a continuum 43

media even under the turbulence circumstances. Considering the continuum character of the fluid flow, conservation laws can be applied for an infinitely small control volume of the flow to derive the governing differential equations of the fluid flow motion. In fact, the procedure results in a set of coupled partial differential equations (PDE), which should be solved numerically. If the conservation of mass is applied to an infinitely small control volume in an incompressible, constant-property Newtonian fluid flow domain, the ‘continuity equation’ is obtained, which can be expressed in a tensor notation, as follows: ui 0 xi

(3.1)

According to the Einstein’s convention, it implies the summation over all three directions representing the space ( i  1, 2,3 ). The other important equations, which govern the fluid particle motions, called ‘Navier-Stokes equations’, are obtained by applying the conservation of momentum to an extremely small control volume. For an incompressible, Newtonian viscous fluid flow, the equations can be stated in the non-conservative form as below: ui u  2u 1 p  uj i    v 2i  xi t x j x j

(3.2)

The above transport equation can be interpreted as the second law of Newton applied to a small control volume of the fluid; in fact, the right hand side states the pressure and viscous forces acting on the control volume. The LHS of the equation represents inertia forces or fluid particle acceleration including unsteady and nonlinear convective terms. The latter term on the LHS is the main source of complexity observed in the fluid flow motion; especially in the turbulent regime, this term via the stretching mechanism plays a crucial role in the energy cascade process producing small scale eddies. This nonlinear term also creates convergence problems in the numerical solution process of the Navier-Stokes equations especially for high Re number flows. If the conservation of energy is applied to the control volume, the ‘energy equation’ is obtained, which is decoupled from mass and momentum equations in the case of incompressible flows. Distribution of the temperature in the computational domain is obtained by solving of the energy equation. In general, Navier-Stokes equations govern both laminar and turbulent flow dynamics. In the case of turbulent flows, some treatments should be applied on equations (3.1) and (3.2) like averaging in the RANS and filtering in the LES as explained in the previous chapter. Performing these operations keeps the shape of the equations as above but adds some source terms in Navier-Stokes equations. Furthermore, some other transport

44

equations associated with turbulent quantities should be coupled to the set of equations as a remedy for the closure problem as discussed in appendices (A1) to (A3). From a mathematical point of view, Navier-Stokes equations, which belong to the second-order PDEs, behave parabolically for unsteady viscous fluid flows and elliptically for steady state cases. According to a mathematical theorem, parabolic PDEs are wellposed if they are expressed in the form of the initial boundary value problems (IBVP). Hence, to ensure the existence of the mathematical solution, providing the initial and boundary conditions is mandatory. It can be shown that under these conditions a unique solution exists, which depends continuously on the initial and boundary conditions. In general, there are different strategies to numerically solve the system of partial differential equations, including finite difference method (FDM), finite volume method (FVM), finite element method (FEM), spectral methods, more recently meshless methods, to name a few. The main objective of the all-aforementioned methods is to approximate the continuous set of PDEs by a discrete set of equations. Except in the case of meshless method, where the discretization is performed on scattered clouds of points, all other methods need a spatial mesh involving node connectivity for discretization of the equations. The discretized equations are solved and, by definition, the solution of the discrete approximation converges to the exact solution of the problem as the mesh is refined. Historically, FDM is the oldest method developed to numerically solve the PDE systems. In this strategy, a computational domain is divided into a number of grid cells and derivatives which appear in the equations are replaced by the aid of truncated local expansion of flow variables based on the Taylor series expansion. The method leads to a set of algebraic equations associated with all grid nodes that can be solved efficiently by iterative methods. Its simplicity makes the method very attractive, although its usage without mapping is restricted to simple geometries [Hirsch 2007]. The FEM method is another popular method developed originally for numerical solution of the solid-mechanic problems. The method relies on the weighted integration of the governing PDEs in a weak form. In this method, the computational domain is divided to a finite number of elements. The variables in each cell are defined using shape functions. The weighted integral equations are estimated based on the Galerkin approximation for each element and assembled in a global stiffness matrix. The procedure ultimately results in a set of algebraic equations for whole domain that can be solved efficiently by sparse solvers. The beauty of FEM method is the ability to easily capture the complex geometrical features of the computational domain [Reddy 1984]. Another popular method, named finite volume method (FVM), is established by directly applying the conservation laws on a control volume. In the present project, FVM is 45

utilized for all simulations of the turbulent draft tube flows. In the next section, the bases of the finite volume method are reviewed in more details.

3.3

Finite volume method (FVM)

Finite volume approach uses the integral form of the transport equations of mass, momentum, energy, turbulent quantities and species. In this method, first the computational domain is divided into a finite number of cells for which the conservation laws are applied. The key step of FVM that ensures the conservation consists of the integration of transport equations over the control volumes (cells) shown in figure (3.1). For a passive scalar quantity  , the general transport equation in an unsteady flow is expressed in the conservative form as below [Versteeg and Malalasekera 2007]:

        U      S t

(3.3)

where the LHS indicates the unsteady (temporal) and convective terms and the RHS represents the diffusive and source terms, respectively, similar to the Navier-Stokes equations. In the above equation,  represents the diffusion coefficient. By performing volume integration on both sides of the above equation and applying the Green’s divergence theorem to the volume integrals of the convective and diffusion terms, the following equation is obtained, which is the skeleton of the FVM formulation:      dV n  U dS   n     dS   S dV     t CV CS CS CV





(3.4)

The second term on the LHS and the first term on the RHS of the equation indicate the convective and diffusive fluxes crossing the cell control surfaces, respectively, and the equation can be interpreted as the flux conservation law; therefore, the conservations are satisfied by default in the FVM. This characteristic of the FVM makes the method an appropriate choice for the fluid flow transport problems in general and especially in the problems involving discontinuities like shock waves; where the flow variables are not differentiable across the discontinuities but mass, momentum and energy are still conserved. The equation (3.4) is therefore exact, and the only errors entering into the problem is the modelling errors in the governing equations, if any. After obtaining the integral form of the transport equation, in the next step of FVM, the derivatives appearing in the integral equation are replaced by some approximate expressions. In this regard, traditionally finite difference approximation based on the truncated Taylor series expansion is utilized, although other possibility such as using shape

46

functions similar to the FEM can also be used for this purpose. The latter creates a class of hybrid methods named ‘finite-volume-based finite-element’ method. By performing the integration over all cells present in the computational domain, finally a set of algebraic equations is obtained which is nonlinear in the case of NavierStokes equations. To treat the nonlinear convective term in the momentum equations and obtain a linear set, usually some lagging or linearization techniques e.g. Newton-Raphson linearization are typically utilized. The resulting set of equations can be solved by welldesigned efficient solvers along with speed-up strategies like multi-grid algorithms and conjugate gradient methods.

Fig 3.1 Control volumes in the finite volume method used for the discretization (modified from [OpenFOAM programmer’s guide]) The method can be applied to the unstructured mesh and more importantly, it can ensure the conservation in all computational cells and, as a result, in the whole domain. It should be mentioned that the accuracy of FVM directly depends on the type of schemes used for the spatial and temporal discretizations of the terms in the integral equations. In general in FVM, the information of physical quantities of the flow field like velocity, pressure and turbulent quantities can be stored on faces, grid nodes or cell centers. As mentioned in the first chapter in the present thesis, OpenFOAM code is used to simulate the flow fields in the BulbT draft tube. In OpenFOAM, physical values of the quantities are stored at cell centers (e.g. P and N in figure 3.1) and all discretized equations ultimately are expressed in terms of cell center values. Discretization of the different terms of the transport equation (3.4) utilized in OpenFOAM is discussed in the following. 3.3.1 Discretization of the Laplacian (diffusive) term

The first term on the RHS of equation (3.4) can be discretized by performing the integration on the control surfaces embracing the control volume, as follows: 47

  n    dS   S    f f     f 

(3.5)

f

CS

where all terms on the RHS are calculated on the control surfaces (figure 3.1). To completely discretize the above equation, the gradient in the above equation should be replaced by an expression in terms of cell center values. As expressed by Jasak [Jasak 1996], for an orthogonal mesh when the length vector connecting the two adjacent cell   centers d is parallel to control surface vector S f , the gradient can be approximated by the following expression:     S f     f  S f N  P d

(3.6)

In the case of non-orthogonal meshes, a non-orthogonal correction is applied to the above equation, which minimizes the error associated with the non-orthogonality as discussed in details by Jasak [Jasak 1996]. 3.3.2 Discretization of the convective term

The convective term in the transport equation (equation 3.4) is a non-linear term providing the most difficulties for numerical solution strategies. By applying the Green’s divergence theorem, the volume integral of the convective term is replaced by a surface integral. By performing the integration on the control surfaces in discrete fashion, the convective term is expressed by the values on the faces as below [Jasak 1996]:     n   U dS  S   U  f   F f (3.7)  f 



CS



f

 

f

f

where F in the above equation is a scalar value and stands for the mass flux. The value of  on the face should be approximated by the cell center values ultimately. On the other hand, the nonlinearity in the above equation should be removed in such a way to result in a linear representation. The lagging (or linearization) technique is typically used to tackle the problem. In the lagging technique, the velocity field of the previous iteration is utilized for an estimation of the fluxes; so providing an initial guess for the velocity field is mandatory right at the beginning of the calculations. In general, to solve the momentum equations in the unsteady flow simulations, two major computational loops exist in the numerical strategies: first, a time loop advances the solution in time using temporal discretization schemes, and second iterations over the nonlinear convective term are performed at each time step. Similar to the Laplacian term, to express the face field  in terms of the cell center values, different interpolation schemes

48

can be used which exhibit different level of accuracy. For example, the ‘central differencing scheme’ is an unbounded second-order accurate scheme and is expressed in terms of cell center values as follows:

f 

fN fP fN fN P  N  P  (1  )N PN PN PN PN

(3.8)

where over-bar stands for the geometrical distance. In other words, fN and fP represent the distance between surface f and the cell centers N and P respectively and PN indicates the distance between cell centers N and P as shown in figure (3.1). By the aid of the Taylor series expansion it can be shown that, the accuracy of the above approximation in space is the second-order. It should be mentioned that the scheme for convection-dominated flow systems can produce unphysical instabilities and results in the solution divergence. The ‘upwinding’ schemes form another class of schemes provide more stability for the convection-dominated flows. For example, the first-order accurate upwind difference scheme approximates the value of  on the face, based on the direction of the flow at the face, as below:   P if S f  U f  0 f     N if S f  U f  0

(3.9)

This scheme provides more stability but at the expense of obtaining lower accuracy, not enough for capturing small features of the flow field in the LES-type simulations. There are also other second order upwind schemes like QUICK, which uses a quadratic weightedinterpolation using three cell center nodes, or LUD which performs a pure upstream differencing using two upstream cell center nodes. These second order upwind schemes bring higher accuracy and less numerical diffusion in the computational procedure in comparison to the first order schemes; but they are still highly dissipative for pure LES computations. In the case of hybrid turbulence treatment, as explained in the previous chapter, two distinct zones exist. Ideally these two zones can be treated differently i.e. low dissipative and high order (at least second order) central difference discretization schemes can be used for the LES zone and high order upwind schemes can be adopted in the near wall zones. In the case of using a continuous discretization scheme across both zones, using the central difference schemes along with adding a local upwinding is desirable, as adopted in this project for the DDES simulations; in this way, a major part of the flow unsteadiness is resolved in the simulations.

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3.3.3 Discretization of the source term

The source term collects all the terms of the governing equation that cannot be expressed as temporal, convective and Laplacian terms. In general, the source term is a function of  . Typically, a linearization is applied before performing the integration ( S     , where

 and  can also be a function of  in general). The volume integration in this case can be simplified, as below:

 S dV  V

P

 PVP

(3.10)

CV

This formulation allows treating the source term implicitly [Jasak 1996]. It can be shown that in the case of negative values of  , the diagonal dominance of the system is increased by the source term’s contribution, which calls for an implicit treatment of the source term. In the case of positive value of  , explicit treatment of the source term and keeping the term on the right-hand side is preferable to avoid decreasing the diagonal dominancy of the system [Jasak 1996]. 3.3.4 Discretization of the unsteady term (temporal derivative)

The temporal derivative in equation (3.4) can be evaluated by the finite difference approximation. Different discretization schemes can be adopted for this purpose including Euler first and second order backward differencing, etc. The Euler scheme is a first order accurate scheme and can be expressed as follows:

   V     PP VP    dV  P P P  t CV t n

n 1

 O(t )

(3.11)

Especially for LES-type simulations, the time accuracy is essential and at least second order schemes are required. The backward differencing utilizes three time levels for estimation of the temporal derivative, which provides second order accuracy in time: 3   PP VP   4   PP VP    dV   t CV 2t n

n 1

   PP VP 

n2

 O(t 2 )

(3.12)

In this research work, a second order backward differencing is utilized to achieve an appropriate accuracy for the DDES simulations. After substitution of all terms in the transport equation (3.4) by the discretized counterparts, the following equation is obtained in the case of an implicit method. The equation is applied for all cells in the computational domain:

50

aPPn   aN Nn  RP

(3.13)

N

where the value of  at the cell center (P) depends on its neighbouring cells at the same level of time for the implicit strategy. In fact, there are two general strategies to numerically solve the discretized transport equation: explicit and implicit approaches. In the explicit method, all spatial derivatives and the source term are calculated in terms of known quantities from the previous time steps. This method is always conditionally stable. In contrast, for the implicit approach, some or all of the aforementioned terms are expressed in terms of unknown quantities at the new time step (n) as observed in equation (3.13). In other words, the value of  at the cell center (P) depends on the values of the neighbouring cells at the same time level. The implicit formulation allows using larger time steps, although larger time step results in a higher truncation error in the computations. On the other, a wide spectrum of events comprising a broad range of time scales exists in the turbulent flows. By increasing the time step, some of the flow events with time scales smaller than the time step size are not seen by the numerical simulation. In other words, by doing so, these events are omitted from the physics of the problem in the simulations. Depending how much these events are important in the prediction of the problem under investigation, larger time steps can result in a noticeable error especially for scaleresolving methods like DNS and LES-type simulations [Davidson 2011]; these physical barriers limit the increase of the time step in practice even for implicit strategies, which are theoretically unconditionally stable. Normally, for solving the set of equations (3.13) obtained by the implicit strategy, iterative solvers are utilized efficiently. 3.3.5 Pressure-velocity coupling

Pressure gradients with respect to the different spatial directions are present in the momentum equations. To solve these equations and to obtain the velocity components, the pressure field should be estimated somehow. For incompressible flows, the density is constant and is not present in the continuity equation (equation 3.1). It should be noted that, although no independent equation exists for the pressure in the case of incompressible flows, the resulting set of equations is mathematically closed and solution exists. In other words, the number of variables (three velocity components and pressure) is equal to the number of equations (three momentum and continuity equations). To construct a pressure equation for incompressible flows, after taking divergence of the momentum equation, the continuity equation is combined with the obtained equation; this ensures the satisfaction of

51

the continuity. The resulting equation is an elliptic (Poisson) equation for the pressure field, which is written as follows: u j ui 2 p    xi2 xi x j

(3.14)

In general, the governing equations of incompressible fluid motions can be solved in two fashions named ‘segregated’ or ‘coupled’ approaches. In the coupled method, the pressure-based continuity and momentum equations are solved simultaneously. The method needs more memory than segregated method and has typically a better convergence rate. The coupled strategy is an appropriate choice especially for simulation of compressible flows including shock waves.

Fig 3.2 Solution procedure for the utilized PIMPLE method

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In the segregated approach, the governing equations are solved sequentially or segregated from one another. In other words, in the strategy, the governing equation for a certain variable is solved for all cells of the computational domain and the process is then repeated. The solution method can be used efficiently for simulations of the incompressible and low-Mach number compressible flows. OpenFOAM solvers are based on the segregated approach. Therefore, the segregated method is utilized to simulate draft tube flows in this project. There are different methods for the pressure and velocity coupling like PISO and SIMPLE algorithms. In this project, the PIMPLE method (merged PISO-SIMPLE) is utilized for the pressure-velocity coupling in the turbulent fluid flow simulations. In this strategy, first, the momentum equations are solved using an initial guess for the pressure field at the beginning of the computation or the pressure field of the previous time step thereafter; then, the pressure-correction equation (continuity) is solved. For the next steps, the pressure field and velocity fields are updated. Next, other transport equations are solved like turbulence equations. At each time step, the aforementioned correction procedure recurs to achieve a predefined level of convergence and the solution procedure proceeds to the next time step until the prescribed running-time is met. Figure (3.2) shows the adopted procedure for the unsteady simulation of draft tube turbulent flows using the PIMPLE approach. 3.3.6 Uncertainty in hydro-turbine turbulent flow simulations

The governing equations of incompressible fluid flow motions (equations 3.1 and 3.2) are based on certain assumptions like the Newtonian fluid, single phase and constant properties especially constant density. Therefore, exactness of the governing equations and its prediction accuracy depends directly on the validity of these underlying assumptions in the certain circumstance. For instance, if the cavitation occurs in the flow field (in the hydro-turbine e.g. due to the local high-speed regions in the runner section) then the validity of the aforementioned equations (i.e. single-phase simulation) becomes doubtful. Otherwise, if the abovementioned assumptions are valid, the incompressible Navier-Stokes equations are considered ‘exact equations’ to numerically reproduce the real fluid flow physics for hydroturbine applications. The numerical methods applied to solve the equation introduce some errors in the solution procedures including the discretization, iterative-process, round-off, mesh-related and turbulence treatment-related errors for turbulent flow simulations [Zikanov 2010]. To solve equation (3.4) numerically, the derivatives should be discretized in time and space,

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e.g. using Taylor series expansions, which creates the ‘discretization error’. This error contributes to the overall numerical simulation error. Using higher order schemes with higher accuracy can mitigate this type of error considerably. Although, it always exists in the numerical solution of PDEs and cannot be totally eliminated. To treat the non-linearity in the Navier-Stokes equations, at each time step an iterative process is adopted to obtain a result satisfying a pre-defined level of convergence. The convergence criteria introduce another type of numerical error in the solution called ‘iterative-process error’. However, using low enough convergence criteria can alleviate the error for practical engineering simulations. Another type of error, called ‘round-off error’ is related to the machine accuracy. In fact, computing systems use a finite number of significant digits to represent a real number, in other words, the real numbers are rounded and truncated by the machine and the computations are performed on the truncated numbers. The difference between approximated value (considered by the machine) and the exact mathematical value is referred to the ‘round-off error’. In a sequential calculation, round-off errors can be accumulated and result in a large computation error. This type of error can be attenuated by using more significant digits in the computations like double precision. The double precession is the default accuracy in OpenFOAM that is used for upcoming simulations in the present thesis. Mesh quality plays a crucial role in accuracy and stability of numerical simulations of fluid flows. Grid irregularities can create serious errors in computations such as an artificial dissipation, which is especially important for LES-type simulations. Different quality measures can be adopted to quantify the befit of a given mesh such as the cell aspect ratio, skewness or angle, orthogonality, smoothness. For example, cells with high aspect ratio, skewness or angle can decrease the accuracy and ultimately destabilize the numerical solution, so should be avoided. Non-orthogonality of the mesh also introduces error, for example, in the calculation of the Laplacian term introduced in subsection (3.3.1). To minimize this type of error, nonorthogonal correction is applied in OpenFOAM. Another important feature in the generation of a mesh is its smoothness. In fact, rapid and sharp volume variations between adjacent cells in the computational grid should be avoided because it increases the truncation error and can result in an artificial vorticity generation at the related region. As a general rule, to minimize the ‘mesh-related errors’, computational grids should be constructed to respect all criteria as much as possible. For complex geometries, ideal values cannot be completely respected and a deviation from the ideal situation always exists. The goal is therefore to minimize the mesh-related errors as much as possible. Furthermore, to capture accurately some important 54

flow features such as flow separations, blade tip vortices, shock waves, the local refinement of the mesh is of great importance. Performing a ‘mesh-independence’ or ‘mesh-convergence’ test is necessary to validate the quality of the mesh for a CFD simulation. In the procedure, the mesh is refined and the simulation is re-run on a finer mesh. The difference between the two solutions is considered as a measure to judge about mesh convergence. The mesh refinement process is repeated until a desired measure is obtained within a predefined tolerance. For practical purposes, a global engineering quantity is typically considered as a measure e.g. the loss or recovery coefficient for a draft tube flow simulation. It is worth to mention that, there is a conceptual difference in interpretation of mesh-convergence test for URANS and LES approaches. In fact, by reducing the grid size in a mesh refinement process, the URANS solution accuracy theoretically increases via reducing truncation errors and interpolation schemes. On the other hand, in LES approach by reducing the mesh size more physical structures (turbulent eddies) with smaller size are directly resolved. In other words, a larger portion of the turbulent spectrum is captured by the computational mesh and less part is modeled. Although the mesh-convergence test can still be applied for LES-type simulations, this fundamental underlying concept should be considered in its interpretation. Limitations of the turbulence treatment strategy can provide another source of uncertainty in numerical results of turbulent fluid flow simulations. As explained in chapter 2 in details, DNS resolves all the scales of the turbulence from smallest to the largest eddy scales; in other words in the CFD world, DNS is considered as reality and only numerical approximation errors reduce its prediction accuracy. In LES method, only a part of spectrum corresponding to large eddies is directly resolved and the rest is modeled via subgrid scale models, which introduces ‘turbulence modeling-error’ in the computations. In URANS approach, the entire turbulence spectrum is modeled and no part is directly resolved; so the modeling-associated error contributes noticeably in the overall error of the numerical simulation for this type of treatment. For draft tube only flow simulations, the inlet section of the simulation domain is placed after the runner component. Rotation of the runner along with its blade wakes makes the flow field very complicated at the entrance of the domain. Uncertainty in the inlet profile, including velocity and turbulence quantities, is another source of error for draft tube flow simulations that considerably affects the quality of the predictions. In fact, inlet profile can be provided for the LES-type simulation either from full-machine flow simulations or from experimental measurements. In the case of experimental inlet profile, measurement errors and uncertainties directly contribute to the accuracy of the simulations. On the other hand, if the inlet profile for the LES-type simulation is obtained from full-machine

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simulations, then feeding the LES region with the pure RANS values at the interface creates another source of error (i.e. lack of inflow turbulence) which will be explained in details in the next chapter. As explained above, there exist several sources of error in the numerical simulation of turbulent fluid flows in general and specially for the draft tube flow simulations. This makes ‘validation process’ an unavoidable part of any attempt to simulate turbulent fluid flows. In fact, to quantify the level of confidence of a simulation and determine how well a numerical strategy can reproduce the real flow physics, simulation results should be compared with experimental data. In this project, at first, a test case (ERCOFTAC conical diffuser) is considered to gain some initial insight into DDES simulations and to validate a portion of the adopted numerical strategy. In addition, to validate the draft tube simulations, results of the experimental measurements on the model bulb turbine conducted at LAMH are utilized [Duquesne 2015]. More details are presented in chapters 5, 6 and 7.

3.4

OpenFOAM CFD platform

In this project, OpenFOAM 1.6-ext code is utilized to simulate the draft tube turbulent flows. OpenFOAM (Open Field Operation and Manipulation) is an appropriate platform to perform flow simulations within hydro-turbine components from the headwater to the tail water. It is an open source environment developed using object oriented C++ language originally in the late 1980s at Imperial College, London. OpenFOAM includes a wide variety of solvers, utilities and libraries making it an appropriate tool to solve arbitrary PDEs arising in continuum mechanics and CFD domains with pre- and post-processing capabilities. This CFD tool has interesting features, which makes it very attractive for academic and industrial researches. First, its full source code is completely available which gives users the flexibility to perform modifications and customize the code for their own needs, even dealing with non-standard continuum mechanic problems beyond the ability of the commercial tools. OpenFOAM is a free-to-use software i.e. with no license cost, in contrast to other relatively expensive commercial software packages. This makes possible a wider international collaboration on its usage and developments among different special interest research groups. On the other hand, due to the large computational scale of the high Re number fluid flow simulations encountered in the industrial applications, usage of the parallel computations is unavoidable. OpenFOAM provides a robust free-of-charge parallel-computing platform with no licence limitation to simulate efficiently the fluid flow problems using OpenMPI library. In this project, OpenFOAM is adopted to perform all

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transient DDES simulations. Figure (3.3) shows a case set-up including the necessary subdirectories utilized for DDES simulations of turbulent flows in the BulbT draft tube.

Fig 3.3 A typical OpenFOAM case structure for transient DDES simulations

3.5 Parallel processing & HPC Nowadays, high performance computing (HPC) is essentially based on the parallel processing technique. In general, to improve computation performance one can rely on faster processors as a first choice. In recent years, CPU technology has been developed very much; in fact, according to Moore’s law, each 1.5 to 2 years the computation power of digital computers has been approximately doubled [Moore 1965]. However, development of the electrical-based processors essentially approaches its theoretical limits due to heatrelated reasons. Currently, maximum CPU clock rate achieved via nitrogen cooling is lower than 9 GHz as appeared in the Guinness world records.

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It seems further improvement in the computer processing technology field needs a change in the computation philosophy by using other developing technologies; like adopting optical, quantum and bio-computing methods, which are not currently available for scientific computations. An alternative promising approach is parallel computations. The ‘Compute Canada’ national HPC platform provides a powerful environment to perform parallel computations for researchers in a broad spectrum of disciplines across Canada. It consists of seven regional consortia in Canada including ACEnet, WestGrid, HPCVL, RQCHP, SciNet, SHARCNET and CLUMEQ. Each consortium serves majorly several academic and research institution members. CLUMEQ (consortium Laval, Université du Québec, McGill and Eastern Québec) involves two major computing servers named Colosse and Guillimin. At present, Colosse consists of 7680 CPU cores and possesses rank 314 among most powerful super-computers worldwide. In fact, the supercomputer has 960 computing nodes and each node has two quad-core Intel processors at 2.8 GHz clock rate. For 936 nodes there exists 3 GB of memory per core and the other nodes have 6 GB of memory per core. The system has a capacity of 500 TB for storage and also utilizes Infiniband interconnect for the network. The Guillimin cluster has also been utilized in the present project; the cluster involves 1200 computing nodes and each node has 12 cores involving two six-core Intel (Xeon X5650) processors. The cluster’s rank is 125 among top supercomputers. On average, the system involves 3.2 GB of memory per core and all nodes are connected via an Infiniband network. For high Re number turbulent flow simulations, computational cost depends on the type of the turbulence treatment anticipated in the simulation and if the simulation is either steady or unsteady. For RANS simulations, a relatively coarse mesh is utilized but LEStype simulations, e.g. hybrid LES/RANS method, needs fine grids to resolve turbulent structures appropriately. In the present project, the intermediate mesh utilized for unsteady DDES simulation of the draft tube flow has number of elements in the order of 7-8 millions. Large number of computational cells, unsteady nature of LES-type simulation along with small physical time step and turbulence treatment procedure utilizing high-order discretization schemes makes the problem very computer-intensive. In fact, a typical simulation of this kind takes few months to run on parallel systems to obtain a statisticalconverged solution on about 120 cores. In this research, ‘Compute Canada’ HPC platform facilities were utilized to run the simulations. For parallel computations, different decomposition approaches can be utilized including data (domain) decomposition, task (function) decomposition and hybrid decomposition methods. In the case of task decomposition, independent tasks of the computation are performed on different cores separately. For data (domain) decomposition,

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the spatial computational domain is divided into a limited number of regions and each subdomain is assigned to a computing core. The common boundary information is exchanged between cores via a network system. Hybrid decomposition method combines both approaches in a unified strategy and takes benefits of both approaches. OpenFOAM provides spatial domain decomposition for parallel computing; for this purpose, four different spatial decomposition techniques can be adopted in OpenFOAM 1.6-ext including simple, metis, hierarchical and manual. In the simple geometric decomposition, the computational domain is divided to predefined number of pieces in different directions. Metis is a load-balanced decomposition strategy that tries to minimize the number of processor boundaries. The hierarchical method is similar to the simple approach, but the user just defines the order of splitting in different directions. Finally, the manual decomposition provides the flexibility to users to allocate each cell to a desired processor. Finally, it is important to notice that one important factor that can affect the OpenFOAM parallel computing performance considerably is the compilation strategy. As reported in [Vincent 2010], by copying pre-compiled version of OpenFOAM on the host super-computer, the efficiency decreases a lot and it is strongly recommended to compile the OpenFOAM locally. For this reason, OpenFOAM was compiled locally on the two super-computers for gaining a maximum performance. Another issue is the C++ compiler used for parallel computing; in fact, OpenFOAM is originally distributed by GCC compiler but Colosse and Guilimin have ICC compiler for computations. As tested by Vincent, both compilers have a comparable performance and ICC even improves slightly the speed-up; that is why ICC module was adopted for compilation in this study. 3.5.1 Speed-up test

To optimize the usage of the available computational resources and to observe the parallel computing performance, a good way is to perform speed-up tests. In this test, the gain obtained by a parallel computation is calculated via the comparison of execution wall clock time for a run on multiple cores (using domain decomposition) with the same program run on a single core on the same cluster. Mathematically, ‘speed-up’ can be defined, as follows: Speed-up =

texecution on one core texecution on NC cores

(3.15)

Ideally, by increasing number of cores, linear speed-up can be obtained although communication between cores limits the parallel computing performance and as a result, linear speed-up is not the case. In the present study, the speed-up test is made using unsteady DDES flow simulations of the ‘basic geometry’ of the BulbT draft tube with 59

about 7.5 million elements. Each Colosse node and Guillimin node has 8 and 12 processors, respectively; therefore, the request cores in the submitted job should be a factor of 8 in the case of the Colosse and 12 in the Gullimin cluster. For the speed-up test, the maximum number of 264 cores are utilized. Then, the execution time is normalized by the single-core run-time (figures 3.4 and 3.5).

Fig 3.4 Speed-up curve for the BulbT draft tube flow simulation on Colosse cluster

Fig 3.5 Speed-up curve for the BulbT draft tube flow simulation on Guillimin cluster

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For domain decomposition two methods of ‘simple’ and ‘metis’ are used for all cases. Due to the importance of the inlet boundary conditions, the inlet plane is preserved and only assigned to one core to avoid possible errors arising from decomposition method. Figures (3.4) and (3.5) indicate that a run time gain is obtained by increasing the number of the cores for all methods of decomposition, but it is not as ideal as the linear speed-up. Simple and metis have approximately the same performance up to 50 cores. For number of the cores more than 50, metis with load-balance between the cores performs better than the simple method. The metis decomposition method is therefore selected in this study for domain decomposition, due to its superior performance. It should be mentioned that to optimize the usage of computational resources, in general, one should decide between the gain in computational time obtained by increasing the number of the cores in terms of the computational time and the available core allocation and also priority of the submitted job on the cluster. In fact, available core allocation limits increasing of number of the cores even in the case of linear speed-up. In this study, normally 120 to 264 cores are utilized for draft tube flow simulations. In the next chapter, the strategy developed to generate inlet boundary condition is discussed in details. 3.6 Summary

The major points of this chapter can be summarized, as follows:



Navier-Stokes equations along with the continuity equation govern turbulent behaviour of the unsteady viscous incompressible flow in general and in the case of the BulbT draft tube.



Navier-Stokes equations, as a second order system of PDEs, behave parabolically for unsteady viscous fluid flows. They form a well-posed system, if they are expressed in the form of IBVP. Under these conditions, existence and uniqueness of the solution are guaranteed, although no closed-form solution is on hand for many practical flows of interest such as in the BulbT draft tube.



In the present project, the governing equations of the turbulent flows inside the BulbT draft tube is numerically solved using the FVM in the OpenFOAM environment.



Different sources of uncertainty exist in CFD simulations, as in the BulbT draft tube simulations, including: discretization, iterative-process, round-off, mesh-related and turbulent treatment-related errors for turbulent flow simulations.

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

Different discretization schemes are applied for different terms of Navier-Stokes equation, including: unsteady, convective, diffusive and source terms.



For LES simulations, the literature mentions that higher order schemes, at least second order, are necessary to discretize the convective term. To avoid numerical dissipation introduced by upwinding, the ‘central difference’ schemes should be also adopted for pure LES simulations. For RANS/URANS cases, the ‘second order upwind’ scheme is typically a good choice. For hybrid RANS/LES methods like DDES, a central difference scheme with local upwinding can be used for convective term discretization.



For temporal term discretization (if any), the ‘second order backward’ scheme is applied for all simulation cases.



Pressure–velocity coupling in OpenFOAM is done with the ‘segregated’ method. In this regard, SIMPLE and PIMPLE methods are utilized for steady and unsteady simulations, respectively.



DDES simulations of BulbT draft tube flows are computer-intensive, therefore ‘Compute Canada’ HPC platform facilities, including Colosse and Guillimin clusters, are used.



OpenFOAM uses spatial domain decomposition method for partitioning. As revealed by the speed-up test on the both clusters, ‘metis’ decomposition is selected for the partitioning applied for the parallel computing.

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Chapter 4

Inflow condition for draft tube flow simulations

4.1

Introduction

Boundary and initial conditions play a crucial rule in solving any PDE. Navier-Stokes equations, as a set of coupled non-linear PDEs, are not exempt from this fact and are even more sensitive to the initial and boundary conditions in the case of turbulence, which is sometimes interpreted as ‘chaos’ [Charles 2013, 2014]. As mentioned in the previous chapters, the Navier-Stokes equations govern all physical fluid phenomena encountered in Newtonian fluid flows in the real world; its applicability also covers a broad range including the laminar and turbulent flows in all flow regimes. It is interesting to notice, regardless of using the same equation i.e. Navier-Stokes equations for all cases, what creates the vast variety of physical phenomena encountered in fluid flows mathematically is the difference in the boundary conditions applied on fluid flow domain boundaries. In this perspective, the importance of providing appropriate boundary conditions for turbulent flow simulations becomes more pronounced. Especially in the case of draft tubeonly simulation, the situation is worse, because the inlet plane is completely located in a turbulence region downstream the runner including different spatial and time scales, which makes the fluid flow simulations very complicated. In this chapter, different aspects of the inlet boundary generation are discussed.

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4.2

RANS versus LES inflow conditions

If the inflow plane is located in a turbulent region, it is essential to prescribe appropriate profiles for the turbulent and velocity quantities as the boundary condition. The most accurate inflow condition would be obtained by imposing the complete turbulent velocity signal at all computational nodes on the inlet plane. This strategy is mandatory for DNS and LES simulations in some applications with spatially evolving flows, like boundary layer and channel flows [Jarrin 2008]. In the case of RANS simulations, because of the dissipative nature of the method and also its inherent characteristic recognized as modeling of all scales of the turbulence spectrum instead of resolving, it is sufficient to provide only the mean quantities, i.e. mean velocity profile and turbulent quantities. On the other hand, for unsteady simulations, temporal variation of the inflow quantities should be considered at the inlet plane, if applicable. In the URANS approach, low frequency variations should be included, whereas for the LES and DNS turbulence treatments, complete turbulent velocity signals should be considered (figure 4.1). In draft tube flow simulations, the runner rotation creates considerable vortical and wake structures, which are transmitted into the draft tube; this creates majorly low frequency variations of the flow field. If the domain contains only the draft tube part, unsteady simulations need the time history of the inflow quantities. If the computational domain involves guide vanes, runner and draft tube or at least runner and draft tube, the inflow can be prescribed by mean flow quantities. As explained in chapter 2, a typical turbulent velocity signal at the inlet plane can be expressed with three constitutive components based on ‘Reynolds triple decomposition’, as below in a tensor notation: ui (t ) 

turbulent velocity signal

 uim  uil (t )    mean

low frequency fluctuations

   coherent part

ui(t ) 

(4.1)

high frequency fluctuations

  

incoherent part

where the index of i  1, 2 and 3 denotes the three spatial directions x, y, z, respectively. The first two terms on the RHS represent the low frequency variation of the inflow condition. In the case of draft tube-only flow simulation, these terms can be obtained by URANS simulation of the upstream components of the draft tube, or by an unsteady 2D rotating profile. The later profile can be obtained by rotation of the mean 2D profile calculated for example from a full-machine k   RANS simulation at the draft tube inlet plane as explained in the following section. The third term on the RHS of equation (4.1), i.e. ui , represents the incoherent part of a turbulent velocity signal, corresponding to the high frequency random fluctuations of the turbulence.

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Fig 4.1 Schematic of a typical turbulent velocity signal

4.3

Inflow condition for draft tube-only flow simulation

For draft tube-only simulations, typically two types of inlet B.C. can be utilized including the steady circumferential averaged-1D profile and the unsteady 2D profile. These profiles can be obtained by the RANS or URANS simulation of the full machine or by experimental measurements. In this project, for DDES simulations of the draft tube flow using OpenFOAM, the inlet profiles are extracted from full-machine k   RANS simulations at selected operating points using ANSYS-CFX software. In the BulbT project, the turbine assembly consists of different parts: intake, bulb section including two pillars, 16 guidevane blades, an adjustable four-blade runner and a horizontal straight draft tube (figure 4.2).

Fig 4.2 Sketch of the hydraulic profile sections in the full-machine simulation

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In the k   full machine RANS simulations, only one blade passage of the guide vanes and one blade passage of the runner is considered [Guénette 2013, Houde et al. 2014]. In other words, periodic boundary conditions are imposed for the guide vane and runner blade. For these simulations, the inflow condition is set in a manner to respect as much as possible the experimental conditions obtained on the LAMH test bench. The inflow boundary condition is imposed as measured mass flow rate corresponding to a defined constant net head. For inflow turbulence quantity at the intake section, the intensity of the 1% and vt v  1 are set. As showed by Guénette, the level of turbulence has almost no effect on the performance prediction of the turbine by k   RANS. For the outlet boundary condition, the average pressure equal to zero is set. To treat the stationary and non-stationary parts, the stage mixing interface is utilized between the runner and the guide vane blades, in addition for the pair of runner and draft tube. As it is visible in figure (4.2), an extension is added for numerical reasons to the geometry. The ‘slip’ boundary condition is applied on the extension wall to avoid convergence problem due to the probable occurrence of back flow at the exit of the draft tube. Figure (4.3) shows the hydraulic profile used for draft tube flow simulations. As explained in the first chapter, the draft tube consists of a conical part and a transition section from a circular to a rectangular section. In the previous project of the LAMH, i.e. AxialT project, the draft tube contains an elbow part to change the direction of the flow leaving the runner from vertical to horizontal axis [Gagnon 2012]. But, in the BulbT project the direction of the flow is already horizontal and only the cross sectional increase criterion is applied in the draft tube geometry design.

Hub

Extension Transition Conical part

Fig 4.3 Final version of the draft tube geometry At the beginning of the project, the draft tube was designed with a relatively mild divergence angle of 6.6 , RANS simulation results at different operating points of the 66

machine showed that there is not a noticeable flow separation in the flow field [Guénette 2013]. Later by advancement of the project, the draft tube was re-designed with a more aggressive divergence angle ( 10.25 ) to generate more physical phenomena interesting to study like large flow separation zones and vortex breakdown, etc. In fact, by increasing the divergence angle of the conical part of the draft tube, adverse pressure gradient increases, which intensifies the tendency for flow separation and vortex breakdown phenomena. In the case of draft tube-only flow simulation, effects of the runner rotation is included in the computational domain via the profiles of velocity and turbulent quantities at the inlet plane. The position of this plane is selected as close as possible to the runner to maximize the influence of the runner and to avoid downstream data diffusion through the dissipative nature of the RANS modeling. The selected position of the plane is also the same as the measurement plane, where LDV measurements are performed. This enables to utilize experimental inlet velocity profiles for draft tube flow simulations as explained in chapter 7. In general, considering unsteady LES-type simulation of the draft tube flow, the important physical phenomena to be studied could be classified, as below:



Flow separation zones



Spiral vortex rope



Runner related vortical structures and wakes



High frequency turbulent fluctuations at the inlet plane



Corner vortices if geometrical corners exist



Horse-shoe vortices if pillars are present in the draft tube geometry (not present in the BulbT configuration)

The presence of the relatively strong pressure gradient intensifies the risk of flow separation in the draft tube especially near the wall. As will be explained in details in chapter 7, dynamics of these separated zones has crucial effects on the behavior of the draft tube. Theoretically, most of the swirl is captured by the runner at the best efficiency point; the turbine is nominally designed to operate at this point. At off-design part-load operating conditions, guide vanes are set to increase the swirl upstream of the runner attempting to absorb the flow swirl. Part of the excess amount of swirl created by the guide vane blades remains and propagates into the draft tube as a continuous physical entity called ‘spiral 67

vortex rope’. Typically, this spiral vortical structure is attached to the runner hub. To capture the vortex rope structure completely, a part of hub is included in the computational domain, as shown in figure (4.3). This enables the simulations to resolve the vortex rope entirely in the draft tube. 4.3.1

Normalization reference scales

For normalization purposes used to represent the results from this chapter on, the reference length ( Rref . ) is considered equal to the runner shroud radius ( RInlet ) measured at the inlet section of the draft tube. The reference velocity is also defined as below: uref . 

Q Measurment f 2  RInlet

(4.2)

where the numerator stands for the measured flow rate at each selected operating point. 4.3.2 Circumferential averaged-1D inflow profile

Depending on the accuracy level desired, different interface treatment methods can be applied between stationary and non-stationary parts like frozen rotor, stage-mixing plane interface and transient rotor-stator interaction as classified in the ANSYS-CFX documentation. In the stage interface approach, the velocity components and turbulent quantities are averaged on different radial circular sections and then the resulting circumferential averaged-1D inlet profile is applied at inlet B.C. for downstream flow simulations. For draft tube-only flow simulation, this profile can be applied at the inlet plane. Figure (4.4) shows circumferential averaged-1D profiles at one operating point, OP.4, for the final draft tube flow simulation coming from the full machine k   simulation. In fact, the circumferential averaged-1D inlet profile only contains the mean part of the turbulent signal. In other words, low and high frequency variations of the turbulent velocity signal are neglected at the inlet plane. By applying this averaging strategy, all seemingly important physical structures at the inlet plane, such as runner related vortical structures and wakes, are smoothed out as will be shown in chapter 6 and 7; although, the global characteristic inflow quantities like the mass flow rate and flow swirl are conserved. In general, in the case of hydro-turbine full machine flow simulations, the selection of rotor-stator interface treatment depends on the type of problem that one is interested to investigate. For example, to study guide vane-runner interaction in details, a transient rotor-

68

stator treatment, which considers relative motion of the components via moving mesh, should be adopted, however this will be at the expense of high computational cost. As mentioned, a stage interface is typically utilized with reasonable computational effort to calculate the full-machine hydro-turbine flow field in the application.

u u Ref.

r RInlet

Fig 4.4 Circumferential averaged-1D inlet profile obtained from full machine k   simulation at OP.4 (  Runner  30.2 , GV  65.4 , N11  170, H  4m ) 4.3.3

Unsteady 2D inflow profile

Unsteady 2D profile keeps the flow structures at the inlet plane and let the draft tube simulation resolve runner related vortical structures and wakes. In contrast to the circumferential averaged-1D profile, in this type of inflow profile, averaging is not applied and the exact 2D variation of velocity and turbulent quantities is imposed at the inlet section. To obtain an appropriate 2D profile, one needs to have the time history of the velocity and turbulent quantity signals at inlet computational nodes obtained from full machine RANS/URANS simulations or from performing experimental measurements. Figure (4.5) shows the global flowchart adopted to apply 2D profiles for DDES simulations of the draft tube flow. In this strategy, first the whole flow field is simulated by k   RANS simulation at the selected operating points as explained in section (4.3). The solution including velocity and turbulent quantity fields are extracted at the experimental inlet plane (called plane A, thereafter). As mentioned previously, one runner-blade passage is only included in the computational domain in the case of full-machine simulations. Therefore, the solution slice 69

is copied 4 times, corresponding to 4 runner-blades, to have a complete profile at the inlet of the draft tube. After this step, all the intended modifications are applied on the inflow profiles like the runner rotation (section 4.3.4), the viscosity treatment (section 4.3.5) and adding the synthetic inflow turbulence (section 4.4), which will be explained in details shortly. Finally, the generated profile is then fed at the inlet section of the draft tube.

Fig 4.5 General sketch on applying a transient 2D inflow profile for the DDES simulation of the draft tube using OpenFOAM 4.3.4

Unsteady 2D-rotating profile

To obtain an unsteady 2D-rotating profile here, the coherent part of the velocity signal is constructed by rotation of the steady 2D velocity profile with the runner rotation speed, which mimics the runner turn. The velocity signal includes the mean and low frequency 70

constitutive terms of the turbulent velocity signal (first two terms on the RHS of the equation 4.1). The underlying assumption of this method can be stated as: the runner rotation is the main factor of the flow field change at the draft tube inlet section. It is also worth to notice that the mean profiles obtained from RANS simulation does not exist physically. In other word, by definition it is an average profile obtained from the summation of all unsteady flow frames at all instant of times at the inlet section. There is a difference between transient real inflow (for example coming from URANS) and the unsteady 2D rotating profile. In this research, however it is considered that this difference to be negligible and it is basically assumed that the major unsteadiness coming from the runner into the draft tube can be captured by the rotation of steady inflow profile coming from the RANS simulations. v log( t ) v

uz uRef. r RInlet

r RInlet

r RInlet

r RInlet

Fig 4.6 2D variation of the axial velocity uZ (left) and turbulent eddy viscosity  t (right) at inlet plane from k   RANS at OP.4 (  Runner  30.2 , GV  65.4 , N11  170, H  4m )

r RInlet

r RInlet

Fig 4.7 Delaunay triangulation at the draft tube inlet plane at one instant of time

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The steady 2D velocity and turbulent quantity profiles are obtained upstream the stage interface by applying the averaging procedure. For all flow simulations in this project, the inlet profiles including the circumferential averaged-1D profile and 2D are obtained by the RANS simulation of the full machine at selected operating points. As an example, figure (4.6) shows the axial velocity and turbulent eddy viscosity variation at the inlet plane at OP.4 for the flow simulations of final version of the draft tube.

Fig 4.8 Rotation of the 2D velocity profiles at the draft tube inlet plane for OP.4 top: radial velocity ( ur ) middle: circumferential velocity ( u ) bottom: axial velocity ( u z ) In the rotating 2D profile procedure, flow quantities, i.e. x, y, z , u , v, w, t (or  ,  ) , are read from the RANS data at all computational nodes on the draft tube inlet plane. In this approach, the rotation is performed by rotating of the inlet grid points and assigning the

72

corresponding velocity and turbulent quantities at each rotating-node. This is done by appropriate interpolation using a ‘Delaunay triangulation’ technique. The Delaunay technique for a set of points ensures that there is no internal node inside the circumcircle of any triangle on the complete triangulation map. It avoids small internal angles and verifies that each point at least belongs to one triangle [Baerentzen et al. 2012]. The constructed map is utilized for flow quantity calculation at inlet plane during the rotation. In the procedure, at each instant of time, a ‘Delaunay map’ is created as shown in figure (4.7). It is worth to mention that the velocity should be expressed in the cylindrical coordinates for the rotation of velocity profile; in this way, the velocity components in cylindrical coordinate at each rotating-node can be considered unchanged during the rotation. Figure (4.8) illustrates the rotation of the radial, circumferential and axial velocity profiles at draft tube inlet between two instants of time corresponding to 35 of the runner rotation. The time history of the velocity signal created at inlet plane nodes by 2D-rotating profile can be analyzed by putting a virtual probe at the inlet plane and performing power spectrum analysis. Figure (4.9) shows power spectrum of a sample velocity signal at one Eulerian probe located at the inlet plane.



5 3

E uz ( f )

f f runner

Fig 4.9 Power spectrum of the axial velocity signal generated by 2D rotating profile at one Eulerian probe placed at the normalized position ( x / RInlet , y / RInlet )  (0.3,  0.55) . It is clear that the minimum frequency with the highest energy corresponds to the blade passing frequency which is equal to 4 times the runner rotation frequency. It should be mentioned that as discussed by applying the 2D-rotating profile for draft tube-only simulation, the fine details of the inflow condition and their evolution in the flow field to 73

some extent are taken into account. In other words, the runner related vortical structures and wakes are feasible to be captured. Although, some upstream flow phenomena signatures, like the guide vane-runner interaction frequency, are lost by the approach. It is reasonable to estimate that it has minor effects on the state of the draft tube flow. For the DDES simulation of the draft tube flow field, some modifications will be applied on the turbulent quantities at the inlet plane, which is explained in the following sections. 4.3.5

Inflow turbulent viscosity treatment

As mentioned, the flow field solution in the all upstream components of the draft tube are simulated by the k   RANS turbulence treatment. Therefore, the turbulent viscosity profile at the inlet plane should be re-estimated with some cautions in the case of DDES simulations of the draft tube flow. In fact, one should keep in mind the difference between the turbulent viscosity ( t ) and sub-grid scale viscosity ( SGS ) concepts. In the case of RANS-type simulations, the turbulent viscosity is responsible to reproduce the behavior of the whole turbulence spectrum as explained in chapter 2, which is really a difficult task as explained in details in chapter 2. But in the case of LES-type simulations, the sub-grid scale viscosity has less responsibility and should only mimic the behavior of the sub-grid scale eddies in the dissipation of the turbulent energy (energy sink). This fundamental difference should be taken into account in the hybrid turbulence treatment approaches.

t t  SGS

Fig 4.10 Schematic of the RANS (B.L.) & LES (core flow) zones at draft tube inlet plane As explained in the DDES as a hybrid turbulence treatment method, the near wall region is modeled by RANS but in the core flow the model switches to the LES mode to resolve the structure appropriately; of course if the mesh is fine enough. The ideal situation for the DDES simulation of the draft tube is to keep the RANS turbulent viscosity ( t ) in the wall region and to lower it in the core flow. In fact, in the core flow (i.e. in the LES zone), it behaves as the sub-grid scale viscosity,  SGS ; high values of this parameter leads to 74

the dissipation of the structures more rapidly. This issue is more pronounced in the case when artificial fluctuations are added to the velocity signal which is explained in details in the upcoming section. As shown in figure (4.10), at the draft tube inlet plane, two separate boundary layer zones are formed close to the hub and the shroud. In these two regions, the RANS turbulent viscosity ( t ) should be fed and in the core flow, sub-grid scale viscosity ( SGS ) should be estimated and provided through a blending strategy. In this regard, first of all one needs a function to recognize the boundary layer regions at the inlet. This can be performed using the f d intermediate function developed by Spalart [Spalart et al. 2006] introduced in chapter 2 (equation 2.12). Figure (4.11) shows the f d field at the inlet plane at one operating point.

fd

r RInlet

r RInlet

Fig 4.11 Function f d at the draft tube inlet plane for OP.4 For all simulations in this study, the Smagorinsky model is used to estimate the subLES grid scale viscosity ( SGS ). The model takes into account the velocity gradient field at the

inlet plane to calculate the sub-grid scale viscosity as below (for more details refer to appendix A1): 2 LES vSGS   CS   S

(4.3)

where CS is the Smagorinsky constant (typically between 0.1  0.2 ) and CS  0.17 based on the decay of isotropic turbulence [Pope 2004]. The strain rate tensor Sij is defined as the following: 1  u u  Sij   i  j  2  x j xi 

(4.4)

75

Figure (4.12) shows the variation of the sub-grid scale viscosity at inlet plane of the draft tube at OP.4. LES  SGS

r RInlet

r RInlet

Fig 4.12 Subgrid scale viscosity treatment using the Smagorinsky model for OP.4 at the draft tube inlet plane Having f d field on hand (equation 2-12) at the inlet plane, one can formulate the blending turbulent-subgrid scale viscosity at the inlet plane for hybrid simulations, as below: H LES  SGS  (1  f d ) tRANS  f d SGS

(4.5)

H ) calculated at one In figure (4.13), the blending sub-grid scale viscosity ( SGS

instant of time at OP.4 is shown in the logarithmic scale. H  SGS

r RInlet

r RInlet

Fig 4.13 Hybrid turbulent/subgrid-scale viscosity treatment for OP.4 By applying 2D rotating profile, the coherent part of the velocity signal is applied at each computational node and in this way important runner related vortical structures and

76

wakes can now be resolved. One remained part from the full turbulent velocity signal, not dealt with so far, corresponds to the high frequency turbulent fluctuations appearing as the last term in the RHS of equation (4.1). In the following section, the methodology developed in this project to generate synthetic turbulent fluctuations is explained in details.

4.4 Synthetic inflow turbulence: Artificial Fluctuation Generation (AFG) techniques As mentioned in the first chapter, one objective of this project is to study the impact of synthetic inflow turbulence on the draft tube flow simulation using a hybrid method, namely DDES. In this regard, appropriate artificial turbulence should be generated by a strategy at the draft tube inlet section. The ideal case to generate these high frequency turbulent fluctuations is to include draft tube upstream components, e.g. runner, in the simulation. In fact, this strategy is still too costly and even not feasible or necessary for the industry most of the time. To reduce the computational effort, one idea is to reconstruct the inflow turbulence via precursor simulations or through some stochastic procedures called ‘artificial fluctuation generation’ methods (hereafter, AFG methods). It is important to note that effectiveness of considering turbulent fluctuations at the inlet plane also depends on the turbulent treatment strategy used for the simulation. In the case of DDES treatment, the core flow is in the LES zone, in this way applying the AFG or precursor simulations can be fruitful. In contrast, near the wall (in the RANS zone) due to dissipative nature of the RANS, adding fluctuations has no meaningful effect and can also lead to doubling the turbulence in the RANS region [Frohlich and Von Terzi 2008]. In this regard, to exclude the RANS zone, a boundary layer detection function ( f d ) can be adopted and multiplied to the generated turbulent signal and the resulting signal is applied at each node on the inlet plane. It is worth to mention that the essence of applying the AFG turbulent signal at the inlet plane is similar to applying artificial fluctuations at the RANS/LES interface in the hybrid methods. In other words for both cases, turbulent fluctuations are added to the RANS profiles to provide better boundary condition for the LES zone. In general, for the generation of inflow fluctuations in the LES-type simulations, the most accurate way is to perform a precursor simulation. In other words, the nonlinear convective term in the momentum equation is physically the most accurate tool to generate the complex statistics of a real turbulent velocity signal, but at the expense of high computational cost. For example, Kaltenbach et al. adopted a precursor channel flow simulation with a periodic boundary condition applied between the inlet and outlet 77

successfully to generate inflow conditions for the LES simulation of a planar asymmetric diffuser [Kaltenbach et al. 1999]. Schluter also obtained good results by a precursor simulation with a periodic boundary condition in the case of a weakly separated diffuser [Schluter et al. 2005]. In general, a periodic boundary condition can only be adopted based on the validity of the fully developed flow assumption, which is not the case for many applications like the draft tube inflow. On the other hand, as can be expected, performing precursor simulations encounters some limitations in practice; for example, in the case of the draft tube inflow, presence of the runner just upstream creates complex turbulent time and length scale fields that should be taken into account. In the case of swirling flow in general, an idea can be performing a precursor simulation with the same mass flow rate and flow swirl along with adding forcing terms in the axial and tangential directions in the momentum equation to compensate friction effects. The strategy was applied for the simulation of relatively simple cases like swirling flow in a cylindrical tube [Pierce and Moin 1998] and for the ERCOFTAC conical diffuser flow [Duprat et al. 2009, 2010]. In the case of draft tube flow, similar to the precursor simulation with no inflow swirl, the method is not able to reproduce the complex turbulent scale fields in the complex situation at the draft tube inlet section, just after the runner. Also for the LES of the swirling flow in a combustor, another approach was adopted by Schluter et al. based on the scaling of the fluctuations of a turbulence database. In the method, the database is created by a precursor LES of a pipe flow with the periodic boundary condition [Schluter et al. 2004]. Then, the turbulent velocity signal is expressed as below, by including a scaling term for the obtained database fluctuations: 2 k 3 m l ui (t )  ui  ui (t )  [ui , DB (t )  ui , DB ]   ui,2DB coherent

(4.6)

where subscript (DB) stands for the database and k represents the inflow turbulent kinetic energy. The method could provide good results in this test case for different swirl levels in comparison to the experimental data [Schluter et al. 2004]. In the case of hydro-turbine draft tube flow simulation, as can be expected the details of the draft tube inflow (downstream of the runner) cannot be taken into account just by setting the mass flow rate and flow swirl at inlet plane or by scaling process. Therefore, other strategies like artificial fluctuation generation (AFG) methods, which are capable of considering inflow details, should be adopted to create the synthetic inflow turbulence. In principle, all AFG strategies involve an input of some random processes (e.g. a phase or an amplitude randomization) in the turbulence signal generation algorithms. In 78

fact, through a stochastic procedure, an artificial velocity signal is constructed which mimics some aspects of a real turbulent velocity signal. In other words, some orders of the real turbulent signal statistics could be captured like, length scale, time scale and Reynolds stress tensor, etc. It is worth to mention that the philosophy behind all AFG methods is to create signals as smart as possible to survive under numerical and viscous dissipation with statistical characteristics that are as much as possible close to the real turbulent velocity signals. As explained in chapter 2, these statistical characteristics of the turbulent signal can be viewed as a fingerprint of the coherent structures in the turbulent flows. 4.4.1 Review of AFG methods

According to the classification of Jarrin, AFG methods can be typically categorized into three major groups: algebraic methods, spectral methods, mixed methods [Jarrin 2008]. In general, the algebraic methods use physical space in its procedure, whereas spectral methods utilize the Fourier space, and mixed methods take benefits of both Fourier and physical spaces for the generation of inflow fluctuations [Jarrin 2008]. Here, to increase the generality and classify appropriately mapping methods like POD, Jarrin’s classification is slightly modified; the AFG methods are classified into three major groups called: physical space-based methods, mapped space-based methods and mixed methods. In the following subsections, each class of methods is explained concisely. 4.4.1 (a) Physical space-based AFG methods

This group of AFG methods uses the physical space to construct turbulent velocity signals; in other words all needed mathematical manipulations are performed in the physical space. The simplest method of this group can be named ‘random noise’ method. This procedure is started by generating an uncorrelated white-noise signal ri with a normal distribution (zero mean and unit variance). The method is capable to reproduce the mean flow and inflow turbulent kinetic energy ( k ), but does not consider turbulent fluctuation correlations. For a homogeneous turbulence, the total turbulent velocity signal can be expressed, as below: ui (t )  uim  uil (t )  ri

2 k 3

(4.7)

In the case of inhomogeneous inflow turbulence, the Reynolds stress tensor can be utilized to break the symmetry of the inflow turbulence. In the procedure, Cholesky decomposition is adopted [Lund et al. 1998] to consider turbulent fluctuation correlations (Reynolds stress) and finally the complete turbulent velocity signal can be stated, as below: 79

ui (t )  uim  uil (t )  aij rj

(4.8)

where the amplitude tensor aij is constructed based on the Reynolds stress tensor elements

 ij (A2.5 formula in appendix A2) as the following: 1/2  11  1/2  aij   21  a11   a 11  31

0

  0  2 2 1/2    a31  a32 )  0

( 22   a )

2 1/ 2 21

( 32   a21a31 ) a22

( 33

(4.9)

The above procedure guarantees fitting the target Reynolds stress tensor for the resulting artificial signal. The disadvantage of this method is the immediate damping of the artificial fluctuations just after the inflow section, due to the lack of energy in low frequencies. In fact, as shown in figure (4.14), a real turbulent velocity signal possess higher energy in the low frequency range (corresponding to the large scale structures) and lower energy in the high frequency range (corresponding to the small scale structures). In contrast, a white-noise signal has a constant energy level or uniform energy distribution over low/high frequencies (figure 4.14). This unphysical characteristic of the constructed artificial random- noise signal results in the re-laminarization of the flow, or in other words it leads to the fast dissipation of the AFG signal right after the inlet plane.

Fig 4.14 Energy spectrum of a real turbulent velocity signal (solid curve) and the white noise signal (horizontal dashed line) The aforementioned shortcoming was reported by different investigators in different applications like LES of a flat plate turbulent boundary layer [Lund et al. 1998], LES of a 80

turbulent jet flow [Holdo and Simpson 2002], DNS of a plane turbulent jet flow [Kelin et al. 2003], LES of a swirling flow in ERCOFTAC diffuser [Duprat 2010], ILES of a sonic jet injection into a supersonic cross flow [Rana et al. 2011]. To remedy this issue, some researchers tried to construct the turbulent velocity signal based on a target turbulent energy spectrum. For example, Klein et al. proposed a strategy in the physical space using digital filters to create a turbulent velocity signal with prescribed statistics like length scale [Klein et al. 2003]. The method first generates a white noise signal taken from normal distribution. Then the signal is convoluted with a digital linear non-recursive filter with Gaussian correlations [Klein et al. 2003] or exponential correlations [Xie et al. 2008] and the coefficients of the filter are determined in such a way to satisfy prescribed length scales. To obtain these filter coefficients, a simplified technique was also developed based on a prescribed autocorrelation function [Di Mare et al. 2006]. The method was successfully applied in different test cases like DNS and LES of planar turbulent jets [Klein et al. 2003, Di Mare et al. 2006], LES of a developing wall boundary layer [Di Mare et al. 2006], DNS of a nozzle flow [Klein et al. 2003] and ILES of a sonic jet injection into a supersonic cross flow [Rana et al. 2011]. The effect of spatially varying turbulence scales at the inlet plane was also investigated for a channel flow problem. The results indicated that considering variation of the scales at inlet boundary condition increases the accuracy of predictions in some important areas of the flow but at the expense of high computational cost of the method [Veloudis et al. 2007]. Another digital filter–based method was proposed by Fathali et al. based on a multicorrelated random velocity field constructed by the linear combination of the uncorrelated random fields fij with zero mean. The method is capable to construct the Reynolds stress tensor and integral length scales. The total turbulent velocity in this method can be stated as below: 3

ui (t )  uim  uil (t )   bij f ij

(4.10)

j 1

where bij stands for the unknown coefficients of the linear combination and is obtained using prescribed statistical quantities i.e. Reynolds stress tensor and known length scales. The uncorrelated random fields fij in expression (4.10) are obtained by the convolution of random white-noise signal with a Gaussian digital filter. The strategy was successfully adopted for the simulation of a homogeneous turbulent shear flows [Fathali et al. 2006, 2008]. Other AFG methods in the physical space use a collection of physical identities e.g. eddies [Jarrin et al. 2006, 2008], attached eddies [Subbareddy et al. 2006], turbulent spots or dipoles [Kornev et al. 2007, 2008] or physical processes e.g. diffusion process [Kempf et 81

al. 2005] to create inflow turbulence. The advantage of these methods over filtering and mathematical manipulation methods is to provide better feeling and understanding about the underlying physical mechanisms responsible for the turbulence generation. Moreover, it makes possible to take additional physical mechanisms into account for the future developments and AFG upgrades. In the diffusion-based method proposed by Kempf et al., a diffusion process is applied on the random noise in the physical space [Kempf et al. 2005]. In this procedure, the first three fields of random white noise ( ri ) are generated and normalized in such a way to have a zero mean and variance equals to unity at each inlet node. Then appropriate diffusion coefficient (  D ), time step ( t ) and the number of iterations to achieve the desired (prescribed) length scale is selected. Afterwards, the diffusion process (equation 4.11) is applied on the white noise to achieve the desired length scale at each node of the inlet plane. ri  2 ri  D t x j 2

(4.11)

The method is capable to consider both uniform and variable length scale distributions at the inlet. For the variable distribution, a different diffusion coefficient at each node is applied. In the final step, to fit the target Reynolds stress tensor, the signal is normalized in a manner to achieve zero mean and unit variance. Finally, Cholesky decomposition is adapted as expressed by equation (4.8). The method was applied with success in LES simulation of a non-premixed jet-flame [Kempf et al. 2005]. Another class of physical space-based methods was proposed by Jarrin et al. in 2006. The method is suitable for complex geometries and irregular meshes. It was developed based on the concept of superposition of coherent structures named ‘synthetic eddy method (SEM)’. In this approach, the synthetic velocity signal is constructed as a sum of induced velocities of a finite number of coherent eddies with random signs and positions which are continuously generated just behind the inlet plane and convected through the domain [Jarrin et al. 2006, 2008]. In this procedure, a shape function is assigned to each turbulent eddy which represents its spatial and temporal characteristics. The resulting complete turbulent velocity signal, including SEM, superimposed synthetic fluctuation signal at a point x on the inlet plane section and at a time t can be expressed as below: ui (t )  uim  uil (t ) 

82

1 Ne

Ne

 j 1

f ( x  x j (t ))

ji i

(4.12)

where  ji is a random step function (with values of +1or -1), N e is the number of eddies at the inlet plane, x j is the random position of the eddies and fi represents a Gaussian shape function. The strategy is capable to satisfy prescribed turbulence statistics (length scale and Reynolds stress tensor). The strategy was applied successfully by different investigators in a wide variety of applications like LES of decaying homogenous isotropic turbulence [Jarrin et al. 2006], LES of a fully developed turbulent channel flow [Jarrin et al. 2006, Patil and Tafti 2012], LES of square duct flow [Jarrin et al. 2009], LES of compressible decaying turbulence [Magagnato et al. 2007], LES of flow over an airfoil trailing edge [Jarrin et al. 2009], LES of flow past a bluff body [Pavlidis et al. 2010], LES of flow over a backward facing step and LES of swirling flow in a combustor [Patil and Tafti 2012]. The later study proves the ability of SEM approach in the case of swirling flows. The results show that time averaged profiles of axial and circumferential velocities and their variance at the sudden expansion section of the combustor are considerably improved by applying the inflow turbulence via SEM. In the special case of turbulent boundary layer flows, Pamies et al. also performed some modifications on the original SEM, by considering inhomogeneity of the turbulent scales in the wall normal direction, and obtained good results [Pamies et al. 2009]. In their procedure, the shape function of the original SEM is divided in some mono-dimensional functions including time, wall-normal and transverse constructing functions. The functionality of these constructing functions is determined based on the physical properties of the coherent structures in different zones of the boundary layer (i.e. viscous sub-layer, buffer zone, logarithmic layer and wake region). The method provides better results than original SEM for the special case of the turbulent boundary layer flow. Kornev et al. also introduced another AFG method based on some physical entities called ‘turbulent spots’ that can be applied either in the case of homogeneous or inhomogeneous turbulent inflows [Kornev et al. 2007, 2008]. The underlying idea is somehow similar to the SEM concept, but in the approach, the inflow turbulence is generated by randomly placed turbulent spots that their internal structure or velocity distribution is determined from some prescribed statistics e.g. auto-correlation. In the procedure, unknown shape functions of the turbulent spots are obtained by solving a nonlinear integral equation based on two-point autocorrelation function. The method was applied for the free decaying turbulence and turbulent boundary layer flow on the flat plate with success [Kornev et al. 2007, 2008]. 4.4.1 (b) Mapped space-based AFG methods

This group of AFG methods utilizes mapped space to construct turbulent velocity signals; in other words, all necessary mathematical manipulations are performed in the mapped 83

space like the Fourier space (spectral method according to the Jarrin’s classification) or any constructed space based on a set of fundamental basis (e.g. the proper orthogonal decomposition, wavelet, etc.). In fact, the idea is to construct a turbulence fluctuation signal based on its building-block modes in the mapped space with prescribed statistics. Kraichnan proposed a method for fluctuation generation based on the spatiotemporal Fourier modes with prescribed spectra in the case of single-particle diffusion in a turbulent incompressible flow [Kraichnan 1970]. Briefly, the complete turbulent velocity signal in this method can be expressed as a sum of harmonics with random phase and amplitudes as below: NK

ui (t )  uim  uil (t )    i j cos( kj xˆkj   kj tˆ)   i j sin( kj xˆkj   kj tˆ) 

(4.13)

j 1

where  i j and i j are random amplitudes.  kj ,  kj and N K are the wave number vector, phase vector and number of modes, respectively. In the above formulation, xˆ and tˆ represent the spatial and temporal coordinates normalized by turbulent length and time scales, respectively. The method was modified by Smirnov et al. to take into account the anisotropy of turbulent shear stresses using some scaling and orthogonal transformations to the Kraichnan’s turbulent fluctuation field. The method was applied successfully in the case of a turbulent boundary layer flow [Smirnov et al. 2001]. Thereafter, Batten et al. simplified the Smirnov’s procedure and applied the Cholesky decomposition to respect the Reynolds stress tensor. The new version was applied in the case of Hybrid RANS/ LES of fully developed channel flow with success [Batten et al. 2004]. Keating and Piomelli also adopted the Batten et al. technique in the LES of a channel flow and a boundary layer flow. They found that the method needs relatively a long distance to generate real statistics of the turbulence [Keating and Piomelli 2004]. Some attempts to use Fourier space in constructing of the turbulent inflow were performed by Moin’s group at center for turbulence research of Stanford University [Lee et al. 1992, Rai and Moin 1993, Le et al. 1997, Na and Moin 1998]. Lee et al. proposed applying an inverse Fourier transformation to a real turbulence energy spectrum; the method was adopted in the case of LES of spatially decaying turbulence [Lee et al. 1992]. Rai and Moin also proposed a strategy based on Fourier modes and random phase numbers; also amplitude of fluctuations was obtained using von Karman spectrum. The method was applied with success to DNS of a spatially evolving boundary layer [Rai and Moin 1993]. Thereafter, for DNS of backward- facing step flow, Le et al. modified the original AFG method (Lee et al.) to take into account inhomogeneity in the wall-normal direction. In this method, the resulting signal is scaled to respect four components of the prescribed Reynolds stress tensor [Le et al. 1997]. In another version of the method, instead of the randomization of phase angle, Rai and Moin adopted an amplitude randomization strategy 84

to generate the synthetic turbulent fluctuation signal. Their motivation originates from this fact that phase angle information is highly correlated to the turbulent structures; therefore, for DNS applications these turbulent structures should be reproduced accurately. The strategy was applied for the case of a separated turbulent boundary layer over a flat plate, and they found that the development length necessary to recover correct statistics is reduced considerably [Na and Moin 1998]. All aforementioned methods utilize Fourier space to generate synthetic fluctuation signals. In principle, other expansion basis can be used for this purpose, like: proper orthogonal decomposition (POD) or wavelet, etc. POD is a mathematical tool capable to extract energetic modes of a field which can be applied to detect the coherent structures in turbulent flows [Lumely 1970]. As an example of this group of AFG methods, Johansson and Andersson proposed a strategy based on the solution of most energetic eddy dynamics. In this procedure, the database for extraction of POD-modes is constructed by performing DNS and LES precursor simulations. Thereafter, the governing ODE equation of the active eddy dynamics (derived from Galerkin projection of the Navier-Stokes equation onto the POD-mode basis) is solved numerically. Finally, random motions of intermediate and small scales are added via randomization of phase and amplitude in the coefficients of the PODmodes. The method was successfully applied in the case of turbulent channel flows [Johansson and Andersson 2003, 2004]. It is worth to mention that to construct the turbulent velocity signal, POD-modes can be obtained by the aid of experimental measurements instead of precursor simulations. In this regard, temporospatial variation of the inflow velocity field can be obatined by any measuring technique like hot wire [Druault et al. 2004] or PIV [Parret et al. 2006, 2008]. However, in contrast to direct numerical simulation, each measurement technique has its own limitation. For example, due to limited number of hot wire adopted for the measurement at the inlet plane, Druault et al. used a linear stochastic estimation technique to spatially generalize the data. In contrast, in the case of non-time resolved PIV, Parret et al. used random time series [Parret et al. 2006]. As a disadvantage which limits the method’s applicability: all POD-based strategies need an apriori database coming from the precursor simulations or experimental measurements for extraction of the POD-modes. Wavelet expansion can also be adopted to construct the synthetic turbulent velocity signal. The complete inflow velocity signal can be expressed as an inverse discrete wavelet transform as below [Tabor and Baba-Ahmadi 2010]:

ui (t )  uim  uil (t )    i , j ,k j ,k j

(4.14)

k

85

where  i , j ,k ,  j ,k , j and k represent the wavelet coefficients, discrete wavelets, scale parameter and shift parameter respectively. For computation, discrete wavelets are expressed in terms of a mother wavelet, which is the Meyer wavelet in this case [Tabor and Baba-Ahmadi 2010]. The method was applied in the case of LES of a channel flow with success. As reviewed so far, there is a wide range of AFG methods but some methods called ‘mixed method’ are not classified purely as physical space-based methods or mapped space-based methods, but take benefits of both. The strategy utilized in this research project belongs to this group of AFG methods. In the following subsection more details are presented. 4.4.1 (c) Mixed AFG methods

In fact, mixed AFG methods utilize both physical space and mapped space to construct turbulent velocity fluctuation signals; in other words, mathematical manipulations are performed in both spaces. One example of this kind of AFG methods proposed by Billson and Davidson utilize the Fourier modes in mapped-space along with a temporal digital filtering in physical space to construct a turbulence signal with the prescribed length and time scales [Davidson 2011]. The strategy is also capable of respecting a prescribed Reynolds stress tensor for turbulence anisotropy at the inlet. The method was originally developed for the turbulent noise generation [Billson 2004] and later was applied for other applications like the inflow turbulence generation [Davidson 2007 1-3, Montorfano et al. 2011] and forcing at LES/RANS interface [Davidson and Billson 2006]. In its original application the signals are generated on fixed grid nodes in space; but in our case, the basic strategy was modified to create the turbulent velocity signals at rotating nodes on the draft tube inlet plane. Furthermore in our problem, 2D variation of inflow turbulent length and time scales are also considered to create more realistic signals. The original procedure is explained in the following based on the detail description found in [Billson 2004, Davidson 2004, 2006, 2007, 2011]. Isotropic fluctuations can be constructed based on Fourier series representation of the turbulent velocity signal including random Fourier modes. In principle, by combination of the ‘sin’ and ‘cosine’ terms in the Fourier expansion, each Fourier mode can be expressed compactly as a cosine term with a phase shift. In the procedure, preliminary turbulent fluctuation velocity signals in three directions at inlet plane in the general form can be expressed as below: Nf

ui, pre (t )  2 uˆ n cos(k nj x j  n )  in n 1

86

(4.15)

where uˆ n , k nj ,  n ,  in and N f denote amplitude, wave number, random phase shift, random direction of the nth Fourier mode and number of Fourier modes, respectively. The random direction of the wave number vector and Fourier modes are obtained based on three other random angles named  n ,  n and  n defined in figure (4.15).

Fig 4.15 Definition of the intermediate random angles, wave number vector and Fourier mode direction In fact, the direction of the wave number vector is random and defined by two random angles including  n and  n . The direction vector of the nth Fourier mode  in is in the plane (1n ,  2n ) and is also random via a random angle  n . The vector system (1n ,  2n .3n ) forms a unit basis for 3D space; therefore it is trivial that the Fourier mode vector  in in the (1n ,  2n ) plane is perpendicular to the 3n (and wave number vector kin ) i.e.

 in kin  0 . As a result, mass conservation is enforced by perpendicularity of  in to kin and the continuity equation is satisfied by definition, as below: Nf

  u   2 uˆ n cos(k nj x j  n )  in kin  0

(4.16)

n 1

Therefore, a solenoidal or divergence-free fluctuation field is generated [Davidson 2011]. If the Reynolds stress is on hand from experimental measurements or upstream simulations, anisotropic inflow turbulence can be generated by considering Reynolds stress tensor [Billson et al. 2004]. The computational procedure adopted in this research project to generate the inflow turbulence for the rotating inlet profile is originally constructed based on the above described formulation, along with considering some modifications to include the 2D 87

variation of the turbulent length and time scale fields at the inlet plane and also to respect the available experimental information for the amplitude of the fluctuations in three spatial directions (explained in the next section). In this manner, the anisotropy of real turbulent inflow velocity signals is considered. In addition, at the final step the near-wall RANS region is excluded in applying fluctuations to avoid doubling the turbulence effects [Frohlich and Von Terzi 2008]. In fact, for the hybrid models in the RANS region, the effects of turbulence are included via turbulent modeling parameters, e.g. the turbulent viscosity in the case of the SpalartAllmaras (S-A) model; applying turbulent fluctuations would double the effect of the inflow turbulence in the near wall RANS region. In contrast, in the LES zone of the hybrid treatment like DDES, as stated before, turbulent viscosity switches to the subgrid-scale viscosity and therefore plays a completely different role, i.e. the dissipation of turbulence energy in the high frequency dissipation range of the turbulence energy spectrum; so the effect of the inflow turbulence should be included by synthetic fluctuations. After these preliminary discussions, the procedure adopted in this study based on [Davidson 2011] can be summarized as below: 1) A uniform and fine-enough equidistance rectangular mesh with grid-spacing  is generated at the inlet plane which overlaps the original draft tube inlet grid. 2) Turbulent time and length scales are calculated at each inflow node based on stationary RANS inflow data using the following expressions (formula A2.10 and A2.11 in appendix A2): Lt  k

3

2

 and Tt  k  .

3) The smallest wave number k1 (marked in figure 4.14) is calculated based on the wave number of the most energetic eddy ke (marked in figure 4.14) as below [Davidson 2011]: k1 

ke 9 1.453  pD 55 Lt pD

(4.17)

In the above formula, pD is a factor which scales the wave number of most energetic eddies to the largest eddies. Typically, pD  2 is a good approximation [Davidon 2006, 2011] and is adopted in the present study. 4) The highest wave number kmax is calculated based on the grid spacing  . In LES-type simulations, the maximum wave number depends on the capacity of the grid to resolve the turbulent structures i.e. the grid resolution. The highest wave or cut-off wave number is typically calculated as below [Davidson 2011]:

88

kmax 

2 2

(4.18)

5) The wave number interval between k1 and kmax is then divided into N m modes with wave number step kn (marked in figure 4.14). 6) Random angles including:  n ,  n ,  n and  n are generated for each random mode; then randomized components k nj are computed (as explained in such a way to satisfy continuity). 7) To obtain the amplitude of fluctuations, the modified von-Karman spectrum is adopted, which is formulated as below: 1

uˆn  ( E ( k nj )kn ) 2

(4.19)

where, E (k s )  1.453

2 urms ( k s ke ) 4 exp  2(k s k ) 2  ke 1  (k k ) 2 17 6 s e  

(4.20)

and intermediate parameters are defined as: ks  k j k j

(4.21)

k   1 4 v 3 4

(4.22)

8) Turbulent velocity fluctuation signals, ui, pre , can be computed using equation (4.15), at the stationary grid nodes of the equidistance mesh. 9) Applying time-correlation on the preliminary signals (equation 4.15) via asymmetric digital time filtering [Davidson 2011]: in fact, the generated turbulent fluctuation signals at step (8) involve no time-correlation in contrast to real turbulent fluctuation signals. At this step, to mimic a real turbulence signal an exponential time-correlation in the form of exp(t / Tt ) is applied through a filter similar to the Klein’s spatial digital filter [Klein et al. 2003], as below: (ui) m  (ui) m 1  (ui, pre ) m

(4.23)

  exp(t / Tt ) ,   1  2

(4.24)

where,

parameter m denotes the time step level. 89

10) The rotation is applied on the inlet grid nodes and corresponding turbulent signals. In this procedure, at each time step the nodes are repositioned at the inlet section and the corresponding signals are interpolated with the aid of Delaunay triangulation technique. 11) Isotropy of the generated inflow turbulence is broken by applying the measured relative fluctuation amplitudes using ANN-artificial neural networks (more details in the next subsection). 12) Fluctuations are imposed in the LES-core flow. In other words, RANS region is excluded in the generation of fluctuation signals, by multiplication of the signal by the f d function (equation 2-12), i.e. f d ui(t ) .

uz uref .

time ( s) uz uref .

time ( s)

Fig 4.16 Isotropic AFG fluctuation signals in axial direction with/without time-correlation before applying runner rotation and inflow anisotropy By performing the above procedure, an individual turbulent fluctuation signal is generated at each computational node at the inlet section for three different directions of the space. The signals involve the prescribed turbulent length-scale and time-scale characteristics. Then the runner rotation is applied on the fluctuation fields. At the final 90

step, as explained in the next section in details, the isotropy of the generated signal is fallen apart with the aid of a designed artificial neural network (ANN) based on experimental relative fluctuation amplitudes stemmed from LDV measurements. It should be mentioned that the generation of fluctuations on the fine inlet mesh is a costly process; for a 100 100 resolution, it takes approximately 30 seconds per time step to create three components of the fluctuations on the whole inlet section. u uref .

Zoom Area

time ( s ) u u ref .

tim e ( s )

Fig 4.17 Isotropic AFG turbulent velocity fluctuation signals with time-correlation and considering runner rotation (top: full signal, bottom: zoomed area) Figure (4.16) depicts turbulent velocity fluctuation signals in the streamwise direction with and without time-correlation generated from the first nine steps listed above at the normalized position ( x / RInlet , y / RInlet )  (0.3,  0.55) placed at the core flow zone for OP.4

91

(  Runner  30.2 , GV  65.4 , N11  170, H  4m ). In fact in figure (4.16), the generated signal without time-correlation (red signal) involves a prescribed length scale and prescribed energy spectrum (i.e. modified von-Karman energy spectrum); so it differs from a random white noise signal with a uniform energy distribution. On the other hand, so far the signal does not possess the effect of the rotation as well as the anisotropy of the inflow fluctuations in the three directions of space. By applying the rotation on the generated isotropic signal which involves timecorrection, one can observe the effect of passing runner-blade wakes on the outcome signal. As shown in figure (4.17), all three components of the turbulent fluctuation velocities are affected in a similar manner by the runner blade wakes. The peaks in the signals correspond to the passage of runner blades, i.e. passage of the relatively higher amplitude fluctuation zones, over the Eulerian probe point positioned at ( x / RInlet , y / RInlet )  (0.3,  0.55) . The top-graph in figure (4.17) shows the generated isotropic signals under rotation in full, in which the zoom area in the bottom illustrates the details of the signals more clearly. As one can see in the zoomed graph of figure (4.17), although some differences exist among the signals in the three different directions of the space as imposed by the random nature of turbulence, they behave similarly due to the isotropy of generated signals based on the strategy explained above up to step 10. For example, as one can see in the figure, the positions of the fluctuation peaks corresponding to the blade passage instants over the Eulerian probe, are similar for all three components of the velocity. As this figure also suggests, the peaks favourably do not have the same amplitude as time proceeds due to the random nature of the generated turbulence at the inlet plane, as intended by default. Fingerprints of the runner blades are also visible in the 3D representation in figure (4.18). This figure presents 2D variation of the streamwise velocity component fluctuations at the inlet plane at one instant of time. There are four zones of high amplitude fluctuations corresponding to the runner blade wakes. In these zones due to the higher turbulent activity, more intensive fluctuations are created. Between two mutual wake zones, a relatively quiescent zone exists which corresponds to the area between the runner blades, possessing much lower amplitude of the fluctuations. It is important to notice that for the stand-alone draft tube flow simulation in this project, the inflow section is positioned slightly upstream the runner/draft-tube stage interface section of the full-machine k   simulation, i.e. exactly at the LDV-experimental measurement plane. At this plane, the effect of the guide vanes on the level of fluctuations are not captured by the AFG method and only the effects of the runner wakes are captured in the generated inflow turbulence (figure 4.18).

92

Runner blade wake zones

uz uref .

r RInlet

r RInlet

u z uref .

r RInlet

Fig 4.18 Inflow AFG fluctuation field of the axial velocity component at t  1.2 s for OP.4 (top: isometric view, bottom: side view)

93

However, as one can see in the next section, the experimental data from LDV measurements clearly shows the effect of the guide vanes on the level of fluctuations at the inlet section (figure 4.19). This difference in the observations is linked to the presence of the stage interface between the guide-vanes and the runner in the case of full-machine k   simulation. The uniformization that happens at the stage interface majorly removes the effect of guide-vane blade on the draft tube inflow profile. That’s why only four high-amplitude fluctuation zones corresponding to the runner blade wakes are captured in the generated synthetic turbulence as one can see in figure (4.18). By contrast, experimental LDV data show regions with relatively high level of fluctuations between the runner-blades, linked to the presence of guide-vanes, in addition to the signatures of the runner-blade wakes for all three components of the velocity (figure 4.19). Bottom-graph in figure (4.18) shows the side view of the generated axial velocity fluctuations at one instant of time. As one can see, the relative amplitude of the fluctuations in the blade-passage zones is much higher than in the inter-blade zones; in other words, these inter-blade zones can be called ‘semi-quiescent’ regions. So far, the generated signals are isotropic; however, in the complex situations right after the runner, the turbulent signals are highly anisotropic. In other words, the level of fluctuations at each computational node at the inlet section of the draft tube is different in the three different directions of space. In the next section, thanks to the availability of the experimental measurement data, a strategy is presented to break the symmetry observed among different components of the turbulent velocity. 4.4.2

Anisotropic inflow turbulence

The strategy described in the previous section generates isotropic inflow turbulence; in other words, the generated signals in three spatial directions have the same amplitude. This condition is far from reality in the core flow especially in the complicated flow situation after the runner, so anisotropy of the fluctuations should be somehow considered. The relative amplitude of the fluctuations can be obtained whether experimentally or numerically. As explained in section (4.4.1), having the Reynolds stresses, one method to break this symmetry is to use the Cholesky decomposition to rescale the fluctuations in three spatial directions. For draft tube-only simulations, this can be performed for example by adopting a RSM technique (which solves transport equations for Reynolds stress tensor components) in runner simulation to obtain the 2D field of the Reynolds stress tensor at the runner exit section. The method is relatively costly and is recommended in the absence of experimental data. On the other hand, the methods like k   (adopted in this thesis to

94

obtain inflow conditions) fails to provide the Reynolds stress tensor or any information about the relative amplitude of the fluctuations in the three dimensions of space. In this study, one can takes advantage of the available experimental data of the core flow turbulent fluctuations measured at the LAMH test bench. In fact, in the experimental procedure, two components of the turbulent velocity i.e. axial and circumferential velocities were measured by LDV techniques on a vastly covering zone on the draft tube inflow section (plane-A) [Vuillemard et al. 2014]. The turbulent radial velocity fluctuations were also measured but only on a narrow zone between 0.65 to 0.75 of the normalized radius ( r / RInlet ). The RMS of the turbulent velocity fluctuations are shown in figure (4.19). As it is clear in the figure, the effects of four runner-blades can be recognized by the four zones (strips) of higher RMS values and also the fingerprints of guide vanes on the RMS are captured by the background spiral structures.

Fig 4.19 Experimental RMS values of the inflow velocity fluctuations at OP.4 (LDV) [top left to bottom right: circumferential, axial, radial velocities and sketch of flow pattern observed experimentally at inlet plane (plane A)] For the purpose, i.e. to obtain the relative amplitude of the velocity fluctuations in the three spatial dimensions, one needs to have a full map of RMS values of the radial component as well. To obtain an educated 2D estimation of RMS value of the radial

95

velocity fluctuation on the full-radius, the idea here is to employ artificial neural networks (ANNs). The ANN is a promising technique for estimation and prediction purposes which can be applied in systems with non-linear dynamics and is also a helpful tool for the data generalization in the case of data sets with missing values. Its concept is based on the brain functionality and how brain learns. In fact, ANN mimics the brain behavior mathematically via non-linear modeling of neurons and theirs synaptic inter-connections. In this study, for each azimuthal position (with angular step size of 1 ), a multi-layer perceptron (MLPR) neural network was designed in supervised (or target oriented) manner and an in-house code was developed based on that to estimate the RMS of radial velocity fluctuations. The details of the MLPR formulation applied in the code are presented in appendix B. Briefly, as shown in figure (4.20), the designed MLPR has the 3-20-1 architecture with 3 neurons in the input layer corresponding to normalized radius, normalized RMS value of circumferential velocity fluctuation and normalized RMS value of axial velocity fluctuation, 20 neurons in the hidden layer and one neuron at the output layer corresponding to an intermediate scaling variable (  radial ) explained in the following. The designed MLPR also involves tangent hyperbolic and linear activation functions in the hidden and output layers, respectively and uses back-propagation learning algorithm for synaptic weight adjustments (appendix B).

back-propagation learning

r / RInlet Norm.

u

2

u z

2

wip, j

bip

 radial (r , )

Norm.

Fig 4.20 Multi-layer perceptron architecture designed for each azimuthal angle It is worth to mention that different versions of the perceptron neural networks with following architecture 4- N h -1 were designed and tested which had an additional neuron in the input layer and also N h in the range of 10-100. Due to the very complex nature of the relation among the radial, circumferential and axial fluctuation amplitudes on the inflow map, the learning’s errors of the designed ANNs were not reduced for different learning rates and exhibited high amplitude undamped oscillations. To solve the problem, a zonal 96

learning approach was adopted, in which the training set size is reduced to a fraction of the original size; here, each azimuthal angle with angular step size of 1 as mentioned (figure 4.21).

Fig 4.21 Zonal training approach for the designed MLPR The ANN application intended here is a kind of extrapolation, which is a challenging task for any kind of prediction techniques. It is a good idea to provide some guidelines from physics of the problem (if applicable) to obtain more reasonable approximations from the designed ANNs. As can be seen in figure (4.22), for RMS of radial velocity fluctuations, in spite of having short window of experimental measurements, the signatures of the four runner blades and the spiral structures due to guide vane effects are also recognizable and exhibits similar pattern with the RMS of circumferential fluctuations. So, an idea is to rewrite the 2D RMS field of the radial velocity fluctuations as below:

ur2   radial (r ,  ). pattern (r , )

(4.25)

where function  pattern presents the guideline pattern for the radial field and is equal to the normalized form of the circumferential velocity RMS field. In addition,  radial is a correction factor field which has 2D variation on the inflow plane (plane A). Therefore, the designed MLPR learns the hidden relation among the normalized RMS of the circumferential and axial velocity fluctuations and correction parameter  radial for the available experimental data set, and then the trained MLPR can be used for the estimation of  radial on the whole inlet plane. In fact, the MLPR does not provide the hidden relation between input and output sets in an explicit manner, rather the relation is implicitly presented via adjustment of the synaptic weights (refer to Appendix B). As explained in the appendix B, the learning 97

process of the ANN depends on many factors like the structure of the learning sets, topology of the ANN and an important parameter called learning ratel . The learning rate in some sense resembles the ‘relaxation factor’ in the discretization of the derivative terms in the context of the numerical solutions of the PDEs. In fact, by increasing the learning rate, speed of the ANN learning increases but it has an optimum, after which the ANN moves towards an unstable/divergence region.

u2 uref .

Fig 4.22 RMS values of the fluctuations OP.4 (LDV measurements) (left: circumferential velocity, right: radial velocity)

Fig 4.23 Learning history of the designed ANN with l  0.005 at   25

98

ur2 uref .

Figure (4.23) indicates a typical learning history of the designed MLPR with the learning rate equals tol  0.005 utilized in this project. The objective function (error) is defined here as the square of norm of deviation vector between the target and MLPR output vectors. In general, with increasing the learning rate, the MLPR may have more rapid learning convergence, but it moves towards the unstable region (l  0.094 ) and ultimately it diverges rapidly, as seen for the learning rate l  0.1 in figure (4.24). After the learning process, the trained-MLPRs can be used to reconstruct the 2D RMS field of the radial velocity fluctuations on the draft tube inlet section.

Fig 4.24 Learning history of the ANN in the unstable region with l  0.1 at   25

For instance, figure (4.25) compares the experimental and predicted values by the MLPR for the scaling parameter  radial for few selected azimuthal positions. As one can see, the trained-ANN output data agree very well to the experimental data, which means an appropriate learning process was set. Figure (4.26) also shows the result of the 2D variation of the  radial field for a quarter of the inlet plane obtained from the final trained MLPR. The corresponding RMS field of the radial velocity fluctuations coming from the MLPR is shown for a quarter and complete profiles in figures (4.27) and (4.28), respectively.

99

 radial

 radial

r RInlet

r RInlet

Fig 4.25 Radial distribution of ANN (MLPR) predictions for the parameter  radial at OP.4 (from top left to bottom right:   10 ,   40 ,   60 ,   90 )

Fig 4.26 Two-dimensional variation of the  radial parameter obtained from ANN (MLPR) at the inlet plane for OP.4

100

Shroud

ur2 uref .

ANN

r RInlet

Hub r RInlet

Fig 4.27 Quarter of the RMS field of radial velocity fluctuations using ANN for OP.4 ur2 uref .

ANN

r RInlet

r RInlet

Fig 4.28 Complete RMS field of the radial velocity fluctuations predicted using ANN for OP.4 As it is clear in figures (4.27) and (4.28), the effect of the runner-blades and guide vanes are visible in the ANN-predicted RMS values of the radial velocity fluctuations as expected. To double-check the ANN predictions, the 1D circumferential average profile of radial fluctuation RMS values, calculated using both 2D data sets (one from the ANN predictions and another from available experimental data in the thin strip), are compared which exhibit very close agreement (figure 4.29). To have an idea about the trend of the experimental fluctuation amplitudes on the full experimental radius, 1D circumferential average of the RMS profile of the circumferential and axial velocity fluctuations-calculated based on the 2D experimental data set- are also calculated, as shown in figure (4.30).

101

u r2 u ref .

r RInlet Fig 4.29 1D circumferential-average of 2D RMS field of radial velocity fluctuations at OP.4 u2 uref .

uz2 uref .

Exp .

Exp.

r RInlet Fig 4.30 1D circumferential average of 2D RMS field of the experimental velocity fluctuations at OP.4 (top: circumferential velocity, bottom: axial velocity) 102

As it is seen in figure (4.29), the amplitude of the radial velocity fluctuations estimated by the ANN increases towards the walls i.e. hub and shroud; similar behavior is also visible for the axial and circumferential components as well (figure 4.30). As seen later in chapter 7, this can be induced by the near wall vortical structures at the shroud and hub which amplifies the level of fluctuations for all three components of the velocity. Interestingly, in addition to the close agreement of the ANN predictions and experimental data, the trained ANN could predict a reasonable overall shape for the radial fluctuation levels on the full experimental radius (figure 4.29). Having on hand, all the fluctuation amplitudes of the turbulent velocity signal in the three spatial directions, one can obtain the inflow anisotropic turbulence. This can be easily performed by the multiplication of three scaling-factors with generated isotropic artificial turbulent velocity signals for all three dimensions of space. The details of the scaling factor calculations are presented in appendix C1. Figure (4.31) shows the scatter plot of the isotropic turbulent velocity fluctuations at a sampling point on the inlet of the draft tube for OP.4 generated by the explained AFG method (section 4.4.3) in the duration of 1 second with 104 samples before applying the rotation and inflow anisotropy (step 11 in page 90). The probe point is initially located in the relatively quiescent zone between two mutual zones of the runner blade wakes; therefore the amplitude of the fluctuations is relatively small.

uy uref .

uz uref . Fig 4.31 Scatter plot of the isotropic turbulent velocity fluctuations generated by AFG method before applying the rotation and scaling factors sampled at a point with the normalized position ( x / RInlet , y / RInlet )  (0.3,  0.55) on the inlet plane 103

As one can see in figure (4.31), in the case of isotropic inflow turbulence the scatter plot exhibits a self-similar and homogeneous pattern in all directions. In other words, there is a semi-uniformity in all directions and there is no preference axis in the cloud of event points.

Fig 4.32 Scatter plot of uz  uy fluctuations generated by AFG method after applying the rotation and scaling factor sampled at a point with the normalized position ( x / RInlet , y / RInlet )  (0.3,  0.55) on the inlet plane Applying the runner rotation and anisotropic scaling factors breaks the inflow symmetry of the fluctuations as expected. Figure (4.32), (4.33) and (4.34) show the scatter plot of the turbulent velocity fluctuations in the u z  u y , u x  u z and u x  u y planes, respectively after applying these two essential factors for duration 1 second with 104 samples. As one can see in all the scatter plot figures, for example figure (4.32), there is a 104

dense zone of events which majorly corresponds to the semi-quiescent inter-blade regions marked by a red rectangle frame in the top graph in the figure. This region forms due to applying the rotation on the generated fluctuations. Outside of this red frame, the events are linked to the runner-blade high-amplitude fluctuation zones. The lower graph in figure (4.32) shows the zoomed vision of the events in the red frame. As one can see, the isotropy is broken and the events are located in a way which forms a preference axis marked by the red oval. Similar behavior is also observed in other planes shown in figures (7.33) and (7.34).

Fig 4.33 Scatter plot of u x  u z fluctuations generated by the AFG method after applying rotation and scaling factor sampled at a point with the normalized position ( x / RInlet , y / RInlet )  (0.3,  0.55) on the inlet plane

105

Figure (4.33) also suggests the presence of a dense region of events in the ux  uz plane. Similar to the previous case the symmetry among the events is broken by applying the rotation and scaling factor on the turbulent fluctuation signals in different directions of the space. As one can see in the zoomed region, there is a preference axis in the cloud of events, marked by the red oval, which extended more in the horizontal axis than in the vertical one (figure 4.33). Similar behavior is also observed in figure (4.34) about the events in the ux  uy plane; as one can see there is a dense region of the events in the core with surrounding events due to applying the runner rotation on the generated inflow fluctuations. Again a preference axis forms in the zoomed region marked by the red oval.

Fig 4.34 Scatter plot of ux  uy fluctuations generated by AFG method after applying the rotation and scaling factor sampled at a point with the normalized position ( x / RInlet , y / RInlet )  (0.3,  0.55) on the inlet plane 106

As observed in figures (4.32) to (4.34), by applying the runner-rotation and the scaling factors on the isotropic fluctuations generated by the base-AFG, anisotropic turbulent fluctuations, which mimic real situation just after the runner, are produced. 

5 3

E uz ( f )

f f runner



5 3

E ux ( f )

f

f runner



5 3

E uy( f )

f

f runner

Fig 4.35 Power spectra of turbulent velocity signals with/without artificial fluctuations sampled at a probe positioned at ( x / RInlet , y / RInlet )  (0.3,  0.55) on the inlet plane 107

To summarize the AFG generation technique, isotropic fluctuations were scaled based on the relative RMS values of the fluctuations in three different directions of the space using scaling factors. These scaling factors are obtained from a supervised targetoriented MLPR neural network trained based on the LDV data. By obtaining these factors and applying rotation, as explained in this section, the final piece of the puzzle i.e. third term or incoherent part in the triple Reynolds decomposition of the turbulent velocity signal is obtained. The resulting inflow turbulence including mean values, low frequency and high frequency contents can now be added together and imposed at the inlet of the draft tube for DDES simulation of the draft tube. Figure (4.35) shows the power spectrum of the turbulent velocity signals indicating the effects of artificial fluctuations. As one can see in the figure, adding the synthetic fluctuations essentially modifies the high frequency content of the turbulent velocity signal for all three components of the velocity. For this Eulerian probe position, the energy is amplified in the case of u x and u z but the energy decreases for u y . The slopes of the spectrums are also getting closer to the -5/3 slope by adding fluctuations. In chapter 7, the results of DDES simulations of the draft tube turbulent flow with and without synthetic turbulent fluctuations are presented and discussed in details.

4.5 Summary

The major points of this chapter can be summarized, as follows:



For the BulbT draft tube flow simulations, the inlet plane is located in a complicated unsteady flow zone just after the rotating runner. Therefore, providing a realistic inflow condition becomes a crucial factor.



Different inflow conditions for DDES simulations are generated, including: ‘1D circumferential-averaged’ profile, unsteady 2D profile with/without AFG.



For unsteady 2D profiles, the coherent part of the velocity signals is obtained from the rotation of a mean 2D profile extracted from a full-machine k   RANS simulation at the draft tube inlet section. The incoherent part of the turbulent velocity signal is also generated by AFG techniques.



The hybrid subgrid scale viscosity applied for DDES simulations is constructed as the blending of the RANS turbulent viscosity in the near-wall zone and the LES subgrid scale viscosity in the core flow. The latter is approximated by the Smagorinsky model.

108



To generate synthetic turbulence, precursor simulations or artificial fluctuation generation (AFG) techniques can be utilized. In the case of BulbT draft tube, AFG technique with capability of imposing 2D variation of turbulent quantity fields was adopted.



AFG techniques are classified into three groups, in general: physical space-based methods, mapped space-based methods and mixed methods. The strategy utilized in this study belongs to the mixed AFG methods [Davidson 2011].



The utilized technique uses Fourier modes in the mapped-space along with a temporal digital filtering in the physical space to construct turbulent velocity signals with prescribed length and time scales. The runner rotation is also applied on the generated signals.



Anisotropy of the inflow turbulence is then considered using the MLPR artificial neural network generalization via scaling factors. The ANNs were designed with a 3-20-1 architecture and were trained by experimental LDV data. The ANN’s prediction for RMS of radial velocity fluctuations shows a similar trend to those of the axial and circumferential velocity components.



In the generated synthetic inflow turbulence, fingerprints of the runner-blade wakes are visible as four high amplitude fluctuation zones. The signature of the guide-vane wakes is not present in the draft tube inflow AFG field, due to the uniformization introduced by the stage interface applied between runner and guide-vane sections.



Scatter plots of the AFG events after applying the anisotropy factors and runner rotation exhibit clustering about some preference axis in the clouds of events.



Slopes of the energy spectra of the turbulent velocity signals in different spatial directions are also getting closer to -5/3 by adding AFG.

109

110

Chapter 5

Inflow condition considerations on the Clausen conical diffuser flow simulations

5.1

Introduction

In the hydraulic-turbine applications, the ERCOFTAC conical diffuser is one of the standard test cases designed to study the effectiveness of turbulence treatment approaches in the case of swirling flows [Clausen et al. 1993, Nilsson et al. 2008, Bounous 2008, Payette 2008, Vincent 2010]. In general, turbulence models are tuned base on simple flows like the flow over a flat plate, but in the hydraulic turbine applications, the flow experiences some complex features like adverse pressure gradient (APG), flow swirl and streamline curvatures that are not present in the simple flows. Especially in the case of the draft tube component of a hydraulic turbine, the flow is strongly affected by an upstream rotating runner and the presence of unfavourable pressure gradient. This makes the draft tube simulation a challenging task. In the present thesis, first of all, simulation of the turbulent flows in the ERCOFTAC conical diffuser is assessed in this chapter as a starting point to observe the effects of inflow conditions on the DDES simulation of draft tube flows. In this regard, the effects of different factors like the radial velocity and turbulent viscosity in the wall-zone are considered.

5.2

Experimental set-up

The experimental measurements for this test case have been performed by Clausen et al. [Clausen et. al. 1993] and are available through the turbo-machinery working group of OpenFOAM [Nilsson et al. 2008]. Figure (5.1) shows the experimental apparatus used for 111

the measurements, which consists of a swirling generator (a rotating cylinder), and a diffuser recognised as the ERCOFTAC conical diffuser. The diffuser involves a 10 opening angle and a length of 0.51 m with the ‘area increase’ ratio of 2.84. In the experiment, the flow swirl has been set on purpose in such a way to prevent the separation on the diffuser’s wall and recirculation in the core flow. In general, adding the swirl stabilizes the boundary layer and prevents the separation. However, increasing the inflow swirl level ultimately creates a recirculation zone in the core flow region. In this test case, airflow with an axial speed of 11.6 m/s passes through the swirling generator, where it gains a circumferential velocity close to a solid-body rotation, with a small radial velocity and a swirl number of 0.3 (defined in equation 1.6), and then enters into the diffuser. The measurements have been performed using the hot-wire technique along different lines at eight different cross sections from S1 to S7 in the diffuser as marked by dashed-blue lines in figure (5.1). The positioning accuracy in the experiment is about ±0.01 mm. Also, the velocity component parallel to the wall ( us ) and circumferential velocity ( u ) were measured along the abovementioned lines with the accuracy of 2%. Due to difficulties in measuring the radial component, this component is not measured in the experiment [Clausen et al. 1993]. The Reynolds number of the flow, based on the inlet diameter i.e. 0.26 m and the air turbulent viscosity of vt  1.5 105 , is Re  2 105 at the inlet plane which falls in the turbulence range. Honey Comb

yn xS

Rotating Part

10 

r z

260 mm

S 1 S1 S 2 S3

100 mm

400 mm

100 mm

S4

S5

S6

S7

510 mm

Fig 5.1 Schematic sketch of the ERCOFTAC conical diffuser with swirling generator [Sketched base on the geometry, used by Clausen et al. 1993]

5.3

Computational domains

In this chapter, the effects of inlet boundary condition on the DDES simulation of the ERCOFTAC conical diffuser flow are principally investigated (base-case in figure 5.2). To 112

this end, two different computational domains are considered: first, the conical diffuser only (base-case) and second, the conical diffuser with swirling generator (extended-case) as shown in figure (5.2).

Zoomed inlet grid

Extension

Extension

Diffuser

Diffuser

Inlet extension

Swirl generator

a) Base-case

b) Extended-case

Fig 5.2 Computational domains adopted for DDES turbulence treatment in the ERCOFTAC diffuser: a) base-case and b) extended-case As one can observe, in the base-case, the swirling generator part is not present, which resembles the ‘draft tube-only’ simulation in the case of the BulbT simulations. Whereas in the extended-case, the swirling generator mimics runner component in the computational field, therefore it resembles the ‘runner and draft-tube’ simultaneous simulation. The extension part is added to avoid the recirculation at the diffuser exit. As shown in the previous studies, the shape of extension has negligible impact on the flow predictions [Paik et al. 2005, Payette et al. 2008]. To optimize the computational cost, the shape of extension is selected as a cylinder with length of 0.59 m. As explained earlier, the goal of the DDES simulation of the draft tube is to resolve accurately turbulent structures far from the walls in the separated zones with a moderate computational cost. In fact, the near-wall is modeled by RANS but the core flow is in the LES region. As mentioned before in the industry, usually, hydraulic turbine flows are simulated by RANS/URANS models to obtain the global performance of the machine. On the other hand, performing the unsteady DDES simulation of the runner is costly, so the inlet boundary condition of the draft tube simulation normally comes from the RANS simulation. This motivates us to perform RANS simulation of the ERCOFTAC diffuser using the extended-case, which enables us to feed the base-case by RANS inflow data. In the following sections, the results of simulations for both configurations are discussed in details.

113

5.4

Numerical simulations of the extended-case

In this section, the turbulent flow in the ERCOFTAC diffuser is simulated using the extended-case configuration to obtain inflow profiles for the base-case simulations. The flow turbulence for the extended-case simulations is treated by the k   , Spalart-Allmaras (S-A) RANS/URANS techniques and also DDES approach. The S-A treatment is considered because the utilized DDES technique is basically formulated based on this turbulence model as explained in details in chapter 2. 5.4.1 Computational grid specifications

The DDES simulation similar to any low-Re turbulence treatment technique (like the SST model) needs a fine mesh with y   1 , because the model has been designed to resolve up to the wall. Figure (5.2) depicts two sets of grid consisting hexahedral cells with y   1 for both base and extended-cases. The inflow grid resolution is the same for both cases and is visible on the left-side of figure (5.2). Table (5.1) presents the specifications of the adopted grids for k   , S-A and DDES simulations of the extended-case arrangement.

Table 5.1 Statistics of the computational grids for the extended-case (including y  on the ERCOFTAC diffuser solid wall) No

Grid (for simulation with)

Number of cells

Min. y 

Max. y 

Avg. y 

1 2 3 4

DDES S-A S-A (with wall-function) k 

825,600 825,600 232,200 232,200

0.81 0.82 31.2 36.0

1.98 1.99 61.6 78.0

1.18 1.19 36.2 54.5

5.4.2 Numerical setups of the extended-case simulations

Among OpenFOAM solvers, the ‘simpleFoam’ solver is adopted for k   simulation, whereas the ‘pimpleFoam’ flow solver is used for S-A and DDES transient simulations. In the case of ‘simpleFoam’ solver, the convergence criteria for the residuals of the velocity and pressure quantities is set to 107 and 106 , respectively. The details of the control setting of the ‘pimpleFoam’ solver is similar to those used for BulbT draft tube simulations in the upcoming chapters and are presented in table F.2 (appendix F). A second-order 114

discretization scheme is applied for the temporal term in the case of transient simulations. The convective term is also discretized by the second-order upwind scheme in the k   and S-A simulations, whereas the linear discretization with filtering of high-frequency staggering modes is adopted for DDES simulations. The details of the discretization schemes applied for other terms of the Navier-Stokes equation are similar to those used for BulbT draft tube simulations and can be found in table F.1 (appendix F). 5.4.3 Inflow and boundary conditions

For all cases, a uniform velocity equal to 11.6 m/s is imposed at the inlet section. This velocity is considered as the reference velocity for normalisations in the rest of this chapter, i.e. uref .  11.6 m/s. In the extended-case topology, the swirl is added on the inflow profile using a tool developed by the turbo-machinery working group of OpenFOAM to achieve solid-body rotation for the walls and internal nodes [Nilsson et al. 2008, Page et al. 2008]. In all cases, a mean pressure equals to zero is applied at the final exit. For all solid walls including swirl generator, diffuser and extension walls, a ‘no-slip’ boundary condition is applied. A Neumann boundary condition is used for all other boundary conditions. For turbulent flow simulations, on top of the velocity and pressure quantities, boundary conditions for turbulent quantities should also be applied. The inflow conditions for turbulent quantities utilized in the different simulations of the extended-case topology can be found in table 5.2.

Table 5.2 Different simulation cases within extended-case configuration Case

Simulation Type

Inflow turbulent quantities

A-1

k 

  896.11 m2 s 3 , k  2.0184 m2 s 2

A-2

S-A

vt v  27

A-3

S-A (with wall-function)

vt v  27

A-4

DDES

vt v  27

A-5

DDES

vt v  104

A-6

k 

  142.25 m2 s 3 , k  2.0184 m2 s 2

The values of inflow turbulence dissipation and turbulent kinetic energy adopted in the (A-1) simulation case are default setting introduced by turbo-machinery working group of OpenFOAM. As shown, these values are appropriately set to result in good predictions for ERCOFTAC diffuser test case [Bounous 2008, Page et al. 2008]. For S-A and DDES 115

simulations, instead of turbulent dissipation rate and turbulent kinetic energy, turbulent viscosity should be fed at the inlet section; its value can be estimated by the following formula [Pope 2004]: vt  C

k2



 0.09 

v (2.0184) 2  4.1104  t  27 896.11 v

(5.1)

It is worth to notice that, as shown for example in the case of DNS of a channel flow at Re = 13, 750 [Kim et al. 1987], there are some deviations from the above formula in reality. The formula is valid in this case for y   80 , although there are some defects even in this range. For y   80 , the constant C  0.09 is not valid and one can observe a semi-linear decrease of the coefficient close to the wall as shown in figure (5.3). However, for all extended-case simulations, C  0.09 was applied at the inlet plane.

C 

vt k2

y   80

y

Fig 5.3 Variation of the C coefficient in DNS of a channel flow [Pope 2004] As described in table (5.2), for S-A and DDES simulations namely (A-2), (A-3) and (A-4), the turbulent viscosity equal to vt v  27 is imposed at the inlet of the extended-case. As explained in the previous chapter, there is a fundamental difference between the turbulent viscosity, t and sub-grid scale viscosity, SGS concepts: the turbulent viscosity should reproduce behavior of the whole turbulence spectrum, but the sub-grid scale viscosity has less responsibility and should only reproduce the behavior of sub-grid scale eddies in the dissipation of turbulent energy. As a result, to perform LES-type simulations,

116

the turbulent viscosity should be lowered appropriately and then fed at inlet as inflow condition. In the (A-5) case, the inflow turbulent viscosity is lowered. Finally in the (A-6) case, the turbulent dissipation rate is lowered compared to the default case. 5.4.4 Extended-case simulation results

All the simulations in the extended-case configuration listed in table (5.2) are well converged. For instance, figure (5.4) depicts the convergence history of the k   simulation (A-1) case. As one can see in the figure, for all velocity components and turbulent dissipation rate, the level of residual decreases to106 ; whereas the pressure and turbulent kinetic energy achieve the level of convergence around 105 .

Residual

ux uy uz p k



Iteration

Fig 5.4 Convergence history of the k   simulation case (i.e. A-1 case in table 5.2) There is not much unsteadiness in the ERCOFTAC test-case, because the case was set experimentally to avoid the formation of separation on the diffuser wall and recirculation zone in the core flow. When the URANS simulations are started from a converged RANS solution in this test-case, no extra unsteadiness is formed and no change happens in the flow field. This issue was also observed in the previous studies of the ERCOFTAC diffuser flow [Payette 2008, Vincent 2010]. All the simulation cases listed in table (5.2) resulted in good predictions of the mean flow fields. Most significantly, no separation is formed on the diffuser wall in agreement with the experimental observations. Because of similarity of the results among different cases, only the mean axial-velocity for (A-1) case is shown in figure (5.5). The details of

117

velocity profiles of the extended-cases are also compared with experimental data in figure (5.6).

uz uref .

Fig 5.5 Mean axial-velocity field on the mid-plane of the diffuser obtained from the k   simulation case (i.e. A-1 case in table 5.2) Figure (5.6) depicts a variation of wall-parallel and circumferential components of the velocity (i.e. us and u , respectively) along hot-wire measurement lines at different sections ranged from S1 to S7 . The measurement lines were already shown in figure (5.1), marked by dashed-blue lines. Positions of the lines on the diffuser wall are presented in table (5.3) expressed in the ( xs , yn ) coordinate system shown in figure (5.1). Table 5.3 Position of the hot-wire measurement lines Coordinates

S1

S1

S2

S3

S4

S5

S6

S7

xs (mm)

-25

25

60

100

175

250

330

405

yn (mm)

0

0

0

0

0

0

0

0

Figure (5.6) suggests a good agreement between the extended-case simulations and the experimental data at measurement sections from S1 to S3 . Although u is a little underestimated at the S1 and S3 sections, an overshoot is present in the us component of the velocity near the centerline of the diffuser. From S4 to S7 sections, some deviations from the experimental data are observed in the us component of the velocity as observed in the previous studies [Payette 2008, Vincent 2010].

118

A  1 case (table 5.2) A  2 case (table 5.2) A  3 case (table 5.2) A  4 case (table 5.2)

A  5 case (table 5.2) A  6 case (table 5.2)

uS uref . , Exp. u uref . , Exp.

uS uref .

uS uref .

S1

S1 u uref .

u uref . r RInlet

r RInlet

uS uref .

uS uref .

S2

S3 u uref .

u uref .

r RInlet

r RInlet

uS uref .

uS uref .

S5

S4 u uref .

u uref . r RInlet

r RInlet

uS uref .

uS uref .

S6 u uref .

r RInlet

S7 u uref .

r RInlet

Fig 5.6 Comparison of the wall-parallel ( us ) and circumferential ( u ) components of the mean velocity at different sections with the experimental data for extended-cases 119

As one can also see in figure (5.6), in general, the obtained velocity profiles of all extended-case simulations are very similar at different sections, although little differences are observed at the S6 and S7 sections. It is also obvious in the figure that for the circumferential component of the velocity u , there is a perfect match for all simulated cases at different cross sections. Whereas, for the wall-parallel velocity component us , small differences are visible growing among the cases by moving from S5 to S7 . The results indicate low-sensitivity of the mean flow field to the inflow turbulent quantities in the case of the extended-case simulations.

5.5

Numerical simulations of the base-case

In the previous section, the ERCOFTAC conical diffuser in the extended-case configuration was simulated using the k   , S-A and DDES techniques. The main aim of this chapter is to assess the impact of different inflow boundary conditions obtained from the extended-case simulations on the base-case DDES simulations. In this regard, the velocity and turbulent viscosity profile are extracted from the k   and S-A extended-case simulations at the S1 section and imposed as the inlet boundary condition for the basecase simulations. This situation is similar to imposing an inflow profile coming from the full-machine k   simulation at the inlet of the BulbT draft tube in the case of ‘draft tubeonly’ simulations. In other words, including swirling generator in the computational domain of the ERCOFTAC test case resembles an inclusion of the ‘runner component’ in the BulbT computations. In the rest of this chapter, the effects of applying inflow conditions on the simulation of the base-case configuration are assessed in details. In this regard, effect of inflow velocity and turbulent quantity profiles are investigated.

Table 5.4 Statistics of the computational grids for the base-case (including y  on the ERCOFTAC diffuser solid wall) No Grid (for simulation with) 1 2

120

DDES S-A

Number of cells

Min. y 

Max. y 

Avg. y 

701,100 701,100

0.93 0.96

1.87 1.92

1.30 1.37

5.5.1 Computational grid specifications

As explained in the case of extended-case simulations, DDES simulations needs a fine mesh with y   1 , because the model has been designed to resolve up to the wall. Table (5.4) presents the specifications of the adopted grids for S-A and DDES simulations of the base-case arrangement. 5.5.2 Numerical setups of the base-case simulations

For S-A and DDES transient simulations, ‘pimpleFoam’ flow solver among OpenFOAM solvers is adopted. Like before, the details of control setting of the ‘pimpleFoam’ solver is similar to those used for BulbT draft tube simulations in the upcoming chapters and are presented in table F.2 (appendix F). A second-order discretization scheme is applied for the temporal term. The convective term is also discretized by second-order upwind scheme in the k   and S-A simulations, while linear discretization with filtering of high-frequency staggering modes is adopted for DDES simulations. The details of the discretization schemes applied for other terms of the Navier-Stokes equation are similar to those used for BulbT draft tube simulations and can be found in table F.1 (appendix F). 5.5.3 Inflow and boundary conditions

In the upcoming simulations, different inflow conditions are tested to see the effect of various factors on the base-case simulations. In this regard, the circumferential-averaged 1D inflow profiles for velocity and turbulent quantities are obtained by averaging the 2D profiles extracted from the extended-case simulations at S1 section.

a) Coarse grid ( k   )

b) Fine grid (DDES)

Fig 5.7 Inflow grid reconstructed by Delaunay triangulation at S1 section

121

The averaging is performed by constructing an unstructured grid using Delaunay triangulation at the S1 section (figure 5.7); the input nodes are ones extracted at S1 section from extended-case simulations. It is clear in figure (5.7) that as expected, the resolution of DDES simulation in the extended–case is higher than the grid used for calculations with the wall-function like k   ; the fine grid also features grid clustering towards the diffuser’s wall. ur uref .

u uref .

uz uref .

a) 2D profile extracted from (A-1) case in table 5.2 (i.e. k   model)

b) 2D profile extracted from (A-2) case in table 5.2, (i.e. S-A model)

c) 2D profile extracted from (A-3) case in table 5.2 (i.e. S-A with wall-function) Fig 5.8 2D-velocity profiles extracted at S1 section from extended-case simulations In all cases, for pressure quantity, mean pressure equal to zero is applied at the outlet of the diffuser extension. For all solid walls including the diffuser and extension walls, ‘no-slip’ boundary condition is applied. A Neumann boundary condition is used for all other boundary conditions. Figure (5.8) shows 2D profiles of the radial, circumferential and axial velocity components extracted from the extended-case simulations with the k   (A-1), S-A (A-2) 122

and S-A with wall function (A-3). For the latter case, a mesh with averaged y   36.2 is adopted as presented in table 5.1; in this case, ‘nutSpalartAllmarasWallFunction’ is applied on the solid walls. As one can see in figure (5.8), the patterns are principally the same, although minor differences in ur fields are observed among these three cases including k   , S-A and S-A with wall-function at S1 section. vt

vt

 104

r RInlet

 104

r RInlet

r RInlet

r RInlet

a)2D profile extracted from (A-1)

b) 2D profile extracted from (A-3)

(i.e. k   extended-case)

(i.e. S-A with wall-function case) vt

 104

r RInlet

r RInlet

c) 2D profile extracted from (A-2) case in table (5.2), (i.e. S-A extended-case) Fig 5.9 Turbulent viscosity extracted at S1 section from extended-case simulations Figure (5.9) also shows the 2D fields of turbulent viscosity at the S1 section extracted from extended-case k   , S-A and S-A with wall-function simulations. As one can see, the observed patterns among cases are relatively similar, although not at the same 123

scale. The cases generate uniform circular patterns. For all base-case simulations, circumferential averaged-1D profiles of velocity and turbulent quantities calculated based on the above 2D-fields are imposed at the inlet section. These 1D profiles are shown in figures (5.10) and (5.11) for the radial, circumferential and axial velocity components and in figure (5.13) for the turbulent viscosity parameter. 5.5.4 1D inflow profiles of mean velocity and turbulent quantities

As explained, 1D inflow profiles utilized for base-case simulations are extracted from extended-case simulations at the S1 section. In this sense, it can be expected that the type of turbulence models adopted for the extended-case simulations has a direct impact on the base-case simulations. Figure (5.10) depicts the 1D-profiles of radial, circumferential and axial components of the mean velocity at S1 section obtained from the extended-case k   simulation and S-A simulation in comparison to the experimental data. : ur uref. ,k -  : u uref. ,k -  : u z uref. ,k - 

: ur uref. , Zero Exp. : u uref. , Exp. : u z uref. , Exp.

: ur uref. ,S  A : u uref. ,S  A : u z uref. ,S  A

uz uref .

uz uref .

r RInlet u uref . ur uref .

u uref .

r RInlet r RInlet

Fig 5.10 Normalized 1D-velocity profiles extracted at S1 section from extended-case simulations (with k   and S-A turbulence models) As shown in figure (5.10), overall shapes of the curves are similar; however as one can see in the zoomed-regions, some differences exist between the curves and experimental 124

data. The main differences stem from the different responses of the k   and S-A turbulence models in the swirling generator part. Figure (5.11) similarly shows the difference between the 1D inflow profiles extracted from the S-A simulation with/without wall-function. As one can see in the figure, although the curves are in close agreement, there are still minor differences between the curves. This indicates the effects of wall-function on the quality of the predictions in the swirling part of the extended-case. : ur uref. , S  A with wall-function

: ur uref. ,Exp Zero.

: u uref. , S  A with wall-function : u z uref. , S  A with wall-function

: u uref. ,Exp. : u z uref. ,Exp.

: ur uref. ,S  A : u uref. ,S  A

: u z uref. ,S  A

uz uref .

uz uref .

r RInlet

u uref .

u uref .

ur uref .

r RInlet

Fig 5.11 Normalized 1D-velocity profile extracted at S1 section from extended-case simulations (with S-A turbulence model with/without wall-function) As it is also seen in the figure, a relatively small amplitude radial velocity is also generated at the S1 section by the simulations. Due to the lack of measurements, this component of the velocity profile was not measured experimentally and is considered equal to zero in the default profile available though the turbo-machinery working group of OpenFOAM as seen in figures (5.10) and (5.11). As shown in the previous studies, the k   simulation results in good predictions even with zero radial velocity [Page et al. 1996]. In fact, applying wall-function in the

125

k   simulations induces some sorts of artificial robustness on the boundary layer, which principally can delay onset of separation. In the next subsections, the effect of radial component of velocity on the S-A simulation of the base-case is considered.

For turbulent flow simulations in addition to the inflow velocity profiles, turbulent quantity profiles should also be provided. In this regard, for the base-case simulations, 1D profile of the turbulent quantities is calculated at S1 section using extended-case simulations. As reviewed, the extended-case simulations involve different turbulence models including k   (A-1 case in table 5.2), S-A (A-2 case in table 5.2) and S-A with wall-function (A-3 case in table 5.2). Figure (5.12) presents the variations of the turbulent dissipation rate and kinetic energy at the S1 section extracted from the extended-case k   simulation. As one can see, they exhibit good agreement compared to their experimental counterparts. It should be mentioned that due to the experimental difficulty to measure the turbulent dissipation rate quantity i.e.  is not measured. In figure (5.12), the experimental  is approximated using experimental kinetic energy values via the   C k 2 vt formula with the assumption of vt v  14.5 . For DDES and S-A simulations of the base-case, instead of a turbulent dissipation rate and turbulent kinetic energy, the turbulent viscosity vt , should be fed at the inlet section of the base-case. The later quantity is directly obtained from S-A simulations and despite shortcomings discussed before, could be approximated via vt  C k 2  in the case of k   simulations. Figure (5.13) shows 1D variation of turbulent viscosity calculated from k   and S-A extended-case simulations.



k 2 uref .

r RInlet

r RInlet

Fig 5.12 Comparison of the 1D turbulent quantity profiles at S1 section obtained from the extended-case k   simulation (A-1 case in table 5.2) with experimental ( k ) and approximated (  ) curves (left: turbulent dissipation rate, right: turbulent kinetic energy) 126

vt

 104

S-A (A-2 case, table 5.2) k   (A-1 case, table 5.2) S-A with wall-function (A-3)

r RInlet

Fig 5.13 1D profile of turbulent viscosity at S1 section coming from the selected extended-case simulations (table 5.2) As one can observe in figure (5.13), response of k   and S-A models are different on this quantity in the extended-case configuration. In fact, after the swirling generator part at S1 section, the level of turbulent quantity in the core flow, i.e. far from the solid walls, is about 4 times higher for S-A simulations than the k   counterpart. In the near-wall zone (marked by dashed-orange oval in the figure), the behaviour of the turbulence models is visible. In the case of k   simulation, a peak in vt curve (black curve) is achieved close to the wall, which decreases later on in the core flow. The level of this maximum is comparable to the S-A one. It is also clear in figure (5.13) as expected a-priori, the response of the S-A with/without wall-function is the same in the core flow i.e. far from the solid walls, but in the near-wall zone some differences are observed between the two profiles (dashed-oval in the figure). As one can recognize, an anomaly in the form of a relatively sudden jump in vt occurs in this region in the case of S-A using wall-function, which is not observed in the pure S-A counterpart. In other words, pure S-A exhibits smoother trend for vt in the near wall zone. 5.5.5 Effect of extended-case turbulence model on the base-case DDES simulations

After calculation of the 1D inflow profiles which are extracted from the extended-case simulations at S1 section, the effect of inflow conditions on the DDES simulation of the base-case can be considered. In the first series of simulations, the inflow profiles are obtained from k   simulation (A-1 case in table 5.2) and S-A simulation (A-2 case in table 5.2) of the extended case as summarized in table 5.5. 127

As explained in chapter 4, in the case of DDES simulations to avoid introducing too much dissipation, the turbulent viscosity is lowered in the core flow (i.e. LES zone) and then fed as the subgrid-scale viscosity for the DDES simulations as one can see in the (B-1) and (B-4) cases in table (5.5). In addition, the 1D-velocity profiles as shown in section (5.5.4), are applied without any modifications at the inlet for DDES simulations of the base-case. Table 5.5 First series of DDES simulations with a base-case configuration (effects of the extended-case turbulence model) Case

Inflow vSGS

Inflow vSGS

(in the core flow)

(in the wall-zone)

B-1

Inflow profile comes from the extended-case simulation with k   (A-1 in table 5.2)

vt 100

vt

B-2

k   (A-1 in table 5.2)

vt

vt

B-3

S-A (A-2 in table 5.2)

vt

vt

B-4

S-A (A-2 in table 5.2)

vt 100

vt

Figure (5.14) shows the mean velocity profiles of the cases listed in table (5.5) at the hot-wire measurement sections, namely S1 to S7 , compared to the experimental data. As one can see in the figure, DDES simulations with an inflow profile coming from the k   simulation (i.e. B-1 and B-2 cases in table 5.5) suffer from the presence of flow separation. That can be recognized by the presence of large deviations of the numerical results from the experimental data. As it is clear in the figure, these deviations are more pronounced for final sections of the measurement i.e. S5 to S7 sections. Figure (5.15) shows the separation zone isosurface (with u z  0 ) formed close to the end of the diffuser covering S5 to S7 sections; that’s why the effects of flow separation are more pronounced on these positions. As one can see in figure (5.14), DDES simulations with inflow conditions coming from S-A extended-case simulations results in a relatively good match to the experiment. The accuracy of numerical predictions are similar to one obtained in the case of the extended-case simulations as shown in figure (5.6). As it is also visible in figure (5.14), lowering the turbulent viscosity in the core flow does not have any impact on the mean profiles for DDES simulations with the k   and S-A inflow profiles.

128

B  1 case (table 5.5)

B  4 case (table 5.5)

B  2 case (table 5.5)

uS uref . , Exp. u uref . , Exp.

B  3 case (table 5.5)

uS uref .

uS uref .

S1

S1 u uref .

u uref .

r RInlet

r RInlet

uS uref .

uS uref .

S2

S3 u uref .

u uref .

r RInlet

r RInlet

uS uref . uS uref .

S4

S5

u uref .

u uref .

r RInlet

r RInlet

uS uref .

uS uref .

S7

S6 u uref .

r RInlet

u uref .

r RInlet

Fig 5.14 Wall-parallel ( us ) and circumferential ( u ) components of the mean velocity obtained from DDES simulations of base-case using different inflow conditions (inflow profiles coming from extended-case k   , S-A simulations listed in table 5.5) 129

uz uref .

uz uref .

a) B-1 case

b) B-2 case

uz uref .

uz uref .

c) B-3 case

d) B-4 case

Fig 5.15 Mean axial-velocity field and separation-zone isosurface (with u z  0 ) obtained from the base-case DDES simulations listed in table 5.5 fd Y x

z

Fig 5.16 Typical f d function variation in the base-case DDES simulation with inflow profile coming from the extended-case k   simulation (B-1 case in table 5.5) [ f d  0 : RANS , f d  1: LES ] To investigate further, the variation of f d function introduced in chapter 2 (equation 2.12) is considered for the B-1 case (in table 5.5) with a flow separation and shown in figure (5.16). As one can see, the wall-zone (i.e. 1.5 cm to the wall) is completely protected by the f d function, which excludes the possibility of grid-induced separation (GIS) as

130

expected a-priori. On the other hand, since the same mesh and numerical setting are adopted for all simulations listed in table (5.5), one can conclude that the DDES simulations are sensitive to the inflow conditions. In addition, the boundary condition obtained from extended-case k   simulations in the (B-1) and (B-2) cases result in flow separation, which is not present in the experiment. In this regard, as shown before in figure (5.10), globally there are minor differences between the inflow velocity profiles coming from the extended-case k   and S-A simulations. On the other hand, there are also clear differences between the turbulent viscosity profiles obtained from these two sets of simulations in both the core flow zone and wall-zone as shown in figure (5.13). However, the difference level of turbulent viscosity in the core flow (as visible in figure 5.13) is not an important factor in the flow separation formation. In fact, as one can observe considering (B-1) and (B-4) cases, by lowering the turbulent viscosity level even by two orders of magnitude in the core flow region, the separation is not formed in the DDES simulation with an inflow profile coming from the S-A simulation (B-4 case). To go further, one should have a look at the close-wall profiles of the base-case inflow extracted from extended-case simulations at S1 section. In this regard, the variation  of utotal versus y  is shown in figure (5.17); utotal is a velocity measure defined as the

resultant of us and u components of the mean velocity, as below: utotal  us2  u2

(5.2)

In fact, just the radial component of the velocity is excluded in the utotal definition. In figure (5.17), utotal and y  are expressed in wall-units defined in section (2.2).  utotal

buffer layer

viscous sub-layer Excess Energy

k   (A-1 case in table 5.2) S-A (A-2 case in table 5.2) S-A with wall-function (A-3) Exp. data (Clausen et al.)

y  Fig 5.17 Variation of utotal versus y  at the base-case inflow section ( S1 section) extracted from various extended-case simulations

131

As one can see, the inflow profiles extracted from the extended-case simulations have close agreement between themselves and with experimental data out of the bufferlayer and the viscous sub-layer. It should be mentioned that the black-curve in the figure extracted from the k   simulation should not be extended below y   70 (blue dashedline in figure 5.17). In fact, the average y  at S1 section in k   simulation is 70 and therefore position of the first computational node is at y   70 . The black curve shown in the figure in the range of 1  y   70 is constructed by a strategy explained shortly in the coming section (5.5.7) and is presented here just for completeness and a comparison to the results of S-A with wall-function.  As one can also see in figure (5.17), there is an offset between utotal obtained from

pure S-A simulation and the one with wall function at the inlet of the base-case configuration. This offset brings a larger energy into the turbulent boundary layer in the buffer layer and viscous sub-layer, which can theoretically postpone the onset of separation. The idea is also confirmed by DDES simulation of the base-case using the inflow profile obtained from S-A with wall-function extended-case simulation (i.e. cyan dashed line in figure 5.17); in fact, a non-physical separation zone is formed similar to the cases with inflow profile coming from k   simulations (i.e. B-1 and B-2 cases in table 5.5). This proves the importance of the inflow profile in the range of y   30 on the downstream flow behavior. One point that should also be kept in mind is that, the classical picture of the boundary-layer consisting viscous, buffer and logarithmic layers is valid for local equilibrium conditions as explained in chapter 2. Due to the lack of experimental data close to the diffuser-wall, it is difficult to justify which curve in figure (5.17) represents the real flow physics in the range of y   30 . 5.5.6 Effect of inflow radial velocity on the base-case simulations

In this section, the effect of including the radial velocity in the inflow profile on the basecase simulations is considered. To see the importance of radial velocity in the pure S-A simulation, different base-case simulations are conducted as listed in table (5.6). In the (C-1) and (C-3) cases, the inflow profile utilized for base-case simulations is the default experimental-based profile introduced by turbo-machinery working group of OpenFOAM. In this inflow profile, the radial component of the velocity is set to zero due to the lack of ur measurements. In the (C-4) case, the inflow profile is extracted from the S-

132

A extended-case simulation (i.e. A-2 case in table 5.2), then the radial velocity is set to zero. Table 5.6 URANS simulations of the base-case configuration (effects of inflow radial velocity) Case C-1

Turbulence model k 

Inflow profile extracted from default profile (Exp.)

Inflow radial velocity ur  0

C-2

S-A

S-A (A-2 in table 5.2)

ur  0

C-3

S-A with wall-function S-A

default profile (Exp.)

ur  0

S-A (A-2 in table 5.2)

ur  0

C-4

As one can see in figure (5.18), k   simulation of the base-case with ur  0 in inflow profile (C-1) avoids the formation of separation zone on the diffuser wall; this outcome is obtained due to the induced artificial robustness of the boundary layer by applying the wall-function as observed in the previous studies [Page et al. 1996, Payette 2008]. As also seen in figure (5.18), applying the wall-function in the S-A simulation with wall-function and ur  0 (C-3) also compensates the shortcomings of the applying zero radial velocity and as a result, no separation is formed on the diffuser wall. This characteristic of the wall-function should not be necessarily considered favorable, because it can cause delay in the prediction of the separation in the complex fluid flows. It is also important to compare behaviour of the (C-2) and (C-4) cases listed in table (5.6), corresponding to the pure S-A simulations with ur  0 and ur  0 in the inflow profile, respectively. As one can see in figure (5.18), the case (C-4), i.e. green curve in figure 5.18, with ur  0 in the inflow profile leads to the flow separation on the diffuserwall, whereas no separation is formed by including ur  0 radial velocity profile. This observation proves that it is mandatory to include the radial velocity profile ( ur ) in the inflow profile for pure S-A simulations resolving to y   1 . This observation for S-A turbulence model is in agreement with the observation of Payette for the k   SST turbulence model resolving to y   1 for the same application [Payette 2008].

133

C  1 case (table 5.6) C  2 case (table 5.6) C  3 case (table 5.6)

C  4 case (table 5.6)

uS uref . , Exp. u uref . , Exp.

uS uref .

uS uref .

S1

S1 u uref .

u uref .

r RInlet

r RInlet

uS uref .

uS uref .

S2

S3 u uref .

u uref .

r RInlet

r RInlet

uS uref . uS uref .

S4

S5 u uref .

u uref . r RInlet

r RInlet

uS uref .

uS uref .

S6

S7 u uref .

u uref . r RInlet

r RInlet

Fig 5.18 Wall-parallel us and circumferential u components of mean velocity obtained from URANS simulations of the base-case indicating effect of inflow radial velocity (different inflow conditions for the cases are presented in table 5.6) 134

5.5.7 Effect of inflow profile near-wall treatment on DDES simulations

In the two previous subsections, the importance of the inflow profile in the region very close to the wall, i.e. in the viscous sub-layer and buffer layer, for pure S-A simulation was discussed in details. It should be emphasized that S-A is the base formulation of the DDES simulations adopted in this study; so one may expect a similar behavior in the case of DDES simulation of diffuser flows. In the present subsection, the aim is to have a closer look at the inflow profile coming from the k   simulation utilized for DDES simulations of the ERCOFTAC diffuser flows to complete the profile along the path followed to get better predictions. As mentioned before, at S1 section the averaged y  is equal to 70, so the first computational node is on y   70 . Lower than y   70 , no data is available from the k   extended-case simulation, which can be extracted at S1 section and then fed to the DDES simulation of the base-case configuration resolving to y   1 . To solve this remedy, it is tried here to reconstruct the inflow profile coming from the k  

extended-case simulation in the region with y   70 based on a simple

assumption of the validity of the classical picture of the boundary layer. In the classical point of view, the boundary layer consists of the viscous ( 0  y   5 ), buffer ( 5  y   30 )  and logarithmic layers ( 30  y   300 ). In the viscous layer, the relation utotal  y  is valid,

 whereas in the logarithmic layer the relation is expressed as u total  (1 kV ) ln y   B , the kV

and B are constants in the equation and with values depending on the flow application. As shown in previous studies, using experimental measurements [Naganto et al. 1998] and DNS data [Lee and Sung 2008], in the case of a turbulent boundary layer subjected to an adverse pressure gradient (APG), the B coefficient is a function of the strength of the adverse pressure gradient. As also shown by Facciolo in the case of rotating pipe flow using experimental and DNS data, the slope of the logarithmic part, i.e. kV , is a function of the flow swirl number [Facciolo 2006]. In the case of diffuser flow, both swirl and adverse pressure gradient are present in the flow field, which implies that the values of the kV and B should be tuned based on the flow situation. In this regard, these coefficients are obtained based on the value of the velocity and slope of the available data at y   70 for y   70 range. Using the inflow profile extracted from the k   extended-case simulation of the ERCOFTAC diffuser at S1 section for y   70 ; the values of the coefficients are obtained as kV  0.15 and B  4.4 .  In the viscous-sub-layer, theoretically, the relation utotal  y  is valid. In the buffer layer, a

135

continuous curve is needed to stitch two ends of the viscous sub-layer and logarithmic layer curves together. This can be done by employing a function with the following form:  utotal  a ( y  ) 2  b( y  )  c  d y 

(5.3)

where the a , b , c and d constants are obtained by solving an algebraic set of equations using the following boundary conditions:  total y  5

u

 utotal

y  30



 5,

1 ln(30)  B, kV

 dutotal dy   dutotal dy 

1

(5.4)

y  5

y  30

 1   kV

 1      30 

(5.5)

 Figure (5.19) shows the complete profile of the utotal compared to the available

experimental data [Clausen et al. 1993]. For radial velocity a linear variation is considered near the wall, as observed by Facciolo in the case of swirling pipe flow [Facciolo 2006].

 utotal

buffer layer

log layer

viscous sub-layer Constructed profile Exp. (Clausen et al.)

y   70

y

 versus y  of the constructed profile at the S1 section Fig 5.19 Variation of utotal

To complete the wall-treated inflow profile for DDES simulation of the base-case configuration, turbulent viscosity should be also provided on top of the velocity profile. Here, the Reichardt’s formula in the wall-zone is adopted. as follows [O’Connor 1995]: vt v  kV   y  [1  (  / y  ) tanh(y  /   )]

(5.6)

where   is a model constant equals to 11, and kV is the von-Karman constant and   is introduced as a scaling factor to be set with a given value of vt at the logarithmic layer end at y   30 . Figure (5.20) shows the complete vt profile as a blending of the equation (5.6) for y   70 and vt profile obtained from k   extended-case simulation for y   70 . 136

vt

y   70

y

Fig 5.20 Variation of vt versus y  of the constructed profile at the S1 section D  1 case (table 5.7) D  2 case (table 5.7) D  3 case (table 5.7)

D  4 case (table 5.7)

uS uref . , Exp. u uref . , Exp.

uS uref .

uS uref .

S7

S6 u uref .

u uref . r RInlet

r RInlet

Fig 5.21 Wall-parallel us and circumferential u components of the mean velocity obtained from DDES simulations of the base-case indicating effect of inflow near-wall treatment (cases are listed in table 5.7) Table 5.7 Second series of DDES simulations with base-case configuration (effect of inflow near-wall treatment) Case

Inflow profile comes from the extendedcase simulation with

Inflow vSGS (in the core flow)

Inflow vSGS (in the wall-zone)

D-1

k   (A-1 in table 5.2)

vt 100

vt

Wall treatment in y   70 No

D-2

k   (A-1 in table 5.2)

vt

vt

No

D-3

k   (A-1 in table 5.2)

vt

vt

Yes

D-4

k   (A-1 in table 5.2)

vt 100

vt

Yes 137

To see the effect of the inflow profile including abovementioned wall-treatment on the DDES simulations of the base-case, 4 different cases are considered as listed in table (5.7). In the (D-1) and (D-2) cases, the original profile extracted from k   extended-case simulation without wall-treatment in y   70 is utilized; it means linear interpolation for y   70 . Whereas, the inflow profiles are extended to y   1 for (D-3) and (D-4) cases using the aforementioned near wall-treatment strategy. As one can see in figure (5.21), in the last two measurement sections, namely S6 and S7 , there is a minor move in us towards the experimental data but in general it seems that the inflow wall-treatment of the k   inlet boundary condition does not solve the separation problem on the diffuser-wall by itself. 5.5.8 Tuning of inflow near-wall turbulent viscosity

So far, the simulations with different inflow conditions were performed and the effects of different inflow factors on the simulations were considered. As shown, S-A and DDES simulations are extremely sensitive to the inflow conditions in contrast to the k   simulations using wall-functions. It was observed that, even a relatively small change in the inflow profile in the viscous sub-layer and buffer-layer could trigger the separation. On the other hand, as it is also shown here, the sensitivity to the inflow profile is not only limited to the velocity profile but also includes the turbulent viscosity. It is worth recalling from chapter 4 that, for DDES simulations of the base-case diffuser flow as a hybrid RANS/LES technique, typically the turbulent viscosity in the core flow is reduced at least by 2 orders of magnitude and applied as vSGS , whereas the turbulent viscosity in the wall-zone (hereafter WZ; 1.5 cm to the solid-walls) is kept untouched and applied in the URANS region. For all DDES simulations in figure (5.22), the turbulent viscosity in the core flow is reduced by 2 orders of magnitude. It is important to notice as well that, the level of turbulent viscosity extracted from the k   extended-case simulation at S1 section depends on the upstream k and  values imposed at the inlet of the extended-case. In fact, the values of turbulent dissipation rate and kinetic energy can be appropriate for the k   simulations due to the robustness induced by the wall-function, but may be just inappropriate for DDES simulations. To see the effect of turbulent viscosity in the wall-region, the turbulent viscosity in the WZ is amplified by a factor, WZ , as below: vt 138

WZ

 WZ  vt

WZ RANS

, (WZ  1, 2, 4, 6,8,10,10 2 )

(5.7)

The effect of tuning the turbulent viscosity in the wall-zone on the DDES simulation of the base-case is shown in figure (5.22). As one can see, by increasing the vt in the WZ, the results move towards the experimental data; for WZ  1 and 2, the separation is present and recognized with its signature on the velocity, which is more pronounced at the final sections. As one can see in figure (5.22), all WZ  4, 6, 8 and 10 remove the separation, also us and u get closer to the experimental data. Finally, WZ  102 over-smooth the curves and gets farther from the experiment. Table 5.8 Global performance quantities obtained from base-case simulations (effect of near-wall turbulent viscosity tuning) No

Simulation

Inflow comes from

Recovery Coefficient ( )

1

DDES, WZ  1

A-1 in table 5.2

0.759

Loss Coefficient ( L ) 0.044

2

DDES, WZ  2

A-1 in table 5.2

0.752

0.046

3

DDES, WZ  4

A-1 in table 5.2

0.770

0.045

4

DDES, WZ  6

A-1 in table 5.2

0.773

0.045

5

DDES, WZ  8

A-1 in table 5.2

0.773

0.045

6

DDES, WZ  10

A-1 in table 5.2

0.774

0.045

7

DDES, WZ  102

A-1 in table 5.2

0.775

0.048

8 9 10

DDES

A-2 in table 5.2 A-2 in table 5.2 Default (Exp.)

0.760 0.760 0.745

0.044 0.044 0.045

S-A

k 

To complete the picture, table (5.8) shows two global engineering quantities of the diffuser, including the recovery and loss coefficients for different simulations. Considering the behaviour of the flow in figure (5.22), the minimum value of the increase-factor that removes the separation is WZ  4 , which exhibits a relatively good agreement with the experimental data. This can be adopted for DDES simulations of the ERCOFTAC diffuser. Considering WZ  4 in table (5.8), one can see that it produces a comparable performance with S-A and k   simulations. As a conclusion, if experimental data are available, adjustment of the near-wall turbulent viscosity to an appropriate value should be considered as an initial step to perform DDES simulations, whenever the inflow profile is extracted from upstream RANS simulations. 139

 WZ  1  WZ  2  WZ  4  WZ  6  WZ  8

 W Z  10  WZ  10 2 u S u ref . , Exp . u u ref . , Exp .

uS uref .

uS uref .

S 1

S1 u uref .

u uref .

r RInlet

r RInlet

uS uref .

uS uref .

S2

S3

u uref .

u uref .

r RInlet

r RInlet

uS uref .

uS uref .

S4

S5

u uref .

u uref .

r RInlet

r RInlet

uS uref .

uS uref .

S7

S6 u uref .

u uref .

r RInlet

r RInlet

Fig 5.22 Wall-parallel us and circumferential u components of mean velocity obtained from DDES simulations of the base-case (effect of tuning of the inflow near-wall vt ) 140

5.6 Summary

The ERCOFTAC conical diffuser with two configurations, i.e. base-case and extendedcase, were selected to study sensitivity of the base-case simulations to the inflow conditions. The base-case configuration with no swirling generator resembles the ‘draft tube-only’ simulation in the BulbT, whereas in the extended-case, the swirling generator mimics the runner and the situation resembles the ‘runner and draft-tube’ simultaneous simulation. The experimental data for the test case is available in [Clausen et al. 1993]. The flow according to the experimental data is at the verge of separation, however was set to have no separation on the diffuser wall. The major points of this chapter can be summarized, as follows:



Results of the extended-case simulations depict an overall good agreement with the experimental data. The simulations exhibit low-sensitivity to the inflow turbulent quantities and the type of turbulence models, including k   , S-A and DDES.



1D Inflow profiles for base-case simulations were extracted from the extended-case SA, k   , and S-A with wall-function simulations. Minor differences exist between the curves of the inflow velocity components for these different extended-case turbulence models (figures 5.10 and 5.11). These differences are more pronounced for y   30 (figure 5.17). The level of inflow turbulent viscosity profile is also different for the S-A than the k   cases (figure 5.13), the later being 4 times lower. In addition, the nearwall behaviour of the turbulent viscosity is different for the extended-case S-A simulations with/without wall-function (figure 5.13), where a peak is observed for the k   case.



The base-case DDES simulation results show that: o The inflow profiles coming from the extended-case k   and S-A simulations (table 5.5) does and does not result in a separation formation on the diffuser wall, respectively. The difference between the velocity (figures 5.10 and 5.11) and turbulent viscosity profiles (figure 5.13) along with the excess energy

existing in the range of y   30 for S-A can be considered as the reasons. o Including the inflow radial component of velocity is necessary for pure S-A simulations (table 5.6). For the k   simulation and S-A simulation with wallfunction, neglecting the inflow radial velocity did not affect the results due to the induced artificial robustness of the boundary layer by applying wallfunctions. This characteristic should not be considered a positive point, because

141

it brings a delay in the prediction of separation, especially for complex fluid flows. o Near wall-treatment of the inflow turbulent velocity and viscosity profiles originally coming from the extended-case k   simulation (table 5.7) in

y   70 , provides a more accurate inflow condition for the DDES simulations, although it does not solve the separation problem on the diffuser-wall in itself (figure 5.21). 

142

Using the available experimental data, the inflow turbulent viscosity in the wall zone was successfully tuned in a manner to postpone the separation formation in the DDES simulations of the diffuser flow.

Chapter 6

BulbT draft tube flow simulations: basic geometry

6.1

Introduction

In this chapter, flow field in the draft tube of the bulb turbine with its ‘basic geometry’ configuration is simulated and the effects of different factors in the flow simulations are considered. Due to the lack of experimental data in this case, the relative comparisons are made among various simulation cases. This enables us to firstly examine the adopted numerical strategy developed in chapter 4 and secondly to better understand the effects of different inflow boundary conditions on the DDES simulations of the BulbT draft tube flow. In this sense, this chapter can be viewed as an application of the simulation strategies on a draft tube geometry with a less aggressive divergence angle compared to the ‘final geometry’ of the BulbT draft tube, which will be more challenging.

6.2

Basic draft-tube geometry and selected operating point

In 2011, the BulbT project was initiated within the framework of an international research partnership in the Hydraulic Machines Laboratory (LAMH) of Laval University [Deschenes et al. 2010]. The project aimed to characterize the flow field in a bulb turbine using both experimental measurements and numerical simulations. In this context, the present Ph.D. study was defined to better understand the details of flow physics via performing hybrid LES/RANS (i.e. DDES) simulations. At the beginning of this project, 143

the basic geometry of the draft tube involved 6.6  and 2.1 divergence angles for the conical and transition parts, respectively; as shown in figure (6.1). In this case, the length of the conical part, transition part and the extension is equal to 3.8 Rref . , 4.9 Rref . and 8.7 Rref . , respectively, where Rref . is the reference length defined in chapter 4. On the other hand as mentioned in chapter 4, after a while, with progression of the project, the geometry of the draft tube was modified to the ‘final geometry’ in a way to have a larger divergence angle, which induces more challenging fluid flow phenomena like a wall-dominant flow separation. The first numerical simulations were all performed on the ‘basic geometry’, for example to obtain the numerical hill-chart [Guenette et al. 2012] and unsteady DDES simulations of the draft tube in the present thesis, whereas the experimental measurements and the final numerical simulations were later conducted on the ‘final geometry’ of the draft tube. In the next chapter, the results of DDES simulation of the ‘final geometry’ of the draft tube are presented and compared to the experimental data.

Y Z

Bulb

6.6



2.1 

Guide Vanes Runner Fig 6.1 Basic geometry of the draft tube with its geometrical divergence angles In the present chapter, all simulations are performed on the ‘basic geometry’ at the best efficiency point (BEP) with inflow obtained from the k   RANS simulations [Guenette 2013]. The specifications of the selected operating point are shown in table (6.1).

Table 6.1 Details of the selected operating point for the numerical simulations OP BEP

144

Guide vane angle  52

Net head Runner blade angle  (m) 5 22.5

Mass flow rate (m3/s) 0.4243

N11 160

6.3

Computational grid specifications

To simulate turbulent flow fields in the draft tube with the ‘basic geometry’, three sets of grids are generated using ICEM-CFD 14.5.7 with hexahedral elements for k   RANS simulations (mesh A) and for DDES or S-A simulations including intermediate grid (mesh B) and fine grid (mesh C). The later grid is adopted to perform a ‘mesh independence’ test. Table (6.2) shows the statistics of these generated computational grids. Ideally, for LEStype simulations, the values of the characteristic parameters named angle, determinant and aspect ratio equals to 90 , 1 and 1 respectively, representing a equidistance edge orthogonal hexahedral element (i.e. cube) introducing minimum numerical error in the computational procedure. In addition, to keep the computational error low, the “element growth factor” is kept lower than 1.2 everywhere in the computational domain. Typically, high aspect-ratio cells are formed in the near-wall zone, where the demand of y   O (1) should be respected. Here, the first node position respects the condition of y   1 on the draft-tube wall with averaged y  of 0.58. In the generated grids for DDES simulations, i.e. grids B and C, the grid in the core flow is close to the cubical shape appropriate to resolve turbulent structures in the LES zone. Furthermore, the face angle of the generated grid cells is higher than 25 , and the determinant parameter of the cells in the generated grids is everywhere higher than the minimum acceptable value of 0.5, essential for the quality in draft tube flow computations. Table 6.2 Grid statistics for k   (mesh A) and S-A, DDES simulations (mesh B & C) Mesh A ( k   ) No. of elements (millions) Min. Angle Min. Determinant Max. Aspect Ratio Max. growth factor

2.15 25.8 0.610 126 1.2

Mesh B (DDES-intermediate grid) No. of elements (millions) Min. Angle Min. Determinant Max. Aspect Ratio Max. growth factor

7.48 32.8 0.503 1540 1.2

145

Mesh C (DDES-fine grid)

6.4

No. of elements (millions)

14.87

Min. Angle Min. Determinant Max. Aspect Ratio Max. growth factor

32.6 0.573 1010 1.2

Numerical setup for turbulent flow simulations

Like the previous chapter, the simulations presented in this chapter are performed using the OpenFOAM 1.6-ext code explained in details in chapter 3. Depending on the type of simulations, different inflow boundary conditions including ‘1D circumferential-averaged’, ‘2D-rotating’ and 2D-rotating+AFG’ profiles for velocity and turbulent quantities are applied. The later inflow, namely ‘2D-rotating+AFG’, involves synthetic inflow turbulence (AFG) added on the inflow velocity profile (see chapter 4). A mean pressure equal to zero ( p  0 ) is applied at the final exit of the draft tube in all simulations. For all solid walls including the hub and draft tube wall, the ‘no-slip’ boundary condition is applied except for the dummy extension wall of the draft tube added to avoid recirculation at the draft tube exit, where ‘slip’ boundary condition is imposed. The details of the boundary conditions for each patches applied in the simulations can be found in table F.3 (appendix F). For RANS simulations, ‘simpleFoam’ solver is adopted whereas for the transient DDES simulations, ‘pimpleFoam’ flow solver among OpenFOAM solvers is used. The details of the control setting of the solver are presented in table F.2 of appendix F. Appendix (F) also presents the details of the numerical discretization schemes and solution procedure adopted for the simulations in more details. A second order discretization scheme is applied for the temporal term (if any). The convective term is discretized by a second order upwind scheme for the RANS simulation, whereas for DDES simulations, a second order linear discretization scheme with local filtering of very high frequency staggering modes, namely ‘filteredLinear2V’, is used unless otherwise stated. The details of the discretization schemes applied for the other terms of the Navier-Stokes equation can be found in table F.1 (appendix F). Furthermore, the specific settings (if any) are mentioned in advance for each simulation group.

146

6.5

Basic geometry RANS simulations

As mentioned in the previous chapters, S-A turbulence model is the base-formulation of the DDES turbulence treatment adopted in this thesis; so its behaviour is important to be considered. Figure (6.2) shows the separated zone obtained from the S-A and k   RANS simulations at BEP characterized in table (6.1). As one can see in figure (6.2), the topology of the separated zone is different: in the case of k   simulation no separation is formed on the draft-tube wall instead a small separation bubble exists in the core-flow, whereas in the S-A case, two regions of separation zone are formed on the two opposite corners of the draft tube wall.

Y x

z

Fig 6.2 Separation zone (blue iso-surface) in the basic geometry of the draft tube obtained from S-A (left) and k   (right) RANS simulations at BEP To investigate further, the circumferential-averaged 1D profile of the velocity components are compared in the conical part of the draft tube at different locations including z Rref .  1.4, 2.4, 3.3 and 4.3, where the latter is the conical part end section. Figure (6.3) compares the profiles obtained from S-A and k   simulations. As it suggests, the radial components of the velocity, ur , is very similar between S-A and k   RANS simulations at all cross-sections. For the circumferential velocity component, u , some differences are observed near the hub region, but the difference decreases by advancing towards the end of the conical section. A similar behaviour is observed for the axial component of the velocity, u z , some differences being present in the near-hub region diminish towards the final section of the conical part. As it is also clear in the figure, in the near-shroud zone both S-A and k   RANS simulations behave similarly for all crosssections. The evolution of draft tube coefficient (  , defined in section 1.2.3 of the first chapter) in the streamwise direction is shown in the left-side of figure (6.4) for S-A and k   RANS simulations. In the k   simulation, as shown in figure (6.2), the draft tube 147

wall is clean with no separation which explains the higher value of the draft tube coefficient compared to the S-A case. u uref .

: ur uref . , inlet : u uref . , inlet

r Rref.

: u z uref . , inlet

I) Inlet profile : ur uref . , k- : u uref . , k-

: ur uref . , S  A : u uref . , S  A

: u z uref . , k- u uref .

: u z uref . , S  A u uref .

r Rref.

r Rref.

II) z Rref.  1.4

III) z Rref.  2.4 u uref .

u uref .

r Rref.

r Rref.

IV) z Rref.  3.3 V) z Rref.  4.3 Fig 6.3 1D circumferential-averaged of mean velocity profiles in the conical part of the basic geometry extracted from S-A (red) and k   (black) RANS simulations at BEP 148



k  S A

Sw

:S A : k 

z Rref.

z Rref.

Fig 6.4 Draft tube recovery coefficient (left) and swirl number (right) evolution in the streamwise direction obtained from S-A and k   RANS simulations at BEP In the S-A, the central reverse–flow bubble disappears and instead, two separation zones are formed on the draft tube wall at two opposite corners (figure 6.2). This explains to some extend the observed reduction in the draft tube coefficient (  ) for the S-A simulation. Another important coefficient to characterize the flow is the swirl number ( S w ) defined in the first chapter (equation 1.6). The right-side of figure (6.4) shows the evolution of the swirl number in the streamwise direction. The swirl number is interpreted as the ratio of the angular momentum flux of the flow to its axial momentum flux calculated at each cross-section. As one can see in the figure, the swirl number monotonically decreases by advancing towards the downstream direction, whereas the absolute value of the swirl number S w increases. In fact, by moving downstream, both the axial velocity level and the circumferential velocity range decrease by increasing the cross-sectional area, as one can see in figure (6.3). Relatively rapid decrease of axial velocity explains the increase of absolute value of the swirl number, i.e. S w , in the streamwise direction. As shown in figure (6.4), the offset of swirl number between S-A and k   simulation increases towards downstream.

6.6

Basic geometry DDES simulations

After performing RANS simulations of the draft tube flow in the ‘basic geometry’, the next step is to study the flow field with DDES turbulence treatment. As mentioned, the purpose of simulations with ‘basic geometry’ is to gain initial insight into flow simulations using DDES. The aim is also to examine the numerical strategy to simulate the draft-tube flow

149

using DDES turbulence treatment and to understand the importance of the various parameters affecting the simulation results. This is achieved by conducting some comparative simulations each of which shows effects of one factor on the obtained results. The following subsections provide more details. All DDES simulations of the draft tube are performed using ‘pimpleFoam’ solver with time step equals to 0.63 runner rotations on the intermediate grid (i.e. mesh B) presented in table (6.2). The details of the control setting of the solver are presented in table F.2 in appendix F. To simulate the draft tube flow, various types of inflow boundary condition are applied at the draft tube inlet including ‘1D circumferential-averaged’, ‘2Drotating’ and ‘2D-rotating+AFG’ profiles. It should be mentioned that in all cases the mean quantities are calculated over a time-period of about 18 runner rotations after achieving a flow statistical convergence (typically after 90 runner rotations). 6.6.1 Effect of wall-zone turbulent viscosity at inflow section

In this subsection, effects of the inflow turbulent viscosity in the wall-zone (WZ) are considered, as it was also found in the case of ERCOFTAC diffuser (chapter 5). In this regard, the turbulent flow field in the draft tube of the BulbT with the ‘basic geometry’ is simulated using the DDES turbulence treatment. The inflow condition for the velocity and turbulent quantity are the ‘1D circumferential-averaged’ inflow profiles presented in figure (6.3). As mentioned before, the convective term is discretized by the second order linear discretization scheme with local filtering of very high frequency staggering modes, namely ‘filteredLinear2V’. The details of the discretization schemes for other terms and solver control settings can be found in appendix F.

fd

Fig 6.5 f d function at the inlet of the draft tube with ‘basic geometry’ at BEP Two simulation cases are considered: first, the original RANS turbulent viscosity at ); second, an amplified turbulent viscosity at the inflow the inflow WZ (i.e. vt WZ  vt WZ RANS 150

WZ (i.e. vt WZ  103  vt

WZ RANS

). Furthermore, in the core flow the vt is reduced two orders of

magnitude as it plays the role of vSGS . It should be noted that as before, the original inflow turbulent viscosity is calculated by vt  C k 2  formula based on the turbulent kinetic energy ( k ) and turbulent dissipation rate (  ) extracted from the full-machine k   simulation at the draft tube inlet section. The wall-zone (WZ) is identified by the f d function defined in chapter 2 (equation 2.12). The narrow region, where f d  1 , is considered as the WZ shown in figure (6.5). The zone is bounded to the region with R R  0.02 with respect to the shroud and hub. As discussed in the previous section, the k   RANS simulation shows no separation on the draft tube wall, whereas the S-A simulation with an inflow condition coming from the full-machine RANS simulation presents two separated regions at two opposite corners on the draft tube wall. The situation is similar to what was observed in the case of the ERCOFTAC diffuser in the previous chapter. In fact, due to the applying wallfunction in the k   RANS computations, the boundary layer is less sensitive to the inflow condition. A similar behavior to the S-A case is expected with the DDES simulation because the underlying formulation of the DDES is S-A in the URANS region. Due to the lack of experimental data in the case of the ‘basic geometry’ configuration, it is not possible to validate which one of the k   simulation or the S-A (or DDES) is closer to reality. Although, according to the experience of industrial partners, no wall-separation is expected in the ‘basic geometry’ at BEP. In these circumstances, amplification of the turbulent viscosity is one way to attenuate the separation for DDES simulations as shown in the case of ERCOFTAC diffuser (chapter 5). In this regard, an amplification factor of 103 is considered here. uz uref .

Y x

a) vt

WZ

 103  vt

WZ RANS

z

b) vt WZ  vt WZ RANS

Fig 6.6 Mean separation-zone (blue iso-surface) and coherent structures ( Q  500 ) obtained from DDES simulations: the effect of inflow vt amplification in the WZ 151

As one can see in figure (6.6), by increasing the inflow turbulent viscosity in the WZ, the mean separation zone extracted from DDES simulations gets smaller (part afigure 6.6) compared to the case with the original inflow turbulent viscosity (part b- figure 6.6). It is also clear that by amplifying the turbulent viscosity in the WZ, the separation zone is limited to the four corners, whereas in the case of original viscosity it covers a waste wall region of the transition part (part b- figure 6.6). Figure (6.7) on the left depicts the evolution of the draft tube coefficient (  ) in the streamwise direction. As one can see, minor differences exist between the two cases. The difference is related to the different flow separation patterns existing in these two cases and can be explained with the aid of mean velocity (figure 6.8). In figure (6.7), it is evident that the draft tube coefficient of the amplified vt case in the range of z Rref .  5 is lower than the original vt case. As one can see in figure (6.8), it is linked to the formation of the low speed region in the core flow in the amplified vt case (part b- figure 6.8), where this semihydraulic obstacle induces a slight increase in the axial velocity and therefore a decrease in the pressure elevation for amplified vt case compared to its original counterpart.



Sw

: Original vt in WZ : Amplified vt in WZ

: Original vt in WZ : Amplified vt in WZ

z Rref.

z Rref.

Fig 6.7 Draft tube recovery coefficient (left) and swirl number (right) evolution in the streamwise direction from DDES simulations at BEP: effect of vt amplification in WZ In the range of 5  z Rref .  8 , the relatively thick separated zone in the case of the original vt case decreases the effective cross-sectional area of the flow passage, which causes a relative increase in the axial velocity and, as a result, a relative decrease in the pressure elevation compared to the amplified vt case. The final increase of  at the real exit of draft tube for the original vt case is explained by the separated zone in the case of the original vt case getting thinner compared to the amplified vt case at the real exit of the 152

draft tube (marked by red dashed-line in figure 6.8). On the other hand, the mean loss coefficient of the draft tube,  L , defined in equation (1.10), is obtained as 0.071 and 0.035 for the original and amplified vt cases, respectively. In fact, the smaller flow separated zone in the case of amplified vt decreases the global loss of the draft tube.

uz uref .

a) Original inflow turbulent viscosity in WZ, vt WZ  vt WZ RANS

b) Amplified inflow turbulent viscosity in WZ, vt WZ  103  vt

WZ RANS

Fig 6.8 Mean axial velocity field on the mid-plane of the draft tube obtained from DDES simulations at BEP: effect of inflow vt amplification in WZ The evolution of the swirl number ( S w ) in the conical part in the steamwise direction is also shown on the RHS of figure (6.7). As one can see, in the range of z Rref .  2.4 , the swirl number evolution is very similar between the two cases, whereas after the point z Rref .  2.4 , the absolute value of the swirl number S w increases more in the case of the amplified vt . As before, this behavior is linked to the distribution of the circumferential and axial component of the velocity at different sections of the conical part. Figure (6.9) shows the circumferential average of the velocity components at two sections of the conical part for instance. As figure (6.9) suggests, the amplification of the inflow turbulent viscosity in the WZ modifies the shape of the velocity profile especially near the draft tube wall and the flow center; the difference between the profiles of two cases grows towards the final section of the conical part z Rref.  4.3 .

153

To study the unsteady effects of inflow vt amplification in the WZ, the energy spectrum of axial velocity signal between two cases is compared at some axial positions on the axisymmetric centerline of draft tube as shown in figure (6.10). It is important to notice that the centerline of the draft tube for both cases lies in the vortex rope region as can be seen in figure (6.6). : ur uref . , Amplified vt in WZ : u uref . , Amplified vt in WZ : u z uref . , Amplified vt in WZ

: ur uref . , Original vt in WZ : u uref . , Original vt in WZ : u z uref . , Original vt in WZ

u uref .

u uref .

r Rref.

a) z Rref.  1.4

r Rref.

b) z Rref.  4.3

Fig 6.9 1D circumferential-averaged of mean velocity profiles in the conical part of the basic geometry extracted from DDES at BEP: effect of vt amplification in WZ As one can see in figure (6.10), closer to the hub at z Rref.  2.4 , two spectra are not matched for the relative large scales and the case with original inflow vt has higher energy than its counterpart, whereas the spectrum of both cases in the high frequency range is matched exhibiting a decrease with slope of  5 3 in the inertial region as expected. The blue dashed-line in the figures shows the cut-off frequency calculated by the f c  2 2    formula where for non-uniform cells,  , is defined based on the grid cell-volume   3 V . Based on the specifications of the mesh utilized for DDES simulations, the cut-off frequency normalized by the rotation frequency of the runner is obtained as f c f runner  60 . By moving towards the exit of the draft tube, i.e. for z Rref.  4.8 and 7.2 , it is clear in figure (6.10) that these two spectrums are better matched on the whole frequency range although minor details of the spectrum are different. In general, it can be concluded that the unsteady detailed features of the flow is also modified by an amplification of the inflow vt in the WZ.

154

5 3

E (u z ) : Original vt in WZ : Amplified vt in WZ

f f runner

a) z Rref.  2.4

5 3

E (u z )

f f runner

b) z Rref.  4.8

5 3

E (u z )

f f runner

c) z Rref.  7.2 Fig 6.10 Energy spectra of the turbulent axial velocity at different axial positions on the axisymmetric center line of the draft tube: effect of inflow vt amplification in WZ To study further, the effect of inflow vt amplification in WZ in the transition part of  the draft tube ( 4.8  z Rref.  9.0 ), the variation of utotal versus y  is considered along the

three specified sampling-lines shown in figure (6.11).

155

Y Z

S1

S2

S3

 versus y  graph in the transition part Fig 6.11 Sampling-lines utilized to obtain utotal

S1

 utotal

: Original vt in WZ : Amplified vt in WZ

y

 utotal

S2

y

 utotal

S3

y

 versus y  on sampling-lines S1 , S2 and S3 extracted Fig 6.12 Variation of utotal from DDES simulations: effect of inflow vt amplification in the WZ

As adopted in the previous chapter, utotal is a measure of velocity defined as the resultant of us and u components of the mean velocity, as below: 156

utotal  us2  u2

(6.1)

where us is the wall-parallel component of the velocity, whereas u is the circumferential component of the velocity. In fact, the wall-normal component of the velocity is excluded in the calculation of utotal . As one can see in figure (6.11), the sampling-lines are perpendicular to the bottom surface of the draft tube and they lie in the mid-symmetry plane i.e. x  0 . The wall nodes of the sampling-lines S1 to S3 are positioned at z Rref.  4.8, 7.2 and 9.0, respectively. Figure (6.12) depicts the variation of

 utotal versus

y  on the sampling-

lines S1 , S2 and S3 . As one can see in the figure, amplification of the turbulent viscosity modifies the dynamics of the core flow. In the wall zone, the behaviour of two cases is similar except for the S2 sampling-line where minor difference is observed between the curves. 6.6.2 Effect of the discretization of the convective term

In this subsection, the effect of discretization of the convective term on the DDES simulation of the draft tube is examined to observe the effect of the upwind and linear schemes (section 3.3.2). The intermediate grid (i.e. mesh B) defined in table (6.2) is still used to perform these simulations at BEP. The draft tube inflow condition for the velocity and turbulent quantity in this case are also imposed as ‘1D circumferential-averaged’ profiles. Furthermore, for both simulation cases, the inflow vt is amplified with a 103 factor in the WZ bounded to the R R  0.02 of the shroud and hub walls. Two types of the discretization schemes are applied: 1) second order upwind scheme, ‘linearUpwindV’; 2) second order linear discretization scheme with local filtering high frequency staggering modes, namely ‘filteredLinear2V’. The details discretization schemes for other terms of the Navier-Stokes equation and solver settings can be found in appendix F.

namely of very of the control

Figure (6.13) shows the turbulent structures in the draft tube flow obtained from DDES simulations at BEP using the two-abovementioned discretization schemes along with the mean separated zone formed on the draft tube wall. The structures are extracted using Q-criteria (appendix E) with Q  500 and are colored by the normalized axial velocity. As one can see in the figure, the global pattern of the wall flow-separation is similar in both cases. Although, some minor differences are present in the separation topology. It is also clear that the type of discretization scheme affects the shape of vortex rope. In fact, for the ‘linearUpwindV’ scheme, the vortex rope is more stable and keeps its 157

integrity almost to the end section of draft tube, whereas in the case of the linear scheme with local filtering, namely ‘filteredLinear2V’, small scale structures are also resolved in the vortex rope region as a result of the vortex breakdown phenomenon. In other words, higher amount of numerical dissipation introduced by the upwind scheme causes damping of the small-scale structures as seen in figure (6.13).

uz uref .

Y x

a)‘filteredLinear2V’ scheme

z

b) ‘linearUpwindV’ scheme

Fig 6.13 Mean separation-zone (blue iso-surface) and coherent structures ( Q  500 ) in the DDES simulations: the effect of discretization scheme of the convective term uz uref .

Low speed region

a) ‘filteredLinear2V’ scheme Low speed spots

b) ‘linearUpwindV’ scheme Fig 6.14 Instantaneous axial velocity field on the draft tube mid-plane obtained from DDES simulations at BEP: the effect of discretization scheme of the convective term In general, it is believed that the unsteady dynamics of the coherent structures in the turbulent flow determines the motion of the fluid particles and equivalently the velocity field. In the case of draft tube flow DDES simulations, the difference in the pattern of structures affects the velocity field mainly in the vortex rope region as shown in figure 158

(6.14). This figure shows the velocity field on the mid-plane of the draft tube (i.e. x  0 ) at one instant of time. As one can see, the velocity field pattern is modified in the core flow in the vortexrope region. In the case of the ‘linearUpwindV’ scheme, successive low speed spots are formed at the inter-section of the spiral vortex-rope and the mid-plane x  0 , which extends farther to the draft tube exit. Whereas, in the case of the ‘filteredLinear2V’ scheme, instead of the individual spots, a single low speed region is formed in the core flow. For this test case, this low-speed region is induced by the complex turbulent structures present in the core flow.



: filteredLinear 2V

Sw

: linearUpwindV

: linearUpwindV : filteredLinear 2V

z Rref.

z Rref.

Fig 6.15 Draft tube recovery coefficient (left) and swirl number (right) evolution in zdirection of DDES simulations at BEP: the effect of convective term discretization scheme As explained in the above discussion, the details of the instantaneous flow field are different depending on the type of convective term discretization scheme and one can imagine that it has an impact on the mean flow quantities. Figure (6.15) depicts the variation of draft tube coefficient and the swirl number evolution in the conical part in the streamwise direction. As one can see in the figure, the evolution of the quantities is very similar for both discretization schemes. This shows that although the details of the unsteady turbulent flows are modified, especially in the core flow, the mean flow draft tube coefficient evolution in the whole draft tube and the swirl number in the conical part are kept unchanged (figure 6.15). Figure (6.16) also shows the 1D circumferential-averaged profiles of the mean velocities at 4 different sections of the conical part for both the ‘filteredLinear2V’ and ‘linearUpwindV’ schemes. As one can see, they are very similar although minor differences between the cases are observed near to the centerline of the conical part, where the resolving resolution of the vortex rope varies.

159

: ur uref . , filteredLinear 2V : u uref . , filteredLinear 2V : u z uref . , filteredLinear 2V

u uref .

: ur uref . , linearUpwindV : u uref . , linearUpwindV : u z uref . , linearUpwindV

u uref .

r Rref.

r Rref.

a) z Rref.  1.4

b) z Rref.  2.4 u uref .

u uref .

r Rref. c) z Rref.  3.3

r Rref. d) z Rref.  4.3

Fig 6.16 1D circumferential-averaged of mean velocity profiles in the conical part in the DDES simulations at BEP: the effect of convective term discretization scheme To further investigate the unsteady nature of the flow, the energy spectrum of the turbulent axial velocity for 3 probes, located at different streamwise positions on the draft tube center-line is plotted in figure (6.17). As the figure suggests, the level of energy resolved by two different schemes is different. The energy level is higher in the case of ‘filteredLinear2V’ scheme as expected a-priori, since it resolves more structures in the vortex-rope region. Both discretization schemes respect the slope of -5/3 in the energy decreasing process in the inertial range of the turbulence spectrum. The frequency of the vortex-rope rotation in the case of the ‘linearUpwindV’ scheme is identified by the first peak in the energy spectrum of the velocity signal around f f runner  0.7 . The peak is also observed in the case of the ‘filteredLinear2V’ scheme but with smaller amplitude; in addition, the scheme smoothes out the other peaks. 160

5 3

E (u z )

: linearUpwindV : filteredLinear 2V

f f runner

a) z Rref.  2.4 E (u z )

5 3

f f runner

b) z Rref.  4.8 E (u z )

5 3

f f runner

c) z Rref.  9.0 Fig 6.17 Energy spectrums of the turbulent axial velocity at different positions on the axisymmetric center-line of draft tube: effect of convective term discretization scheme As before, the effects of discretization scheme of the convective term in the  transition part of the draft tube ( 4.8  z Rref.  9.0 ) is also considered. The variation of utotal

versus y  along three specified sampling-lines (shown in figure 6.11) are plotted in figure (6.18). As one can see at all three sections, the behavior of the total mean velocity in wallunits is the same in the wall region and the core flow and some negligible differences are observed for y   103 ; therefore, the effect of the convection scheme type is mostly limited to the core flow region.

161

Overall, the results in this section show that the core flow unsteady details are considerably modified by the choice of convective term discretization scheme. In general, for DDES simulations like pure LES simulations, upwind schemes should be avoided due to the high numerical dissipation that these methods introduce in the solution.  utotal

S1 : linearUpwindV : filteredLinear 2V

y

 utotal

S2

y

 utotal

S3

y

 Fig 6.18 Variation of utotal versus y  on sampling-lines S1 , S2 and S3 extracted from the DDES simulations: the effect of convective term discretization scheme

6.6.3 Grid independence test

In this section, the effect of grid resolution on the solution of DDES numerical simulations is considered. Due to the high computational cost of the DDES simulations to perform grid convergence test, only two sets of intermediate and fine grids are adopted i.e. mesh B with 7.48 millions cells and mesh C with 14.87 millions cells, respectively, as listed in table 162

(6.2). In these simulations, the draft tube inflow condition for velocity and turbulent quantity are also imposed as ‘1D circumferential-averaged’ profiles. Furthermore, in both simulation cases, the inflow vt is amplified with a 103 factor in the WZ bounded to R R  0.02 from the shroud and hub walls.

The convective term is discretized by the second order linear discretization scheme ‘filteredLinear2V’. The details of discretization schemes for other terms of the NavierStokes equations and the solver control settings are similar to the previous cases and can be found in appendix F. uz uref .

Y x

a)Mesh C- fine grid

z

b) Mesh B- intermediate grid

Fig 6.19 Mean separation-zone (blue iso-surface) and coherent structures ( Q  500 ) in the DDES simulations: grid independence test It is worth mentioning that the mesh convergence concept in the case of LES-type simulations is different than with RANS simulations. In the case of LES-type simulations, by increasing the grid resolution, finer structures are resolved and certainly flow details become different. In fact, for well-converged grids, one can be satisfied having small variations in the global features of the flow especially for engineering purposes, like draft tube recovery coefficients. Figure (6.19) shows the turbulent structures resolved by the intermediate and fine grids colored by the normalized axial velocity. The figure also depicts the mean separated zone formed on the draft tube wall. Despite the similarity observed in the resolved turbulent structures in the vortex-rope region for both cases, there are some differences in the pattern of the core structures. In addition, the mean flow separation in both cases is limited to the four corners. The overall pattern of the separated zones is quite similar although minor differences are also observed between grids B and C. To investigate further, the flow field for both cases, the evolution of draft tube coefficient and the swirl number are plotted in figure (6.20). As it is clear in the figure, only 163

a minor difference exists between the draft tube recovery coefficient (  ) curves for the two cases; the maximum offset in  quantity between the two curves is 2.5%. Furthermore, the mean loss coefficient of the draft tube i.e.  L is equal to 0.035 and 0.037 for the intermediate and fine grids, respectively, which represents a difference of 5% between the two cases for the loss coefficient. Considering BEP, the losses in the draft tube is 20% of the global turbine loss. Therefore, the difference between the two grid sets is small for an engineering point of view as about 1% of the global turbine loss.

 : Fine grid : Interm. grid

Sw

: Fine grid : Interm. grid

z Rref.

z Rref.

Fig 6.20 Draft tube recovery coefficient (left) and swirl number (right) evolution in the streamwise direction of the DDES simulations at BEP: grid independence test The RHS of figure (6.20) shows that the evolutions of swirl number in the conical part of the draft tube are very similar except in the range of 1.4  z Rref .  3.3 , where a minor difference between the two cases is observed. To investigate the unsteady details of the flow, as in the previous sections, the energy spectrum of turbulent axial velocity at three selected probe positions on the axisymmetric centerline of the draft tube is shown in figure (6.21). As one can see in the figure, some differences are observed between the meshes B and C. It is clear that in general the level of energy in the case of fine mesh is a little higher than in its counterpart. This behavior being linked to this fact that, more structures are resolved in the simulation with the finer mesh. The difference in the topology of turbulent structures resolved by the intermediate and fine grids in the core flow also causes a mismatch of the relative peaks between the energy spectra of the both cases. To compare the behavior of flow in the trumpet part of draft tube, the mean velocities expressed in wall-unit are also compared on the sampling-lines S1 , S2 and S3 in figure (6.22). As one can see in the figure, the two solutions are almost identical on the S1 sampling-line in the range of y   103 and for S2 and S3 sampling-lines in the range

164

of y   102 , whereas outside these ranges, some minor deviations exist between the two cases. Considering the minor differences observed in the global and detail behavior of the draft tube simulations with the grids B and C and also taking into account the high computational cost of the DDES simulations of the draft tube, mesh B is adopted for the upcoming simulations in this chapter.

5 3

E (u z ) : Fine grid : Interm. grid

f f runner

a) z Rref.  2.4

5 3

E (u z )

f f runner

b) z Rref.  4.8

5 3

E (u z )

f

f runner

z Rref.  7.2

Fig 6.21 Energy spectra of turbulent axial velocity at different axial positions on the axisymmetric center-line of the draft tube: grid independence test 165

 utotal

S1 : Fine grid : Interm. grid

y  utotal

S2

y

 utotal

S3

y

 versus y  on the sampling-lines S1 , S2 and S3 extracted Fig 6.22 Variation of utotal from the DDES simulations: grid independence test

6.6.4 Unsteady 2D inflow profile

In this section, the turbulent flow in the draft tube with ‘basic geometry’ is simulated using unsteady 2D inflow profiles. As mentioned before, one of the objectives of this project is to resolve runner-related vortical structures and wakes and to study its effect on the fluid flow details. As explained in chapter 4, these structures can be captured by taking into the account the unsteady 2D variation of the inflow velocity and turbulent quantity profiles. The unsteady 2D inflow profiles can be obtained by the rotation of 2D mean profiles extracted from the steady RANS simulation at the inlet of draft tube with the rotation frequency of runner. This approach is called ‘2D rotating’ inflow profile, hereafter. As also discussed in details in chapter 4 for the ‘final geometry’ at OP.4, synthetic artificial fluctuations which mimic the inflow turbulence can be generated based on the 2D 166

fields of the integral turbulent quantities using artificial fluctuation generation (AFG) techniques. The procedure adopted to generate the inflow was explained in details in section (4.4.3). As mentioned before the explained strategy generates isotropic inflow turbulence; in other words, the generated signals in three spatial directions have the same amplitude. In the case of the ‘basic geometry’, due to the lack of experimental data to scale the fluctuations, the following assumption is applied for the relative amplitude of fluctuations in the different directions of the space: u x2  u y2  0.1 u z2

(6.2)

By adding these synthetic turbulent velocity fluctuations on top of the ‘2D rotating’ inflow profile, the most complete inflow condition for the velocity in the present thesis is obtained involving unsteady synthetic fluctuations in the LES zone. This profile is called hereafter, ‘2D rotating+AFG’ inflow profile. Figure (6.23) shows the generated synthetic inflow turbulence frozen at one instant of time on the grid with 100  100 resolution at the inflow section. As one can see in figure (6.23), the level of fluctuations in the z-direction is 10 times higher than in the x and y directions as imposed by equation (6.2); it was also explained in details in chapter 4 that in the case of ‘final geometry’ the generated turbulent fluctuations are scaled based on the experimental data using ANN technique. Four regions with relatively high amplitude of the turbulent fluctuations are present in figure (6.23) exhibiting high turbulence activity in the runner-blade wakes. To observe the time-evolution of the generated AFG turbulent velocity signals, a probe is placed at ( x Rref . , y Rref . )  (0.39, 0.07) at the inlet plane. Figure (6.24) shows the obtained turbulent fluctuation signals in the x, y and z directions of space at the probe location. As intended the signals exhibit the unsteady nature of the inflow turbulence with different amplitude levels. As also explained in chapter 4, the successive peaks in the fluctuation levels of the signals are linked to the fact that by the runner rotation, the highamplitude runner-wake zones pass successively over the probe point. In this section, the turbulent flow in the draft tube with the ‘basic geometry’ is simulated by DDES turbulence treatment using unsteady 2D inflow profiles at BEP. The 2D unsteady inflow profiles are fed into the draft tube by ‘timeVaryingMappedFixedValue’ boundary condition for velocity and turbulence quantities at the inlet patch. The details of boundary conditions on the other patches can be found in appendix F. The base 2D inflow fields utilized for the rotation are extracted from full-machine RANS simulation. Here, the inflow turbulent viscosity, vt , is not amplified in the WZ and kept untouched as its original

167

values, whereas the viscosity in the core flow is reduced by the turbulent viscosity method explained in chapter 4; as shown vt is reduced 2 orders of magnitude in the core flow.

103

103

u y

u x uref .

uref .

r Rref .

r Rref .

r Rref .

r Rref .

u z uref .

r Rref .

r Rref .

Fig 6.23 A snapshot of the generated artificial fluctuations in different spatial directions in the DDES simulation of ‘basic geometry’ using ‘2D rotating+AFG’ inflow profile Similar to the case with the ‘1D inflow profile’, the governing equations are solved using ‘pimpleFoam’ solver with time step equals to 0.63 runner rotations on the intermediate grid (i.e. mesh B) presented in table (6.2). As shown in the previous sections, the ‘linearUpwindV’ scheme introduces too much dissipation in the solution and should be avoided. For comparison purposes, here also the convective term is discretized using both ‘filteredLinear2V’ and ‘linearUpwindV’ schemes to see the effects in the case of unsteady 168

2D inflow profiles. The details of the discretization schemes for other terms of the NavierStokes equations and the solver control settings are similar to the previous cases and can be found in appendix F. u uref .

z y x

time (s)

0.006 u uref .

0.006 4

time (s)

4.1

Fig 6.24 Time-evolution of the turbulent fluctuation signals in different spatial directions adopted for the DDES simulation of ‘basic geometry’ with ‘2D rotating+AFG’ profile Figure (6.25) shows the coherent structures and the mean wall-separation zone for both discretization schemes. Too much dissipation in the upwind scheme damps the structures more rapidly after the inlet section. The coverage and position of the mean separation zones for both cases are comparable although some minor differences are observed between two cases. Since the inflow vt is not amplified in the WZ, a vast separation zone is formed on the draft tube wall as previously observed in the case of ‘1D inflow profile’ (figure 6.6-part b). The instantaneous separation zones defined by u z  0 are observed in figure (6.26) for both discretization cases. Some strips of separated flow ( u z  0 ) are formed on the draft tube wall close to the inlet section, marked by red dashed-circles on the figure. These structures are induced by the proximity of runner blade-tip vortical structures to the draft tube wall. By moving downstream, the distance between tip-vortices and the wall increases, which in turn avoids formation of the separation later on. It is interesting to notice that, in the mean flow field, no mean separation exists close to the inlet plane (figure 6.25), whereas in the instantaneous field, flow separation is observed close to the inlet plane (figure 6.26). In fact, by rotation of the runner, these separated strips also rotate and in average, no mean separation forms close to the draft tube inlet.

169

uz uref .

Y z

x

a) ‘filteredLinear2V’ scheme

Y z

x

b) ‘linearUpwindV’ scheme 6.25 Coherent structures with Q  350 (left) and mean separation zone (right) obtained from DDES simulations with ‘2D rotating+AFG’ inflow profile

Y x

a)‘linearUpwindV’ scheme

z

b) ‘filteredLinear2V’ scheme

6.26 Instantaneous flow separation obtained from DDES simulations of the ‘basic geometry’ with ‘2D rotating+AFG’ inflow profile

170

Figure (6.27) shows the axial velocity field on the symmetry plane ( x  0 ) of the draft tube obtained from the simulation with ‘2D rotating+AFG’ inflow profile for both discretization schemes. The results for the simulations with ‘1D’ inflow profile are replotted here for comparison purposes. As one can see in the figure, the runner-related structures and wakes resolved in the case of simulations with ‘2D rotating+AFG’ modify the velocity field considerably compared to the cases with ‘1D’ inflow profiles. The difference between ‘2D rotating+AFG’ and ‘1D’ cases is more pronounced in the initial part of the conical section of draft tube, where some vertical strips are formed in the axial velocity field in the case of ‘2D rotating+AFG’ case. These patterns, induced by resolved vortical structures, are removed in the case of ‘1D’ inflow profile. uz uref .

a) ‘2D rotating+AFG’ inflow; ‘linearUpwindV’ scheme

b) ‘2D rotating+AFG’ inflow; ‘filteredLinear2V’ scheme uz uref .

c) ‘1D’ inflow; ‘linearUpwindV’ scheme

d) ‘1D’ inflow; ‘filteredLinear2V’ scheme Fig 6.27 Instantaneous axial velocity field on the draft tube mid-plane ( x  0 ) extracted from the DDES simulation using ‘2D rotating+AFG’ and ‘1D’ inflow profiles at BEP The generated strip pattern by the unsteady 2D profile is linked to the presence of spiral vortices generated by the runner. In the literature, a similar pattern was also observed

171

in the axial velocity field in the wake of a marine propeller [Muscari et al. 2013], although the propeller in that study is located in a free flow, i.e. with no surrounding walls. The case is different with the case of BulbT configuration, where the runner blades are confined by the shroud wall. To further study the effect of applying unsteady 2D inflow profiles on the global behavior of the draft tube, the evolutions of draft tube coefficient (  ) and swirl number ( S w ) in the streamwise direction are plotted in figure (6.28) . Four different cases are compared including ‘1D’, ‘2D rotating’ and ‘2D rotating+AFG’ inflow profiles with ‘filteredLinear2V’ scheme, and ‘2D rotating+AFG’ inflow profile with the upwind scheme. As one can see in the figure, the draft tube coefficient evolution is similar for all cases, although minor differences are observed between ‘1D’ and ‘2D’ cases. It is also clear that considering AFG has a minor effect on the draft tube recovery coefficient as an integral quantity in the case of the ‘basic geometry’ at BEP.

: "2D rotating" , scheme : filteredLinear 2V : "2D rotating  AFG" , scheme : filteredLinear 2V : "2D rotating  AFG" , scheme : linearUpwindV : "1D profile" , scheme : filteredLinear 2V

 Sw

z Rref.

z Rref.

Fig 6.28 Draft tube recovery coefficient (left) and swirl number (right) evolution in the streamwise direction of the DDES simulations at BEP: ‘2D’ vs. ‘1D’ profiles The RHS of figure (6.28) depicts the evolution of the swirl number in the conical part of the draft tube. The trends for ‘1D’ and ‘2D’ cases are different. In fact, S w increases more in the case of ‘1D’ profile, due to the relative decrease of the axial velocity in the core flow (figure 6.29). As one can also see, the type of convective term discretization scheme, and adding AFG, have negligible effect on the swirl number evolution in the case of DDES simulations of the ‘basic geometry’ with unsteady 2D inflow profiles. 172

Figure (6.29) shows the 1D-circumferential averaged profile of axial, radial and circumferential components of the velocity at the two cross-sections in the conical part. It is apparent that the difference among the profiles is distributed over the full-radius. : ur : u : uz : ur : u : uz

uref . , "2D rotating" , scheme : filteredLinear 2V uref . , "2D rotating" , scheme : filteredLinear 2V uref . , "2D rotating" , scheme : filteredLinear 2V uref . , "2D rotating  AFG" , scheme : filteredLinear 2V uref . , "2D rotating  AFG" , scheme : filteredLinear 2V uref . , "2D rotating  AFG" , scheme : filteredLinear 2V

: ur : u : uz : ur : u : uz

uref . , "2D rotating  AFG" , scheme : linearUpwindV uref . , "2D rotating  AFG" , scheme : linearUpwindV uref . , "2D rotating  AFG" , scheme : linearUpwindV uref . , "1D profile" , scheme : filteredLinear 2V uref . , "1D profile" , scheme : filteredLinear 2V uref . , "1D profile" , scheme : filteredLinear 2V

u uref .

u uref .

r Rref.

a) z Rref.  1.4

r Rref.

b) z Rref.  3.3

Fig 6.29 1D circumferential-averaged of mean velocity profiles in the conical part obtained from DDES simulations at BEP: ‘2D’ vs. ‘1D’ profiles To observe the unsteady details of the flow, the energy spectrum of the axial velocity signal at three different probe positions located on the axisymmetric centerline of the draft tube are plotted in figure (6.30). In the low frequency range corresponding to the large scale phenomena, there is a difference between simulation cases with ‘1D’ and ‘2D’ inflow profiles: in fact, the level of energy is lower in the case of ‘1D’ inflow profile. This behavior is linked to the fact that more flow structures are resolved in the case of ‘2D’ inflow profile compared to the ‘1D’ case. In the relatively high frequency range i.e.10  f f runner  60 , the energy among cases better matches especially for the two later probe positions. As before, the red dashedline on the graphs indicates the grid cut-off frequency. As mentioned before, in the case of ‘basic geometry’, due to the lack of experimental data comparative simulation cases were only considered to grasp an initial insight into the crucial parameters affecting the DDES simulations of draft tube; furthermore, the numerical strategies and developed tools to perform simulations with ‘1D’, ‘2D rotating’ and ‘2D rotating+AFG’ inflow profiles were tested. In the next chapter, for ‘final geometry’ of the draft tube more details of the unsteady features like two-point 173

correlations along with performing the comparisons to the LDV and PIV experimental data are presented in details. : "2D rotating" , scheme : filteredLinear 2V : "2D rotating  AFG" , scheme : filteredLinear 2V : "2D rotating  AFG" , scheme : linearUpwindV : "1D profile" , scheme : filteredLinear 2V

5 3

E (u z )

f f runner

a) z Rref.  2.4

5 3

E (u z )

f f runner

b) z Rref.  4.8 5 3

E (u z )

f f runner

c) z Rref.  7.2 Fig 6.30 Energy spectra of the turbulent axial velocity at different axial positions on the axisymmetric center-line of the draft tube: ‘2D’ vs. ‘1D’ profiles

174

6.7 Summary

This chapter presented the comparative simulation results of the ‘basic geometry’ of the draft tube at BEP. The geometry involves 6.6  and 2.1 divergence angles for the conical and transition parts, respectively. In this case, no experimental data is available, however according to the k   simulation results at BEP, no wall-separation is expected for this configuration with relatively mild opening angles. The aim of this chapter is to examine the proposed simulation strategy as well as to test different factors including inflow conditions. In addition, the goal is to see their impacts on the global quantities like the recovery coefficients and swirl number in a draft tube geometry with less aggressive divergence angles compared to the final geometry. The major points of this chapter can be summarized, as follows:



In the conical part of the draft tube, the absolute value of the swirl number S w increases in the streamwise direction for all simulations, including RANS and DDES. This is linked to the higher decrease rate of axial velocity compared to the attenuation rate of circumferential velocity.



Topology of the separated zone is different for S-A and k   RANS simulations. No separation is formed on the draft-tube wall for the k   case, whereas in the SA case, separation zones are formed on the two opposite corners of draft tube wall. The recovery coefficient is slightly under-predicted in the case of the S-A.



Amplification of the inflow vt in the WZ attenuates the wall-separation in the DDES simulations and limits it to four corners of the transition part. The recovery coefficient is slightly modified by the amplification. As revealed by the energy spectrum analysis of turbulent velocity signals, the detail features of the flow in the core are also modified by an amplification of the inflow vt in the WZ.

 Discretization scheme of the convective term affects the vortex rope pattern. The filtered linear scheme, ‘filteredLinear2V’, introduces lower amount of numerical dissipation in the solution resolving more coherent structures compared to the upwind scheme ‘linearUpwindV’. The recovery coefficient, swirl number and mean flow profiles are practically the same for both schemes, although details of the flow are considerably different, as shown in the energy spectrum.  Considering variations of the global engineering quantities and local features of the flow in the grid independence test, the intermediate mesh with about 7.5 million elements was adopted for further DDES simulations of the draft tube. The 175

difference between the recovery coefficient for the intermediate and fine grids at BEP is about 2.5%, whereas it is about 1% of the global turbine loss for the loss coefficient.  Applying unsteady 2D inflow profiles provides a more realistic representation of inflow boundary conditions and allows resolving the runner-related vortical structures and wakes. In this regard, two sets of unsteady 2D profiles with and without synthetic inflow turbulence were utilized for the DDES simulations including ‘2D rotating+AFG’ and ‘2D rotating’, respectively. By comparing the results of the simulations with ‘1D’ and ‘2D’ inflow profiles, the following observations were made: o Both mean and instantaneous velocity fields are affected by the type of inflow boundary conditions. Essentially, the structures ejected by the runner into the draft tube are resolved in the cases with ‘2D’ inflow profiles, whereas ‘1D’ profile unrealistically smooth out these structures. o The evolution of the draft tube recovery coefficient is practically insensitive to the type of inflow profiles due to its integral nature. However, the swirl number exhibits dependency on the inflow type. o Detail features of the flow as revealed by the energy spectrum analysis also depend on the inflow type. On the other hand, the upwind scheme produces lower energy level in the spectrum as expected.

176

Chapter 7

BulbT draft tube flow simulations: final geometry

7.1

Introduction

In this chapter, final version of the draft tube geometry is adopted to simulate the flow fields. It involves more aggressive divergence angles in the conical and transition parts in comparison to the basic version. In contrast to the ‘basic geometry’, for the final version of the geometry, PIV and LDV measurements performed at LAMH test bench are available, which enables us to adjust and validate the simulations as explained in details in the following sections. At the beginning, the RANS/URANS simulations are performed aiming to have a general overview about the global performance of draft tube and also to study the capability of the aforementioned techniques to capture the correct topology of the fluid flow. Then the major part of the chapter is dedicated to the DDES simulations. Different types of boundary conditions with and without experimental corrections are considered and their effects on the draft tube flow simulations are studied. Furthermore, the impact of the inflow synthetic turbulence is studied in details.

7.2

Final draft tube geometry and selected operating points

In the final version of the geometry, the divergence angles are 10.25 and 5 in the conical and transition parts, respectively (figure 7.1). As it is obvious, these divergence angles are higher than the corresponding angles in the basic geometry of the draft tube which is 177

directly translated to higher adverse pressure gradients (APG). As a result, one can expect more dominant separation regions and APG-linked phenomena like vortex break down in the final version of the draft tube in comparison to the basic one. Due to the high computational cost of each DDES simulation and large number of simulations that should be performed to study the effects of different types of inflow conditions, two operating points are selected in this study including OP.1 and OP.4.

Y

Z

Bulb

10.25 5

Guide Vanes Runner Fig 7.1 Final geometry of the draft tube with its geometrical divergence angles Figure (7.2) shows the selected operating points. It is worth mentioning that the results, both experimentally [Duquesne et al. 2014-1] and numerically obtained from full machine simulations of the turbine using k   RANS [Houde et al. 2014] showed that the turbine exhibits a hysteresis behavior through opening and closing of the guide vanes. In figure (7.2), the CFD curve is the averaged curve of opening and closing curves obtained from k   full machine simulations. Table (7.1) presents the details of the aforementioned chosen operating points. Table 7.1 Details of the selected operating points for numerical simulations

OP

GV 

1 4

61.3 65.4

Net head (m) 4.008 4.011

Runner rotation speed (1/s) 1001.4 1002.4

Mass flow rate (m3/s) 0.5302 0.5482

N11 170.07 170.21

The first operating point (OP.1) is selected to validate the simulation results in the case of no separation. In this case, no separation was observed experimentally using PIV measurements on some pre-defined planes [Duquesne et al. 2014-3] and tuft visualization

178

covering a vast portion of draft tube wall-surface in both the conical and transition parts [Duquesne et al. 2014-2]. As it is clear in figure (7.2), this operating point, i.e. OP.1, is an operating point just before the best efficiency point (BEP). In fact, the BEP was not selected because at BEP, a small zone of separation was observed experimentally using tuft visualizations [Duquesne et al. 2014-2]. As seen shortly in this chapter, the main challenging problem here, in the DDES simulations of draft tube flow, is to reproduce the separation zone on the draft tube walls; that is why it is important to consider OP.1 as an important check point and as a ‘tuning’ case as well. At the other operating point, i.e. OP.4, break-off has already happened in the turbine power and efficiency and a dominant separation zone is formed on the draft tube wall, as observed experimentally [Duquesne et al. 2014-2]. As stated earlier, the reference coordinate system is considered as a runner-defined coordinate system (figure 7.1), where the origin of the coordinate is positioned on the runner at a predefined location, the Z axis is in the streamwise direction pointing downstream, Y axis is upward or in other words in opposite direction of the gravity force and finally X axis is set to form a right-handed coordinate system. In the following sections, details on the computational techniques adopted to simulate the draft tube flow and obtained results are presented.

OP.1 OP.4



Guide vane opening angle o

Fig 7.2 Selected operating points on the turbine efficiency curve

179

7.3

Available experimental data

Flow field inside the draft tube of bulb turbine in the BulbT project was studied in details experimentally on the test bench available at LAMH [Duquesne 2015, Vuillemard 2015]. The test loop is a test-rig suitable to perform measurements on different types of reaction hydro-turbine models up to 0.5 m in diameter with a maximum head of 50 m, a maximum discharge rate of 1 m3 s and a maximum runner rotation speed of 2000 RPM. The maximum acceptable torque and power outputs of the turbines on the test bench are limited to 1100 NM and 225 KW, respectively. These specifications make possible to study the model scale BulbT turbine experimentally. In one experimental measurement campaign, turbulent velocity fields at two different locations for the five different selected operating points of the project, were measured using the LDV technique: one location at the inlet of the draft tube (i.e. plane A with Z Rref .  0.38 ) and the other, right after the hub (i.e. plane B with Z Rref .  0.96 ) [Vuillemard 2015]. In the procedure, unsteady turbulent velocity signals are measured on two different axes with azimuthal 0 and 180 (figure 7.3). By acquisition of the runner blade position at the instants of the sampling and also considering this assumption that the only factor responsible for changes in the flow is the runner, one can reconstruct the phase average 2D profiles. Also for all five selected operating points, pressure signals were measured experimentally using wall pressure sensors [Duquesne et al. 2014-2]; in figure (7.3) the positions of pressure sensors are indicated. The measured pressures are used to calculate the recovery coefficient of draft tube at different operating points.

Pressure sensors

Plane A

Plane B

Az.0

Fig 7.3 Pressure sensor positions and LDV measurement axis

180

Az.180

In another measurement campaign, the presence of separation zones at the selected operating points was detected using the tuft visualization technique [Duquesne et al. 20142]. Furthermore, to investigate the dynamics of flow in the draft tube, turbulent velocity signals were measured on 4 different 2D-planes using PIV techniques close to the walls at the separation zone [Duquesne et al. 2014-3]. Figure (7.4) shows the position of the PIV measurement planes, namely Plane B1, B2, S3 and S4. As it is depicted in the figure, planes B1 and B2 are located at the bottom of the draft tube, although B1 is closer to the wall than B2. Also plane S3 and S4 are positioned on the side wall of the draft tube where the experimental separation was observed. It is worth to notice that each plane consists of an upstream part and a downstream part on which the turbulent velocity measurements are performed separately.

Fig 7.4 PIV-measurement planes namely B1, B2, S3 and S4 [Duquesne et al. 2014-3] The measurement data coming from the PIV-planes and LDV data are used for validation and in the case of DDES simulations are also adopted as check points to define the simulation strategy using OP.1 data. It is also important to notice that these PIV-planes are further downstream in comparison to the LDV plane (i.e. plane B); they are better quantities to examine the quality of simulations in the downstream flow and provide deeper insight into the dynamics of the simulated draft tube flow. Moreover, these planes, especially the downstream parts of the planes, are close to the separated zone or in the separation depending on the operating point selected. Therefore, they can show the flow behavior more clearly in these complicated zones. For OP.1, no separation on the side walls or any specific fluid dynamic phenomenon was observed, so the PIV-measurements were limited to the plane B1. But, at OP.4, the turbulent velocity fields were measured for all four aforementioned planes. At

181

this operating point, the measurements show a dominant separation on the draft tube side wall starting from the cone section. Due to the lack of measurements in the core flow especially at the draft tube exit for OP1 and OP4, it is not possible to validate the presence of central separation bubble, experimentally. In the following sections and subsections, the numerical details of the strategy applied for the simulations of turbulent flow in the BulbT draft tube along with the obtained results and discussions are presented.

7.4

Computational grid specifications

In general, different simulation techniques dictate their own demands in the design step of the computational mesh. As mentioned, some wall resolving techniques like SpalartAllmaras (S-A) and DDES need fine mesh in wall normal direction i.e. y   O(1) . In the case of pure LES and DNS also some limitations should be applied in other direction ( x  , z  ) on the mesh generation to resolve important turbulent structures properly (see chapter 2). In this study, two sets of grids are generated with hexahedral elements: mesh A for k   RANS/URANS simulations and mesh B for S-A or DDES simulations. In the computational domain as explained in chapter 4, a part of the rotating hub is included in the computational domain. An O-grid is utilized near the hub to capture the curve wall of the hub properly, while avoiding creation of low quality elements, for example, with a small internal angle. As well for better capturing the details of the internal hydraulic surface of draft tube, different blocks with different resolutions and cell distributions are used in the streamwise direction for different parts of the draft tube, including the initial and second parts of the conical section, the first and second mid of the transition part (trumpet) and finally the extension part. In appendix D, one can find details of the generated grid topology for the k   and DDES simulations. The main difference between these two sets of mesh is the resolution in the near wall zone dictated by the y  demand for each of turbulence treatment techniques. For the mesh used for k   simulations, the first off-wall node should be placed in the logarithmic layer of the boundary layer ( 30  y   300 ); in fact, the turbulence treatment method belongs to the high-Reynolds number models. In these models, the low-speed viscous and buffer layers are not computed through the main computational procedure, but are estimated with a wall function. On the other hand, low-Reynolds number models like S-A integrates to the wall and needs much finer mesh in the area close to the solid walls y   O(1) ; this considerably increases the number of cells and in turn the computational

effort. In contrast to pure LES or DNS, in the case of DDES and S-A, only the wall normal 182

direction is of main importance in clustering process near the wall. In principle, less restriction is imposed for other two directions [Spalart 2000]. By turning our attention to the core flow, it is worth mentioning that in the case of DDES simulations of the draft tube, this zone, far from the boundary layer, is already in the LES mode after the runner. On the other hand, it is clear that the ideal cell shape for pure LES simulations is a cube (appendix A1). Therefore, to have an appropriate mesh for DDES simulations in this case one should have a continuous transition from clustered nodes near the wall to cubic elements in the core. In this study, the grid generated for DDES simulations (mesh B) respects both characteristics consisting of a condensed mesh close to the wall respecting y  demands and a large percentage of cubic elements in the core flow (appendix D). Table 7.2 Grid statistics for k   (mesh A) and S-A, DDES simulations (mesh B)

Mesh A No. of elements (Millions) Min. Angle Min. Determinant Max. Aspect Ratio Max. growth factor

25 0.755 62.8 1.1

Mesh B No. of elements (Millions)

7.26

Min. Angle Min. Determinant Max. Aspect Ratio Max. growth factor

25.2 0.739 965 1.2

2.5

Table (7.2) shows the statistics of the computational grids for the upcoming performed simulations in this chapter. As mentioned, the values of the characteristic parameters such as angle, determinant and aspect ratio should ideally be equal to 90 , 1 and 1 respectively, representing an equidistance edge orthogonal hexahedral element (i.e. cube) which introduces minimum numerical error in the computational procedure. It is evident that for complex geometries and especially in the boundary layer region, there are always some deviations from this ideal situation. However, the designed grid should still respect some constrains on these qualification criteria to minimize the computational errors depending on the type of solver. For example, high aspect ratio cells are normally formed

183

in the node clustering process in the wall normal direction within regions close to the walls, especially in the case of low-Reynolds number models. In the case of OpenFOAM 1.6-ext code, having double precision calculations, one can have an aspect ratio limit up to 1000, the angle for complex geometries can be higher than 25 and the minimum acceptable value of the determinant parameter for draft tube calculations is bounded to 0.5. In addition, limiting the cell growth factor to 1.2 ensures avoiding sharp increase in the cell sizes between neighbor cells, which in turn avoids generation of false local flow features like artificial vorticity. By respecting the entire aforementioned criteria, in addition to respecting y  and performing mesh convergence test, one can be confident about the limited level of numerical error introduced by the grids.

7.5

Numerical setup for turbulent flow simulations

The numerical setup adopted here for turbulent flow simulations of the final geometry is similar to the one adopted in the case of the ‘basic geometry’, given in chapter 6. Appendix (F) depicts the details of the numerical discretization schemes and the solution procedure used for all simulations in this thesis. Depending on the type of simulations, three different inlet boundary conditions, including circumferential-averaged 1D profile, 2D-rotating profile and 2D-rotating profile plus synthetic inflow turbulence, are applied for velocity and turbulent quantities. In all cases, mass flow rate is kept constant equal to experimental one. In all cases, a mean pressure equals to zero is applied at the final exit of the draft tube. For all solid walls including the hub and draft tube wall, a ‘no-slip’ boundary condition is applied except for the dummy extension wall. The extension is added to avoid the recirculation at the draft tube exit section; on this wall, a ‘slip’ boundary condition is imposed. In the following sections, the specific settings applied for the discretization schemes or boundary conditions (if any) are mentioned in advance for each simulation.

7.6

Circumferential-averaged 1D inflow profile variants

As explained in chapter 4, to simulate the turbulent flows in the bulb turbine draft tube individually, the base inflow velocity and turbulent quantity profiles come from the full machine k   RANS simulations of the turbine [Houde et al. 2014]. It is worth to recall that the first off-wall node of the grid for high-Re turbulence models like k   is positioned farther from the wall in comparison to the one for low-Re turbulence treatments

184

like S-A or DDES. Therefore, the first issue is to reconstruct the inflow profile close to the wall. In the following subsection, the details of the reconstruction procedure are presented. On the other hand, by comparison of the draft tube inflow velocity profiles obtained from the experimental measurements (LDV) at LAMH and the base inflow profiles stemmed from the full-machine simulations, a mismatch was found at both selected operating points, OP1 and OP4. To remedy this situation, variant versions of the base profiles at the plane A are constructed. The details are presented in the following. 7.6.1 Inflow profile near-wall treatment

In the case of the full machine k   RANS simulation, the first grid node off the wall should respect the constraint 30  y   300 as explained earlier; while for low-Re turbulence treatments in the case of S-A and DDES simulations of draft tube, the first offwall node should rely on y   O(1) . Therefore, it can be postulated that feeding the draft tube with the base 1D profile stemmed from k   RANS simulation could be considered as a probable source of error for low-Re turbulence treatment methods demanding low y  like S-A and DDES (figure 7.5). In other words, although in the case of the draft tube just after the inflow section the near-wall profiles in the computational domain are adjusted accordingly while advancing the numerical computations; but one can reasonably have a concern that the anomaly created by a linear interpolation using 1D profile in the near wall zone at the inlet section near the shroud and hub can be considered as an unphysical numerical disturbance which in turn can affect the near-wall flow physics especially the separation onset (triggering/delaying). To compensate this shortcoming of the profile stemmed from k   RANS simulations, two different strategies can be adopted:

1. Simulation of the upstream component of draft tube including runner component, with a turbulence model resolving to the wall like S-A. This can theoretically provide an appropriate inflow profile up to y   1 . Although, as observed in chapter 4, the S-A is not completely validated for this purpose. 2. Reconstruction of the velocity and turbulent quantities (here turbulent viscosity vt for S-A and DDES) in the near-wall zone lower than the runner-shroud or runnerhub y  (figure 7.5-part b). In this study, the second approach is adopted. In this regard, the inflow profile gap between y   1 to the runner-shroud (or runner-hub) y  is reconstructed to feed the draft

185

tube simulations using inflow 1D profile (figure7.5). Figures (7.6) and (7.7) depict the 2D variation of the y  quantity on the runner-shroud and hub respectively, for one-blade passage just upstream of the draft tube component obtained from full machine k   RANS simulations at the two selected operating points. Table (7.3) shows the average value of

y  at the inlet section of the draft tube. As one can observe, the y  range is much higher than y   1 , so the velocity and turbulent quantity in the gap should be reconstructed. First off-wall node

y   30

y   30 Inflow profile treatment First off-wall node

y  1

b) Draft tube inlet section (in low-Re model zone)

a) Runner outlet section (in high-Re model zone)

Fig 7.5 Near-wall zone of inflow profile treatments for draft tube turbulent flow simulations using low-Re treatments e.g. S-A and DDES For the velocity component in the streamwise direction, different mathematical formulations in different constitutive layers of boundary layer- namely, viscous sub-layer, buffer and logarithmic layers- are applied. For viscous sub-layer and logarithmic layer,   utotal  y  and u total  (1 kV ) ln y   B is applied, respectively; where kV and B are

constants. Table 7.3 Average y  on the hub and shroud at inlet section of the draft tube

OP 1 4

Averaged y  Shroud 73.69 75.04

Hub 108.23 113.839

Due to the lack of experimental data close to the wall, a similar approach explained in chapter 5 is adopted. For the buffer layer ( 5  y   30 ), there is no explicit formula, but one knows a-priori that the velocity profile should be continuous with the same slope as other layers at the two ends. A polynomial function is introduced for the buffer layer with the following form: 186

y

y

o OP.1

z

x

OP.4

Fig 7.6 2D variation of y  quantity on the runner-shroud surface i.e. the upstream component of draft tube stemmed from full-machine k   RANS simulations y

y

o OP.1

OP.4

z

x

Fig 7.7 2D variation of y  quantity on the runner-hub surface i.e. the upstream component of draft tube stemmed from full-machine k   RANS simulations

187

 utotal

buffer

 a ( y  ) 2  b( y  )  c 

d y

(7.1)

where the constants are obtained by solving a simple algebraic equation using the following boundary conditions:  utotal

y  5

 dutotal dy 

  5, utotal

 1, y  5

y   30

 dutotal dy 



y   30

1 ln(30)  B kV

(7.2)

 1  1       kV   30 

(7.3)

where kV and B are constants; the equations in the LHS and RHS of the (7.2) and (7.3) formula are obtained from the curve continuity and slope conservation at two ends of the curve corresponding to the viscous sub-layer and logarithmic layer, respectively.

u

Buffer layer

y Fig 7.8 Buffer-layer velocity profile variants In figure (7.8), the curve in black shows the variation of formula (7.1) with the two other counterparts formulated here by the following equations:

u u

buffer

buffer

 a ( y  ) 3  b( y  ) 2  c ( y  )  d

(7.4)

 a ( y  ) 4  b( y  )3  c( y  ) 2  d( y  )

(7.5)

As it is evident in figure (7.8), formula (7.4) and (7.5) produce belly-shape curves with local maxima unsuitable for the intended purpose. As a conclusion, to reconstruct the velocity in the buffer layer at inlet section, formula (7.1) is adopted in this study.

188

For S-A and DDES simulations, the turbulent eddy viscosity in the close-wall region can also be reconstructed based on some empirical models like those proposed by Reichardt, Deissler, Van-Driest and Rannie models [O’Connor 1995]. Here, the Reichardt formula for the close wall zone is adopted which is expressed as follows:

vt  kV   y  [1  (  / y  ) tanh(y  /   )] v

(7.6)

where   is a model constant equals to 11, kV is the von-Karman constant equals to 0.41 and   is introduced as a scaling factor to be set with the given value of vt at the logarithmic layer coming from full machine k   RANS simulations. It is worth mentioning that at the limit, in the vicinity of solid walls ( y   0 ), formula (7.6) is reduced to the cubic variation (i.e. vt v  y  3 ), whereas for example, Deissler and VanDriest models are reduced to the fourth order variation (i.e. vt v  y  4 ). As proved by LDV measurements and DNS data in the viscous sub-layer limit 

( y  0 ), the variation of the Reynolds stress is cubic; it is the same for the eddy viscosity, since it is proportional to the Reynolds stresses in the viscous sub-layer [O’Connor 1995]. This justifies the choice of Reichardt model in the present study. Figure (7.9) shows the variation of the turbulent viscosity close to the draft tube wall at the two selected operating points.

vt v

y Fig 7.9 Turbulent viscosity in the near-wall zone of the draft tube using Reichardt model for two selected operating points, OP.1 and OP.4

189

uz uref .

OP.1

OP.4

OP.1

OP.4

OP.1

OP.4

u uref .

vt v

r RInlet

r RInlet

Fig 7.10 1D inflow profiles of u z and u velocities and turbulent viscosity vt at plane A used for draft tube flow simulations resolving to the wall at OP.1 (left) and OP.4 (right) (solid black line: reconstructed profile, solid red line with circles: LDV measurements) 190

Figure (7.10) shows the normalized complete inflow profiles for the axial and circumferential velocity components and the turbulent viscosity at the draft tube inlet section. As one can observe in this figure, the axial velocity profile compares appropriately with experimental LDV-data. However, for the circumferential velocity in the core flow there is an offset between the profile coming from the full machine k   RANS simulation and the experimental LDV data. As seen later, this offset plays an important role in the prediction of separation in the draft tube. Furthermore, as is evident in the figure, experimental data is not available close to the draft tube wall due to the difficulties of measurements in this area caused by the reflection effects on the LDV window access [Vuillemard et al. 2014]. It is worth mentioning that due to experimental limitations, the radial velocity component was not measured explicitly over the full radius, but this component was approximated by finite difference solution of the continuity equation. Figure (7.11) shows the variation of radial velocity profile with its counterpart approximated by the solution of continuity equation.

ur uref .

OP.4

OP.1

r RInlet

r RInlet

Fig 7.11 1D inflow profiles of radial velocity at OP.1 (left) and OP.4 (right) at plane A (solid black line: reconstructed profile, solid red line with circles: approximation profile) As also observed in figure (7.11), a mismatch also exists for the radial velocity component between the numerical and experimental-based approximation profiles [Vuillemard et al. 2014]. In general, the observed deviations in the circumferential and radial velocity components can be linked to different factors, like: the level of inflow turbulence, geometrical feature simplifications in the hydraulic profile used for numerical simulations and finally shortcoming of the k   turbulence treatment and stage interface to

191

handle the rotating and non-rotating part interactions. In the following subsection, different variants of the inflow velocity profiles including different corrections are presented. 7.6.2

Circumferential and radial velocity profile corrections

As observed in the previous subsection (figure 7.10), in the range of available experimental data, the axial velocity obtained from the full machine k   RANS simulation shows a relatively good agreement with the experiment; except in a small region to the hub ( 0.33  r / RInlet  0.4 ), where there is a bump in the numerical simulation, whereas the experimental counterparts show more flat variations. This difference can be related to the gap between the runner blades and hub which is present in the experimental model but is removed in the numerical simulations. In fact, for all numerical simulations here, the gap between the runner blades and the shroud (or draft tube wall) is considered but the gap between the hub and runner blades was omitted.

u uref .

ur uref .

r RInlet

r RInlet

Fig 7.12 Circumferential and radial velocity inflow profile correction for OP.1 (left: original profile, right: corrected profile based on experimental data i.e. red line) 192

On the other hand, one should keep in mind that modification of the axial velocity is a tricky task in the present application, because u z distribution defines the mass flow rate entering the draft tube. Furthermore, by taking into account the lack of a complete experimental profile on the full radius and also considering close agreement of the numerical and the experimental axial velocity profiles, the best choice is probably to keep the numerical axial inflow velocity profile untouched for all draft tube simulations, as performed in this study.

u uref .

ur uref .

r RInlet

r RInlet

Fig 7.13 Circumferential and radial velocity inflow profile correction for OP.4 (left: original profile, right: corrected profile based on the experimental data i.e. red line) By correcting the offsets for the two other velocity components, u and ur , variants of the inflow profiles are obtained. Figures (7.12) and (7.13) depict the corrected 1D profiles of the circumferential (named hereafter, u -correction) and radial velocity components (named hereafter, ur -correction) for the selected operating points OP.1 and OP.4, respectively. In the case of the circumferential components, the offset is corrected by 193

vertical translation, whereas in the case of the radial components, the slope is adjusted to respect the experimental slope. As it is clear in the figures, the outcome profiles follow closely the experimental and approximation profiles. In the following subsections, the variant versions of the profiles discussed here are applied for draft tube flow simulations and evaluated to find the best configuration to represent the flows.

7.7

URANS simulations of the BulbT draft tube turbulent flows

In this section, turbulent flow in the draft tube of the bulb turbine in the BulbT project is simulated using RANS turbulence treatments with different 1D inflow boundary conditions involving the above mentioned corrections. Due to the low computational cost of the k   RANS simulations, the technique is widely used for global performance prediction of the hydraulic turbines in the industry. Although the URANS technique captures only part of the flow unsteadiness, it is too dissipative suppressing major portion of the unsteadiness in the flow field. Furthermore, using wall-function in this model can result in a low accuracy in the prediction of separation region. Presented in this section, the turbulent flows in the draft tube are simulated using k   and Spalart-Allmaras (S-A) methods. As explained in chapter 2, the base formulation of DDES adopted in this thesis is basically S-A technique; that is why it is important to assess the S-A behavior here. The SA model was originally developed for the external aero-applications; here the intension is to use the method for the internal hydrodynamic applications. As discussed in chapter 2, in the case of external aerodynamic applications, most of the time the turbulent viscosity is set to a very low value corresponding to the calm ambient situation. In the case of internal hydrodynamics of draft tubes, the situation is more complicated especially in the complex flow situation just after the runner, where the boundary conditions come from the full machine k   RANS simulations. Therefore, inflow turbulent viscosity should be estimated based on the k   RANS simulation at the draft tube inlet plane. It is important to also notice that the computational cost of URANS simulations depends on the type of turbulence model adopted; for example, in the case of the low-Re number techniques like S-A, in which the mesh is of the order of DDES mesh resolving to the wall, URANS simulations need relatively high computational efforts. 7.7.1 Simulation numerical setup

The statistics of the mesh utilized for k   and S-A simulations, i.e. Mesh A and B respectively, have been presented in table (7.2) in section (7.4). For inflow condition at the inlet section of the draft tube, different variants of the circumferential-averaged 1D profiles 194

of velocity and turbulent quantities are applied as explained in subsection (7.6.2). In all cases, as mentioned, a mean pressure equals to zero is applied at the final exit. For all solid walls including hub and draft tube wall, ‘no-slip’ boundary condition is applied except for the extension wall of draft tube, where a ‘slip’ boundary condition is imposed. The details of the boundary conditions applied for each patch in the simulations can be found in table F.3 (appendix F). For transient simulations, ‘pimpleFoam’ flow solver among OpenFOAM solvers is used, for which the details of control setting are presented in table F.2 (appendix F). The time steps used are 1.2 and 0.6 runner rotations for k   and S-A simulations, respectively. For the discretization schemes, a second order backward scheme is applied for the temporal term and the convective term is also discretized by a second order upwind scheme. The details of the discretization schemes applied for other terms of the NavierStokes equation can be found in table F.1 (appendix F). 7.7.2 Inflow velocity profile correction effects

As a matter of fact, any anomaly in the inflow profiles feeding the draft tube has the potential to create consequential errors in the predictions, especially in the case of very sensitive flow in the BulbT turbine. Here, the results of k   and S-A URANS simulations using different variants of inflow corrected profiles are presented. 7.7.2 (a) Comparison to the LDV-measurements data

Figure (7.14) presents the mean profile at plane B obtained by applying three different inflow profiles including the original profile, u -correction and u , ur -correction at OP.1. After statistical convergence about 40 runner rotations, the mean is calculated. As seen in the figure, by feeding the draft tube by u -corrected inflow profile at OP.1, the mean u profile at plane B (blue line) is getting closer to the experiments (part a - figure 7.14), although the u z deviates a little more at the center of hub in comparison to the two other variants (part b – figure 7.14). It is also visible in the part (a) of the figure that the original inflow profile, with no corrections, presents an offset between the obtained u profile (green line) and the experiment. Whereas, by correcting the u velocity at the inlet of draft tube (plane A) in the two other cases, the profile at plane B are also matched with the experiment in the core flow (blue and orange lines).

195

u uref .

Exp. LDV original profile u θ corrected u θ , u r corrected

x Rref.

a) Mean circumferential velocity variation at plane B, just after the hub

uz uref .

Exp. LDV original profile u θ corrected u θ , u r corrected

x Rref.

b) Mean axial velocity variation at plane B, just after the hub Fig 7.14 Effect of variant inflow profiles on the mean flow obtained from k   URANS simulations at OP.1 compared to the LDV-experimental data In figure (7.15), the effect of inflow corrections on the mean velocity profile at plane B at OP.1 is considered in the case of S-A URANS simulations. For the SpalartAllmaras (S-A) turbulence treatment (figure 7.15), similar trend is observed in comparison to the k   model (figure 7.14) at OP.1, although the overall performance of S-A specially in the hub zone is lower for the u z component of the velocity, where there is the most deviation from the experiment under the hub. Like for the k   results, by feeding the draft tube by u -corrected inflow profile at OP.1, the mean u profile at plane B (blue line) is getting closer to the experiments (part a - figure 7.15), although the u z deviates from the experimental data at the center of hub ( 0.1  x Rref .  0.1 ) in comparison to the other

196

counterparts i.e. green and orange lines (part b – figure 7.15). It seems that all variants of the S-A simulations fails to appropriately capture the real physics under the hub. u uref .

Exp. LDV original profile u θ corrected u θ , u r corrected

x Rref.

a) Mean circumferential velocity variation at plane B, just after the hub

uz uref .

Exp. LDV original profile u θ corrected u θ , u r corrected

x Rref.

b) Mean axial velocity variation at plane B, just after the hub Fig 7.15 Effect of variant inflow profiles on the mean flow obtained from S-A URANS simulations at OP.1 compared to the LDV-experimental data For another operating point at higher flow rate, OP.4, simulations are also performed to investigate the flow prediction. Figure (7.16) shows the results of k   URANS simulations in comparison to the experimental data for OP.4. The trend observed in the mean velocity profile obtained from the k   URANS simulations in figure (7.16) for OP.4 is similar to the one for OP.1. The closest curves to the experimental counter parts for u z and u profiles are obtained by the u -correction of

197

the inflow velocity field. Finally, figure (7.17) depicts the variation of circumferential and axial velocity components coming from the S-A URANS simulations at OP.4. u uref .

Exp. LDV original profile u θ corrected u θ , u r corrected

x Rref.

a) Mean circumferential velocity variation at plane B, just after the hub

uz uref .

Exp. LDV original profile u θ corrected u θ , u r corrected

x Rref.

b) Mean axial velocity variation at plane B, just after the hub Fig 7.16 Effect of variant inflow profiles on the mean flow obtained from k   URANS simulations at OP.4 compared to the LDV-experimental data As it is evident in figure (7.17), the behavior of the numerical simulations in the case of S-A simulations for OP.4 is similar to the k   URANS simulations. It seems that the u -correction provides better agreement with experimental data than other counterparts, although in this case some deviations from the experimental data are visible in the u -curve in the hub region ( 0.3  x Rref .  0.3 ). Furthermore, some differences are present between

198

the experimental curve and numerical simulation results for the u z -curve in the range of 0.2  x Rref .  0.2 . u uref .

Exp. LDV original profile u θ corrected u θ , u r corrected

x Rref.

a) Mean circumferential velocity variation at plane B, just after the hub

uz uref .

Exp. LDV original profile u θ corrected u θ , u r corrected

x Rref.

b) Mean axial velocity variation at plane B, just after the hub Fig 7.17 Effect of variant inflow profiles on the mean flow obtained from S-A URANS simulations at OP.4 compared to the LDV-experimental data 7.7.2 (b) Separation topology

Up to now, comparisons to the experimental data at plane B let us have an initial insight into the behavior of the RANS type turbulence treatment, i.e. k   and S-A simulations, close to the inlet plane. Another important check point is to investigate the separation topology induced by different turbulence treatment methods and variant inflow profile corrections. As mentioned earlier, at first with the aid of tuft visualization and later with PIV techniques, wall separation was experimentally visualized for all selected operating 199

points including the operating points considered for numerical simulations, i.e. OP.1 and OP.4 [Duquesne et al. 2014-2]. Table (7.4) summarizes the separation topology obtained by URANS simulations in comparison to the experimental data. Table 7.4 Mean separation topology for different inflow profile variants using k   and S-A RANS simulations

Type

Inflow profile correction original profile

k 

u correction u , ur correction original profile

S-A

u correction u , ur correction

Experiments ( tuft & PIV)

….….

Separation type OP.1 Central bubble; No wall separation Central bubble; No wall separation Central bubble; No wall separation Small central bubble; Opposite corner wall separation Central bubble; No wall separation No central bubble; Large wall separation No wall separation

Separation type OP.4 Central bubble; Corner wall separation Central bubble; No wall separation No central bubble; Corner wall separation No central bubble; Large separation on the side wall No central bubble; Opposite corner wall separation No central bubble; Large separation on the side wall Separation on the side wall (at x  )

As summarized in table (7.4), for S-A turbulence model, the original profile does not provide a good prediction of the separation for both operating points, a probable cause can be presence of a clear offset in u velocity component between the experimental data and the profile extracted from full-machine k   simulations at draft tube inlet plane (plane A). In the case of S-A turbulence treatment, the result obtained from the ur -correction does not agree with experiments; in other words, at OP.1 a separation zone forms on the wall which is not observed experimentally. The best match to the experiments for S-A simulations are obtained considering an inflow profile with u -correction. This is consistent to the previous observation in the case of u -correction in the comparison of circumferential and axial velocity profiles at plane B, just after the hub. For instance, figure (7.18) shows the average mean separation zone obtained from S-A and k   URANS simulations using inflow profile with u -correction at OP1. As it is seen in the figure, they are generally in agreement, although there are minor 200

differences between the two cases. The reverse–flow bubble obtained in the k   URANS result is bigger than the one obtained with the S-A URANS simulation. Furthermore, there are two negligible small zones of separation in the corners in the case of S-A simulation which are not present in the k   results. Formation of the separation bubble after the hub decreases the effective flow passage area which does accelerate the flow which in turn energizes the boundary layer in the conical part and beginning of the draft tube transition from circular to rectangular section. Therefore, theoretically the process abates possibility of the flow separation on the draft tube walls. To verify which separation topology is closer to the reality, further experimental measurements are needed at the core flow or at the real exit of the draft tube. Furthermore, the formation and size of the opposite corner separation zones observed in the case of S-A (figure 7.18) can not be also confirmed, because no measurements were performed on the corners even with the tuft visualizations. It is worth mentioning, the length of draft tube extension with ‘slip’ boundary condition is reduced in the case of S-A model compared to the k   case, to reduce the fluid flow passing time and therefore to save computational time spent achieving statistical convergence (figure 7.18). This will be also helpful in the case of DDES simulations which are highly demanding in terms of the computational cost.

Y x

z

a) S  A

b) k  

Fig 7.18 Mean separation zone (blue iso-surface) obtained from S-A (top) and k   (bottom) URANS simulations with inflow u -correction at OP.1 7.7.2 (c) Recovery coefficient

Another important parameter that is often used to evaluate the draft tube behavior is the draft tube ‘recovery coefficient’ which was defined in the first chapter. Here, to compare 201

the recovery coefficient obtained from the simulations with the experimental counterparts, a definition based on pressure obtained from the pressure sensors is used, as below:

 

p Inlet  p z 1  Qf    2  AInlet 

(7.7)

2

where p Inlet is the average of four azimuthal sensor mean pressures positioned at inlet section, p z is the average of four azimuthal sensor mean static pressures positioned at any other sections defined by sensor position at different (z) coordinates,  is the fluid density, Q f is the flow rate and finally AInlet is the area of inlet section.



 k  Exp. data original profile u θ corrected u θ , u r corrected

SA Exp. data original profile u θ corrected u θ , u r corrected

z Rref.

z Rref.

a) OP.1; k   (left), S-A (right)



 k  Exp. data original profile u θ corrected u θ , u r corrected

SA

z Rref.

Exp. data original profile u θ corrected u θ , u r corrected

b) OP.4; k   (left), S-A (right) Fig 7.19 Draft tube recovery coefficient evolution in the streamwise direction for different variants of the URANS simulations

202

z Rref.

Figure (7.19) compares the evolution of draft tube coefficient along the streamwise axis for k   and S-A turbulence models to the experimental counterparts for both OP.1 and OP.4. In this figure, all draft tube coefficients have been normalized based on the overall  value at OP.1. As seen in the figures, in overall it seems that for both operating points, the  quantity is better predicted by k   rather than S-A in comparison to the experiment. In the case of k   , considering the u plus ur correction provides the closest curve to the experiment but this correction fails to provide the good agreement for S-A simulations. In the case of S-A simulations, the u -correction provide better agreement at OP.4 where the slope of the numerical and experimental curves is almost identical except in the range of 0  z Rref .  1 . Overall, there is an offset between experimental and numerical counterparts for all variants of the numerical curves. This offset is mainly created in the range 0  z Rref .  1 , that is the first part of the cone. In general, the recovery coefficient as an average (global) quantity  is not a very good measure to guaranty a good agreement between details of the flow in the experiment and the simulations. For instance, in the case of k   simulations all the inflow profiles do a relatively similar job in the prediction of  , but as discussed before the details of the flow at plane B and separation topology are different in those cases. Considering the observations on the recovery coefficient and also separation topology, one can come to the conclusion that in the case of k   , the best agreement with experimental data is obtained by applying the u , ur -correction, whereas for the SpalartAllmaras turbulence treatment, the best predictions are obtained by u -corrected profile. However, based on the observations on the velocity profiles obtained from the LDV data at plane B, u -inflow correction provides better agreement with the experiments especially under the hub zone for both k   and S-A simulations. In the following subsection, these observations are further examined by PIV measurement data at other checking positions in the draft tube. 7.7.2 (d) Comparison to the PIV-measurements data

As mentioned earlier, in addition to the LDV measurements performed at plane-B, PIV measurements were also done at four different planes further downstream in the draft tube (defined in section 7.3). These PIV-data can be adopted to provide a more complete picture about the behaviour of the applied turbulence treatment methods using different variants of the inflow conditions. On the PIV-planes, the velocity normal to the planes is not measured 203

due to the limitations of the measurements and two components of the velocity are only measured which are bounded to the plane surface, namely (u x , u z ) for B1, B2 and (u y , u z ) for S3 and S4. The coordinates ( x, z ) and ( y, z ) are defined on planes B1, B2 and S3, S4, respectively. Therefore, there is a transformation angle to change the ( x, y, z ) coordinate system to the ( x, y, z ) counterpart. To compare the numerical results and experimental data, the complete numerical velocity vector (u x , u y , u z ) is transformed to the (u x , u z ) and (u y , u z ) for planes B1, B2 and S3, S4, respectively. The following measures are defined to compare the experimental 2D fields to the numerical counterparts for both components of the mean velocity. In the case of B1 and B2 planes, the difference measures in percentage are defined for u x , u z , as below: u Num.  u Exp. E (u x )  x Exp. x 100 u x

(7.8)

u Num.  u Exp. E (u z )  z Exp. z  100 u z

(7.9)

It should be mentioned that the experimental mean is calculated considering a duration of about 40 minutes with a sampling frequency of 4 Hz (time step equals to 0.25s). Whereas, in the numerical simulation, the total duration of sampling is equal to 20 runner rotations or 1.25 second with sampling frequency of 100 Hz (time step equals to 0.01s). For comparison purposes, here it is essentially assumed that these two pictures are the same. Figures (7.20) and (7.21) show the difference between the numerical ( k   ) and experimental 2D fields for plane B1 positioned close to the bottom wall of the draft tube including upstream and downstream parts, for OP.1 and OP.4, respectively. As seen in figure (7.20), in the case of k   simulations at OP.1 in the downstream part of the PIVplane B1 and considering the percentage of coverage on the whole field, u , ur correction case generates the minimum field error to the experiment for both component of the mean velocity (u x , u z ) . In the case of u -correction, an even higher deviation from the experiment is obtained compared to the original inflow profile (the profile with no correction). At the upstream part of plane B1, the original profile seems to exhibit the minimum deviations for both components of the velocity. To draw a general conclusion, the other operating point with a higher flow rate should be considered. At OP.4, as seen in figure (7.21) for both the upstream and downstream sections of the PIV-plane, u , ur correction case again provides the lowest

204

relative error. However, the level of deviation from the experiment generally is higher for OP.4 in comparison to OP.1. This observation is consistent with the previous observations using the draft tube recovery coefficient and also separation topology as discussed before.

%

upstream   E (u x ) 

upstream   E (u z  ) 

downstream   E (u x ) 

downstream   E (u z ) 

a) Original profile

b) u -corrected profile c) u , ur -corrected

Fig 7.20 Difference measure of the velocity field between k   URANS simulation results and the PIV experimental data at OP.1, plane B1 (upstream/downstream) One important point that is visible for the both operating points is that the level of difference increases while passing from upstream to downstream of the draft tube. In other words, by advancing in the streamwise direction from the conical section of draft tube towards the rectangular outlet section in the transitional zone, the accuracy of predictions

205

decreases. As seen shortly, similar trend is observed for S-A and also DDES turbulence treatments. In general, limitations of the turbulence models in the complicated situation presented here in the case of turbulent swirling flow in the draft tube can be considered as a plausible explanation of the observations.

 %

upstream   E (u x ) 

upstream   E (u z  ) 

downstream   E (u x ) 

downstream   E (u z ) 

a) Original profile

b) u -corrected profile c) u , ur -corrected

Fig 7.21 Difference measure of velocity field between k   URANS simulation results and the PIV experimental data at OP.4, plane B1 (upstream/downstream) Another way to look at the problem is to compare the streamlines obtained from experimental and numerical velocity fields. For instance, figure (7.22) depicts the streamlines obtained from k   and S-A simulations at both OP.1 and OP.4 operating

206

points. As it is clear in this figure, in the case of k   simulations, the best agreement is obtained using u , ur -correction; whereas for the two other inflow profiles, the original and u -corrected profiles, streamlines with more deviations from the experiment are generated.

k    OP.1

S  A  OP.1 

k    OP.4 

S  A  OP.4 

a) Original profile

b) u -corrected profile

c) u , ur -corrected

Fig 7.22 Mean streamlines for k   and S-A simulations at OP.1/OP.4 on plane B1: downstream; (red: experimental measurements, black: numerical simulations) In the case of S-A simulations, in contrast to the k   turbulence treatment, u correction of the inflow profile provides the best agreement with the experimental streamlines for both operating points. As one can observe in figure (7.22), in the case of S-A method, u , ur -correction fails to capture the real physics and deviates from the experiment, especially at OP.4. Figures (7.23) and (7.24) depict the difference measure between the numerical S-A and experimental 2D fields on plane B1 for OP.1 and OP.4, respectively. At OP.1, for the 207

upstream section of plane B1, it seems that the original profile results in a better agreement with the experiment; but on the downstream section of the plane, for u z velocity component, the u -corrected profile agrees better with the experiment overall, whereas the lateral velocity i.e. u x is better predicted by the original profile.

%

upstream   E (u x ) 

upstream   E (u z  ) 

downstream   E (u x ) 

downstream   E (u z ) 

a) Original profile

b) u -corrected profile c) u , ur -corrected

Fig 7.23 Difference measure of the velocity field between S-A URANS simulation results and the PIV experimental data at OP.1, plane B1 (upstream/downstream)

208

It is also seen in figure (7.24), considering both component of the velocity for both the upstream and downstream sections of plane B1, in general the u -corrected profile provides better agreement with the experiment at OP.4. As mentioned before and as it is clear in figures (7.23) and (7.24), it seems that in the case of BulbT draft tube flow S-A simulations, considering ur -correction based on the approximated profile does result in a lower accuracy in predictions. For example, at OP.4, u , ur -correction results in an error higher than 50% on the downstream part of plane B1.

 %

upstream   E (u x ) 

upstream   E (u z  ) 

downstream   E (u x ) 

downstream   E (u z ) 

a) Original profile

b) u -corrected profile c) u , ur -corrected

Fig 7.24 Difference measure of the velocity field between S-A URANS simulation results and the PIV experimental data at OP.4, plane B1 (upstream and downstream) 209

As discussed in details, all of these observations made on the URANS simulations of BulbT draft tube turbulent flows at OP.1 and OP.4 using the PIV and tuft visualization data, hang to one conclusion: to obtain better agreement with the experimental data for the k   turbulence treatment, u , ur -corrected profile should be adopted. By contrast, for SA simulations, u , ur -correction fails to improve the predictions, whereas u -correction provides the best results. It is worth to emphasize that in the case of k   , although the velocity profile at plane B, just after the hub, is better predicted in the hub zone by the u -correction; further investigations downstream of the draft tube at the PIV-plane B1, revealed that the details of flow are not predicted as accurately as the u , ur -correction case. Considering the lessons learned from the k   URANS simulations as the base simulation strategy and the S-A model as the underlying formulation of the DDES turbulence treatment, the following section is dedicated to the different scenarios applied to perform DDES simulations of turbulent flow passing through the BulbT draft tube.

7.8

DDES simulations of the draft tube turbulent flows

As explained in details in the previous chapters, the S-A based DDES approach as a hybrid turbulence treatment models the turbulent flow in the wall zone via S-A URANS technique, while it resolves simultaneously the structures in the core flow via its LES mode. In the case of draft tube flow, one of the main challenging issues is to provide an appropriate inflow boundary condition, which mimics the real inflow within the framework of the applied hybrid turbulence treatment. In chapter 4, details of the strategy adopted to generate the various inflow boundary conditions for draft tube flow simulations were discussed, including 1D profile, unsteady 2D rotating profile and 2D rotating profile plus AFG, which the latter tries to generate the real turbulence at the inflow plane. In this section, the effects of applying the aforementioned boundary conditions are studied in details. To do so, first of all, the numerical simulation set-up is reviewed in the following subsection. 7.8.1 Simulation numerical setup

The statistics of the mesh utilized for DDES simulations (i.e. mesh B) is presented in table (7.2) of section (7.4). For the inflow condition at the inlet section of draft tube, the circumferential averaged-1D profile and 2D rotating profiles with/without the artificial inflow turbulence for velocity and turbulent quantities are applied; the details of inflow generation were presented in chapter 4. 210

In all cases, a mean pressure equals to zero is applied as boundary condition at the draft tube final exit. For all solid walls including hub and draft tube walls, ‘no-slip’ boundary condition is applied except for the extension wall of the draft tube where a ‘slip’ boundary condition is imposed. The details of boundary conditions for each patches applied in the simulations can be found in table F.3 (appendix F). For transient simulations, ‘pimpleFoam’ flow solver in OpenFOAM is used. The details of control setting of solver are presented in table F.2 (appendix F). The time step used for all DDES simulations after reaching the statistical convergence is considered as

0.6 runner rotations. All simulations are started from the initial conditions equal to the converged S-A URANS mean solution at the corresponding selected operating points (i.e. OP.1 or OP.4). In the start-up period of simulations, for stability reasons and to let the solution develop and avoid divergence, the time step is set as small as 0.006 runner rotation at the beginning of the computations and is gradually increased to 0.6 runner rotation in approximately one second of the physical wall-clock time. Due to the high computational cost of DDES simulations after achieving the statistical converged state, about 20 runner rotations is considered to calculate the mean flow properties. Considering the mean flow speed, this duration is roughly sufficient for turbulent flow to pass the draft tube about 3 times. In the case of LES simulations of flows involving pressure-induced separation like in the diffuser flows (as opposed to separation induced by sharp edges/ obstacles), highorder temporal and spatial discretization schemes should be adopted. In fact, prediction of separations and reattachments in general are very sensitive to any disturbance that could come from an inaccurate (e.g. low-order) discretization scheme [Ohlsson et al. 2010]. For pure LES simulations, at least second-order temporal and linear spatial discretization scheme without addition of any upwinding factor can be adopted [Davidson 2011]. In the case of hybrid LES/RANS methods like DDES applied to simulate convection-dominated flows, a linear scheme can cause a growth in the amplitude of instabilities and ultimately leads to a numerical divergence. Here, similar to the ‘basic geometry’, the convective term is discretized by the second-order linear discretization scheme with local filtering, namely ‘filteredLinear2V’ scheme for DDES simulations. This scheme introduces low dissipation locally via adding a small amount of upwind where necessary in the computational domain to remove staggering and avoid divergence in the case of convection-dominated flows, whereas it does not have an adverse effect on the LES statistics. In addition, a second-order backward scheme is applied for the discretization of temporal term. The details of discretization schemes applied for all terms of Navier-Stokes equation can be found in table F.1

211

(appendix F). In the following, the strategy applied to simulate the turbulent flow in the BulbT draft tube and the obtained results are discussed in details. 7.8.2 Simulation results using 1D-inflow profiles

The type of inlet boundary conditions applied for draft tube simulations should have an important effect on the resolved structures by DDES methods. In the case of circumferential averaged-1D profile applied at the inlet of draft tube, similar to the stage interface applied for rotor-stator interaction treatment, the dependency of inlet profile in the  -direction is removed and the profile varies only in the radial direction. Therefore, the runner-related vortical structures like tip-vortices and wakes, which depend on the 2D variation of inflow profile, are not resolved. In this study, a systematic path is followed for DDES simulations. Before considering 2D inflow profiles, first of all, different variants of the 1D profiles are utilized to establish a strategy for DDES simulations of the draft tube flows. In the following subsection, the effect of 1D-inflow velocity corrections is considered. 7.8.2 (a) Effect of inflow velocity corrections

As explained in chapter 4, in the present study the original profile utilized for DDES simulations of draft tube flow basically stemmed from k   simulations of full machine BulbT assembly. It should be also recalled that in the case of DDES simulations of draft tube, the inlet section is divided in two different zones including a SA-URANS region near the solid walls and a LES region in the core flow. For the first region, i.e. the URANS inflow region, in the case of the ‘original profile’, turbulent viscosity is estimated based on the turbulent kinetic energy and turbulent eddy dissipation rate values at the inlet plane extracted from the full machine k   simulations using the formula vt  C k 2  . In the second region, i.e. the core flow, the role of turbulent viscosity switches to the subgrid scale viscosity, which is two or three times lower in order of magnitude than the base turbulent viscosity coming from RANS simulations (chapter 4). In fact, as explained before, in URANS simulations all effects of the flow turbulence is considered via the turbulent viscosity field; but in the LES turbulence treatment, the role of subgrid scale viscosity is to dissipate the turbulence energy. In the case of the ‘original profile’ and its variants, the turbulent viscosity for all simulations is lowered 100 times in the core flow, whereas the turbulent viscosity coming from RANS is kept untouched in the near wall-zone (for simplicity ‘WZ’ hereafter) unless otherwise stated. This reduction in the order of magnitude in the core flow is consistent with the 2D

212

turbulent viscosity field estimated based on the Smagorinsky subgrid scale model (figure 4.13). Figure (7.25) shows the streamline patterns obtained from DDES simulations using different variants of the ‘original’ inflow profile on the PIV-plane B1 at upstream and downstream sections compared to the experimental counter parts. It is obvious in the figure that for all operating points even on the upstream section of plane B1, clear deviation from the experiments is present at upstream section and as a result, on the downstream section. On the upstream section of the plane, in the case of the ‘original’ profile at OP.1, a deviation from the experiment is being formed at the bottom of the section. Whereas at OP.4 an unstable spiral repeller exists indicating the presence of separation which is not matched with the experimental counterpart.

upstream   OP.1 

downstream   OP.1 

upstream   OP.4 

downstream   OP.4 

a) Original profile b) u -corrected profile c) u , ur -corrected Fig 7.25 Mean streamline patterns for DDES simulations using different variants of the ‘original’ inflow profile on plane B1: upstream/downstream (red: experimental measurements, black: numerical simulations)

213

In general, for all cases especially on the downstream section of the plane, the deviation from the experiment is more serious at OP.4 in comparison to OP.1. In figure (7.25), the separation topology varies by applying different inflow boundary conditions. In the figure, a separation can be identified by the presence of curved streamlines with a direction opposite to the flow direction. In the case of ‘original’ inflow profile on the downstream section of the plane at OP.1, the stable/unstable spiral attractor/repeller can be seen. Whereas, presence of a saddle point is clear in the case of u -corrected inflow profile on the downstream section of the plane at OP.4. Figure (7.26) shows the separation zone identified using iso-surfaces of the negative axial velocity for the case with ‘original’ inflow profile along with other cases involving the different inflow velocity corrections at OP.1. As shown in the figure, for all cases a separation zone is formed on the draft tube wall; which is not present in the experimental measurements on the test bench at OP.1 using tuft visualization technique. Also as discussed in the previous sections, the results of S-A URANS simulations depict that applying ur -correction on the inflow profile reduces the prediction accuracy of simulations. Therefore, it can be expected a-priori that including ur -correction for DDES simulations does not improve the simulation results. This concept is proved in figure (7.26). By considering the ur -correction on the profile, the separation extends and covers a larger portion of the draft tube walls on both sides. It is also important to notice that the DDES technique was theoretically developed to remedy the grid-induced separation (GIS) in the DES97 method. As explained in chapter 2, the f d function is introduced in the original DES97 formulation to protect the boundary layer and to ensure that the boundary layer is treated by URANS. By this method, switching to LES is hindered to the outer region of the boundary layer. The f d function, as defined as equation (2.12), is rewritten below:





0 RANS 3 f d  1  tanh 8rd    1 LES

(7.10)

where rd is a function of velocity field and distance to the wall defined in equation (2.11). Figure (7.27) shows the variation of f d function in the draft tube obtained from the DDES simulation with u -corrected profile at the inlet plane for OP.1. As it is obvious in the figure, the boundary layer region is fully protected on the whole draft tube wall, which guarantees avoiding grid-induced separation. Therefore, the observed flow separation should come from other factors, which will be investigated in the following subsections.

214

X

Y Z

u z uref .

Z

Y

X

a) ‘Original’ inflow velocity profile, original turbulent viscosity in WZ

b) u -corrected inflow profile, original turbulent viscosity in WZ

c) u , ur -corrected inflow profile, original turbulent viscosity in WZ Fig 7.26 Mean flow separation zone topology obtained from DDES simulations with different inflow profiles at OP.1 Another important issue which is visible in figure (7.26) is that, there is no separation bubble in the core flow of the draft tube for all cases; instead the separation is formed on the draft tube walls. As mentioned before, the presence of central bubble of the reverse flow decreases the effective cross sectional area of the passing fluid flow which delays onset of the separation on the draft tube walls and yield a more realistic flow at OP.1.

215

To further investigate the reason of separation zone formation at OP.1 by all the variants of DDES simulations, the variation of u z and u velocity profiles at plane-B are considered in figure (7.28) compared to the experimental counterpart coming from LDV measurements.

fd

Fig 7.27 f d function variation extracted from DDES simulation of the draft tube with u -corrected inflow profile at OP.1 As shown in figure (7.28), regardless of the similarities observed, deviations from the experiment are visible in the axial velocity on the full radius for all variants of the inflow corrections with ‘original’ viscosity. In the last case however, the turbulent viscosity has been amplified in the near-wall zone (WZ) close to the both hub and shroud walls ( R R  0.09 ). In fact, as one can see in the dashed circle on the graph, for all cases except the last case, unphysical overshoots exist in the u z profiles on the left and right radiis; which in turn cause a deficiency in the u z profile in the region close to the hub respecting conservation of mass (continuity equation) on the full radius. The details of turbulent viscosity tuning in the near-wall zone (WZ) using PIV-planes are presented in the coming section. As shown in the part (b) of figure (7.28), without applying u -correction there exists an offset in the circumferential velocity profile at plane B section (green line). The impressive agreement of the u z velocity component to the experimental data by turbulent viscosity amplification in the WZ is obtained at the expense of the presence of undershoots in the u profile under the hub. In the following it is proved that in spite of this shortcoming, the details of flow simulation results are also better matched by the modification in many aspects.

216

uz uref .

Exp. LDV original velocity, original vt u θ corrected, original vt u θ , u r corrected, original vt u θ corrected, 103× vt (in WZ )

x Rref.

a) Mean axial velocity variation at plane B, just after the hub u uref .

x Rref.

Exp. LDV original velocity, original vt u θ corrected, original vt u θ , u r corrected, original vt u θ corrected, 103 × vt (in WZ )

b) Mean circumferential velocity variation at plane B, just after the hub Fig 7.28 Effect of ‘original’ inflow profile variants on the mean flow obtained from DDES simulations at OP.1 compared to the LDV-experimental data As a global measure, figure (7.29) compares the evolution of the draft tube recovery coefficient (defined in equation 7.7) in streamwise direction obtained from the DDES numerical simulations to the experimental counterpart at both operating points. As one can see, despite the presence of an offset between the experimental and numerical curves, the slope of the curves is better predicted over z Rref .  1 range by an amplification of turbulent viscosity in the WZ. All other pure velocity corrections including original, u -corrected and u , ur -corrected inflow profiles fail to reproduce the curve slope in agreement with the experiment. This shows that the detail physics of the flow predicted by the numerical simulations is different from the experiment. 217





OP.4

OP.1 Exp. data Original velocity, original vt u θ corrected, original vt u θ , u r corrected, original vt u θ corrected, 103× vt (in WZ )

z Rref.

Exp. data Original velocity, original vt u θ corrected, original vt u θ , u r corrected, original vt u θ corrected, 103× vt (in WZ )

z Rref.

Fig 7.29 Draft tube recovery coefficient evolution in streamwise direction obtained from DDES simulations for both operating points

uz uref .

x Rref.

Exp. LDV original velocity, original vt u θ corrected, original vt u θ , u r corrected, original vt u θ corrected, 103× vt (in WZ )

a) Mean axial velocity variation at plane B, just after the hub u uref .

Exp. LDV original velocity, original vt u θ corrected, original vt u θ , u r corrected, original vt u θ corrected, 103 × vt (in WZ )

b) Mean circumferential velocity variation at plane B, just after the hub Fig 7.30 Effect of ‘original’ inflow profile variants on the mean flow obtained from DDES simulations at OP.4 compared to the LDV-experimental data 218

x Rref.

This observation is also confirmed in figure (7.26), where an extensive flow separation forms on the draft tube wall for all DDES simulations, whereas there is no separation on the wall at OP.1 experimentally using tuft visualizations. A similar conclusion is also made from figure (7.29) for OP.4, where the best curve slope is obtained by applying u -correction along with the turbulent viscosity amplification in the WZ. In figure (7.30), a similar behavior to OP.1 is also observed at OP.4 for velocity variation at plane B. As seen in the dashed circle and its left-side counterpart, overshoot deviations in the axial velocity exist from the experiments at Op.4. In this case, similar to OP.1, the best agreement with the experiment is obtained in the latter case i.e. the DDES simulation with including a turbulent viscosity amplification in the WZ. This improvement is obtained but at the expense of higher deviation in the u component of the velocity in the hub region 0.3  x Rref .  0.3 .

k

k   RANS Exp. LDV

r Rref.

Fig 7.31 Turbulent kinetic energy ( k ) variation at the inlet plane at OP.1 Before switching to the DDES simulations with 2D inflow profiles, the next reasonable step to proceed is to adjust the turbulent viscosity in the near wall zone (WZ). First of all, one should answer this fundamental question: how is it justified to increase the turbulent viscosity in the WZ? In fact, in this research the idea was initially conceived while performing a comparison of the turbulent kinetic energy extracted from the k   full machine RANS simulation at the draft tube inflow plane (plane A) to the experimental LDV-measurements. Figure (7.31) depicts the variation of the turbulent kinetic energy on the full radius compared to the experiment. As one can see, the numerical turbulent kinetic energy ( k ) is about 1-2 orders of magnitude lower than the experimental counterpart. On the other hand, due to the difficulties present in the measurement of the dissipation rate (  ); this quantity is 219

not on hand experimentally. Therefore, the turbulent viscosity can not be calculated using the formula: vt  C k 2  . As an example, by assumption of a constant  , the difference of 1-2 orders of magnitude in the turbulent kinetic energy quantity can bring a difference of 2-4 orders of magnitude in the vt quantity. This abstract example triggers the idea of amplification of the turbulent viscosity to adjust the insufficient turbulent viscosity in the WZ as already observed in the case of ERCOFTAC and ‘basic geometry’ of the BulbT draft tubes. It should be also kept in mind that in the core flow the turbulent viscosity (i.e. subgrid scale viscosity in this context) is already reduced about 2 orders of magnitudes in the LES zone of the DDES simulations as shown in chapter 4. In the next section, the procedure followed to adjust the turbulent viscosity in the WZ is discussed in details. 7.8.2 (b) Adjustment of inflow near wall turbulent viscosity using OP.1

The methodology adopted here to adjust the inflow vt in the WZ is to start with the circumferential averaged-1D profile as the inlet B.C. at OP.1, where no separation is observed experimentally. Obtaining a correct behaviour of numerical DDES simulations at this checkpoint is essential before switching to the OP.4, where a large flow separation zone forms on the draft tube side-wall experimentally and also before switching to the 2D inflow profile. In other words, OP.1 is utilized here for tuning purposes and a simulation strategy is established based on that; then the setting procedure is exactly applied to simulate the flow at OP.4 for the validation purpose. In this regard, available LDV and PIV data measured on plane B and plane B1 are crucial in tuning of the turbulent viscosity in the WZ. A typical way is to amplify the turbulent viscosity by a factor ( WZ ) ranging from 101 to 104, as below: vt

WZ

 WZ  vt

WZ RANS

WZ  10 , n  1,.., 4 n

(7.11)

Figure (7.32) depicts the effects of amplification of the inflow turbulent viscosity ( vt ) in WZ for both the hub rotating-wall and shroud wall on the mean velocity profile at plane B just after the hub. For all cases, the u -correction was also applied on the inflow profiles (plane A). As one can notice in the region marked by a dashed circle, for n  1, 2 overshoots still exist in the axial velocity in comparison to the experiment. In addition, for n  4 , the curve is getting flatter than necessary and deviates a lot from the experiment. The best agreement with the experiment is obtained with n  3 , where an impressive agreement exists in u z component of the velocity, although a deviation in u component is present under the hub. 220

uz uref . Exp. LDV WZ , n  1 WZ , n  2 WZ , n  3 WZ , n  4 original vt

x Rref.

a) Mean axial velocity variation at plane B, just after the hub u uref .

Exp. LDV WZ , n  1 WZ , n  2 WZ , n  3 WZ , n  4 original vt

x Rref.

b) Mean circumferential velocity variation at plane B, just after the hub Fig 7.32 Effect of amplification of turbulent viscosity in the WZ on the mean flow obtained from DDES simulations with u -corrected inflow profile at OP.1 One important observation which can be made for all the experimental LDVmeasurements presented up to here is that there is a dissymmetry between left and right wings of the profile as a fingerprint of upstream components like intake, guide-vanes, etc. which is not captured by the numerical simulations due to the boundary condition uniformization applied at the draft tube inlet using the circumferential-averaged 1D profiles. As one can also see in part (b) of figure (7.32), for all resulting profiles there is no offset between the numerical and experimental u curves at the LDV measurement plane (plane B) indicating necessity of applying u -correction on the velocity profile at the draft tube inlet plane (plane A). 221

a) n  1

b) n  2

c) n  3

d) n  4

Fig 7.33 Mean streamline patterns for DDES simulations with amplification of vt ( WZ  10n ) at OP.1 on plane B1: downstream (red: measurements, black: simulations) To better understand the quality of flow predictions by the amplification of turbulent viscosity in WZ, PIV-planes (i.e. plane B1 at OP.1) can be adopted. In this regard, both components of the velocity namely u x and u z are compared on the both upstream and downstream sections of plane B1. Figure (7.33) depicts the effect of the amplification of turbulent viscosity in the WZ on the streamlines predicted by DDES simulations. For all cases, the u -correction has been applied on the inflow profile. As one can see in the figure, for n  1, 2 deviations from the experiment exist on the downstream section; the best agreement is obtained by n  3 . As it is also clear in the figure for the case of n  4 , the error of mismatch again increases. The same conclusion can be made by checking details of the flow using PIV measurement data of u x and u z velocity profiles on both plane sections (appendices G.3.1 and G.3.2). Figure (7.34) depicts the difference measure of the velocity fields for u x and u z (defined in equations 7.8 and 7.9) on both upstream and downstream sections of plane B1. As one can see in the figure, for both components of the velocity on the both sections of the plane, the original vt shows the maximum difference in comparison to the experiment. Especially for the downstream section of the plane, the difference on the whole plane is higher than 10%. By increasing the power factor to n  3 , the difference (error) decreases for both components of the velocity on both sections of the PIV-plane. Through jumping from n  3 to n  4 , the error again increases especially for u z component of the velocity on the downstream section of the PIV-plane. This is consistent to the previous observation in the above about the streamlines (figure 7.33). In fact, the main conclusion that can be made here is summarized as follows: by amplification of the turbulent viscosity with a factor of n  3 on the hub and shroud, not only the 1D variation of the u z component of the velocity is improved but also the details of flow on the PIV-planes are better predicted. 222

% Upstream, E (u x )

Upstream, E (u z )

Downstream, E (u x )

Downstream, E (u z )

a) Original vt

b) n  1

c) n  2

d) n  3

e) n  4

Fig 7.34 Difference measure of the velocity field obtained from DDES simulation results with amplification of vt in the WZ ( WZ  10n ) at OP.1 at plane B1 (upstream/downstream) As mentioned before, an important feature of the turbulent flow in the draft tube of BulbT turbine at OP.1 is the absence of separation on the draft tube wall. Figure (7.35) shows the effect of increasing the inflow turbulent viscosity in the WZ on the topology of the mean separated zone. As one can see in the figure, in the case of n  1 , there is still a large separated zone formed on the draft tube wall similar to the case of the original profile (refer to figure 7.26-part b). By increasing the turbulent viscosity by a power factor of n  2 , the separation zone is reduced but still persists. One important point that should be noticed is that for both

223

aforementioned cases ( n  1, 2 ) no reverse–flow bubble is formed in the solution instead, one can observe separation formation on the draft tube wall.

X

Y Z

u z uref .

Z

Y

X

a) u -corrected inflow profile, WZ  10n , n  1

b) u -corrected inflow profile, WZ  10n , n  2

c) u -corrected inflow profile, WZ  10n , n  3

d) u -corrected inflow profile, WZ  10n , n  4 Fig 7.35 Mean flow separation topology obtained from DDES simulations with turbulent viscosity amplification in the WZ applied on both hub and shroud zones at OP.1 By increasing of the turbulent viscosity to n  3 , a reverse-flow bubble is formed in the center of draft tube and the wall separation is effectively eliminated. As one can also 224

observe in figure (7.35) by further increasing the factor from n  3 to n  4 , although no separation exists on the draft tube wall, the central bubble is over-suppressed in the core flow and nearly vanished. In addition to this observation, as discussed based on the LDV and PIV data in figure (6.34), the case with n  4 deviated considerably from the experiments; so it seems that n  4 is off and the best agreement with the experiment using all checking points is obtained by setting n  3 . Thus far, all of the observations compared to the LDV and PIV results hang together towards the selection of n  3 . Figure (7.36) shows the evolution of the draft tube recovery coefficient for different amplification factors. As one can observe in the figure, despite presence of an offset between the experimental and numerical curves, arising in the range of 0  z Rref .  1 , the curve slope is well predicted by n  3 over z Rref .  1 range.



Exp. data  WZ , n  1  WZ , n  2  WZ , n  3 original v t

z R ref.

Fig 7.36 Effect of amplification of vt in the WZ ( WZ  10n ) on the draft tube recovery coefficient evolution obtained from DDES simulations at OP.1 Another important quantity which helps to have a more complete picture about the behaviour of draft tube is the loss coefficient  L , defined in equation (1.10) and rewritten as below:      . PU n dS PU   .n dS   m S  m Soutlet  inlet 

L  

 1      U .U dS    2Sinlet S  inlet  

(7.12)

where P stands for the total pressure. As one can see in table (7.5), the case with n  4 is over-estimated, due to the high value of turbulent viscosity and the negative work done on the fluid by the shear stress. The other cases present a more reasonable loss coefficient

225

consistent with the mean loss coefficient equals to 0.0798 obtained from a full machine k   simulation using ANSYS and 0.0799 obtained from k   URANS simulation of the draft tube using OpenFOAM. To summarize the observations thus far, it was shown that in the case of DDES simulation of BulbT draft tube, for OP1, application of the u -correction along with amplification of turbulent viscosity with n  3 in the WZ of the inflow profile presents the best agreement with the global and detail features of the flow using the available experimental data. These corrections will be applied for OP.4 as well. Table 7.5 Draft tube loss coefficient obtained from DDES simulations with an amplification of vt in the WZ and applying inflow u -correction at OP.1 Type of Simulation

Solver

DDES

OpenFOAM

k 

OpenFOAM ANSYS

vt in WZ original vt n 1 n2 n3 n4 Original Original

Loss Coefficient ( L ) 0.0909 0.0685 0.0610 0.0830 0.4226 0.0799 0.0798

To obtain the time evolution of the global draft tube coefficient, the formula defined in equation (1.9) is adopted. The formula is rewritten as below:      pU .n dS  pU .n dS     m S  m Sinlet  outlet 

 

 1      U .U dS    2 Sinlet S  inlet  

(7.13)

It represents the mass flow rate integration of the static pressure on the inlet and outlet of draft tube. Figure (7.37) shows the time evolution of draft tube coefficient  for the DDES simulation with the u -corrected inflow profile along with amplification of turbulent viscosity with n  3 in the WZ. As one can see, there is an initial transient step to let the flow develop sufficiently and then a statistical-converged zone is reached. In this zone, the flow still involves unsteadiness, which produces limited cycle oscillation around the mean (the horizontal dashed line in the figure). The red line shows the approximate border of the initial transient and achieved statistical converged zones.

226



Transient zone

Statistical converged zone

t (s)

Fig 7.37 Time evolution of the global draft tube coefficient (  ) for DDES simulation applying inflow u -correction and amplification of vt in the WZ ( n  3 ) at OP.1

uz uref .

Exp. LDV u θ -corrected, n=3: hub &shroud u θ -corrected, n=3: just shroud u θ -corrected, n=3: just hub n=3: hub & shroud

x Rref.

a) Mean axial velocity variation at plane B, just after the hub u uref .

Exp. LDV u θ -corrected, n=3: hub & shroud u θ -corrected, n=3: just shroud u θ -corrected, n=3: just hub n=3: hub & shroud

x Rref.

b) Mean circumferential velocity variation at plane B, just after the hub Fig 7.38 Effect of amplification of vt in the WZ applied just for hub/shroud with/without u -correction at OP.1 227

Before proceeding to OP.4, some questions need to be answered. First, can amplification of vt in the WZ but without u -correction of the inflow profile solve the problem? In this manner the inflow velocity is kept untouched. On the other hand, in all studied cases thus far, the inflow turbulent viscosity amplification was applied equally on both the hub and shroud wall-zones. But, there is a difference between the hub and shroud walls: one wall is rotating (hub) and the other (shroud) is not. Despite the presence of swirl in the inflow, theoretically, the rotation of the hub wall can affect the curvature of the near-wall streamlines and hence the turbulent production. To some extent, this process may justify a difference in the turbulent viscosities applied for the near-wall zones of the hub and shroud. Due to the intensive computational cost of DDES simulations, which limits the number of simulations to perform, an on/off situation is considered here. In one scenario the turbulent viscosity amplification is only made on the shroud-zone and in another case, just on the hub. For the first three cases, the u -correction is applied on the inflow profile but for the latter case the velocity profile is kept untouched. As one can see in figure (7.38), if amplification of vt is removed in the hub zone, a large deviation in the axial component of velocity exists (blue line), although the observed undershoots in the u profile in the hub zone are removed as well. As concluded by all simulations reviewed so far, it seems that a correct variation of u z component of the velocity is essential for the formation of the central reverse-flow bubble and its consequent effects as one can observe in figure (7.39). On the other hand, although amplification of vt just on the hub (not shroud) zone corrects the axial velocity distribution on a large range of the plane-B span (purple line in part a – figure 7.38), but a deviation is formed near the shroud wall as marked by dashed circle on the top of the curve. As shown in part (b) of figure (7.39), this deviation is amplified further downstream and finally a large separation is formed on the draft tube wall. It is also visible in figure (7.38) that by applying vt amplification just on the hub, u curve is very similar to the case with the amplification on both hub and shroud. To summarize considering the findings so far, it seems that it is necessary to amplify the turbulent viscosity in the WZ for both the hub and shroud zones and also to correct u . This is achieved at the expense of presence of overshoots in u under the hub (figure 7.38). Figure (7.39) also shows that by amplification of vt in the WZ ( n  3 ) applied only on the hub wall-zone or just on the shroud, the central reverse–flow bubble is not formed. Instead, a separation zone appears on the draft tube wall, which is more intense in the case named ‘just for hub’. 228

X

Y Z

u z uref .

Z

Y

X

a) u -corrected, n  3 for both hub & shroud

b) u -corrected, n  3 just for hub

c) u -corrected, n  3 just for shroud

d) Original velocity profile, n  3 for both hub & shroud Fig 7.39 Effect of amplification of vt in the WZ applied just for hub/shroud with/without u -correction on the mean flow separation topology at OP.1 On the other hand, as part (d) of figure (7.39) depicts the amplification of inflow vt on both the hub and shroud wall-zones but without u -correction of the inflow velocity profile, makes the separation on the draft tube wall grow in comparison to the case with

229

inflow u -correction (part a-figure 7.39). It seems that u -correction based on the experimental LDV-data is an essential step to capture an appropriate behavior of the flow and consequently a correct prediction of the separation zone.

% Upstream, E (u x )

Upstream, E (u z )

Downstream, E (u x )

Downstream, E (u z )

a) b) c) d) 7.40 Effect of amplification of inflow vt in the WZ applied just for hub/shroud on the difference measure of the velocity field with/without u -correction at plane B1 at OP.1 a) u -corrected, n  3 for both hub & shroud ; b) u -corrected, n  3 just for hub , c) u corrected, n  3 just for shroud ; d) Original velocity profile, n  3 for both hub & shroud The effects of the different scenarios explained above can be further investigated with the aid of PIV-measurement data. In this regard, figure (7.40) shows the ‘error 230

measure’ of the velocity field defined by equations (7.8) and (7.9) based on the experimental data for upstream and downstream sections of the PIV-plane B1 ( u x and u z velocity profiles on the both plane sections can be found in appendix G.3.3 and G.3.4). As one can expect a-priori, and also confirmed in all simulations presented in this chapter, in general the difference measure on the downstream section is higher than the upstream one for both components of velocity u x and u z . This fact is related to the ability of the turbulence treatment to correctly simulate the challenging situation of the BulbT draft tube flow i.e. high Re-number turbulent swirling fluid flow within a relatively complex hydraulic profile involving a transition from circular to rectangular sections. Figure (7.40) also confirms the conclusion previously made based on the available LDV-experimental data at plane B (figure 7.38) and separation zone topology. As one can see in the figure, the minimum error on the PIV-plane B1 for both upstream and downstream sections is obtained by applying u -correction of the inflow velocity profile along with the amplification of inflow vt on the both hub and shroud wall-zones with n  3 . This main conclusion is considered for all upcoming simulations through the rest of thesis. In other words, the aforementioned corrections are applied on the inflow profile for all upcoming DDES simulations using 1D profile, 2D rotating profile and 2D rotating profile plus AFG at OP.4. a)

b)

Separation Bubble

z x

Separation Bubble Vortex Breakdown

Y

uz uref .

7.41 Coherent structures formed in the BulbT draft tube turbulent flow using DDES at OP.1 a) reverse-flow bubble, b) coherent structures visualized with Q  2000 At OP.1, a separation bubble of reverse-flow is formed at the center of BulbT draft tube, as seen in the case of DDES simulation using the u -corrected inflow profile and an amplification of the turbulent viscosity with n  3 for both hub and shroud wall-zones (figure 7.41). The reverse-flow zone in this case is generated by the turbulent coherent structures created by the vortex breakdown phenomenon. Figure (7.41) shows the vortex rope formation and onset of the vortex breakdown phenomenon extracted using Q-criterion

231

at one instant of time colored by the instantaneous normalized axial velocity (for the definition along with the details of Q-criterion technique in comparison to the other vortex identification schemes, refer to appendix E). As one can see in the part (a) of the figure, a separated reverse-flow zone is formed in the core of the vortex rope. This zone is formed as a result of numerous interactions of local reverse flows induced by small/large coherent structures in the vortex breakdown zone (part b). The phenomenon is classified under a spiral type vortex breakdown [Lucca-Negro and O'Doherty 2001]. In the next section, the results of the simulations at OP.4 are investigated in details. To confirm the presence of a reverse–flow bubble at OP.1, further measurements in the core are needed. 7.8.3 DDES simulations at OP.4

After setting the DDES simulations at OP.1, the same strategy is applied to simulate the flow field in the draft tube of BulbT turbine at OP.4. The same mesh and the same numerical set-up as explained in section (7.8.1) are utilized to simulate the flow field. Considering the lessons learned from the various DDES simulations made at OP.1 in the previous sections, two factors are included in all upcoming simulations at OP.4: first, the u -correction of the inflow velocity profile based on the experimental data, and second, an amplification of the turbulent viscosity with a factor of n  3 in both the hub and shroud wall-zones (WZ). Before proceeding to the DDES simulations using 1D and 2D inflow profiles at OP.4, first of all, a mesh convergence test is performed. In this regard, due to the high computational cost of DDES simulations for draft tube flow simulations, only two sets of mesh including a fine mesh (mesh C) and the original mesh (mesh B) are considered. Table 7.6 Draft tube mesh convergence test at OP.4 No.

Mesh

Type of mesh

No. of elements (millions)

Mean recovery coefficient (  )

1 2

B C

Intermediate Fine

7.26 15.6

0.77 0.76

Mean loss coefficient ( L ) 0.078 0.086

Table (7.6) shows the mean recovery and loss coefficients of the draft tube flow simulations using circumferential average-1D profiles at OP.4. As it is seen in this table, the difference between the recovery coefficients (  ) for both grids is about 1% and is well-predicted. The difference for the loss coefficient ( L ) is higher and is about 9%. To further investigate and confirm that the intermediate mesh is fine enough, the details of the flow on the PIV-planes and global separation topology are considered for both intermediate and fine grids at OP.4 in the following.

232

X

Y Z

u z uref .

Z

Y

X

a) Intermediate grid (grid B)

b) Fine grid (grid C) 7.42 Topology of the mean separated region for intermediate/fine girds at OP.4 visualized by iso-surface of the negative velocity Figure (7.42) shows the mean separation topology for both cases. There are some minor differences in the topology of the separated region; for instance, in the case of the fine grid, a very small separated bubble formed in the center of draft tube which is not present in the intermediate grid counterpart. However, the position of the separation on the

x  side wall and the separated zone pattern are generally in good agreement. To grasp a better insight about the details of the flow, PIV-data are utilized (the compared velocity profiles can be found in G.3.5 to G.3.8 in appendix G). Figures (7.43) and (7.44) depict the error measure between the experimental data and the numerical counterpart on the different PIV-planes. For OP.4, measurements were performed for four planes namely: B1, B2, S3 and S4, each including upstream/ downstream sections. As one can see in these figures, the overall error between experimental and numerical DDES simulations is higher for OP.4 than for OP.1. Although, there are some minor differences between difference (error) measures of intermediate and fine grids, but in general there is a very good match between the details of the flow for both grids on the planes.

233

Another important observation stemming from figures (7.43) and (7.44) is that in general, the error in the numerical prediction of mean axial velocity ( u z ) is lower than the lateral velocity ( u x ). This observation is valid for all PIV-planes and for both upstream and downstream sections of each plane.

 % B1: upstream

B1: downstream

B 2 : upstream

B 2 : downstream

a) E (u x ) , grid B

b) E (u x ) , grid C

c) E (u z ) , grid B

d) E (u z ) , grid C

Fig 7.43 Difference measure obtained from DDES simulations at OP.4 for both the intermediate (mesh B) and fine (mesh C) grids on planes B1, B2 (upstream/downstream) At this point, it is worth to mention that the computational cost of the DDES simulations, even in the case of applying the circumferential averaged-1D profile, is high; it typically takes few months to have a statistical-converged solution in the case of the intermediate mesh on 120 cores. As discussed later, the situation is worse in the case of 2D rotating profile. Considering these facts and also taking into account all of the aforementioned measures to evaluate the behavior of the intermediate/fine grids including flow details on all PIV-planes, one can reasonably conclude that the ‘intermediate’ mesh 234

exhibits a good compromise between the level of accuracy and computational cost. Therefore, it is adopted for the upcoming simulations with 1D and 2D inflow profiles.

 % S3 ( x  ) : upstream

S3 ( x  ) : downstream

S 4 ( x  ) : upstream

S 4 ( x  ) : downstream

a) E (u y ) , grid B

b) E (u y ) , grid C

c) E (u z ) , grid B

d) E (u z ) , grid C

Fig 7.44 Difference measure obtained from DDES simulations at OP.4 for both the intermediate (mesh B) and fine (mesh C) grids on planes S3, S4 (upstream/downstream) As discussed around figure (7.42), at OP.4 a large separation zone is formed on the side-wall. The position of separation for the simulations is predicted on the x- side-wall, whereas the separation position observed experimentally at Op.4 is on the x+ side-wall. Before discussing about plausible reasons, first of all it is worth to have a brief look on the physics of separation in this case. As mentioned in the first chapter, BulbT draft tube is not symmetric in the vertical direction; in fact in the transition range from the circular to rectangular, both side-walls of the draft tube involve a divergence angle of about 9 , whereas the upper wall is nearly flat and the bottom wall of the draft tube has a 5

235

divergence-angle. These specifications induce higher adverse pressure gradients on the side walls in comparison to the bottom and upper walls of the draft tube. As an immediate result, it is expected that the separation should occur on one of these side-walls or both. Furthermore, it is important to notice that the experimental measurements and numerical simulations unambiguously showed that BulbT draft tube flow is highly unsteady, and experiences a complex chaotic separation on the side-wall at high flow rates like OP.4. Any deviation from the real experimental situation can trigger the separation on the opposite-wall in the numerical simulations like any difference in the hydraulic geometry compared to the model on the test bench. On the other hand, although the mass flow rate is kept constant on the mean in the experiment (or in reality), the experimental data show oscillations with a maximum of about 2% around the mean of the averaged mass-flow rate signal, equivalent to the maximum deviation of about 12% on the total head [Duquesne 2015]. Considering physics of the problem, as a hypothesis, the mass-flow rate variation can affect the pressure gradient field in the draft tube and the pressure gradient can affect the flow separation in turn. In this way, the experimental mass flow rate variation can affect the separation zone. This justification can provide a plausible explanation for the difference observed on the position of the separation on the side-walls between the numerical and experimental counterparts. In fact, in all numerical simulations the inflow mass-flow rate is kept constant at both selected operating points even in the case of the transient simulations. Further investigation on numerical and experimental side is needed on this problem in the future work. As shown in figure (7.42), iso-surfaces of the negative mean axial velocity can be the first representative of the separation zone behavior. To further investigate the dynamics of the separation zone, the concept of ‘reverse-flow intermittency’ can be very helpful. The quantity (  Intermittency ) is defined as a ratio of the time that the flow is in the opposite direction of the streamwise flow to the total time [Broeren 2006]. The values 0 and 1 indicate that flow is always in the streamwise direction and flow is always in the oppositestreamwise direction, respectively. In this regard, the reverse-flow intermittency at OP.4 on the different slices perpendicular to the side walls with y Rref . equals to -0.96, -0.48, 0, 0.48 and 0.96, and also on one slice cutting the separated zone nearly parallel to the side-wall with x Rref . equals to -1.44, are considered. Figure (7.45) shows the intermittency parameter on these defined planes obtained from the DDES simulation of the draft tube flow using circumferential averaged-1D profile.

236

As it is clear in the figure, the separation is formed on the side-wall (x-). It is also visible that the separation is more frequent near the bottom; in other words, the depth of the separated region in x-direction, is also larger in the bottom-half of the draft tube (planes with a negative y Rref . ) in comparison to the top-half of the draft tube (planes with a positive y Rref . ).  Intermittency y  0.96 Rref .

x

x z

z y 0 Rref .

x z

z

y  0.48 Rref .

x z

y  0.96 Rref .

x

y  0.48 Rref .

x  1.44 Rref .

y z

Fig 7.45 Reverse-flow intermittency on the different slices extracted from DDES simulation using 1D-inflow profile at OP.4 On the semi-parallel plane to the side wall ( x Rref .  1.44 ), which cuts the separated zone laterally, a chaotic pattern is formed in the intermittency field which shows the complicated dynamics of the flow in the separated region. As it is also clear in this figure, the separation intermittency is higher near the bottom surface than the top surface of the draft tube in y-direction. Due to the relatively direct link between the separation intermittency and mean separation zone size, these observations in x and y directions indicate that the separation zone should be larger in both directions at the bottom of the draft tube. This conclusion is consistent with previous observations made by extracting isosurfaces of the reverse-flow zone in figure (7.42). It is important to mention that due to the high computational cost of DDES simulations, the total duration of the sampling to calculate the 3D intermittency is limited to 18 runner rotations (i.e. about 1 second) after a statistical convergence. Figure (7.46) shows the variation of the reverse-flow intermittency quantity along with the axial velocity for two

237

probe points located in the separated zone in terms of time. The probe coordinates are presented in table (7.7). Table 7.7 Probe locations to study reverse-flow intermittency quantity convergence No. 1

Coordinates x Rref .

Probe A -1.7

Probe B -1.4

2

y Rref .

0

0

3

z Rref .

5.4

5.4

As one can see in table (7.7), probe A is located closer to the wall, in other words, located deeper in the separated zone which experiences higher frequent reverse-flow phenomena indicated by a higher intermittency value. It is also interesting to notice that the intervals with constant intermittency in the reverse-flow intermittency curves versus time correspond to positive axial velocity events, i.e. u z  0 . uz uref .

uz uref .

time (s)

time (s)

 Intermittency

 Intermittency

time (s)

time (s)

a) Probe A

b) Probe B

Fig 7.46 Evolution of the axial velocity (top) and reverse-flow intermittency (bottom) for two probes extracted from the DDES simulation using 1D inflow profile at OP.4 As it is also visible in the axial velocity signal, variation in the streamwise velocity at both probe positions evolves with a low frequency. This issue can be partially stemmed 238

from the highly chaotic nature of the separated zone dynamics, as well from a relatively short duration of the total sampling time. By increasing the sampling duration time, a better converged state would be obtained for the axial velocity signal and hence for the reverseflow intermittency. After this preliminary discussion about the results of DDES simulations with 1D profile at the inlet, the following sections present the strategy applied to perform DDES simulations using 2D inflow profiles along with the obtained results. It is expected that richer simulated flow dynamics will be obtained. 7.8.3. (a) 2D inflow velocity profile modification

As explained in details in chapter 4, to resolve the runner-related vortical structure and wakes, 2D variation of the inflow profile should be included at the inlet section of the draft tube. To summarize the procedure explained in the aforementioned chapter, following steps are adopted to apply a 2D-inflow profile for DDES simulations: 1) Extraction of the 2D profiles of the turbulent quantities (i.e. k and  quantities) and three components of velocity at the inlet section of the draft tube for one runner-blade passage from full-machine k   simulations. 2) Copying the fields on the one slice (corresponding to only one-blade passage) four times to consider the full-runner arrangement. 3) Performing modifications on the 2D inflow profile to make it compatible to the hybrid LES/RANS needs as well as special manipulations required to correct the velocity profiles based on the experimental data in this case. These modifications partially come from the lessons learned from the OP.1 simulations. As explained in chapter 4, the modifications are as below:

i. Treatment of the 2D inflow turbulent viscosity profile, ii. Treatment of the 2D inflow velocity profile, if any, iii. Application of the rotation on the profile to mimic runner rotation effects, iv. Addition of rotating synthetic inflow turbulence (AFG) at the inlet plane.

4) Feeding the draft tube with the generated inflow profiles in terms of time.

239

Considering all the lessons learned from the previous DDES simulations with 1Dinflow profile at both operating points i.e. OP.1 and OP.4, one can expect that the simulations with 2D profiles should take into account both the u -correction of the inflow profile and amplification of the turbulent viscosity in the WZ simply with a factor of n  3 . In the above list of steps, these two modifications are classified under the sub-items (i) and (ii) in the item (3). Considering an assumption which states that the runner is the only factor capable to make changes on the flow at the draft tube inlet plane, one can reconstruct the 2D phaseaveraged field based on the LDV measurements on one axis while the runner is rotating. This can be done by simultaneously recording the velocity components and also the position of the runner with respect to a reference axis [Vuillemard et al. 2014]. In the present study, to correct the 2D field of the u profile stemmed from the full machine RANS simulation, the above-mentioned reconstructed 2D experimental field obtained from the LDV measurements at OP.4 (part b, figure 7.47) is utilized. In this regard, a blending field is reconstructed based on the experimental and RANS 2D- u fields. Due to the limitations present in the LDV-measurements like the window/wall reflection problem, experimental data are not available on the full radius in regions very close to the walls. In figure (7.47), ‘cyan’ dashed-circles indicate the boundaries of the PIV-measurement zone at the inlet section ( 0.31  r Rref .  0.88 ), whereas the ‘black’ dashed–circles show the core experimental field boundaries ( 0.42  r Rref .  0.8 ) adopted here for the blending purpose.

u uref .

a) RANS

b) Exp. (PIV)

c) Blending

Fig 7.47 Reconstruction of the u profile utilized for DDES simulations with unsteady 2D-inflow profiles

240

Due to some deviations previously observed in the case of experimental 1D- u inflow profile close to the boundaries compared to the RANS 1D- u inflow profile counterpart (figure 7.13), only a fraction of whole experimental field bounded to the ‘black’ dashed-circles in the core flow are used for the blending. Outside of this zone, the RANS u -inflow profile is normalized and blended in a way to create a continuous field especially at the stitching boundary of the RANS and experimental regions (‘black’ dashed-circles), as one can see in part (c) of figure (7.47). In this manner, the u -offset observed in the RANS profile is corrected based on the LDV-experimental data for the case of 2D inflow profile similar to the offset correction previously performed for the 1D-inflow profiles (section 7.6.2). To double check the blended 2D profile of u -component of the velocity, circumferential averaging is applied on the field and the result is compared with the 1D experimental profile at OP.4 in figure (7.48). One can see in the figure that an almost perfect match has been obtained between two sets of data. u uref .

r RInlet

Fig 7.48 Circumferential-averaging of the blended 2D u -profile at OP.4 (black: averaged curve of the 2D profile, red: experimental data) This test along with the above discussion shows that the blended 2D inflow u profile is a well-representative 2D field compared to the experiment. It takes into account r and  variations of the inflow field, which is essential to capture runner-related vortical structure and wakes as shown shortly. 7.8.3 (b) Simulation results using unsteady 2D inflow profiles

As explained in the procedure list in the previous section, for DDES simulations in the case of unsteady 2D inflow profiles, necessary modifications are applied on the inflow profile 241

using the lessons learned from the simulations with 1D profile. In this regard, the u correction of 2D-inflow profile and amplification of the turbulent viscosity in the WZ simply with a factor of n  3 are applied, whereas the axial and radial (i.e. u z and ur ) 2D inflow velocity fields stemmed from the full machine k   simulation are kept untouched. Two types of unsteady 2D inflow profiles are considered in the present study based on the Reynolds triple decomposition which was defined in chapter 4 and rewritten here, as below: ui (t )  uim  uil (t )  ui(t )

(7.14)

As explained before, the first, second and third terms on the RHS of the equation indicate the mean, low frequency (or phase-averaged part) and high frequency (or random part) contents of the turbulent signal, respectively. The two types of 2D inflow profiles adopted hereafter are:

I.

‘2D rotating’ inflow profile, which involves the two first terms of the RHS of above equation. In other words, in this case, the effect of runner rotations on the draft tube inflow is included by the rotation of modified (blended) 2D inflow profiles with the rotation speed of the runner at each operating point. The profiles originally come from the k   full-machine simulations along with applying the blending strategy as described in section (7.8.3 (a)).

II.

‘2D rotating+AFG’ inflow profile, which encompasses all of the terms of the RHS of the above equation. In other words, the effect of inflow turbulence ui(t ) generated by the adopted artificial fluctuation generation (AFG) method, explained in chapter 4, is added on top of the ‘2D rotating’ inflow profile in the LES portion of the inlet section.

It is worth to remember that to avoid doubling the effects of turbulence in the wallregion, which is modeled by RANS method in theory, AFG content is only applied in the core flow, i.e. LES zone of the DDES method. By the aid of the abovementioned 2D inflow boundary conditions, not only the important runner-related vortical structures are resolved, but also the effects of applying the generated synthetic turbulence on the behaviour of simulated turbulent flow in the draft tube can be studied. Figure (7.49) depicts complete turbulent flow structures resolved by DDES simulation at OP.4 applying ‘2D rotating+AFG’ inflow boundary condition colored by the axial velocity quantity.

242

uz uref .

Runner spiral tip-vortices Runner-blade wakes

x Small coherent structures induced by inflow AFG

Hub-induced vortex

y

z

Separated zone coherent structures

Fig 7.49 Turbulent flow coherent structure anatomy resolved by Q  350 coming from DDES simulation using ‘2D rotating+AFG’ inflow profile at OP.4 As one can see in the figure, a complicated flow structure forms in the draft tube at OP.4 involving runner-blade wakes, runner spiral tip-vortices, hub-induced vortex and coherent structures in the separated zone formed on the side wall. Thanks to the adopted low-dissipation scheme, runner spiral tip-vortices are captured in full, which extend to the exit of the draft tube. Another interesting observation about the tip-vortices is that they closely follow the shape of hydraulic passage. In other words, in the conical section, the external shape of the tube of tip-vortical structure is circular, but it is getting changed to a nearly rectangular shape at the exit of the draft tube. It is worth mentioning that specifications of the spring-like tube of tip-vortices depends on the runner specifications, e.g. number of runner-blades, runner-blade geometry and the rotation speed of the runner. As it is also visible in figure (7.49), the opening angle of the spiral tip-vortices, or in other words the distance between two successive circles in the tube of spiral-tip vortices, is relatively constant. At the first sight, one would expect to have more dense circles in the tip-vortex tube towards the exit of draft tube by decreasing the mean axial velocity towards the exit due to increasing the cross-sectional area. In contrast to this initial intuition, in fact, the separation zone effectively creates a hydraulic obstacle in the flow passage which contracts the effect of the overall cross-section increase. As a result, the effective cross-sectional area changes in a way to keep the opening angle of the spiral tip-vortices relatively constant.

243

a) 1D

b) 2D rotating

c) 2D rotating+AFG

Fig 7.50 Effect of different inflow profiles on the resolved coherent structures at one instant of time obtained from DDES simulations at OP.4 (colorbar: u z uref . ) Figure (7.50) shows the top-view of the vortical structures obtained from applying different inflow boundary conditions. As seen in the figure, due to the presence of separated zone, the hub-induced vortex and tip-vortical structure tube (if any) are deflected at OP.4 for all types of inflow boundary conditions, which breaks the symmetry previously observed at OP.1. As one can also see in the figure, applying circumferential averaged-1D inflow profile removes all of the runner-related vortical structures and wakes, and only the hub-induced vortex and separated zone coherent structures are directly resolved. To investigate the flow in more details, first of all, figure (7.51) shows the 1D circumferential-averaged variations of the mean axial and circumferential velocity components at plane B (just downstream the hub) in comparison to the experimental LDVdata. As one can see in the figure, overall there is a mutually good agreement at plane B between all the axial velocity profiles including ‘1D’, ‘2D rotating’ and ‘2D rotating+AFG’ inflow profiles and the experiment, except small deviations near the shroud wall-zone. It is also seen in the mean circumferential profile, as a positive point, applying the 2D variation of inflow profile for both cases including with/without artificial turbulent fluctuations, attenuate the undershoot deviations from the experiment in the hub-zone identified with the range of 0.3  x Rref .  0.3 .

244

uz uref .

Exp. LDV 1Dinflow profile 2D rotating profile 2D rotating+AFG

x Rref.

a) Mean axial velocity variation at plane B, just after the hub u uref .

Exp. LDV 1Dinflow profile 2D rotating profile 2D rotating+AFG

x Rref.

b) Mean circumferential velocity variation at plane B, just after the hub Fig 7.51 Effect of type of the inflow profile on the mean flow obtained from DDES simulations at OP.4 compared to the LDV-experimental data It is also important to notice that the difference in the mean axial or circumferential velocity profiles obtained from the two simulations including/excluding artificial fluctuations, i.e. ‘2D rotating+AFG’ and ‘2D rotating’ inflow profiles respectively, is negligible at plane B. As one can see in figure (7.52), the effect of inflow profile type is minor on the recovery coefficient curves (defined in equation 7.7). In addition, at least in the case of DDES simulations of the BulbT draft tube flow at OP.4, adding artificial fluctuations does

245

not bring major effect on the recovery coefficient. Figure (7.53) shows the time evolution of the global draft tube coefficient (  ) defined in equation (7.13).



Exp. data 1D inflow profile 2D rotating profile 2D rotating+AFG

z Rref.

Fig 7.52 Effect of inflow profile type on the draft tube recovery coefficient obtained from DDES simulations compared to the experiment at OP.4



0.8

0.7

t (s)

2.05

2.35

Fig 7.53 Effect of inflow profile type on the time evolution of the global draft tube coefficient (  ) obtained from DDES simulations at OP.4 As one can see in this graph, the recovery coefficient curves in the case of unsteady 2D profiles reasonably oscillate around the curve with 1D inflow profile, a result linked to resolving incoming flow structures from the runner. Figure (7.53) zoomed window also suggests that there are some minor differences between curves with and without inflow artificial fluctuations (AFG), although, in general the two curves closely follow each other. 246

This confirms the minor effect of AFG on the global quantities like the recovery coefficient. However, as shown shortly in the following sections, AFG affects the details of the flow physics like the unsteady separation dynamics, two-point correlations and fluid mixing. 7.8.3 (c) Effects of the synthetic inflow turbulence on separation topology

To investigate further the effect of adding artificial inflow turbulence (AFG) on the behaviour of DDES simulations, the mean separation topology is shown in figure (7.54). To calculate the mean quantities, 20 runner rotations after statistical convergence are considered. Both simulations (with/without AFG) start from the same fully-developed case with a 1D-inflow profile at the corresponding operating point, i.e. OP.4.

X

Y Z

u z uref .

Z

Y

X

a) ‘2D rotating’ inflow profile

b) ‘2D rotating+AFG’ inflow profile 7.54 Topology of the mean separated region for different unsteady 2D inflow profiles at OP.4 visualized by iso-surface of the negative velocity As one can see in figure (7.54), separation zones obtained from ‘2D rotating’ and ‘2D rotating+AFG’ are similar, although there are small differences between them. For both cases, separation forms on the side-wall at x- in contrast to the experiment, which is at

247

side-wall x+. Therefore, considering the 2D variation of inflow profile along with imposing the inflow turbulence in the LES zone (core flow) does not solve the problem of separation position.  %

B1: upstream

B1: downstream

B 2 : upstream

B 2 : downstream

a)

b)

c)

d)

e)

f)

Fig 7.55 Difference measure obtained from DDES simulations with different inflow at OP.4 on planes B1, B2 (up/downstream) [ a) E (u x ) ,1D; b) E (u x ) ,2D rotating; c) E (u x ) , 2D rotating+AFG; d) E (u  ) ,1D; e) E (u  ) , 2D rotating; f) E (u  ) ,2D rotating +AFG] z

z

z

As explained before, as a hypothesis, this deviation from reality could be probably linked to the oscillations of the real mass flow rate as revealed in the experiment, with maximum about 2% around the mean, equivalent to the maximum deviation of about 12% on the total head [Duquesne 2015]. Whereas, the mass flow rate for all types of inflow profiles are kept constant in terms of time for numerical simulations. It could also be linked to the small differences existing between the hydraulic profiles on the test bench and the CAD constructed model used for CFD simulations. In addition, some triggering factors like vibrations could be involved in the experiment, which is not present in the simulations. As mentioned, this shortcoming should be investigated further in the future studies. It is also worth to notice that some minor differences in the topology of the mean separation zones exist between the two sets of 2D inflow boundary conditions as shown in 248

figure (7.54). It implies an underlying difference in the dynamics of separated zone in terms of time for these two cases. As before, PIV-measurement data can be used to further investigate the effects of inflow turbulence on the flow details. In this regard, planes B1, B2, S3 and S4 are utilized to investigate the mean flow variation in comparison to the PIV-measurement data. The original velocity fields on these planes can be found in sections (G.3.9) to (G.3.16) in appendix G. Here, figures (7.55), (7.56) and (7.57) depict the difference (error) measure obtained from DDES simulations at OP.4 using different inflow profiles in comparison to the experimental data.

 %

S 3 x  : upstream

S 3 x  : upstream

S 3 x  : downstream

S 3 x  : downstream

a)

b)

c)

d)

e)

f)

Fig 7.56 Difference measure obtained from DDES simulations using different inflow at OP.4 on planes S3 x+/x- (up/downstream) [ a) E (u y ) ,1D; b) E (u y ) ,2D rotating; c) E (u y ) , 2D rotating+AFG; d) E (u  ) ,1D; e) E (u  ) , 2D rotating; f) E (u  ) ,2D rotating+AFG] z

z

z

249

As mentioned, the position of separation is at x- side-wall in the simulations, whereas the corresponding position in the experiment is at x+ side-wall. In the experiment, S3 and S4 planes are defined at x+ to study the separation dynamics on the x+ side-wall (here called, S3 x+ and S4 x+). In the simulations, to study the separation zone details, in addition to these two surfaces, the mirror of S3 and S4 on the x- side-wall of the draft tube is created (called hereafter S3 x- and S4 x-). In other words, the S3 x- and S4 x- planes is the mirror of the S3 x+ and S4 x+ planes with respect to the vertical symmetry plane passing the y-direction of the global reference frame of the BulbT assembly. As figures (7.55), (7.56) and (7.57) suggest, the overall error of velocity fields obtained from the DDES simulations for all types of inflow boundary conditions on the downstream planes is higher than the upstream planes. The figures also depict that for all measurement planes including upstream and downstream sections, the error for the lateral components (i.e. u x for B1, B2 planes or u y for S3, S4) is always higher than the error of axial velocity (i.e. u z ). In other words, the simulation predictions are more accurate for the axial velocity component than for the lateral one. As one can see in figure (7.55), considering both components of the velocity field and both B1 and B2 planes including upstream and downstream sections, despite the similarity observed, the DDES simulation with ‘2D rotating+AFG’ inflow profile exhibits the lowest error in comparison to the two other inflow profiles. This is important in itself because it means that AFG still has an impact far away from the inlet. Another visible conclusion stemmed from figure (7.55) is that a similar pattern is visible in the u x error field for all inflow profiles, but the pattern changes in the case of 1D or 2D for the u z error field. In the case of 2D inflow profile, the highest error occurs at the bottom of the plane, whereas for 1D inflow profile, it happens at the top. Figures (7.56) and (7.57) show the error measures on the S3 and S4 planes, respectively. Similar to planes B1 and B2, for both figures the level of error for downstream sections, is higher than for the upstream section; especially for the u y component of the velocity. By comparing figures (7.56) and (7.57), one can also observe that the overall error in the numerical predictions decreases from planes S3 to S4. In other words, by getting closer to the wall, the error increases for both components of the velocity i.e. u y and u z . As it is visible in the figures, u y error indicates a similar pattern on both x+ and xside-walls; whereas the u z error pattern is different for x- and x+ side-walls. Considering especially the downstream sections of planes S3 and S4, despite the presence of different patterns, overall the u z error between the DDES results and the experimental data is lower 250

on the x- side-wall. This issue is related to the presence of flow separation on the x- sidewall in the numerical simulation results.  % 

S 4 x : upstream

S 4 x  : upstream

S 4 x  : downstream

S 4 x  : downstream

a)

b)

c)

d)

e)

f)

Fig 7.57 Difference measure obtained from DDES simulations using different inflow at OP.4 on planes S4 x+/x- (upstream/downstream) [a) E (u y ) ,1D; b) E (u y ) ,2D rotating; c) E (u  ) ,2D rotating+AFG;d) E (u  ) ,1D; e) E (u  ) ,2D rotating;f) E (u  ) , 2Drotating+AFG] y

z

z

z

Although the mean separation zone can be considered as the first indication of the separation zone dynamics, to better understand the effect of applying AFG on the unsteady dynamics of the separated zone, the signature of the reverse-flow over a time period should be studied. To calculate the intermittency (  Intermittency ) quantity, the 3D flow data snapshots are recorded during 18 runner rotations with sampling frequency of 100 Hz. Figure (7.58) shows the variations of intermittency quantity on the different slices cutting the separation zone at different vertical levels, including y Rref . equals to -0.96, -0.48, 0, 0.48 and 0.96, and a slice nearly-parallel to the side-wall with x Rref . equals to -1.44. As one can see in the figure, AFG affects the separation dynamics considerably. 251

Although separation dynamics visualized by the intermittency quantity is different for both cases, size and location of the separated zone extracted by mean velocity looks similar in figure (7.54). That is why the recovery coefficient is relatively insensitive to the AFG. In fact, the details of flow like dynamics of the flow separation, or as shown shortly turbulence spectrum and flow transport barriers are modified by applying AFG, although global quantities like the recovery coefficient do not see these features locally experienced by the flow due to their integral nature.  Intermittency y  0.96 Rref .

x

y  0.96 Rref .

x

z

z y  0.48 Rref .

x

y  0.48 Rref .

x

z

z y 0 Rref .

x

y 0 Rref .

x

z

z y  0.48 Rref .

x

y  0.48 Rref .

x

z

z y  0.96 Rref .

x z

y  0.96 Rref .

x z

x  1.44 Rref .

x  1.44 Rref .

y

y z

a) ‘2D rotating’

z b) ‘2D rotating+AFG’

Fig 7.58 Reverse-flow intermittency on the different slices extracted from DDES simulations using 2D inflow profiles at OP.4 252

 Intermittency

uz uref .

time (s)

time (s)

a) ‘2D rotating+AFG’  Intermittency

uz uref .

time (s)

time (s)

b) ‘2D rotating’ Fig 7.59 Evolution of the axial velocity (left) and reverse-flow intermittency (right) at probe A; extracted from DDES simulations using 2D inflow profiles at OP.4 Figures (7.59) and (7.60) depict the variation of reverse-flow intermittency quantity along with the axial velocity for two probe points located in the separated zone in terms of time for unsteady 2D profiles including ‘2D rotating’ and ‘2D rotating+AFG’. The probe coordinates are the same as before in the case of 1D inflow profile (table 7.7). It is important to notice that probe A is located deeper in the separated zone and experiences higher frequent reverse-flow phenomena, which is translated to the higher intermittency value. As it is also visible in the axial velocity signal, variation in the streamwise velocity in the case of ‘2D rotating’ inflow profile still continues with low frequency but with relatively small amplitude, however in the case of ‘2D rotating+AFG’ inflow profile the signal reaches to a limit-cycle oscillation state. As explained before, this behavior could be stemmed partially from the highly chaotic nature of the separated zone dynamics, as well partially from a relatively short duration of total sampling time. By increasing the sampling duration time, a better convergence on this quantity would be achieved, but due to the high 253

computational cost of DDES simulations especially in the case of unsteady 2D inflow profiles, this became prohibitive.

 Intermittency

uz uref .

time (s)

time (s)

a) ‘2D rotating+AFG’  Intermittency

uz uref .

time (s)

time (s)

b) ‘2D rotating’ Fig 7.60 Evolution of the axial velocity (left) and reverse-flow intermittency (right) at probe B; extracted from DDES simulations using 2D inflow profiles at OP.4 Figure (7.61) shows the LES content of DDES simulations in the case of applying unsteady 2D inflow profiles. The parameter relative sgs-viscosity index can be adopted [Celik 2005] to quantify sufficiency of the adopted grid resolution for LES-like i.e. DDES simulations, as below: 1 (7.15) LES  IQv  veff . nv 1 v ( ) v where nv and  v are two constants equal to 0.53 and 0.05, respectively [Celik 2009]. The effective viscosity is obtained by veff .  vsgs  vnum  v where the first, second and the third

254

terms are subgrid-scale, numerical and fluid flow kinematic viscosity, respectively. The numerical viscosity is also estimated by the following equation if one assume the same LES-filter size and grid spacing: vnum  cv  k sgs , where cv  0.1 . The LES-index should be higher than 0.8 for a good LES simulation and higher than 0.95 in the case of DNS simulations [Celik 2009]. LES  IQv

a)

‘2D rotating’ inflow profile

b) ‘2D rotating+AFG’ inflow profile

Fig 7.61 LES-content identified via LES  IQv obtained from DDES simulations with different unsteady 2D inflow profiles at OP.4 As one can see in figure (7.61), a similar behavior is observed for ‘2D rotating’ and ‘2D rotating+AFG’ inflow profiles, as expected a-priori due to the same computational grid for both cases which means the same resolution. More importantly, the figure also suggests that the core flow is majorly well-resolved by LES mode of the DDES technique with LES-index as high as 0.85, although LES  IQv partially decreases near the centerline of the draft tube. 7.8.3 (d) Effects of the inflow synthetic turbulence on the turbulence energy spectrum at plane B

To deepen the analysis, here the effect of synthetic turbulence on the details of the flow physics is further investigated via computing the turbulence spectrum compared to the LDV-experimental data at plane-B section, just after the hub. Figure (7.62) shows the turbulence spectra of the u and u z components of the turbulent velocity signal for experimental LDV data and DDES simulations using three different types of the inflow boundary conditions, namely ‘1D’, ‘2D rotating’ and ‘2D rotating+AFG’. In the figure, the passing frequency of the runner blades is also marked with an orange arrow.

255

E u ( f )

E uz ( f )

Cut-off frequency 

f f runner

Cut-off frequency

5 3



Blade passing frequency

f f runner

5 3

Blade passing frequency

a) x Rref .  0.64 E uz ( f )

Cut-off frequency

E u ( f )



f f runner

Cut-off frequency

5 3

Blade passing frequency

f f runner



5 3



5 3

Blade passing frequency

b) x Rref .  0.77 E uz ( f )

Cut-off frequency

E u ( f )



f f runner

Cut-off frequency

5 3

Blade passing frequency

f f runner

Blade passing frequency

c) x Rref .  0.88 Fig 7.62 Energy spectra of the turbulent velocity signals obtained from DDES simulations with different inflow profiles at different x-positions of plane B

256

It is also important to notice that depending on the resolution of the grid used for LES-type simulations, there is a so-called ‘cut-off frequency’ for which no frequency can be captured by the mesh higher than this Nyquist frequency [Sagaut 2006]. Theoretically, the highest frequency on a grid can be achieved by assuming a wave in which the wave length is equal to twice grid spacing ( 2 ). Then, the highest achievable frequency is obtained by the following formula f c  2 2    , where for non-uniform cells  is defined based

on the grid cell-volume as   3 V . Based on the specifications of the mesh utilized for the DDES simulations, the cut-off frequency normalized by a rotation frequency of runner is f c f runner  60 . The violet solid-line on the graph indicates the cut-off frequency for the obtained spectra. As one can see in figure (7.62), all obtained spectra including the experimental and numerical simulation ones, up to the cut-off frequency could reproduce the (-5/3) decay slope in the power spectrum. In addition, as it is clear in the figure, 1D inflow profile generates the farthest curve compared to the experiment. By including the unsteady 2D variation of the inflow profile, the numerical energy curves are getting closer to the experimental curve although there is still a gap in the level of energy between the numerical and experimental spectra. This issue can also be linked to the variation of mass-flow rate in the experimental measurements as previously mentioned, which can in general affect the level of fluctuations by unsteady energizing the flow. The figure also shows that there are some low frequency phenomena in the range of 4

4  10  f c f runner  4 102 , which are captured in the experiment but not present in the simulations. This issue might also be linked to the signature of fluid phenomena in the upstream components of the draft tube which are excluded in the stand-alone draft-tube flow simulations. This could also be related to the mass flow rate variation in the experiment and the short duration of sampling of the turbulent velocity signal in the numerical simulations. The velocity is sampled with a frequency of 104 over the total sampling time equals to 20 runner rotations, which does not let the numerical signal see all the low frequency phenomena present in the experiment. Figure (7.62) also suggests that by taking into account the artificial fluctuations (AFG) on the inflow profile in the DDES simulation, details of the flow are modified and are getting closer to the experimental counterpart, although, as previously discussed, global quantities like the draft tube recovery coefficients do not necessarily take into account these details. It is important to notice that for global measures like the recovery coefficient, these details would become important, if they trigger the separation or other phenomena in the draft tube affecting the global dynamics of the flow; otherwise, like in the BulbT draft tube flow case at OP.4, only the details of the flow physics is modified by including AFG.

257

7.9 Effect of synthetic inflow turbulence: two-point correlation analysis In this section, to grasp a better insight into the details of flow physics, the effects of AFG on the draft tube flow are investigated via the concept of two-point correlation. From a physical point of view, two-point correlation is a measure that indicates how much two points sense each other in the flow field. In other words, the quantity shows the influence of a point on the other point in the field. To define the quantity utilized here, it is worth to refer back to the Reynolds triple decomposition concept, which is rewritten as below: ui (t )  uim  uil (t )  ui(t )  uim  ui† (t )   mean

(7.16)

unsteady part

where the unsteady term ui† (t ) exhibits the time varying part of the turbulent velocity signal. This term involves both the low frequency uil (t ) and high frequency ui(t ) contents of the signal. To study the effects of AFG, two sets of so-called normalized two-point correlation are defined. First, for the total unsteady term i.e. ui† (t ) , as below: 

†, norm ii

ui† ( z I )ui† ( z I  zˆI )  † ui ,RMS ( z I )ui†,RMS ( z I  zˆI )

(7.17)

where z I is the base-point position located at draft tube inflow plane with the same radial coordinate to the other point located at ( z I  zˆI ), and zˆI indicates the separation distance between the two-points of interest on a plane. The bar sign on the quantities also shows the temporal mean operator. The ui†,RMS also stands for the root mean square (RMS) of the total unsteady velocity fluctuations. In a similar manner for the semi-random turbulent fluctuation term ui(t ) , the quantity is defined as below:  iinorm 

ui( z I )ui( z I  zˆI ) ui,RMS ( z I )ui,RMS ( z I  zˆI )

(7.18)

In the above formulations, i (= 1, 2, 3) stands for the different components of the velocity namely: radial, circumferential and axial components, respectively. To calculate the aforementioned quantities, a set of probe nodes on three different planes are defined in the domain to record the turbulent velocity signals coming from the DDES simulations. Figure (7.63) depicts the positions of the probes in the computational domain. As one can see in the figure, the probes are located on the three different planes, namely planes 1, 2 and 3, having mutual difference angle of 120 which allows to investigate the symmetry of the defined quantities on a full-circle. As it is also obvious in the figure, to better capture the effects of AFG, the probe resolution is denser close to the 258

inlet section of draft tube, whereas the grid is expanding towards the exit. On each plane, the node set consists of 10 rows (ic=1,..,10) and 47 columns of probe locations as shown in the figure. The planes are spread in the axial direction as 0.38  z / Rref .  4.14 and in the radial direction over 0.36  r / Rref .  0.94 .

y x

z

Plane 2

Plane 1

ic  1 ic  10

ic  10 ic  1

Plane 3

Plane 2 Plane 2 Plane 1 Plane 3

Plane 1

y

y

z x

Plane 3

Fig 7.63 Triple view of the probe positions adopted to calculate two-point correlation (top: isometric view, bottom-left: side-view, bottom-right: front-view) First of all, in figure (7.64) the normalized two-point correlation for the total †, norm unsteady term of radial component of the velocity is considered, i.e. 11 . As one can see

in the figure, a ‘zebra-like’ pattern is created by successive positive and negative two-point correlation zones; as a result, a cyclic pattern is created. As shown in figures (7.66) and (7.67), similar patterns are also obtained for the other components of the velocity i.e. circumferential and axial ones, respectively. Considering individual pairs of (a,b), (c,d) and (e,f) in figure (7.64), it is clear that adding AFG on the inflow profile does have negligible effect on the two-point correlation quantity defined for the total unsteady term of the radial †, norm , on all three planes 1, 2 and 3. component of the velocity, 11

Another interesting observation that can be made from the figure is the symmetry †, norm fields obtained on all three defined planes despite the minor which exists among the 11 †, norm differences present. It means that 11 quantity can be effectively considered independent

of the circumferential direction. Similar conclusion is still valid for two other components of the velocity i.e. circumferential and axial components of the velocity as shown in figures (7.66) and (7.67), respectively. 259

†, norm 11

r Rref.

z Rref.

a) Plane1, no AFG r Rref. z Rref.

b) Plane1, with AFG r Rref. z Rref.

c) Plane2, no AFG r Rref. z Rref.

d) Plane2, with AFG r Rref. z Rref.

e) Plane3, no AFG r Rref. z Rref.

f) Plane3, with AFG Fig 7.64 Normalized two-point correlation field for the radial velocity unsteady term i.e. †, norm 11 extracted from DDES simulations at OP.4 with/without AFG on planes 1, 2, 3 It is a normal outcome because the low-frequency content of the turbulent signal, uil (t ) , dominates the high-frequency content ui(t ) , in terms of the magnitude; in other words, uil (t ) suppresses the effects of ui(t ) in the  †,ii norm term. As a result, the first defined

260

set of two-point correlation i.e.  †,norm in equation (7.17) is not a good measure to study the ii effects of AFG. Table 7.8 Probe locations on plane 1 to study u1† (t ) No. 1

Coordinates x Rref .

Probe C 0.68

Probe D 0.68

2

y Rref .

0

0

3

z Rref .

0.4

0.64

To understand how the ‘zebra’ pattern is generated, the radial velocity signal for two probe points located at two successive positive and negative two-point correlation zones are plotted in figure (7.65). The probes are located on plane 1 with specifications listed in table (7.8). The two-point correlation of unsteady term ui† (t ) of the radial turbulent †, norm velocity signal i.e. 11 , is equal to 0.81 (positive) and -0.28 (negative) for the probes C

and D, respectively.

u1† uref .

time( s )

u1† uref .

time( s )

Fig 7.65 Variation of the unsteady portion of turbulent radial velocity signal u1† (t ) in terms of time obtained from DDES simulations with ‘2D rotating +AFG’ inflow profile at OP.4 at two probe positions presented in table 7.8 (red: probe C, blue: probe D)

261

 †,22norm r Rref. z Rref.

a) Plane1, no AFG r Rref.

z Rref.

b) Plane1, with AFG r Rref. z Rref.

c) Plane2, no AFG r Rref. z Rref.

d) Plane2, with AFG r Rref. z Rref.

e) Plane3, no AFG r Rref. z Rref.

f) Plane3, with AFG Fig 7.66 Normalized two-point correlation field for circumferential velocity unsteady term,  †,22norm , extracted from DDES simulations at OP.4 with/without AFG on planes 1, 2, 3 As an example, if one consider two sinusoidal signals with mutual shifting phase angle of  , i.e. sin(t) and sin(t +  ) , the two-point correlation quantity would be equal to -1 over any time interval. This is an ideal example, in which two signals are completely mirror of each other with respect to the time axis. As one can also see in figure (7.65), time variation of u1† (t ) for probe C (red curve) is somehow opposite to the one obtained for 262

probe D (red). In other words, typically, when one of the signals is at maximum, another one is at its minimum, which results in positive (0.81) and negative (-0.28) two-point correlation values for probes C and D, respectively. This example clearly shows how the ‘zebra’ pattern is generated in the two-point correlation of the ui† (t ) signals. It is also linked to the applying unsteady 2D inflow profile in which the runner-related vortical structure and wakes are resolved.  †,33norm r Rref. z Rref.

a) Plane1, no AFG r Rref.

z Rref.

b) Plane1, with AFG r Rref. z Rref.

c) Plane2, no AFG r Rref. z Rref.

d) Plane2, with AFG r Rref. z Rref.

e) Plane3, no AFG r Rref. z Rref.

f) Plane3, with AFG Fig 7.67 Normalized two-point correlation field for axial velocity unsteady term i.e.  †,33norm extracted from DDES simulations at OP.4 with/without AFG on planes 1, 2, 3 263

A similar explanation is also valid for the circumferential and axial components of the velocity as shown in figures (7.66) and (7.67), respectively.  norm 11 r Rref. z Rref.

a) Plane1, no AFG r Rref. z Rref.

b) Plane1, with AFG r Rref.

z Rref.

c) Plane2, no AFG r Rref. z Rref.

d) Plane2, with AFG r Rref. z Rref.

e) Plane3, no AFG r Rref. z Rref.

f) Plane3, with AFG  norm Fig 7.68 Normalized two-point correlation field of radial velocity turbulent term i.e. 11 extracted from DDES simulations at OP.4 with/without AFG on planes 1, 2, 3 Thus far, only the two-point correlation for the total unsteady term was considered is not a good measure to see the effect of inflow and it was shown that the quantity  †,norm ii AFG. To proceed further, two-point correlation for the random turbulent term i.e.

264

iinorm defined in equation (7.18) is adopted. Figures (7.68) to (7.70) depict the 2D field of

the iinorm quantity for the radial, circumferential and axial components of velocity, respectively. As one can see in the figures, in general the two-point correlation iinorm is maximum at the inlet section and decreases gradually as moving away from the inlet section as expected. 22norm r Rref. z Rref.

a) Plane1, no AFG r Rref. z Rref.

b) Plane1, with AFG r Rref.

z Rref.

c) Plane2, no AFG r Rref. z Rref.

d) Plane2, with AFG r Rref. z Rref.

e) Plane3, no AFG r Rref. z Rref.

f) Plane3, with AFG Fig 7.69 Normalized two-point correlation field of circumferential velocity turbulence term i.e. 22norm extracted from DDES simulations at OP.4 with/without AFG on planes 1, 2, 3 265

As one can see in figures (7.68) and (7.69) considering pairs of (a,b), (c,d) and (e,f), by adding the inflow AFG, the pattern of two-point correlation iinorm is modified. It increases in overall and diffuses further downstream in the streamwise direction.

33norm r Rref.

z Rref.

a) Plane1, no AFG r Rref. z Rref.

b) Plane1, with AFG r Rref. z Rref.

c) Plane2, no AFG r Rref. z Rref.

d) Plane2, with AFG r Rref.

z Rref.

e) Plane3, no AFG r Rref. z Rref.

f) Plane3, with AFG Fig 7.70 Normalized two-point correlation field of axial velocity turbulence term i.e. 33norm extracted from DDES simulations at OP.4 with/without AFG on planes 1, 2, 3

266

The other important observation is that the symmetry previously observed in the is not observed for the iinorm anymore. In other words, the symmetry breaks case of  †,norm ii among planes 1, 2 and 3 for the iinorm quantity. This also confirms that AFG influences the details of flow physics at the downstream of the draft tube.

 norm 11

 norm 11

 norm 11

22norm

22norm

 22norm

33norm

33norm

z Rref.

z Rref.

a) Plane 1

z Rref.

z Rref.

z Rref.

33norm

z Rref.

z Rref.

z Rref.

b) Plane 2

z Rref.

c) Plane 3

Fig 7.71 1D plot of iinorm , i  1, 2,3 extracted from the DDES simulation with inflow AFG at OP.4 plotted at different rows (ic=1,..,10) of the probe locations on planes 1, 2, 3

267

The most noticeable effects of AFG happen for the axial component of the velocity as shown in figure (7.70). As one can see, 33norm is deeply affected by considering the AFG which expands almost to the end of the region; whereas without AFG, the correlation rapidly decreases just after the inlet section for all three defined planes. It is also visible that the symmetry also breaks for 33norm among the defined planes. To better analyze the normalized two-point correlation field of the random turbulent term iinorm , the 1D variation of the 2D field is plotted along different rows (ic=1,…,10) in figure (7.71). One can see that the symmetry breaks among the defined planes, plausibly due to the dissymmetry created by the separation zone in the draft tube. In addition, for all three components of the velocity on the three defined planes, iinorm gradually decreases from unity to zero. In the case of radial and circumferential  norm and 22norm curves components of the velocity, there exist serious oscillations in 11

starting approximately from z / Rref .  2 . This issue is plausibly linked to the influence of rotating hub-vortical structures on the turbulent fluctuations as one can see in figure (7.72). In fact, around z / Rref .  2 , the probe planes are getting very close to the hub-vortex  norm and circumferential 22norm two-point structure; the oscillation observed in the radial 11

correlations is the finger-print of approaching to this rotating vortical structure. It is interesting to notice that no oscillation is observed in the two-point correlation of axial velocity 33norm , which means that fluctuations in the z-direction is not very much affected by the hub-vortex. This issue is probably linked to the higher relative magnitude of the axial velocity compared to the circumferential and radial velocities, which keeps minimum the influence of hub-vortex rotation on the turbulent fluctuations. As shown shortly in the following with the aid of scatter plots, the fluctuations in the z-direction are not affected by the presence of the hub-vortex. z / Rref .  2 S8 S7 S6

S1

S2 S3 S4

S5

x z Fig 7.72 Sketch of plane 1 and the hub-vortex at OP.4 indicating positions of probes S1 to S8 utilized for the scatter plots 268

To investigate further, two different paths including axial and radial directions, cutting through the hub-vortex are considered on the three defined planes as shown for plane 1 chosen as an example in figure (7.72). On each plane, 8 different probes on both paths with coordinates presented in table (7.9) are defined as shown in figure (7.72). Table 7.9 Probe locations to study the turbulent fluctuations Coordinates r Rref .

S1 0.36

S2 0.36

S3 0.36

S4 0.36

S5 0.36

S6 0.55

S7 0.74

S8 0.94

z Rref .

1.12

2.69

2.94

3.42

4.14

4.14

4.14

4.14

As one can see in the case of ( ur  u z ) scatter plot in figure (7.73), the fluctuations span in the z-direction is relatively the same ( 4  uz uz ,rms  4 ) for all probe positions along the horizontal line for all of three defined planes; whereas, ur ur, rms decreases when moving downstream along the horizontal axis from S1 to S5. As a result, the slope of diagonal axis of the fluctuation clouds decreases and ultimately stands horizontally as marked with red arrows. For ( u  uz ) scatter plot in figure (7.73), despite anomalies observed for plane 2, u u ,rms decreases by moving downstream along the horizontal axis from S1 to S5; whereas the fluctuation span in the z-direction is relatively maintained the same ( 4  u z uz , rms  4 ) as in the previous ( ur  uz ) scatter plot. This also confirms the previous observation that only the circumferential and radial velocity fluctuations, hence  norm and 22norm , are affected by approaching to the hub-vortex, not the corresponding 11 axial velocity fluctuations or 33norm . Figure (7.74) also shows the variation of the fluctuation clouds by moving from S5 (in the hub-vortex zone) to S8 (outside of hub-vortex zone) in the radial-outward direction. As one can see in figure (7.74), depending on the position of the plane cutting the hubvortex, the turbulent fluctuation dynamics is different. This issue is linked to the complicated 3D skeleton of the coherent structures in the rotating hub-vortex. For example, for the ( ur  uz ) scatter plot in figure (7.74), the maximum ur ur, rms events happens at probe S5 on plane 1, whereas it happens at S7 on plane 3. In addition, in the ( u  uz ) scatters, the maximum u u ,rms events occurs at S6 on plane 1, at S5 on the plane 2 and at S7 on plane 3. The  -dependent behavior of the turbulent fluctuations here is due to the proximity to the hub-vortex, although, as observed in the latter discussion and also the two-point correlation curves (i.e. iinorm ) in figure 269

(7.71), fluctuations in the z-direction are not affected. The influence of hub-vortex is limited to the circumferential and radial components of the velocity. In the next subsection, to further investigate the effects of AFG on the flow, the concept of Lagrangian coherent structures (LCS) is introduced.

ur ur, rms

u u , rms

u z uz, rms

u z uz, rms

a) Plane 1

ur ur, rms

u u ,rms u z uz, rms

u z uz, rms

b) Plane 2

u  u ,rms

ur ur, rms u z uz, rms

u z uz, rms

c) Plane 3 Fig 7.73 Scatter plot of the turbulent fluctuation clouds along the axial path corresponding to the probes (S1: blue; S2: green; S3: magenta, S4: red; S5: black)

270

ur ur,rms

u u ,rms

u z uz, rms

u z uz, rms

a) Plane 1 u u ,rms

ur ur, rms

u z uz, rms

u z uz, rms

b) Plane 2 ur ur, rms

u u ,rms

u z uz, rms

u z uz, rms

c) Plane 3 Fig 7.74 Scatter plot of the turbulent fluctuation clouds along the radial path corresponding to the probes (S5: blue; S6: green; S7: red, S8: black)

7.10 Flow skeleton: Lagrangian Coherent Structure (LCS) analysis In general, coherent structures are persistent entities so-called ‘material surfaces/lines’ in the flow domain, which play a crucial role in the unsteady dynamics and fluid-mixing 271

characteristics of the fluid flows [Shadden 2012]. As opposed to the Eulerian structure detection techniques (appendix E), which basically rely on the instantaneous velocity field and its gradients, in the Lagrangian perspective the hidden coherent structures in the flow emerge from considering the fluid flow changes over a given time interval by applying some kind of integration operators. In this section, the flow skeleton in the BulbT draft tube is extracted and studied based on the concept of the Lagrangian coherent structures (LCS). 7.10.1 Introduction

The Lagrangian coherent structure (LCS) is a mathematical tool to detect flow skeleton in the Lagrangian point of view; in fact, it shows the Lagrangian boundaries within the fluid flow system in the context of dynamical systems [Kent et al. 2008]. The technique has been widely adopted in the recent years to identify transport barriers in the fluid flow systems for various fields of applications. For example, the concept was successfully applied for extracting coherent structures and transport barriers of 3D turbulent fluid flows, for example, in the DNS of boundary layer [Green et al. 2007] or in the experimental quasitwo dimensional turbulent flows [Mathur et al. 2007], to name a few. The technique has been also for the detection of fluid flow coherent structures in the environmental and biological fluid flows [Samelson 2013] for detecting dynamical barriers in the bio-inspired fluid flows [Green et al. 2010], in aortic valve jets [Shadden et al. 2010] and in the analysis of the 3D flow in jellyfish feeding process [Peng and Dabiri 2009]. It was also utilized to study the inertial particle dynamics in a hurricane [Sapsis and Haller 2009], to understand the transport barriers in the case of river current flows [Harrison and Glatzmaier 2012] and as well in the astrophysical applications e.g. the internal dynamics of the solar photospheric turbulent flows [Chain et al. 2014] and to study the dynamical barriers of the strong jet stream existing on Jupiter [Hadjighasem and Haller 2014]. In all the above-mentioned cases, the dynamical barriers or so-called separatrices in the fluid flow system are extracted as relatively long-lived structures by which separate sets within the flow field with different fates are identified. The basic concept of the LCS applied to study fluid flow systems was initially introduced in the theory of non-linear dynamical systems [Haller and Yaun 2000]. In this theory, which first introduced by Poincare, a dynamical system is defined as a set of equations governing the time evolution of a system [Holmes 1990]. In this context, a system is analysed both quantitatively and more importantly qualitatively; in fact in this field of research, the destiny of zonal sets within the system domain with distinct asymptotic behaviour is studied [Kent 2008]. In this regard, the concept of invariant manifolds in particular, stable and unstable manifolds corresponding to the repelling and attracting LCS are adopted to extract the transport barriers in the dynamical system theory 272

(figure 7.75). As one can see in figure (7.75), two particles positioned initially close together are getting far apart in the proximity of a stable manifold (repelling LCS); as opposite, near an unstable manifold (attracting LCS) two separated particles approaching the material surface. It is worth to emphasize that the particles in the different sides of a stable or unstable manifold have different asymptotic fates. In fluid flow systems, the LCS material surfaces (or lines) are the places in the flow field that the maximum stretching rate exists compared to its neighbor points [Tallapragada 2010]. Considering the later characteristic, one should also expect maximum concentration of the particle tracers on these manifolds in the flow field. Tallapragada showed in his Ph.D. thesis that for the specific case of the atmospheric transport of spores, the maximum concentration of spores measured at the probe points is linked to the passage of timedependent atmospheric LCS structures over these probe positions [Tallapragada 2010]. Inspired by this characteristic, the idea to extract these material LCS surfaces (lines) qualitatively in the LDV or PIV measurements, in which the flow is seeded by tracer particles, is to find zones (surfaces or lines) within the flow with maximum concentration of the tracer particles. Unstable manifold

Stable manifold

a) Attracting LCS

b) Repelling LCS

Fig 7.75 Flow trajectory in the proximity of the two types of dynamical manifolds (solid red circle: initial position; hollow red circle: final position of fluid particles) In the case of turbulent flows, stable and unstable manifolds forms a complex weft pattern so called ‘Lagrangian skeleton of the turbulence' in which a fluid particle simultaneously repels and attracts by the repelling and attracting LCS [Mathur et al .2007]. In this perspective, the complicated fluid particle motions in the turbulence can be interpreted based on the effects induced by the underlying LCS web on the fluid particles. Furthermore in the energy cascade process, the skeleton of turbulent flows causes stretching of the incoherent vortices which leads to energy transfer from large to small scales and amplification of the vorticity [Bourgeois et al. 2012]. One way to extract the LCS structures in application is to find the ridges of the finite-time Lyapunov exponent (FTLE) field. The following subsection presents the mathematical formulation and the 273

computational strategy applied to extract the LCS in the DDES simulation results of the turbulent fluid flows in the BulbT draft tube. 7.10.2 Mathematical formulation

To find the separatrices in the non-autonomous (i.e. time-dependent) dynamical systems like fluid flow systems, an indirect approach is to look at the behaviour of the trajectories in the proximity of these important structures. In this sense, the concept of finite-time Lyapunov exponent (FTLE) that measures the maximum divergence of trajectories over a given period of time is an appropriate tool [Shadden et al. 2005]; in the literature, it is equivalently called direct Lyapunov exponent (DLE) after Haller’s work [Haller 2001]. Mathematically, by writing the Taylor series expansion for a neighbor point of an arbitrary point X in the system domain over a given period of time T , the rate of stretching of two neighbor particles is obtained as below [Shadden et al. 2005, 2006]:

  t00 1 1 C G ( C G ))  ln  ln( max t0  T T T  X t0 t T



FTLE ( X ,t0 ,T)

(7.19)

C G is the maximum eigenvalues of the finite-time Cauchy-Green deformation where max

tensor (  C G ) which is a symmetric matrix, defined as below:  C G

   tt00 T    X tt0 T 0 

*

   

   tt00 T    X tt0 T 0 

   

(7.20)

where ( * ) sign denotes the transpose operator. The infinitesimal difference (   tt0 ) in the FTLE field also depicts the differential distance between stretched points in the time interval, T . It is worth emphasizing that the calculation of FTLE is performed both in the forward-time fashion ( T  0 ) corresponding to the stable manifold (or repelling LCS) and also in the backward sense ( T  0 ) equivalent to the unstable manifold (or attracting LCS). That is why the time interval in equation (7.19) is written as T instead of T [Shadden et al. 2005]. For autonomous (time-independent) dynamical systems, the concept called maximal Lyapunov exponent is successfully utilized [Barreira and Pesin 2002]. It is typically defined as equation (7.19) but at the limit T   . Thus, it needs to know the behaviour of system at T   , which is only the case for steady-state systems. It cannot be applied for unsteady fluid flow systems in which the flow field data are known for a given period of time only (i.e. simulation or measurement duration). As explained above, for the time-

274

dependent fluid dynamical systems, the finite-time Lyapunov exponent (FTLE) tool is successfully adopted. Having on hand the FTLE field corresponding to a fluid flow system, the LCS structures can be distinguished as ridges of the FTLE field indicating high FTLE values. This can be simply performed by applying a threshold limit on the FTLE field. It is worth to notice that according to a theorem derived by Shadden, flux across a LCS as a ridge in the FTLE is negligible [Shadden et al. 2005]. This issue is an important property of the LCS structures, which defines the dynamical transport barriers in fluid flow systems. In the case of turbulent flows, it should be also noted that the transport barriers are the convective-barriers, whereas the turbulent transport across the Lagrangian manifolds via diffusion-type mechanism can still be present [Bourgeois et al. 2012]. Thus far, the details of the method adopted to extract LCS structures within the flow field was introduced; as explained the LCS in the flow filed is identified by the ridges of high FTLE values using a kind of thresholding operation. In the next section, the effects of time interval T on the extracted LCS structures are studied in the case of lid-driven cavity flow as a simple and fast running test case. Then, the turbulent flow inside draft tube coming from the DDES simulations is studied using LCS concept in the upcoming sections. 7.10.3 Effect of integration time on the LCS structures: Lid-driven cavity test case

In general, the value of integration time T in equation (7.19) affects the level of the spatial resolution of the LCS structures [Lipinski et al. 2008]. Theoretically, more details of the flow structures are captured in the FTLE field by increasing T , but to a specific limit. After this limit, increasing T causes obscuring essential LCS structures in the flow field by resolving unimportant flow features. To exemplify the effect of integration time on the LCS structures captured in the FTLE field and also have a closer feeling about the LCS structures, the flow in a lid-driven cavity test case is analysed. 7.10.3 (a) Numerical simulation of the lid-driven cavity flow

In this test case, the flow trapped in a cavity is moving due to the constant motion of its upper lid, which is continuously pulled in x  direction with a constant speed. Here, the Reynolds number is about 1.5 106 based on the hydraulic diameter of the cavity (equal to 0.15). At this Reynolds number, the flow in the cavity is turbulent. To simulate the velocity field and extract the LCS structures in this case, the flow field is simulated by the Piso solver of OpenFOAM. The turbulence is treated by k   URANS. The wall-function is applied on the cavity walls, which removes the necessity of the grid clustering in the 275

proximity of the wall. As a result, the mesh adopted here has the resolution of the 120  40  1 cells. The time step is set equal to 0.001 second. Figure (7.76) depicts the velocity vector field along with streamlines stemmed from the k   URANS simulation at steady-state converged state. As one can see in the figure, two counter-rotating vortical zones, one in the core and another in the corner are formed in this fluid flow system indicated by presence of circular streamlines.

Y X Fig 7.76 Streamlines and velocity vector field of the cavity fluid flow stemmed from k   URANS simulation at Re  1.5 106 7.10.3 (b) LCS structures of the cavity flow

As explained in the section of the FTLE field formulation, a direct way to extract repelling and attracting LCS structures is to calculate FTLE field over a period of time in the forward directions, respectively. Here, the FTLE field ( T  0 ) and backward ( T  0 ) corresponding to the cavity flow is calculated for both time-directions using a computational LCS code developed in the biological propulsion laboratory of Caltech [Peng and Dabiri 2009]. In the following, effects of the integral time T on the extracted LCS structures are studied. Figures (7.77) and (7.78) depict the effect of integration time on the revealing of the repelling and attracting LCS structures, respectively. In these figures, the integral time is expressed as a factor of a reference time i.e. T   nCTC , where TC  2 s . Figure (7.77) shows the hidden repelling LCS of the cavity flow distinguished by red/yellow colors corresponding to high FTLE values. The structure forms a spiral towards the vortex core on the right part of the cavity. The internal structure of the spiral is more revealed by increasing the integration time, as one follows the sub-figures from (a) to (f) in the figure. It is also clear that by increasing the integration time, on the left side of the cavity, boundaries of a secondary corner vortex are also detected. Figure (7.78) is the FTLE field for the backward time integration of the cavity flow, as one can see, similar to the forward integration fashion, by increasing the integration time T more structures emerge in the FTLE field. 276

a) nC  1

b) nC  2

c) nC  3

d) nC  4

e) nC  6

f) nC  10

Fig 7.77 FTLE field of the velocity field in the cavity flow case obtained from forward time integration ( T  nCTC , TC  2s ) As one can see about the secondary vortex in the cavity in both forward and backward integrated FTLE fields, the outer most boundaries of a vortex in the Lagrangian sense can be captured by the repelling or attracting LCS (see FTLE fields with nC  10 in figures 7.77 and 7.78). As mathematically proved by Farazmand and Haller, these material LCS boundaries possess characteristics of maximum coherence life-time in the flow field. In other words, any other closed curve exterior to these curves loses its coherence more rapidly under advection [Farazmand and Haller 2014, Farazmand 2014]. In this sense, the boundaries of the vortex identified by LCS can be viewed as one of the most long-lived structures in the flow field in the Lagrangian perspective.

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a) nC  1

b) nC  2

c) nC  3

d) nC  4

e) nC  6

f) nC  10

Fig 7.78 FTLE field of the velocity field in the cavity flow case obtained from backward time integration ( T   nC TC , TC  2 s ) 7.10.4 Definition of the LCS planes for draft tube flow analysis

Thus far, the aforementioned discussions aimed to create a backbone about the concept of the LCS structures. In the following subsections, turbulent flow fields in the draft tube of BulbT turbine is analysed with the aid of the LCS concept. In this regard, the flow field data under investigation come from the DDES simulations of the draft tube flow using different inflow profiles at OP.4. The core reverse-flow bubble at OP.1 visualized in the DDES simulation is also analysed by the LCS concept. Similar to the cavity case, FTLE fields on different 2D planes in the 3D computational domain are computed using the computational LCS code [Peng and Dabiri 2009]. Figure (7.79) shows the different sliceplanes utilized for the LCS analysis of draft tube flow at OP.4.

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z xy3

x

xz 2 xy1

xz1

xy 2

x

y z

yz1

xz 2

yz1

xz1

xy3

xy1

xy3

xy 2

yz1

xz

xy 2

y z

xy1

Fig 7.79 LCS planes defined to study flow structures in the draft tube at OP.4 The planes xy1, xy2 and xy3 are vertical to the streamwise direction cutting the hub-vortex laterally with z / Rref .  1.2, 2.99 and 5.98, respectively (figure 7.79). Plane yz1 cuts the separated region semi-parallel to the sidewall with x / Rref .  1.44 . Planes xz1 and xz2 are also adopted, to study the LCS structures in the separated region and in the core flow in hub-vortex wake with y / Rref .  0 , respectively. All of the aforementioned planes in figure (7.79) are utilized to study the Lagrangian coherent structures at OP.4. At OP.1, as observed there is no wall-separation and the interesting phenomenon at this operating point is the reverse–flow bubble in the core flow, which is analysed here by LCS concept. Figure (7.80) shows the planes cutting the reverse–flow bubble at the different angles ranging C = 0 o ,45o ,90 o and 135 o , which are utilized to study transport barriers at OP.1. C  135

C  90 C  45

C  0

x

y z

Fig 7.80 LCS planes defined to study reverse–flow bubble using LCS at OP.1 279

7.10.5 Core reverse–flow bubble analysis at OP.1 using LCS

First of all, the interesting fluid flow phenomenon of the turbulent flow in the draft tube of BulbT turbine at OP.1, i.e. reverse–flow bubble, is studied here with the aid of LCS concept. This will provide a better understanding about the flow physics at OP.1 via the hidden coherent structures, where a vortex breakdown happens. To calculate the attracting and repelling Lagrangian coherent structures at this operating point, as defined in figure (7.80), 4 planes are adopted, which provide a complete picture about the reverse-flow bubble dynamics over 360 . Figure (7.81) depicts the axial velocity field at one instant of time on the planes with C = 0 o ,45o ,90 o and 135 o .

uz uref .

r Rref.

 C  0

r Rref.

C  45

r Rref.

C  90

r Rref.

C  135

z Rref.

z Rref.

z Rref.

z Rref.

Fig 7.81 Axial velocity field on the LCS planes with C = 0 o ,45o ,90 o and 135o from top to bottom, at one instant of time, obtained from DDES simulation applying ‘1D’ inflow profile with u -correction and amplification of vt in the WZ ( n  3 ) at OP.1

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As one can see in figure (7.81), due to the presence of vortex breakdown phenomenon, a complex pattern exists in the flow field, which are well resolved by the LES mode of DDES simulations. To see the integral dynamics of the separated region over time in the Lagrangian point of view, the LCS hidden structures are obtained by the calculation of FTLE fields and detecting the high-value ridges as explained in section (7.10.2). As shown in the case of the lid-driven cavity flow, by increasing the integration time the more details of fluid flow barriers emerge. So, to calculate FTLE field in forwardtime and backward-time, 100 snapshot of the velocity field after statistical convergence are adopted with time step of 0.01 second. Figure (7.82) and (7.83) shows the FTLE fields obtained from forward and backward-time integrations at t  0.5 s , respectively.  C  0

C  45

C  90

C  135

Fig 7.82 FTLE field with forward-time integration (repelling LCS) on the LCS planes with C = 0 o ,45o ,90 o and 135o from top to bottom, at one instant of time, at OP.1 As one can see in figure (7.82), a complicated web of transport barriers forms at each instant of time due to the complexity of core flow at this operating point, especially due to the presence of vortex breakdown. As it is clear in the figure, there is no symmetry between the FTLE fields on the different planes defined with different cutting angles C . 281

This observation is expected a-priori due to the complexity of the reverse–flow bubble dynamics over time. As one can see in figures (7.82) and (7.83), there are some holes or openings in the transport barrier lines for both the repelling and attracting LCS manifolds. This pattern indicates some kind of transport of the fluid material into or out of the reverse–flow bubble in time. This Lagrangian picture is in contrast to the Eulerian picture of the separation bubble, which is defined theoretically based on the zero mass-flux in the time-averaged point of view (figure 7.41). The latter observation is consistent with the observation of Lipinski et al. in the case of dynamics of the separation bubble on an airfoil with vortex shedding. In that case, the openings in the manifolds also cause penetration of mass into or out of the separation bubble and, as a result, deviating from the classical definition of the separation bubble with zero mass-flux criteria [Lipinski et al. 2008].

 C  0

C  45

C  90

C  135

Fig 7.83 FTLE field with backward-time integration (attracting LCS) on the LCS planes with C = 0 o ,45o ,90 o and 135 o from top to bottom, at one instant of time, at OP.1

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Figure (7.83) also suggests dissymmetry of the FTLE field in the case of backward integration on the four different planes with different cutting angle C . It is again linked to the complicated and chaotic dynamics of the separation bubble in the core flow. As one can also see in the figure, the attracting LCS structures are getting more pronounced as they are convected downstream; in other words, the width of structures grows by advancing in the axial direction. Motion of an individual fluid particle in the reverse-flow separation bubble region is majorly governed by the complex underlying web, as the superimposing of repelling and attracting LCS structures. In fact, the fluid particles are continuously and simultaneously pulled and pushed by these hidden LCS structures; as a result, the process induces a complicated dynamics for the fluid particle motions. 7.10.6 Near-wall flow skeleton at OP.4

As discussed in the previous sections, at OP.4 the topology of turbulent flow in the draft tube dramatically changes compared to the OP.1. Despite variation observed in the core flow comparted to OP.1, which leads to the removing of vortex breakdown, a dominant separation zone also forms on the side-wall. To study the effect of different inflow boundary conditions on the global flow dynamics obtained from DDES simulations at this operating point, the Lagrangian perspective can be very useful because it shows the integral effects over a period of time.

uz uref .

y Rref.

z Rref.

Fig 7.84 Axial velocity field on the LCS plane yz1 obtained from DDES simulation using ‘2D rotating+AFG’ inflow condition at t  0.5 s , at OP.4

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In this regard, first the effect of different inflow conditions on the LCS structures in plane yz1 is considered. Plane yz1 cuts the separated zone nearly parallel to the side-wall (figure 7.79). Figure (7.84) shows the axial velocity field at one instant of time i.e. t  0.5 s in the statistical converged state. As one can see in the figure, the region consists of some lakes of positive speed ( u z  0 ) in the dominant negative speed, separated region ( u z  0 ). As time advances, these positive and negative lakes emerge and disappear continuously exhibiting a complicated dynamics due to the complexity of separation zone evolutions in terms of time. To have a global view about the dynamics of separated flow on the plane over a given time interval, LCS is a fruitful approach. Figures (7.85) and (7.86) depict the forward-time and backward-time FTLE field for different inflow boundary conditions including ‘1D’, ‘2D rotating’, ‘2D rotating+AFG’ profiles. As one can see in the figures, the overall pattern of the dynamical transport barriers is different for different inflow conditions. It is interesting to notice that for both cases of the forward and backward-time integrations, AFG changes the Lagrangian fingerprint of the separated flow dynamics. In fact, as explained in chapter 4, the synthetic turbulence is only added to the core flow not in the boundary layer, but downstream its effect is even felt in the dynamics of the separated region, as also seen in the LCS structures.

a) ‘1D profile’

b) ‘2D rotating’

c) ‘2D rotating+AFG’

Fig 7.85 FTLE field with forward-time integration (repelling LCS) on the LCS plane yz1 calculated from DDES simulations using different inflow conditions at OP.4 As seen in figure (7.85), in the case of ‘2D rotating+AFG’ profile, a large LCS structure emerges in the middle as the ridges with high FTLE values, showing a clear separation in the fates of fluid particles on its two sides. In contrast, in the case of ‘2D rotating’ profile, a medusa-shape structure appears just on the bottom of plane yz1 . Few transport barriers are seen on the top of the figure, while the central LCS lines disappear which indicates more mixing compared to the ‘2D rotating+AFG’ case. On the other hand, 284

applying 1D profile changes the pattern of LCS structures in comparison to the 2D inflow profiles as well.

a) ‘1D profile’

b) ‘2D rotating’

c) ‘2D rotating+AFG’

Fig 7.86 FTLE field with backward-time integration (attracting LCS) on the LCS plane yz1 calculated from DDES simulations using different inflow conditions at OP.4 Figure (7.86) also suggests that the type of inflow condition affects the pattern of attracting manifolds on the LCS plane yz1 . Again by comparing the (b) and (c) parts of the figure, one can see the effects of synthetic turbulence on the hidden unstable manifolds. In the case of ‘2D rotating+AFG’ inflow profile, the attracting LCS structures creates more complex pattern in comparison to the ‘2D rotating’ inflow case which is more dense on the left of plane yz1 . On the other hand, ‘1D’ inflow profile generates unstable manifolds covering middle to the top of plane yz1 .

uz uref .

x Rref.

x Rref.

z Rref.

z Rref.

Fig 7.87 Axial velocity field on the LCS plane xz1 obtained from DDES simulation using ‘2D rotating+AFG’ inflow (top) and ‘1D’ inflow (bottom), at t  0.5 s , at OP.4 285

Another plane that cuts the separation zone vertically in respect to the side-wall is the LCS plane xz1 (figure 7.79). Figure (7.87) shows the typical axial velocity field on this plane at t  0.5 s obtained from the DDES simulation with ‘2D rotating+AFG’ inflow profile at OP.4 on the top part of the figure, also with ‘1D’ inflow profile on the bottom. As one can see in the figure, the separated zone forms close to the wall. Also, a repetitive inclined-strip pattern is visible on the top-half of the figure in the case of ‘2D rotating+AFG’ inflow as a signature of resolved spiral tip-vortical structures on the velocity field which is not present in the case of ‘1D’ inflow profile. The stable and unstable manifolds on the plane xz1 are shown in figures (7.88) and (7.89), respectively. As one can see in the figures, the type of inflow profiles affects the LCS structures defined, as the ridges of the FTLE field with high FTLE values. Adding AFG on the inflow profile has also an important contribution in the dynamics of the separated zone. For example, as shown in figure (7.88), the level of the horizontal repelling LCS lines in the case of ‘2D rotating+AFG’ inflow profile is lower than the ‘2D rotating’ counterpart. Also the density of the stable LCS structures is higher in the case of ‘2D rotating’ inflow profile compared to the ‘2D rotating+AFG’ inflow counterpart.

“2D rotating+AFG”

“2D rotating”

“1D”

Fig 7.88 FTLE field with forward-time integration (repelling LCS) on the LCS plane xz1 calculated from DDES simulations using different inflow conditions at OP.4 (bottom: ‘1D’, middle: ‘2D rotating’, top: ‘2D rotating+AFG’)

286

Figure (7.89) also shows the edge of separated region captured in the Lagrangian sense by attracting unstable manifolds. In reality, the edge of separated region has a complicated dynamics in terms of time; its signature over a period of time calculated by the backward-time integration clearly defines two separate regions with distinct-fates, one inside and another one outside of the separated region. It is also obvious in figures (7.88) and (7.89) that applying ‘1D’ inflow profile affects the pattern of LCS structures in the case of both forward and backward integrations. As it is seen in figure (7.88), the horizontal repelling LCS are getting separated in the case of ‘1D’ inflow profile compared to the 2D inflow profiles. In addition, the density of LCS structures in the right-half zone of the figure is reduced in the case of ‘1D’ inflow profile. The edge of the separated zone is also getting closer to the wall in the case of ‘1D’ inflow profile compared to the two other cases.

“2D rotating+AFG”

“2D rotating”

“1D”

Fig 7.89 FTLE field with backward-time integration (attracting LCS) on the LCS plane xz1 calculated from DDES simulations using different inflow conditions at OP.4 (bottom: ‘1D’, middle: ‘2D rotating’, top: ‘2D rotating+AFG’) 7.10.7 Core flow skeleton at OP.4

In this subsection, flow skeleton in the core flow region of the draft tube at OP.4 is extracted using the ridges of FTLE field on planes xz 2 , xy1 , xy 2 and xy3 . Figure (7.90)

287

depicts the axial velocity field at one instant of time on plane xz 2 in the core flow of draft tube, cutting the deflected hub-vortex, at OP.4. As discussed in the previous sections, due to the formation of separated region on the side-wall at this operating point, the hub-vortex is deflected from the horizontal axis, as shown before in figure (7.50). That is why the lowspeed steak in figure (7.90) created by the hub-vortex coherent structures on plane xz 2 is tilted upward induced by the hub-vortex deflection. In figure (7.90) also the effects of resolved spiral runner-tip vortical structures on the axial velocity field can be recognized by the repetitive inclined-strip pattern on the top part of the figure. As shown recently, the effect is also visible on plane xz1 as shown in figure (7.87). uz uref .

x Rref. z Rref.

Fig 7.90 Axial velocity field on the LCS plane xz 2 obtained from DDES simulation using ‘2D rotating+AFG’ inflow profile, at t  0.5 s , at OP.4 Figure (7.91) shows the repelling and attracting LCS manifolds corresponding to the forward-time and backward-time integrations, respectively, identified by the red-ridges in the FTLE field. As one can see in the figure at this operating point (i.e. OP.4), fewer Lagrangian transport barriers appear in the hub-vortex zone compared to the OP.1. If one recalls figures (7.82) and (7.83) considering OP.1, a much more complicated pattern of the transport barriers is generated due to the presence of vortex breakdown phenomenon. In contrast, here at OP.4 in figure (7.91), fewer number of separatrices emerge which implies higher level of fluid flow mixing in the hub-vortex zone in comparison to the OP.1 counterpart. As one can also see in figure (7.91), concentration of the repelling manifolds in the hub-vortex wake at OP.4 is limited to the left region of plane xz 2 , whereas in the case of the attracting LCS, the structures are dominating somehow in the right portion of plane xz 2 . It is also visible in figure (7.91) that a different inflow boundary condition contributes to the pattern of LCS structures. In particular despite the relative similarity that exists between the LCS structures in the case of ‘2D rotating+AFG’ and ‘2D rotating’ inflow

288

profiles, it is clear that adding AFG also modifies the LCS patterns in the core flow for both forward-time and backward-time integration fashions.

“2D rotating+AFG”

“2D rotating”

“1D”

“2D rotating+AFG”

“2D rotating”

“1D”

a) Forward-time integration

b) Backward-time integration

Fig 7.91 FTLE field with forward and backward-time integrations on the LCS plane xz 2 calculated from DDES simulations using different inflow conditions at OP.4 (bottom: ‘1D’, middle: ‘2D rotating’, top: ‘2D rotating+AFG’) After studying the LCS structures in the hub-vortex wake on plane xz 2 , it is interesting to have a look at the hub-vortex from another angle; from this perspective, the flow is considered on the different lateral cross-sectional planes xy1 , xy 2 and xy3 with z / Rref .  1.2, 2.99 and 5.98, respectively (figure 7.79).

Figure (7.92) shows the axial velocity field on plane xy1 positioned after the runner-hub at one instant of time. As it is clear in the figure, a low speed core forms in the middle of the region. On the other hand, figure (7.93) depicts the formation of closed-loop Lagrangian transport barriers in the forward-time integration case for all inflow conditions. It means that the fluid particles inside the circular closed-loops in the core flow are isolated from the surrounding fluid particles outside the circular closed-loops.

289

The effects of inflow profile type on the repelling LCS structures on plane xy1 are observed in the (a) - (c) parts of figure (7.93). In the case of ‘2D’ inflow profiles, adding AFG exhibits minor effects on the so-called sun-like pattern created, whereas ‘1D’ inflow profile generates a perfect circular transport barrier. It is also important to emphasize that the area inside the sun-like closed-loop transport barrier in the case of ‘2D’ inflow profiles is clearly larger than the area embedded by the circular transport barrier in the case of ‘1D’ inflow profile. Despite this effect, resolving the runner-related vortical structures in the case of ‘2D’ inflow profiles causes deviation from the perfect circular shape of repelling LCS and adds some sunray-like structures in the radial directions as one can see in the part (b) and (c) of figure (7.93). Figure (7.94) also shows the attracting unstable manifolds topology on plane xy1 for different inflow boundary conditions. In the case of ‘2D’ inflow profiles, spiral attracting LCS forms due to resolving the vortical structures and wakes, which is not present in the case of ‘1D’ inflow profile. uz uref .

y Rref.

x Rref.

Fig 7.92 Axial velocity field on the LCS plane xy1 obtained from DDES simulation using ‘2D rotating+AFG’ inflow profile, at t  0.5 s , at OP.4 By moving further downstream in the streamwise direction, as one can see in figure (7.96), although the transport barrier still exist, but the closed-loop transport barrier previously observed on plane xy1 gets open on plane xy 2 . This means that the fluid particles are now getting mixed in the core flow, in contrast to plane xy1 , in which the fluid particles were isolated in the core flow from the surrounding flow-stream by the circular LCS. One can also observe in figure (7.96) that the type of inflow profile affects the formation of LCS structures at plane xy 2 . In particular, adding AFG changes the transport barrier considerably and reduces the extension of the LCS tail, which means more fluid 290

mixing in the case of inflow profile with AFG compared to the profile without AFG. Figure (7.97) also depicts the attracting LCS obtained on plane xy 2 which also shows importance of the inflow condition type on the formation of the unstable manifold structures in the Lagrangian sense. To grasp a better understating about the vortical fluid flow on plane xy 2 , the repelling and attracting LCS structures are extracted by applying a thresholding technique on the FTLE field. Figure (7.95) depicts the axial velocity field on plane xy 2 superimposed by the skeletons of the repelling LCS (black structure) and attracting LCS (cyan structure). In the figure, the structures are extracted as the ridges with FTLE>70% of maximum value and FTLE>79% of maximum value for repelling and attracting LCS, respectively.

a) ‘1D profile’

b) ‘2D rotating’

c) ‘2D rotating+AFG’

Fig 7.93 FTLE field with forward-time integration (repelling LCS) on the LCS plane xy1 calculated from DDES simulations using different inflow conditions at OP.4

a) ‘1D profile’

b) ‘2D rotating’

c) ‘2D rotating+AFG’

Fig 7.94 FTLE field with backward-time integration (attracting LCS) on the LCS plane xy1 calculated from DDES simulations using different inflow conditions at OP.4 291

As one can see in figure (7.95), the central vortical region can be identified in the core flow embracing by repelling and attracting manifolds. The intersection of repelling and attracting LCS manifolds shown by the dashed rectangle in figure (7.95) also shows a time-dependent saddle point on the vortex boundaries as discussed by Green et al. [Green et al. 2010]. Further downstream on plane xy3 , the flow topology radically changes as shown in figure (7.98). In contrast to planes xy1 and xy 2 , a separation zone forms on the side wall. The structure of the transport barriers are also getting more complex. As one can see in figure (7.99), on this cross-section, the simulation results obtained from ‘2D rotating+AFG’ inflow profile generates a more pronounced repelling transport barrier in the core flow compared to the case with ‘2D rotating’ inflow profile, another indication that the AFG affects the downstream flow dynamics in the draft tube. In addition, the case with ‘2D rotating+AFG’ inflow profile shows more mixing in the separated region on the left portion of plane xy3 identified by less number of internal repelling LCS structures compared to the case with ‘2D rotating’ inflow profile.

uz uref .

y Rref.

x Rref.

Fig 7.95 Axial velocity field on the LCS plane xy 2 superimposed with repelling LCS (black structure) and attracting LCS (cyan structure) obtained from DDES simulation using ‘2D rotating+AFG’ inflow profile, at t  0.5 s , at OP.4 As it is visible in figure (7.99), ‘1D’ inflow profile also modifies the repelling LCS structures compared to the ‘2D’ inflow profile counterparts and mostly removes the repelling LCS structures in the core, in addition to the right-portion of plane xy3 . In figure (7.100), the attracting LCS structures calculated by the backward-time integration are shown. As one can see in the figure, the separatrice of the separated flow region in the Lagrangian sense emerges more clearly by adding AFG to the ‘2D’ inflow profile 292

compared to the two other cases. In the case of ‘2D rotating’ without AFG, the attracting LCS which forms in the case of ‘2D rotating+AFG’ majorly disappears between the separated and non-separated regions, indicating higher level of exchange of fluid flow particles between these two regions or, in short, higher level of ‘fluid mixing’.

a) ‘1D profile’

b) ‘2D rotating’

c) ‘2D rotating+AFG’

Fig 7.96 FTLE field with forward-time integration (repelling LCS) on the LCS plane xy 2 calculated from DDES simulations using different inflow conditions at OP.4

a) ‘1D profile’

b) ‘2D rotating’

c) ‘2D rotating+AFG’

Fig 7.97 FTLE field with backward-time integration (attracting LCS) on the LCS plane xy 2 calculated from DDES simulations using different inflow conditions at OP.4

293

uz uref .

y Rref.

x Rref.

Fig 7.98 Axial velocity field on the LCS plane xy3 obtained from DDES simulation using ‘2D rotating+AFG’ inflow profile, at t  0.5 s , at OP.4

a) ‘1D profile’

b) ‘2D rotating’

c) ‘2D rotating+AFG’

Fig 7.99 FTLE field with forward-time integration (repelling LCS) on the LCS plane xy3 calculated from DDES simulations using different inflow conditions at OP.4

a) ‘1D profile’

b) ‘2D rotating’

c) ‘2D rotating+AFG’

Fig 7.100 FTLE field with backward-time integration (attracting LCS) on the LCS plane xy 3 calculated from DDES simulations using different inflow conditions at OP.4

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7.11 Summary

This chapter presented the simulation results of the ‘final geometry’ of draft tube at two selected operating points, namely OP.1 and OP.4. The ‘final geometry’ involves 10.25 and 5 divergence angles for the conical and transition parts, respectively. Considering the lessons learned from the previous chapters, the emphasize was put in this chapter on analyzing the effects of different inflow boundary conditions including ‘1D’ and ‘2D’ profiles on the DDES simulations of the draft tube flow. This chapter also presents the results of the detail analysis of the fluid flow phenomena as well as the comparisons to the available LDV and PIV measurement data. The experimental data shows that there is no wall-separation at OP.1, whereas a large separation zone is formed on the sidewall at OP.4. The major points of this chapter can be summarized, as follows:  For ‘1D’ inflow profiles, modifications were applied to the original u and ur

inflow profiles coming from full-machine k   RANS simulations, in order to correct the mismatch observed between these profiles and experimental-LDV counterparts.  Considering the draft tube recovery coefficient, separation topology, streamline patterns and PIV and LDV data: o For k   simulations, u , ur -correction of the inflow profile results in the

best agreement with the experiment for both OP.1 and OP.4. o For S-A simulations, u -correction of the inflow profile results in the best

agreement with the experiment for both OP.1 and OP.4. The size of reverseflow bubble predicted by S-A is smaller than its counterpart by k   at OP.1. Confirmation of the bubble presence needs further experimental measurements in the core flow at OP.1 in the future.  S-A and DDES simulation results showed that WZ treatment of the inflow profile does not solve the problem of false separation by itself at OP.1.  DDES simulation of the draft tube flow at OP.1 revealed that: o Inflow velocity corrections can not solve the unphysical separation problem by itself. o Considering the separation topology, streamline patterns and PIV and LDV data, the best agreement with the experimental data is obtained by including u -correction and amplification of vt in the WZ for both the hub and

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shroud with n  3 . In this case, a reverse-flow bubble is formed in the core flow as a result of vortex breakdown phenomenon.  DDES simulations of the draft tube flow at OP.4 were performed using 3 different inflow conditions including ‘1D’, ‘2D rotating’ and ‘2D rotating+AFG’ profiles. o The inflow velocity and turbulent viscosity corrections similar to OP.1 were applied for all simulations at OP.4. o For the ‘2D’ inflow profile, a blending u -field is reconstructed based on

the experimental-LDV and the full-machine k   RANS data. o DDES computations at OP.4 predict a flow separation on the opposite wall that was observed experimentally for all types of the inflow conditions.  DDES simulations using ‘unsteady 2D’ profiles at OP.4 could resolve the runnerrelated vortical structures and wakes appropriately.  Adding AFG to the ‘2D rotating’ inflow profiles at OP.4, is not practically felt by the global quantities of the flow like the recovery coefficient. However, unsteady details of the flow are modified as energy spectrum and two-point correlation. It was also interesting to find that the effects of AFG are also seen downstream far away from the inlet section on the PIV-planes for example and also in the separated zone dynamics, e.g. reverse-flow intermittency.  Energy spectrum analysis (at plane B) showed that adding inflow AFG modifies the energy spectrum in a way to get closer to the experimental-LDV data, although an offset still exists between the numerical and experimental curves.  As revealed by the two-point correlation analysis, iinorm is a good measure to see

the effects of inflow AFG on the flow field. By adding AFG, iinorm increases in overall and diffuses further downstream. Oscillations in the iinorm curves are also observed for radial and circumferential components of velocity induced by getting close to the hub-vortex, but not for the axial component of velocity. In other words, the hub-vortex region is transparent for the axial velocity fluctuations. The circumferential symmetry also breaks for iinorm at OP.4.  LCS analysis illustrated the transport barriers on some pre-defined planes placed in the core flow and separated flow zone. As revealed by LCS: o There is a transport of fluid into and out of the reverse-flow bubble at OP.1, due to the presence of openings in repelling and attracting LCS. o The type of inflow boundary conditions modifies the transport barriers or the fluid mixing in the different part of the flow field at OP.4. 296

“The purpose of computing is insight, not numbers.”

Richard Hamming (1915-1998)

Chapter 8 Conclusion and future directions In the framework of the BulbT project initiated in 2011 at LAMH, the ability of the delayed detached eddy simulation (DDES) formulated based on Spalart-Allamaras (S-A) was assessed in the predication of internal fluid flow phenomena in the draft tube of a low head bulb turbine. Although the method was originally developed for the external flow applications, the present investigation highlighted that the method can be successfully applied in a relatively reasonable time for the internal high-Re confined flow applications by taking into account some essential cautions. On a general point of view, hybrid LES/RANS simulations, such as DDES, are capable to resolve a portion of turbulence spectrum; therefore, they take into account more unsteadiness and their LES-content provides more information about the unsteady details of the flow. Furthermore, time-dependent chaotic flow phenomena in the flow field, like separation dynamics as well as the onset of separation are sensitive to the unsteadiness level in the flow. On the other hand, in contrast to the LES methods, major portion of the flow unsteadiness is lost in the case of URANS methods, due to their dissipative nature. Considering above reasoning, better predictions are expected by moving towards the LES simulations. In addition, the LES-content of simulations provides deeper insight into the underlying physics of the problem, at least in the LES-core. In this work, thanks to the availability of large parallel-computing CLUMEQ clusters, the DDES turbulence treatment was utilized to simulate the draft tube flows in OpenFOAM environment. The later code allowed using a large number of cores on the clusters for free. Three different cases were selected for the simulations including ERCOFTAC diffuser, ‘basic’ and ‘final’ geometries of the BulbT draft tube. Due to the high computational cost of the DDES simulations, for the ‘basic geometry’, the best efficiency point was selected, whereas for the ‘final geometry’ of the BulbT draft tube, two operating points were selected to explore the flow fields including OP.1 and OP.4. 297

To pursue objectives of the project, a numerical strategy was devised for the standalone DDES simulations of BulbT draft tube flows. The procedure consists of two steps: at first, flow fields in the hydro-turbine are simulated with the widely used k   RANS technique in the full-machine arrangement, which brings the inflow boundary conditions (i.e. velocity and turbulent quantities) for the stand-alone draft tube simulations; second, the draft tube flows are simulated using DDES approach. Thanks to the availability of the experimental data, the top priority was put on the quality of the inflow boundary condition profiles for stand-alone DDES simulations of the draft tube flows. Generally speaking, the inflow profiles for DDES simulations of the draft tube can be obtained from two sources: one, RANS/URANS simulations of the upstream components of the draft tube, another from the experimental measurements. Due to the measurement limitations especially in the near-wall zone, a complete inflow profile on the full radius is not typically available; therefore, as devised in the proposed procedure, one should rely on the simulations of the upstream components to obtain the inflow profile for the draft tube DDES simulations. There usually exist some mismatches between the numerical and experimental profiles at the draft tube inlet. In the BulbT project, these deviations in the inflow velocity profiles from the experiments were negligible for the axial velocity, but pronounced for the circumferential component of the velocity and less pronounced for the radial component. This shortcoming was the first source of uncertainty entering into the stand-alone numerical simulation of the draft tube. Another important unknown that has a crucial effect on the DDES simulations of the draft tube, especially on the onset of separation is the turbulent viscosity parameter ( vt ) in the wall zone (WZ). For DDES simulations, as a hybrid model, the vt coming from the upstream RANS simulations is reduced by about 2-3 orders of magnitudes using Smagorinsky model in the core flow. On the other hand, as proposed in the thesis, the vt should be adjusted in the WZ of the inflow section in a way to get numerical results as close as possible to the reality considering all available experimental data, e.g. separation topology, PIV and LDV data. In this sense, the availability of experimental data is essential to perform reasonable DDES simulations of the draft tube flows. In the present research, different inflow profiles including ‘1D’ and unsteady ‘2D’ profiles were tested to simulate the turbulent flows in BulbT draft tube. The present study illustrated that the circumferential-averaged 1D profile smoothes out all the incoming structures from the runner and does not provide realistic fluid phenomena as long as the flow details is of interest. To resolve the physical structures like vortical structures and wakes ejected by the runner in the low head BulbT, applying unsteady ‘2D’ inflow profiles is mandatory.

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The effects of applying the artificial turbulence, AFG, on the DDES simulations of the draft tube were also investigated in details in this project. To this end, the synthetic inflow turbulence was generated using a mixed AFG technique with capability of imposing 2D variation of turbulent quantity fields [Davidson 2011]. The technique uses both the physical and mapped spaces in the procedure to generate the artificial fluctuations. Anisotropy of the inflow turbulence was also considered using MLPR artificial neural network (ANN) generalization via scaling factors. The ANNs were trained by experimental LDV data. In the generated synthetic inflow turbulence, fingerprints of the runner-blade wakes are visible as four high amplitude fluctuation zones. As shown, the slopes of energy spectra of the sample turbulent velocity signals in different spatial directions were also getting closer to -5/3 by adding AFG. To see the effects of AFG on the simulated flow fields, different comparisons were made using LDV and PIV experimental data as well as different techniques were applied to analyze the flow details like two-point correlation, energy spectrum and LCS. Analyses such as dynamics of the flow separation, energy spectrum and flow transport barriers, showed that the details of the flow are modified by applying AFG, although the global quantities, such as the recovery coefficient do not see these variations in the local flow features, due to their integral nature. The results of present project suggest that DDES simulations of the draft tube considering 2D variation of the inflow profile can be successfully adopted to resolve the structures coming out of the runner. In this sense, the method provides closer picture to the reality, although it seems that the onset of separation and its dynamics are essentially governed by the URANS mode of the hybrid method in the near-wall zone. As a result, the global quantity predictions are not necessarily improved in comparison to the URANS techniques in the engineering point of view. The recovery factor for example remained short of the measured one by around 10%: this needs further investigations. Considering these points and relatively high computational cost of the DDES simulations for high-Re draft tube flows and dependency to the experimental data for set-up the simulation, usage of the technique for the industrial needs has its pros and cons; although, it still provides valuable information about the underlying physics of the problem.

8.1 Summary of the results for three simulation cases The main observations of the present thesis in chapters 5, 6 and 7 are reviewed here in details. In chapter 5, a standard test-case, the well-known ERCOFTAC conical diffuser was selected with two different configurations to study the sensitivity of DDES simulations to the inflow conditions. The experimental data for this test-case was available [Clausen et al. 299

1993]. The test-case has the particularity of resembling a hydro-turbine with two configurations: the base-case and the extended-case with upstream swirl generator. The extended-case simulations depicted overall good agreement with the experiments and showed low-sensitivity to the inflow turbulent quantities. DDES simulations of the ‘base-case’ showed that inflow profiles coming from the k   leads to the false flow separation in contrast to the inflow profile coming from the SA, due to the less energy existing in y   30 in the case of k   compared to S-A. It was also found that inclusion of the inflow ur is necessary for pure S-A simulations, whereas it can be neglected for simulations with wall-function, due to the induced artificial boundary layer robustness by applying wall-function in the latter case. This characteristic of applying wall-function should not be considered positive, because it can lead to a delay in the prediction of separation for complex fluid flows. The results of the ERCOFTAC simulations also illustrated that although near walltreatment of the inflow velocity and turbulent viscosity inflow profiles provides more accurate inflow condition for the DDES simulations, it does not solve the false separation problem on the diffuser-wall by itself. Finally, in the case of availability of the experimental data, the inflow turbulent viscosity in the WZ can be tuned in a manner to postpone separation formation in DDES simulations of the diffuser flows. In chapter 6, the comparative simulations were performed on the ‘basic geometry’ of the draft tube at BEP. The geometry involved mild divergence angles of 6.6  and 2.1 for the conical and transition parts, respectively. There is no experimental data available for this case, although no wall-separation is expected from k   simulations. It was considered to gain an initial insight into the effects of inflow conditions on the DDES simulations of BulbT draft tube with a less aggressive divergence angle. The results showed that the topology of separated zone is different for S-A and k   RANS simulations. The recovery coefficient was slightly lower in the case of the S-A. The results also showed that amplification of the inflow vt in the WZ attenuates the wall-separation in the DDES simulations, while the recovery coefficient is very slightly modified by the amplification. Discretization scheme of the convective term also affects the vortex rope pattern; in fact, more structures are resolved by the filtered linear scheme compared to the upwind scheme. The recovery coefficient, swirl number and mean flow profiles are practically the same for both schemes, although details of the flow are modified, as revealed by the energy spectrum analysis. The proposed methodology was shown to be able to resolve the vortical structure and wakes ejected by the runner. In chapter 7, the turbulent flows in the ‘final geometry’ of the draft tube were studied using URANS and DDES simulations at OP.1 and OP.4. The ‘final geometry’ 300

involved the divergence angles of 10.25 and 5 for the conical and transition parts, respectively. Experimental measurements showed that no separation forms on the draft tube wall at OP.1, whereas a large separation forms on the side-wall at OP.4. To pursue objectives of the project, different inflow boundary conditions including ‘1D’ and unsteady ‘2D’ profiles were considered for numerical simulations of the draft tube. Intensive comparisons were made using available LDV and PIV experimental database measured at LAMH. The results showed that the inflow boundary conditions play a crucial role in the numerical simulations of the turbulent flows in the BulbT draft tube. Considering the recovery coefficient, separation topology, streamline patterns and PIV and LDV experimental data comparisons at both operating points, u plus ur -correction and u correction leads to the best agreement with the experiment for k   and S-A simulations, respectively at both operating points. DDES simulation results showed that the inflow WZ treatment and inflow velocity corrections at OP.1 do not solve the problem of false separation in itself. Considering the separation topology, streamline patterns and PIV and LDV data, the best agreement of the DDES simulations to the experimental data at OP.1 is obtained by including u -correction and amplification with a power factor of n=3 of vt in the WZ for both the hub and shroud. In this case, a reverse-flow bubble is formed in the core flow as a result of the vortex breakdown phenomenon. Confirmation of the bubble presence needs further experimental measurements in the core flow in the future. DDES simulations of the draft tube flow were also performed at OP.4 using three different inflow conditions including ‘1D’, ‘2D rotating’ and ‘2D rotating+AFG’ profiles. The inflow velocity and turbulent viscosity corrections similar to the OP.1 were applied for all simulations at OP.4. For the ‘2D’ inflow profile, a blending u -field was reconstructed based on the experimental-LDV and the full-machine k   RANS data. The hybrid subgrid scale viscosity was also applied for DDES simulations. DDES simulations using ‘unsteady 2D’ profiles at OP.4 could resolve the runnerrelated vortical structures and wakes appropriately. By comparing the simulation results to the PIV-data it was shown that the accuracy of numerical predictions decreases in the axial directions. The separation, located at the junction between the cone and the trumpet, is obtained on the opposite wall compared to the experiment with all tried inflow boundary conditions; this still needs further investigations. Adding AFG to the ‘2D rotating’ inflow profiles at OP.4, is not practically felt by the global quantities of the flow like the recovery coefficient, however unsteady details of the flow are modified as the energy spectrum and two-point correlation. The effects of AFG are also felt downstream far away from the inlet 301

section at the location of the PIV-planes and also in the separated zone dynamics, e.g. through the reverse-flow intermittency. In addition, adding the inflow AFG modifies the energy spectrum (for example, at plane B) in a way to get closer to the experimental-LDV data. Furthermore, by adding AFG, two-point correlation of the turbulent fluctuations increases in overall and diffuses further downstream. Oscillations in the two-point correlation curves of the velocity incoherent part are observed for radial and circumferential components of velocity when getting close to the hub-vortex, but not for the axial component of velocity. The symmetry breaks for two-point correlations among the planes. Finally, the LCS analysis illustrated the transport barriers on some pre-defined planes placed in the core flow and separated flow zone. The results at OP.1 showed that there is a transport of fluid into and out of the reverse-flow bubble as identified by the presence of openings in the repelling and attracting LCS. At OP.4, it was shown that the type of inflow boundary conditions modifies transport barriers (or fluid mixing) in different parts of the flow field.

8.2 Future directions The present thesis shed light on different factors crucial for stand-alone DDES simulations of the draft tube, like the sensitivity of solutions to the inflow profiles or the effects of adding synthetic inflow turbulence on the detail flow features. As shown in this thesis, DDES simulations of the turbulent flows in the draft tube are not ideally independent from the experiment, still far from the ultimate goal of the CFD as a ‘virtual test-rig’. In other words, the simulation strategy needs the availability of experimental data for both validation as well as setting-up the case (e.g. in the tuning of turbulent viscosity). It is a reasonable outcome, because in the hybrid models, WZ is left to the URANS treatment and therefore vt parameter (in DDES or S-A) should model the whole turbulence effect in that region. As a direct result, the prediction accuracy of separation will majorly depend on the URANS behaviour. On the other hand, appropriate estimation of the vt in the middle of the flow, in the complicated situation just after the runner, is not straightforward and as discussed in details, serves as an important source of uncertainty in the stand-alone simulation of the draft tube. As a conclusion, an estimation of this parameter independent of the experimental data for draft tube simulations still remains as a challenging task. To proceed further, as perspective of the present project, the following directions are suggested for the near future: 

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For stand-alone simulation of the draft tube, switching to the pure LES without wall modeling (instead of hybrid treatment) along with applying the high order

discretization schemes for the convective term (as least, a pure second order central difference scheme) would be fruitful to directly resolve the detail flow features in the WZ. In fact, in the pure LES, the role of vt is replaced by vSGS , which has obviously a lower responsibility acting as a dissipation sink in the energy cascade process. In this way: o AFG would be applied not only for the core flow, but also in WZ. o One pays a higher computational cost to perform pure-LES simulations compared to the hybrid techniques, however the estimation of vSGS will be

less critical than vt at the inlet section of draft tube. 

Another feasible path is to include the upstream components of the draft tube in the DDES simulations, ideally using full-machine arrangements. The high computational cost of unsteady DDES techniques, induced by the necessity to model up to the wall ( y   1 ) and a small time-step makes this practically prohibitive. One way to tackle the problem could be to adopt DDES simulation with wall-function [Paik and Sotiropoulos 2012] for draft tube upstream components and to solve the draft tube flows using original DDES with y   1 . In this regard, a ‘scalable wallfunction’ concept should be adopted along with the transient rotor-stator interface for the stationary-rotating part treatments. This strategy removes the necessity of applying AFG.



In the present project, as shown the vt in the inlet wall-zone for hybrid RANS/LES simulations was adjusted considering all available experimental controlling checkpoints including the separation topology, PIV and LDV data. Another method that could be adopted to adjust inflow vt in WZ, is to use directional wall shearstress sensors at the inlet section of the draft tube. A project to develop this kind of sensors has been started a few years ago at LAMH.



For all numerical simulations in this project, the mass flow rate was kept constant in all variants of the inflow boundary conditions including ‘1D’, ‘2D rotating’ and ‘2D rotating+AFG’ inflow profiles. However, as revealed in the experiment, the mass flow rate varies in terms of time with a maximum about 2% around the mean [Duquesne 2015]. The next generation of the DDES simulations of the draft tube flow can be conducted considering the experimental variation of inflow mass flow rate. As a hypothesis, the mass flow rate variation can be linked to the difference observed between the positions of the separation zone on the side walls between the experimental and numerical results.

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

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Different analysis techniques have been applied in this thesis to determine the integral and detail unsteady features of the draft tube flow like two-point correlations and LCS. In general, the database generated from LES-like simulations is so rich, any advance statistical technique can be adopted to explore the data; the list is practically endless.

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APPENDICES: Appendix A1 Large eddy simulation (LES) formulation The method is constructed base on the idea of separation of scales. In fact, small scale eddies are removed in the strategy by the aid of spatial filtering technique and only large scale, anisotropic geometry-dependent motions are resolved directly by the mesh. Hereupon large eddy simulation (LES) is a well-representative name for the method. On the other hand, effect of small scale eddies on the overall turbulence dynamics is also considered via sub-grid scale (SGS) models. To obtain a governing equation to resolve large-scale eddy motions, typically, a low-pass filter is applied to the Navier-Stokes equation and the obtained equations are solved instead of original equations. The filter can be defined as follow:   u ( x , t )   G ( ,  f ) u ( , x , t ) d  (A1.1) f

where  f denotes the filter width. In general, filter width can be different from the grid spacing, although to take advantage of maximum capacity of the grid resolution, the filter width (  f ) is typically considered equal to the grid spacing measure, or  f   . Ideally, LES should be performed on an equi-spacing (cubic) grid but in application, this ideal situation on the whole computational domain is far from achievement. In practice, a filter width base on a measure of grid spacing is then adopted, e.g. cubic root of cell volume ( xyz ) or maximum edge spacing ( max(x, y, z ) ) for hexahedral grids. Using the filtering concept, the flow variables can be decomposed into the resolved ( u ) and sub-grid scale (SGS) motions ( us ) respectively, for turbulence velocity signal: u  u  us

(A1.2)

By applying the filtering (equation A1.1), the continuity and Navier-Stokes equations for incompressible flows can be rewritten as follows in terms of the filtered quantities and sub-grid scale stress tensor: ui 0 xi S ui  (ui u j )  2u 1  ij 1 p    v 2i  t x j x j  x j  xi

(A1.3)

323

where  ijS denotes the sub-grid scale (residual) stress tensor, which involves the effect of subgrid- scale motions on the filtered (resolved) motions and can be expressed as below:   ijS   (u i u j  ui u j )

(A1.4)

The above equation (A1.3), which has the same form as the original Navier-Stokes equation is called ‘filtered Navier-Stokes’ equations, and can be adopted to resolve largescale eddy motions in the LES method. In the case of DNS, as explained in chapter 2, all the scales are directly resolved from Kolmogorov smallest scale to large scale eddies, so the sub-grid scale tensor vanishes. To close the system of filtered Navier-Stokes equations,  ijS should be expressed in terms of the filtered quantities. This is achieved by the aid of subgrid scale (SGS) closure models. In general, the SGS models can be physically interpreted as energy sink in the energy cascade, which provides appropriate dissipation for annihilation of turbulent energy at small-scale eddy level, where viscous effects are extremely important. There are different strategies to construct a SGS model; typically, Boussinesq’s eddy viscosity concept is utilized to relate filtered quantities and SGS-stress tensor, as below: 1 3

 ijS   kkS  ij  2  vSGS Sij

(A1.5)

where  ij and vSGS denote Kronecker delta and sub-grid scale viscosity, respectively. Filtered strain rate tensor Sij is also defined as follows: 1  u u  Sij   i  j  2  x j xi 

(A1.6)

The oldest sub-grid scale model was proposed by Smagorinsky based on the aforementioned Boussinesq assumption [Smagorinsky 1963]. In this model, the sub-grid scale viscosity is obtained by the following relation based on Prandtl mixing length theory with ( lm  CS  ): 2 2 vSGS   CS   S   CS   2Sij Sij

(A1.7)

where CS is Smagorinsky constant (typically between 0.1  0.2 ) and CS  0.17 based on decay of isotropic turbulence [Pope 2004]. It should be mentioned that in the near wall region, a damping function should be utilized to avoid high unphysical values of SGS viscosity. Furthermore, the value of CS is not really constant and is flow-dependent. To overcome this shortcoming, other methods like the dynamic model have been developed. In

324

the later method, CS is not constant but calculated as a function of time and space, using two different grid sets on the same computational domain [Germano et al. 1991]. In the LES method large scale fluctuations are directly resolved by the method and the relatively universal behaviour of small-scale fluctuations and energy dissipation are fully considered through the SGS modeling. It is worth mentioning that ideally, the utilized discretization schemes should not add any extra dissipation on the solution. In other words, except in the case of implicit LES (ILES) where the dissipation is exclusively produced by the discretization schemes [Grinstein et al. 2007], generally, dissipative schemes (e.g. upwind scheme) should be avoided to obtain accurate results.

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Appendix A2 Unsteady Reynolds Averaged Navier-Stokes (URANS) formulation In the method, turbulent eddies are not directly resolved and the whole turbulence behaviour is modeled. In fact, by the aid of time averaging technique, only relatively low frequency variations of the flow field are captured by URANS approach in contrast to high frequency turbulent fluctuations that are totally modeled. In this regard, the following time averaging operator is applied on the flow variables and Navier-Stokes equations:

1 t0 T    ( x , t )    ( x , t ) dt T t0

(A2.1)

where T is a measure of slow variation time scale of the flow field. By applying the abovementioned time averaging operator on turbulent velocity signal and using triple decomposition of the flow variable fields [Hussain and Reynolds 1972], one can obtain: ui (t )  uim  uil (t )  ui(t )  

(A2.2)

coherent part

where uim represents the mean velocity; uil (t ) and ui(t ) indicate the low frequency (periodic) and high frequency fluctuating parts of the turbulent velocity signal, respectively. Equivalently, ui (t )  ui (t )  ui(t )

(A2.3)

where ui (t ) stands for coherent part of the velocity signal in the turbulent flow which involves all organized low frequency motions in the flow field and excludes semi-random turbulence background. The triple decomposition strategy can be applied for pressure signal as well. By applying time averaging operator (equation A2.1) on Navier-Stokes equations and substitution of triple decomposed velocity and pressure signals, URANS equations for incompressible flows read as follows: ui 0 xi ui  (ui u j )  2u 1  ij 1 p    v 2i   xi t x j x j  x j

(A2.4)

where  ij stands for Reynolds stress tensor. It is worth mentioning that the origin of this term is the non-linear term of momentum on the LHS and acts as a stress on the fluid elements. It is physically interpreted as correlations of velocity fluctuations and can be stated as follows: 326

u2 uv uw     ij   uiuj   uv v2 vw    uw vw w2   

(A2.5)

The first term on the LHS of the momentum equation (A2.4) represents temporal variation of the velocity field; in RANS formulation (in contrast to URANS), this temporal term is omitted. Furthermore, presence of  ij in the RANS and URANS equation system (A2.4) leads to the ‘closure problem’, which means that the number of variables and equations are not balanced. To remedy the problem and make the equations solvable  ij should be expressed based on coherent (mean) flow variables; different turbulence models have been developed. Eddy viscosity models (including linear and non-linear approaches) tries to relate Reynolds stress tensor to a new-defined variable named turbulent viscosity vt , in contrast to Reynolds stress model (RSM) [Launder et al. 1975] which solves transport equations for independent components of Reynolds stress tensor. Linear eddy-viscosity models are constructed on the Boussinesq’s assumption which states that the Reynolds stress tensor depends linearly on the mean strain rate tensor; mathematically: 2 3

 ij   k ij  2  vt Sij

(A2.6)

where  ij and vt denote Kronecker delta and turbulent eddy viscosity, respectively. The mean strain rate tensor Sij and turbulent kinetic energy per mass unit k can also be defined, as follows: 1  u u  Sij   i  j  2  x j xi 

(A2.7)

1 1 k  uiui  (u 2  v2  w2 ) 2 2

(A2.8)

By the aid of Boussinesq’s eddy viscosity assumption, unknown Reynolds stress components are replaced by a single unknown variable vt ; so the aim of turbulence modeling is reduced to an appropriate approximation of turbulent viscosity as a function of time and space in the whole computational domain. RANS turbulence models based on linear eddy-viscosity can be classified based on the number of equations solved to approximate vt , including zero equation (algebraic) models (mixing length based models e.g. Baldwin-Lomax model), one equation models (e.g. Spalart-Allmaras model; for more 327

details, see appendix A3) and two equation models (e.g. k   , k   or SST models) [Wilcox 1993]. For example, in the case of the well-known k   model, two transport equations are solved for the turbulent kinetic energy k and turbulent eddy dissipation  . Then, the turbulent viscosity is calculated at each point of the computational domain by the following relation obtained by dimensional analysis [Pope 2004]: vt  C

k2



(A2.9)

where C is an empirically determined coefficient (typically  0.09 ). Then the time and length scales of the representative eddy at each position in the computational domain can be approximated as follows [Pope 2004]: Lt  Tt 

k

3

 k



2

(A2.10) (A2.11)

In turbulent flows, as explained, a wide range of scales exists, but in RANS models only one representative length and one time scale are obtained by the method, which can be viewed as a fundamental limitation of the RANS techniques. On the other hand, as discussed by Spalart, the Reynolds stress tensor in URANS model can also be affected by the coherent (organized) motions in practice, which is undesirable [Spalart 2000] and for the cases where there is not a clear distinction between time scales of the organized motions and semi-random turbulent fluctuations, URANS strategy cannot improve the quality of the results compared to RANS solution by definition. It is worth noticing that the general form of the system of equations (A2.4) is similar to the original Navier-Stokes equations (e.g. in the case of low-Re number laminar flows) except the last term (Reynolds stress tensor) which includes the whole turbulence contribution. That is why it is so difficult to construct a universal turbulence model to reproduce turbulent flows in a broad range of applications. In practice, each turbulence model is efficient in certain limited conditions and its validity is certainly flow-dependent. It is also obvious by comparison of the LES equations (A1.3) and URANS equations (A2.4) that both sets have the same form; although in the case of LES, the filtered fields are calculated but in URANS, coherent (mean) fields are obtained. Furthermore, in LES method, the last term is responsible for sub-grid scale contributions but in URANS, the last term includes the whole effects of the turbulence. This same pattern can be viewed as the origin of the unified hybrid LES/RANS methods.

328

Appendix A3 Spalart-Allmaras one equation model formulation Spalart-Allmaras model is classified under one-equation RANS models. It was originally developed for simulation of external aerodynamic flows. In the strategy, one transport equation is solved for an intermediate variable v (called modified turbulent viscosity) which is directly related to the turbulent viscosity vt . The length scale is included by a simple algebraic formula. In the development of the governing equation, Galilean invariance property and dimensional analysis are applied and the model coefficients are adjusted based on experimental data [Spalart and Allamaras 1994, Shure et al. 2000]. The empirical based transport equation reads as below:   v  cb1 v  v v   1    Dv  cb1 (1  f t 2 ) Sv v    cb 2 (v  v )   cw1 f w  2 ft 2    (A3.1)       xk  xk xk     xk  Dt k    d Production V    2

Destruction

Diffusion

In the production and destruction terms, ft 2 is defined as follows: ft 2  ct 3 exp(ct 4  2 )

(A3.2)

The production term has a linear dependency to the local mean vorticity, as below: v Sv  v  2 2 f v 2 kV d

(A3.3)

where d is the distance to the nearest wall. The wall-damping function f v 2 and vorticity magnitude v are also defined as: fv 2  1 

 1   f v1

v  2ijij

(A3.4) (A3.5)

where f v1 is a wall-damping function and expressed in terms of   v v with the following relation: f v1 

3  3  cv31

(A3.6)

The function f w is applied to provide a faster decay of destruction term in the outer region of the boundary layer and is mathematically expressed as: 329

 1  c6  f w  g  6 w36   g  cw 3 

1

6

(A3.7)

where intermediate variables g and r are constructed as below: g  r  cw 2 (r 6  r )

(A3.8)

 v  r  min   2 2 ,10   SkV d 

(A3.9)

By solving the modified turbulent eddy equation (A3.1), ultimately the eddy viscosity can be obtained by the following formula: vt  v f v1

(A3.10)

Its value tends to one and zero at high Reynolds number and the wall, respectively. The constants of the model are adjusted based on empirical data, as follows:

cb1  0.1355, cb 2  0.622, cv1  7.1, ct 3  1.2, ct 4  0.5, cw1  3.2390, cw 2  0.3, cw3  2, kV  0.41,   2 3

330

Appendix B Multi-Layer Perceptron Neural Network (MLPR) formulation As explained in chapter 4, for scaling of the artificial turbulent velocity signals at inlet section of the draft tube, a multi-layer perceptron (MLPR) artificial neural network is designed and adopted in this study to estimate RMS of the radial velocity fluctuations on the full-radius. Here, the mathematical background of the Perceptron neural network adopted to develop the ANN in-house code is presented. A multi-layer perceptron consists of an input layer and an output layer with one or more hidden layers. Each layer consists of information-processing units called neurons. The neuron is a simplified mathematical model of a real neuron in biological sciences. The nonlinear model of a neuron can be summarized as seen in figure (B.1).

Fig B.1 Nonlinear model of a neuron In fact, in the case of a single neuron, the weighted summation of the input signals is added to a bias term (threshold). The result is then fed to an activation function and in this way outcome signal of the neuron is produced. In mathematical term, the procedure can be expressed as below: N

yip   ANN (vip )   ANN ( wip, j x jp  bip )

(B.1)

j 1

where vip and yip is the activation potential and neuron output signal, respectively. As shown in figure (4.16) in chapter 4, in a typical MLPR neural network, neurons are organized in some distinct layers and are also inter-connected to each other through some 331

synaptic weights in different layers. During the computation (i.e. learning process), signals are passed in two opposite directions: forward direction and backward. In the forward direction, as explained by formula (B.1), the input signals passing via the inter-connecting net are weighted and added to the bias and the result coming from the activation function is transmitted to the next layer. Mathematically, the net output of neuron i in layer ( s  1) can be written as follows: N

s 1 s 1 yis 1   ANN (vis 1 )   ANN ( wis,j1 x sj  bis 1 )

(B.2)

j 1

During the forward propagation of the signal, all the synaptic weights are fixed and their values come from the previous iteration. At the final layer (output-layer), the deviation of the output signals and target signals are computed and a cost function is defined as the norm of this deviation. Then, the backward propagation phase of the signal transferring is started. During this phase, the synaptic weights of the ANN are adjusted using delta learning rule in a way to decrease the cost function. This process is called ‘learning of the neural network’. Mathematically, MLPR output deviation signal for neuron i at output layer and at iteration number n is defined as below: O



n ANN

1N ( wi , j )   [ yi (n) - Ti (n)]2 2 i 1

(B.3)

where N O denotes the number of ANN outputs. Furthermore, T is the target values for the neural network; the aim is to minimize the deviation between the output produced by the ANN and the target values. Based on the delta learning rule, the synaptic weights in backward propagation phase are adjusted based on the steepest descent method as follows [Haykin 1999]: old winew , j  wi , j  l

n  ANN wi , j

(B.4)

where l is the learning rate factor. The above formula can be expressed based on the local gradient (  is+1 ) as below: old s+1 s winew , j  wi , j  l   i ( n) yi (n)

(B.5)

i

For hidden layers the local gradient can be expressed as: 1  is+1 (n)  i,sANN (vis 1(n))  ks+2 (n) wk,is  2 (n)

k

But for output layer the gradient is as below:

332

(B.6)

1  is+1 (n)  i,sANN (vis 1(n)) [ yis 1(n) - Ti s 1(n)]

(B.7)

It is obvious in the above formulation that the activation functions should be continuous and differentiable. The shape of these functions in the hidden and output layers is one of the ANN architecture parameters for the ANN design. Learning rate l is another important parameter which determines the rate of change of the synaptic weights at each iteration. This parameter is similar in many aspects to the relaxation factor for the discretization schemes in solving PDEs. By lowering the learning rate, typically the speed of convergence decreases but on the other hand, one will have smoother trajectory in the ANN weight space created by the steepest descent method, and as a result it reduces the risk of divergence of the ANN learning process.

333

Appendix C1 Scaling of the artificial fluctuations for anisotropic inflow turbulence In the case of anisotropic inflow turbulence, amplitude of the fluctuations is different in different spatial directions. As explained in chapter 4, to consider the anisotropy of the inflow turbulence, RMS values of the turbulent velocity fluctuations obtained from LDV experimental measurements are utilized. This is simply performed by multiplying scaling factors to isotropic signals generated by the AFG technique explained in chapter 4. The turbulent kinetic energy in the cylindrical coordinate system is defined as below (refer to appendix C2): 1 k  (ur 2  u 2  w2 ) 2

(C1.1)

In the case of the isotropic turbulence, fluctuations in the three different spatial directions are the same and one can have an estimation of RMS of the fluctuations as below: 2k 3

ur 2  u 2  w2 

(C1.2)

In the scaling process, the relative amplitude of the fluctuations is important. For this purpose square root of RMS values can be a good measure to calculate the mean amplitude of the fluctuations, so one can write: u  u2



ur  u  w 

2k 3

(C1.3)

In the case of anisotropic turbulence, the relative amplitude of the fluctuations is no longer the same for different directions. As explained in chapter 4, having the experimental RMS values of the fluctuations in three spatial directions using ANN, two factors are defined at each inlet node. These two factors are good measures of the relative amplitude of the fluctuations in different directions:

 s1 

ur 2 w

2

, s2 

u 2 w

2

(C1.4)

The idea is to scale the fluctuations considering its anisotropy in a way to keep the turbulent kinetic energy invariant. Therefore, one can re-calculate the approximation of the velocity fluctuation amplitudes based on (C1.4) as below:

334

k

1 2 1 (ur  u 2  w2 )  ( s21 w2   s22 w2  w2 ) 2 2 1  w2 ( s21   s22  1) 2

(C1.5)

Therefore, the axial fluctuation amplitude is obtained as below: 2k (   s22  1)

w  w2 

2 s1

(C1.6)

For the two other directions, one can simply obtain: ur  ur2 

2k s21 ( s21   s22  1)

(C1.7)

u  u2 

2k s22 ( s21   s22  1)

(C1.8)

By comparing equations (C1.6) to (C1.8) with its isotropic counterparts in equation (C1.3), the scaling factors of the turbulent velocity signals are obtained as below: r   scaling

   scaling

z  scaling 

3 s21 ( s21   s22  1)

(C1.9)

3 s22 ( s21   s22  1)

(C1.10)

3 (   s22  1) 2 s1

(C1.11)

By applying the abovementioned factors on the isotropic turbulent velocity signals generated by the AFG method, in three different spatial directions, the anisotropy of the fluctuations based on the experimental data are considered. Furthermore, as intended the turbulent kinetic energy field remains invariant under this manipulation. It is worth noticing that, as a simple checkpoint, in the case of the same velocity fluctuations in three dimensions of space (i.e. isotropic inflow turbulence with  s1   s 2  1 ) all the above scaling factors are simplified to the unity as expected.

335

Appendix C2 Turbulent kinetic energy in the cylindrical coordinate Although it seems trivial by a-priori physical sense but for completeness and due to the lack of an explicit proof in the literature, here a theorem is proved that the shape of turbulent kinetic energy formulation is invariant under coordinate transformation from the Cartesian to the cylindrical coordinate systems. The latter coordinate system is typically adopted in the turbo-machinery applications. In the Cartesian system, turbulent kinetic energy is defined as below: k

1 2 (u  v2  w2 ) 2

(C2.1)

By transformation from Cartesian coordinate ( x, y,z ) to cylindrical coordinate

(r , ,z ) , the velocity vector is transformed from (u,v,w) to (ur ,u ,w) . One can define the fluctuations in Cartesian coordinate as below: u  u  u   v  v  v  w  w  w 

(C2.2)

where the quantities with () denote the coherent parts of the velocity signals and the second part stands for the turbulent fluctuations. Similarly, in the cylindrical coordinate, one can write: ur  ur  ur  u  u  u  w  w  w 

(C2.3)

For transformation of the instantaneous velocity signals from cylindrical to Cartesian coordinates, one can write: u  ur cos   u sin   v  ur sin   u cos 

(C2.4)

By substitution of equations (C2.3) and (C2.4) into equation (C2.2) for u component, one can write: u  u   (ur  ur ) cos   (u  u ) sin 

336

(C2.5)

By an easy mathematical manipulation considering the relation for the mean flow ( u  ur cos   u sin  ), one can write a similar relation for the u component fluctuation: u  ur cos   u sin 

(C2.6)

Similarly, for the v component fluctuation, one can simply have: v  ur sin   u cos 

(C2.7)

By substitution of equations (C2.6) and (C2.7) into equation (C2.1), turbulent kinetic energy is obtained by some mathematical manipulation as follows: 1 k  [(ur cos   u sin  ) 2  (ur sin   u cos  ) 2  w2 ] 2 1  (ur 2 cos 2   u2sin 2   2uru sin  cos  )  2



(ur 2 sin 2   u2 cos 2   2uru sin  cos  )  w2



The above formula can be simplified further as below: 1 k  [(ur 2 cos 2   u2sin 2   2uru sin  cos  )  (ur 2 sin 2   u2 cos 2   2uru sin  cos  )  w2 ] 2 1  [ur 2 cos 2   u2sin 2   2uru  sin  cos   ur 2 sin 2   u2 cos 2   2uru  sin  cos   w2 ] 2 1 2 2 2   )  u2 (sin    )  w2 ]  [ur (cos  sin 2  cos 2   2 1 1 1  [ur 2  u2  w2 ] 2

So finally, in the orthogonal cylindrical coordinate system, turbulent kinetic energy can be formulated similar to the Cartesian coordinate system as with an average square of the fluctuations around the mean in three different spatial directions, as below: k

1 2 (ur  u 2  w2 ) 2

(C2.8)

337

Appendix D Computational grid details of the draft tube Here, some aspects of the created mesh for draft tube k   RANS/URANS simulations and Spalart-Allmaras and DDES simulations are presented. In section (7.3), statistics of the generated grids are presented. For k   simulations the grid is much coarser than SpalartAllmaras-type simulations due to the higher y  demand on the solid walls ( 30  y   300 ). For DDES simulations y  should be of the order of unity ( y   1 ). Figure (D.1) shows different views of the mesh used for draft tube k   simulations.

Fig D.1 Computational grid for draft tube k   RANS/URANS simulations (top: Full domain draft tube mesh, bottom left: Top view of middle plane mesh intersection, bottom right: zoomed area near the hub of the middle plane intersection) 338

Fig D.2 Computational grid for draft tube DDES simulations. (top: Full domain draft tube mesh, bottom left: Top view of middle plane mesh intersection, bottom right: zoomed area near the hub of middle plane mesh intersection) By comparing figures (D.1) and (D.2), one can observe that the mesh for DDES simulations is much more refined than the k   simulation mesh and much more uniform, as requested for LES simulations. In fact, in the core flow just after the runner, the flow is in the LES zone and the mesh needs to be as uniform as possible. Near the wall also, the

339

DDES mesh is much more refined than the k   mesh, due to the y  demands for these two different turbulence treatment techniques. As it is clear in figures (D.1) and (D.2), a circular dense region is formed after the hub region. This pattern is formed due to handling of the required y  on the hub wall and trying to keep the cell surfaces as orthogonal as possible to the flow direction which minimizes the numerical errors. In other words, the refined mesh layer region around the hub with given y  propagates in the mesh domain to avoid sharp jump (i.e. discontinuities) in the grid cell size and results in a gradual increase in the grid size all over the computational domain. Also as it is visible with comparison of figures (D.1) and (D.2), the extension length of the computational domain is reduced for the DDES mesh. This reduces the running time of DDES simulations to get statistical convergence, by reducing the time required for fluid particles to pass whole length of the draft tube.

340

Appendix E

Vortex identification schemes Vortical flow structures play a crucial role in the time-dependent behaviour of flow dynamics. Turbulent flows contains a tangle of vortices with broad range of time and length scales and some hidden structures; as a matter of fact, their dynamics and generation mechanisms are not fully understood. Especially for draft tube turbulent flows, in spite of presence of small-scale vortices, sometimes there are some dominant large vortical structures, i.e. vortex ropes, which have great influence on flow field topology. Moreover, presence of adverse pressure gradient in draft tube may lead to a flow separation. The large structures and separated flow structures are resolved directly in the case of DDES turbulence treatment. To visualize these coherent structures (organized fluid flow motions), various techniques have been developed, which can be classified under the name ‘vortex identification’ methods. In fact, there is no common accepted definition of a ‘vortex’ among researchers. Generally, one can express that a vortex involves a relatively low-pressure core and a border, in other words, a vortex is an identity that comprises rotating fluid particles around a 3D centerline, which is called vortex core. The definition of Robinson is more clarifying [Robinson 1991]: “A vortex exists when instantaneous streamlines mapped onto a plane normal to the vortex core exhibit a roughly circular or spiral pattern, when viewed from a reference frame moving with the center of the vortex core”. There are different methods to capture these identities within the flow fields, each with its different advantages and disadvantages. Generally, the methods can be classified based on Eulerian and Lagrangian points of view [Green et al. 2007]. Eulerian methods rely on velocity gradient at a local point or value of pressure, such as iso-surface of static pressure (threshold on pressure), iso-surface of vorticity magnitude, Q -criterion [Hunt et al. 1988], 2 -criterion [Jeong and Hussain 1995], swirling strength

ci -criterion [Zhou et al. 1999] and  -criterion [Chong et al. 1990], etc. As showed by Chakraborty, in spite of some differences in performance and results of the aforementioned methods, they are related to each other [Chakraborty et al. 2005]. It is worth mentioning that all the methods, which rely on the local velocity gradient, suffer from shear contamination. In other words, the error contribution due to the shear portion of velocity gradient is always present, as showed for example by Maciel et al. for the ci - criterion method in the detection of turbulent vortices in 2D cross-sections [Maciel et al. 2012].

341

They developed a vortex detection method based on the triple decomposition of kolar [kolar 2007] for velocity gradient in 2D cross-sections. Yet there is no 3D Eulerian vortex detection strategy, which removes shear effects for detecting vortices in general. Lagragian methods (such as LDE and FTLE), on the other hand, rely on the artificial seeding concept and fluid particle trajectories. These methods are less sensitive to the noises and are favourably invariant under frame changes, although they involve more computational costs [Green et al. 2007]. As a matter of fact, pressure drops in the vortex core region; therefore, iso-surface of pressure can be theoretically adopted to capture a vortex, however in most real complex fluid flows, there is not a unique appropriate level to capture all vortices simultaneously. Viscous and unsteady effects can also lead to noticeable errors in detection of vortices [Jeong and Hossain 1995]. Furthermore, as showed by Dubief and Franck in the study of coherent vortices in some test cases, such as isotropic homogeneous turbulence, mixing layers backward facing step and channel flows, the pressure iso-surface criterion fails to capture details of the vortical structures in the regions of high concentration of vortices [Dubief and Franck 2000]. In general, vortices are commonly linked with high vorticity regions. Iso-surface of high-vorticity can also be utilized to detect vortices in the flows, but the method has severe shortcomings for example in the case of parallel shear flows with high vorticity, in which the method can lead to false detection of vortices [Haller 2005]. Furthermore, as showed by Dubief and Franck in the case of LES and DNS simulations of plane channel flows, highvorticity criterion is somehow useless. In fact, streak vortices are merged so that the method cannot provide clear representation of vortical structures [Dubief and Franck 2000]. Jeong and Hossain proposed a detection method named ‘ 2 -criterion’ base on the Hessian of pressure ( p,ij   2 p xi x j ) which can provide more reliable information about minimum pressure by neglecting the effects of unsteady straining and viscosity [Jeong and Hossain 1995]. By taking the gradient of Navier-Stokes equation and using the vorticity transport equation, the following equation can be obtained: D 1 Sij  vSij ,kk  ik  kj  Sik S kj   p,ij  Dt

(E.1)

where  and S denote the anti-symmetric and symmetric parts of the velocity gradient tensor, respectively. The temporal term on the LHS comprises a material derivative defined as below: D     U  Dt t

342

(E.2)

The RHS of the above equation also involves Hessian of the pressure. The first two terms of the LHS represent the unsteady straining and viscosity effects, respectively. By removing these two first terms, the remained expression ( ik  kj  Sik S kj ), which is a symmetric tensor, can be considered to find the minimum extrema of pressure. Jeong and Hossain define the vortex core as a connected region, where pressure Hessian has two positive eigenvalues or equivalently two negative eigenvalues for the expression ( ik  kj  Sik S kj ). Since the expression is a real and symmetric tensor, it has only real eigenvalues that can be expressed and ordered as ( 1  2  3 ). Vortex core regions can be defined where 2  0 . The 2 -criterion has been being widely adopted for extraction of vortical structures in the literature. Another important vortex identification technique was introduced by Hunt et al. The ‘ Q -criterion’ is based on the second invariant of the velocity gradient tensor Q [Hunt et al. 1988]. The second invariant for an incompressible flow (where, ui xi  0 ) can be defined as below: Q

1  ij ij  Sij Sij  2

(E.3)

In fact, it is a measure that quantifies the dominance of rotation rate over strain rate or vice versa. A vortex region can be defined in a region where Q  0 , in other words, where rotation rate dominates the strain rate, a potential vortex can be detected. For incompressible flows, it can be shown that the Q quantity is proportional to the Laplacian of pressure. As showed by Dubief and Franck, the Q  0 criterion is a necessary condition for presence of low-pressure vortex cores [Dubief and Franck 2000]. Similar to the 2 -criterion method, the Q -criterion has also been intensively utilized for extraction of vortical structures in the literature. If the values of the iso-surfaces are appropriately set, both methods capture vortical structures similarly in the vast majority of applications. According to the reported results, however Q -criterion provides less sensitivity to noises (numerical and experimental) than the 2 -criterion counterpart [Dubief and Franck 2000, Paik et al. 2009]. It should be emphasized that the chosen threshold value has great influence on the captured structures in all aforementioned Eulerian methods. In the case of Q -criterion, by increasing the threshold value, density of the captured structures decreases. Finally, it is worth mentioning that ideally a desirable vortex detection scheme should be ‘objective’, in other words, detected vortices should be independent of the choice of Newtonian frames of reference or equivalently should be invariant under translation and rotation. Both 2 -criterion and Q -criterion methods are Galilean invariant, but as showed 343

by Haller on a simple vortex, they are not invariant under rotational transformation [Haller 2005]. This may create some ambiguities in the validity of these schemes especially for rotational flows. The solution seems to be a switch towards the Lagrangian methods based on particle trajectories, which are ‘objective’ by definition but at the expense of intensive computational cost.

344

Appendix F

Numerical setup for OpenFOAM simulations This appendix presents details of the numerical setup of the discretization schemes (fvSchemes-dictionary) and solution procedure details (fvSolution-dictionary) as well as the boundary conditions adopted for transient URANS and DDES simulations in OpenFOAM. F.1 Discretization schemes and numerical solution algorithms

For steady RANS simulations, ‘simpleFoam’ solver is used. For URANS and DDES simulations, SIMPLE mode of ‘pimpleFoam’ solver is adopted which involves one pressure correction step at each sub-iteration (table F.2) and continuing iterations on the momentum, pressure and turbulent quantity equations until achieving convergence at each time step. Table (F.1) presents an overview of the discretization schemes utilized for transient simulations. Table F.1 Discretization schemes in fvSchemes dictionary Sub-dictionary ddtSchemes gradSchemes

divSchemes

Keyword default default default div(phi,U)

div(phi,nuTilda) div(phi,ψ c ) laplacianSchemes default interpolationSchemes default snGradSchemes default

Discretization scheme Backward Gauss linear Gauss linear Gauss filteredLinear2V 0.2 0 Gauss linearUpwindV cellLimited Gauss linear 1 Gauss linearUpwind cellLimited Gauss linear 1 Gauss limitedLinear 1 Gauss limitedLinear 1 (where ψc : B, k, ε ) Gauss linear corrected Linear Corrected

As it is obvious in the above list of the discretization schemes, second order accurate temporal and spatial schemes are adopted in this thesis for all transient simulations. For URANS simulations, a second-order backward differencing is used for the convective term. For LES-type simulations, as explained in chapter 2, the ideal discretization for the convective term is the central difference scheme, which avoids 345

introducing too much numerical dissipation in the solution and let turbulent fluctuations survive. In contrast, upwind schemes like quadratic upwind differencing (QUICK), vanLeer scheme, normalized variable diminishing (NVD), total variation diminishing (TVD), etc. add too much dissipation in the solution and should be prohibited for LES-type simulations. On the other hand, due to the lack of numerical dissipation, central differencing schemes considerably increase the risk of unphysical oscillations growth in the solution domain that can lead ultimately to a diverged solution, especially in the case of convection-dominated flows. Table F.2 Details of fvSolution dictionary Sub-dictionary

Quantity

p

pFinal

Solvers

U

k, ε

nuTilda

Sub-dictionary

solverControls

relaxationFactors

346

Keyword solver preconditioner tolerance relTol solver preconditioner tolerance relTol solver preconditioner tolerance relTol solver preconditioner tolerance relTol solver preconditioner tolerance relTol

Keyword nCorrectors (max) nPISOCorrectors nNonOrthogonalCorrectors U relU p relp p U, k, ε, nuTilda

Entry PCG DIC 106 102 PCG DIC 106 0 PBiCG DILU 106 0 PBiCG DILU 105 0 PBiCG DILU 106 0 Entry 100 1 0 106 0 105 0 0.3 0.7

In the case of hybrid URANS/LES techniques, e.g. DDES, there are two different regions of RANS and LES zones. Ideally, the best strategy is to use different discretization schemes for these two different regions, which does not exist by default in OpenFOAM and would need code developments. Another way is to follow a compromised solution by using linear scheme with local filtering of very high frequency staggering modes like the ‘filteredLinear2V’ scheme, which is available in OpenFOAM. The scheme introduces local low dissipation via adding small amount of upwind in the computational domain where necessary. This removes staggering and avoids divergence in the case of convectiondominated flows, with almost no adverse effect on the LES statistics. As seen in chapters 6 and 7, in the case of simulation of turbulent flows in the draft tube, it is proved that ‘filteredLinear2V’ is able to resolve fine structures of the turbulence and let them considerably survive in comparison to the upwind schemes. As seen in table (F.1), for the temporal term an implicit backward differencing scheme named ‘backward’ is utilized. It uses three time-levels and result in a second order temporal accuracy. Also as one can see in table (F.1), the schemes belong to the Gaussian finite volume integration class, which need values on the cell faces using different available interpolation schemes like linear (i.e. central differencing), etc. [OpenFOAM user guide 2009]. For the convective term of turbulent quantities, a limited version of central differencing or upwind differencing is typically adopted as here. For discretization of the Laplacian term in the equations, an unbounded, conservative Guassian central differencing scheme with second order accuracy is used. As well, an explicit non-orthogonal correction scheme is applied for calculation of the normal gradient term on the cell faces. After discretization of the governing equations and obtaining a linear algebraic system of equations of the fluid flow system, one should choose a solution strategy to solve the resulting system of equations. The entries in fvSolution dictionary determine the pressure-velocity coupling strategy and the solver controls used to solve the equations. Table (F.2) depicts the entries of fvSolution dictionary. The type of linear solvers is specified based on the symmetry/asymmetry of the coefficient matrix, which is introduced by the structure of the equation being solved. For pressure quantity, a preconditioned conjugate gradient solver (PCG) is adopted, while for the asymmetric matrices corresponding to the other quantities like U, k, ε and v , a preconditioned bi-conjugate gradient solver (PBiCG) is used. Furthermore, to boost-up the convergence, a preconditioning technique is applied, including a diagonal incomplete-Cholesky (DIC) and a diagonal incomplete-LU factorization (DILU) for symmetric and asymmetric matrices, respectively.

347

F.2 Boundary conditions

As explained and pursued in details in the different chapters of the thesis, the main objective of this research is to study the effects of applying different inflow conditions for draft tube-only DDES simulations. To do so, different types of velocity and turbulent quantity inflow conditions should be imposed at the inlet plane of the draft tube.

Table F.3 Boundary conditions on patches Quantity

U [m/s]

P [m2/s2]

nuTilda [m 2 /s]

nuSgs [m 2 /s]

  m 2 / s3 

K [m2/s2]

348

Patch Rotating hub Draft tube wall Extension Inlet

Outlet Rotating hub Draft tube wall Extension Inlet Outlet Rotating hub Draft tube wall Extension Inlet Outlet Rotating hub Draft tube wall Extension Inlet Outlet Rotating hub Draft tube wall Extension Inlet Outlet Rotating hub Draft tube wall Extension Inlet Outlet

Type rotatingWallVelocity (hub rotation rate) fixedValue (zero) slip profile1DfixedValue timeVaryingMappedFixedValue zeroGradient zeroGradient zeroGradient slip zeroGradient fixedMeanValue (zero) fixedValue (zero) fixedValue (zero) slip profile1DfixedValue timeVaryingMappedFixedValue zeroGradient fixedValue (zero) fixedValue (zero) slip profile1DfixedValue timeVaryingMappedFixedValue zeroGradient epsilonWallFunction epsilonWallFunction slip profile1DfixedValue zeroGradient kqRWallFunction kqRWallFunction slip profile1DfixedValue zeroGradient

Table (F.3) shows the boundary conditions applied for different simulations performed in this thesis. As one can notice, for inflow section two different sets of boundary conditions are adopted depending on the type of the simulations. For imposing circumferential average 1D profile and unsteady 2D inflow profile of velocity and turbulent quantities at the inlet plane, ‘profile1DfixedValue’ and ‘timeVaryingMappedFixedValue’ utilities are used, respectively. For velocity boundary conditions on the rotating (hub) and stationary (draft tube) walls, ‘no-slip’ boundary condition is imposed whereas for the extension part, ‘slip’ boundary condition is applied. To consider the effect of rotation of the hub, a ‘rotatingWallVelocity’ type boundary condition is considered, for which rotation speed of the runner and the direction of rotation axis (here, z direction) are specified in the subdictionary. For pressure field, mean pressure at the exit is fixed to zero resembling the effects of the tailrace. It should be mentioned that the absolute value of pressure is not important in the calculation, because only the pressure gradient is present in the NavierStokes equations. As a result, the pressure difference (gradient) is calculated appropriately by the numerical simulations, then the pressure field can be adjusted by adding an offset to the pressure field which can be obtained based on an available experimental data at a point in the computational domain.

349

Appendix G Mean velocity field comparison with PIV-data In this appendix, mean velocity fields obtained from the transient numerical simulations of the draft tube flow are compared with the experimental data on eight PIV sub-planes, i.e. upstream and downstream sections of PIV-planes for both OP.1 and OP.4. G.1 k - ε URANS simulations

In this section, the results of different inflow profile corrections on k   simulations of the draft tube are presented including: original profile (no correction included, untouched profile coming from the full machine k   RANS simulation), u -corrected profile and ultimately u , ur -corrected profile. G.1.1 k - ε URANS simulations, OP1, plane B1, upstream

 u x uref .

a) Exp.

b) Original profile

c) u -corrected inflow d) u , ur -corrected

b) Original profile

c) u -corrected inflow d) u , ur -corrected

 u z uref .

a) Exp.

Fig G.1 Comparison of the simulation mean velocity field to the PIV experimental data k   URANS simulations, OP.1, plane B1: upstream

350

G.1.2 k - ε URANS simulations, OP.1, plane B1, downstream

 u x uref .

a) Exp.



b) Original profile

c) u -corrected inflow d) u , ur -corrected

b) Original profile

c) u -corrected inflow d) u , ur -corrected

u z  uref .

a) Exp.

Fig G.2 Comparison of the simulation mean velocity field to the PIV experimental data k   URANS simulations, OP.1, plane B1: downstream G.1.3 k - ε URANS simulations, OP.4, plane B1, upstream

 u x uref .

a) Exp.

b) Original profile

c) u -corrected inflow d) u , ur -corrected

b) Original profile

c) u -corrected inflow d) u , ur -corrected

 u z  uref .

a) Exp.

Fig G.3 Comparison of the simulation mean velocity field to the PIV experimental data k   URANS simulations, OP.4, plane B1: upstream 351

G.1.4 k - ε URANS simulations, OP.4, plane B1, downstream

 u x uref .

a) Exp.

b) Original profile

c) u -corrected inflow d) u , ur -corrected

b) Original profile

c) u -corrected inflow d) u , ur -corrected

 u z  uref .

a) Exp.

Fig G.4 Comparison of the simulation mean velocity field to the PIV experimental data k   URANS simulations, OP.4, plane B1: downstream G.1.5 k - ε URANS simulations, OP.4, plane B2, upstream

 u x uref .

a) Exp.

b) Original profile

c) u -corrected inflow d) u , ur -corrected

b) Original profile

c) u -corrected inflow d) u , ur -corrected

 u z  uref .

a) Exp.

Fig G.5 Comparison of the simulation mean velocity field to the PIV experimental data k   URANS simulations, OP.4, plane B2: upstream 352

G.1.6 k - ε URANS simulations, OP.4, plane B2, downstream

 u x uref .

a) Exp.

b) Original profile c) u -corrected inflow d) u , ur -corrected

 u z  uref .

a) Exp.

b) Original profile c) u -corrected inflow d) u , ur -corrected

Fig G.6 Comparison of the simulation mean velocity field to the PIV experimental data k   URANS simulations, OP.4, plane B2: downstream G.2 S-A URANS simulations

In this section, the results of different inflow profile corrections on S-A simulations of the draft tube are presented including: original profile (no correction included, untouched profile coming from the full machine k   RANS simulation), u -corrected profile and ultimately u , ur -corrected profile. G.2.1 S-A URANS simulations, OP1, plane B1, upstream

 u x uref .

a) Exp.

b) Original profile c) u -corrected inflow d) u , ur -corrected Fig G.7 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.1, plane B1: upstream 353

 u z  uref .

a) Exp.

b) Original profile

c) u -corrected inflow d) u , ur -corrected

Fig G.7 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.1, plane B1: upstream (- continue) G.2.2 S-A URANS simulations, OP1, plane B1, downstream

 u x uref .

a) Exp.

b) Original profile

c) u -corrected inflow d) u , ur -corrected

b) Original profile

c) u -corrected inflow d) u , ur -corrected

 u z uref .

a) Exp.

Fig G.8 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.1, plane B1: downstream G.2.3 S-A URANS simulations, OP4, plane B1, upstream

For OP.4, the results of S-A simulations applying u , ur -corrected profile considerably deviates from the experimental data, therefore, hereafter the graphs are rescaled to better visualize the S-A results.

354

 u x uref .

a) Exp.

b) Original profile

c) u -corrected inflow

 u z  uref .

a) Exp.

b) Original profile c) u -corrected inflow Fig G.9 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.4, plane B1: upstream u x uref .

u z  uref .

a) Exp.

b) u , ur -corrected a) Exp. b) u , ur -corrected Fig G.10 Simulation mean velocity field in the case of u , ur - inflow corrected profile S-A URANS simulations, OP.4, plane B1: upstream G.2.4 S-A URANS simulations, OP4, plane B1, downstream

 u x uref .

a) Exp.

b) Original profile c) u -corrected inflow Fig G.11 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.4, plane B1: downstream 355

 u z uref .

a) Exp.

b) Original profile

c) u -corrected inflow

Fig G.11 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.4, plane B1: downstream (-continue) u x uref .

u z  uref .

a) Exp.

b) u , ur -corrected

a) Exp.

b) u , ur -corrected

Fig G.12 Simulation mean velocity field in the case of u , ur -inflow corrected profile S-A URANS simulations, OP.4, plane B1: downstream G.2.5 S-A URANS simulations, OP4, plane B2, upstream

 u x uref .

a) Exp.



b) Original profile

c) u -corrected inflow

u z  uref .

a) Exp.

b) Original profile

c) u -corrected inflow

Fig G.13 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.4, plane B2: upstream 356

u x uref .

u z  uref .

a) Exp.

b) u , ur -corrected

a) Exp.

b) u , ur -corrected

Fig G.14 Simulation mean velocity field in the case of u , ur -inflow corrected profile S-A URANS simulations, OP.4, plane B2: upstream G.2.6 S-A URANS simulations, OP4, plane B2, downstream

 u x uref .

a) Exp.

b) Original profile

c) u -corrected inflow

 u z  uref .

a) Exp.

b) Original profile

c) u -corrected inflow

Fig G.15 Comparison of the simulation mean velocity field to the PIV experimental data S-A URANS simulations, OP.4, plane B2: downstream u x uref .

u z uref .

a) Exp.

b) u , ur -corrected

a) Exp.

b) u , ur -corrected

Fig G.16 Simulation mean velocity field in the case of u , ur -inflow corrected profile S-A URANS simulations, OP.4, plane B2: downstream 357

G.3 DDES simulations

In this section, the original velocity profiles obtained from different DDES simulations of the draft tube turbulent flows are presented. G.3.1 DDES simulations: vt amplification effect; OP1, plane B1, upstream

Effects of turbulent viscosity amplification in the WZ ( vt

WZ

 WZ  vt

WZ RANS

with WZ  10n ,

n  1,.., 4 ) on the velocity fields at PIV-plane B1 are presented below through u x and u z velocity profiles.

 u x uref .

a) Exp.

b) n  1

c) n  2

d) n  3

e) n  4

 u z uref .

b) n  1 c) n  2 d) n  3 e) n  4 Fig G.17 Effects of vt amplification on the simulation mean velocity fields DDES simulations, OP.1, plane B1: upstream

a) Exp.

G.3.2 DDES simulations: vt amplification effect; OP1, plane B1, downstream

 u x uref .

b) n  1 c) n  2 d) n  3 e) n  4 Fig G.18 Effects of vt amplification on the simulation mean velocity fields DDES simulations, OP.1, plane B1: downstream

a) Exp.

358

 u z  uref .

a) Exp

b) n  1

c) n  2

d) n  3

e) n  4

Fig G.18 Effects of vt amplification on the simulation mean velocity fields DDES simulations, OP.1, plane B1: downstream (-continue) G.3.3 DDES simulations: vt amplification inclusion/exclusion hub/shroud; OP1, plane B1, upstream

Effects of turbulent viscosity amplification in the WZ applied just for hub or shroud zone ( WZ  10n , n  1,.., 4 ) on the velocity fields at PIV-plane B1 are presented below through the u x and u z  velocity profiles.

 u x uref .

a)

b)

c)

d)

e)

b)

c)

d)

e)

 u z  uref .

a)

a) Exp., b) u -corrected, n  3 for both hub & shroud ; c) u -corrected, n  3 just for hub , d) u -corrected, n  3 just for shroud ; e) Original velocity profile, n  3 for both hub and shroud

Fig G.19 Effects of vt amplification with inclusion/exclusion hub/shroud on the simulation mean velocity fields obtained from DDES simulations, OP.1, plane B1: upstream 359

G.3.4 DDES simulations: vt amplification inclusion/exclusion hub/shroud; OP1, plane B1, downstream

 u x uref .

b)

a)

c)

d)

e)

 u z uref .

b) c) d) e) a) Exp., b) u -corrected, n  3for both hub & shroud ; c) u -corrected, n  3 just for hub , d) u -corrected, n  3 just for shroud ; e) Original profile, n  3 for both hub and shroud a)

Fig G.20 Effects of vt amplification with inclusion/exclusion hub/shroud on the simulation mean velocity fields obtained from DDES simulations, OP.1, plane B1: downstream G.3.5 DDES simulations: mesh convergence test, OP4, plane B1

Effects of grid resolution (mesh B: intermediate mesh; mesh C: fine mesh) on the velocity fields at PIV-plane B1 is presented below through u x and u z  velocity profiles.

 u x uref . B1, upstream

B1, downstream

a) Exp. b) Mesh B c) Mesh C a) Exp. b) Mesh B c) Mesh C Fig G.21 Effects of mesh resolution on the simulation mean velocity fields obtained from DDES simulations using 1D profile, OP.4, plane B1 360



u z  uref .

B1, upstream

a) Exp.

B1, downstream

b) Mesh B c) Mesh C

a) Exp.

b) Mesh B c) Mesh C

Fig G.21 Effects of mesh resolution on the simulation mean velocity fields obtained from DDES simulations using 1D profile, OP.4, plane B1 (- continue) G.3.6 DDES simulations: mesh convergence test, OP4, plane B2

Effects of grid resolution (mesh B: intermediate mesh; mesh C: fine mesh) on the velocity fields at PIV-plane B2 is presented below through u x and u z  velocity profiles.



u x uref .

B2, upstream

a) Exp. 

b) Mesh B

B2, downstream

c) Mesh C

a) Exp.

b) Mesh B c) Mesh C

u z  uref .

B2, upstream

a) Exp.

b) Mesh B

B2, downstream

c) Mesh C

a) Exp.

b) Mesh B

c) Mesh C

Fig G.22 Effects of mesh resolution on the simulation mean velocity fields obtained from DDES simulations using 1D profile, OP.4, plane B2 361

G.3.7 DDES simulations: mesh convergence test, OP4, plane S3 x+

Effects of grid resolution (mesh B: intermediate mesh; mesh C: fine mesh) on the velocity fields at PIV-plane S3 is presented below through u y and u z  velocity profiles. As one can see in the below figures for both planes S3 and S4, deviations from the experimental data increase for both components of the velocity on these planes especially for the downstream section of the plane; this issue is expected a-priori due to the difference of the separated flow region position, which is on the opposite side-walls in the numerical (x-) and experimental (x+) results.



u y uref .

S3 x+, upstream

a) Exp. 

S3 x+, downstream

b) Mesh B

c) Mesh C

a) Exp.

b) Mesh B

c) Mesh C

u z  uref .

S3 x+, upstream

a) Exp.

b) Mesh B

c) Mesh C

S3 x+, downstream

a) Exp.

b) Mesh B

c) Mesh C

Fig G.23 Effects of mesh resolution on the simulation mean velocity fields obtained from DDES simulations using 1D profile, OP.4, plane S3 x+ G.3.8 DDES simulations: mesh convergence test, OP4, plane S4 x+

Effects of grid resolution (mesh B: intermediate mesh; mesh C: fine mesh) on the velocity fields at PIV-plane S4 is presented below through u y and u z  velocity profiles.

362



u y uref .

S4 x+, upstream

a) Exp.  u z  uref .

b) Mesh B

S4 x+, downstream

c) Mesh C

a) Exp.

S4 x+, upstream

a) Exp.

b) Mesh B

b) Mesh B

c) Mesh C

S4 x+, downstream

c) Mesh C

a) Exp.

b) Mesh B c) Mesh C

Fig G.24 Effects of mesh resolution on the simulation mean velocity fields obtained from DDES simulations using 1D profile, OP.4, plane S4 x+ G.3.9 DDES simulations: unsteady 2D-inflow profiles, OP4, plane B1: upstream

In this section, effects of the type of inflow boundary conditions applied for DDES simulations including ‘1D’, ‘2D rotating’ and ‘2D rotating+AFG’ inflow profiles on the velocity fields on PIV-plane B1 are studied via comparing the obtained u x and u z  velocity profiles to the experiment.



u x uref .

a) Exp.

b) 1D c) 2D rotating d) 2D rotating+AFG Fig G.25 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane B1: upstream 363



u z  uref .

a) Exp.

b) 1D

c) 2D rotating

d) 2D rotating+AFG

Fig G.25 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane B1: upstream (- continue) G.3.10 DDES simulations: unsteady 2D-inflow profiles, OP4, plane B1: downstream



u x uref .

a) Exp.



b) 1D

c) 2D rotating

d) 2D rotating+AFG

u z  uref .

a) Exp.

b) 1D

c) 2D rotating

d) 2D rotating+AFG

Fig G.26 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane B1: downstream G.3.11 DDES simulations: unsteady 2D-inflow profiles, OP4, plane B2, upstream



u x uref .

a) Exp. 364

b) 1D

c) 2D rotating

d) 2D rotating+AFG



u z  uref .

a) Exp.

b) 1D c) 2D rotating d) 2D rotating+AFG Fig G.27 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane B2: upstream

G.3.12 DDES simulations: unsteady 2D-inflow profiles, OP4, plane B2, downstream



u x uref .

a) Exp. 

b) 1D

c) 2D rotating

d) 2D rotating+AFG

u z  uref .

a) Exp.

b) 1D c) 2D rotating d) 2D rotating+AFG Fig G.28 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane B2: downstream

G.3.13 DDES simulations: unsteady 2D-inflow profiles, OP4, plane S3 x+/x-, upstream



u y uref . , x+

a) Exp.

b) 1D

c) 2D rotating

d) 2D rotating+AFG 365



u z  uref . , x+

a) Exp.

b) 1D

c) 2D rotating

d) 2D rotating+AFG

Fig G.29 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane S3 x+: upstream 

u y uref . , x-

a) Exp. 

b) 1D

c) 2D rotating

d) 2D rotating+AFG

b) 1D

c) 2D rotating

d) 2D rotating+AFG

u z  uref . , x-

a) Exp.

Fig G.30 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane S3 x-: upstream G.3.14 DDES, unsteady 2D-inflow profiles, OP4, plane S3 x+/x-, downstream



u y uref . , x+

a) Exp.

b) 1D

c) 2D rotating

d) 2D rotating+AFG

Fig G.31 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane S3 x+: downstream 366



u z  uref . , x+

a) Exp.

b) 1D

c) 2D rotating

d) 2D rotating+AFG

Fig G.31 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane S3 x+: downstream (- continue) 

u y uref . , x-

a) Exp. 

b) 1D

c) 2D rotating

d) 2D rotating+AFG

b) 1D

c) 2D rotating

d) 2D rotating+AFG

u z  uref . , x-

a) Exp.

Fig G.32 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane S3 x+: downstream G.3.15 DDES simulations: unsteady 2D-inflow profiles, OP4, plane S4 x+/x-, upstream



u y uref . , x+

a) Exp.

b) 1D

c) 2D rotating

d) 2D rotating+AFG

Fig G.33 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane S4 x+: upstream 367



u z  uref . , x+

a) Exp.

b) 1D

c) 2D rotating

d) 2D rotating+AFG

Fig G.33 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane S4 x+: upstream (- continue) 

u y uref . , x-

a) Exp. 

b) 1D

c) 2D rotating

d) 2D rotating+AFG

b) 1D

c) 2D rotating

d) 2D rotating+AFG

u z  uref . , x-

a) Exp.

Fig G.34 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane S4 x-: upstream G.3.16 DDES simulations: unsteady 2D-inflow profiles, OP4, plane S4 x+/x-, downstream



u y uref . , x+

a) Exp.

b) 1D

c) 2D rotating

d) 2D rotating+AFG

Fig G.35 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane S4 x+: downstream 368



u z  uref . , x+

a) Exp.

b) 1D

c) 2D rotating

d) 2D rotating+AFG

Fig G.35 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane S4 x+: downstream (- continue) 

u y uref . , x-

a) Exp. 

b) 1D

c) 2D rotating

d) 2D rotating+AFG

b) 1D

c) 2D rotating

d) 2D rotating+AFG

u z  uref . , x-

a) Exp.

Fig G.36 Effects of the type of inflow profile on the DDES simulation results, OP.4, plane S4 x-: downstream

369

Appendix H Error of experimental measurements To validate and tune the numerical simulations in this project, some state of the art PIV and LDV measurements were used. In general, level of the error depends on the method adopted as well as the plane on which a specific measurement is done. In the following, the experimental error levels for the different compoenets of velocity at two selected operating points are summarized. As one can see, the level of errors is quite small. H.1 Error of LDV measurements on planes A and B at OP.1 and OP.4:



u uz ,    0.0017 uref . uref .

(H.1)

H.2 Error of PIV measurements on planes B1, B2, S3 and S4 at OP.1 and OP.4:

B1: 

u x u  0.015,  z  0.014 uref . uref .

(H.2)

B2 : 

u x u  0.021,  z  0.022 uref . uref .

(H.3)

S3 :  S4 : 

370

u y uref . u y uref .

 0.026, 

u z  0.029 uref .

(H.4)

 0.036, 

u z  0.038 uref .

(H.5)