Modélisation numérique d’une hydrolienne à axe horizontal

UNIVERSITÉ DE SHERBROOKEFaculté de génie

Département de génie mécanique

Modélisation numérique d’une hydrolienne àaxe horizontal de type Darrieus en eau peu

profonde

Thèse de doctoratSpécialité : génie mécanique

Alla Eddine Benchikh Le Hocine

Sherbrooke (Québec) Canada

Mars 2019

MEMBRES DU JURY

R. W. Jay LaceyDirecteur

Sébastien PoncetCodirecteur

Hachimi FellouahÉvaluateur

Julien FavierÉvaluateur

Bruce MacVicarÉvaluateur

RÉSUMÉ

Avec en toile de fond la question du changement climatique, les contraintes énergétiquessont de plus en plus importantes : population et consommation mondiales croissantes, fluc-tuations des prix des énergies fossiles et diminution des ressources disponibles et exigencesenvironnementales toujours plus contraignantes. Dans un contexte mondial où la questionénergétique est donc devenue centrale, le besoin de se tourner vers des alternatives renou-velables est devenu primordial. L’énergie électrique produite par des hydroliennes est l’unedes sources alternatives les plus exploitées. Parmi ces hydroliennes, les turbines de typeDarrieus à axe vertical qui ont été largement considérées dans la littérature. Au contraire,la configuration horizontale pour des applications de faible puissance en rivière n’a jamaisété étudiée jusqu’à présent. Cette thèse a donc pour objectif d’optimiser une hydroliennede rivière à axe horizontal de type Darrieus, tant sur le profil hydrodynamique que surle nombre de pales utilisées, et de quantifier ses performances pour différentes conditions(hauteur d’eau) opérationnelles réalistes.

Un benchmark numérique de modèles de turbulence RANS (Reynolds-Averged Navier-Stokes) à deux équations et de modèles de sous mailles LES (large Eddy Simulation,simulation des grandes échelles) a d’abord été effectué dans le cadre d’un écoulementturbulent autour d’un obstacle fixe et submergé en forme de D. Une comparaison a étéfaite entre les résultats numériques et expérimentaux obtenus par PIV (Particle ImageVelocimetry) 2D et 3D stéréoscopiques. Le modèle k-ω SST à bas nombre de Reynoldsprédit le mieux la couche de cisaillement au-dessus de l’obstacle. Par contre, le modèlek-ε à haut nombre de Reynolds est plus performant dans la zone de recirculation enaval de l’obstacle. Les résultats produits par la LES Wale sont meilleurs que ceux dumodèle Smagorinsky qui s’avère trop dissipatif. L’analyse spectrale ne montre aucun picdistinct dans la région de sillage. La méthode POD (Proper Orthogonal Decomposition)a finalement été appliquée dans le sillage de l’obstacle pour en étudier la dynamique enextrayant les différents modes dominants.

Dans la seconde étude sur l’hydrolienne de type Darrieus à axe horizontal, une approche2.5 D RANS instationnaire a été adoptée en utilisant un maillage très raffiné et le modèlek-ω SST à bas nombre de Reynolds. La turbine est placée dans un canal ouvert sans surfacelibre. Quatre profils de pales ont été testés. L’approche numérique a été validée avec moinsde 13 % d’erreur par rapport aux résultats expérimentaux obtenus sur une éolienne. Leprofil S1046 a permis d’accroître les performances produites par le NACA0018. Dans lesrégions de décrochage dynamique et de transition, le S809 a été le moins performant.Pour des hautes vitesses de rotation, le FXLV152 est le plus performant. En variant lenombre de pales sur l’hydrolienne de type Darrieus équipée du profil S1046, les meilleuresperformances ont été produites à basses et hautes vitesses de rotation pour un nombre depales égal à 4 et 2, respectivement.

Dans la dernière étude, des simulations 2.5 D multiphasiques ont été accomplies en utilisantle modèle VOF (volume of fluid) afin de quantifier l’influence de la hauteur d’eau sur les

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performances de l’hydrolienne de type Darrieus à axe horizontal. Deux configurations ontété considérées : hydrolienne partiellement (configuration 1) ou complètement submergée(configuration 2). L’approche VOF a été validée avec une erreur moyenne de 0.6 % parrapport aux résultats expérimentaux obtenus dans le cadre de la rupture d’un barrage.Pour une turbine totalement immergée, le coefficient de puissance est supérieur de 36.8% à la configuration partiellement immergée. Le nombre de Froude calculé en amont dela turbine croît progressivement pour des hautes vitesses de rotation. Quand la turbineest complètement submergée, elle extraie les plus grandes quantités de mouvement à deshautes vitesses de rotation. L’aptitude de la turbine à produire de la puissance dans lesdeux configurations assurera son efficacité dans les rivières peu profondes.

Mots-clés : Hydrolienne, Darrieus, Modélisation numérique, Turbulence, LES, POD.

ABSTRACT

Numerical modeling of a Darrieus horizontal axis hydrokinetic turbine in shal-low water

Energy constraints are becoming increasingly important as global consumption continuesto increase day by day. The price of oil fluctuates continuously tending yet is predicted toincrease substantially under more stringent environmental requirements. The idea of turn-ing to and developing renewable alternatives has become paramount. The electric energyproduced by hydrokinetic turbines is one of the most exploited alternative sources. Amongthese hydrokinetic turbines, the Darrieus vertical axis hydrokinetic turbine DVAHT hasbeen largely considered. On the contrary, the horizontal Darrieus configuration for riverapplications has never been studied numerically. Moreover, no geometrical optimizationof blade profile has been carried out to improve its performances and no research on theinfluence of the water height has yet been done. To answer these problems, three differentstudies are performed increasingly progressively the complexity of the flow configuration.

A numerical benchmark of the RANS (Reynolds-Averaged Navier-Stokes) turbulence clo-sure models with two equations and LES (large eddy simulation) subgrid scale models isfirst carried out on the turbulent flow around a bed mounted D-section. A comparison ismade between the numerical results and 2D planar and 3D stereoscopic PIV (particle im-age velocimetry) measurements. In comparison with the other models, the low-Reynoldsk-ω SST model correctly reproduces the shear layer above the D-section. On the otherhand, the high-Reynolds number k-ε model is more accurate in the recirculation regiondownstream the obstacle. The results produced by the Wale LES model appear betterthan those obtained by the Smagorinsky model which is more dissipative. Spectral anal-ysis shows no distinguishable peaks in the wake region. The POD (proper orthogonaldecomposition) is applied in the wake region in order to extract the different modes.

In the second study, a Darrieus horizontal axis hydrokinetic turbine DHAHT is modeledusing unsteady 2.5D RANS simulations based on the low-Reynolds k-ω SST turbulencemodel. Four blades profile are tested for the fully submerged configuration. The numericalapproach is first validated and shows less than a 13 % error against experimental datapublished on a wind turbine. The S1046 profile produces higher performances than theNACA0018 profile. In the dynamic stall and transition regions, the S809 profile showedthe poorest performance. For high tip-speed ratios, the FXLV152 produces the highestpower coefficient. The effect of blade number N on the DHAHT with the S1046 has alsobeen investigated. For N = 4 and 2, the best performance is obtained at low and hightip-speed ratios, respectively.

In a last study, instantaneous 2.5D multiphase simulations are run using the VOF (vol-ume of fluid) method in order to quantify the influence of the free surface and water levelon DHAHT performance. The DHAHT is tested in two configurations: partially (con-figuration 1) or completely submerged (configuration 2). The VOF model is validatedwith an average error of 0.6 % against experimental results for the breaking of a dam.

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The total immersion of the DHAHT in configuration 2 increases by 36.8 % the powercoefficient compared with configuration 1. The calculated Froude number downstreamof the turbine increases gradually with increased tip-speed ratios. The quantification ofthe momentum loss shows that the DHAHT in configuration 2 extracts higher values ofmomentum loss with higher tip-speed ratios. The ability of the turbine to produce powerin both configurations will ensure its efficiency in shallow rivers.

Keywords: Hydrokinetic turbine, Darrieus, Numerical modelling, Turbulence, LES,POD.

«Le génie est fait d’un pour cent d’inspira-tion et de quatre-vingt-dix-neuf pour cent detranspiration.»

Thomas Edison

REMERCIEMENTS

Je voudrais tout d’abord remercier grandement mes directeur et co-directeur de thèse, JayLacey et Sébastien Poncet, pour toute leur aide et confiance. Je suis ravi d’avoir travailléen leur compagnie car outre leur appui scientifique, ils ont toujours été là pour me souteniret me conseiller au cours de l’élaboration de cette thèse. Leur disponibilité et simplicitéont permis d’avoir un excellent climat de travail.

J’adresse tous mes remerciements à Monsieur Hachimi Fellouah, de l’honneur qu’il m’afait en acceptant d’être rapporteur de cette thèse.

Je tiens à remercier Julien Favier du laboratoire M2P2 de l’Université d’Aix-Marseillepour avoir accepté de participer à mon jury de thèse et pour sa participation scientifiquesur la méthode POD.

Je remercie également Bruce MacVicar de l’Université de Waterloo pour l’honneur qu’ilme fait d’être dans mon jury de thèse.

Je tiens à remercier mes deux amours Asma et Hacene, pour leurs encouragement, soutienet présence car sans eux je n’aurai jamais accompli ce travail. À l’amour de ma vie Asma,je te remercie pour ta patience, présence et tes sacrifices pour moi et Hacene.

Un énorme merci à ma mère Soraya Boumghar et mon père Raouf qui m’ont instruit etgravé en moi la persévérance et la science depuis mon jeune âge. Cet accomplissementvous est offert.

À ma petite sœur Nihad (Ninou), pour ton support et encouragement.

Un grand merci à mes beaux-parents Lahcene et Fatima pour leurs confiance et encoura-gement.

À tout mes collègues de travail (Sergio, Yu, Ibai, Junior, Kamel, Nidhal) et de sport(Sébastien, Hachimi, Hacene, Taoufik ...) pour leur soutien et encouragement. À Amrid,pour nos longs débats scientifiques.

TABLE DES MATIÈRES

1 INTRODUCTION 11.1 Introduction générale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectifs de la thèse et originalités . . . . . . . . . . . . . . . . . . . . . . 21.3 Plan de la thèse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Turbulent flow over a D-section bluff body : a numerical benchmark 52.1 Avant-propos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Numerical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1 Geometrical modeling . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Turbulence modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.3 Numerical parameters . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.4 Experimental database . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5.1 Mean flow field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5.2 Coherent structures in the wake flow . . . . . . . . . . . . . . . . . 222.5.3 POD analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Numerical modeling of a Darrieus Horizontal Axis shallow-Water Tur-bine 333.1 Avant-propos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Characteristics of the Darrieus Horizontal Axis Water Turbine (DHAHT) . 373.5 Numerical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5.1 Geometrical modeling . . . . . . . . . . . . . . . . . . . . . . . . . 403.5.2 Numerical method and turbulence closure . . . . . . . . . . . . . . 413.5.3 Boundary conditions and numerical parameters . . . . . . . . . . . 41

3.6 Validation of the flow solver . . . . . . . . . . . . . . . . . . . . . . . . . . 443.7 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.7.1 Comparison between blade profiles . . . . . . . . . . . . . . . . . . 463.7.2 Influence of the blade number N . . . . . . . . . . . . . . . . . . . 53

3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Multiphase modeling of the free surface flow through a Darrieus hori-zontal axis shallow-water turbine 574.1 Avant-propos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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4.4 Characteristics of the Darrieus Horizontal Axis Hydrokinetic Turbine (DHAHT) 614.5 Numerical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5.1 Geometrical modeling . . . . . . . . . . . . . . . . . . . . . . . . . 644.5.2 Numerical method and turbulence closure . . . . . . . . . . . . . . 644.5.3 Boundary conditions and numerical parameters . . . . . . . . . . . 66

4.6 Validation of the flow solver . . . . . . . . . . . . . . . . . . . . . . . . . . 674.7 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.7.1 General performances . . . . . . . . . . . . . . . . . . . . . . . . . . 704.7.2 DHAHT’s influence on flow regime and momentum loss . . . . . . . 75

4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 CONCLUSION FRANÇAISE 83

6 ENGLISH CONCLUSION 89

LISTE DES RÉFÉRENCES 95

LISTE DES FIGURES

1.1 Nombre d’installations hydroélectriques par région. Adapté de [75]. . . . . 3

2.1 Sketch of the computational domain (not to scale), and views of the D-section bluff body with its main dimensions. . . . . . . . . . . . . . . . . . 11

2.2 Mesh distributions for the different approaches. Coarse grid for the high-Reynolds number models (top) and fine mesh for the low-Reynolds numbermodels and the LES (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 2D maps around the D-section obstacle in the channel midplane of the meanvelocity components u∗ and v∗ and turbulence kinetic energy k∗. Compa-risons between the PIV measurements, three two-equation RANS modelsand two LES models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Distributions of the two main mean velocity profiles (a,c) u∗ and (b,d) v∗along the streamwise direction X/D. Results obtained at (a,b) Y/D = 1and (c,d) Y/D = 0.5 in the median plane of the channel. . . . . . . . . . . 20

2.5 Distributions of the mean streamwise velocity component u∗ along the span-wise direction 2Z/b. Results obtained at Y/D = 0.5 for two X/D locations :(a) X/D = 0.8 and (b) X/D = 1.6. . . . . . . . . . . . . . . . . . . . . . . 21

2.6 2D views of the instantaneous vorticity around the D-section body obtainedby the LES Smagorinsky (a,c,e) and the LES Wale (b,d,f) models. (a,b)Side views in the mid-plane of the channel (2Z/b = 0) ; Top views at (c,d)Y/D = 0.5 and (e,f) Y/D = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7 Side and top views of the Q-criterion (Q=0.029) distribution over the D-section obstacle colored by the normalized averaged longitudinal velocity(u∗). Results obtained by the LES Wale (a,b) and the LES Smagorinsky(c,d) models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.8 PSD distributions (m2/s) of the mean velocity components u and v downs-tream of the D-section obstacle at four positions X/D. Comparison bet-ween the results obtained by PIV and the LES Wale model at (Y/D = 0.5,2Z/b = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.9 Relative and cumulative energy contributions of the POD modes obtainedby the LES Wale model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.10 First six POD modes from (a) to (f) extracted from the instantaneous LESWale results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.11 Velocity field reconstructed (m/s) using : (a) only the first mode ; (b) modes2 to 10. Results obtained by the LES Wale model. . . . . . . . . . . . . . . 29

2.12 Contour map of longitudinal velocity field reconstructed U/U0 using the firstmode with the (a) minimum and (b) maximum time coefficients. Resultsobtained by the LES Wale model. . . . . . . . . . . . . . . . . . . . . . . . 29

3.1 CAD geometry of the DHAHT. . . . . . . . . . . . . . . . . . . . . . . . . 383.2 2D cross-sections of the four blade profiles. . . . . . . . . . . . . . . . . . . 39

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3.3 2D sketch of the computational domain with its main dimensions and theboundary conditions.Note that for the current numerical experiments theturbine rotates in the counter-clockwise direction. . . . . . . . . . . . . . . 40

3.4 Boundary layer velocity profile imposed at the inlet. . . . . . . . . . . . . . 423.5 2D views of the numerical mesh distribution of a) entire domain, b) rotor

domain, and c) turbine blade . . . . . . . . . . . . . . . . . . . . . . . . . 433.6 Performance curve of a three-blade H-rotor Darrieus wind turbine versus λ.

Comparison with former CFD [24, 41, 84] and experimental results [24]. . . 453.7 Comparisons of the averaged power coefficient CP versus λ for the S1046,

S809, FXLV152 and NACA0018 profiles. Results obtained for the 3 bladeturbine (N = 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.8 Comparison of the averaged torque coefficient CT for the S1046, S809,FXLV152 and NACA0018 profiles. Results obtained for the 3 blade tur-bine N = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.9 Comparison of the instantaneous torque coefficient CTiof one blade over

one rotation (degrees) between the S1046, S809, FXLV152 and NACA0018profiles. Results obtained for the 3 blade turbine N = 3 and λ = 1.8. . . . 48

3.10 Polar distributions of the instantaneous torque coefficient CTi. Results ob-

tained for the 3 blade turbine N = 3. . . . . . . . . . . . . . . . . . . . . . 513.11 Distributions of the mean streamwise velocity U∗ along the vertical direction

Y/H for the 3 blade turbine N = 3 and λ = 1.8. Results obtained for fourX/D locations : (a) X/D = 1, (b) X/D = 2, (c) X/D = 3 and (d) X/D = 4. 52

3.12 Influence of the blade number N on the averaged (a) power CP and (b)torque CT coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.13 Influence of the blade number N on the polar distributions of the instanta-neous torque coefficient CTi

for four tip speed ratios λ. . . . . . . . . . . . 54

4.1 CAD geometry of the DHAHT. . . . . . . . . . . . . . . . . . . . . . . . . 624.2 2D sketch of the computational domain with its main dimensions and the

boundary conditions. The turbine rotates in the counter-clockwise direction.Note that the water level HW and the height of the air layer Ha are fixedto HW = 0.65 m, Ha = 1.14 m and HW = 0.82 m, Ha = 0.97 m forconfigurations 1 and 2, respectively. . . . . . . . . . . . . . . . . . . . . . . 64

4.3 (a) Example of a 2D view of the mesh distribution for configuration 1 (Fr =0.625) ; (b) 2D views of the numerical mesh distribution in the rotor regionand around the blades. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 VOF validation against the experimental data of Koshizuka et al. [57]. (a)Normalized evolution of the water level H∗ versus time t∗. (b) 2D sketch ofthe configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5 Performance curve of a three-blade Darrieus vertical axis wind turbine(VAWT). Comparison between the present predictions and the experimen-tal results of Castelli et al. [24]. . . . . . . . . . . . . . . . . . . . . . . . . 69

LISTE DES FIGURES xv

4.6 Distributions of the averaged power coefficient CP as a function of the tipspeed ratio λ for configurations 1, 2 and the single-phase (slip wall) case.The multiphase configurations are calculated with (w/) and without (w/o)correction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.7 Distributions of the averaged torque coefficient CT as a function of the tipspeed ratio λ for configurations 1, 2 and in the single-phase (slip wall) case. 72

4.8 Instantaneous snapshots of the water volume fraction α around the DHAHTduring the first 3 rotations. Results obtained for λ = 1.8 in configuration 1. 74

4.9 Instantaneous snapshots of the water volume fraction α around the DHAHTat the 15th rotation. Results obtained for λ = 1.8 in configurations 1 and 2. 74

4.10 Instantaneous vorticity fields around the blade for θ = 90 (a,b,c), 150

(d,e,f) and 210 (g,h,i). Comparisons between configuration 1 (a,d,g), 2(b,e,h) and the single-phase (slip wall) case (c,f,i). Results obtained forλ = 1.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.11 Comparison of the polar distributions of the instantaneous torque coefficientCT for configurations 1 and 2 for four tip speed ratios λ. . . . . . . . . . . 77

4.12 Froude number Fr distributions downstream (X/D = 11D) the DHAHTfor configurations 1 and 2, with the corresponding linear regressions. . . . . 78

4.13 Distributions of the momentum loss LM for configurations 1 and 2, withthe corresponding linear regressions. . . . . . . . . . . . . . . . . . . . . . . 79

xvi LISTE DES FIGURES

LISTE DES TABLEAUX

2.1 Mesh grid parameters for the high- and low-Reynolds number models andthe LES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Main geometrical and operating parameters for the flows through the DHAHT. 383.2 Mesh grid parameters for the three-bladed configuration. . . . . . . . . . . 433.3 Instantaneous vorticity fields for θ = 90, 150, 210, 270, 330. Compari-

sons between the S1046, S809, FXLV152 and NACA0018 profiles for the 3blade turbine N = 3 and λ = 1.8. . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 Main geometrical and operating parameters for the flows through the DHAHT. 624.2 Mesh grid parameters for the three-bladed configuration. . . . . . . . . . . 67

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CHAPITRE 1

INTRODUCTION

1.1 Introduction générale

La demande mondiale en énergie ne cesse de croître, alors que les ressources en énergiesdites fossiles ne cessent de décroître. Afin de faire face à ce problème, il est nécessaired’améliorer les rendements des différents procédés alimentés par des énergies fossiles, ouse tourner vers l’exploitation des énergies renouvelables. L’exploitation d’une seule sourced’énergie renouvelable ne permet souvent pas de répondre à la demande de consommation,la solution consiste à faire un mix de ressources énergétiques. Couplées à une solution destockage d’énergie, les énergies renouvelables offrent également une alternative intéressanteaux régions isolées des réseaux électriques principaux. Il existe quatre principales sourcesnaturelles pour produire de l’énergie électrique : solaire, biomasse, l’éolien et l’hydrolien.La situation géographique du Canada et celle du Québec, en particulier, favorisent plu-tôt l’exploitation des énergies éolienne et hydraulique. L’hiver pouvant durer jusqu’à 6mois, l’énergie solaire n’est pas une source fiable, à cause de l’accumulation de neige surles panneaux solaires, ce qui peut réduire notablement la production d’énergie électrique.Néanmoins, le potentiel solaire reste toujours une source non-négligeable durant l’été. Ence qui concerne l’énergie éolienne, plus de 12 796 MW sont produits sur tout le territoire ca-nadien, grâce aux 297 parcs éoliens [73]. Cette production d’énergie couvre autour de 6% dela demande totale du Canada, ce qui représente la consommation de 3.8 millions de foyers.L’intermittence du vent cause cependant une discontinuité dans la production d’électricitépar les éoliennes. Cette limite n’existe pas avec l’énergie hydraulique, qui est une sourcecontinue soit via des hydroliennes dans les rivières, des barrages hydroélectriques et deshydroliennes ou centrales marémotrices près des cotes maritimes. Le potentiel canadien enénergie hydraulique est immense dû au fait que toutes les provinces ont accès à au moinsune de ces ressources hydrauliques. Cela favorise la production hydroélectrique avec 63% (76 000 MW) de la production canadienne totale, et un potentiel non exploité de 160000 MW et autour de 4 400 MW dans la région du Québec [74]. Avec cette dépendance àl’énergie hydroélectrique, le Canada est le troisième au monde en nombre d’installationshydroélectriques (Fig.1.1) et au deuxième rang en termes de production avec 9.6 % de laproduction mondiale [50]. Cette énergie est majoritairement produite par des hydroliennesdans des barrages avec retenue ou au fil de l’eau.

1

2 CHAPITRE 1. INTRODUCTION

L’entraînement des turbines dans la majorité des stations hydroélectriques au Canada estassuré par l’effet potentiel dû à une chute d’eau. Dans quelques endroits fluviaux, où lesprofondeurs d’eau sont importantes, comme dans le fleuve St-Laurent, des turbines sontinstallées dans le fond et entraînées par les courants d’eau. Des milliers de réseaux derivières peu profondes restent néanmoins inexploités mais ils se trouvent la plupart dutemps dans des régions reculées non raccordées au réseau électrique principal à cause ducoût prohibitif d’un possible raccordement. L’idée d’installer des turbines dans ces coursd’eau peu profonde permettra d’assurer une production électrique continue pour un utili-sateur local (ex : chalet). Le choix du modèle de turbine est une tâche complexe car lesconditions d’écoulement dans ces rivières sont instables : hauteurs d’eau faibles et inter-mittentes, faibles vitesse d’écoulement (1-3 m/s), présence de rochers sur le lit des rivières,débris de végétation ou glaces en surface . . . autant d’éléments qui perturbent l’écoulementen amont des turbines. Peu de turbines peuvent être utilisées dans ces conditions d’écou-lement. Néanmoins la turbine de type Darrieus à axe vertical permet de surmonter unepartie de ces limites, par son indépendance à la direction de l’écoulement et son efficacitéà des faibles vitesses d’eau et de rotation.

Depuis 1926, la turbine de type Darrieus à axe vertical DVAT (Darrieus vertical axisturbine) a été largement étudiée. De nombreuses études ont été faites pour optimiserses performances et réduire son bruit dans le cas d’éoliennes. Appliquer ces turbines enrivières (DVAHT) est plus problématique du fait des faibles niveaux d’eau généralementobservés et des variations de ces niveaux potentiellement importantes. Elles ont donc reçupeu d’attention dans la littérature [26]. Néanmoins, le placement de la turbine de typeDarrieus dans une position horizontale permettrait d’obtenir un meilleur rendement mêmepour des faibles hauteurs d’eau. C’est dans ce cadre que s’inscrit cette thèse : optimiserles performances d’une turbine DHAHT (Darrieus horizontal axis hydrokinetic turbine)existante et simuler ses performances dans des conditions réalistes d’écoulement.

1.2 Objectifs de la thèse et originalités

L’objectif principal de cette thèse est de mettre en place un guide méthodologique numé-rique pour optimiser et étudier la sensibilité des performances des hydroliennes et plusprécisément la turbine de type Darrieus à axe horizontal. Pour atteindre cet objectif, dessous-objectifs allant par ordre croissant de difficulté ont été également définis :

1. Modélisation numérique de l’écoulement autour d’un obstacle fixe (en forme de D)et submergé.

1.2. OBJECTIFS DE LA THÈSE ET ORIGINALITÉS 3

Figure 1.1 Nombre d’installations hydroélectriques par région. Adapté de [75].

• Comparaison entre plusieurs modèles de turbulence RANS et de sous-mailles LESavec des mesures PIV.• Identifier le ou les modèles RANS et/ou LES offrant le meilleur compromis préci-

sion/coût de calcul.• Identification des différentes structures tourbillonnaires instationnaires qui se forment

autour de l’obstacle (vorticité, critère Q) et compréhension de leur dynamique(méthode de décomposition en modes propres).

2. Optimisation des performances de la turbine de type Darrieus à axe horizontal com-plètement submergée avec confinement.

• Modélisation numérique monophasique (eau) autour de la turbine de type Darrieussubmergée.• Validation du modèle de turbulence choisit à partir du benchmark (objectif 1).• Étude paramétrique de plusieurs profils de pales afin d’optimiser les performances

de la turbine.• Étude de l’influence du nombre de pales sur la turbine de type Darrieus à axe

horizontal équipée avec le profil le plus performant.

3. Étude de l’interaction entre la surface libre et la turbine de type Darrieus à axehorizontal dans deux configurations : 1) partiellement submergée ; 2) complètementsubmergée

• Modélisation numérique multiphasique de la turbine de type Darrieus équipée avecle profil le plus performant sélectionné à partir du deuxième objectif.• Validation de l’approche multiphasique sur la déformation de la surface libre.• Étude de l’influence de la hauteur d’eau sur les performances de la turbine et les

mécanismes de détachement tourbillonnaire autour des pales.

4 CHAPITRE 1. INTRODUCTION

L’originalité de cette thèse consiste à accomplir des simulations numériques d’une turbinede type Darrieus à axe horizontal, afin de quantifier et optimiser ses performances maisaussi de comprendre l’interaction de la surface libre avec la turbine et son influence surles différents coefficients.

1.3 Plan de la thèse

Le manuscrit de cette thèse de doctorat est présenté sous format d’articles. Chaque chapitreest un article qui répond à un des objectifs énumérés ci-dessus. Dans le second chapitre,un benchmark numérique des modèles de turbulence RANS et des modèles de sous maillesLES est accompli pour un écoulement turbulent autour d’un obstacle fixe en forme de D.Par la suite, dans le troisième chapitre des simulations numériques monophasiques ont étéfaites autour de la turbine de type Darrieus à axe horizontal en testant plusieurs profilsde pales afin d’accroître ses performances. Dans le quatrième chapitre, des simulationsnumériques multiphasiques ont été accomplies sur l’interaction de la surface libre et laturbine de type Darrieus à axe horizontal, et aussi sur l’influence de la hauteur d’eau surses performances. La thèse se termine par des conclusions sur les principaux résultats etdes perspectives de recherche.

CHAPITRE 2

Turbulent flow over a D-section bluff body :a numerical benchmark

2.1 Avant-propos

Auteurs et affiliation :A. E. Benchikh Le Hocine : étudiant au doctorat, Université de Sherbrooke, Faculté degénie, Département de génie mécanique.R. W. J. Lacey : professeur, Université de Sherbrooke, Faculté de génie, Département degénie civil.S. Poncet : professeur, Université de Sherbrooke, Faculté de génie, Département de géniemécanique.Date d’acceptation : 6 octobre 2018État de l’acceptation : version en ligne publiéeRevue : Journal of Environmental Fluid MechanicsRéférence : Benchikh Le Hocine et al. [13].Titre français : Écoulement turbulent autour d’un obstacle en forme de D : benchmarknumériqueContribution au document : Cet article constitue la première étape de validation né-cessaire vers la simulation d’une hydrolienne de rivière et consiste à simuler, dans unpremier temps, l’écoulement autour d’un obstacle fixe submergé. Le but est de mettre enplace une ligne directrice sur le choix des approches numériques grâce à un benchmarkdes modèles de turbulence de type RANS et LES validés par des mesures expérimentalesobtenues par PIV. D’autre part, cet article permet une compréhension détaillée des struc-tures tourbillonnaires qui se forment autour d’obstacle submergé en utilisant différentesapproches pour le post-traitement comme le calcul du critère Q et la POD, qui permettentd’identifier les structures cohérentes et de caractériser leur dynamique, respectivement.Résumé français : Un benchmark numérique des différents modèles de turbulence a étéréalisé pour étudier l’écoulement turbulent derrière un obstacle submergé en forme de D.Les modèles de fermeture incluent des modèles RANS (Reynolds-Averaged Navier-Stokes)à deux équations et des modèles de simulation des grandes échelles (LES). Ils sont com-parés aux mesures PIV planes et stéréoscopiques. Le modèle k-ω SST à bas nombre de

5

6CHAPITRE 2. TURBULENT FLOW OVER A D-SECTION BLUFF BODY : A

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Reynolds s’avère mieux adapté pour capturer la couche de cisaillement intense au-dessusde l’obstacle par rapport aux autres modèles à deux équations. Pourtant les modèles k-εet k-ω SST à haut nombre de Reynolds présentent des performances supérieures dans larégion de recirculation derrière l’obstacle. Des LES ont également été réalisées sur l’écou-lement autour de l’obstacle afin de déterminer l’influence des modèles de sous mailles surla prédiction des structures tourbillonnaires. Le modèle Wale combiné avec un schéma auxdifférences centrées a montré un meilleur accord global par rapport au modèle standardde Smagorinsky, qui est plus dissipatif. Une analyse spectrale a été réalisée dans la régiondu sillage, mais aucune fréquence distincte n’a pu être trouvée. Une décomposition ortho-gonale aux valeurs propres (POD) a été appliquée aux résultats de la LES pour extrairela dynamique de l’écoulement et les structures cohérentes.

2.2. ABSTRACT 7

2.2 Abstract

A numerical benchmark of different turbulence closures was performed to investigate theturbulent wake flow behind a submerged D-shaped bluff body. The numerical models in-cluded steady two-equation Reynolds-Averaged Navier-Stokes (RANS) models and LargeEddy Simulations (LES), which were compared to planar and stereoscopic particle imagevelocimetry (PIV) measurements. The k-ω SST low-Reynolds number model was foundto be better adapted to capture the intense shear layer at the top of the D-section com-pared to the other two-equation models. However, the k-ε and k-ω SST high-Reynoldsnumber models demonstrate higher performance in the recirculation region. LES was alsocompleted over the D-section to determine the influence of the sub-grid scale models onthe prediction of the vortical structures. The Wale model together with central differenceschemes showed a better overall agreement over the standard Smagorinsky model, whichappears too dissipative. A spectral analysis was performed in the wake region, yet no dis-tinct shedding frequencies could be found. A proper orthogonal decomposition (POD) wasapplied to the LES results to extract the mean flow dynamics and the coherent structures.

2.3 Introduction

Fishways allow fishes to travel across anthropogenic obstructions [53].The flow in fishwayshas interested researchers for decades [15, 18, 22, 71, 93, 95, 108, 120, 124], yet the flowstructure is complex because of the three dimensional geometry, the elevated turbulenceintensity, and the interactions between the coherent vortices produced by successive obs-tacles. The fish’s progress through passes depend on a multitude of mean and turbulentflow variables [121] and an accurate flow description is required for the improved design offuture fishways. The objectives of the present work are to perform a numerical benchmarkof different turbulence closures for the flow behind a D-shaped bluff body and to displaythe capability of more advanced large eddy simulations (LES) in predicting the coherentstructures appearing in the wake. The present flow configuration represents a first steptowards a more realistic flow simulation of fishways and a better understanding of flowstructures shedding from bluff bodies. An indepth study of flow over a wall mounted bluffbody also has implications for other fields such as : shape optimization in the automobileindustry [38], the prediction of microclimate in urban [119] and mountainous regions [21],and the prediction of dust emissions due to industrial stockpiles [33].Experimental and numerical approaches have been performed to understand the flow infish passes and to correlate the flow features with fish behavior. Different numerical studieswere performed for fishways without any experimental validation [2, 71, 95]. Sometimes,


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the numerical tool is clearly used as a black box [43]. When experimental validation ispossible, the numerical predictions are not always satisfactory and/or not carefully valida-ted. Bombac et al. [18] compared the results of an experimental and numerical study for avertical slot fishway. A 2-D depth-averaged shallow-water numerical model PCFLOW2Dcoupled with three different turbulence models was used for comparisons with physicalacoustic Doppler velocimeter (ADV) measurements. All turbulence closures provided sa-tisfactory results. However, as all the comparisons were performed at cross-sections whereflow complexity is reduced, it is difficult to draw definitive conclusions on model perfor-mance. Tran et al. [120] conducted a numerical and experimental study in which rocksof the natural fishway were replaced by cylinders while keeping the same arrangement.An ADV was used to measure the velocity and the turbulent kinetic energy and resultswere compared to a depth averaged model obtained with the Telemac-2D software. Alarge discrepancy between the numerical and experimental results was found upstreamand downstream of the cylinders, which was attributed to the numerical approach andthe choice of the k-ε turbulence model. Many other numerical studies of turbulent flowand optimization of fish passes demonstrate that the choice of the appropriate flow sol-ver and/or turbulence closure remains an open question in the literature [15, 29, 108].Research is needed to develop and validate an optimal flow solver on a simple (thoughrelevant) geometry where wake flow characteristics are similar to the those in fishways.Compared to a systemic approach used to optimize fish passes, it is believed that a betterunderstanding of the flow dynamics and vortical structures produced by the wake of asingle obstacle will in the not too distant future lead to improved understanding of theirinteractions with fishes [66]. In addition, computational resources being limited, the ac-curacy of the calculations with a refined mesh around a single obstacle are improved incomparison to using larger cells [15, 18, 108].The turbulent flow around a single obstacle/boulder in a nature like fishway can resembleflows around more canonical obstacles such as a cube, cylinder or sphere. This resem-blance is found in the different vortex structures formed in the near wake region [12],in the shear-layer developed at the top of the obstacles and in the separation and recir-culation zones observed both upstream and downstream. The stability and transition tounsteadiness in the wake behind bluff bodies have received much attention up to the 90’sas shown in the review of Williamson [129] and Sumner [115]. For a wall-mounted cube,Hussein and Martinuzzi [49] performed lased Doppler anemometry (LDA) measurementsto determine the turbulence kinetic energy (k) budgets in the wake. More recently, theeffect of the free-stream turbulence (FST) on the characteristics of the wake flow behindbluff bodies has been considered experimentally. For example, Khabbouchi et al. [54] in-

2.3. INTRODUCTION 9

vestigated the influence of the FST on the development of a separating shear layer inthe near wake of a circular cylinder for Reynolds numbers based on cylinder diameterup to ReD = 4.7 × 104 by hot-wire measurements. The shear layer shedding frequencyand its harmonics became broader as FST level increases to finally disappear for a FSTlevel equal to 6.2%. The authors suggest that the shear layer behavior can be regardedas a mixing layer for FST intensities lower than 6.2%. Using multiple ADVs, Lacey andRennie [61] studied the turbulent wake past a submerged bed-mounted cube for a bulkReynolds number (based on the cube height h) Reh = 46000 and three water depth tocube height ratios. By decreasing the water depth, the shedding vortical structures aremore confined and the turbulent shear stresses are modified close to the bed, which af-fects the local transport of bed sediments. Hearst et al. [42] conducted particle imagevelocimetry (PIV) and hot-wire measurements to investigate the influence of the FST onthe flow around a wall-mounted cube at a boundary layer development Reynolds numberRex = 1.8× 106. They showed that the stagnation point on the upstream side of the cubeand the reattachment length in the wake do not depend on the FST. Contrarily, Son et al.[111] showed that FST level triggers boundary layer instability above a sphere and delaysthe separation for Reynolds numbers up to ReD = 2.8 × 105. The authors found that asFST intensity increases, the critical Reynolds number known as the drag crisis (where theboundary layer over the sphere becomes turbulent and drag decreases rapidly) decreases.The authors demonstrated that the main mechanism for the drag evolution is linked tothe presence of a separation zone which is controlled by the FST level. Khan et al. [55]conducted Particle image Velocimetry (PIV) measurements around a suspended cube in awater tunnel to investigate the influence of the Reynolds number, Re, over the range [Re= 500 to 55000]. Khan et al. [55] found that the recirculation length decreases with in-creasing Re before reaching an asymptotic value at (Re ≥20000). An asymptotic behaviorwas also observed for the wake width at (Re ≥ 2654). Furthermore, the authors show thatthe mean vorticity is independent of Re at X/D ≥ 2 downstream of the cube. Sadeque etal. [104] studied the flow patterns in the near-wake behind bed-mounted cylinders, in ashallow turbulent channel flow with smooth and rough beds. They focused both on wallwake similarity and on the region away from the bed, which was found to be well modeledby a Law of the wall.Numerically, Richmond-Bryant and Flynn [98] used the discrete vortex method to simulatethe time-averaged flow fields past a circular cylinder at ReD = 1.4 × 105. Though goodagreement was obtained against literature experimental data [23],the unsteady simulationsdon’t converge numerically. Palau-Salvador et al. [92] conducted LES of the flow aroundfinite-height cylinders for two height-to-diameter ratios H/D = 2.5 and H/D = 5, with


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Reynolds numbers based on the both cylinder diameters D were ReD = 22000 and 43000,respectively. For the shortest cylinder, the vortex shedding is observed close to the ground,while for the longest one, it appears over the entire height of the cylinder. The length ofthe recirculation zone gets larger when H/D increases. Saeedi and Wang [105] performedLES on a wall-mounted rectangular block with H/d = 4, H/d = 9 and Red = 12000

(where d the width and H the height). They studied the interactions between the tipvortices produced at the top of the block, the Karman vortices from the side walls andthe developing boundary layer over the bed. Elkhoury [30] compared the performance ofthe Scale Adaptive Simulation (SAS) turbulence model with the predictions of both theSpalart-Allmaras and the k-ω SST models for a square block and a wall-mounted cube(ReH = 4× 104), though discrepancies over 10% are obtained for the position of the recir-culation zone compared with physical measurements, the author recommended the use ofthe SAS model. To the best of our knowledge, D-shaped bluff bodies have not been consi-dered so far in the literature and offer a good opportunity to perform a detailed numericalbenchmark of different turbulence closures. Testing various closure models is seldom donein the literature, yet can have a great impact on the quality/accuracy of the numericalsimulations, especially in complex flows such as those encountered in fishways.The purpose of the current study is to clarify the advantages and trade-offs of using va-rious numerical approaches for modeling complex flows around obstacles in order to bettermodel nature like fishways in the future. In this study, the flow over a bed mounted D-section is modeled in order to describe the general flow topology. A benchmark of differentRANS turbulence closure models is performed around the D-section while taking into ac-count the appropriate mesh resolution for each approach. LES is also conducted with 2different sub-grid scale models and resolution schemes. Numerical results are compared toPIV measurements for validation purpose. A Proper Orthogonal Decomposition (POD) isfinally applied to LES results to extract the most energetic modes within the flow and tofurther characterize the bluff-body wake flow structure. To the best of the authors’ know-ledge, such a careful numerical benchmark validated by PIV measurements for a surfacemounted bluff body has not been done before.

2.4 Numerical modeling

A 3D Navier-Stokes incompressible flow solver based on the finite-volume method wasused. Two levels of turbulence closures were considered to model the turbulent flow overthe submerged D-shaped bluff body, namely steady-state two-equation Reynolds AveragedNavier-Stokes (RANS) models and Large Eddy Simulations (LES).

2.4. NUMERICAL MODELING 11

2.4.1 Geometrical modeling

A sketch of the computational domain is shown in Figure 2.1. It consists in an open channelflume whose dimensions are : length L = 1.4 m, width b = 0.15 m and height H = 0.1

m. A D-section bluff body was positioned at mid-width of the channel with the flat sideface way downstream at a distance of 0.22 m (' 1.8Dh, Dh the hydraulic diameter, Dh =

2Hb/(H + b)) from the inlet. The D-section diameter, height and width are respectivelyD = 0.025 m, h = 0.025 m and w = 0.0125 m (Fig.2.1). During the experiments, the meanflow depth (H = 0.1 m) and mean inlet streamwise velocity (U0 = 0.175 m/s) were keptconstant leading to a bulk Reynolds number equal to ReH = 17500.

Figure 2.1 Sketch of the computational domain (not to scale), and views of theD-section bluff body with its main dimensions.

2.4.2 Turbulence modeling

Three two-equation RANS models were compared to determine which model is more or lessadapted to accurately reproduce the flow structure around the bluff body. The standardk-ε model of Jones and Launder [51] was used in its high-Reynolds number formulationand compared to the Shear Stress Transport k-ω (k-ω SST) model developed by Menter[82]. The k-ω SST model used in both its high- and low-Reynolds number formulation,combines the robust and accurate formulation of the k-ω Wilcox model [127] in the nearwall region and the free stream independence of the k-ε out from boundary layer. Blen-ding functions are introduced in the transport equation of k and ω. When available, a


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production limiter is applied to avoid any possible overproduction of the turbulence kine-tic energy (k) in low velocity regions. Moreover, pressure gradient effects are included inthe resolution of the effective velocity distribution close to the wall. These well-establishedmodels are fully described in the monograph of Wilcox [128] or in the detailed numericalwork of Elkhoury [30].Large Eddy Simulations (LES) are currently applied in a wide variety of engineering appli-cations and appear to be a good compromise between accuracy and computational costs[131]. Compared to direct numerical simulations, the smallest length scales are ignored inLES via a low-pass filtering of the Navier-Stokes equations. This concept was first intro-duced by Smagorinsky [110] in 1963, who developed the so-called standard Smagorinskymodel. Subgrid scales, which are any scales that are smaller than a cutoff filter width ∆,need to be modeled via an empirical turbulent viscosity νt. The turbulent viscosity in theSmagorinsky model is expressed as follows :

νt = (Cs∆)2√

2SijSij (2.1)

where Cs ' 0.18 is the standard Smagorinsky constant and ∆ = (∆x∆y∆z)1/3 is the cutoff

filter width. ∆x, ∆y and ∆z are respectively the size of the local mesh element in the x,y, and z directions, respectively. Sij represents the filtered strain tensor defined as :

Sij =1

2(∂ui∂xj

+∂uj∂xi

) (2.2)

A dynamic version of the Smagorinsky model has been later developed to overcome the toodissipative nature of the standard model [35]. However, Poncet et al. [94] showed, for theturbulent flow in a Taylor-Couette-Poiseuille system, that the dynamic version providedsimilar results compared to the Wall-Adapting Local Eddy Viscosity (Wale) model whilerequiring about 12% of extra computational time.Thus, herein the Wale subgrid-scale model developed by Nicoud and Ducros [88] has alsobeen used for comparison. The Smagorinsky model is based on the second invariant of thesymmetric part of Sij. The main drawbacks are that this invariant is of order O(1) closeto a wall and it is not related to the rotation rate of the turbulent structures. To avoidthat, Nicoud and Ducros [88] developed the Wale model based on the gradient velocitytensor gij, which is a good candidate to represent the velocity fluctuations at the lengthscale ∆. The turbulent eddy-viscosity is then modeled by :


νt = (Cm∆)2(Sd

ijSdij)

32

(SijSij)52 + (Sd

ijSdij)

54

(2.3)

where Cm = 0.4929 and Sij corresponds to the filtered strain tensor. The Wale modelemploys the traceless symmetric part of the square of the velocity gradient tensor gij, asfollows :

Sdij =

1

2(gikgkj + gjkgki)−

1

3gkigikδij gij =

δuiδxj

(2.4)

The Wale model behaves well near the wall with good approximation to the assumedphysics of the flow, and is defined to handle with transitional parietal flows.

2.4.3 Numerical parameters

All calculations have been performed using the software CFX ANSYS 16.2 based on afinite-volume method. For the RANS calculations, a second-order high resolution advec-tion scheme was used to avoid dissipation and ensure a better accuracy. For the LESsimulations, a second-order Backward-Euler scheme is employed for the temporal discreti-zation together with an implicit time-stepping scheme. For the spatial discretization, bothhigh-resolution and central difference schemes were used. The velocity-pressure couplingwas performed by a Rhie Chow fourth-order coupling algorithm, which guarantees thatthe dissipation term vanishes rapidly under mesh refinement.Concerning the boundaries conditions, the mean streamwise velocity profile imposed at theinlet corresponds to the PIV measurements, leading to an average streamwise velocity ofU0 = 0.175 m/s at the channel inlet. The imposed turbulence intensity I0 = u′/U0 = 10%

agrees also with the experimental value and is similar to the value I0 = 0.16(ReDH)−1/8 =

5.5% recommended by Elkhoury [30] (DH the hydraulic diameter). No slip is imposed atthe two side walls and at the bottom wall, while free slip is imposed at the upper boun-dary to account for the flat free surface in the experiments. The computational domainbeing long enough to suppress the sensitivity of the flow to the outlet condition, a simplepressure outlet condition was selected. A verification was performed by imposing outflowor convective condition wich lead to similar results.As discussed previously, steady-state RANS models in their high- or low-Reynolds num-ber formulation and LES were considered. Two unstructured mesh grids were constructedusing the software Centaur (Fig.2.2). They are composed of tetrahedral elements in the


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core of the flow and prismatic layers along the walls. The coarser grid used for the high-Reynolds number RANS models gathers 12.35 million cells and the maximum value ofthe wall coordinate in the whole domain reaches y+ = 40. To better account for the flowdynamics close to the D-shaped bluff body and in the near wall regions, a finer mesh hasbeen used for the low-Reynolds number models and the LES. The finer mesh is compo-sed of 22.17 million cells with 10 prismatic layers along the walls to satisfy the conditionmax(y+) = 1 (see Table 2.1). A stretching factor of 1.2 was used to avoid any numericaldissipation of possible coherent structures. A mesh refinement is also imposed in the wakeof the bluff body to capture the recirculation zone and the vortices in the shear layer.

Figure 2.2 Mesh distributions for the different approaches. Coarse grid for thehigh-Reynolds number models (top) and fine mesh for the low-Reynolds numbermodels and the LES (bottom).

Turbulence closure Number of cells (x106) Number of Nodes (x106) max(y+) GCI [%]High-Reynolds number RANS 12.4 2.8 40 0.011Low-Reynolds number RANS & 22.2 4.5 0.95 0.0011LES

Tableau 2.1 Mesh grid parameters for the high- and low-Reynolds numbermodels and the LES.

The Grid Convergence Index (GCI) provides an uniform measure of convergence for gridrefinement studies [99]. It is based on the estimated fractional error derived from thegeneralization of the Richardson’s extrapolation. The GCI value represents the resolutionlevel and how much the solution approaches the asymptotic value. The GCI can be writtenas follows :

GCIi+1,i = FS|εi+1,i|rp − 1

(2.5)

The safety factor FS selected for this study is fixed to 1.25, according to [99]. The order-of-accuracy (p) can be estimated by using the following equation :

p = ln

(f3 − f2f2 − f1

)/ln(r) (2.6)


where fi represents the numerical solution of the ith mesh and r represents the grid refi-nement ratio. The relative error εi+1,i writes :

εi+1,i =fi+1 − fi

fi(2.7)

The GCI method has been applied using the low and high Reynolds number meshes and athird coarse mesh with 22.2, 12.4 and 6.9 million elements, respectively. A grid refinementratio of approximately 1.8 was applied between the three grids, while keeping the numberof prismatic layers constant. The GCI is calculated by considering the magnitude of themean velocity.

As shown in Table 2.1, the GCI for the finer and coarser meshes are relatively low (below1), indicating that the dependency of the numerical simulation on the cell size has beenalready achieved for both meshes.The time step in the LES is fixed to δt = 0.0028 s to ensure a CFL number lower than 1.The convergence is reached when all residuals get lower than 10−8 and the mass imbalanceis lower than 10−6. The calculations were run using the cluster MP2 provided by CalculQuébec. The CPU time for the low-Reynolds, high-Reynolds number models and LES wasrespectively 26 hours, 18 hours and more than 20 days, using 32 processors for the RANSmodels and 96 processors for the LES. The LES calculations have been initialized usinga converged RANS calculations. Then, statistics have been cumulated after the elapse oftwo other flow-through times L/U0 and continued until reaching 4.5 flow-through times.

2.4.4 Experimental database

Model validation data was obtained from previous experiments conducted in a small tiltingglass-walled open-channel at Wageningen University (The Netherlands). The experimen-tal section of the channel was L = 1.4 m long and b = 0.15 m wide. The D-section wasmounted on a sharp-edged flat plate with dimensions of 0.75 m long and 0.15 m wide.The plate was raised off the bed of the channel by 0.1 m. The D-section was made fromblack anodized aluminum. As stated previously the D-section diameter, height and widthwas D = 0.025m, h = 0.025m and w = 0.0125m, respectively. During the experiments themean flow depth was kept constant (H = 0.1 m) and the mean velocity of the incomingflow was U0 = 0.175 m/s. The experimental set-up is fully described in [59].Planar PIV was used to characterize the flow field over different vertical (2 component pla-nar) and horizontal (3 component stereoscopic) planes. A Nd :YAG laser (Laser QuantumLtd, United Kingdom) of 2.3 W continuous wave was used. The field of view (FOV) was


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approximately 0.1× 0.1 m2, 0.06× 0.06 m2 for the planar and stereoscopic PIV, respecti-vely. Perpendicularly to the side of the channel, a high-speed 1 megapixel CMOS digitalcamera (FASTCAM 1024 PCI, Photron Ltd., Japan) was positioned, for the planar PIVmeasurements. For the stereoscopic PIV configuration, two high-speed CMOS cameraswere positioned above the channel. The acquisition frequency of the camera (both PIVmeasurements) was set at 500 frames per second (fps). 3200 images were taken, which cor-responds to 6.4 s of recording. The duration was limited due to onboard camera memory.The camera shutter speed was set to 1/4000s, which was sufficient to avoid streaking ofthe seeding particles. Silver coated hollow microspheres with a mean diameter of 0.013mm (Potters Industries, USA) were used to seed the flow.The software Davis (LaVision Inc., USA) was used to produce the instantaneous velocityvectors. For the extraction of the velocity field, a multipass algorithm was used. The finalinterrogation area (IA) was 16 × 16 px2 with a 50% overlap, while the first and secondIA were composed of 64× 64 px2 and 32× 32 px2 respectively. The resulting matrix hasa dimension of 128 × 128 px2 for each velocity vector at each time step. The resolutionof the final IA was 1.63 × 1.63 mm2 for planar PIV and 1.5 × 1.5 mm2 for stereoscopicPIV. Three filters were applied to the raw vector maps to remove erroneous vectors : 1) asignal-to noise ratio filter, 2) global histogram operator and 3) a median filter. An interpo-lation of the nearest neighbor vectors was done to replace identified spurious vectors. Anaverage of 9% of the raw vector field was replaced by interpolated values for the planarand stereoscopic PIV.

2.5 Results and discussion

The results presented below have been obtained for an aspect ratio (relative roughness)h/H = 0.25 and a Reynolds number ReH = 17448. The streamwise u and vertical vvelocity components and turbulence kinetic energy k are normalized by the mean inletvelocity U0 and its square respectively, such that : u∗ = u/U0, v∗ = v/U0, k∗ = k/(0.5 ×U0

2). The origin of the reference (X = Y = Z = 0) is set at the bottom wall in the medianplane behind the D-section bluff body.

2.5.1 Mean flow field

The flow around the D-section body contains complex three dimensional turbulent vorticalstructures [60]. Two recirculation regions are observed at the front of the D-section andin its wake (Fig.2.3). The advection of the primary recirculation region induces a complexhorseshoe vortex downstream. Flow separates overtop and along the sides of the D-section

2.5. RESULTS AND DISCUSSION 17

producing a reattachment zone in the lee (Fig.2.3). The shear layer above the obstaclegenerates intense coherent structures, which are shed downstream.According to Figure 2.3, all the RANS turbulence models in their high- or low-Reynoldsnumber formulations predict the existence of the recirculation zone. Nevertheless, the sizeof the recirculation differs from one model to another, reflected in a displacement of thereattachment region. The prediction of the recirculation length seems to be linked to theflow field in the near-wall regions. The extent of the recirculation zone for the low-Reynoldsnumber k − ω SST (Xr/D ≈ 2.4) and the two LES models (Xr/D ≈ 2.5) compare fairlywell with the PIV measurements (Xr/D ≈ 1.9). On the contrary, the high Reynoldsformulations of k−ω SST and k− ε RANS models, which do not solve the flow field closethe walls, strongly underestimate the length of the recirculation zone behind the bluffbody, Xr/D ≈ 1.2 and Xr/D ≈ 1.3, respectively.

Figure 2.3 2D maps around the D-section obstacle in the channel midplaneof the mean velocity components u∗ and v∗ and turbulence kinetic energy k∗.Comparisons between the PIV measurements, three two-equation RANS modelsand two LES models.

Regarding the subplots of the vertical velocity component v∗ (Fig.2.3), the two LES mo-dels and the low-Reynolds number k − ω SST appear to reproduce correctly the negativevertical velocity (v∗ ≈ −0.3) region located on the stoss side of the obstacle, where thehorseshoe vortex develops. This region is less apparent with the two high-Reynolds num-ber models. In general, the two high-Reynolds number models (k − ω SST and k − ε)give a poorer representation of the flow structure around the obstacle. Comparisons with


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the experimental measurements show differences in the recirculation region. Furthermore,within the recirculation zone along the back wall of the obstacle, strong regions of elevatedpositive vertical velocity (v∗ ≈ 0.4) are apparent which are not observed in the experimen-tal data. Possible reasons for the overpredictions could be the coarseness of the grid andthe use of a wall functions. The main difference between the low-Reynolds number k − ωSST and the LES lies in the abrupt plunging of the shear layer to the bed for the RANSmodel, whereas it is much more progressive in the LES and PIV results. This mismatchcould be due to the unsteady nature of the shear layer and the appearance of possiblecoherent structures, which can not be captured by steady-state RANS models. All turbu-lence models predict quite well the vertical acceleration due to the obstacle blockage atthe top of the bluff body.The flow behind the D-section bluff body is highly turbulent. High values of k are lo-cated essentially in the shear layer and vortex shedding area according to the PIV andLES results (Fig.2.3) ; a similar k spatial distribution has been observed in the wake re-gion of a cube [49, 72]. It is noteworthy that, for the PIV measurements, k has beencalculated using the fluctuating streamwise and vertical velocity components only. Thehigh k (k∗ ≈ 0.001 − 0.005) region extends more or less the same distance for the PIV(X/D ≈ 0.1 − 4), and both LES models (X/D ≈ 0.15 − 4) suggesting a good fit withthe PIV. This good agreement is likely due to both a direct calculation of the large- andintermediate-scale eddy vortices and to the unsteady nature of the flow, which is accountedfor. The difference between the results obtained via the two LES approaches is not strikingapart from a minor discrepancy in the region X/D ≈ 1.5− 2. The high-Reynolds numberk− ε model strongly overpredicts k just after the obstacle (X/D ≈ 0−1.8) with very highvalues (k∗ ≈ 0.005), which extend to the back wall of the bluff body (X/D ≈ 0 − 0.2).This suggests that the dissipation rate ε is too low in this region. This is perhaps becausethe estimated ε does not account for the rotational motion of the fluid particles and is notcorrectly modeled in the near-wall region. The high Reynolds k − ω SST model is muchmore dissipative than the k − ε model though they have comparable wall functions. Thelow-Reynolds number k−ω SST is more dissipative than the LES models, yet has a similark distribution due a correct near-wall resolution. Therefore, the specific dissipation rateω is a better candidate compared to ε to determine flow structure and the characteristicscale of turbulence. Moreover, the use of a production limiter improves the predictionsof the turbulence intensities (Fig.2.3,k∗). However, the low-Reynolds number k − ω SSTmodel in its steady-state version does not capture the high turbulence levels during thedestabilization and the plunging of the shear layer as it is not able to predict the smallest3D unsteady coherent structures in that flow region.


Figure 2.4 shows the distributions of the dimensionless longitudinal and vertical meanvelocities u∗ and v∗ along the streamwise direction X/D. The results have been obtainedat two positions Y/D = 1 and Y/D = 0.5 in the median plane of the channel (where Yis the vertical distance from the bed). At Y/D = 1, the high-Reynolds number modelscompletely fail to capture the u∗ distribution until X/D ≈ 2, likely because of the useof a wall functions based on a logarithmic law of the mean velocity [63]. For X/D > 2

, the high-Reynolds models give very good agreement with the PIV results (outside theshear-layer region). Conversely, the v∗ profiles predicted by the high-Reynolds models arein poor agreement with the experimental values over the whole range of X/D positionsconsidered here. For example, the maximum negative velocity, associated with the plun-ging shear layer, is v∗=-0.175 instead of v∗=-0.25 in the experiments. Furthermore, theposition of minimum v∗ is shifted closer to the lee of the obstacle. The use of wall functionssignificantly affects the flow separation and the shear layer formation on the D-section trai-ling edge and the model is unable to achieve a no-slip condition at the back wall of theobstacle. The choice of ε or ω to determine the scale of turbulence has no noticeable in-fluence on the velocity distributions for this particular position Y/D = 1. The agreementbetween the LES results, low-Reynolds number k − ω SST and the PIV measurements isgenerally good in terms of the streamwise velocity u∗ distribution (Fig.2.4a). A slight shifttowards higher magnitude u∗ is observed in the LES results and could be attributed to amisrepresentation of the experimental 3D inlet flow conditions. At the channel entrance,2D planar PIV measurements were obtained along the centerline, missing then the thirdvelocity component. So, a 2D PIV velocity profile and a turbulent intensity of 10% werenumerically imposed. In a previous study, Baetke et al. [7] demonstrated the influenceof inlet conditions on flow topology around a cube. The authors showed that a small va-riation of the inlet boundary layer profile resulted in the appearance/disappearance of aseparation region on the top of the cube and a variation of the recirculation length be-hind the cube. The low-Reynolds number k − ω SST is the only model able to capturewell the vertical velocity overshoot centered around X/D ' 0.4 (Fig.2.4b). The overshootrepresent the entrainment of the fluid from the recirculation zone by the shear layer. TheLES Smagorinsky and Wale models underestimate the flow acceleration in this region, andprovide a more extended recirculation zone. When the shear layer plunges to the bed, theLES Wale conforms with the PIV measurements in terms of the peak value v∗ ≈ −0.24

and peak position X/D ≈ 2.5, compared to v∗ ≈ −0.25 and X/D ≈ 1.9, respectively. TheSmagorinsky model under and overestimates the peak value v∗ ≈ −0.2 and its positionX/D ≈ 3.0, respectively. In general, the LES Wale provides a better overall performancecompared to the LES Smagorinsky. The Wale approach is based on a more appropriate


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turbulent viscosity definition, which takes into account wall effects. The Smagorinsky mo-del is less suited for wall bounded flows, the subgrid scale model being unable to reproducean asymptotic variation of the turbulent viscosity near the wall. As the flow near the wallis not correctly modeled, the flow within the shear layer is also affected.

(a) (b)

(c) (d)

Figure 2.4 Distributions of the two main mean velocity profiles (a,c) u∗ and(b,d) v∗ along the streamwise direction X/D. Results obtained at (a,b) Y/D = 1and (c,d) Y/D = 0.5 in the median plane of the channel.

Figure 2.4c,d presents the u∗, v∗ velocity distributions at Y/D = 0.5 (half way up theobstacle). This vertical position in theory coincides with the center of the recirculationregion, as can be observed by the negative values of u∗. The length of the recirculationregion at Y/D = 0.5 can be estimated by considering the position for which the u∗ is equalto 0. From the PIV, the extent of the recirculation zone at Y/D = 0.5 is aroundX/D = 1.8.The high-Reynolds number k−ε and k−ω SST models predict a recirculation length equalto around X/D = 1.3. The low-Reynolds number k − ω SST model and the LES Walepredict a recirculation length equal to around X/D = 2.5 and the LES Smagorinsky model


equal to around X/D = 2.8. While two high-Reynolds number models show an acceptableestimation of the u∗ outside the shear layer regionX/D > 2.6, they fail to accurately modelthe vertical velocity component v∗ distribution in the near wake. The wall functions useddon’t allow the flow to recover the no-slip condition for v∗ on the back wall of the obstacle.The k − ω SST model is based on the k − ε model in the core region of the flow and thek − ω in the boundary layer. Using the k − ω SST model in its high Reynolds numberformulation on a coarser mesh makes the influence of the underlying k − ω model lessimportant. It is thus not surprising that the results are very close to those provided bythe high Reynolds number k − ε model, while remaining slightly better.

(a) (b)

Figure 2.5 Distributions of the mean streamwise velocity component u∗ alongthe spanwise direction 2Z/b. Results obtained at Y/D = 0.5 for two X/D loca-tions : (a) X/D = 0.8 and (b) X/D = 1.6.

Figure 2.5 shows the distributions in the spanwise direction 2Z/b (where b = 0.15m

is the flume width) of u∗ at Y/D = 0.5 and two X/D locations, namely X/D = 0.8

and X/D = 1.6, these latter are inside the recirculation region. As presented in themethodology, the PIV for the horizontal measurement plane (presented here) is 3C .Asthe 3C PIV field of view is equal to 0.06× 0.06 m2 and the flume width b is fixed to 0.15

m, the PIV profiles do not reach both side walls. As expected, there is a clear deficit in u∗

around 2Z/b ' 0 highlighting the recirculation zone behind the bluff body. By conservationof mass, u∗ gets higher than 1 closer to the side walls as it accelerates around the obstacle.The profiles u∗ predicted by the two high-Reynolds number models do not recover a zerovelocity at the side walls, which is inherent to the use of wall functions. Generally, thehigh-Reynolds number k−ω SST and k− ε turbulence models predict acceptable velocitydistributions for bothX/D positions. The LES and low-Reynolds number k−ω SST modelsestimate more correctly the backward flow in the recirculation region behind the bluff body


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at X/D = 1.6 (Fig.2.5b). Outside the wake region (|2Z/b| ≥ 0.25), the flow is acceleratedto reach a peak value u∗ ' 1.23. All models predict well the velocity distribution there atX/D = 0.8 (Fig.2.5a) ; while at X/D = 1.6, the high-Reynolds number approaches betterpredict velocities compared with the low Reynolds and LES models.

2.5.2 Coherent structures in the wake flow

The LES models in general compare more favorably with the experimental results (Fig.2.3).This is likely because they are able to reproduce the unstationary 3D coherent structuresshed from the obstacle. In order to get a better appreciation for these structures, Figure2.6 displays 2D views of the instantaneous vorticity around the D-section obtained by theLES Wale and the LES Smagorinsky models. As can be seen, the flow topology does notdiffer significantly between these two sub-grid scale models. The undulation of the shearlayer before the onset of the vortex shedding is more apparent in the Wale model, whichalso predicts a slightly longer shear layer and so a delayed shedding position compared withSmagorinsky model (Fig.2.6a,b). The most noticeable differences between the two subgrid-scale models are more visible in the plane at Y/D = 0.5 (Fig.2.6c,d). The shear layersproduced on both sides of the bluff body destabilize much faster in the LES Smagorinsky,resulting in a complex turbulent flow containing vortical structures at X/D ' 2.2. For theLES Wale, the vortex interactions lead to similarly complex flow structure at X/D ' 3.5.The vortical structures are advected mainly in the streamwise direction for the LES Wale,whereas a spanwise velocity component also induces a motion of the coherent structures inthe transverse direction (Fig.2.6c,d) for the LES Smagorinsky. At Y/D = 1, the wake flowmodeled by the LES Wale is more stable, while vortex shedding is much more apparentfor the Smagorinsky model with a larger wake (Fig.2.6e,f).Investigations of the temporal evolution of the vorticity field (not shown), revealed aflapping of the shear layer which appears to be related to the interaction between coherentstructures within the recirculation region and the shear layer. In other words, some of thecoherent structures generated within the shear layer are injected in to the recirculationregion, causing a slow interaction with the underside of the shear layer and implying aflapping movement. This flapping has already been reported by Castro and Robins [25]for the flow behind a mounted cube and is observed here for both LES models.

The differences between the two subgrid scale models results may be explained by thedifferent turbulent viscosity definitions. As previously stated in Section 2.4.2, the invariantused in the Smagorinsky model is not related to the rotation rate of the turbulent structurescontrary to the Wale model, which is based on the gradient velocity tensor. The Wale model


Figure 2.6 2D views of the instantaneous vorticity around the D-section bodyobtained by the LES Smagorinsky (a,c,e) and the LES Wale (b,d,f) models.(a,b) Side views in the mid-plane of the channel (2Z/b = 0) ; Top views at (c,d)Y/D = 0.5 and (e,f) Y/D = 1.

is then more adapted to the present flow configurations where intense vortical structuresare observed in the wake of the obstacle. It is directly reflected in the maps of the turbulentviscosity (not shown), which reveal that the Smagorinsky model is much more dissipativethan the Wale one.

The 3D complex vortices surrounding the D-section are visible using the Q-criterion pre-sented in Figure 2.7, the isovalues are colored by the normalized averaged longitudinalvelocity (u∗) on top and side view. A horseshoe vortex is distinguishable in front of theobstacle for both LES models, and is advected by the mean flow downstream. The horse-shoe vortex breakdown at X/D ≈ 2− 3 gives birth to smaller vortices, which merge withthe coherent structures produced within the shear layer. Hairpin vortices are visible in thewake once the shear layer destabilizes. The horseshoe vortex breakdown and the sheddingphenomenon within the shear layer occur closer to the obstacle with the Smagorinskymodel at X/D ≈ 2 than the Wale model at X/D ≈ 3 (Fig.2.6a,b).

Compared to the central difference schemes used in the present case, LES Wale and Smago-rinsky models calculations have also been performed using high-resolution 2nd order spatialschemes (not shown here). The high resolution 2nd order schemes lead to an overprediction


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Figure 2.7 Side and top views of the Q-criterion (Q=0.029) distribution overthe D-section obstacle colored by the normalized averaged longitudinal velocity(u∗). Results obtained by the LES Wale (a,b) and the LES Smagorinsky (c,d)models.

of the numerical dissipation, such that the model captures only the largest vortices. Giventhe fair agreement between the experimental values and the results based on the LES Walemodel with central difference schemes, the following analysis and discussion focuses on theLES Wale model results.

Figure 2.8 presents the power spectral density (PSD) of the u and v velocity components(autospectra) in m2/s extracted from the PIV and LES Wale at four X/D locations inthe shedding region (Y/D = 0.5, 2Z/b = 0). The instantaneous velocities obtained byPIV and LES were acquired at frequencies of 500 Hz and 357 Hz, respectively, for a totalrecording time of 6.4 s and 20 s, respectively. In the wake, the PIV u and v componentvelocity autospectra contain similar amounts of energy, and in general contain less energyover all frequencies than the LES velocity spectra. The magnitudes of energy observed viaPIV and LES compare well with energy extracted on the wake flow in a gravel-bed river[113]. The discrepancy between the energy presented in the PIV and LES spectra at highand low frequencies is perhaps due to sampling frequency (500 Hz, 357 Hz), duration (6.4s and 20 s) and sampling volume. The sampling volume of the PIV and the LES is around4 × 10−9m3 and 3 × 10−12m3, respectively, so the LES autospectra likely contains moreenergy from the small turbulent structures due to spatial averaging over the samplingvolume [112]. With distance downstream from the bluff body, increased energy is observedat higher frequencies - especially noticeable in the LES results and is likely related to


(a) X/D = 0.8 (b) X/D = 1.6

(c) X/D = 2.4 (d) X/D = 3.0

Figure 2.8 PSD distributions (m2/s) of the mean velocity components u andv downstream of the D-section obstacle at four positions X/D. Comparisonbetween the results obtained by PIV and the LES Wale model at (Y/D = 0.5,2Z/b = 0).

the breakdown/cascade of shedding vortices. Portions of the velocity spectra, for both thePIV and LES follow a −5/3 slope in the suggested inertial subrange, in agreement withthe theoretical value reported by Kolmogorov [56]. By using the Strouhal number St =0.2 [126], the predicted shedding frequency is 1.4 Hz. At this frequency, no distinct peakin energy can be observed on the PIV and LES spectra for the different X/D positions.The fact that no distinct peak in the energy spectra is observed is due to bottom walleffects. As the D-section is wall mounted, the wake flow is a 3D, chaotic and non-periodicdue to the interactions between the coherent structures released from the shear layer, thehorseshoe vortex and the wall. The vortices deform by impacting the bottom wall, whichleads to a reduction in flow periodicity. For that reason, no distinguishable peak is observed


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compared to a periodic case where the obstacle is suspended. This is supported by Rubinand McDonald [102] who showed the spectral signature of flow around a bed-mounted,horizontal cylinder.

2.5.3 POD analysis

Proper Orthogonal Decomposition (POD) was applied to the velocity field obtained fromthe LES Wale model to determine the different energetic structures composing the flowaround the D-section bluff body. As described in detail in [46], the POD technique de-composes a velocity or scalar field into a sum of weighted, linear, basis functions or modesφk(~x). The basis functions are weighted by time coefficients ak(t). A velocity field U(~x, t),can be decomposed as follows :

U(~x, t) = U(~x) +∞∑k=1

ak(t)Φk(~x) ' U(~x) +N∑k=1

ak(t)Φk(~x) (2.8)

where U(~x) denotes the time-averaged part of the velocity field. The velocity field maybe approximated by a finite number N of modes. The Snapshot POD method proposedby Sirovich et al. [109] was used since the number of spatial locations Nxy in the velocityfield is greater than the number Nt of instantaneous snapshots.The POD theorem is equivalent to the matrix eigenvalue problem :

CAk = λkAk (2.9)

where C is the covariance or the spatial correlation matrix of the velocity field and λk andAk are its eigenvalues and eigenvectors, respectively. The C covariance matrix writes :

C =1

Nt

(SuSTu + SvS

Tv ) (2.10)

The velocity fluctuations u′ and v′ calculated with the LES Wale in the middle plan behindthe bluff-body (Fig.2.3) were recorded into Nt×Nxy matrices of snapshots Su and Sv. ST

u

and STv are the transposed matrices of Su and Sv, respectively. The basis functions Φk

u(x)

and Φkv(x) representing the coherent structures are obtained as follows :

Φku(~x) = ST

uAk (2.11)


Φkv(~x) = ST

v Ak (2.12)

The time coefficients ak(t) are then given by :

ak(t) = SuΦku(~x) + SvΦ

kv(~x) (2.13)

Knowing the basis functions Φku(~x) and Φk

v(~x) and the coefficients ak(t), the velocity fieldU(~x, t) may be reconstructed using Equation (2.8).Figure 2.9 represents the contribution of the first 200 POD modes to the turbulence kineticenergy k. The calculation of the k is based on the both basis functions Φk

u(~x) and Φkv(~x).

The first 100 modes capture more than the 90% of the k contained within the flow. Thefirst POD mode taken by it self contributes to 14% of the k, whereas the individualcontribution from higher modes is lower and decreases rapidly. Above 40 POD modes, theindividual contribution is less than 1%, a comparable distribution was observed by Taifouret al. [116] for a geometry-induced turbulent separated bubble (TSB), where the first modecontributes to 31% of the k, and above 30 POD modes the individual contribution is lessthan 2%.

Figure 2.9 Relative and cumulative energy contributions of the POD modesobtained by the LES Wale model.

The first sixth normalized Φk=1:6(~x) modes are represented in figure 2.10. The first PODmode is dominated by the shear layer and some of the wake region, which are the mostenergetic (14% of the k) physical mechanisms. From the second to the sixth modes, thecoherent structures appear clearly, localized in distinct zones. A harmonic behavior canbe distinguishable by passing from one mode to another. This harmonic behavior was also


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Figure 2.10 First six POD modes from (a) to (f) extracted from the instanta-neous LES Wale results.

observed in the first sixth POD modes of a cylinder wake flow by Qingshan et al. [96].A velocity field reconstruction based on the basis functions Φk

u(~x) and Φkv(~x) was performed

to further show the flow structures associated with the different modes : 1) using the firstmode ; and 2) using the second to the tenth modes. For comparison, the time-averagedvelocity field U(x) was subtracted from the reconstructed velocity field such that coherentstructures could be more clearly discerned. The reconstruction is given by :

U(~x, t)− U(~x) =

N2∑k=N1

ak(t)Φk(~x) (2.14)

where N1 and N2 represent the first and last modes considered.

The velocity field reconstructed shows that the first POD mode represents well the shearlayer mechanism ; while the velocity field reconstruction obtained by summing equation2.14 over modes N1 = 2 to N2 = 10 shows only the coherent part of the field (Fig.2.11)and contains more than 40% of the k. Thus, the reconstruction of the field over a limitednumber of modes allows to separate energetically the main physical mechanisms of theflow, namely the shear layer and the shedding coherent structures.


Figure 2.11 Velocity field reconstructed (m/s) using : (a) only the first mode ;(b) modes 2 to 10. Results obtained by the LES Wale model.

Figure 2.12 Contour map of longitudinal velocity field reconstructed U/U0

using the first mode with the (a) minimum and (b) maximum time coefficients.Results obtained by the LES Wale model.


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Figure 2.12 shows the longitudinal velocity field reconstructed U based only on the firstmode (eq.2.8), and normalized by the mean inlet velocity U0. This reconstruction usesspecific values of the time coefficient in the range [a1min(t),a1max(t)]. This technique hasbeen successfully used by [48, 117] for separated flows and elucidates the temporal flowvariation. The presented velocity fields can be considered as a conditional average intime between the expanded recirculation region obtained for a1(t) = a1min(t) and thecontracted one obtained for a1(t) = a1max(t). Figure 2.12 shows that the reattachment ofthe overlying boundary layer fluctuates temporally between X/D = 3.2 to X/D = 4.0.This pulsating recirculation dynamic is clearly visible on the instantaneous fields obtainedeither through PIV or the LES models (not shown). Between the two extreme values ofthe time coefficients (i.e.,a1(t) = 0), the reconstructed velocity field is approximately theaverage velocity field presented in Fig.2.3.

2.6 Conclusion

A numerical benchmark of the most commonly used two-equation RANS models, as wellas, two LES based on the Smagorinsky and Wale subgrid scale models has been performedto analyse the turbulent flow dynamics behind a D-shaped bluff body. PIV measure-ments have served as reference data. The flow is characterized by a bulk Reynolds numberReH = 17448 and an aspect ratio (relative roughness) h/H = 0.25.All models predict a recirculation region behind the obstacle, as well as, the formation ofa shear layer at the top of the D-section, which destabilizes and plunges towards the bed.In general, two main conclusions can be drawn from the RANS models approaches : (i)the specific dissipation rate ω is a better candidate to determine the scale of turbulencecompared to the standard dissipation rate ε (Fig.2.3) ; (ii) modelling the near-wall regionsaround the obstacle using the low-Reynolds number model enables a better prediction ofthe horseshoe vortex (negative v∗ region) at the foot of the obstacle (stoss side) and avoidsthe over prediction of v∗ along the back wall of the obstacle. While none of the selectedturbulence closures provide velocity and turbulence kinetic energy distributions in perfectagreement with the PIV results at all locations in the wake, the LES Wale has shown toperform slightly better.For both LES Smagorinsky and Wale subgrid scale models, a horseshoe vortex is observedat the stoss side of the D-section, which is advected downstream along the sides of the bodyand interacts with the coherent structures released by the shear layer in the wake. Theseinteractions lead to smaller vortices in the wake. The horseshoe vortex breakdown and theshedding phenomenon occur farther from the obstacle with the Wale model than the Sma-

2.6. CONCLUSION 31

gorinsky model. No distinct shedding frequencies could be observed via the autospectra ofthe u and v velocity components for different X/D positions in the wake, yet the PIV andthe LES autospectra follow the theoretical −5/3 slope in the inertial subrange. The PODanalysis revealed that the first and the most energetic mode represents the shear-layer andmain recirculation dynamics, while the other modes account for more localized coherentstructures within the wake flow. The u component velocity reconstruction based on thefirst mode showed a pulsating dynamic in the recirculation region.In general, the LES Wale model results gave a better agreement with the PIV experimentsthan the Smagorinsky model. Given that the LES Wale model is less computationally ex-pensive than the standard LES Smagorinsky model, the Wale model offers a fair compro-mise between model accuracy and CPU time. We believe that it would be feasible to usethe LES Wale model on subset fishway configurations (e.g., one or two pool weir sections)to characterizes the instantaneous turbulence structures. This characterization could aidin the design of fishway structures and eventually give insight on the interactions betweenturbulence and fish behaviour.

Acknowledgements

All calculations have been done using the computational resources of the Compute Canadanetwork, which is here gratefully acknowledged. The authors would like to acknowledgeProfessor J. Favier from Aix-Marseille University for fruitful discussions about the PODanalysis.


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CHAPITRE 3

Numerical modeling of a Darrieus HorizontalAxis shallow-Water Turbine

3.1 Avant-propos

Auteurs et affiliation :A. E. Benchikh Le Hocine : étudiant au doctorat, Université de Sherbrooke, Faculté degénie, Département de génie mécanique.S. Poncet : professeur, Université de Sherbrooke, Faculté de génie, Département de géniemécanique.R. W. J. Lacey : professeur, Université de Sherbrooke, Faculté de génie, Département degénie civil.Date de soumission : 22 Août 2018Revue : Journal of EnergyTitre français : Modélisation numérique d’une hydrolienne de type Darrieus à axe hori-zontal pour des applications en rivièreContribution au document : Ce deuxième article représente la deuxième étape vers lasimulation du problème complet. Il consiste de simuler l’écoulement autour d’une hydro-lienne de rivière de type Darrieus à axe horizontal complètement immergée. Une approchemonophasique URANS 2.5 D est couplée à un modèle de turbulence de type k-ω SST.Dans cet article, on sélectionne le meilleur profil de pale et le nombre de pales optimal afind’optimiser les performances de cette turbine. D’autre part, il apporte une compréhensiondétaillée sur la formation et le lâcher des structures tourbillonnaires autour des pales dela turbine.Résumé français : Pendant des décennies, les hydroliennes de type Darrieus ont dé-montré leur avantage par rapport aux autres hydroliennes dans la production d’énergiehydroélectrique. La plupart des études ont tendance à être effectuées sur une configura-tion à axe vertical, tandis que peu examinent la configuration à axe horizontal. Dans cetarticle, nous effectuons une analyse CFD qui étudie plusieurs profils de pale afin d’amé-liorer l’efficacité de la turbine Darrieus à axe horizontal à trois pales dans des conditionsd’écoulement réalistes. Un profil de couche limite et un domaine confiné sont imposés pourmodéliser des conditions d’écoulement fluvial. Quatre profils de pales sont étudiés pour

33

34CHAPITRE 3. NUMERICAL MODELING OF A DARRIEUS HORIZONTAL AXIS

SHALLOW-WATER TURBINE

différentes vitesses de rotation. L’approche actuelle basée sur le modèle de turbulence k-ωSST est tout d’abord validée par rapport aux résultats expérimentaux existants publiésdans la littérature et révèle une erreur inférieure à 13 %. Le profil de pale S1046 améliorele coefficient de puissance Cp de 14% par rapport au profil NACA0018. Le profil S809présente les performances les plus faibles dans les régions de décrochage dynamique et detransition. Pour des vitesses de rotation élevées, le profil FXLV152 produit le coefficient depuissance le plus élevé. L’influence du nombre de pales est également quantifiée. Les confi-gurations S1046 à quatre et deux pales permettent de produire les coefficients de puissanceles plus élevés dans les régions de vitesses de rotation faibles et élevées, respectivement.

3.2. ABSTRACT 35

3.2 Abstract

For decades, Darrieus water turbines have shown their advantage over other water turbinesfor providing hydro power. Most studies tend to be conducted on a vertical axis configu-ration of the turbine while few examine the horizontal axis configuration. The horizontalaxis configuration allows the Darrieus turbine to produce power in shallow water. In thispaper, we undertake a CFD analysis that investigates blade profile alternatives to improvethe efficiency of the horizontal three-blade Darrieus turbine under more natural flow condi-tions. An inlet boundary layer profile and a confined domain are imposed to model morerealistic river conditions. We consider four blade profiles under various tip-speed ratios.The present model based on an unsteady k − ω SST turbulence closure associated with avery fine grid mesh is first validated against existing experimental results published in theliterature and shows less than 13% discrepancy. The S1046 blade profile is shown to im-prove the power coefficient Cp by 14% compared to the NACA0018 blade profile. The S809blade profile exhibits the lowest performance in the dynamic stall and transition regions.For high tip-speed ratios, the FXLV152 profile produces the highest local efficiency powercoefficient. The influence of the blade number is also quantified. The four and two bladedS1046 configurations achieve the highest power coefficients in the low and high tip-speedratio regions, respectively.

3.3 Introduction

The total world energy consumption will rise from 17622 TW in 2017 to 21616 TW in 2040,and is expected to reach more than 294000 TW in 2060 [100]. In 2016, fossil fuel energyrepresented more than 84.7% of the total world energy consumption [75]. The human de-pendency on fossil fuels causes many environmental problems including oil spill, acid rain,air pollution and climate change. So the need for new renewable, ecological and durablesources of energy has become crucial. Several green energy sources have been developed :solar, wind, and water. Hydropower is a promising alternative as energy can be producedcontinuously, without relying too much on short term climatic conditions such as for solarenergy or wind. Moreover, hydropower systems can be operated for several decades dueto a longer life cycle and low failure rates [52].Hydrokinetic water turbines can be classified according to the orientation of the rotor axisrelative to the direction of the mean water flow. In a river, when the mean water direc-tion is parallel to the turbine rotation axis, the water turbine is called axial or horizontal.The vertical water (cross flow) turbine is that wherein the rotation axis is perpendicularto the flow direction. In situ vertical axis hydro turbines (VAHTs) are popular and have



received much interest due to their ability to extract power independently of the waterflow direction (a necessity for tidal/marine deployment) [123] and of the flow intensity[106]. VAHTs can as well operate in a larger range of flow conditions than horizontal tur-bines [83]. VAHTs can be divided into two types, namely the Savonius and the Darrieus.The Savonius turbine operates under a differential drag between the buckets. However,the Darrieus turbine is based on lift force. The Darrieus turbines are more widely used[36, 114] due to their ability to provide maximum power over a large tip-speed ratio range[40].Experimental and numerical approaches have been done to investigate the flow aroundvertical Darrieus turbines and to determine their efficiency. Gorle et al. [37] investigatednumerically and experimentally a 3 bladed Darrieus VAHT under confined flow (bounded)conditions. Numerically, 2D unsteady RANS simulations were performed using the k − ωSST turbulence model, which were compared to phase locked 2D-2C PIV measurements.The authors validated the instantaneous vortex dynamics and the global performance ofVAHT with a discrepancy of 6% for low tip-speed ratio λ values [1 − 2], and highlightedthe correlation between the coherent structures around the blades and the rotor torque.McLaren etal. [80] studied numerically the correlation between the tip-speed ratio λ (orTSR) and the flow physics on a 3 bladed Darrieus vertical axis wind turbine (VAWT)using 2D URANS simulations and the k−ω SST turbulence model. A good fit was obser-ved between the numerical and experimental force coefficients with an error of 15%. Theauthors found that low λ results in complex flow/blade interaction mechanisms and dyna-mic stall causing vortex shedding, vortex impingement and reduction in power production.The solidity effect on Darrieus VAWT performance was studied by Sabaeifard et al. [103]numerically using 2D URANS simulations with a standard k − ε turbulence model. Anexperimental assessment of the numerical calculations was performed in a low speed opencircuit wind tunnel. A discrepancy of 10% was observed by the authors between the nu-merical and experimental power coefficients. The authors also showed that the turbinepower coefficient Cp was maximized with a solidity within the range [0.3− 0.5]. However,above this range, Cp decreases drastically. Lee and Lim [64] showed also that increasingthe solidity [0.4− 0.8] leads to Cp improvement specially for low λ [1.2− 2]. On the otherhand, an inverse relationship was observed between solidity and Cp at higher values of λ[2.4− 3.2]. The effect of the blade number N was also discussed by Sabaeifard et al. [103].For 2 blades, Cp reaches high values at high λ and for 4 blades, the opposite is observed.The three bladed configuration recorded the highest power coefficient in comparison withthe 2 and 4 bladed cases. Different studies have been conducted in order to improve theDarrieus VAWT performance [11, 65, 84, 85, 107]. Mohamed [84] tested numerically 20 dif-

3.4. CHARACTERISTICS OF THE DARRIEUS HORIZONTAL AXIS WATERTURBINE (DHAHT) 37

ferent symmetric and non-symmetric airfoils. The author conducted 2D simulations usingthe standard k− ε turbulence model and a mesh of around 95000 cells. He found that theuse of symmetrical airfoils delays the stall, and allows the wind turbine to operate overa wider range of operating conditions. Moreover, the S1046 airfoil increases the powercoefficient by 28% over the common NACA 0018 airfoil. Many other experimental andnumerical studies have been done to investigate different Darrieus VAWT mechanisms :dynamic stall [1, 31, 32, 62], self starting characteristics [4, 6, 47, 101, 122], fluctuatingand skewed inflow conditions [11, 16, 27, 125].In contrast, Darrieus horizontal axis hydro turbines (DHAHTs) have evoked little interestamong researchers to date [76–79]. The DHAHT was introduced by McAdam et al. [79]as a variant of the vertical Darrieus turbine, with the advantage to extract power in lowdepth flow. McAdam et al. [76, 78] investigated experimentally the effect of Froude num-ber Fr on the power coefficient Cp for a 6 straight-bladed DHAHT in an open channel.The increase of the Froude number Fr [0.10 − 0.17] leads to an improvement of Cp. Theauthors observed an overpass of the Betz’s limit when Fr = 0.10. The authors also foundthat by reducing the solidity and increasing the rotor blockage ratio, Cp could be furtherincreased. The study of McAdam et al. [77] tested experimentally two configurations ofDHAHTs and showed that truss configuration achieves a 0.72 lower Cp than a straightconfiguration.To the best of the authors’ knowledge, the present work is the first to present a detailedinvestigation on the influence of blade profile and number on performance for a DHAHT.Numerically, different turbine configurations/blade profiles are considered to improve theglobal performance of an existing DHAHT. A 2.5D URANS approach is used based on ak−ω SST turbulence closure and an unstructured fine mesh. A boundary layer profile anda confined domain are imposed to model more realistic open channel flow conditions. Thenumerical approach is validated using an experimental Cp curve obtained for a DarrieusVAWT found in the literature. The main performance coefficients and the flow topology(trailing/leading edge vortices) around the blades are compared between four blade pro-files. Finally, the influence of the blade number is investigated.

3.4 Characteristics of the Darrieus Horizontal Axis Wa-

ter Turbine (DHAHT)

The DHAHT geometry used in this study was based on an actual full scale 3-blade riverturbine currently in use. From this base, we adjusted the blade profile and curvaturein order to be more generally applicable. Figure 4.1 and Table 4.1 provide the general



specifications of the modeled DHAHT. The solidity σ is calculated according to Arayaet al.’s definition [5] as follows :

σ =Nc

πD(3.1)

Where N is the number of blades, c the blade chord and D the rotor diameter. In thepresent case, σ = 0.13, 0.2, 0.26 for N= 2, 3 and 4, respectively.

Tableau 3.1 Main geometrical and operating parameters for the flows throughthe DHAHT.

Rotor diameter D [m] 0.45Rotor width Wrotor [m] 0.025 (2.5D simulation)Number of blades N [-] 2 to 4Blade profile S1046, S809, FXLV152, NACA0018Blade chord length c [m] 0.094Rotor solidity σ [-] 0.13, 0.2, 0.26Tip-speed ratio λ [-] 1.8 to 5Water speed U0 [m/s] 1.58Reynolds number ReD [-] 711000Froude number Fr [-] 0.56

Figure 3.1 CAD geometry of the DHAHT.

The profiles of the 4 blades used in our study are presented in Fig. 3.2and are a) NACA0018,b) S1046, c) S809 and d) FXLV152. The NACA0018 was used by Idénergie inc. and the

3.4. CHARACTERISTICS OF THE DARRIEUS HORIZONTAL AXIS WATERTURBINE (DHAHT) 39

three latter profiles were selected following an airfoil comparison by [84] of a DarrieusVAWT. Of the 20 profiles investigated in the authors study, these three profiles producedthe highest power coefficient Cp. The geometrical features and the mean inlet water speed(U0 = 1.58m/s) are fixed for the four configurations. However, the tip-speed ratio (TSR)or λ is varied within the range λ = [1− 5], in order to evaluate the turbine performance.The λ is the ratio between the tip blade speed and the free stream water velocity :

λ =ΩD

2U0

(3.2)

where Ω corresponds to the turbine rotation rate (angular velocity) (rad/s).

Figure 3.2 2D cross-sections of the four blade profiles.

The power coefficient Cp and the torque coefficient CT were used to compare the perfor-mance of the four blade profiles. These two coefficients are estimated as follows :

Cp =TΩ

12ρAU3

0

(3.3)

CT =T

14ρADU2

0

(3.4)

where A (A = πDWrotor) is the swept area (m2) of the rotor, ρ is the water density(kg.m−3) and T is the instantaneous rotor torque (N.m). The overbar indicates timeaveraging. All the water properties are evaluated at 20C. Cp is a ratio of the mechanicalpower obtained by the turbine over the water power available, while CT is the ratio of thetorque generated by the turbine over a theoretical value derived from the mean flow.



For the current study, the ratio between the rotor diameter and the flow depth is D/H =

0.55. This elevated ratio causes an acceleration of fluid trough the turbine, which influencesthe Cp and CT coefficients. The value of U0 should be adjusted by the solid blockagecorrection factor εb [20]. The corrected velocity Uc is calculated as follows :

Uc = (1 + εb)U0 (3.5)

εb =1

4

D

H(3.6)


2.5D unsteady-state Reynolds Averaged Navier-Stokes (URANS) calculations were usedto model the flow around the DHAHT.


A sketch of the computational domain is shown in Figure 4.2. It is divided into two mainregions : the rotor domain and the rest of the channel. The rotor domain corresponds to therotating turbine region, while the outer region is stationary and represents the surroundingflow. The turbine of diameter D = 0.45 m is composed of N straight blades and rotatescounterclockwise. The domain length and height are fixed to x = 9.5D and y = 1.8D

(mean flow depth of H = 0.8 m), respectively. The center of the rotor domain is located atx = 2.5D from the inlet and the distance between the bed and the bottom of the rotatingdomain, known as the turbine stem height, is 0.64D. The width of the domain in thelateral (z ) direction is z = 25 mm (2.5D calculations), to satisfy the software constraints.

Figure 3.3 2D sketch of the computational domain with its main dimensionsand the boundary conditions.Note that for the current numerical experimentsthe turbine rotates in the counter-clockwise direction.


3.5.2 Numerical method and turbulence closure

All calculations have been performed using the software CFX ANSYS 18.1 based on afinite-volume method. For the spatial and temporal discretizations, a second-order highresolution advection scheme, and a second-order Backward-Euler scheme with an implicittime-stepping scheme were used, respectively. The second-order high resolution schemewas selected to avoid numerical dissipation. The velocity-pressure coupling was overcomeusing a Rhie Chow fourth-order coupling algorithm, which guarantees that the dissipationterm vanishes rapidly under mesh refinement. The gradients were evaluated through theGreen-Gauss Cell-Based method.The Shear Stress Transport k − ω (k − ω SST) turbulence model developed by Menter[82] is a commonly used two-equation eddy viscosity model and was used herein in itslow-Reynolds number formulation as the turbulence closure model for the flow around theDHAHT. The k − ω SST model combines the robust formulation of the k − ω Wilcoxmodel [127] in the near wall region and the k− ε away from the wall. A blending functionensures a smooth transition between the two models. The resulting model exhibits thenless sensitivity to free stream conditions, while the shear stress limiter helps the k−ω modelavoiding excessive turbulent kinetic energy levels near stagnation points. Bardina et al.[10] rated the k−ω SST model as the most accurate model for aerodynamic applications.Different authors [28, 37] obtained very satisfactory results compared to experimental datausing this turbulence closure for modeling the flows across a Darrieus turbine. Moreover,in our study, we successfully validated the k − ω SST model against the experimental Cp

results of Castelli et al. [24], as it will be shown hereafter.

3.5.3 Boundary conditions and numerical parameters

The main boundary conditions are shown in Figure 4.2. The velocity profile at the inlet ofthe domain was estimated using the law of the wall (log-law) with rough-bed conditions(roughness height, ks= 0.05 m) (Fig. 3.4). A rough-bed logarithmic profile was used sothat the numerical experiment would be more representative a river flow conditions wherethe turbine is meant to be installed. The inlet velocity profile leads to an average (timeand space) streamwise velocity of U0 = 1.58 m/s.

A simple pressure outlet condition was selected, but 2 other conditions (i.e., outflow andconvective) were verified and gave similar results. A no slip condition was imposed onthe turbine blades and channel bed, while a free slip condition was imposed on the top(upper boundary) of the channel to represent the water free-surface. This simplified free-surface boundary condition was used, given that modeling the free-surface deformation is



0 0.2 0.4 0.6 0.8 1 1.2

u/u0 [-]

0

0.2

0.4

0.6

0.8

1

Y/H

[-]

Figure 3.4 Boundary layer velocity profile imposed at the inlet.

not the focus of the current study and would require extensive computation resources. Atranslation periodicity is applied in the lateral, z, dimension on the channel sidewalls.An interface is set between the rotor and the outer domain as a ’transient rotor/stator’condition, which accounts for flux continuity and transient interaction effects betweenthe rotor and outer domain. This interface condition showed a good agreement with PIVresults for an impeller-diffuser-volute interaction in a centrifugal fan [81] and rotor-statorinteractions in an axial turbine [34].

An unstructured fine grid mesh has been generated using the commercial software Centaur(Austin, USA) (Fig. 3.5). The mesh is composed of tetrahedral elements in the outer regionand rotor domain, with 20 prismatic layers around the blades. A stretching factor of 1.15

was used to avoid any numerical dissipation of possible coherent structures. The meshwas refined in the wake of the rotor to capture the wake flow vortices and close to thewalls. For the 3 bladed configuration, the total number of elements is approximately 8.34million cells and the maximum value of the wall coordinate in the whole domain satisfiesmax(y+) < 0.9 (see Table 4.2). A grid sensitivity analysis was performed, using two coarsermeshes of 2 and 4.1 million cells. Despite the 1% discrepancy of the Cp and CT betweenthe 8.35 and 4.1 million cell meshes, the finer mesh was selected to model accurately theflow around the blades.

The selected time step for all simulations corresponds to the time that the DHAHT needsto rotate by 20 (2π/180). The choice of the time step was based on the former work by


Figure 3.5 2D views of the numerical mesh distribution of a) entire domain,b) rotor domain, and c) turbine blade

.

Tableau 3.2 Mesh grid parameters for the three-bladed configuration.Grid type Unstructured Tetrahedral/PrismaticTotal Number of Cells 8.34 x 106Total Number of Nodes 3.12 x 106Number of Cells in the Rotor domain 6.14 x 106Number of Cells in the Outer domain 2.20 x 106Stretching factor 1.15Maximum wall coordinate (y+) 0.9



Ma et al. [69] who investigated the power performance of a high solidity VAWT. Ma et al.[69] tested three time steps corresponding to the time that their VAWT needed to rotateby 2π/90 (∆t1), 2π/180 (∆t2) and 2π/360 (∆t3). They got identical results using ∆t2 and∆t3, such that ∆t2 has been selected in the present case for its good trade-off betweenaccuracy and computational cost.The calculations were initialized using converged steady-state RANS calculations and wererun using the Mammouth cluster MP2 provided by Calcul Québec. The CPU time for theURANS calculations to obtain a single λ value for a given configuration was approximately10 days using 50 processors (AMD Opteron 6172). Given that approximately 8 TSR valueswere obtained per configuration (i.e., blade profile or number), the total calculation timewas over 480 days. In order to shorten this time, multiple calculations with different TSRvalues were run simultaneously using Mammouth. More than 10 rotations of the turbinewere necessary for a converged solution, which was attained when 1) the total torquedeviation between two subsequent revolutions was below 0.5% ; 2) all residuals were lowerthan 10−8 ; and 3) the mass imbalance is lower than 10−6. The power and torque coefficientsare then averaged over two additional periods after reaching the convergence.

3.6 Validation of the flow solver

The validation of the numerical model was achieved by comparing with the publishedexperimental data of Castelli et al. [24] for a VAWT. The authors studied numerically andexperimentally a three-bladed open H-rotor Darrieus wind turbine under a 9 m/s windspeed. The blade profile was a NACA 0021. The rotor diameter and the blade chord werefixed to 1030 mm and 85.8 mm, respectively. For our comparison, similar grid criteria andresolution discussed in Section 4.5.3 were applied to the Castelli et al.’s Darrieus VAWTgeometry. Contrary to Castelli et al. [24], the blockage effect was taken into account inthe present case. Figure 4.5 shows the Cp distribution for different λ. The current study’smodel is compared with numerical/experimental curves obtained by Castelli et al. [24],and other CFD results by [41, 84]. [41, 84] conducted 2D URANS calculations using ak−ε turbulence closure and standard wall functions associated with an unstructured meshcomposed of 153200 [41] and 85000 − 95000 [84] cells elements. In comparison with theCFD results from previous authors [24, 41, 84], the present CFD results show an excellentagreement with the experiment values of Castelli et al. [24]. The location of maximum CP

around λ = 2.6 is modeled correctly, while the highest discrepancy with the experimentsis about 13% for λ = 2.5.


1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

TSR, [-]

0

0.1

0.2

0.3

0.4

0.5

0.6

Pow

er C

oeffi

cien

t, C

p [-

]

Experimental (Castelli et al., 2011)CFD (Castelli et al., 2011)Present CFD modelCFD (Mohamed et al., 2018)CFD (Mohamed, 2012)

Figure 3.6 Performance curve of a three-blade H-rotor Darrieus wind turbineversus λ. Comparison with former CFD [24, 41, 84] and experimental results[24].

This good agreement with the published experimental data is due to the accuracy ofthe low-Reynolds number formulation of the k − ω SST model [28, 37] and the highspatial resolution especially in the blade region where the average wall coordinate is aroundy+ = 0.87. The comparison with Castelli et al. [24] confirms the ability of the present modelto accurately predict the complex flows around a VAWT and provides a validation on themesh configuration and size used for Darrieus turbine modeling, which was then used toinvestigate the flow around the DHAHT.


The corrected velocity Uc is used hereon for the calculation of Cp and CT and is equalto 1.8 m/s, which represents an increase of 14% compared to the depth averaged inletvelocity U0. The results presented below have been obtained for fixed conditions of theincoming flow characterized by a Reynolds number ReD = UcD/ν = 711000 and a Froudenumber Fr = Uc/

√gH = 0.56. The origin of the axes (x = y = z = 0) is located at the

rotor center.



3.7.1 Comparison between blade profiles

In this subsection, we investigate the effect of turbine blade profiles on CP and CT valuesand the characteristics of flow over the 4 blade profiles. The number of blades on theturbine is fixed at N = 3, which corresponds to a rotor solidity σ = 0.2 (see Eq.4.1).The S1046, S809, FXLV152 and NACA0018 straight profiles are compared over a widerange of λ (λ = [1.8 − 5]). Beyond this range, the power coefficient Cp is usually verysmall as attested by preliminary tests. Figure 3.7 shows the distributions of the averagedpower coefficient CP of the selected blade profiles. Their shape is quite comparable tothat obtained for the Darrieus VAWT studied numerically by Maitre etal. [70], yet theoperating λ range is larger for wind turbines ([2 − 10]) due to the difference in workingfluid density. As with the Darrieus VAWT, three regions can be distinguishable in all fourblade profiles [19] :

1. Dynamic stall with high angle of attack (λ < 2) ;

2. Transition region where an equilibrium occurs between the viscous and dynamic stalleffects (2 < λ < 3) ;

3. High TSR region (λ > 3) where the drag forces are dominant.

1.5 2 2.5 3 3.5 4 4.5 5

TSR, [-]

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

CP

[-]

S1046

S809

FXLV152

NACA0018

Figure 3.7 Comparisons of the averaged power coefficient CP versus λ for theS1046, S809, FXLV152 and NACA0018 profiles. Results obtained for the 3 bladeturbine (N = 3).

In the range λ = [1.8−2], the S1046 and NACA0018 profiles provide the highest CP valuescompared to the S809 and FXLV152. Maximum Cp is produced for all profiles over the


range of λ = [2.1 − 3]. The S1046, FXLV152 and NACA0018 profiles peak at a values ofCP = 0.33, 0.30, 0.31 at λ = 2.3, 2.6, 2.4, respectively, while the S809 profile, has a lowermaximum Cp value (CP = 0.25) occuring at a higher λ (λ = 3).The poorer performanceon the S809 is surprising as in Darrieus VAWT applications, the S809 profile producedone of the highest Cp = 0.5 for σ = 0.1 [84]. The difference with our study, is essentiallydue to the density and the solidity variation. In higher values of λ (λ > 4), the FXLV152profile provides slightly higher CP values compared with the other 3 profiles investigatedand a slight shift is observed between the different profiles. The negative Cp values beyondλ = 5 are unrealistic as they reflect that the DHAHT provides power to the flow.

The distributions of the averaged torque coefficient CT obtained for the different profilesare presented in Figure 3.8. The values of CT for each profile follow similar trends as thosediscussed above for CP . The S1046 and S809 profiles produce respectively the maximum(CT = 0.15) and minimum (CT = 0.09) values, respectively. For comparison in DarrieusVAWT applications (σ = 0.1), the NACA0018 and FXLV152 profiles produced the highest(CT = 0.07) and lowest (CT = 0.05) values [84], respectively. As for the CP distributions,the S809 profile exhibits the overall lowest CT . The maximum CT peaks occur at λ = 2.1

for the S1046, NACA0018 and FXLV152 profiles. However, for the S809, this peak is shiftedto λ = 2.6. A shift was already observed for the power coefficient as shown in Figure 3.7.

1.5 2 2.5 3 3.5 4 4.5 5

TSR, [-]

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

CT [-

]

S1046S809FXLV152NACA0018

Figure 3.8 Comparison of the averaged torque coefficient CT for the S1046,S809, FXLV152 and NACA0018 profiles. Results obtained for the 3 blade turbineN = 3.



The instantaneous torque coefficient CTiobtained for the four blade profiles for one com-

plete turbine rotation is presented in Figure 3.9. The origin of the azimuth position (θ = 0)corresponds to the top blade position (12 o’clock). Qualitatively, the four blade profilesexhibit similar curves. In general, the curves of CTi

of our study compare well with pre-vious research (e.g., the vertical Darrieus VAHT study of Gorle et al [37]). During thefirst quarter of rotation (θ = [0 − 90]), the torque coefficient CTi

increases to reach amaximum as the blade comes near in perpendicular with the oncoming flow. The increasein CTi

is related to the tangential force rise, similar to what is observed for a DarrieusVAHT and VAWT. The peaks of CTi

for the S1046 (CTi,max = 0.231) and NACA0018(CTi,max = 0.225) profiles are reached at θ = 90. However, the S809 (CTi,max = 0.210) andFXLV152 (CTi,max = 0.165) profiles peak earlier for θ between 75 and 80 and is due toincreased flow separation and stall along the pressure side (inner side) of these two profiles(see Table 3.3). The second quarter of rotation (θ = [90−180]) is characterized by a fastCT decay, due to the tangential force loss by a complete stall of the four profiles. In thesecond half of rotation (θ = [180− 360]), the torque coefficient obtained for the differentprofiles increases again after a minimum CTi,min = −0.03 at θ = 200. The maximumCTi,max ' 0.02− 0.03 at θ = 330 is due to the blade coming parallel with oncoming flow.

0 30 60 90 120 150 180 210 240 270 300 330 360

Azimuth position, [0]

-0.05

0

0.05

0.1

0.15

0.2

0.25

CT

i [-]


Figure 3.9 Comparison of the instantaneous torque coefficient CTiof one blade

over one rotation (degrees) between the S1046, S809, FXLV152 and NACA0018profiles. Results obtained for the 3 blade turbine N = 3 and λ = 1.8.

The instantaneous vorticity fields around the four profiles for different azimuthal positionsare presented in Table 3.3. The investigation of the flow around the different profiles


allows a better understanding of the curves shown in Figure 3.9. The vorticity fields areextracted for λ = 1.8. As discussed previously, at θ = 90, the flow is separated along thepressure side of the S809 and FXLV152 profiles and leads to loss of CT (Fig.3.9). At thisazimuthal position, the NACA0018 and S1046 profiles have only a small separation regionnear the trailing edge and thus produced their CP,max. At θ = 150, the flow around thefour profiles is completely separated, thus all profiles are stalled. A leading edge vortex isclearly distinguishable along the pressure side of the FXLV152 and S809 profiles. On thecontrary, a long shear layer is visible along the inner side of the S1046 and NACA0018profiles, without any apparent rolling. The presence of the leading edge vortex or theshear layer along the pressure side leads to a negligible torque production CT ≈ 0 (seeFig.3.9). At θ = 210, a vortex is apparent in the trailing edge region of the four profiles.This vortex corresponds to the leading edge vortex which is convected along the chord.The S809 profile’s vortex occupies a larger region in comparison with the other cases. Atθ = [270 − 330], the flow reattaches along the pressure side for the four profiles, leadingto a re-increase of the torque coefficient CT (Fig.3.9).

The polar distribution of the total instantaneous CTi, for different tip speed ratios λ is

presented in Figure 3.9. The total torque coefficient CTiis obtained by summing the

contribution of the three blades. The four profiles have a comparable polar distribution,which corresponds to a 3 lobed rosette. The three lobes are located at near θ = 90, 210

and 330. The 3 lobed rosette distribution was already observed for a 3 blade DarrieusVAHT by Maitre et al. [70], and for a Darrieus VAWT by Mohamed [84]. By increasingλ, the 3 lobed rosette distribution rotates slightly for the S809 and FXLV152 profiles andCTi

values for all profiles decrease consistent with the CTicurve presented in Figure (3.8).

The shift for the S809 profile is on the order of θ = 10 between λ = 1.8 and λ = 3.1. Thisangular phase shift is probably due to a delay in the stall angle caused by increasing theflow chord Reynolds number Rec.

The distribution in the vertical direction Y/H of the dimensionless averaged streamwisevelocity U∗(U∗ = U/U0), at four X/D locations downstream of the rotation center, namelyX/D = 1, X/D = 2, X/D = 3 and X/D = 4 for N = 3 and λ = 1.8 is shown in Figure3.11. These X/D locations were selected in order to investigate wake evolution along thestreamwise (longitudinal) direction for the different blade profiles. As presented in Figure4.2, the Darrieus rotor is located between Y/H = 0.4 and Y/H = 0.9. In the very nearwake (at X/D = 1), the wake width corresponds approximately to the rotor diameter(Y/H = [0.4−0.9]) with U∗ = 0.35. As the downstream distance from the rotor increases,the velocity deficit increases due to flow diffusion. The width of the wake increases and a



Tableau 3.3 Instantaneous vorticity fields for θ = 90, 150, 210, 270, 330.Comparisons between the S1046, S809, FXLV152 and NACA0018 profiles forthe 3 blade turbine N = 3 and λ = 1.8.

S1046 S809 FXLV152 NACA0018

θ = 90

θ = 150

θ = 210

θ = 270

θ = 330


0

30

60

90

120

150

180

210

240

270

300

330

-0.05

0

0.05

0.1

0.15

0.2

0.25

(a) λ = 1.8

0

30

60

90

120

150

180

210

240

270

300

330

-0.05

0

0.05

0.1

0.15

0.2

0.25

(b) λ = 2.3

0

30

60

90

120

150

180

210

240

270

300

330

-0.05

0

0.05

0.1

0.15

0.2

0.25

(c) λ = 3.1

0

30

60

90

120

150

180

210

240

270

300

330

-0.05

0

0.05

0.1

0.15

0.2

0.25

(d) λ = 4.0

Figure 3.10 Polar distributions of the instantaneous torque coefficient CTi. Re-

sults obtained for the 3 blade turbine N = 3.



hump/convex-shaped curve appears after the downstream locationX/D = 3. AtX/D = 4,the minimum velocity in the wake is located at Y/H ' 0.6−0.7 with a velocity of U∗ = 0.1.At the bottom of the turbine (at Y/H = 0.4), and at a downstream distance of X/D = 1,the flow accelerates to U∗ = 1.6 due to rotor obstruction of the flow. The traces of thevortices shedding from the different profiles are revealed by ripples in the velocity profileat Y/H = 0.5 (X/D = 1). This shedding corresponds to the downstream convection of theleading edge vortex developed for θ = [150− 210] (see Table 3.3). The same phenomenahave been observed in the wake flow of a Darrieus VAWT by Bianchini et al. [17]. Thedifference in the ripple distribution (Y/H = 0.5,X/D = 1) between the different profiles, isprobably associated to the difference in the leading edge vortex patterns around θ = 210.

0 0.5 1 1.5 2

U*

0

0.2

0.4

0.6

0.8

1

y/H


(a) X/D = 1

0 0.5 1 1.5 2

U*

0

0.2

0.4

0.6

0.8

1

y/H


(b) X/D = 2

0 0.5 1 1.5 2

U*

0

0.2

0.4

0.6

0.8

1

y/H


(c) X/D = 3

0 0.5 1 1.5 2

U*

0

0.2

0.4

0.6

0.8

1

y/H


(d) X/D = 4

Figure 3.11 Distributions of the mean streamwise velocity U∗ along the verticaldirection Y/H for the 3 blade turbine N = 3 and λ = 1.8. Results obtained forfourX/D locations : (a)X/D = 1, (b)X/D = 2, (c)X/D = 3 and (d)X/D = 4.


3.7.2 Influence of the blade number N

The influence of the blade number N on the performance of the DHAHT was investigatedusing the S1046 profile only, as it exhibited the best overall performances in the previousanalysis (i.e., Cp = 0.33). Three blade numbers, N = 2, 3 and 4 were modeled. The meshwas built based on the same criteria as the previous analysis with N = 2, 3, and 4 having4.2, 6.2 and 8.3 million cell elements in the rotor domain, respectively. The outer domainfor all three cases had an additional 2.2 million cell elements. In addition, the maximumvalues of the wall coordinates in the whole domains satisfy max(y+) < 0.92 (N = 2),max(y+) < 0.9 (N = 3) and max(y+) < 0.89 (N = 4).

The averaged power Cp and torque CT coefficients for the three assessed blade numbers arepresented in Figure 3.12. In general, with increasing blade number, N , maximum CP andCT values occur at lower values of λ and the peak values of CP are reduced. For example,the configuration with 2 blades produces the highest CP (CP = 0.34) which occurs atλ = 3. The 4 blade turbine has a CP = 0.29 at λ = 2.

1.5 2 2.5 3 3.5 4 4.5 5

TSR, [-]

-0.25

-0.15

-0.05

0.05

0.15

0.25

0.35

CP

[-]

2 Blades

3 Blades

4 Blades

1.5 2 2.5 3 3.5 4 4.5 5

TSR, [-]

-0.05

0

0.05

0.1

0.15

0.2

CT [-

]

2 Blades3 Blades4 Blades

(a) (b)

Figure 3.12 Influence of the blade number N on the averaged (a) power CP

and (b) torque CT coefficients.

The configurations with 2 and 4 blades allows to reach maximum CT values at higher andlower values of λ, respectively. The 3 bladed turbine has a peak CP = 0.32 at λ = 2.3.Contrary to the results shown here, the study by [39, 103] on a Darrieus VAWT showedthat between N = 2, 3 and 4, the highest CP was produced with a 3 bladed rotor.

The polar distributions of the total instantaneous torque coefficient CTifor different λ

values is shown in Figure 3.13. The total coefficient CTiis obtained by summing the

contribution of the N blades for each configuration.



0

30

6090

120

150

180

210

240270

300

330

-0.1

0

0.1

0.2

0.3

(a) λ = 1.8

0

30

6090

120

150

180

210

240270

300

330

-0.1

0

0.1

0.2

0.3

(b) λ = 2.3

0

30

6090

120

150

180

210

240270

300

330

-0.1

0

0.1

0.2

0.3

(c) λ = 3.1

0

30

6090

120

150

180

210

240270

300

330

-0.1

0

0.1

0.2

0.3

(d) λ = 4.0

Figure 3.13 Influence of the blade number N on the polar distributions of theinstantaneous torque coefficient CTi

for four tip speed ratios λ.

3.8. CONCLUSION 55

For the 2 and 4 bladed rotor, a dipole and a circular distribution are obtained, respectively,while with the 3 blades configuration, a 3 lobed rosette is observed. For λ = 1.8, themaximum CTi

' 0.3 value is produced for the dipole distribution and is located aroundθ = 90 and 270. On the contrary, for N = 4, an homogeneous CTi

distribution isinsured, with a lower maximum torque value CTi

= 0.16 for λ = 1.8. The shape of thepolar distribution for the three configurations are independent of the tip speed ratio.

3.8 Conclusion

This paper reported the first numerical simulations of a DHAHT, using an unsteady RANSsimulation based on the k − ω SST turbulence model. A parametric study was underta-ken in order to improve the turbine performance, covering four blade profiles, namely theS1046, S809, FXLV152 and NACA0018 profiles. The streamwise flow conditions were cha-racterized by fixed Froude number Fr = 0.56 and Reynolds number ReD = 711000.The numerical approach was validated against the experimental CP curve of a DarrieusVAWT equipped with three NACA 0021 blades. A maximum discrepancy of 13% comparedto the experimental values of Castelli et al. [24] has been obtained. This small discrepancyshows a marked improvement in comparison to the previous numerical results of Castelliet al. [24] and Mohamed et al. [41, 84], highlighting the importance of well resolving theflow in the near blade regions.For the three blade DHAHT, the best performance (CP = 0.33) was obtained by the S1046profile for λ = 2.3. In the dynamic stall region (low tip speed ratios), the NACA0018 andS1046 profiles achieved the highest CP and CT values. For high λ values (λ > 3.5), theFXLV152 had the highest CP values. Poor performance was with the S809 profile over thewhole range of λ values considered here. The low performance of the S809 profile may bemainly attributed to the early flow separation along the pressure side at θ = 90, followedby the development of a strong leading-edge vortex at θ = 150 which occupies a largeflow region. For the 4 profiles tested, vortex shedding traces were observed in the nearwake region (X/D = 1) as ripples in the U∗ profiles at y/H = 0.5. An increase of thevelocity deficit and in the wake width was noticed downstream.The DHAHT sensitivity to blade number (N = 2, 3 and 4) was performed. An inverserelationship between blade number and CP and CT was observed. Smaller blade numberswere as well associated with higher λ values. The best performance was obtained by the2 blade turbine where maximum CP = 0.34 at a λ = 3. The 3 bladed rotor offers the bestoverall compromise (homogeneous distribution) comparing to the configurations with 3and 4 blades.



In general, the 3 straight-bladed DHAH based on the S1046 profile is the suitable configu-ration for those river flow conditions (Fr,ReD). A 3D LES simulation of the whole DHAHTwould be interesting to investigate in detail the development of the leading edge vortexand the different 3D vortical structures around the stem. Accounting for the deformationof the free surface and the possibility that the turbine may be not entirely submerged aretwo research perspectives to model even more realistic flow conditions.

Acknowledgements

All calculations have been done using the computational resources of the Compute Canadanetwork, which is here gratefully acknowledged. The authors would also like to thank theNatural Sciences and Engineering Research Council of Canada (NSERC) for their financialsupport (grant 514264-17 with the company Idénergie).

CHAPITRE 4

Multiphase modeling of the free surface flowthrough a Darrieus horizontal axis shallow-water turbine

4.1 Avant-propos

Auteurs et affiliation :A. E. Benchikh Le Hocine : étudiant au doctorat, Université de Sherbrooke, Faculté degénie, Département de génie mécanique.R. W. J. Lacey : professeur, Université de Sherbrooke, Faculté de génie, Département degénie civil.S. Poncet : professeur, Université de Sherbrooke, Faculté de génie, Département de géniemécanique.Date de soumission : 25 Janvier 2019Revue : Journal of Renewable EnergyTitre français : Modélisation multiphasique de l’écoulement à surface libre à travers uneturbine de type Darrieus à axe horizontal dans l’eau peu profonde.Contribution au document : Ce troisième article contribue à mettre en évidence l’in-fluence de la modélisation de la surface libre en utilisant l’approche VOF sur les per-formances de l’hydrolienne de type Darrieus à axe horizontal. Cela permet, en outre, decomprendre l’impact de la turbine sur la déformation de la surface libre et par la suitel’influence de cette dernière sur les structures tourbillonnaires générées dans le sillage despales.Résumé français : Les turbines de type Darrieus à axe horizontal ont montré leur ca-pacité à fournir de l’énergie hydroélectrique dans des rivières peu profondes. L’étude desperformances de la turbine de type Darrieus et de son interaction avec la surface libreest de grande importance ce qui fait l’objet de cette étude numérique. La surface libreest modélisée en utilisant un solveur multiphasique basé sur la méthode du volume defluide (VOF) qui est associée au modèle de turbulence k-ω SST. Un profil de couche limitedéveloppée est imposé à l’entrée du domaine. Deux configurations de submersion de laturbine sont testées en variant le niveau d’eau : partiellement immergée (configuration

57

58CHAPITRE 4. MULTIPHASE MODELING OF THE FREE SURFACE FLOW

THROUGH A DARRIEUS HORIZONTAL AXIS SHALLOW-WATER TURBINE

1) et complètement immergée (configuration 2). Le solveur a été validé par rapport auxdonnées expérimentales disponibles dans la littérature pour une éolienne de type Darrieusà axe vertical à trois pales et pour une expérience de rupture de barrage. L’immersiontotale de la turbine de type Darrieus entraîne une amélioration de 36.8 % du coefficientde puissance CP par rapport à la configuration 1. Pour des vitesses de rotation élevées, lenombre de Froude en aval de la turbine augmente. La quantité de mouvement extraite parla turbine de type Darrieus est également quantifier. La quantité de mouvement maximaleest extraite lorsque la turbine de type Darrieus est complètement immergée

4.2. ABSTRACT 59

4.2 Abstract

Previous research on the Darrieus horizontal axis hydrokinetic turbines (DHAHT) haveshown their ability to provide hydro-power in shallow rivers. Investigation of DHAHTperformance and interactions with the free surface are of great importance and are inves-tigated numerically in the present work using a multiphase CFD solver. The free surfaceis modeled using the volume of fluid (VOF) method associated with an unsteady k − ωSST turbulence closure. Fully developed boundary layer conditions are imposed at theinlet. Two submergence configurations are considered by varying the water level : partiallysubmerged (configuration 1) and fully submerged (configuration 2). The flow solver hasbeen carefully validated against experimental data available in the literature for a three-blade vertical axis Darrieus wind turbine and a dam break experiment. Total immersionof the DHAHT leads to an improvement in the power coefficient CP by 36.8% comparedto configuration 1. For high tip-speed ratios, the Froude number downstream the DHAHTis observed to increase. The momentum extracted by the DHAHT from the flow is alsoquantified. The submerged DHAHT extracts the most momentum at high tip-speed ratios.

4.3 Introduction

Delivering electricity to remote consumers is an often a costly and arduous endeavor. Asan alternative, the idea of self-generating green power is becoming attractive. Differentnatural sources of energy are available depending on the consumers’ location and weatherconditions : solar, wind and water. For locations close to rivers, hydropower offers someadvantages over solar and wind sources by offering continuous power generation. Asidefrom dams, hydropower may be produced by placing a water turbine (Darrieus, Savonius,axial turbine . . . ) [3, 87] directly in a river. The performances of river water turbines aredependent mainly on the water depth, velocity, and turbulent intensity, which may varywith time.Horizontal- (HAHT) and Vertical-Axis Hydrokinetic Turbines (VAHT) (Savonius, Dar-rieus, pelton) are widely used to extract power from water currents. Different experimentaland numerical studies have been done in order to quantify the effect of the free surface onthe turbine’s performance. Nishi et al. [89, 90] investigated numerically the performanceof a 3 bladed HAHT under variable free surface conditions in a shallow water channel.For model validation, the authors placed the HAHT in an open channel with a FroudeNumber (Fr = U/

√gH)) equal to 0.415, and recorded the output power using a torque

sensor. Numerically, 3D unsteady RANS (Reynolds Averaged Navier-Stokes) multiphaseand single-phase calculations were performed using the k− ω SST turbulence model. The



Volume Of Fluid (VOF) method was used to model the free surface in the multiphase ap-proach. In the single-phase runs, the flow was assumed to be uniform and the free surfacewas considered to be a free slip wall. An excellent agreement was observed between themultiphase approach and the experimental power coefficient CP , with a discrepancy of 5%

[89, 90]. However, the single-phase approach overpredicted CP and the torque coefficientCT by a factor of 2. The multiphase approach resulted in an increase and decrease in thewater level, upstream and downstream of the turbine, respectively. Yan et al. [130] alsostudied numerically the influence of the distance between the free surface and a 3 bladedHAHT on the generated power. 3D unsteady RANS runs using the level set method wereperformed to model the multiphase flow around the HAHT. The authors validated theCP and CT values for deep (Fr = 0.72) and shallow (Fr = 1.22) depths against Bahajet al.’s [8] experimental values, and obtained a very good agreement (discrepancy of 3%).For deep tip immersion, the authors observed an increase of CP and CT values comparedto shallow tip immersion. The same tendency was observed under the influence of waves,where the CP (resp. Ct) values increased from 0.3919 (resp. 0.8513) in shallow tip im-mersion to 0.4144 (resp. 0.8794) in deep tip immersion. Contrary to Nishi et al. [89, 90],Yan et al. [130] noticed that their single-phase model correctly reproduced the CP and CT

coefficients in the deep tip immersion. Predictably, in shallow tip immersion where the freesurface effect is important, the single-phase model failed to reproduce the hydrodynamicmechanisms. The influence of the distance between the free surface and the turbine onthe CP and CT coefficients was also investigated experimentally for a 2 bladed Savoniushorizontal axis hydrokinetic turbine (SHAHT) (Nakajima et al. [87]). The authors pla-ced the SHAHT in an open-channel flume with a Reynolds number (ReD = (U0D)/ν) ofRe = 1.1× 105 and Fr = 0.4. The torque generated by the SHAHT was recorded using atorque meter for different rotor relative submergence values, and two rotation directions,clockwise (CW) and counterclockwise (CCW). A clearance ratio (HG/D) was introducedto characterize the rotor position, and is defined as the ratio of the distance rotor-bottomHG and the rotor diameter D. When the SHAHT rotates in the CCW direction withflow from left to right, a decrease in the CP values within the range [0.25 − 0.21] wasobserved by increasing the clearance ratio from 0.2 to 1.1. However, for the CW rotationdirection, the CP coefficient increases from 0.2 to 0.23. The CCW rotation allows to reachthe maximum CP = 0.25 for a tip-speed ratio (λ = ΩD/(2U0)) λ = 1.1. The water leveldownstream the SHAHT decreases as the rotor is closer to the free surface, and a maxi-mum difference of 50 mm between the upstream and downstream water levels was reachedfor HG/D=1.08. McAdam et al. [76, 78] investigated experimentally the effect of Fr onthe power coefficient Cp for a 6 straight-bladed Darrieus Horizontal Axis Hydrokinetic

4.4. CHARACTERISTICS OF THE DARRIEUS HORIZONTAL AXISHYDROKINETIC TURBINE (DHAHT) 61

Turbine (DHAHT) in an open-channel flow. An improvement of CP from 0.7 to 0.9 isobserved by increasing the Froude number Fr from 0.1 to 0.17. Moreover, an overpass ofthe Betz’s limit was recorded for Fr = 0.1. A comparable behavior was also observed for a6 truss-bladed DHAHT by [77]. Many experimental and numerical studies have been doneon HAHT [9, 97, 118] and VAHT [37, 44, 58] but without introducing the free surface effect.In contrast, numerically modeling the interactions between the free surface and the DHAHThave not been considered so far. In a former paper [14], the current authors investigatednumerically different blade profiles and blade numbers on a DHAHT, in order to maximizeCP and CT coefficients in a water channel without a free surface. The 3 blade configura-tion associated with the S1046 profile was selected as it led to a 10% improvement of theCP and CT values. All the runs were performed using a single-phase model without freesurface.The present study, goes further to model more realistic river conditions, 2.5 D URANSmultiphase (air/water) runs using the VOF method and the k−ω SST turbulence closureare conducted, in the present paper, on a 3 bladed DHAHT. The DHAHT is confrontedin the river conditions to water depth variations, in order to investigate its influence onthe turbine’s performance, two configurations are tested : (i) 3/4submergence of the rotordiameter ; and (ii) submerged rotor. The flow solver is first validated against experimen-tal results for a Darrieus VAWT (Vertical Axis Wind Turbine) and the breaking of adam. The values of CP , CT and the flow topology are analyzed for both configurations. Aperformance comparison between the single/multiphase model is also performed for thesubmerged DHAHT. Finally, a particular emphasis is put on the rotor wake flow for thetwo water levels.

4.4 Characteristics of the Darrieus Horizontal Axis Hy-

drokinetic Turbine (DHAHT)

The 3 bladed DHAHT geometry used in this study is based on the former work of BenchikhLe Hocine et al. [14]. Figure 4.1 and Table 4.1 provide the general specifications of themodeled DHAHT. The solidity σ is calculated according to Araya et al.’s definition [5] asfollows :

σ =Nc

πD(4.1)

where N = 3 is the number of blades, c the blade chord and D the rotor diameter. In thepresent case, σ = 0.2.



Tableau 4.1 Main geometrical and operating parameters for the flows throughthe DHAHT.

Rotor diameter Drotor [m] 0.45Rotor width Wrotor [m] 0.025 (2.5D simulation)Number of blades N [-] 3Blade profile S1046Blade chord length c [m] 0.094Rotor solidity σ [-] 0.2Tip-speed ratio λ [-] 1.4 to 5.7Water speed U0 [m/s] 1.58Turbulence intensity I [%] 10

Figure 4.1 CAD geometry of the DHAHT.

4.4. CHARACTERISTICS OF THE DARRIEUS HORIZONTAL AXISHYDROKINETIC TURBINE (DHAHT) 63

In order to investigate the effect of the free surface on the performance of the DHAHT,two configurations were tested by changing the water levels HW and keeping the turbineposition constant. In configuration 1, three quarters of the DHAHT are submerged (HW =

0.65 m), and in configuration 2, the DHAHT is completely submerged (HW = 0.82 m).

For the two configurations, the geometrical features and the mean inlet water velocity(U0 = 1.58m/s) are fixed. However, the tip-speed ratio (TSR or λ) is varied within therange λ = [1.4 − 5.7]. The coefficient λ is the ratio between the tip blade speed and thefree stream water velocity defined as :

λ =ΩD

2U0

(4.2)

where Ω corresponds to the turbine rotation rate (angular velocity) (rad/s).

The power coefficient Cp and the torque coefficient CT are used to compare the performancefor the different configurations. They are estimated as follows :

Cp =TΩ

12ρAU3

0

(4.3)

CT =T

14ρADU2

0

(4.4)

where A (A = πDWrotor) is the swept area (m2) of the rotor, ρ is the water density(kg.m−3) and T is the time averaged rotor torque (N.m). All the water properties areevaluated at 20C. Cp is a ratio of the turbine’s mechanical power over the water poweravailable, while CT is the ratio of the generated turbine’s torque over a theoretical valuederived from the mean flow.

The ratio between the rotor diameter and the flow depth HW = 0.65 m (respectivelyHW = 0.82 m) is D/H = 0.7 (respectively D/H = 0.55). In order to correct the elevatedvalues of the Cp and CT coefficients due to the influence of the blockage, the value of U0

should be adjusted by considering the solid blockage correction factor εb [20]. The correctedvelocity Uc is calculated as follows :

Uc = (1 + εb)U0 (4.5)

εb =1

4

D

H(4.6)




2.5D multiphase unsteady Reynolds Averaged Navier-Stokes (URANS) calculations areperformed to model the flow around the DHAHT for two different water depths. In addi-tion, for configuration 2 a comparison is made between the single and multiphase results.


A sketch of the computational domain is shown in Figure 4.2. The computational domainis divided into two main regions : the rotor domain and an outer region for the rest ofthe channel. The rotor domain corresponds to the rotating turbine region, while the outerregion is stationary and represents the surrounding flow (air and water). The turbine ofdiameter D = 0.45 m is composed of 3 straight blades and rotates counterclockwise. Theprofile of the 3 blades is a symmetric S1046. The domain length and height for the differentconfigurations are fixed to L = 17.5D and H = HW + Ha = 4D, respectively. The centerof the rotor domain is located at x = 6D from the inlet and the distance between the bedand the bottom of the rotating domain, known as the turbine stem height, is 0.64 D. Thewidth of the domain in the lateral (z ) direction is z = 25 mm (2.5D calculations).

Figure 4.2 2D sketch of the computational domain with its main dimensionsand the boundary conditions. The turbine rotates in the counter-clockwise di-rection. Note that the water level HW and the height of the air layer Ha arefixed to HW = 0.65 m, Ha = 1.14 m and HW = 0.82 m, Ha = 0.97 m forconfigurations 1 and 2, respectively.

4.5.2 Numerical method and turbulence closure

All calculations are performed using the software CFX ANSYS 18.1 based on a finite-volume method. For the spatial and temporal discretizations, a second-order high resolu-tion advection scheme, and a second-order Backward-Euler scheme with an implicit time-stepping scheme are used, respectively. The second-order high resolution scheme avoidsexcessive numerical dissipation. The velocity-pressure coupling is overcome using a Rhie-


Chow fourth-order coupling algorithm, which guarantees that the dissipation term vanishesrapidly under mesh refinement. The gradients are evaluated through the Green-Gauss Cell-Based method.The volume of fluid (VOF) method is selected to model the free surface (air/water inter-face). The VOF method is capable of accurately reproducing the free surface deformationfor different configurations [67, 68, 86, 91]. Moreover, the VOF method has been validatedagainst the experimental results of Koshizuka et al. [57] for the breaking of a dam with1% of discrepancy, as it will be shown hereafter. The VOF method consists of explicitlyintroducing a volume fraction α for water and air. The conservation property must besatisfied in each cell of the domain by the two phases’ volume fractions as follows :

2∑i=1

αi = 1 (4.7)

where the indexes 1 and 2 correspond to air and water respectively. The physical propertiesof the fluid are obtained by volume phase averaging :

ρ = αρwater + (1− α)ρair (4.8)

µ = αµwater + (1− α)µair (4.9)

where ρ and µ are the density and viscosity, respectively. α corresponds to the volumefraction of the primary phase, herein water (as an example α = 1 corresponds to 100% ofwater within the cell). A single set of momentum and volume fraction continuity equationsis solved for both phases by introducing α in the whole domain. As a result, a sharedvelocity field for water and air is obtained. A more detailed description of the VOF methodcan be found in [45] .

The Shear Stress Transport k−ω (k−ω SST) turbulence model developed by Menter [82] isa commonly used two-equation eddy viscosity model and is used herein in its low-Reynoldsnumber formulation. The k − ω SST model combines the robust formulation of the k − ωWilcox model [127] in the near wall region and the k − ε away from the wall. A blendingfunction ensures a smooth transition between the two models. The resulting model exhibitsless sensitivity to free stream conditions, while the shear stress limiter helps the k−ω modelavoid excessive turbulence kinetic energy levels near stagnation points. Bardina et al. [10]rated the k − ω SST model as the most accurate model for aerodynamic applications.Different authors obtained very satisfactory results compared to experimental data using



this turbulence closure for modeling the flows across a Darrieus turbine [28, 37] or theturbulent flow over a mounted D-shaped bluff body in a water channel [14]. Moreover, aprevious study by the current authors [14], successfully validated the k − ω SST modelagainst the experimental Cp results of Castelli et al. [24].

4.5.3 Boundary conditions and numerical parameters

The main boundary conditions are shown in Figure 4.2. The inlet corresponds to theplane (x = 0, 0 ≤ y ≤ HW , z). The water volume fraction is fixed to α = 1 and thevelocity profile is estimated using the law of the wall (log-law) with rough-bed conditions(roughness height, ks = 0.05 m) on the bottom plane. A rough-bed logarithmic profileis used to mimic more realistic river flow conditions. The inlet velocity profile leads toan average (in time and space) streamwise velocity of U0 = 1.58 m/s. The boundarylayer profile height at the inlet varies for each water level condition, in order to maintainthe same value of U0 in all cases. An opening condition (fixed atmospheric pressure) isimposed for air in the plane (x = 0, HW ≤ y ≤ H, z) as well as at the top of the domain(x, y = H, z). A pressure outlet condition is imposed in the plane (x = L, 0 ≤ y ≤ H, z).Preliminary runs showed that outflow or convective conditions at the outlet led to similarresults. A no slip condition is imposed on the turbine blades and the channel bed. On thelateral channel sidewalls, translation periodicity is applied. For the single phase runs, thefree-surface is replaced by a free slip wall condition, following the set-up carried out in[14].Between the rotor and the outer domain, an interface is set as a ’transient rotor/stator’condition, which accounts for flux continuity and transient interaction effects between therotor and the outer domain. This interface condition has led to very satisfactory resultscompared to PIV results [34, 81].An unstructured fine grid mesh was generated using the software Centaur (Austin, USA)(Fig. 4.3a). The mesh is composed of tetrahedral elements in the outer region and rotordomain, with 20 prismatic layers around the blades (Fig. 4.3b). A stretching factor of1.15 is used to avoid any numerical dissipation of possible coherent structures. The meshis refined in the rotor wake to capture the wake flow vortices, close to the walls andaround the initial free surface along the whole domain. The total number of elements isapproximately 11.6 million cells and the maximum value of the wall coordinate in thewhole domain satisfies max(y+) < 0.87 (see Table 4.2). For the single phase approach thetotal number of elements is around 8.34 million cells (more details are available in [14]).A grid sensitivity analysis was performed, using two coarser meshes of 6.1 and 8.4 millioncells. The maximum discrepancy between the 8.4 and 11.6 million cell meshes remains

4.6. VALIDATION OF THE FLOW SOLVER 67

lower than 0.8% on the Cp and CT values. Yet the finer mesh has been then selected tomodel accurately the turbulent flow around the turbine.

Tableau 4.2 Mesh grid parameters for the three-bladed configuration.Grid type Unstructured Tetrahedral/PrismaticTotal Number of Cells 11.59 x 106Total Number of Nodes 3.42 x 106Number of Cells in the Rotor domain 6.53 x 106Number of Cells in the Outer domain 5.06 x 106Prismatic layers along the blades 20Stretching factor 1.15Maximum wall coordinate (y+) 0.87

The selected time step for all runs corresponds to the time that the DHAHT needs torotate by 2 (2π/180). This choice is based on the former work by Ma et al. [69] whoinvestigated the power performance of a high solidity vertical axis wind turbine (VAWT).Ma et al. [69] tested three time steps corresponding to the time that their VAWT neededto rotate by 2π/90 (∆t1), 2π/180 (∆t2) and 2π/360 (∆t3). They got identical results using∆t2 and ∆t3, such that ∆t2 has been selected in the present case for its good trade-offbetween accuracy and computational cost.The calculations are initialized using a converged steady-state multiphase RANS calcu-lation and are run using the Mammouth cluster MP2 provided by Calcul Québec. TheCPU time for the multiphase URANS calculations at a given λ value is approximately 20days using 90 processors (AMD Opteron 6172). Approximately 8 TSR values are obtainedper configuration. More than 25 rotations of the turbine are generally necessary to geta converged solution. Convergence is reached when i) the total torque deviation betweentwo subsequent revolutions gets below 0.5% ; ii) when all residuals are lower than 10−8 ;and iii) when the mass imbalance is lower than 10−6. The power and torque coefficientsare then averaged over three additional revolutions after reaching convergence.

4.6 Validation of the flow solver

The validation of the multiphase numerical approach is performed in two steps : i) valida-tion of the multiphase VOF method ; ii) validation of the turbulence closure model k − ωSST.

The validation of the VOF model is achieved by comparing the present predictions with theexperimental data of Koshizuka et al. [57] for the breaking of a dam. The authors studiedthe fall of a water column under the influence of gravity inside a reservoir. Initially, the



(a)

(b)

Figure 4.3 (a) Example of a 2D view of the mesh distribution for configuration1 (Fr = 0.625) ; (b) 2D views of the numerical mesh distribution in the rotorregion and around the blades.

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Figure 4.4 VOF validation against the experimental data of Koshizuka et al.[57]. (a) Normalized evolution of the water level H∗ versus time t∗. (b) 2D sketchof the configuration.

4.6. VALIDATION OF THE FLOW SOLVER 69

water column in their experiments was supported on the side by a vertical plate, andis then drawn up rapidly at t = 0 s. The free surface location is recorded regularly totrack the evolution of the water column. Numerically, the experimental configuration hasbeen reproduced using an unstructured mesh with 1.2 million cells. In order to captureaccurately the evolution of the water column fall versus time, a time step of 0.012 s isselected. Figure 4.4 shows the normalized evolution of the water level H∗ versus time t∗

(t∗ = t√g/H). The water level is normalized by the initial water column height (H = 0.292

m). An excellent agreement is observed regarding the temporal evolution of H∗ betweenthe VOF method and the experimental data, with an average discrepancy of 0.6% whichremains within the experimental error bar. The maximum error of 1% was observed att∗ = 0.7.

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

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er C

oeffi

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]

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Figure 4.5 Performance curve of a three-blade Darrieus vertical axis wind tur-bine (VAWT). Comparison between the present predictions and the experimen-tal results of Castelli et al. [24].

The turbulence closure model k−ω SST has been validated in a former work [14] againstthe experimental data of Castelli et al. [24] for a three-blade VAWT. The authors studiednumerically and experimentally the H-rotor Darrieus wind turbine under a 9 m/s wind.The rotor diameter and the blade chord were fixed at 1030 mm and 85.8 mm, respectively,and the blade profile is a NACA 0021 profile. Figure 4.5 shows the Cp distribution fordifferent tip speed ratios λ. In the present case, contrary to Castelli et al. [24], the blockageeffect is taken into account. The present CFD results show an excellent agreement with the



experiment values of Castelli et al. [24], and a more accurate prediction of the maximumCP ’s location around λ = 2.6 compared to Castelli et al.’s numerical results [24]. Thehighest discrepancy with the experiments is about 13% for λ = 2.5. This good agreementwith the published experimental data is due to the increased accuracy of the low-Reynoldsnumber formulation of the k − ω SST model [28, 37] and the high spatial resolutionespecially in the blade region where the average wall coordinate is around y+ = 0.87.The comparisons with the experimental results of Koshizuka et al. [57] and Castelli et al.[24] show the ability of the present model to accurately predict the free surface deformationand the turbulent flow around rotating blades, respectively. Therefore, the present modelcan be used confidently to investigate the interactions between the free surface and theDHAHT.


4.7.1 General performances

In this subsection, the global performances of the DHAHT in terms of CP and CT coeffi-cients are compared for configurations 1 and 2. In addition, the influence of the free surfaceis investigated by reproducing configuration 2 using a single-phase model, for which thefree surface is replaced by a slip wall condition. The values of CP , CT and λ are calculatedby two different ways : (i) including a correction due to the blockage effect through thecorrected velocity UC ; (ii) without correction by using the average (in time and space)streamwise velocity U0. The CP and CT values with correction are more realistic, dueto the high blockage ratio in configurations 1 (0.7) and 2 (0.55). The without correc-tion values are included here for comparison as several previous published studies presentincorrected values which can lead to an overestimation of CP and CT (e.g., [76]). Theincoming flow is characterized by a Reynolds number based on the DHAHT’s diameterReD = UcD/ν = 80.1×104 (resp. 80.7×104) and a Froude number Fr = Uc/

√HWg = 0.62

(resp. 0.55) for configuration 1 (resp. 2).

Figure 4.6 shows the distribution of the averaged power coefficient CP for the differentconfigurations. For configurations 1 and 2, the CP and λ values are calculated with (w/)and without (w/o) using the blockage correction. Only corrected values are presented forthe single-phase (slip wall) approach, because CP exceeds 1 if the correction is not applied.The different curves have the same general shape, where the three standard main regionsare distinguishable : dynamic stall, transition and high λ regions. The same distributionwas also observed by Hashem et al. [41] for a DVAWT. The application of the correction


factor shifts the curves towards lower λ values and decreases CPmax by 45% and 32% forconfigurations 1 and 2, respectively. The curve displacement is due to a higher UC valuecompared to U0.

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Configuration 1 w/o correction



Figure 4.6 Distributions of the averaged power coefficient CP as a functionof the tip speed ratio λ for configurations 1, 2 and the single-phase (slip wall)case. The multiphase configurations are calculated with (w/) and without (w/o)correction.

A difference of 42.4% is observed between the single-phase case (CPmax = 0.33) and theconfiguration 2 (w/) (CPmax = 0.19). In addition, the position of the CPmax peak is shiftedfrom λ = 2.3 to λ = 1.8 between the single-phase case and configuration 2 (w/). The CP

loss is due to the multiphase modeling, where a strong deformation of the free surfaceis observed, which affects the flow around the blades. Modeling the free surface has animportant impact in the prediction of the DHAHT’s CP distribution. The overpredictionof the CP values by a single-phase model has already been reported for a HAHT by Nishiet al. [89], where the single-phase modeling overestimated by 38% the experimental andthe multiphase CP values.Decreasing the water level HW (configuration 1) leads to a decrease by 36.8% for themaximum power coefficient (CPmax = 0.12) compared to configuration 2, and a shift of itslocation to λ = 2.1. For λ = [3 − 4], identical CP values are obtained for both configura-tions. However, above λ = 4, the DHAHT in configuration 1 is more efficient.



The averaged torque coefficient CT curves for both configurations as shown in Figure4.7 show similar trends as for the CP comparisons. For λ = 1.8, a loss of 26.6% is dis-tinguishable between the single-phase case (CTmax = 0.15) and configuration 2 (W/)(CTmax = 0.11). For configuration 1 (w/) at lower λ values (λ < 2), the DHAHT pro-duces a minimum torque coefficient (CT = 0.075) for λ = 1.8, which represents a loss of32% compared to configuration 2. Similar to the CP distributions, configurations 1 and 2provide similar CT values in the region λ = [3− 4], and configuration 1 gets slightly moreefficient for λ > 4. Thus the water level (HW ) has no significant impact on the CP and CT

distributions in the region λ = [3− 4]. However, in deeper water, the maximum values ofCPmax and CTmax increase. Similar trends were observed for HAHT by Yan et al. [130].CPmax (resp. CTmax) increases from 0.39 (resp. 0.85) in shallow tip immersion (0.19D) to0.42 (resp. 0.88) in deep tip immersion (0.55D), respectively.

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Figure 4.7 Distributions of the averaged torque coefficient CT as a function ofthe tip speed ratio λ for configurations 1, 2 and in the single-phase (slip wall)case.

Positive CP are reported for the DHAHT in configuration 1 (w/) (Fig.4.6) for λ < 3.5,though its maximum remains 36.8% lower than CPmax obtained for configuration 2. TheDHAHT’s ability to produce power in configuration 1 is due to the backwater inducedre-immersion of the whole turbine when rotating, despite the initial conditions, whereonly three quarters of the turbine is submerged. In order to understand the mechanisms


of the flow variations around the DHAHT, snapshots of the instantaneous water volumefraction α field are extracted for λ = 1.8 and at various rotor azimuthal positions (Fig.4.8).URANS runs are initialized by the water level HW = 0.65m (Fig.4.8a) and a hydraulicpressure profile.

The major free surface adjustments occur during the primary rotation. At the first halfof a rotation, the blade splashes the free surface and generates a wave. In the immediateaftermath, the second blade hits the generated wave, and the third blade follows justafter (Fig.4.8b). When the DHAHT starts to rotate, the free surface is unable to reco-ver the initial water level HW , due to the high tangential velocity of the turbine (3.28m/s=2.07×U0) compared to the incoming flow velocity U0. At the end of the first rota-tion, blade 1 is partially submerged (Fig.4.8c). However, beyond the second rotation, theDHAHT is completely submerged, and a bump in the water surface is formed above therotor (Fig.4.8e). As the DHAHT rotates, the bump advects upstream as a wave (Fig.4.8dto f). The generated wave continues to progress upstream until the inlet region, wherean opening condition is imposed in order to allow its evacuation from the calculatingdomain.The same wave generation mechanism is observed in configuration 2 (Fig.4.9b).The extent of the elevated stage upstream of the DHAHT is partially due to the artificiallateral confinement of the model and is likely much greater than it would be in a naturalsetting where flow could move to the sides of the turbine. For configuration 1, at the firstblade 1’s splash, an air pocket is attracted underneath by the blade motion (Fig.4.8b).The air pocket is fragmented progressively by interacting with the blades in multiple airbubbles and then advected downstream in the wake of the DHAHT (Fig.4.8e,f). Downs-tream of the DHAHT, a continuous decrease in the water level is observed while the rotoris rotating (Fig.4.8d to f).

Figure 4.9 displays the distribution of the instantaneous water volume fraction α after 15revolutions for both configurations. The water level continues to decrease downstream ofthe DHAHT in configuration 1, and a comparable distribution is observed in configuration2. A decrease of water level in the wake was also observed for a Savonius HAHT byNakajima et al. [87] and a HAHT by Nishi et al. Overtop of the turbine, the water levelincreases locally in both configurations, (Fig.4.9a,b), showing the importance of the freesurface modeling in order to correctly model the flow.

The deformation of the free surface considerably influences the produced CP and CT

coefficients, which are directly related to the flow around the blades. In order to gain a morefundamental understanding, the instantaneous vorticity fields obtained for configurations 1and 2 and the single-phase case at specific azimuthal blade positions θ = 90−150−210



(a) θ = 0 (t=0s) (b) θ = 180 (t=0.2142s)

(c) θ = 360 (t=0.4285s) (d) θ = 540 (t=0.6428s)

(e) θ = 720 (t=0.8571s) (f) θ = 1080 (t=1.2857s)

Figure 4.8 Instantaneous snapshots of the water volume fraction α around theDHAHT during the first 3 rotations. Results obtained for λ = 1.8 in configura-tion 1.

(a) Configuration 1 (b) Configuration 2

Figure 4.9 Instantaneous snapshots of the water volume fraction α around theDHAHT at the 15th rotation. Results obtained for λ = 1.8 in configurations 1and 2.


are shown in Figure 4.10. The three azimuthal positions are selected according to theauthors’ previous investigation [14] without a free surface, where when looking at a singleblade, at θ = 90 a maximum torque CT is produced, at θ = 150 the inner side of the bladeis completely separated, and at θ = 210 the leading edge vortex is shed. At θ = 90, thevorticity fields are comparable whatever the water level and the presence of the free surface(fig.4.10 a to c). However, the boundary layer for the single-phase case is slightly thicker inthe trailing edge region. The influence of the free surface deformation in the flow aroundthe blades is clearly distinguishable at θ = 150−210. At θ = 150, 40 % and 50 % of theblade inner side are separated in configurations 1 and 2 (fig.4.10d,e), respectively. Withoutfree surface modeling, the inner side of the blade is completely separated (fig.4.10f). Thisdifference is related to the perturbation of the local relative velocity vector by the freesurface plunging in configuration 1 and 2. The perturbation increases when the blade isclose to the plunging region θ = [180 − 330] (not shown). At θ = 210, the leading edgevortex observed for the slip wall case is replaced by a long shear layer in the wake of theblade in configurations 1 and 2, (fig.4.10g to i). The wake shear layer in configuration 1 isuniform and follows the blade motion (arc). However, in configuration 2, the shear layer isscattered and little uniform pattern is distinguishable. In an animation (not shown here),the wake shear layer in both configurations is transformed into a periodic vortex shedding,created by a combination of the underneath flow suction and the blade rotating motion.

The polar distributions of the total instantaneous torque coefficient CT for different tipspeed ratios λ are presented in Figure 4.11. The total torque coefficient CT is obtainedby summing the contribution of each blade. A comparable polar distribution is observedbetween configurations 1 and 2, which corresponds to a 3 lobed rosette. The location ofthe three lobes is identical in all configurations : θ = 90, 210 and 330. Similar polardistributions were observed in a previous single-phase approach [14] and for the DVAWTinvestigated by Mohamed [84]. An important decrease in CT in configurations 1 and 2is distinguishable with increased tip speed ratio λ (fig.4.11a to c). At λ = 4, the polardistributions of configuration 1 and 2 are approximately identical, which is consistent withthe averaged CT curve (Fig.4.7).

4.7.2 DHAHT’s influence on flow regime and momentum loss

In order to further investigate the influence of the DHAHT on the flow regime, the Froudenumber Fr is presented downstream at X/D = 11D (from the center of the rotor) forvarious λ values (fig.4.12). Fr is calculated by using the local average (in space andtime) mean velocity and water level. Positive relationships are observed between λ and



(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 4.10 Instantaneous vorticity fields around the blade for θ = 90 (a,b,c),150 (d,e,f) and 210 (g,h,i). Comparisons between configuration 1 (a,d,g), 2(b,e,h) and the single-phase (slip wall) case (c,f,i). Results obtained for λ = 1.8.


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Figure 4.11 Comparison of the polar distributions of the instantaneous torquecoefficient CT for configurations 1 and 2 for four tip speed ratios λ.



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Configuration 1Linear fit configuration 1Configuration 2Linear fit configuration 2

Figure 4.12 Froude number Fr distributions downstream (X/D = 11D) theDHAHT for configurations 1 and 2, with the corresponding linear regressions.


Fr for both configurations (fig.4.12). The trend appears quite linear, with coefficients ofdetermination of R2 = 0.91 and 0.90, for configuration 1 and 2, respectively. The slopeof the regression in configuration 2 (m = 0.05) is lower than that for configuration 1(m = 0.5), where Fr increases drastically with λ. In configuration 1, the flow regimedownstream switches from sub-critical to super-critical at λ = 1.8. For λ > 1.8, the flowis super-critical and Fr continues to increase until a maximum value Fr = 2.6 for λ = 5.The flow in configuration 2 is always super-critical and a maximum Fr = 2.15 is achievedfor λ = 5. The transition between the sub-critical regime at the inlet and the super-critical regime downstream, in both configurations, is due to the restriction imparted bythe rotor. When the flow passes over the DHAHT, it is forced to plunge and passes throughthe critical water level (Fr = 1), followed by a switch to the super-critical regime.

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LM

Configuration 1Configuration 2Linear fit configuration 1Linear fit configuration 2

Figure 4.13 Distributions of the momentum loss LM for configurations 1 and2, with the corresponding linear regressions.

The quantification of the linear momentum loss LM caused by the presence of the DHAHTin shallow water is important. The averaged value of LM (kg.m/s2) is evaluated betweenthe inlet and the outlet of the domain as follows :

LM =ρg

2(Hw.inAw.in −Hw.outAw.out) + ρ Q(U0 − Uw.out) (4.10)



whereHw.in = HW andHw.out are the water depths at the inlet and the outlet, respectively.Aw.in/out is the cross-sectional area at the inlet or the outlet, and Q is the water flowrate(m3/s). U0 and Uw.out are the average (in time and space) velocity magnitude at the inletand outlet. The distributions of LM for configurations 1 and 2 are displayed presented inFigure 4.13 as a function of the tip speed ratio λ.

Clear trends between LM and λ are observed in both configurations. Linear regression R2values are 0.29, 0.23 for configurations 1 and 2, respectively. The evolution of LM followsa slope of 0.03 and 0.02 for configurations 1 and 2, respectively. In configuration 2, the LM

values are on average × 2.18 higher than in configuration 1, due to a deeper inlet waterlevel. For λ = [1.4−3.5], LM is a combination of the drag and the extracted momentum ofthe DHAHT. Above λ = 3.5 the drag forces are more dominant, and contribute essentiallyto the momentum loss LM .

4.8 Conclusion

This paper reports numerical results of the interactions between the free surface and aDarrieus horizontal axis hydrokinetic turbine (DHAHT), using a multiphase unsteadyflow solver based on the k − ω SST turbulence model and the volume of fluid (VOF)approach. The influence of the free surface on the power and torque coefficient, CP andCT , prediction was investigated and compared to the single phase (w/o free surface) re-sults. In addition, the influence of the water level has been investigated by testing twoconfigurations : a partially and totally submerged DHAHT. The streamwise flow condi-tions are characterized by a Froude number Fr = 0.62 (resp. 0.55) and Reynolds numberReD = 80.1× 104 (resp. 80.7× 104) for configuration 1 (resp. 2).The numerical approach has been validated in two steps against experimental resultsavailable from the literature. The validation of the VOF model has been achieved by com-parison with the experimental results for the break of a dam obtained by Koshizuka et al.[57]. The evolution of the normalized water level H∗ curve was reproduced with a maxi-mum discrepancy of 1% compared to the experimental values. The k − ω SST turbulencemodel has shown its ability to accurately predict the power coefficient CP distribution fora Darrieus vertical axis wind turbine. The maximum discrepancy is 13% compared to themeasurements of Castelli et al. [24] and the present predictions improve significantly theprevious numerical results of Castelli et al. [24].The case without free surface provided CP and CT values higher by 42.4% and 26.6%,respectively, compared to configuration 2. The totally submerged DHAHT (configuration2) produced maximum CP = 0.19 and CT = 0.11 values compared to configuration 1

4.8. CONCLUSION 81

(CP = 0.12, CT = 0.075). The high counter clockwise rotation of the DHAHT in configu-ration 1 causes a significant blockage and forces the flow to overtop the turbine despite theinitial water level submerging the turbine by only 3/4. The overprediction of CP and CT

by the single phase model is essentially due to the disregard of the important free surfacedeformation around the DHAHT. The free surface plunging influences directly the flowaround the blade, where the leading edge vortex in the single-phase approach without freesurface was replaced by a long shear layer in the blade wake.A CT polar plot distributions of a 3 lobed rosette was observed for all configurations,with an identical location of the three lobes (θ = 90, 210, 330). A linear trend of λwith Froude number Fr was distinguishable downstream the DHAHT. The flow regimeswitches from sub-critical at the inlet to super-critical downstream for both configurations1 and 2 over most values of λ with the exception of λ = [1.4−1.8] for configuration 1. Themomentum loss LM was also quantified between the inlet and the outlet, where higher LM

values were recorded in configuration 2 (LM=0.76 kg.m/s2) compared to configuration 1(LM=0.32 kg.m/s2). For low λ = [1.4− 3.5] values, LM is a combination of the DHAHT’sdrag and the extracted power. At higher tip speed ratios, the drag forces become predo-minant.Future work should investigate the interactions between the deformable free surface andthe whole DHAHT including its support using large eddy simulation to show in detailthe deformation of the free surface and the blade vortex shedding. Furthermore, the ap-plication of the Proper Orthogonal Decomposition (POD) method around the blade andin the turbine wake would allow for a better understanding the different modes and theirinteractions.

Acknowledgements

All calculations have been done using the computational resources of the Compute Canadanetwork, which is here gratefully acknowledged. A.E.B.L., S.P. and J.L. would also liketo thank the Natural Sciences and Engineering Research Council of Canada (NSERC) fortheir financial support (Discovery Grant RGPIN-2015-06512 and RGPIN-2017-31147).



CHAPITRE 5

CONCLUSION FRANÇAISE

Dans cette thèse, plusieurs simulations numériques ont été menées pour des écoulementsturbulents autour d’obstacles fixe puis tournant. Dans le Chapitre 2, un benchmark demodèles de turbulence a été accompli sur l’écoulement turbulent autour d’un obstacle enforme de D submergé caractérisé par un nombre de Reynolds égal à ReH = 17448. Deuxmodèles de sous mailles LES ont été comparés à des résultats expérimentaux obtenus parPIV. Concernant l’obstacle tournant, deux études numériques ont été faites sur une turbinede type Darrieus à axe horizontal DHAHT. Dans le Chapitre 3, 4 profils de pales, S1046,S809, FXLV152 et NACA0018, ont été testés dans un écoulement monophasique caractérisépar ReD = 711000 et Fr = 0.56. Le modèle de turbulence choisi pour cette étude a étésélectionné à partir du premier benchmark numérique. Le but de cette étude paramétriqueest d’accroître les coefficients de puissance CP et de couple CT de la turbine DarrieusDHAHT. Par la suite, l’influence du nombre des pales a été étudié sur la turbine équipéedu profil S1046, qui est le plus performant. Dans le chapitre 4, des calculs multiphasiquesont été accomplis sur l’interaction de la surface libre et la turbine Darrieus. Deux hauteursd’eau équivalentes à Fr = 0.62 et 0.55 ont été testées. L’influence de la hauteur d’eau surles coefficient CP et CT et ainsi que sur le nombre de Froude en aval et sur les pertes dela quantité de mouvement a été quantifiée.

Dans le deuxième chapitre, la plupart des modèles de turbulence RANS testé reproduisentla zone de recirculation et la couche de cisaillement qui se forment en aval et au-dessus del’obstacle, respectivement. Néanmoins, les longueurs de réattachement restent différentesentre les modèles et les données expérimentales. Deux points peuvent être conclus surle choix du modèle de turbulence et l’utilisation ou non d’une loi de paroi : le premierpoint concerne la modélisation de la dissipation, où l’approche basée sur le calcul de ωest plus pertinente que celle avec ε car cette dernière ne prend pas en considération lemouvement rotationnel des particules fluides ce qui engendre une sur production de k(Fig.2.3). Le deuxième point concerne la modélisation de l’écoulement proche de la paroiet l’utilisation ou non d’une loi de paroi. L’approche à bas nombre de Reynolds préditmieux le tourbillon en fer à cheval qui se forme au bas en amont de l’obstacle. Cette bonneprédiction est visible sur le champ de v∗. D’autre part, l’approche à bas nombre de Reynoldsa permis aussi d’éviter l’apparition d’une zone de surprediction de v∗ sur la face arrière de

83

84 CHAPITRE 5. CONCLUSION FRANÇAISE

l’obstacle (Fig.2.3). Parmi les modèles de turbulence RANS, le modèle k-ω SST est celui quipermet d’avoir une meilleure prédiction globale. Néanmoins aucun modèle est totalementfidèle aux résultats PIV. Concernant la modélisation des grandes échelles (LES), les deuxmodèles de sous mailles ont permis une meilleure prédiction des différents profils parrapport à la PIV. Le modèle Wale a permis une meilleure prédiction par rapport au modèleSmagorinsky, qui est trop dissipatif. Ces deux modèles ont prédit correctement la formationdu tourbillon en fer à cheval qui se forme au pied de l’obstacle. Le tourbillon se fait advecterpar l’écoulement et contourne l’obstacle pour interagir par la suite avec les structurescohérentes générées par la couche de cisaillement, pour donner naissance à de plus petitesstructures tourbillonnaires. Un décalage dans la position d’interaction est observé entreles deux modèles LES, où le modèle Smagorinsky predit une position d’interaction plusproche de l’obstacle que le modèle Wale. L’analyse spectral n’a pas permis de distinguerdes pic caractérisant une fréquence de lâcher tourbillonnaire pour les différentes positionsX/D en aval de l’obstacle. L’inexistance de ces pics est essentiellement due a une pertede la périodicité suite à l’interaction entre le tourbillon en fer à cheval, les structurescohérentes issues de la couche de cisaillement et le lit. Par contre, la courbe théorique en−5/3 dans la zone inertielle est retrouvée par la PIV et la LES. L’analyse POD a permisde révéler que le mode le plus énergétique est représenté par la zone de recirculation et lacouche de cisaillement, et les autres modes représentent plutôt les structures cohérentes.Cette analyse a permis aussi de mettre en évidence le mécanisme de pulsation de la zonede recirculation en aval de l’obstacle.

Dans le troisième chapitre, le modèle de turbulence k-ω SST à bas nombre de Reynoldsa été utilisé pour simuler l’écoulement autour de la turbine Darrieus à axe horizontal. Cemodèle a été validé avec 13% d’erreur sur le coefficient CP expérimental obtenu pour uneéolienne de type Darrieus verticale équipée de trois pales NACA0021 par Castelli et al. [24].Le modèle améliore également nettement les résultats numériques antérieurs de Castelliet al. [24] et Mohamed et al. [41, 84]. Cela confirme le bon choix du modèle de turbulence etla nécessité de modéliser l’écoulement en proche paroi. Concernant l’hydrolienne DarrieusHAHT à trois pales, les profils S1046 et NACA0018 sont les plus performants dans larégion de décrochage dynamique à faibles valeurs de λ. La valeur maximale de CP = 0.33

a été produite par le profil S1046 pour λ = 2.3. Par contre pour des valeurs plus grandesde λ > 3.5, le profil FXLV152 est celui qui offre les meilleures valeurs de CP et CT .Contrairement à ce qui a été observé pour le profil S809 sur l’éolienne Darrieus verticale[84], pour l’hydrolienne, ce profil produit les plus basses valeurs de CP et de CT pourtoutes les valeurs de λ. Cette performance modeste est reliée directement à l’écoulementautour des pales, où une séparation précoce de la couche limite a été observé sur l’intrados

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à θ = 90, suivi par la formation d’un large tourbillon de bord d’attaque à θ = 150, et quioccupe une large région de l’intrados. En plus, un tourbillon secondaire est généré suite aupassage du tourbillon de bord d’attaque dans la région du bord de fuite, ce qui expliquecette perte de performance par rapport aux autres profils où un plus petit tourbillon debord d’attaque est observé. Les traces du lâcher tourbillonnaire sur le profil de U∗ dansle sillage à X/D = 1 sont observées sous forme de petites ondulations à y/H = 0.5.L’influence du nombre de pales sur la Darrieus HAHT équipée avec du profil S1046 arévélé que pour N = 2 les valeurs maximales de CP et CT sont produites à des grandesvaleurs de λ. Par contre, pour N = 4, le contraire est observé. Pour 2.1 < λ < 2.6,la configuration avec trois pales fournit la meilleure performance. Néanmoins, la valeurmaximale de CP = 0.34 a été produite lorsque N = 2 et pour λ = 3. La distribution polairedu coefficient de couple CT varie avec le nombre de pales. Pour N = 2, la distribution estdipolaire avec deux lobes situés à θ = 90 et 270. Pour N = 3, la distribution prend laforme d’une rosette avec trois lobes à θ = 90, 210 et 330. Pour terminer lorsque N = 4,le profil est plutôt pseudo-circulaire avec aucun lobe apparent.

Dans le quatrième chapitre, des simulations URANS multiphasiques ont été accompliespour étudier l’interaction de la Darrieus HAHT et la surface libre. Deux hauteurs d’eauont été considérées correspondant à une turbine partiellement ou totalement submergées(nombres de Froude Fr = 0.62 et 0.55). La surface d’eau a été modélisée en utilisant lemodèle VOF qui a été validé dans le cas de la rupture d’un barrage. Une erreur moyennede 0.6% est observée entre les simulations et les mesures expérimentales de Koshizukaet al. [57] en termes d’hauteur d’eau normalisée H∗. Dans cette étude, le modèle de tur-bulence k-ω SST a été choisi suite aux résultats des deux chapitres précédents. L’approchemonophasique du troisième chapitre surestime de 42.4% le coefficient de puissance CP etde 26.6% le coefficient de couple CT calculés par l’approche multiphasique dans le castotalement immergé. Cette surestimation est due au modèle monophasique qui ne prendpas en compte l’importante déformation de la surface libre causée par la turbine dontl’effet potentiel est important. Cette déformation est caractérisée par une décroissance dela hauteur d’eau en aval de la turbine, et l’apparition d’une vague. Cette vague remontel’écoulement et engendre l’augmentation de la hauteur d’eau localement. La déformationde la surface libre influence également l’écoulement autour des pales où le tourbillon debord d’attaque observé dans le chapitre 3 est remplacé par une longue couche de cisaille-ment dans le sillage de la pale. Cette dernière est par la suite transformée en un tourbillonsuite à la combinaison de l’aspiration de l’écoulement primaire et le mouvement de la pale.Le même mécanisme est observé dans les deux configurations. La turbine submergée estplus performante avec CP = 0.19 et CT = 0.11, ce qui est à comparer aux valeurs obtenues


CP = 0.12 et CT = 0.075 pour la turbine partiellement immergée. Pour une hauteur d’eauinitiale où les trois quarts du diamètre sont submergés, la hauteur d’eau évolue durant larotation de la turbine et converge vers un submergement complet de la Darrieus HAHT.Comme la vitesse tangentielle des pales est 2 fois plus grande que la vitesse de l’écoule-ment U0, l’eau est forcée de passer au-dessus de la turbine. Pour les deux configurations,les distributions polaires du coefficient CT ressemblent à une rosette avec trois lobes situésà θ = 90, 210 et 330. Un changement dans le régime d’écoulement est observé suite aupassage à travers la turbine. Le régime d’écoulement dans la configuration 2 en aval dela Darrieus HAHT est toujours torrentiel indépendamment de la valeur de λ. Le mêmerégime d’écoulement est observé dans la configuration 1, sauf dans la région λ = [1.4−1.8]

où l’écoulement est fluvial. Une régression linéaire a été appliquée sur la distribution dunombre de Froude Fr en fonction de λ dans les deux configurations, la pente de régressionest plus grande dans la configuration 1 (0.5) que dans la configuration 2 (0.05). La quan-tification de la quantité de mouvement perdue LM entre l’entrée et la sortie du domainepermet de mettre en évidence les forces de frottements dues à la turbine et la puissanceextraite par cette dernière. Pour des faibles valeurs de λ = [1.4 − 3.5], LM représente lamajeure partie de la puissance extraite par la turbine Darrieus HAHT. Au-delà de cesvaleurs (λ > 3.5), ce terme est plutôt dominé par les forces de frottement. Les valeurs deLM sont plus importantes lorsque la turbine est totalement submergée.

Perspectives : Pour compléter ce travail, deux perspectives de recherche sont suggéréesci-dessous :

• Une campagne expérimentale de la turbine Darrieus HAHT en entier est en coursde préparation dans un grand canal hydraulique (6× 0.5× 0.45 m3) du départementde génie civil afin d’investiguer l’influence des effets 3D sur les performances dela turbine. Le couple généré par la turbine à différentes vitesses de rotation seramesuré. Plusieurs hauteurs d’eau seront testées afin de confirmer les conclusions duchapitre 4. Différents diamètres seront testés pour étudier l’influence de la soliditéσ. Le champ de vitesse en aval de la turbine sera étudié en utilisant l’ADV (AcousticDoppler Velocimetry) puis par la suite la PIV (Particle Image Velocimetry).• Une série de simulations 3D URANS multiphasiques de la turbine complète est

envisagée afin de comparer les valeurs de CP et CT avec les résultats expérimentaux.Par la suite, une comparaison avec les calculs 2.5D est nécessaire afin de mettre enévidence les possibles effets 3D (structures tourbillonnaires qui se forment le long del’envergure des pales).

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• Des calculs 3D LES sur la turbine Darrieus HAHT permettraient une meilleureconnaissance de l’hydrodynamique et notamment de l’apparition de structures cohé-rentes de plus petites échelles. Afin d’optimiser le nombre d’éléments, qu’une partiedu canal en amont, et au moins 8 D en aval du rotor sera résolue. La valeur deλ choisit correspondra au point de fonctionnement optimal issu du calcul URANS3D et validé par l’expérience. Ainsi, les valeurs de CP et CT du calcul LES serontcomparées aux résultats URANS et aux expériences. L’analyse POD selon plusieursplans (XY,XZ) pourra être faite dans le sillage de la turbine Darrieus HAHT, pourdistinguer les différents modes de lâchers tourbillonnaires. Elle pourra être appliquéeégalement sur un champ plus étroit autour d’une pale pour comprendre plus en détaille mécanisme de génération des différents tourbillons.• Examiner les performances d’un groupe de turbines placé dans une ferme. Les critères

de performance à l’échelle d’une seule turbine (CP et CT ) doivent être remplacés pardes grandeurs plus globales qui représentent les performances de toutes les turbinesde la ferme. Par la suite, plusieurs paramètres peuvent être étudiés : espacemententre les turbines, positionnement dans le canal . . .• À cause des conditions hivernales extrêmes au Québec (qui représentent jusqu’à 6

mois de l’année), il peut s’avérer intéressant d’étudier l’impact du frasil (couche deglace superficielle qui se crée sur la surface libre) sur les performances de l’hydro-lienne. Pour cela, une machine à coulis de glace disponible à l’UdeS sera utilisée pourproduire le frasil et faire varier l’épaisseur de cette couche.• Finalement, une étude future focalisera sur l’impact d’une seule turbine Darrieus

HAHT ou d’une ferme de turbines sur l’environnement (habitat des poissons etgrenouilles). En effet, avant de commercialiser une turbine, une étude environne-mentale est nécessaire portant notamment sur l’impact d’une turbine sur le passagedes poissons (taux de blessure ou de mortalité). En deuxième partie, les risques d’in-nondation causée par la ferme de turbines doivent être analysés car elles engendrentune augmentation de la hauteur d’eau localement en amont et donc un risque dedébordement.


CHAPITRE 6

ENGLISH CONCLUSION

In this PhD thesis, several numerical simulations were conducted for the turbulent flowsover fixed then rotating obstacles. In Chapter 2, a benchmark of RANS (Reynolds Avera-ged Navier-Stokes)turbulence models was first performed on the turbulent flow around asubmerged D-shaped obstacle characterized by a Reynolds number equal to ReH = 17448.Two LES (Large Eddy Simulation) subgrid scale models were compared and validatedagainst experimental PIV (Particle Image Velocimetry) measurements. Concerning therotating obstacle, two numerical studies were done on a Darrieus HAHT water turbine. InChapter 3, 4 blade profiles, namely S1046, S809, FXLV152 and NACA0018, were testedin a channel under turbulent flow conditions at ReD = 711000 and Fr = 0.56. The turbu-lence model chosen for this study was selected based on the previous numerical benchmarkfor the D-section obstacle. The goal of the Darrieus HAHT (horizontal axis hydrokineticturbine) study was to optimize blade shape in order to increase the CP and CT coefficients.Subsequently, the influence of the number of blades was studied on the Darrieus HAHTequipped with the most efficient profile (S1046). In Chapter 4, multiphase calculationswere performed on the interactions between the free surface and the Darrieus HAHT. Twowater levels equivalent to Fr = 0.62 and 0.55 were tested. The influence of the waterlevel on the CP and CT coefficients, the Froude number and on the momentum losses wasquantified.

In Chapter 2, a benchmark of different RANS turbulence closure models and LES sub-grid scale models was performed around a D-shaped bluff body. The results show thatmost RANS turbulence models reproduced the recirculation region and the shear layerformed downstream and above the fixed D-shape obstacle, respectively. Nevertheless, thereattachment distance was not consistent between these models and the PIV results usedas validation. Two points can be concluded on the choice of the turbulence model andthe near wall treatment : 1) the turbulence model which calculates ω is more accuratethan that of ε because the latter does not take into account the rotational motion of thefluid particles, which induces an over production of k (Fig.2.3) ; 2) concerns the modelingof the near-wall flow explicitly (low Reynolds number) or the use of the law of the wall(high Reynolds number). The low Reynolds number approach was found to be far better atpredicting the horseshoe vortex that forms on the stoss side of the D-section as can be seen

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90 CHAPITRE 6. ENGLISH CONCLUSION

on the v∗ fields where the negative region corresponds to horseshoe vortex location. Thelow Reynolds number approach also avoided the overprediction of v∗ on the rear face ofthe D-section (Fig.2.3). Amongst the RANS turbulence models, the k-ω SST (shear stresstransport) model allowed for a better prediction of the overall flow field, yet, no model wastotally faithful to the PIV results. The two LES sub-grid scale models allowed a betterprediction of the different profiles, where the Wale model was closer to PIV results thanthe Smagorinsky model. Both models correctly predicted the horseshoe vortex that formsat the stoss side of the obstacle. The models show that the horseshoe vortex is advected bythe flow and passes to the sides of the obstacle to interact subsequently with the coherentstructures generated by the shear layer giving rise to smaller secondary vortex structures.An offset in the downstream distance to the interaction region was observed between thetwo LES models, where the Smagorinsky model predicted it closer to the obstacle than theWale. Contrary to what was expected, the autospectra of the velocity components u andv, revealed no distinct peak characterizing a vortex shedding for the different streamwiseX/D positions downstream the D-section. The absence of a peak is mainly due to a loss ofperiodicity following the interactions between the horseshoe vortex, the coherent structuresreleased from the shear layer and the bed. On the other hand, the theoretical −5/3 curvein the inertial region was recovered by the PIV and LES results. The POD analysis hasrevealed that the most energetic modes are represented by the recirculation region andthe shear layer, while the other modes represent the coherent structures in the wake. Thisanalysis highlighted the pulsation mechanism of the recirculation region downstream theobstacle.

In Chapter 3, the low Reynolds number k-ω SST turbulence model, which gave reasonableresults in the previous chapter, was used to model the flow around a Darrieus HAHT. Thelow Reynolds number k-ω SST model was validated with a discrepancy of 13% against theexperimental CP values of a 3 bladed NACA 0021 Darrieus VAWT obtained by Castelliet al. [24]. A clear improvement in the numerical predictions was achieved by comparingthe actual results with previous numerical results by Castelli and al. [24] and Mohamedand al. [41, 84]. This improvement supports the choice of the turbulence model and thelow Reynolds number approach used in our study. For the three-bladed Darrieus HAHTturbine, the S1046 and NACA0018 profiles performed the best in the dynamic stall regionfor low tip speed ratios λ. The maximum value of CP = 0.33 was produced by the S1046profile for λ = 2.3. On the other hand, for λ > 3.5, the FXLV152 profile was the mostefficient attaining higher values of CP and CT . Contrary to what has been observed for aS809 profile on the Darrieus VAWT [84], the S809 profile produced the lowest values of CP

and CT for all values of λ. This poor performance is directly related to the flow around the

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S809 profile blades, where an early separation of the boundary layer was observed on theinner side at θ = 90, In addition, the loss in performance is related to a large leading edgevortex formation occurred at θ = 150 and a secondary vortex was generated following thepassage of the leading edge vortex in the trailing edge region. Traces of vortex sheddingwere observed in the vertical, near-wake U∗ profile at X/D = 1 as small ripples (variationsin U∗ ) at y/H = 0.5. The influence of blade number N on the Darrieus HAHT with theS1046 profile revealed that, for N = 2, the maximum values of CP and CT are producedfor large values of λ, whereas, for N = 4, the opposite was observed. For 2.1 < λ < 2.6,the configuration with three blades provided the best overall performance. However themaximum value of CP = 0.34 was produced when N = 2 at λ = 3. The polar distributionof the instantaneous torque coefficient CT varies depending on the blade number. ForN = 2, the distribution is di-polar with two lobes located at θ = 90 and 270. For N = 3,the distribution is a 3-lobed rosette located at θ = 90, 210 and 330. Finally, when N = 4

the polar distribution is rather pseudo-circular with no apparent lobes.

Chapter 4 furthers the work of the previous chapters. Multiphase URANS simulationswere performed on the interactions between the Darrieus HAHT and the free surface.Two configurations 1 and 2, characterized by Froude numbers Fr = 0.62 and 0.55 wereconsidered corresponding to two water levels, totally or partially submerged. The freesurface was modeled using the VOF method which was validated against the experimentalstudy by Koshizuka and al. [57] for the breaking of a dam. An average error of 0.6% wasobserved between the experimental and numerical curves of the normalized water depthH∗. In this study, the low Reynolds number k-ω SST turbulence model was also selectedfollowing the validations in the two previous chapters. The single-phase approach of thethird chapter overestimates by 42.4% the CP values and by 26.6% the torque coefficientCT calculated by the multiphase approach in configuration 2. The single-phase model doesnot indeed take into account the significant deformation of the free surface caused by theturbine, which has a significant effect. This deformation was characterized by a decreasein the water level downstream of the turbine, the appearance of a wave or bulking abovethe turbine, and an increased water level upstream. The deformation of the free surfaceinfluenced the flow around the blades, where the leading edge vortex observed in Chapter3 was replaced by a long shear layer in the wake of the blade. The latter was subsequentlytransformed into a vortex following the combination of the primary flow suction and themovement of the blade. Similar flow structure was observed in configurations 1 and 2. Thesubmerged turbine in configuration 2 was more efficient with CP = 0.19 and CT = 0.11,compared to the first configuration where CP = 0.12 and CT = 0.075. In configuration1, the initial water level condition where three quarters of the diameter are submerged,


evolved during the rotation of the turbine and converged to a complete submergence of theDarrieus HAHT. As the tangential speed of the blades is 2 times greater than the incomingvelocity U0, the water was forced to pass over the turbine. For both configurations 1 and2, the polar distributions of the instantaneous CT was a 3 lobed rosette located at θ = 90,210 and 330. A switch in the flow regime was observed before and after the DHAHT.The flow regime in configuration 2 downstream of Darrieus HAHT was always supercriticalregardless of the λ values. The same flow regime was observed in configuration 1, exceptfor λ = [1.4− 1.8] where the flow was subcritical. Linear regression was applied to the Frdistributions in both configurations, and the regression slope was steeper in configuration 1(0.5) than in configuration 2 (0.05). The quantification of the momentum loss LM betweenthe inlet and the outlet highlighted the contributions of drag and extracted power bythe DHAHT. For low values of λ = [1.4 − 3.5], LM represented most of the momentumextracted by the Darrieus HAHT (based on a comparison of CP values). For λ > 3.5, LM

was dominated by the turbine’s drag. The values of LM were larger when the DHAHTwas placed in configuration 2 because the water level was deeper.

Perspectives : to complement this work, two research perspectives are suggested in thefollowing :

• An experimental study of the entire Darrieus HAHT is being prepared in a largehydraulic channel (6× 0.5× 0.45 m3) in the Civil Engineering Department to inves-tigate the influence of the 3D effects on the performance of the turbine. The torquegenerated by the turbine at different tip speed ratios λ will be measured by a torquemeter. Several water levels will be tested to compare with the results of Chapter 4.Different turbine diameters will also be tested to study the influence of the solidity σ.The velocity field downstream of the turbine will be measured by an ADV (acousticDoppler velocimetry) and using PIV (particle image velocimetry) measurements.• Further multiphase URANS simulations of the entire DHAHT (full 3D geometry)

should be performed and compared with the experimental results. A comparisonwith the 2.5D calculations is necessary in order to determine the affect of 3D turbinegeometry on the CP and CT coefficients. For example, vortex structures that formalong the blade span.• In the future, 3D LES calculations should be performed also on the Darrieus HAHT.

In order to optimize the number of elements, only part of the upstream channel,and at least 8D downstream of the rotor should be solved. The value of λ chosenshould correspond to the optimal operating point resulting from the 3D URANS

93

calculations and validated by the experimental results. Thus, the CP and CT valuesof the LES calculation could be compared to the URANS and experimental results.A POD analysis could be applied on several planes (XY, XZ) in the wake of theDarrieus HAHT, in order to distinguish the different modes of vortex shedding. Theapplication of the POD analysis on a narrower field around one blade will allow usto understand in more detail the generation mechanism of different vortices.• The electrical power generated by a single turbine being rather limited, it could be

interesting to consider a group of DHAHT placed on a farm. The single turbinescale performance criteria (CP and CT ) should be replaced by global quantities thatrepresent the performance of all the turbines on the farm. Thus, several parameterscould be considered (spacing between turbines, positioning on the river), in order toquantify their influences on the global performance.• Due to severe winter conditions in Québec, which correspond to approximately half

a year, it may be of interest to quantify the performance of the DHAHT turbinewhen ice forms at the river surface and frazil is advected by the river current. Toinvestigate this, the ice slurry machine available at UdeS could be used to impose alayer of slush at the top of the water surface and then characterize its impact on theDHAHT performances for different thicknesses of the frazil/slush layers.• A final study could focus on the impact of a single Darrieus HAHT or a group of

turbines on the surrounding environment. They may have a large impact on thebiological environment of the river (e.g., fish habitat, frog survival). For example,to be commercialized, fish mortality due to fish passing accidentally through theDHAHT should be quantified. Furthermore, the added bed roughness and increasedstage during floods caused by a turbine farm should be quantified.


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