Modélisation numérique d’une hydrolienne à axe horizontal de type Darrieus en eau peu profonde

Faculté de génie

Département de génie mécanique

Modélisation numérique d’une hydrolienne à

axe horizontal de type Darrieus en eau peu

profonde

Thèse de doctorat

Spécialité : génie mécanique

Alla Eddine Benchikh Le Hocine

Sherbrooke (Québec) Canada

R. W. Jay Lacey

Sébastien Poncet

Hachimi Fellouah

Julien Favier

Bruce MacVicar

Avec en toile de fond la question du changement climatique, les contraintes énergétiques sont de plus en plus importantes : population et consommation mondiales croissantes, fluc-tuations des prix des énergies fossiles et diminution des ressources disponibles et exigences environnementales toujours plus contraignantes. Dans un contexte mondial où la question énergétique 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’une des sources alternatives les plus exploitées. Parmi ces hydroliennes, les turbines de type Darrieus 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 jamais été étudiée jusqu’à présent. Cette thèse a donc pour objectif d’optimiser une hydrolienne de rivière à axe horizontal de type Darrieus, tant sur le profil hydrodynamique que sur le 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 écoulement turbulent autour d’un obstacle fixe et submergé en forme de D. Une comparaison a été faite entre les résultats numériques et expérimentaux obtenus par PIV (Particle Image Velocimetry) 2D et 3D stéréoscopiques. Le modèle k-ω SST à bas nombre de Reynolds prédit le mieux la couche de cisaillement au-dessus de l’obstacle. Par contre, le modèle k- à haut nombre de Reynolds est plus performant dans la zone de recirculation en aval de l’obstacle. Les résultats produits par la LES Wale sont meilleurs que ceux du modèle Smagorinsky qui s’avère trop dissipatif. L’analyse spectrale ne montre aucun pic distinct 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 en extrayant les différents modes dominants.

Dans la seconde étude sur l’hydrolienne de type Darrieus à axe horizontal, une approche 2.5 D RANS instationnaire a été adoptée en utilisant un maillage très raffiné et le modèle k-ω SST à bas nombre de Reynolds. La turbine est placée dans un canal ouvert sans surface libre. Quatre profils de pales ont été testés. L’approche numérique a été validée avec moins de 13 % d’erreur par rapport aux résultats expérimentaux obtenus sur une éolienne. Le profil S1046 a permis d’accroître les performances produites par le NACA0018. Dans les régions 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 le nombre de pales sur l’hydrolienne de type Darrieus équipée du profil S1046, les meilleures performances ont été produites à basses et hautes vitesses de rotation pour un nombre de pales égal à 4 et 2, respectivement.

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

(configuration 2). L’approche VOF a été validée avec une erreur moyenne de 0.6 % par rapport 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 de la turbine croît progressivement pour des hautes vitesses de rotation. Quand la turbine est complètement submergée, elle extraie les plus grandes quantités de mouvement à des hautes vitesses de rotation. L’aptitude de la turbine à produire de la puissance dans les deux configurations assurera son efficacité dans les rivières peu profondes.

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

Energy constraints are becoming increasingly important as global consumption continues to increase day by day. The price of oil fluctuates continuously tending yet is predicted to increase substantially under more stringent environmental requirements. The idea of turn-ing to and developturn-ing renewable alternatives has become paramount. The electric energy produced by hydrokinetic turbines is one of the most exploited alternative sources. Among these hydrokinetic turbines, the Darrieus vertical axis hydrokinetic turbine DVAHT has been largely considered. On the contrary, the horizontal Darrieus configuration for river applications has never been studied numerically. Moreover, no geometrical optimization of blade profile has been carried out to improve its performances and no research on the influence of the water height has yet been done. To answer these problems, three different studies 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 is first carried out on the turbulent flow around a bed mounted D-section. A comparison is made between the numerical results and 2D planar and 3D stereoscopic PIV (particle im-age velocimetry) measurements. In comparison with the other models, the low-Reynolds k-ω SST model correctly reproduces the shear layer above the D-section. On the other hand, the high-Reynolds number k- model is more accurate in the recirculation region downstream the obstacle. The results produced by the Wale LES model appear better than those obtained by the Smagorinsky model which is more dissipative. Spectral anal-ysis shows no distinguishable peaks in the wake region. The POD (proper orthogonal decomposition) 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 modeled using unsteady 2.5D RANS simulations based on the low-Reynolds k-ω SST turbulence model. Four blades profile are tested for the fully submerged configuration. The numerical approach is first validated and shows less than a 13 % error against experimental data published on a wind turbine. The S1046 profile produces higher performances than the NACA0018 profile. In the dynamic stall and transition regions, the S809 profile showed the poorest performance. For high tip-speed ratios, the FXLV152 produces the highest power coefficient. The effect of blade number N on the DHAHT with the S1046 has also been investigated. For N = 4 and 2, the best performance is obtained at low and high tip-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 level on DHAHT performance. The DHAHT is tested in two configurations: partially (con-figuration 1) or completely submerged (con(con-figuration 2). The VOF model is validated with an average error of 0.6 % against experimental results for the breaking of a dam.

of the turbine increases gradually with increased tip-speed ratios. The quantification of the momentum loss shows that the DHAHT in configuration 2 extracts higher values of momentum loss with higher tip-speed ratios. The ability of the turbine to produce power in both configurations will ensure its efficiency in shallow rivers.

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

tion et de quatre-vingt-dix-neuf pour cent de transpiration.»

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

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

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

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

Je tiens à remercier mes deux amours Asma et Hacene, pour leurs encouragement, soutien et 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 et gravé en moi la persévérance et la science depuis mon jeune âge. Cet accomplissement vous 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.

1 INTRODUCTION 1

1.1 Introduction générale . . . 1

1.2 Objectifs de la thèse et originalités . . . 2

1.3 Plan de la thèse . . . 4

2 Turbulent flow over a D-section bluff body : a numerical benchmark 5 2.1 Avant-propos . . . 5 2.2 Abstract . . . 7 2.3 Introduction . . . 7 2.4 Numerical modeling . . . 10 2.4.1 Geometrical modeling . . . 11 2.4.2 Turbulence modeling . . . 11 2.4.3 Numerical parameters . . . 13 2.4.4 Experimental database . . . 15

2.5 Results and discussion . . . 16

2.5.1 Mean flow field . . . 16

2.5.2 Coherent structures in the wake flow . . . 22

2.5.3 POD analysis . . . 26

2.6 Conclusion . . . 30

3 Numerical modeling of a Darrieus Horizontal Axis shallow-Water Tur-bine 33 3.1 Avant-propos . . . 33

3.2 Abstract . . . 35

3.3 Introduction . . . 35

3.4 Characteristics of the Darrieus Horizontal Axis Water Turbine (DHAHT) . 37 3.5 Numerical modeling . . . 40

3.5.1 Geometrical modeling . . . 40

3.5.2 Numerical method and turbulence closure . . . 41

3.5.3 Boundary conditions and numerical parameters . . . 41

3.6 Validation of the flow solver . . . 44

3.7 Results and discussion . . . 45

3.7.1 Comparison between blade profiles . . . 46

3.7.2 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 57 4.1 Avant-propos . . . 57

4.2 Abstract . . . 59

4.3 Introduction . . . 59 xi

4.4 Characteristics of the Darrieus Horizontal Axis Hydrokinetic Turbine (DHAHT) 61

4.5 Numerical modeling . . . 64

4.5.1 Geometrical modeling . . . 64

4.5.2 Numerical method and turbulence closure . . . 64

4.5.3 Boundary conditions and numerical parameters . . . 66

4.6 Validation of the flow solver . . . 67

4.7 Results and discussion . . . 70

4.7.1 General performances . . . 70

4.7.2 DHAHT’s influence on flow regime and momentum loss . . . 75

4.8 Conclusion . . . 80 5 CONCLUSION FRANÇAISE 83

6 ENGLISH CONCLUSION 89

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-high-Reynolds number models and the LES (bottom). . . 14 2.3 2D maps around the D-section obstacle in the channel midplane of the mean

velocity components u∗ and v∗ and turbulence kinetic energy k∗. Compa-risons between the PIV measurements, three two-equation RANS models and 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 = 1 and (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 obtained

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. . . 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 obtained

by the LES Wale model. . . 27 2.10 First six POD modes from (a) to (f) extracted from the instantaneous LES

Wale results. . . 28 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. . . 29 2.12 Contour map of longitudinal velocity field reconstructed ˜U /U0using the first

mode with the (a) minimum and (b) maximum time coefficients. Results obtained by the LES Wale model. . . 29 3.1 CAD geometry of the DHAHT. . . 38 3.2 2D cross-sections of the four blade profiles. . . 39

3.3 2D sketch of the computational domain with its main dimensions and the boundary conditions.Note that for the current numerical experiments the turbine rotates in the counter-clockwise direction. . . 40 3.4 Boundary layer velocity profile imposed at the inlet. . . 42 3.5 2D views of the numerical mesh distribution of a) entire domain, b) rotor

domain, and c) turbine blade . . . 43 3.6 Performance curve of a three-blade H-rotor Darrieus wind turbine versus λ.

Comparison with former CFD [24, 41, 84] and experimental results [24]. . . 45 3.7 Comparisons of the averaged power coefficient CP versus λ for the S1046,

S809, FXLV152 and NACA0018 profiles. Results obtained for the 3 blade turbine (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 CTi of one blade over

one rotation (degrees) between the S1046, S809, FXLV152 and NACA0018 profiles. 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. . . 51 3.11 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 four X/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. . . 62 4.2 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 fixed

to HW = 0.65 m, Ha = 1.14 m and HW = 0.82 m, Ha = 0.97 m for

configurations 1 and 2, respectively. . . 64 4.3 (a) Example of a 2D view of the mesh distribution for configuration 1 (F r =

0.625) ; (b) 2D views of the numerical mesh distribution in the rotor region and 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 of the 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

4.6 Distributions of the averaged power coefficient CP as a function of 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. . . 71 4.7 Distributions of the averaged torque coefficient CT as a function of the tip

speed 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 DHAHT

during 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 DHAHT

at 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 for λ = 1.8. . . 76 4.11 Comparison of the polar distributions of the instantaneous torque coefficient

CT for configurations 1 and 2 for four tip speed ratios λ. . . 77

4.12 Froude number F r distributions downstream (X/D = 11D) the DHAHT for configurations 1 and 2, with the corresponding linear regressions. . . 78 4.13 Distributions of the momentum loss LM for configurations 1 and 2, with

2.1 Mesh grid parameters for the high- and low-Reynolds number models and the LES. . . 14 3.1 Main geometrical and operating parameters for the flows through the DHAHT. 38 3.2 Mesh grid parameters for the three-bladed configuration. . . 43 3.3 Instantaneous vorticity fields for θ = 90◦, 150◦, 210◦, 270◦, 330◦.

Compari-sons between the S1046, S809, FXLV152 and NACA0018 profiles for the 3 blade turbine N = 3 and λ = 1.8. . . 50 4.1 Main geometrical and operating parameters for the flows through the DHAHT. 62 4.2 Mesh grid parameters for the three-bladed configuration. . . 67

INTRODUCTION

1.1

Introduction générale

La demande mondiale en énergie ne cesse de croître, alors que les ressources en énergies dites fossiles ne cessent de décroître. Afin de faire face à ce problème, il est nécessaire d’améliorer les rendements des différents procédés alimentés par des énergies fossiles, ou se tourner vers l’exploitation des énergies renouvelables. L’exploitation d’une seule source d’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 de stockage d’énergie, les énergies renouvelables offrent également une alternative intéressante aux régions isolées des réseaux électriques principaux. Il existe quatre principales sources naturelles 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 6 mois, l’énergie solaire n’est pas une source fiable, à cause de l’accumulation de neige sur les 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. En ce 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% de la 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’électricité par les éoliennes. Cette limite n’existe pas avec l’énergie hydraulique, qui est une source continue soit via des hydroliennes dans les rivières, des barrages hydroélectriques et des hydroliennes ou centrales marémotrices près des cotes maritimes. Le potentiel canadien en énergie hydraulique est immense dû au fait que toutes les provinces ont accès à au moins une 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 160 000 MW et autour de 4 400 MW dans la région du Québec [74]. Avec cette dépendance à l’énergie hydroélectrique, le Canada est le troisième au monde en nombre d’installations hydroélectriques (Fig.1.1) et au deuxième rang en termes de production avec 9.6 % de la production mondiale [50]. Cette énergie est majoritairement produite par des hydroliennes dans des barrages avec retenue ou au fil de l’eau.

L’entraînement des turbines dans la majorité des stations hydroélectriques au Canada est assuré par l’effet potentiel dû à une chute d’eau. Dans quelques endroits fluviaux, où les profondeurs d’eau sont importantes, comme dans le fleuve St-Laurent, des turbines sont installées dans le fond et entraînées par les courants d’eau. Des milliers de réseaux de rivières peu profondes restent néanmoins inexploités mais ils se trouvent la plupart du temps dans des régions reculées non raccordées au réseau électrique principal à cause du coût prohibitif d’un possible raccordement. L’idée d’installer des turbines dans ces cours d’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 les conditions 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’écoulement en 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 une partie de ces limites, par son indépendance à la direction de l’écoulement et son efficacité à des faibles vitesses d’eau et de rotation.

Depuis 1926, la turbine de type Darrieus à axe vertical DVAT (Darrieus vertical axis turbine) a été largement étudiée. De nombreuses études ont été faites pour optimiser ses performances et réduire son bruit dans le cas d’éoliennes. Appliquer ces turbines en rivières (DVAHT) est plus problématique du fait des faibles niveaux d’eau généralement observés et des variations de ces niveaux potentiellement importantes. Elles ont donc reçu peu d’attention dans la littérature [26]. Néanmoins, le placement de la turbine de type Darrieus dans une position horizontale permettrait d’obtenir un meilleur rendement même pour des faibles hauteurs d’eau. C’est dans ce cadre que s’inscrit cette thèse : optimiser les 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 plus précisément la turbine de type Darrieus à axe horizontal. Pour atteindre cet objectif, des sous-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.

Figure 1.1 Nombre d’installations hydroélectriques par région. Adapté de [75]. • Comparaison entre plusieurs modèles de turbulence RANS et de sous-mailles LES

avec 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 Darrieus submergée.

• 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 axe horizontal dans deux configurations : 1) partiellement submergée ; 2) complètement submergée

• Modélisation numérique multiphasique de la turbine de type Darrieus équipée avec le 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

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

1.3

Plan de la thèse

Le manuscrit de cette thèse de doctorat est présenté sous format d’articles. Chaque chapitre est 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 mailles LES 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 été faites autour de la turbine de type Darrieus à axe horizontal en testant plusieurs profils de pales afin d’accroître ses performances. Dans le quatrième chapitre, des simulations numériques multiphasiques ont été accomplies sur l’interaction de la surface libre et la turbine de type Darrieus à axe horizontal, et aussi sur l’influence de la hauteur d’eau sur ses performances. La thèse se termine par des conclusions sur les principaux résultats et des perspectives de recherche.

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 de génie, Département de génie mécanique.

R. W. J. Lacey : professeur, Université de Sherbrooke, Faculté de génie, Département de génie civil.

S. Poncet : professeur, Université de Sherbrooke, Faculté de génie, Département de génie mécanique.

Date d’acceptation : 6 octobre 2018

État de l’acceptation : version en ligne publiée Revue : Journal of Environmental Fluid Mechanics Référence : Benchikh Le Hocine et al. [13].

Titre français : Écoulement turbulent autour d’un obstacle en forme de D : benchmark numérique

Contribution 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 un premier temps, l’écoulement autour d’un obstacle fixe submergé. Le but est de mettre en place une ligne directrice sur le choix des approches numériques grâce à un benchmark des modèles de turbulence de type RANS et LES validés par des mesures expérimentales obtenues par PIV. D’autre part, cet article permet une compréhension détaillée des struc-tures tourbillonnaires qui se forment autour d’obstacle submergé en utilisant différentes approches pour le post-traitement comme le calcul du critère Q et la POD, qui permettent d’identifier les structures cohérentes et de caractériser leur dynamique, respectivement. Résumé français : Un benchmark numérique des différents modèles de turbulence a été réalisé 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

Reynolds s’avère mieux adapté pour capturer la couche de cisaillement intense au-dessus de l’obstacle par rapport aux autres modèles à deux équations. Pourtant les modèles k- et k-ω SST à haut nombre de Reynolds présentent des performances supérieures dans la région 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 sur la prédiction des structures tourbillonnaires. Le modèle Wale combiné avec un schéma aux différences centrées a montré un meilleur accord global par rapport au modèle standard de Smagorinsky, qui est plus dissipatif. Une analyse spectrale a été réalisée dans la région du 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 extraire la dynamique de l’écoulement et les structures cohérentes.

2.2

Abstract

A numerical benchmark of different turbulence closures was performed to investigate the turbulent wake flow behind a submerged D-shaped bluff body. The numerical models in-cluded steady two-equation Reynolds-Averaged Navier-Stokes (RANS) models and Large Eddy Simulations (LES), which were compared to planar and stereoscopic particle image velocimetry (PIV) measurements. The k-ω SST low-Reynolds number model was found to 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- and k-ω SST high-Reynolds number models demonstrate higher performance in the recirculation region. LES was also completed over the D-section to determine the influence of the sub-grid scale models on the prediction of the vortical structures. The Wale model together with central difference schemes showed a better overall agreement over the standard Smagorinsky model, which appears 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) was applied 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 fishways has interested researchers for decades [15, 18, 22, 71, 93, 95, 108, 120, 124], yet the flow structure is complex because of the three dimensional geometry, the elevated turbulence intensity, 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 turbulent flow variables [121] and an accurate flow description is required for the improved design of future fishways. The objectives of the present work are to perform a numerical benchmark of different turbulence closures for the flow behind a D-shaped bluff body and to display the capability of more advanced large eddy simulations (LES) in predicting the coherent structures appearing in the wake. The present flow configuration represents a first step towards a more realistic flow simulation of fishways and a better understanding of flow structures shedding from bluff bodies. An indepth study of flow over a wall mounted bluff body also has implications for other fields such as : shape optimization in the automobile industry [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 in fish passes and to correlate the flow features with fish behavior. Different numerical studies were performed for fishways without any experimental validation [2, 71, 95]. Sometimes,

the numerical tool is clearly used as a black box [43]. When experimental validation is possible, 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 a vertical slot fishway. A 2-D depth-averaged shallow-water numerical model PCFLOW2D coupled with three different turbulence models was used for comparisons with physical acoustic Doppler velocimeter (ADV) measurements. All turbulence closures provided sa-tisfactory results. However, as all the comparisons were performed at cross-sections where flow 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 rocks of 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 results were compared to a depth averaged model obtained with the Telemac-2D software. A large discrepancy between the numerical and experimental results was found upstream and downstream of the cylinders, which was attributed to the numerical approach and the choice of the k-ε turbulence model. Many other numerical studies of turbulent flow and 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 (though relevant) 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 better understanding of the flow dynamics and vortical structures produced by the wake of a single obstacle will in the not too distant future lead to improved understanding of their interactions 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 in comparison to using larger cells [15, 18, 108].

The turbulent flow around a single obstacle/boulder in a nature like fishway can resemble flows 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 to unsteadiness in the wake behind bluff bodies have received much attention up to the 90’s as shown in the review of Williamson [129] and Sumner [115]. For a wall-mounted cube, Hussein and Martinuzzi [49] performed lased Doppler anemometry (LDA) measurements to determine the turbulence kinetic energy (k) budgets in the wake. More recently, the effect of the free-stream turbulence (FST) on the characteristics of the wake flow behind bluff bodies has been considered experimentally. For example, Khabbouchi et al. [54]

in-vestigated the influence of the FST on the development of a separating shear layer in the near wake of a circular cylinder for Reynolds numbers based on cylinder diameter up to ReD = 4.7 × 104 by hot-wire measurements. The shear layer shedding frequency

and its harmonics became broader as FST level increases to finally disappear for a FST level equal to 6.2%. The authors suggest that the shear layer behavior can be regarded as a mixing layer for FST intensities lower than 6.2%. Using multiple ADVs, Lacey and Rennie [61] studied the turbulent wake past a submerged bed-mounted cube for a bulk Reynolds number (based on the cube height h) Reh = 46000 and three water depth to

cube height ratios. By decreasing the water depth, the shedding vortical structures are more 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 image velocimetry (PIV) and hot-wire measurements to investigate the influence of the FST on the flow around a wall-mounted cube at a boundary layer development Reynolds number Rex = 1.8 × 106. They showed that the stagnation point on the upstream side of the cube

and 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 delays the separation for Reynolds numbers up to ReD = 2.8 × 105. The authors found that as

FST intensity increases, the critical Reynolds number known as the drag crisis (where the boundary layer over the sphere becomes turbulent and drag decreases rapidly) decreases. The authors demonstrated that the main mechanism for the drag evolution is linked to the 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 a water 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 behavior was also observed for the wake width at (Re ≥ 2654). Furthermore, the authors show that the mean vorticity is independent of Re at X/D ≥ 2 downstream of the cube. Sadeque et al. [104] studied the flow patterns in the near-wake behind bed-mounted cylinders, in a shallow turbulent channel flow with smooth and rough beds. They focused both on wall wake similarity and on the region away from the bed, which was found to be well modeled by a Law of the wall.

Numerically, Richmond-Bryant and Flynn [98] used the discrete vortex method to simulate the time-averaged flow fields past a circular cylinder at ReD = 1.4 × 105. Though good

agreement was obtained against literature experimental data [23],the unsteady simulations don’t converge numerically. Palau-Salvador et al. [92] conducted LES of the flow around finite-height cylinders for two height-to-diameter ratios H/D = 2.5 and H/D = 5, with

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 of the recirculation zone gets larger when H/D increases. Saeedi and Wang [105] performed LES 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 tip vortices produced at the top of the block, the Karman vortices from the side walls and the developing boundary layer over the bed. Elkhoury [30] compared the performance of the Scale Adaptive Simulation (SAS) turbulence model with the predictions of both the Spalart-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 of the 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 numerical benchmark of different turbulence closures. Testing various closure models is seldom done in the literature, yet can have a great impact on the quality/accuracy of the numerical simulations, 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 better model 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 different RANS 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 2 different sub-grid scale models and resolution schemes. Numerical results are compared to PIV measurements for validation purpose. A Proper Orthogonal Decomposition (POD) is finally applied to LES results to extract the most energetic modes within the flow and to further 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 surface mounted bluff body has not been done before.

2.4

Numerical modeling

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

2.4.1

Geometrical modeling

A sketch of the computational domain is shown in Figure 2.1. It consists in an open channel flume 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 side face 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 respectively D = 0.025 m, h = 0.025 m and w = 0.0125 m (Fig.2.1). During the experiments, the mean flow depth (H = 0.1 m) and mean inlet streamwise velocity (U0 = 0.175 m/s) were kept

constant leading to a bulk Reynolds number equal to ReH = 17500.

Figure 2.1 Sketch of the computational domain (not to scale), and views of the D-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 less adapted to accurately reproduce the flow structure around the bluff body. The standard k- model of Jones and Launder [51] was used in its high-Reynolds number formulation and 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 near wall 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

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 in the resolution of the effective velocity distribution close to the wall. These well-established models are fully described in the monograph of Wilcox [128] or in the detailed numerical work 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 in LES 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 Smagorinsky model. 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 the

Smagorinsky model is expressed as follows : νt= (Cs∆)2

q

2 ¯SijS¯ij (2.1)

where Cs' 0.18 is the standard Smagorinsky constant and ∆ = (∆x∆y∆z)1/3is 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 too dissipative nature of the standard model [35]. However, Poncet et al. [94] showed, for the turbulent flow in a Taylor-Couette-Poiseuille system, that the dynamic version provided similar results compared to the Wall-Adapting Local Eddy Viscosity (Wale) model while requiring about 12% of extra computational time.

Thus, herein the Wale subgrid-scale model developed by Nicoud and Ducros [88] has also been used for comparison. The Smagorinsky model is based on the second invariant of the symmetric part of Sij. The main drawbacks are that this invariant is of order O(1) close

to a wall and it is not related to the rotation rate of the turbulent structures. To avoid that, Nicoud and Ducros [88] developed the Wale model based on the gradient velocity tensor gij, which is a good candidate to represent the velocity fluctuations at the length

νt= (Cm∆)2 (S_ijdS_ijd)32 ( ¯SijS¯ij) 5 2 + (Sd ijSijd) 5 4 (2.3) where Cm = 0.4929 and ¯Sij corresponds to the filtered strain tensor. The Wale model

employs the traceless symmetric part of the square of the velocity gradient tensor gij, as

follows : S_ijd = 1 2(gikgkj+ gjkgki) − 1 3gkigikδij gij = δ ¯ui δxj (2.4) The Wale model behaves well near the wall with good approximation to the assumed physics 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 a finite-volume method. For the RANS calculations, a second-order high resolution advec-tion scheme was used to avoid dissipaadvec-tion and ensure a better accuracy. For the LES simulations, a second-order Backward-Euler scheme is employed for the temporal discreti-zation together with an implicit time-stepping scheme. For the spatial discretidiscreti-zation, both high-resolution and central difference schemes were used. The velocity-pressure coupling was performed by a Rhie Chow fourth-order coupling algorithm, which guarantees that the dissipation term vanishes rapidly under mesh refinement.

Concerning the boundaries conditions, the mean streamwise velocity profile imposed at the inlet corresponds to the PIV measurements, leading to an average streamwise velocity of U0 = 0.175 m/s at the channel inlet. The imposed turbulence intensity I0 = u0/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 at

the 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 domain being long enough to suppress the sensitivity of the flow to the outlet condition, a simple pressure outlet condition was selected. A verification was performed by imposing outflow or 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 constructed using the software Centaur (Fig.2.2). They are composed of tetrahedral elements in the

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 of the wall coordinate in the whole domain reaches y+ = 40. To better account for the flow dynamics close to the D-shaped bluff body and in the near wall regions, a finer mesh has been 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 condition max(y+_{) = 1 (see Table 2.1). A stretching factor of 1.2 was used to avoid any numerical}

dissipation of possible coherent structures. A mesh refinement is also imposed in the wake of 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 the high-Reynolds number models (top) and fine mesh for the low-Reynolds number models and the LES (bottom).

Turbulence closure Number of cells (x106_{) Number of Nodes (x10}6_{) max(y}+_{) GCI [%]}

High-Reynolds number RANS 12.4 2.8 40 0.011 Low-Reynolds number RANS & 22.2 4.5 0.95 0.0011 LES

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

The Grid Convergence Index (GCI) provides an uniform measure of convergence for grid refinement studies [99]. It is based on the estimated fractional error derived from the generalization of the Richardson’s extrapolation. The GCI value represents the resolution level and how much the solution approaches the asymptotic value. The GCI can be written as 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− f2

f2− f1



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 a third coarse mesh with 22.2, 12.4 and 6.9 million elements, respectively. A grid refinement ratio of approximately 1.8 was applied between the three grids, while keeping the number of prismatic layers constant. The GCI is calculated by considering the magnitude of the mean velocity.

As shown in Table 2.1, the GCI for the finer and coarser meshes are relatively low (below 1), indicating that the dependency of the numerical simulation on the cell size has been already 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 imbalance is lower than 10−6. The calculations were run using the cluster MP2 provided by Calcul Québec. The CPU time for the low-Reynolds, high-Reynolds number models and LES was respectively 26 hours, 18 hours and more than 20 days, using 32 processors for the RANS models and 96 processors for the LES. The LES calculations have been initialized using a converged RANS calculations. Then, statistics have been cumulated after the elapse of two 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 tilting glass-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 was mounted 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 from black anodized aluminum. As stated previously the D-section diameter, height and width was D = 0.025m, h = 0.025m and w = 0.0125m, respectively. During the experiments the mean flow depth was kept constant (H = 0.1 m) and the mean velocity of the incoming flow 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 Quantum Ltd, United Kingdom) of 2.3 W continuous wave was used. The field of view (FOV) was

approximately 0.1 × 0.1 m2_{, 0.06 × 0.06 m}2 _{for the planar and stereoscopic PIV,}

respecti-vely. Perpendicularly to the side of the channel, a high-speed 1 megapixel CMOS digital camera (FASTCAM 1024 PCI, Photron Ltd., Japan) was positioned, for the planar PIV measurements. For the stereoscopic PIV configuration, two high-speed CMOS cameras were positioned above the channel. The acquisition frequency of the camera (both PIV measurements) 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 of the seeding particles. Silver coated hollow microspheres with a mean diameter of 0.013 mm (Potters Industries, USA) were used to seed the flow.

The software Davis (LaVision Inc., USA) was used to produce the instantaneous velocity vectors. For the extraction of the velocity field, a multipass algorithm was used. The final interrogation area (IA) was 16 × 16 px2 with a 50% overlap, while the first and second IA were composed of 64 × 64 px2 and 32 × 32 px2 respectively. The resulting matrix has a dimension of 128 × 128 px2 for each velocity vector at each time step. The resolution of the final IA was 1.63 × 1.63 mm2 _{for planar PIV and 1.5 × 1.5 mm}2 _{for stereoscopic}

PIV. Three filters were applied to the raw vector maps to remove erroneous vectors : 1) a signal-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. An average of 9% of the raw vector field was replaced by interpolated values for the planar and 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 v

velocity components and turbulence kinetic energy k are normalized by the mean inlet velocity U0 and its square respectively, such that : u∗ = u/U0, v∗ = v/U0, k∗ = k/(0.5 ×

U02). The origin of the reference (X = Y = Z = 0) is set at the bottom wall in the median

plane behind the D-section bluff body.

2.5.1

Mean flow field

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

producing a reattachment zone in the lee (Fig.2.3). The shear layer above the obstacle generates intense coherent structures, which are shed downstream.

According to Figure 2.3, all the RANS turbulence models in their high- or low-Reynolds number formulations predict the existence of the recirculation zone. Nevertheless, the size of the recirculation differs from one model to another, reflected in a displacement of the reattachment region. The prediction of the recirculation length seems to be linked to the flow field in the near-wall regions. The extent of the recirculation zone for the low-Reynolds number k − ω SST (Xr/D ≈ 2.4) and the two LES models (Xr/D ≈ 2.5) compare fairly

well with the PIV measurements (Xr/D ≈ 1.9). On the contrary, the high Reynolds

formulations of k − ω SST and k −  RANS models, which do not solve the flow field close the walls, strongly underestimate the length of the recirculation zone behind the bluff body, Xr/D ≈ 1.2 and Xr/D ≈ 1.3, respectively.

Figure 2.3 2D maps around the D-section obstacle in the channel midplane of the mean velocity components u∗ and v∗ and turbulence kinetic energy k∗. Comparisons between the PIV measurements, three two-equation RANS models and 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 negative vertical velocity (v∗ ≈ −0.3) region located on the stoss side of the obstacle, where the horseshoe vortex develops. This region is less apparent with the two high-Reynolds num-ber models. In general, the two high-Reynolds numnum-ber models (k − ω SST and k − ) give a poorer representation of the flow structure around the obstacle. Comparisons with

the experimental measurements show differences in the recirculation region. Furthermore, within the recirculation zone along the back wall of the obstacle, strong regions of elevated positive 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 and the use of a wall functions. The main difference between the low-Reynolds number k − ω SST and the LES lies in the abrupt plunging of the shear layer to the bed for the RANS model, whereas it is much more progressive in the LES and PIV results. This mismatch could be due to the unsteady nature of the shear layer and the appearance of possible coherent 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 at the 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 and LES 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 been calculated using the fluctuating streamwise and vertical velocity components only. The high 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 with the PIV. This good agreement is likely due to both a direct calculation of the large- and intermediate-scale eddy vortices and to the unsteady nature of the flow, which is accounted for. The difference between the results obtained via the two LES approaches is not striking apart from a minor discrepancy in the region X/D ≈ 1.5 − 2. The high-Reynolds number k −  model strongly overpredicts k just after the obstacle (X/D ≈ 0 − 1.8) with very high values (k∗ ≈ 0.005), which extend to the back wall of the bluff body (X/D ≈ 0 − 0.2). This suggests that the dissipation rate  is too low in this region. This is perhaps because the estimated  does not account for the rotational motion of the fluid particles and is not correctly modeled in the near-wall region. The high Reynolds k − ω SST model is much more dissipative than the k −  model though they have comparable wall functions. The low-Reynolds number k −ω SST is more dissipative than the LES models, yet has a similar k distribution due a correct near-wall resolution. Therefore, the specific dissipation rate ω is a better candidate compared to  to determine flow structure and the characteristic scale of turbulence. Moreover, the use of a production limiter improves the predictions of the turbulence intensities (Fig.2.3,k∗). However, the low-Reynolds number k − ω SST model in its steady-state version does not capture the high turbulence levels during the destabilization and the plunging of the shear layer as it is not able to predict the smallest 3D unsteady coherent structures in that flow region.

Figure 2.4 shows the distributions of the dimensionless longitudinal and vertical mean velocities u∗ and v∗ along the streamwise direction X/D. The results have been obtained at two positions Y /D = 1 and Y /D = 0.5 in the median plane of the channel (where Y is the vertical distance from the bed). At Y /D = 1, the high-Reynolds number models completely fail to capture the u∗ distribution until X/D ≈ 2, likely because of the use of 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 the shear-layer region). Conversely, the v∗ profiles predicted by the high-Reynolds models are in poor agreement with the experimental values over the whole range of X/D positions considered 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, the position of minimum v∗is shifted closer to the lee of the obstacle. The use of wall functions significantly 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 the obstacle. The choice of  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 agreement between the LES results, low-Reynolds number k − ω SST and the PIV measurements is generally good in terms of the streamwise velocity u∗ distribution (Fig.2.4a). A slight shift towards higher magnitude u∗ is observed in the LES results and could be attributed to a misrepresentation of the experimental 3D inlet flow conditions. At the channel entrance, 2D planar PIV measurements were obtained along the centerline, missing then the third velocity component. So, a 2D PIV velocity profile and a turbulent intensity of 10% were numerically imposed. In a previous study, Baetke et al. [7] demonstrated the influence of 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 a separation 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 capture well the vertical velocity overshoot centered around X/D ' 0.4 (Fig.2.4b). The overshoot represent the entrainment of the fluid from the recirculation zone by the shear layer. The LES Smagorinsky and Wale models underestimate the flow acceleration in this region, and provide a more extended recirculation zone. When the shear layer plunges to the bed, the LES 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. The Smagorinsky model under and overestimates the peak value v∗ ≈ −0.2 and its position X/D ≈ 3.0, respectively. In general, the LES Wale provides a better overall performance compared to the LES Smagorinsky. The Wale approach is based on a more appropriate

turbulent viscosity definition, which takes into account wall effects. The Smagorinsky mo-del is less suited for wall bounded flows, the subgrid scale momo-del being unable to reproduce an asymptotic variation of the turbulent viscosity near the wall. As the flow near the wall is 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 = 1 and (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 the obstacle). This vertical position in theory coincides with the center of the recirculation region, as can be observed by the negative values of u∗. The length of the recirculation region at Y /D = 0.5 can be estimated by considering the position for which the u∗ is equal to 0. From the PIV, the extent of the recirculation zone at Y /D = 0.5 is around X/D = 1.8. The high-Reynolds number k − and k −ω SST models predict a recirculation length equal to around X/D = 1.3. The low-Reynolds number k − ω SST model and the LES Wale predict 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 acceptable estimation of the u∗outside the shear layer region X/D > 2.6, they fail to accurately model the vertical velocity component v∗ distribution in the near wake. The wall functions used don’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 −  model in the core region of the flow and the k − ω in the boundary layer. Using the k − ω SST model in its high Reynolds number formulation on a coarser mesh makes the influence of the underlying k − ω model less important. It is thus not surprising that the results are very close to those provided by the high Reynolds number k −  model, while remaining slightly better.

(a) (b)

Figure 2.5 Distributions of the mean streamwise velocity component u∗ along the 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 the methodology, the PIV for the horizontal measurement plane (presented here) is 3C .As the 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 conservation of 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 zero velocity at the side walls, which is inherent to the use of wall functions. Generally, the high-Reynolds number k − ω SST and k −  turbulence models predict acceptable velocity distributions for both X/D positions. The LES and low-Reynolds number k−ω SST models estimate more correctly the backward flow in the recirculation region behind the bluff body

at X/D = 1.6 (Fig.2.5b). Outside the wake region (|2Z/b| ≥ 0.25), the flow is accelerated to reach a peak value u∗ ' 1.23. All models predict well the velocity distribution there at X/D = 0.8 (Fig.2.5a) ; while at X/D = 1.6, the high-Reynolds number approaches better predict 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 structures shed from the obstacle. In order to get a better appreciation for these structures, Figure 2.6 displays 2D views of the instantaneous vorticity around the D-section obtained by the LES Wale and the LES Smagorinsky models. As can be seen, the flow topology does not differ significantly between these two sub-grid scale models. The undulation of the shear layer before the onset of the vortex shedding is more apparent in the Wale model, which also predicts a slightly longer shear layer and so a delayed shedding position compared with Smagorinsky 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 layers produced 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 the LES 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 in the transverse direction (Fig.2.6c,d) for the LES Smagorinsky. At Y /D = 1, the wake flow modeled by the LES Wale is more stable, while vortex shedding is much more apparent for the Smagorinsky model with a larger wake (Fig.2.6e,f).

Investigations of the temporal evolution of the vorticity field (not shown), revealed a flapping of the shear layer which appears to be related to the interaction between coherent structures within the recirculation region and the shear layer. In other words, some of the coherent structures generated within the shear layer are injected in to the recirculation region, causing a slow interaction with the underside of the shear layer and implying a flapping 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 the different turbulent viscosity definitions. As previously stated in Section 2.4.2, the invariant used in the Smagorinsky model is not related to the rotation rate of the turbulent structures contrary 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 body obtained 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 structures are observed in the wake of the obstacle. It is directly reflected in the maps of the turbulent viscosity (not shown), which reveal that the Smagorinsky model is much more dissipative than 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 longitudinal velocity (u∗) on top and side view. A horseshoe vortex is distinguishable in front of the obstacle 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 with the coherent structures produced within the shear layer. Hairpin vortices are visible in the wake once the shear layer destabilizes. The horseshoe vortex breakdown and the shedding phenomenon within the shear layer occur closer to the obstacle with the Smagorinsky model 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 spatial schemes (not shown here). The high resolution 2nd order schemes lead to an overprediction