Caractérisation et instabilités des tourbillons hélicoïdaux dans les sillages des rotors

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1 AIX-MARSEILLE UNIVERSITÉ Institut de Recherche sur les Phénomènes Hors Equilibre -UMR CNRS THÈSE pour obtenir le grade de DOCTEUR DE AIX-MARSEILLE UNIVERSITÉ Spécialité : Mécanique & Physique des Fluides École doctorale : Physique, modélisation et sciences pour l ingénieur Préparée à l Institut de Recherche sur les Phénomènes Hors équilibre Présentée par Mohamed ALI Caractérisation et instabilités des tourbillons hélicoïdaux dans les sillages des rotors Dirigée par Malek ABID Soutenue le Avril 24 devant le jury composé de : Jens N. SØRENSEN, Professeur, Technical University of Denmark, (Rapporteur) Ivan DELBENDE, Maître de Conférence, LIMSI, UPMC, Paris 6, (Rapporteur) Maurice ROSSI, Directeur de Recherche CNRS, IJLRDA, Paris 6, (Examinateur) Thomas LEWEKE, Directeur de Recherche CNRS, IRPHE, Marseille, (Examinateur) Malek ABID, Maître de Conférence, IRPHE, Aix-Marseille Université (Directeur de thèse)

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3 AIX-MARSEILLE UNIVERSITÉ Institut de Recherche sur les Phénomènes Hors Equilibre -UMR CNRS THÈSE pour obtenir le grade de DOCTEUR DE AIX-MARSEILLE UNIVERSITÉ Spécialité : Mécanique & Physique des Fluides École doctorale : Physique, modélisation et sciences pour l ingénieur Préparée à l Institut de Recherche sur les Phénomènes Hors équilibre Présentée par Mohamed ALI Characterization and instability of helical vortices in rotor wakes Dirigée par Malek ABID Soutenue le Avril 24 devant le jury composé de : Jens N. SØRENSEN, Professeur, Technical University of Denmark, (Rapporteur) Ivan DELBENDE, Maître de Conférence, LIMSI, UPMC, Paris 6, (Rapporteur) Maurice ROSSI, Directeur de Recherche CNRS, IJLRDA, Paris 6, (Examinateur) Thomas LEWEKE, Directeur de Recherche CNRS, IRPHE, Marseille, (Examinateur) Malek ABID, Maître de Conférence, IRPHE, Aix-Marseille Université (Directeur de thèse)

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5 3 Dedicated to The memory of my father and brother Rafik, My mother, brothers and sister, All my family, and friends,

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7 5 Acknowledgements I start by praising ALLAH, the Creator, and thank Him for this achievement. I want to express my sincerest appreciation and gratitude to my supervisor Malek ABID for his beneficial advices, encouragement, and patience throughout this work. I thank Professors Jens N. SØRENSEN and Ivan DELBENDE for agreeing to be a referees of my PhD thesis. I thank also Professors Thomas LEWEKE and Maurice ROSSI for their honourable participation in the jury. I thank the members of the Aerodynamics team for their collaborations, especially Professor Stéfan LE DIZÈS, Francisco BLANCO-RODRIGUEZ and Umberto QUARANTA. A special thanks go to Hadrien BOLNOT for the many fruitful discussions and for providing the experimental data. The help and support of all administrative staff of the Institute is acknowledged. I thank especially Mireille, Lucienne, Saïda, Sadia and George. I am thankful to my colleague Bastien DI PIERRO for his encouragement and his helpful assistance for the numerical and computational problems during my PhD. I am also thankful to my PhD-colleagues with them I spent good time, especially Alexander, Greg, Edward, Lionel, Wael, Moncef, Marjorie and all others, thank you for the discussions and the coffees. Outside the Institute, I want to express my thank to Professor Aziz CHIKHAOUI for his support and encouragement during my PhD. A special thank go to my Tunisian friends in Marseille: Khaled, Ibrahim, Gaith, Tarek, Bassem, with them I spent very good moments. I appreciate their support. My heartfelt thanks and gratitude go to my family for their love, encouragement and support during my PhD. My success today is a fruit of their patience. Marseille, April 24 Mohamed Ali

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9 7 Abstract This present work is aimed to study helical vortices encountered in the wakes of rotating elements. For this, the generation of a helical wake of a one-bladed-rotor in a laminar velocity field, is simulated by the actuator line method for different Reynolds numbers and different Tip Speed Ratios. This method is a coupling of a Navier-Stokes solver with the Actuator Line Method where the blade is replaced, in the computational domain, by the body forces that it applies on the fluid. A local two-dimensional aerodynamic approach is used to calculate the forces using tabulated aerodynamic coefficients. This method has been implemented in a finite difference code, that we have written in parallel to solve the 3D incompressible Navier Stokes equations written in cylindrical coordinates. The incompressibility is applied using the fractional step technique as a projection method. The order of accuracy of the method is two both in time and space. The Navier-Stokes solver was validated comparing growth rates of an unstable jet, found numerically, and those of linear instability theory. A good agreement was found. A good agreement was also found comparing numerical results to analytical formulations and experimental data. It was shown that the method predicts well the aerodynamic parameters and the induced velocities. A particular attention was, then, given to the helical tip vortex. It was characterized for different Reynolds numbers and Tip Speed Ratios. The vorticity and the azimuthal velocity were found self-similar and the vortex core follows asymptotically the linear two-dimensional diffusion law. A simple model for the helical vortex core was proposed. The presence of an axial velocity inside the vortex core was highlighted. Then, a stability study of the helical tip vortex was done using an angular velocity dependent on time to perturb the flow. The largest growth rates were found in good agreement with those of the (2D) pairing instability. Three types of modes were identified based on the perturbation frequency. The results are similar to those found in previous analytical and experimental works. Résumé Les tourbillons hélicoïdaux générés derrière les rotors sont étudiés. Pour les générer, une méthode basée sur le couplage entre la technique de l actuator line et un solveur des équations de Navier-Stokes (NS), incompressibles et tridimensionnelles, a été développée. Elle consiste à modéliser la pâle par son équivalent de forces volumiques qu elle applique sur le fluide et qui sont injectées dans les équations de NS. Ces forces sont calculées par une approche aérodynamique localement bidimensionnelle en se basant sur des polaires aérodynamiques prédéfinies. Les équations, écrites en coordonnées cylindriques, sont résolues par un schéma de différences finies, écrit en parallèle. Une méthode de pas fractionnaire a été utilisée pour assurer l incompressibilité. La méthode est d ordre deux en temps et en espace. Le solveur des équations de NS a été validé par la reproduction des taux de croissance d un écoulement de jet, instable, trouvés par la théorie d instabilité linéaire. Un bon accord a été trouvé entre les résultats des simulations numériques et ceux des formulations analytiques pour les vitesses induites. La comparaison avec des données expérimentales a montré que la méthode

10 8 prédit bien les propriétés aérodynamiques des pâles. Ensuite, le tourbillon de bout de pâle a été, en particulier, caractérisé pour différents nombres de Reynolds et rapports de vitesse de bout de pâle. La vorticité et la vitesse azimutale ont été trouvées auto-similaires et la taille du cœur suit asymptotiquement la loi de diffusion linéaire à deux dimensions. Un modèle simple du cœur du tourbillon a été proposé. La présence d une vitesse axiale dans le cœur du tourbillon a été montrée et a été caractérisée en fonction du rapport de vitesse (rotation/translation) au bout de la pâle. Finalement, une étude de stabilité du tourbillon du bout de pâle a été faite en utilisant une vitesse angulaire variable pour perturber l écoulement. Les taux de croissances des modes les plus instables sont en bon accord avec celui de l instabilité d appariement (bidimensionnel) des tourbillons. Trois types de modes ont été identifiés en fonction de la fréquence des perturbations et ont été trouvés similaires aux modes décrits par la théorie et à ceux obtenus, précédemment, par l expérience.

11 Contents Resumé 3. Introduction Contexte général L objectif de cette thèse Méthode numérique et validations Les équations de Navier-Stokes Implémentation numérique Validation de la méthode Caractérisation d un tourbillon hélicoïdal généré par un rotor mono-pale Méthode de caractérisation Taille du cœur et auto-similarité de la vorticité et de la vitesse azimutale Un modèl du cœur du tourbillon hélicoïdal Etude de stabilité du tourbillon hélicoïdal Perturbation et taux de croissance Résultats Introduction 3 2. General context Wake model and helical vortex characterization Stability analysis of a helical vortex system The present work Thesis outline Numerical method and Validations 4 3. Governing equations Discretized computational domain Boundary conditions Fractional step method Time-stepping procedure: non-solenoidal velocity field calculation Adopted method Inversion along the θ-direction Pressure Poisson equation solving Fast and direct method

12 CONTENTS 3.7 Body force computing Parallel Computing Navier-Stokes solver validation: 3D perturbations Actuator Line Validation: comparison with Biot-Savart law and analytical solutions Velocity field induced by a finite-length helical vortex-filament Velocity field induced by an infinite helical vortex-filament Comparisons between finite and infinite length cases Comparison between numerical simulations and analytical models Actuator Line validation: comparison with blade design parameters and an experimental work Conclusion Characterization of the wake behind a one-bladed rotor Vortex wake geometry Vortex core properties Characterization procedure Core size Core shape Maximum vorticity in the core Vorticity and velocity profiles Vorticity in the vortex core Azimuthal velocity in the vortex core A vortex core model for a helical vortex Vorticity profile in the vortex core Azimuthal velocity profile in the vortex core Axial velocity profile in the vortex core Conclusions Instability of a helical vortex Perturbation Growth rate Results The in-phase modes The out-of-phase modes Other modes Conclusion Conclusions & Perspectives 2 7 Appendix Paper presented in the International Conference on Aerodynamics of Offshore Wind Energy Systems and Wakes, Denmark, Paper 2 published on Journal of Fluid Mechanics,

13 CONTENTS 7.3 Paper 3 submitted to Physics fo Fluids

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15 Chapter Resumé. Introduction.. Contexte général Pour la mécanique des fluides, la dynamique des tourbillons est jusqu à nos jours, un domaine de recherche intense. Les tourbillons sont observés dans des écoulements dans la nature et dans plusieurs applications industrielles, comme la tâche rouge du Jupiter, les sillages des corps et les processus du mélange. Un tourbillons peut exister sous plusieurs formes, il peut être: circulaire plan, ligne (ou tube) droite, ou bien un anneau. Une forme intéressante des tourbillons est la forme hélicoïdale. Ces tourbillons sont rencontrés principalement dans les sillages des rotors tels que les éoliennes ou les hélicoptères (figure.). Dans ces cas, le sillage est principalement constitué de deux parties: (i) le sillage proche où l écoulement est influencé par les propriétés aérodynamiques des pales du rotor, et (ii) le sillage lointain où le rotor n influe plus sur l écoulement et il est caractérisé par la formation d une zone turbulente. Le sillage d un rotor est formé d un système de tourbillons: un tourbillon au bout de la pale et Figure.: Sillage hélicoïdal: (gauche) rotor d hélicpothère, (droite) éolienne. un tourbillon au niveau du moyeu avec la présence d une nappe de vorticité reliant les deux. Dans le cas des hélicoptères, l étude de ce type d écoulement est intéressante pour une meilleure 3

16 4 CHAP. : RESUMÉ prédiction des forces aérodynamiques des pales et ainsi un meilleur désign. Dans certains cas particuliers, l instabilité de cet écoulement peut provoquer l apparition d un gros anneau de vorticité pouvant causer de grands dégâts pour l hélicoptère (allant jusqu à sa chute). Donc, l étude de cet écoulement est critique pour analyser ce phénomène ( Vortex Ring State ) et améliorer les conditions du vol. Par conséquent, d un point de vue fondamental, il est important de répondre aux deux questions élémentaires concernant ce type de tourbillon: Quelles sont les propriétés physiques (taille du cœur, profils de vitesse et de vorticité dans le cœur...) du sillage généré derrière les rotors, et en particulier celles du tourbillon du bout de pale? Comment se fait la transition du sillage proche vers le sillage lointain? quels sont les mécanismes de la déstabilisation du sillage?..2 L objectif de cette thèse Cette thèse a, donc, pour objectif principal la caractérisation du sillage d un rotor mono-pale dans une configuration relativement simple afin de mieux comprendre la dynamique des tourbillons hélicoïdaux. Pour cette raison, des solutions numériques des équations de Navier-stokes, tridimensionnelles et incompressibles, en géométrie cylindrique sont à trouver. La pale du rotor est remplacée dans les équations par son équivalent de forces volumiques qu elle exerce sur le fluide. Ces forces sont obtenues par la méthode de la ligne active ( Actuator Line method ) développée par Sørensen & Shen 22 (47). J ai mis au point un code de différences finies centrées, développé à partir de zéro. Je l ai parallélisé en utilisant le language OpenMP et ainsi, les simulations numériques sont faites sur des machines à mémoire partagée. Les ressources de calcul du Centre Informatique National de l Enseignement Supérieur (CINES) de France ont été utilisées. Le solveur Navier-Stokes est validé par la reproduction des taux de croissance, d un écoulement de jet instable, comparés à ceux trouvés en utilisant la théorie d instabilité linéaire par Abid & Brachet 993 (2) et Abid & Brachet 998 (3). La validation de l implémentation de la méthode de la Ligne Active a été faite, dans un premier temps, par comparaison des vitesses induites obtenues numériquement avec celles obtenues par des formulations analytiques. Une étude semi-analytique est faite pour trouver la différence entre les formulations tourbillon de longueur infinie et celle de longueur finie, rencontrées dans les études des sillages. Dans un deuxième temps, l utilisation de la méthode de la Ligne Active a été validée en comparant les propriétés aérodynamiques, le long de la pale, obtenues par les simulations numériques et celles mesurées expérimentalement. J ai montré que la méthode prédit bien les forces aérodynamiques malgré que les effets de la couche limite et le décrochage ne sont pas pris en compte. En résumé, une partie importante de la thèse est dédiée à la caractérisation du tourbillon hélicoïdal du bout de pale en tenant en compte des effets visqueux et de la longueur finie du tourbillon, caractérisation obtenue à partir des solutions numériques des équations de Navier-

17 . Introduction 5 Stokes. Le rotor, avec une vitesse de rotation, Ω, est placé dans un écoulement axial et uniforme de vitesse caractéristique W et de viscosité ν. La pale a une longueur caractéristique R b. Les simulations numériques ont permis de caractériser, pour plusieurs tours du rotor, le sillage généré et l évolution spatio-temporelle du tourbillon hélicoïdal du bout de pale. Plusieurs combinaisons de différents nombres de Reynolds, R e W R b /ν =(5;;5;2), et différents rapports de vitesse au bout de pale, λ ΩR b /W =(5;6;9;2), ont été simulées. Cette étude a montré que la vorticité et la vitesse azimutale du cœur du tourbillon sont autosimilaires. Elle a montré aussi que le tourbillon du Lamb-Oseen est une bonne approximation des profils de vorticité et de vitesse dans le cœur du tourbillon hélicoïdal. Par conséquent, un modèle original des sillages des rotors est présenté. De plus, la présence d une vitesse axiale dans le cœur du tourbillon est prouvée. Elle diminue lorsque le rapport de vitesse au bout de pale, λ, augmente. Ces profils peuvent être utilisés pour définir un écoulement de base plus réaliste afin d améliorer les études de stabilité des tourbillons hélicoïdaux existantes où une taille du cœur et des profils de vitesse arbitraires sont utilisés (Hattori & Fukumoto 24 (23), Okulov & Sørensen 27 (34), Fukumoto & Okulov 25 (6), Okulov 24 (35), Widnall 972 (54)...). Après la caractérisation du cœur du tourbillon, une étude de sa stabilité a été faite pour des perturbations contrôlées. Dans cette étude, peu de paramètres interviennent ce qui permet d avoir des conclusions plus précises: les propriétés du tourbillon sont solutions des équations de Navier-Stokes. C est une étude visqueuse d un tourbillon de longueur finie. Les résultats sont trouvés en bon accord avec ceux de Widnall 972 (54), obtenus par une étude analytique, non-visqueuse, d un filament infini de vorticité et représentant des résultats de référence pour les études de stabilité de ce type d écoulement. Deux familles principales de modes ont été trouvées: les modes en phase qui sont stables et les modes en opposition de phase qui sont les plus instables. Une relation entre les (angles) déphasages et les positions spatiales des zone de formation de l appariement local est établie. Elle est liée aux fréquences des perturbations. Les travaux réalisés dans le cadre de cette thèse ont fait l objet des publications suivantes: Mohamed Ali and Malek Abid, Helical vortex: how far is the infinity?, The 23 International Conference on Aerodynamics of Offshore Wind Energy Systems and Wakes (ICOWES23),7-9 June 23, Copenhagen, Denmark. (Appendix (4)) Mohamed Ali and Malek Abid, Self-similar behaviour of a rotor wake vortex core, Journal of Fluid Mechanics 74, R doi:.7/jfm (Appendix 2 (5)) L article suivant est soumis à la revue Physics of Fluids: Mohamed Ali and Malek Abid, Voticity and axial velocity in the core of a rotor-wake helical-vortex. (Appendix 3) Un article est en cours de rédaction:

18 6 CHAP. : RESUMÉ Mohamed Ali and Malek Abid, The stability study of a helical tip vortex generated by a single blade rotor..2 Méthode numérique et validations Les modèls utilisés pour l étude des sillages des rotors sont, principalement, basés sur des hypothèses de la symétrie de l écoulement (Hansen et al. 26 (2), Vermeer et al. 23 (5)) et modélisent le rotor par un disque actif. Leur principal inconvénient est que les forces sont distribuées uniformément le long du disque (direction radiale et azimutale). Pour franchir cette limite, une méthode plus adéquate a été développée par Sorensen et Shen 22 (47). Cette méthode consiste à coupler les équations tridimensionnelles de Navier-Stokes à une approche aérodynamique bidimensionnelle pour le calcul des forces appliquées par la pale sur le fluide. Ces forces sont calculées pour différentes positions radiales le long d une ligne fictive en se basant sur les données aérodynamiques du profil de pale utilisé (figure.2). Ces dernières sont fonctions de l angle d attaque et du champ de vitesse local et elles sont données sous forme de tableaux de données. Pour contourner les singularités numériques, les forces sont distribuées dans le domaine de z U z U z F z L θ U θ φ γ α Ωr D F θ Figure.2: Section 2D de la pale: paramètres aérodynamiques calcul en utilisant un produit de convolution entre les forces calculées et un noyau gaussien bidimensionnel η ε : Le noyau gaussien est donné par: f ε = f η ε. (.) η ε (p) 2D = ε 2 π e( p/ε)2, (.2)

19 .2 Méthode numérique et validations 7 où p est la distance entre les points d application de la force linéique et les centres des cellules de calcul appartenant aux plans orthogonaux à la pale (figure.3). Le paramètre ε est choisi de l ordre de quelques mailles pour ne pas influencer la structure du sillage généré. Ω r z θ Figure.3: Distribution des forces volumiques dans le domaine du calcul..2. Les équations de Navier-Stokes L écoulement est décrit par les équations de Navier-Stokes, incompressibles ( ρ =, ρ densit e du fluide) et tridimensionnelles: U t +(U. )U= p+ Re 2 U+f(U,λ),.U=, où U est le champ de vitesse, p est la pression, Re est le nombre de Reynolds et f représente les forces volumiques, le rapport de vitesse au bout de la pale est λ = ΩR b /W, où Ω est la vitesse de rotation. Ces équations sont résolues en coordonnées cylindrique. Les singularités introduites par les dérivées premières et secondes sur l axe (r = ) sont contournées en utilisant des grilles décalées et des variables primitives définies par Verzicco et Orlandi 996 (5) (q r = rv r,q θ = v θ,q z = v z ), où v r, v θ et v z sont les composantes de vitesse radiale, azimutale et axiale, respectivement. La vitesse d entrée axiale W est utilisée comme échelle de vitesse et le rayon de la pale R b est utilisé comme échelle de longueur. L échelle de temps est donné par R b /W.

20 8 CHAP. : RESUMÉ.2.2 Implémentation numérique Les équations décrites précédemment sont résolues en utilisant un schéma de différences finies, centré d ordre deux. L équation de Poisson tridimensionnelle qui garantit l incompressibilité de l écoulement est résolue en deux étapes: (i) une inversion rapide (Fast Fourier Tronsform) selon la direction azimutale pour obtenir N θ problèmes de Helmholtz bidimensionnels, (ii) la résolution de ces problèmes est réalisée utilisant une technique de réduction cyclique. Pour le calcul du champ de vitesses solénoïdal (à divergence nulle), un schéma de Runge-Kutta d ordre trois et un schéma implicite Crank-Nicholson sont combinés. Cette méthode est d ordre deux pour les termes visqueux et d ordre trois pour les termes convectifs. Elle est d ordre deux globalement en temps. Le code est parallélisé suivant le langage OpenMP, et les simulations sont donc faites sur des machines à mémoire partagée. Un bon accord a été trouvé entre le Speed-up de notre code et la formule d Amdhal reliant le Speed-up et le nombre de processeurs utilisés (figure.4), pour une fraction séquentilelle, s, du code. 2 Speed Up Linear Amdahl s=4% Our Code 8 T /T N Number of Threads Figure.4: Speed up: comparison avec la loi thd Amdahl avec une fraction séquentielle du code s=4%..2.3 Validation de la méthode Le solveur développé pour résoudre les équations de Navier-Stokes a été validé en comparant les taux de croissance d un jet instable obtenu numériquement à ceux obtenus à l aide de la théorie d instabilité linéaire par Abid et Brachet 993 (2) et Abid et Brachet 998 (3). Un bon accord a été trouvé (figure.5). L implémentation de la méthode de la Ligne Active est validée en comparant:

21 .3 Caractérisation d un tourbillon hélicoïdal généré par un rotor mono-pale 9 Figure.5: Evolution temporelle de l énergie: comparaison avec la théorie linéaire. les vitesses induites obtenues numériquement avec celles calculées par des formulations analytiques pour des filaments de longueur finie (Saffman 992 (4) et de longueur infinie (Fukumoto et Okulov 25 (6)) de tourbillons (figure.6). les variables aérodynamiques obtenues le long de la pale par les simulations numériques, et celles mesurées expérimentalement pour le même modèle de pale utilisé (Hadrien Bolnot 22 (8)). Les résultats ont montré que la méthode utilisée prédit bien l aérodynamique de la pale du rotor (figure.7)..3 Caractérisation d un tourbillon hélicoïdal généré par un rotor mono-pale.3. Méthode de caractérisation La caractérisation du sillage, en particulier le tourbillon du bout de pale, est basée sur le calcul des moments de vorticité. Elle est faite dans un plan r z non orthogonal à l axe local du cœur faisant un angle χ avec cet axe. Cet angle, comme le montre la figure.8-gauche, est donné par χ = arctan(h/(2π))=arctan(/λ), (.3) avec h=2π/λ est la pas d hélice (du tourbillon). Pour les rapports de vitesses au bout de pale, λ, considérés dans cette étude, cos(χ) est proche de (.98 < cos(χ) <.9965). Par conséquent,

22 2 CHAP. : RESUMÉ Figure.6: Moyenne azimutale des vitesses axiales au niveau du plan du rotor. En vert: tourbillon de longueur infinie; en bleu: tourbillon de longueur finie L 6R; en rouge: résultat de la simulation numérique pour un sillage de longueur L 6R. l erreur survenant du fait que la caractérisation est faite dans le plan r z est inférieur à 2%. Un système de coordonnées local est défini par (ρ,θ,ζ ), son origine est placé au centre du tourbillon (R c,z c ) qui correspond à la position de la vorticité azimutale maximale (figure.8- droite). Vu que la position spatiale pour une hélice est définie par z=hθ/(2π), sa relation avec le temps peut être donnée par z=hω(t t )/(2π). Ainsi, l étude de l évolution temporelle des propriétés du tourbillon est analogue à celle de son évolution spatiale. Les temps considérés sont exprimés en fonction de nombre n de rotation de la pale donné par n = Ω(t t )/(2π). Dans cette étude, les âges du tourbillon considérés sont n 7. La procédure de la caractérisation est définie par les étapes suivantes: le centre du vortex (R c,z c ) qui correspond à la vorticité maximale est déterminé par ω θ (R c,z c ) = maxω θ pour le domaine de calcul D défini par z Z c [ h/2,h/2] et r R c [ R b /2,R b /2]; la taille du cœur du tourbillon ainsi que son ellipticité sont calculées en se basant sur les moments de vorticité:

23 .3 Caractérisation d un tourbillon hélicoïdal généré par un rotor mono-pale 2 = D =8.34 =9.6 /(W R b ) /(W R b ) r/r b r/r b.5 =2.3.4 /(W R b ) r/r b Figure.7: Circulation le long de la pale: comparaison des résultats des simulations numériques et les données expérimentales pour différents rapports de vitesse au bout de pale λ: numérique, expérimental, désign de la pale. 6 5 θ (rad) z Figure.8: (Gauche) Présentation d un tourbillon hélicoïdal dévelopé dans un plan(θ z) pour λ = 5 et Re=2: plan du cœur du tourbillon, plan (r z). (Droite) Forme elliptique du cœur.

24 22 CHAP. : RESUMÉ la circulation Γ est: Γ= ω θ dd, (.4) D les moments d ordre deux sont: I = ω θ r z dd, I 2 = D l angle d ellipticité est: D ω θ z 2 dd, I 3 = ω θ r 2 dd. (.5) D tan(2θ e )= 2I I 2 I 3, (.6) la taille du cœur selon ses deux axes principaux est telle que: σ 2 z = Γ [(2sinΘ e cosθ e )I + (cos 2 Θ e )I 2 + (sin 2 Θ e )I 3 ], (.7) σ 2 r = Γ [( 2sinΘ e cosθ e )I + (sin 2 Θ e )I 2 le rayon du dispersion du cœur est donc donné par: + (cos 2 Θ e )I 3 ], (.8) a 2 c = σ 2 z + σ 2 r = I 2+ I 3 Γ. (.9).3.2 Taille du cœur et auto-similarité de la vorticité et de la vitesse azimutale Un résultat principal de cette étude est la détermination de l évolution de la taille du cœur a c en fonction de l âge du tourbillon n. La plupart des études précédentes, principalement expérimentales, ont déterminé l évolution de la taille du cœur pour une seule rotation de pale (n=). Dans notre étude, la taille est calculée pour une période de six rotation de pale (n=6) avant apparition des zones de turbulence. Pour le cas bidimensionnel, un fluide incompressible et de viscosité homogène est caractérisé par sa circulation constante Γ comme l avait montré Poincaré 89 (38). Ainsi, l évolution temporelle de la taille du cœur est donnée par la relation suivante: (a c (t) 2 a c (t ) 2 )= 4(t t ). (.) Re Malgré que l écoulement étudié est tridimensionnel, la dynamique du tourbillon dans un plan axial (r z) est asymptotiquement bidimensionnelle (figure.9). Pour certains nombres de Reynolds, Re, et rapports de vitesses au bout de la pale, les interactions nonlinéaires entre les

25 .3 Caractérisation d un tourbillon hélicoïdal généré par un rotor mono-pale (a c (t) 2 a c (t ) 2 )Re 3 2 (a c (t) 2 a c (t ) 2 )Re 3 2 (a) Ω(t t )/(2π) 4 (b) Ω(t t )/(2π) 4 (a c (t) 2 a c (t ) 2 )Re 3 2 (a c (t) 2 a c (t ) 2 )Re 3 2 (c) Ω(t t )/(2π) (d) Ω(t t )/(2π) Figure.9: Influence du nombre de Reynolds sur l évolution temporelle de la taille du cœur: Re=5, Re=, Re=5, Re=2 pour (a) λ = 5, (b) λ = 6, (c) λ = 9, (d) λ = 2. La loi de diffusion linéaire, bidimensionnelle, est tracée en ligne discontinue. tourbillons adjacents sont importantes et l évolution du cœur s éloigne du comportement 2D. Ces résultats sont importants parce que les études analytiques précédentes considèrent le cœur du tourbillon comme circulaire et de taille constante. Il est intéressant de noter que la taille initiale a c (t = ) est trouvée égale à 4% de la taille de la pale R b. D autre part, la vorticité ω et la vitesse azimutale u Θ dans le cœur du tourbillon sont trouvées auto-similaires selon chaque direction Θ (figures. -.). Pour la vorticité, la variable de similarité est donnée par: tandis que la vorticité normalisée est donnée par: ω(ξ,t) ω s (t) ξ = ρ ρ /2, (.) = ω(ξ). (.2) Elle est trouvée indépendante du temps et donc, tous les profils de vorticité (pour différents

26 24 CHAP. : RESUMÉ instants) se superposent. Ainsi, la vorticité ω est donnée par: ω(ξ)=exp( ln(2)ξ 2 ). (.3) Ce résultat est similaire au cas du tourbillon du Lamb-Oseen (symétrique, 2D et circulaire). Pour la vitesse azimutale, la variable de similarité est donnée par: 5.8 ω 5 ω (a) ρ/a c (b) 2 3 ξ Figure.: (a) Profils de vorticité dans le cœur du tourbillon en fonction du nombre n de tours du rotor obtenus pour λ = 2 et Re = 2. Discontinu n =, n = 3, n = 5, n = 7. (b) Forme auto-similaire de la vorticité en fonction de la variable d auto-similarit é ξ (equation.). L équation.3 est tracée en ligne continue. 2ρ η = ρ/2 a +, (.4) ρb /2 avec ρ/2 a et ρb /2 qui correspondent à la mi-hauteur de la vitesse maximale notée us θ. La vitesse normalisée est, ainsi, donnée par: u Θ (η,t) u s Θ (t) = ũ Θ (η). (.5) Elle est indépendante du temps et peut être écrite sous la forme générale suivante: ũ Θ (η)=a( exp( (η/b) 2 ))/η. (.6) Les valeurs de a et b sont proches de ceux trouvées pour le cas du tourbillon Lamb-Oseen comme le montre le tableau.. Ces profils peuvent être utilisés pour améliorer les études de stabilité des tourbillons hélico daux à courte longueur d onde (proche de la taille du cœur a c ).

27 .3 Caractérisation d un tourbillon hélicoïdal généré par un rotor mono-pale 25 Table.: Les valeurs (obtenues pour différents Re) de a et b de l équation.6. Le cas 2D correspond à la vitesse azimutale dans le cœur du tourbillon Lamb-Oseen. λ D a.94,.97.86,.9.87,.89.84,.88.9 b.58,.63.54,.55.55,.57.52, ũθ (a) (b) Figure.: (a) Profils de vitesse azimutale dans le cœur du tourbillon en fonction du nombre n de tours du rotor obtenus pour λ = 2 et Re=2. Discontinu n=, n=3, n=5, n=7. (b) Forme auto-similaire de la vitesse en fonction de la variable d auto-similarité η (equation.4). L équation.6 est tracée en ligne continue..3.3 Un modèl du cœur du tourbillon hélicoïdal Plus généralement, un modèle du cœur du tourbillon hélicoïdal est déterminé en utilisant les profils des valeurs moyennes de vorticité et de vitesse azimutale, ŵ=(2π) 2π ωdθ et Û θ = (2π) 2π u θ dθ, respectivement. Ces fonctions sont auto-similaires (figure.2-.3) et la variable de similarité est donnée par: où ρ /e est le rayon du cœur du tourbillon déterminé par η = ρ ρ /e, (.7) ŵ(ρ /e,t)= e max(ŵ(ρ,t)).

28 26 CHAP. : RESUMÉ La vorticité et la vitesse normalisées sont données par: Ũ θ (η)= w(η) = ŵ(η, t) max(ŵ(η,t)) = exp( η2 ), (.8) Û Θ (η,t) max(û Θ (η,t)) = a( exp( η2 ))/η, (.9) avec a=η /( exp( (η ) 2 ), où η correspond à la position radiale de la valeur maximale de la vitesse (Ũ θ (η )=). Nous avons monté que ces profils sont ceux d un tourbillon de Lamb-Oseen avec la même ω 5 5 ω ρ η Figure.2: (gauche) Profils de vorticité moyenne dans le cœur du tourbillon obtenus pour Re = 2 et λ = 8.4. (droite) Forme auto-similaire de la vorticité en fonction de la variable d auto-similarité η (equation.7). L équation.8 est tracée en ligne dicontinue..4.5 Ûθ.3.2. UΘ ρ η Figure.3: (gauche) Profiles de vitesse azimutale moyenne dans le cœur du tourbillon obtenus pour Re=2 et λ = 8.4. (droite) Forme auto-similaire de la vitesse en fonction de la variable d auto-similarité η (equation.7). L équation.9 est tracée en ligne discontinue. procédure de mise en échelle et que l erreur relative provenant de la moyennisation ne dépasse pas le 7%. D autre part, la taille du cœur ρ /e suit la loi de la diffusion linéaire bidimensionnelle (figure

29 .4 Etude de stabilité du tourbillon hélicoïdal 27.4). Par conséquent, le tourbillon de Lamb-Oseen est une bonne approximation de la vorticité du 25 2 (ρ /e (t) 2 ρ /e (t ) 2 )/ν Ω(t t )/(2π) Figure.4: Evolution temporelle de la taille du cœur ρ /e pour Re=2 et λ = 8.4. La loi de diffusion bidimensionnelle est tracée en ligne discontinue (8πn/Ω). cœur du tourbillon hélicoïdal. Ce modèle peut être utilisé comme un profil de base pour les études d instabilité des tourbillons hélicoïdaux. Finalement, la présence d une vitesse axiale dans le cœur du tourbillon a été prouvée, et contrairement à la vorticité et la vitesse azimutale, cette composante de vitesse n est pas autosémilaire. Elle diminue lorsque le rapport de vitesses au bout de pale, λ, augmente. Le rapport de vitesse q (azimutale/axiale) est donné en fonction du λ comme suite: pour λ < 8.4, q 2; pour λ 8.4, q 6; pour λ > 8.4, q..4 Etude de stabilité du tourbillon hélicoïdal.4. Perturbation et taux de croissance Dans cette partie, la stabilité d un tourbillon hélicoïdal généré par un rotor mono-pale est étudiée. L écoulement est perturbé en utilisant une vitesse de rotation dépendante du temps Ω= f(t) ce qui permet d avoir une variation du pas d hélice h. La perturbation est contrôlée par son amplitude d, normalisée par le pas d hélice h, et sa fréquence /T normalisée par la vitesse de rotation du cas non perturbé Ω D. 2πΩ D Ω(t)= 2π d p θ(t) T h sin( T ) (.2)

30 28 CHAP. : RESUMÉ avec θ(t) la position angulaire de la pale. Un taux de croissance est calculé pour différents nombres d onde et est normalisé par la quantité Γ/(2h 2 ), où Γ est la circulation du cœur du tourbillon dans le cas non perturbé. Considérant trois tourbillons adjacents dans le plan(r z), la distance entre le premier tourbillon et le deuxième est notée d 2 et celle entre le deuxième et le troisième est notée d 23. Le calcul du taux de croissance revient à déterminer la croissance temporelle de la différence entre ces deux distances d = d 2 d 23 lors de sa phase linéaire..4.2 Résultats L étude d instabilité est faite pour différentes fréquences de perturbation et pour une amplitude d p /h=3%. Les résultats (figure.5) montrent que les taux de croissance des modes les plus instables sont en bon accord avec le taux de croissance de l instabilité d appariement dans le cas bidimensionnel et celui trouvé pour le cas d une allée infinie d anneaux de vorticité (Levy et Forsdyke 928 (29), Bolnot et al 2 (9), Bolnot 22 (8)). Les modes sont classés en trois familles de modes selon les fréquences de perturbation: 2.5 Growth rate σ frequency /T Figure.5: Taux de croissance normalisé (par Γ/(2h 2 )) en fonction de la fréquence de perturbation /T comparé à celui de l instabilité d appariement bidimensionnelle σ 2D = π/2 (ligne discontinue). Les modes en phase: Ils correspondent aux fréquences en nombre entier (/T = n, avec n entier). Ces modes sont stables sur la période de simulation considérée. Pour ce type de mode, les tourbillons adjacents se déplacent en phase, leur distance est constante le long du sillage. Un exemple de ce type de mode est illustré par la figure.6.

31 .4 Etude de stabilité du tourbillon hélicoïdal 29 Ils correspondent aux modes stables identifiés dans le travail analytique de Widnall 972 (54) qui a considéré un filament infini d un tourbillon hélicoïdal à taille du cœur constante. Les formes ondulées des modes restent stables jusqu à l apparition d une zone de turbulence due: à la fusion des tourbillons adjacents comme le cas non perturbé; La destruction du tourbillon à cause des grands rayons (locaux) de courbure. Ces rayons sont grands pour des fréquences plus grandes..8 Rbθ/(2π) z Figure.6: Mode en phase stable /T = 2. (gauche) Présentation dévelopée du mode; tourbillon en ligne bleu continue, tourbillon 2 en ligne rouge discontinue et tourbillon 3 en ligne verte pointillée. (droite) Présentation 3D du mode (le champ de vorticité ω = ω est montré). Les modes en opposition de phase: Ce sont les modes les plus instables et ils correspondent aux modes les plus instables prédits par Widnall 972 (54) et Bolnot 22 (8). Ils correspondent aux fréquences en demi-entiers /T = n+/2,n =,,2,... Ils sont caractérisés par le déplacement en opposition de phase des tourbillons adjacents qui provoque un appariement de tourbillon local comme le cas bidimensionnel et le cas d une allée infinie d anneaux de vorticité. Leurs taux de croissance sont proches de la valeur σ 2D = π/2, taux de croissance de l appariement 2D (Lamb 932 (28)). Ils sont mesurés sur la partie linéaire de la croissance de la distance d. La figure.7 illustre le mode /T = 3/2. Autres modes: Pour les autres fréquences, les modes sont, aussi, instables et leurs taux de croissance sont inférieurs (localement) à ceux des modes d appariement. Le déphasage entre les tourbillons adjacents pendant leur déplacement (Φ ±π et Φ ) amplifie la perturbation jusqu à la destruction de la forme hélicoïdale du tourbillon du bout de pale. Le mode /T = 5/4 est présenté dans la figure.8. Il est important de noter que la zone de croissance linéaire de l instabilité, cruciale pour la mesure des taux de croissance, est limitée par la fusion des tourbillons sous les effets visqueux.

32 3 CHAP. : RESUMÉ.8 Rbθ/(2π) z Figure.7: Mode en opposition de phase, instable /T = 3/2. (gauche) Présentation dévelopée du mode; tourbillon en ligne bleu continue, tourbillon 2 en ligne rouge discontinue et tourbillon 3 en ligne verte pointillée. (droite) Présentation 3D du mode. La norme de la vorticité est montrée..8 Rbθ/(2π) z Figure.8: Mode instable /T = 5/4 avec un déphasage, π. (gauche) Présentation dévelopée du mode; tourbillon en ligne bleu continue, tourbillon 2 en ligne rouge discontinue et tourbillon 3 en ligne verte pointillée. (droite) Présentation 3D du mode. La norme de la vorticité est montrée. Ces effets sont dus au faible nombre de Reynolds utilisé dans les simulations qui est limité par la plus haute résolution que nous avons pu atteindre (23 millions de points de maillages) et qui est nécessaire pour la convergence des solutions des équations du Navier-Stokes. Par conséquent, l une des perspectives importantes du présent travail, est l amélioration de la capacité du calcul du code développé en utilisant des méthodes plus performantes pour la parallélisation tel que le langage MPI. Ceci permet d utiliser le code comme un outil de simulations numériques directes pour l étude des configurations plus complexes des sillages des rotors: écoulement non-uniforme, rotor à plusieurs pales, interaction des sillages des rotors (parc éolien). Ainsi, l étude d instabilité peut être généralisée à des configurations plus complexes que le sillagede la simple pale de rotor utilisée dans cette étude.

33 Chapter 2 Introduction 2. General context In fluid mechanics and fields related to it, the vortex dynamics remains until today a subject of intense research (Meleshko & Aref 27(32)). Vortices at several scales are observed in natural and industrial flows as such in the Jupiter Great Red-Spot, in body wakes or during the mixing process. They can be in different forms as such in planar vortices, straight lines or vortex tubes (wakes of tip wings) and also vortex rings (fuel injection in engines). An important form of vortices is the helical vortex where, as evidenced by its name, the flow is mostly a spinning motion about a helical axis. The importance of the study of this kind of flow is its presence in many industrial applications, especially in rotor wakes as such behind in wind turbines and helicopters (figure 2.). The wake is generally divided into two major parts: Figure 2.: Illustration of the helical vortex wake: (left) helicpter wake (Eurocopter), (right) wind turbine. (i) the near wake where the flow dynamics is strongly influenced by the aerodynamics of the rotating blade(s), and (ii) the far wake, where the rotor characteristics do not influence the wake dynamics and where the flow breaks down and turbulence features are formed (Vermeer et al. 23 (5), Hansen et al. (2)). For a general description, the wake generated by a rotating-blade consists of two helical-vortex structures at the root of the blade (the hub) and at the tip section. These complex structures 3

34 32 CHAP. 2: INTRODUCTION result from the rapid roll-up of the vortex sheet continuously generated at the trailing edge of the rotating blade as the consequence of the variable circulation along the rotor. Obviously, for a multi-bladed rotor a system formed by many pairs of these helical vortices is obtained in the wake of the rotor. For the helicopter rotor, the wake dynamics is very important for predicting aerodynamic loads on the blades. The instability of the wake can generate critical flight conditions up to the appearance of the Vortex Ring State which is very dangerous for the helicopter (illustrated in figure 2.2). For wind turbine industry, the most important source of renewable energy (Sørensen 2 (45)), Figure 2.2: Vortex Ring State in a helicopter rotor wake (figure taken from Drees & Hendal 95 (4)). turbines are organised in, increasingly, larger parks where turbines may be in the wake of others (figure 2.3). The power production of wind turbines, in such grouping, is reduced (Sanderse 29 (4)). It has been, also, proven that the fatigue loading is more severe when the turbine is located in a wake consisting of stable tip vortices than if the vortices are unstable and have broken down (Sørensen 2 (46)). Therefore, the knowledge of the mechanism of the wake dynamics and the understanding of the transition from near wake to far wake, due to its destabilization, are very important in such applications because they allow the improvement of the rotor design tools and the resolution of problems caused by vibrations and by structural fatigue (or reduce their effects). Therefore, it is an important scientific challenge to answer these two elementary questions: What are the properties of the wake, in particular those of the helical tip vortex, the most important component? How do the transition from the near wake to the far wake is established?

35 2.2 Wake model and helical vortex characterization 33 Figure 2.3: Turbine wake interaction in the Horns Rev offshore wind turbines park ( 2.2 Wake model and helical vortex characterization Despite many research works, until now, there is no a universal model for the flow dynamics of rotor wakes. In particular, the properties of the helical vortex generated at the tip of blades remain an open research subject. Many theories have been developed to understand the wake properties and so to improve design methods. The flow behind rotor system, in most studies, is basically modelled by a system of helical vortex filaments or tubes (Levy & Forsdyke 928 (29), Widnall 972 (54), Hardin 982 (2), Okulov 24 (35), Fukumoto & Okulov (6), Okulov & Sørensen (34)). These models are not solutions of Navier-Stokes equations and, ad-hoc, circular vortex cores are used. The wake is considered as a system of an array of infinite helical-vortex lines (or tubes) with constant, small, circular core and the vorticity inside the core itself is imposed (such as the Rankine vortex). A few models relating the helical system properties to those of the blade rotor were developed. They are mainly: the Joukowsky model (92), where the circulation along the blades is considered constant. The other one is the Betz model (99) considering that the optimum (rotor) efficiency is obtained when the distribution of circulation, along the blades, generates a rigidly helical wake (2.4). Later, analytical and numerical investigations of the rotor wake were based on the actuator disc theory where the Blade Element Momentum (Glauert 935 (8)) theory is used to model the blades. The actuator disc is developed later to give the generalized actuator disc (Mikkelsen 23 (33)), the actuator line method (Sørensen & Shen (47), Troldborg 28 (49), Ivanell 29 (24)) and the actuator surface Method (Shen & al. 29 (43)). These methods allow to generate more realistic rotor wakes (wind turbine) but little attention have been devoted to the vortex core properties. During the last two decades, there were many studies performed to achieve a better understanding of the wake vortex system. Thompson et al. 988 (48), using a laser Doppler velocimeter, compared between tip vortex properties of a rotating blade and a fixed one. They

36 34 CHAP. 2: INTRODUCTION Figure 2.4: Optimum rotor concept: (a) the ideal propeller of Jokowsky, (b) the ideal propeller of Betz (figure taken from Okulov & Sørensen 27 (34)). found that the two vortices differ considerably in structure. This proved the limits of the previous approach based on a fixed-wing tip vortex data to analyse the helicopter rotor wakes. Young 23 (56) in his study derived a number of simple analytical expressions for estimating the core size of tip vortices in the near wake of rotors in hover and axial-flow flight. He found that the vortex core size in the near wake is indeed a function of rotor thrust and induced power losses. This work is, almost, the only parametric investigation of the effect of rotor characteristics and operating conditions on vortex structure, strength, and core size but it needs more experimental measurements to refine the proposed analysis. Generally, most analytical studies carried out on rotor wake studies were based on models of known vortices (Rankine vortex, Batchelor vortex, etc) or more simply a vortex with constant core and circulation. Drobev et al. 28 (2) ((3)) succeeded to reconstruct the 3D averaged velocity field and kept track of the created tip vortices using PIV measurements. They analyzed the vortex induced velocities and characterized the tip vortices. But, unlike many other studies, they found that the vortex core radius remains constant with the vortex age. Sherry et al. 23 (44) characterized the wake behind a Tjaereborg turbine using planar particle image velocimetry for different operating conditions. Results were compared to the numerical results of Ivanell 29 (24) and Troldborg 28 (49). The meander phenomena was shown and well characterized, and was found as an important indication of instabilities. Although the phase-averaged data, used in this work, give accurate general properties of the wake, they can mask the true properties of the vortex. Recently, also, a high-resolution PIV system was used by Yang et al. 2 (55) to characterize the wake structure downstream a horizontal axis wind turbine placed in an atmospheric boundary layer wind-tunnel. The transient behaviour of the helical tip vortex and turbulence flow structures in the near wake were quantified. The dynamic loads acting on the blade were analyzed in correlation with detailed flow field measurements. Wake properties were found to vary significantly as a function of the tip speed ratio of the wind turbine tested. View the important computing time and resources needed, there was little numerical studies in literature that are devoted to details of the tip helical vortex. Whale et al. 2 (53) compared results from experimental data and simulations from ROVLM (Rotor Vortex Lattice Method) code. This study allowed more details about the vortex wake geometry but the numerical method used

37 2.3 Stability analysis of a helical vortex system 35 was inviscid, and therefore the numerical analysis was limited. There were many other studies performed to achieve a better understanding of the wake vortex system. However, all these studies are limited to a vortex age of two or three blade revolutions (Ivanell 29 (26), Dobrev et al. 28 (3), Troldborg 28 (49), Young 23 (56), Whale et al. 2 (53), Thomspson et al. 988 (48)). Furthermore, the axial velocity in the vortex core, which is an important feature for the stability of trailing line vortices from a fixed wing, is a matter of debate for the helical vortex system from a rotor. For some authors, the axial velocity in the core is a sharp negative (with respect to the direction of the blade rotation) peak. For others, the axial core velocity distribution is positive and self-similar. In contradiction with the two previous studies, McAlister 24(3) has found that the axial flow in the core is composed of two concentrated zones moving in opposite directions. Recently, direct numerical simulations, based on helical symmetry assumptions, were done for a single helical vortex and for an array of helical vortices. They show that the core size follows the 2D diffusion law for a small helical pitch h. The core size leaves this behavior as the pitch increases (Piton 2 (36), Piton et al. (37), Delbende et al. 22 ()). The main limitations of the previous studies, are their use of an infinite helical-vortex system or their inviscid character. Therefore, the properties of the helical vortex core are still an open subject needed to more understand, and the effect of different parameters are to be determined such as the Reynolds number Re, the pitch h, the number of the blades... Therefore, more investigations about helical vortices generated behind rotors are needed to develop more realistic vortex wake model. In the present study, the wake of a single blade rotor is studied for different values of Reynolds number Re and pitch h. To better understand the wake properties and especially those of the vortex core, the rotor is embedded in a uniform axial flow. Solutions of the full, incompressible, 3D Navier-Stokes equations are considered, so no ad-hoc properties are used here and the generated helical vortex is of finite length and dependent on the rotor blade aerodynamics. 2.3 Stability analysis of a helical vortex system Levey and Forsdyke 928 (29) were among the first who studied the stability of a infinite helical vortex filament of finite core. They showed that such a vortex is moving forward in the direction of its axis with a steady speed and rotates about its axis with a uniform angular velocity. They proposed a criterion of transition from stability to instability related to the pitch of the helix h: the filament is unstable as long as its reduced pitch h/2π is less than.3r b, where R b is the helix radius. Their results were found to be in error as proven by Widnall 972 (54) in her stability study of the infinite helical-vortex filament submitted to small sinusoidal displacements of its center-line. Her inviscid analysis considers the effect of the entire perturbed filament on the selfinduced motion of each element. She demonstrated that the helical vortex filament is unstable to this type of perturbation and the unstable modes were listed based on their nondimensional

38 36 CHAP. 2: INTRODUCTION wavenumber (γ/k ), representing the number of waves per cycle of the helix, into three types of modes: short-wave instability (figure 2.5-), long-wave instability and mutual induction instability. The most unstable ones were found to correspond to half-integer number of waves per cycle of the helix: γ/k = /2,3/2,5/2... (figure 2.5-(2,3,4)). Their dynamics is similar to the vortex pairing instability since adjacent vortices are 8 out-of-phase. This study was extended to an array of multiple interdigitated, right circular, infinite helical Figure 2.5: Instability mode shapes: the short-wave instability, the mutual-inductance modes with γ/k = 5/2 and 3/2 and the long-wave instability with γ/k = /2 (Widnall 972 (54)). vortices by Gupta & Loewy 974 (9). They demonstrated that the growth rates of the unstable modes increase as the helical pitch h decreases, increase as the number of considered helices increases and decrease as the wavenumber (γ/k ) increases. The modes with integer wavenumbers are the least (locally) unstable ones, except for wavenumber and in the single helix case which are stable (figure 2.6). These results are found to be mainly the same in the case of helicopter rotor wakes investigated by Bhagwat & Leishman (7) in their eigenvalue analysis using a numerical wake model. A more general analysis study was done by Okulov 24 (35) and Okulov & Sørensen 27 (34), where the tip vortices were considered embedded in a wake flow generated by the trailing vortex sheet of the rotor and the presence of a hub vortex was taken into account. It was found that the stability of the helical vortex system is strongly dependent on the (wake) vorticity extension. The principle limits of these works are that: (i) the helical vortices studied are of infinite length (except in Bhagwat & Leishman (7)), (ii) the vortex core vorticity is taken with an ad-hoc profile and (iii) all the studies are inviscid, so the core radius is considered constant. In numerical studies, the viscous effects were considered, for the first time, by Walther et al. 27 (52) in the stability analysis of the helical vortex system. Later, Ivanell et al. 2 (25) studied the stability properties of the Tajaereborg wind turbine. In these numerical investi-

39 2.3 Stability analysis of a helical vortex system 37 Figure 2.6: Instability grwoth rate as a function of the wavenumber for a single helix (Gupta 974 (9)). gations, the wake was generated using Large Eddy Simulations of the Navier-Stokes equations combined with the actuator line method (Sørensen & Shen 22 (47)). Applying a controlled harmonic perturbations near the tip of the the blades, they found the instability arises only for some specific frequencies and type of modes. Excellent agreement was found in similar experimental studies. Felli et al. (5) could follow the growth of some instabilities in the wake. They demonstrated that the instability of the hub vortex is weakly related to that of tip vortices and they highlighted, through visualisations, a multi-step grouping mechanism, driven by a mutual induction, between adjacent spirals. Using a multiblade rotor, it was difficult to separate the different, superimposed, unstable modes described by Widnall 972 (54) unlike the case of a single blade studied by Bolnot (8). In his work, Bolnot showed that different unstable Widnall-modes can be generated separately. It was the first time that these modes are seen experimentally (hence in a finite length configuration), especially the pairing instability whose growth rate was found in a good agreement with analytical predictions (figure 2.7). The previous analytical, numerical and experimental studies of the stability of the helical vortex system allow to highlight some instability mechanisms and phenomena related to it. Obviously, they demonstrated that many parameters influence in the instability generation and growth such as the properties of the flow where the rotor is embedded and the number of the blades. Then, it is important to discriminate between these parameters in order to better understand the instability appearance (Sørensen 2 (46)). Therefore, a part of the present study is devoted to study the stability of the wake of a single blade rotor placed in a uniform axial flow (difficult to control in tunnel), subject to controlled (simple) perturbations (an angular velocity dependent in time). Direct numerical simulations of the Navier-Stokes equations are used to study the evolution of the perturbed helical wake (no turbulence model is introduced).

40 38 CHAP. 2: INTRODUCTION Figure 2.7: Local pairing instability of a tip helical vortex (Bolnot 22 (8)). 2.4 The present work The purpose of the present thesis is to study the properties of the rotor wake in a configuration relatively simple to better understand the dynamics of this type of flow. Special attention is given for the helical tip vortex, most important component of the wake system. Therefore, a numerical solution of the full three dimensional (3D), incompressible, Navier- Stokes (NS) equations in cylindrical coordinates are obtained. The body forces, exerted by the blade on the encountering fluid are obtained using a (local) two dimensional (2D) aerodynamic approach (the Actuator Line Method by Sørensen & Shen 22 (47)). The, finite difference, home-made, code is parallelized using OpenMP language, and the direct numerical simulations are done on shared-memory machines: computational facilities of Centre Informatique National de l enseignement supérieur-france (CINES) are used. The NS solver is validated by reproducing growth rates of an unstable jet flow as done by Abid & Brachet 993 (2) and Abid & Brachet 998 (3). The validation of the actuator line method implementation is done, firstly, by comparing the numerical simulations results with some analytical formulations (Hardin 982 (2), Saffman 992 (4), Fukumoto & Okulov 25 (6)). A semi-analytical investigation is done to highlight the difference between the infinite length formulation and the finite length one. Secondly, the comparison between aerodynamic properties along the blade obtained numerically and those obtained by a previous experimental work (Bolnot 22 (8)) show that this method predicts well the aerodynamic properties along the blade (although the boundary layer and the stall problems are avoided). An important part of the present work is devoted to better understand the dynamics of the wake and its properties, taking in account both the viscous effects and the finite length of the generated vortex. There is no imposed condition on the vortex core properties: size and vorticity profile are obtained as solutions of the NS equations. The rotor, with an angular velocity Ω, is placed in a uniform axial inflow with a characteristic velocity W and a viscosity ν. The blade has a characteristic radius R b. The numerical inves-

41 2.4 The present work 39 tigation is conducted to characterize (for many blade s revolutions) the wake structure and the evolution of the helical tip vortex. The computations are done for different combinations between four Reynolds numbers, R e W R b /ν =(5;;5;2), and four different Tip Speed Ratios, λ ΩR b /W = (5;6;9;2), which represent large operational conditions of the rotor. An interesting result found is that vorticity and azimuthal velocity profiles inside the tip helical vortex-core are self-similar. The Lamb-Oseen vortex is found to be a good approximation for the vorticity in the core of a helical vortex, in the wake of a rotor. Thus, an original model for wind turbine (near) wake vortex is presented. Furthermore, the presence of an axial velocity in the core is highlighted. This velocity decreases when the tip speed ratio increases. These profiles should be used to ameliorate instability studies like those done in Fukumoto & Okulov 25 (6), Okulov 24 (35), Okulov & Sørensen 27 (34) and Widnall 972 (54) where ad-hoc vorticity and velocity profiles in the core are used, and should be used especially as next order basic-profiles in the study of short wavelength (compared to the vortex core length) instabilities ( Hattori & Fukumoto 24 (23)). After the characterization of the vortex core, a stability analysis of the generated helical tip vortex, under controlled perturbations, is done. In this study, few parameters are involved and so more defined conclusions can be made. There is no imposed ad-hoc properties of the vortex core, which is freely determined as a solution of the full NS equations: it is a viscous study of a, finite length, helical tip vortex. It is shown mainly that results of Widnall 972 (54), done for inviscid study of an infinite helical filament, are recovered for finite Reynolds numbers and vortex length. The presence of two, principal families of modes is highlighted: the stable in-phase modes and the most unstable, outof-phase, modes. A relation between the phase shift of adjacent vortices and the spatial positions where the (local) vortex pairing instability occurs is found, and it is related to the perturbation frequency. The work presented in this thesis was the subject of the following publications: Mohamed Ali and Malek Abid, Helical vortex: how far is the infinity?, The 23 International Conference on Aerodynamics of Offshore Wind Energy Systems and Wakes (ICOWES23),7-9 June 23, Copenhagen, Denmark. (Appendix (4)) Mohamed Ali and Malek Abid, Self-similar behaviour of a rotor wake vortex core, Journal of Fluid Mechanics 74, R doi:.7/jfm (Appendix 2 (5)) The following paper is submitted to Physics of Fluids: Mohamed Ali and Malek Abid, Voticity and axial velocity in the core of a rotor-wake helical-vortex. (Appendix 3) The following paper is in preparation: Mohamed Ali and Malek Abid, The stability study of a helical tip vortex generated by a single blade rotor.

42 4 CHAP. 2: INTRODUCTION 2.5 Thesis outline The outline of the thesis is the following: In Chapter 2, the numerical method implemented (from the scratch) during this work is described: the governing equations, mathematical formulation and different numerical aspects are presented. The validation of the method is then discussed using comparison with analytical formulations for finite (Saffman 992 (4)) and infinite length helical filaments (Fukumoto & Okulov 25 (7)) and experimental data (Bolnot 22 (8)). In Chapter 3, the numerical method is used to generate a helical tip-vortex behind a single-blade rotor and to characterize it for different operational conditions (Tip speed ratios λ and Reynolds numbers Re). The global properties are studied such as the pitch of the helix and the expansion of the wake. A special attention is given to the vortex core properties (its size and the profiles of vorticity and velocity inside it). Finally, in Chapter 4, the stability of the helical tip-vortex is studied. The method used to perturb the flow and the way to measure the growth rate are presented. The results are discussed and compared with previous works (Widnall 972 (54), Bolnot 22 (8)).

43 Chapter 3 Numerical method and Validations The numerical method used in the present investigation is presented. The method consists on solving the three dimensional (3D), incompressible, Navier-Stokes (NS) equations in cylindrical coordinates where the body forces are obtained using a (local) two dimensional (2D) aerodynamic approach (Sørensen & Shen 22 (47). In the present chapter, we present the governing equations and mathematical formulation of the problem (how to avoid the singularity at the origin, boundary conditions...) and different numerical aspects encountered in this work (parallel computing, spatial discretization, time-stepping...). The validation of the NS solver was done comparing the numerical growth rate of a 3D instability with that obtained using linear instability theory. The full method (NS solver and Actuator Line) was validated comparing numerical results with analytical models and experimental data. 4

44 42 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS 3. Governing equations We consider a flow governed by the incompressible three-dimensional Navier-stokes equations in cylindrical coordinates. The singularities introduced by the first and second derivatives around the axis (r = ) are bypassed using a staggered grid and using primitive variables defined by Verzicco & Orlandi 996 (5) (q r = r.u r,q θ = U θ,q z = U z ) where U r,u θ and U z are the velocity components in the radial, azimuthal and axial directions, respectively. In terms of q i, the equations are written in a dimensionless form as follows: The continuity equation (. v=) The momentum equations in conservative form q r r + q θ θ + r q z z =. (3.) Dq θ Dt Dq r Dt Dq z Dt = p r θ + [ ( Re r r r q θ r = r p r + [ r ( q r Re r r r = p z + [ ( r q z Re r r ) q θ r q θ r 2 θ q θ z q r r 3 θ ) + r 2 2 q r θ q r ) + 2 q z r 2 θ q z z 2 q θ θ ] + f θ, (3.2) ] + f r, (3.3) z 2 2 r ] + f z, (3.4) with Dq θ Dt Dq r Dt q θ t q r t + rq θ q r r 2 + q 2 θ r r θ + q θ q z, (3.5) z + ( ) q 2 r + ( qθ q ) r + q rq z r r θ r z q2 θ, (3.6) Dq z Dt q z t + r q r q z r + r q θ q z θ + q2 z z, (3.7) where the following identities have been used for the q θ equation development: ( r r r q ) θ q θ r r 2 = ( ) rq θ, r r r (3.8) rq θ q r r 2 = q θ q r + q θ q r r r r r 2. (3.9) The Reynolds number is defined as Re = W R b /ν, with W is the maximum injection velocity (along the z direction), R b is the blade length and ν is the kinematic viscosity. Time is made non-dimensional using the time scale R b /W.

45 3.2 Discretized computational domain 43 The evolution equation for a passive scalar: Dχ Dt = Pe [ ( r χ ) + 2 χ r r r 2 θ ] χ z 2, (3.) with Dχ Dt χ t + q r χ + q θ χ r r r θ + q zχ z. (3.) The Peclet number is defined as a function of the molecular diffusivity D m of the passive scalar: Pe= W.R b D m = Re.Sc, Sc=ν/D m (Schmidt number) (3.2) 3.2 Discretized computational domain The computational domain is cylindrical (Fig 3.) and defined by a longitudinal length Lz and a maximum radius R max. z(k) r(i) p,χ qr qθ p,χ qθ qz qr qz Cell (i,j,k) Cell (,j,k) Figure 3.: (Left) Computational domain. (Right) 3D representation of computational cells. In the present work, we have considered a uniform staggered grid in all directions but nonuniform grids or multi grid algorithms can be, also, considered to solve elliptic equations in complex regions (such as along the blade). We have considered (N r,n θ,n z ) points. The primitive variables q i are calculated at different points located at the center of the surfaces of each cell while the pressure and the passive scalar are calculated at the center of cells. The locations of the different variables, using the indexes of

46 44 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS discretization, are defined as follows: with q r (i, j+ /2,k+ /2), q θ (i+/2, j,k+ /2) (3.3) q z (i+/2, j+ /2,k), p,φ, χ (i+/2, j+ /2,k+ /2), r i =(i ) r, i=..(n r ), r=r max /(N r ), θ j =( j ) θ, j=..(n θ ), θ = 2π/(N θ ), z k =(k ) k, k=..(n z ), z=l z /(N z ). Staggered grids were first used by Harlow & Welch 965 (22), and they became popular since then. In such grids, the scalar variables (pressure, density, and etc.) are located in the center of the neighbouring cells, and the velocity components are placed in the middle of the corresponding sides, as shown in Fig 3.. The staggered grids enable the pressure difference to be represented with second-order accuracy at the grid center using velocity components at adjacent grids instead of alternate grid points, and therefore the decoupling between pressure and velocities does not take place. In fact, the strong coupling between pressure and velocities is the biggest advantage of staggered grids and it is the main reason for the popularity of the method. This helps to overcome convergence problems and oscillations in pressure and velocity fields. However because the variables are not defined at the same grid point, accurately evaluating the nonlinear convective terms becomes difficult. The discretization of the region around r = is the most important feature of the present scheme. It might seems that in equations ( ) there are lots of singularities. However, they are only apparent. In fact the advantage of using a staggered grid is that only the component q r is evaluated at the grid point i = (r = ), and there q r = by definition. The equation for q r can be then discretized for all i 2. The equations for q z and q θ at i=3/2 require the evaluation of radial derivatives in the region around r =. The fact that q r = at i = avoids the equations of q θ and q z at i = and the radial derivatives of the convective terms can be discretized without any approximation. 3.3 Boundary conditions The boundary conditions used are as follows: Along θ direction We have considered a condition of periodicity along the azimuthal direction. Then in the interval[θ,θ N ], this condition is written as

47 3.3 Boundary conditions 45 Along the radial direction q r (r,θ,z)=q r (r,θ N,z), q θ (r,θ,z)=q θ (r,θ N,z), q z (r,θ,z)=q z (r,θ N,z). For the treatment of the axis r=, we have used the fact that q r = by definition: q r (r=,θ,z)=, q θ (r=,θ,z)=, q z r (r=,θ,z)=. We chose to take q θ (r=,θ,z)= because: a single helical filament vortex with an infinite length has only two motions: an axial displacement with a constant axial velocity V and a rigid-body rotation along its axis (r=) with a constant angular velocity Ω as shown in many previous studies (Levy and Forsdyke 928 (29), Widnall 972 (54)...), and so there is no flow through the axis; in the experiment done by Bolnot (8) and that we try to reproduce numerically (figure 3.2), the rotating blade is mounted on a motor shaft such that the shaft is located in the wake, and so there is no flow through the axis of the cylindrical domain (r = ) (eliminated by the presence of the shaft); this type of boundary condition is simple to implement (compared to the Neumann conditions for example) since there is no flow through the axis. It is important to note that if the inflow is not perpendicular to the rotor plane, a flow through the axis of the cylindrical domain is present, and so this boundary condition is not valid and other conditions must be implemented. For the treatment of the boundary r = Rmax, we have used a Neumann conditions for the different variables q i r =, i=r,θ,z.. Along the axial direction For z= we have imposed a Dirichlet condition for each variable q i (r,θ,)=q i(r,θ), i=r,θ,z. This condition allows us to study the phase of fluid injection in the computational domain and to generate different types of flows (axial flows, helical flows,...).

48 46 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS Figure 3.2: The experimental setup mounted by Bolnot (8) (U canal W ) For z=l z we have used a Neumann boundary condition q i z =, i=r,θ,z. Figure 3.3: Boundary conditions (r-z) representation. 3.4 Fractional step method Fractional step method together with projection method advance the momentum equation and enforce the continuity condition in separate steps. These methods use Hodge decomposition

49 3.4 Fractional step method 47 theorem, which states that any vector can be decomposed into the sum of a vector with zero divergence (solenoidal) and a vector of zero curl (irrotational). The incompressible Navier-Stokes equations in vector form are: with boundary conditions (to illustrate the method) v t +(v. )v = p+/re 2 v+f, (3.4).v =, (3.5) v Ω = v b. (3.6) The Hodge theorem states a vector field v can be decomposed into a divergence-free part v and a gradient of a potential φ. That is, with.v=. v = v+ φ, (3.7) The divergence-free part of the fictitious velocity vector v, that does not satisfy the continuity constraint, can be obtained by a projection onto the orthogonal subspaces of divergence-free vectors and its complement Symbolically the projection operator and its complement are: v = P(v ), (3.8) Φ = Q(v ). (3.9) P = I [( 2 ).], (3.2) Q = [( 2 ).]. (3.2) The projection operator P and its complement Q are purely representations for the principle and numerical steps needed to solve the governing equations for the velocity and pressure variables. Using this principle, the projection method first approximates the momentum equation and the boundary condition v t +(v )v = p + Re 2 v + g, (3.22) B(v )=, (3.23) with an intermediate velocity v, not satisfying the continuity constraint in general. Here p is a quantity which gives an approximation of the pressure. After the intermediate velocity v is solved by advancing in time Eq(3.22), the Hodge de-

50 48 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS composition allows to write v = v+ φ. (3.24) Taking divergence of Eq 3.24 and substituting.v= gives a Poisson equation for φ with the boundary condition, 2 φ = v, (3.25) ˆn φ Ω =. (3.26) Then the divergence-free velocity v and the pressure can be recovered with The pressure can be written into the gradient form: v = v φ, (3.27) p = p + f(φ). (3.28) p= p + f( φ), (3.29) here f represents the dependence of p on φ, it commutes with. We discuss later the method used to do these calculations at each time step. In the literature, we can found various projection methods. They differ principally in the way used to approximate the pressure-like quantity p, the function f(φ), the boundary condition B(v ), the advective terms(v )v, and the advancement in time (explicit or implicit). In Kim and Moin s (985 (27)), pressure-free, projection method, p is set to zero in Eq(3.22), and v is obtained at each time step t by: Then, the pressure, if needed, is given by the relations v = v+ t φ. (3.3) p= p + f(φ), f(φ)=φ ( t/2re) 2 φ. To sum-up, the fractional step method consists on the numerical calculation of the velocity field in two steps: calculate a non-solenoidal velocity field v by the resolution of the momentum equation; correct the velocity field to satisfy the continuity equation, this step is based on a Poisson equation for the pressure.

51 3.5 Time-stepping procedure: non-solenoidal velocity field calculation Time-stepping procedure: non-solenoidal velocity field calculation 3.5. Adopted method In the present work, the Runge-Kutta method was used for the convective terms because of their stability compared with other method. For the viscous terms, we have used an implicit Crank- Nicholson scheme (Kim & Moin 985 (27), Rai & Moin 99 (39)). This method is second-order accurate in time for the viscous terms and third-order accurate in time for the convective terms, the overall accuracy being second order in time. The momentum equations can be written in a time-discretized form as: where ˆq i q l i t [ = γ l Hi l + ρ l Hi l α l G i p l + α l (A iθ + A ir + A iz ) ( ˆq i q l i ) ], (3.3) 2 ˆq i is the non-solenoidal velocity field. H i contains the convective terms and those viscous terms with a single velocity derivative: H θ = H r = [ rq θ q r r 2 + q 2 θ r r θ + q ] θ q z 2 q r z Re r 3 θ, [ ( ) q 2 r + ( qθ q ) r + q ] rq z r r θ r z q2 θ + 2 q θ Re r θ, (3.32) H z = r q r q z r + r q θ q z θ + q2 z z, the operator A i j denote discrete differential relations for the viscous terms: A θr A rr A zr A θθ = A rθ = A zθ A θz = A rz = A zz = [ ( Re r r r ) ] r r 2, = [ r ], Re r r r = [ ], (3.33) Re r r r = [ 2 ] Re r 2 θ 2, = [ 2 ] Re z 2, G is the gradient operator:

52 5 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS ( G= r r, r θ, ), z α l, γ l and ρ l are the coefficients of the time advancement scheme (Runge-Kutta coefficients)(rai & Moin 99 (39)) α = 8/5 α 2 = /5 α 3 = /3, γ l = 8/5 γ 2 = 5/2 γ 3 = 3/4, ρ = ρ 2 = 7/6 ρ 3 = 5/2. (3.34) We stress here that since ρ = each time step is self-starting. Then the advancement by a time step t = t l+ t l is done by an integration over the intermediate intervals (α t), the intermediate instances are: τ = t l, τ = t l + α t, τ 2 = t l +(α + α 2 ) t, τ 3 = t l + Σα l t = t l+. By using the increment ˆq i = ˆq i q l i, the equation (3.3) can be written as: ( β l (A iθ + A iz + A ir )) ˆq i = (3.35) t[γ l H l + ρ l H l α l G i p l + α l (A iθ + A iz + A ir )q l i], where β l = α l t/2. The implicit treatment of the viscous terms, in this case,requires for the solution the inversion of large sparse matrices. To obtain the solution, these are reduced to three tridiagonal matrices by a factorization procedure with an error O( t 3 )(Beam & Warming 976 (6)) ( β l (A iθ + A iz + A ir )) = ( β l A iθ )( β l A iz )( β l A iz ) Omitting the t 2 and higher order terms, we obtain β 2 l (A iθ A iz + A iθ A ir + A iz A ir )+β 3 l A iθ A iz A ir. ( β l A iθ )( β l A iz )( β l A ir ) ˆq i = (3.36) t[γ l H l + ρ l H l α l G i p l + α l (A iθ + A iz + A ir )q l i]. We note here the presence of the pressure in the equation in a different way from the free pressure projection and the fractional step method. The introduction of the pressure q l in equation (3.3) simplifies the need for the boundary conditions for ˆq. The resolution of the discretized equation (3.36) consists on the inversion of three matrices successively: first inversion: along the azimuthal direction ( β l A iθ ) ˆq i = t[γ l H l + ρ l H l α l G i p l + α l (A iθ + A iz + A ir )q l i], (3.37)

53 3.5 Time-stepping procedure: non-solenoidal velocity field calculation 5 second inversion: along the axial direction third inversion: along the radial direction ( β l A iz ) ˆq i = ˆq i, (3.38) ( β l A ir ) ˆq i = ˆq i. (3.39) The intermediate velocity fields ˆq i and ˆq i have not any physical signification, then the boundary conditions for these fields are the same as those applied for ˆq i. In the next paragraph, we describe the terms of the equation along the θ-direction Inversion along the θ-direction The equation to solve is ( β l A iθ ) ˆq i = t[γ l H l + ρ l H l α l G i p l + α l (A iθ + A iz + A ir )q l i], the A iθ represents the viscous operators for each direction i: A θθ = A rθ = A zθ = [ 2 ] Re r 2 θ 2. The matrices of these operators are obtained by spatial discretization and are given by: A iθ = with the coefficients given by: b i () c i ()... β a i (2) b i (2) c i (2) b i (N θ 2) c i (N θ 2) α... a i (N θ ) b i (N θ ) for the radial velocity field { a r ( j)= Re( θ) 2 r 2 ( j), c r( j)= } Re( θ) 2 r( j), b r( j)= (a r ( j)+c r ( j)),, for the azimuthal velocity field { a θ ( j)= Re( θ) 2 r 2 ( j), c θ( j)= } Re( θ) 2 r( j), b θ( j)= (a θ ( j)+c θ ( j)),

54 52 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS for the axial velocity field { } a z ( j)= Re( θ) 2 r 2 ( j+ 2 ), c z( j)= Re( θ) 2 r( j), b z( j)= (a z ( j)+c z ( j)). The coefficients α and β are obtained by applying the periodicity boundary condition q(i,,k)= q(i,n θ,k), which gives α = a i (N θ ) and β = c i (). For inverting the matrix( β l A iθ ), we have used a fast algorithm based on the cyclic reduction which its operation count is O(N θ log 2 (N θ )). 3.6 Pressure Poisson equation solving The pressure field in an incompressible fluid flows is described by Poisson equation. This equation is a consequence of the projection method. After computing the non-solenoidal velocity field ˆq i, the solenoidal velocity field for the next Runge-Kutta step is given by the correction equation (obtained from the Hodge decomposition theorem) q l+ i ˆq i = α tg i φ l+, (3.4) where φ is a scalar calculated by enforcing the free divergence condition q l+ =, and given by the equation Lφ l+ =+ D ˆq, (3.4) α l t where L=DG is the Laplace operator in cylindrical coordinates and D is a modified divergence operator in cylindrical coordinates: D= r θ + r r + z, L= 2 r 2 θ 2 + r r r r + 2 z 2. As shown in equation (3.4), the calculation of the scalar φ needs an inversion of the Laplace operator. A method based on a cyclic reduction is used. The boundary conditions needed for the equation must be consistent with those imposed for the non-solenoidal field ˆq: periodicity condition along the azimuthal direction: φ(i,,k)=φ(i,n θ,k), Neumann condition along the radial condition:

55 3.6 Pressure Poisson equation solving 53 φ r r=,r max =, Neumann condition along the axial condition: 3.6. Fast and direct method φ z z=,lz =. The method used to solve the 3D Poisson equation is based on three steps: first step: a fast Fourier transform along the azimuthal direction. The Poisson equation of the present problem (equation 3.4) can be rewritten as r The Fourier transformation in θ of equation (3.42) gives r r r φ r + 2 φ r 2 θ φ = rhs. (3.42) z2 r r φ r + 2 φ z 2 + λ r φ 2 = rhs, (3.43) where φ is the Fourier transform of φ. This equation is a 2D Helmholtz problem. Second step: solution of the 2D Helmholtz problem. We have used for this step a method developed by Schumann & Sweet 976 (42). This method is based on the use of a cyclic reduction to solve a modified Helmholtz problem in cylindrical coordinates with Neumann boundary conditions on a staggered grid. For an N r xn z grid the operation count of this algorithm is proportional to N r N z log 2 (N r N z ) and about N r N z storage locations are required. Third step: inverse Fourier transform along the azimuthal direction. Once the solution φ is found, we apply an inverse Fourier transform to obtain the unknown φ in the physical space. For the Fourier transform and the inverse Fourier transform, a Fast Fourier Transforms are used. We have tested this method by solving the equation f = g, g= f exact (3.44) where f exact is given by: ( r 3 f exact (r,θ,z)= 3 )( z 2 (R + R 2 )r 2 3 +(R R 2 )r 3 ) 2 (H + H 2 )r 2 +(H H 2 )z cosθ.(3.45) The function f obtained numerically is compared to f exact given by the equation As shown by Figure 3.4, this method is a second order in space and its operational count is proportional to Nlog 2 N, with N = N r N θ.

56 54 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS Figure 3.4: Space accuracy order of the method used for Poisson s equation solving. 3.7 Body force computing The Blade Element Momentum (BEM) method developed by Glauert 935 (8) consists on dividing each blade into N elements. Each element will experiences a slightly different flow as they have a different rotational speed (Ωr), a different chord length (c), a different twist angle (γ) and different induced velocities. BEM method involves dividing up the blade into a sufficient number of elements and calculating the flow at each one. Overall performance characteristics are determined by numerical integration along the blade span. The local fluid velocity relative to the rotating blade is determined from the velocity triangle z U z U z F z L θ U θ φ γ α Ωr D F θ Figure 3.5: 2D Airfoil cross section.

57 3.7 Body force computing 55 (Figure 3.5) U rel = Uz 2 +(Ωr U θ ) 2. (3.46) Here, Ω denotes the angular velocity and U z and U θ are the velocities in the axial and tangential directions, respectively. The flow angle between U rel and the rotor plane is determined as: The local angle of attack is given by: U z φ = tan ( ). (3.47) Ωr U θ α = φ γ, (3.48) where γ denotes the local pitch angle. Knowing the angle of attack and the relative velocity, the lift and the drag forces (per spanwise length) are found as f 2D =( L, D)= 2 ρu 2 rel c(c L e L,C D e D ), (3.49) where c denotes is the chord length, C L and C D are the lift and drag coefficients (and are functions of the angle of attack α and the Reynolds number Re), and are given as tabulated data. The unit vectors e L and e D are defined in the direction of the lift and drag respectively. The Force per spanwise unit length is written as the vector sum: F = L + D = Fz e z + F θ e θ, (3.5) with The loading for an annular area of differential size is: F z = Lcosφ + Dsinφ, (3.5) F θ = Lsinϕ Dcosφ. (3.52) f = df rdθ = 2 ρu rel 2 cb(c L e L +C D e D ), (3.53) 2πr where B is the number of blades. The force f is defined in cylindrical coordinates as: and the resulting volume force: f =( f r, f θ, f z), (3.54) f = f dz. (3.55)

58 56 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS To avoid singular behaviour of the aerodynamic forces distributed radially along a line, the forces are distributed among the neighboring node points using a Gaussian kernel. This is obtained by taken the convolution of the computed load f rθz and the regularization kernel η ε : f b ε = f b rθz η ε, (3.56) where b is the blade number index. The regularization kernel can be defined as a 3D Gaussian or a 2D Gaussian: η ε (p) 3D = ε 3 π 3/2 e( p/ε)2, (3.57) η ε (p) 2D = ε 2 π e( p/ε)2, (3.58) where p is the distance between cell centered grid points and points at the actuator line. The parameter ε is a constant introduced as a smearing factor that serves to adjust the concentration of the regularization load and its size is related to the local grid size. According to some previous works (Mikkelsen 23 (33), Troldborg 28 (49), using the 3D Gaussian smoothing results in inconsistencies near the tip region. So, the 2D Gaussian distribution is favored and it on distributing the forces on the blade in the directions normal to it (in a plane orthogonal to the actuator line) (Figure 3.6). The 2D Gaussian distribution is controlled Ω r z θ Figure 3.6: Distribution of the applied forces in the actuator line model. by the parameter ε. The choice of the value of ε is critical and have important impact on the wake structure (because it affects the numerical discontinuity at the blade tip), and should be as

59 3.7 Body force computing 57 small as possible in order to minimize the influence the wake-structure. The algorithm used to determine the distribution of body forces can be described as follows (Figure 3.7): Blade r P B θp P B P S M2 S M θb O r O zb O zp z Figure 3.7: Distribution of the forces. For each cell center P, compute the dot product SP =( OP OO ). O B=r P cos(θ P θ B ). For the point where s p R B, compute the distance d P = SP = rp 2 sin2 (θ P θ B )+(z P z B ) 2, which is used to calculate the body force at P. Determine the points (on the blade) M and M 2 limiting S along the blade: O M O S O M 2 (r r s r 2 ) and ζ =(r s r )/ r. Determine the body force at S using a linear interpolation: Compute the body force at the cell center P: f z (r S ) = ζ f z (r )+( ζ) f z (r 2 ), (3.59) f θ (r S ) = ζ f θ (r )+( ζ) f θ (r 2 ). (3.6) f z (r P ) = f z (r S )ηε 2D (d P ), (3.6) f θ (r P ) = f θ (r S )ηε 2D (d P ), (3.62)

60 58 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS where η 2D ε (d P )= ε 2 π exp[ (d P/ε) 2 ]. (3.63) General algorithm i). Initialization (U, Ω, θ B ). ii). For each radial position and for each blade, determine local velocities (U z,u θ ) (solution of the Navier-Stokes equation). iii). Determine the aerodynamic forces on each blade element: Determine the relative flow characteristics U rel = Uz 2 +(Ωr U θ ) 2, (3.64) Determine the aerodynamic coefficients: U z φ = tan ( ), Ωr U θ (3.65) α = φ γ. (3.66) Using the local angle of attack α, read C L (α) and C D (α) from tabulated data. Compute C z and C θ. Determine body forces in each blade element: f z (r) b = ρcu 2 rel 2rdθdz (C L cosφ +C D sinφ), (3.67) f z (r) b = ρcu 2 rel 2rdθdz (C L sinφ C D cosφ). (3.68) iv). Determine the body forces distribution in the full computational domain: For each cell center P, compute the dot product SP =( OP OO ). O B=r P cos(θ P θ B ). For the point where s p R B, compute the distance d P = SP = rp 2 sin2 (θ P θ B )+(z P z B ) 2, which is used to calculate the body force at P. Determine M and M 2 which limiting S along the blade: O M O S O M 2 (r r s r 2 ) and ζ =(r s r )/ r.

61 3.8 Parallel Computing 59 Determine the body force at S using an interpolation: Compute the body force at the cell center P: where v). Solve the Navier Stokes equations: f z (r S ) = ζ f z (r )+( ζ) f z (r 2 ), (3.69) f θ (r S ) = ζ f θ (r )+( ζ) f θ (r 2 ). (3.7) f z (r P ) = f z (r S )ηε 2D (d P ), (3.7) f θ (r P ) = f θ (r S )ηε 2D (d P ), (3.72) η 2D ε (d P )= ε 2 π exp[ (d P/ε) 2 ]. (3.73) U t + U. p U = ρ + µ 2 U + f,. U =. vi). Rotate the blade. vii). Return to step (ii). 3.8 Parallel Computing The implementation of the method was parallelized using the Open-MP libraries. The speed-up is presented in figure 3.8. It follows Amdahl s law, gievne by equation 3.74, with a sequential fraction of the program estimated as s=4% T N = T N +(N )s. (3.74) 3.9 Navier-Stokes solver validation: 3D perturbations In this section, we make a simulation that can be considered as a quantitative validation of our Navier-Stokes solver. It consists on the study of the spatio-temporal evolution of a disturbance of an unstable stationary flow. An immediate validation is obtained using the comparison between the spatial wave number and temporal frequency of the disturbance evolution with those of the initial excitation, and it is very interesting to compare the growth rate of the instability with that given by the linear instability

62 6 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS 2 Speed Up Linear Amdahl s=4% Our Code 8 T /T N Number of Threads Figure 3.8: Speed up: comparison with the Amdahl s law for a sequential fraction of the program s=4%. theory (in the linear regime). Considering the velocity field given by Eq 3.75 as the stationary profile initially imposed in the entire domain (Batchelor vortex): q r = q r (t =,r,θ,z) =, q θ = q θ(t =,r,θ,z) =.5( e r2 )/r, (3.75) q z = q z (t =,r,θ,z) = e r2. This profile is perturbed using the excitation of an azimuthal mode of instability at z= using the following profile: where the disturbance function f d is: q z (t,r,θ,)=(+ f d (t,θ))q z, (3.76) f d (t,θ)=a p.cos(ωt mθ), (3.77) with a p =.5 m=4 ω =.824

63 3.9 Navier-Stokes solver validation: 3D perturbations 6 In this case, the azimuthal mode exited is the 4 th with an excitation frequency f defined as ω = 2π f =.824. The simulation parameters are R max = 4, L z = 5 N r N θ N z = Re= T f = 3 t =. As it is illustrated by Figure 3.9, we can observe the evolution (in space and time) of the radial velocity. Figure 3.9: 3D representation of the time evolution of radial velocity for an excited azimuthal mode,(a) t=.5, (b) t=.5, (c) t=7.5, (d) t=5. Figure 3. shows a 2D representation of the time evolution of the radial velocity at (z = 2). It is clear that the code conserves the axisymmetry of the flow. To validate quantitatively the results obtained, we have measured the characteristics parameters of the spatio-temporal evolution of the disturbance. Time frequency: Figure 3. shows that the radial velocity has a temporal oscillatory behavior. It is characterized

64 62 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS Figure 3.: 2D representation of the time evolution of radial velocity for an excited azimuthal mode at z=2,(a) t=7.5, (b) t=5, (c) t=22.5, (d) t=3. by its frequency ω DNS = 2π/T where T is the period shown in the figure. We have found ω DNS =.852 which represents an error of.26% compared to the excitation frequency ω =.824. We remark that we have a good agreement between the excitation and the frequency obtained in the simulation. Axial wave number According to the linear instability theory (Abid 28 (), Abid & Brachet 998 (3), Delbende et al. 998 (), Abid et Brachet 993 (2)), the present case corresponds to the most unstable azimuthal mode which is characterized by the axial wave number k z = 2.2 and a growth rate σ =.3322 As shown figure 3.2, the axial wave number obtained from the DNS code is equal to(k z ) DNS = 2π/L = 2.32 (L = 2.67 is the wave length measured). Compared to the theoretical case, the simulated results seems to be acceptable (relative error 7%). Growth rate of the disturbance Figure 3.3 represents a comparison between the time evolution of the radial kinetic energy obtained from the linear instability theory and that obtained from our DNS code simulations. It

65 3. Actuator Line Validation: comparison with Biot-Savart law and analytical solutions 63 Figure 3.: Time evolution of the radial velocity for an excited azimuthal mode at the points,(a)(r= 2/2,θ =,z=), (b) (r= 2/2,θ =,z=2). Figure 3.2: 2D representation of the time evolution of the radial velocity: axial wave number L estimation. is shown that there are three phases for the disturbance evolution. The first phase is a transient regime. The second phase is characterized by a linear evolution. For this phase we have estimated a rate growth σ DNS =.33, so a result with a relative error equal to.67%. This good agreement between theoretical and simulated results validates the DNS code developed. For the third phase, there is a nonlinear saturation of the disturbance time evolution. 3. Actuator Line Validation: comparison with Biot-Savart law and analytical solutions To validate the implementation of the Actuator Line Method, we compare, in a first time, the velocity fields generated behind a blade using the present method with those given by some analytical model (filament and tube vortex). In this part, we discuss also the assumption that the helical vortex wake, such as the wind turbine wake, can be considered as an infinite vortex filament or tube. In the real life, vortices in helical

66 64 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS Figure 3.3: Time evolution of the energy: comparison with linear instability theory. wakes are of finite length. We compare the velocity field induced by an infinite helical vortexfilament and the velocity field induced by a finite length one. The analytical solution given by Fukumoto & Okulov 25 (6) is used for the infinite case and a Biot-Savart (Saffman 992 (4)) formulation is used for the finite case. Then we compare analytical formulations with numerical results. Figure 3.4: Helical vortex system: (left) an infinite helical vortex-tube, (right) a semi-infinite helical vortex-tube. The rotor plane is defined as the z= plane. 3.. Velocity field induced by a finite-length helical vortex-filament We consider a right-handed helical vortex-filament (figure 3.5) of circulation Γ and constant pitch h, with a finite length L=2πhN c (N c is the number of coils considered), the axis of which

67 3. Actuator Line Validation: comparison with Biot-Savart law and analytical solutions 65 is that of a cylinder with radius R. The position of the vortex filament is given, in a cylindrical L=2hN c z 2 e θ e r R y 2 e θ M(r,θ,z) e r R 2 2 Figure 3.5: Geometry of the helical vortex filament coordinate system,(r,θ,z), z being the axial direction, by the following equation: r (θ )=R e r (θ )+hθ e z. (3.78) The velocity induced by the finite filament on a point M ( r (θ)=r e r (θ)+z e z ) is given by the Biot-Savart relation (Saffman 992 (4)) as follows: V F ( r) = Γ t (s ) ( r r (s )) 4π Helix r r (s ) 3 ds, r / Helix, (3.79) where t is the unit tangent vector to the helical vortex and s is the (curvilinear) abscissa along t and is given by : t d r dθ = d r dθ = d r ds, (3.8) ds dθ = d r dθ, (3.8) r (θ ) = R e r(θ )+kθ e z, (3.82) r(r,θ,z) = r e r (θ)+z e z, (3.83) (3.84) with e r = cos(θ θ ) e r + sin(θ θ ) e θ, (3.85) e θ = sin(θ θ ) e r + cos(θ θ ) e θ. (3.86)

68 66 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS This allows us to write : ds = t = R 2 + k 2 dθ, (3.87) R 2 + k 2(R e θ + k e z), (3.88) and so, t ( r r )= R 2 + k 2 Rsin(θ θ ) Rcos(θ θ ) k r Rcos(θ θ ) Rsin(θ θ ) z kθ. (3.89) Finally, the velocity induced by the finite length filament is given as follows: for r / Helix. V F ( r) = Γ 4π [θ ] ( (r Rcos(θ θ )) 2 + R 2 sin 2 (θ θ )+(z kθ ) 2) 3/2 Rcos(θ θ )(z kθ ) Rk sin(θ θ ) k(r Rcos(θ θ )) (z kθ )Rsin(θ θ ) R 2 rrcos(θ θ ) dθ, (3.9) 3..2 Velocity field induced by an infinite helical vortex-filament For the velocity induced by an infinite filament, V I, we consider the solution given by Fukumoto & Okulov 25 (6): u r (r, χ)= Γ 2πrk [ Im 4 (k 2 + r 2 )(k 2 + R 2 ) (3.9) ( 2k 2 + 9R 2 (k 2 + R 2 ) 3/2 2k2 + 9r 2 ) ] (k 2 + r 2 ) 3/2 ln( e ±ξ+iχ ), e iχ e ξ e iχ ± k 24 u θ (r, χ)= ( Γ 2πr ( 4 k 2 + R 2 4 k 2 + r 2 N b ) (3.92) N b Re n= [ ±e iχ n e ξ e iχ n + k ( 3r 2 2k 2 24 (k 2 + r 2 ) 3/2 + 9R2 + 2k 2 ) ] (k 2 + R 2 ) 3/2 ln( e ±ξ+iχ ) ),

69 3. Actuator Line Validation: comparison with Biot-Savart law and analytical solutions 67 u z (r, χ)= ( Nb Γ 2πr ( 4 k 2 + R 2 4 k 2 + r 2 ) + (3.93) N b Re n= [ ±e iχ n e ξ e iχ n + k ( 3r 2 2k 2 24 (k 2 + r 2 ) 3/2 + 9R2 + 2k 2 ) ] (k 2 + R 2 ) 3/2 ln( e ±ξ+iχ ) ), where χ = θ z/k, (3.94) = χ+ 2π(n ), N b (3.95) χ n e ξ = r(+ +R 2 /k 2 )exp( +r 2 /k 2 ) R(+ +r 2 /k 2 )exp(+ +R 2 /k 2 ), (3.96) with Re[.] and Im[.] indicate the real and imaginary parts of the complex expression, respectively. For the notations ± and (:), the upper sign or symbol in parenthesis corresponds to r< R and the lower to r R Comparisons between finite and infinite length cases a/ Radial induced velocity at the rotor plane In what follows the axial velocity of the translation of the infinite helical vortex is chosen as a velocity scale. One effect of breaking the symmetry of the wake is the creation of a radial velocity in the rotor plane. Figure 3.6 shows that the radial velocity in the rotor plane has a magnitude with the same order as the axial induced velocity (only the h/r= case is presented here, but the same behavior is found for different h/r). The presence of the radial velocity brings into question many theoretical approaches that neglect the radial velocity effect on the wind turbine wakes or does not consider it in the blade design process. The non-zero radial velocity at the rotor plane is the most important difference between the two approaches considered here (infinite and finite vortex filament). b/ Axial induced velocity As the axial induced velocity is the important component for the wake modeling and blade design, we discuss here its behavior for the two considered configurations. As shown in Figure 3.7, the axial velocity induced by a finite helical vortex-filament depends on the pitch-radius ratio (h/r) and the number of coils which form the filament considered. It s found that, for small h/r values, a high number of coils is needed to consider that V F is equivalent to V I (i.e., ( V F V I )/ V I =ε, ε ), contrarily to the cases with high h/r where few coils are sufficient. In what follows, unless otherwise stated, V F and V I are evaluated at r= and z=l/2, and only results for their axial components, Vz F, and Vz I are shown. From an experimental point of view, the helical tip vortex generated during few rotor revolutions

70 68 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS.2 <V r > θ Finite case Inifinite case r/r Figure 3.6: Radial induced velocity, averaged in the azimuthal direction, V r θ : comparison between velocity induced by the finite helical vortex and the infinite helical vortex at the rotor plane and for h/r=. <V z F Vz I >θ /<V z I >θ h/r=. h/r=.5 h/r=. <V z F Vz I >θ /<V z I >θ h/r=. h/r=.2 h/r=.5 h/r= h/r= Number of coils N c Number of coils N c Figure 3.7: Comparison between a finite vortex filament (Biot-Savart formulation) and an infinite one (Fukumoto and Okulov analytical solution) for different h/r: the azimuthal average of the axial-velocity relative-difference is presented as a function of the number of coils N c. is equivalent to an infinite helical vortex with the same parameters (Γ, R, h) when h/r O() which is the case of wind turbine rotors. But for small h/r which is the case of helicopter rotors, the helical vortex generated can not be considered equivalent to an infinite helical vortex until it is formed by a large number of coils. Based on our calculations, the relative difference between the two velocities Vz I and Vz F, for a

71 3. Actuator Line Validation: comparison with Biot-Savart law and analytical solutions 69 given h/r, follows the following law: V F z Vz I Vz I N c = = exp( N c /N c), α β + h/r. (3.97) We find α 2 and β 2 3, two coefficients approximating the characteristic number of coils Nc. Note that this law is the same (data not shown) for r if Vz F θ and Vz I θ are considered with. θ the average in the azimuthal direction θ. It is interesting to note that, the number of coils can be transformed to a non-dimensional axial distance d, behind the rotor plane, given by: d = N c h/r. (3.98) As shown in Figure 3.8, the difference between the two axial induced-velocities has a similarity behavior, i.e., all the curves collapse when the variable d is used, and V F z Vz I Vz I = exp( d/d ), d = N c h/r= α +βr/h. (3.99) It is shown in Figure 3.9 that d converges quickly to a constant value equal to α (i.e.,d = 2), it is weakly related to h/r, unlike Nc, ans so d is weakly related to the pitch-radius ratio. It is shown in the Figure 3.2 that the axial velocity field induced by the finite helical filament is equivalent to that given by an infinite helical filament for a nondimensional axial position d 8 from the rotor plane (with less than % relative difference). So the length, L, of the finite helical vortex-filament, that must be considered to ensure that the axial induced velocity coincides (near L/2) with that of an infinite vortex-filament prediction, is at least L/R = 2d 6 as sketched in Figure 3.2. Furthermore, this study justifies that the axial velocity induced by a finite helical vortex at the rotor plane is the half of the velocity induced by an infinite helical vortex if L 6R. Figure 3.22 shows the difference between the averaged axial velocities at the rotor plane (d = ) and at axial position d = Comparison between numerical simulations and analytical models In this paragraph we present a comparison of azimuth averaged velocities, of infinite and finite helical (filament) vortices, with those of a finite helical vortex (with a core) in the wake of a rotating blade, obtained in our numerical simulation using the actuator line method. A helical vortex filament, having the circulation and radius of the hub vortex, is also used in the finite and infinite computations of vortex filament velocities (see figure 3.23). The azimuth averaged velocities, in the rotor plane, are presented in figure In the finite cases, the lengths of vortices are L 6R. It is shown that average velocities of finite length vortices are in good agreement, however they are different from those of the infinite case. Note, however, the factor

72 7 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS θ )/<V I >θ z.8.6 (<V z F >θ V z I d Figure 3.8: Comparison between the averaged axial velocity induced by the finite filament and that induced by the infinite filament at r = as a function of the axial distance d, for different h/r : h/r=., h/r=., h/r=.5, h/r=.8, h/r=, h/r=2,+h/r=,. exp( d/.65), exp( d/2). 2.9 d * Figure 3.9: Characteristic distance d converge rapidly to α = 2 for increasing pitch-radius ratio h/r h/r two in magnitude between the finite and infinite cases for azimuthal and axial velocities. 3. Actuator Line validation: comparison with blade design parameters and an experimental work To test, and validate again our numerical method for the computation of a helical vortex system we have compared its results with those of the experimental work done by Bolnot 22 (8). In his work, to minimize the role of the vortex sheet and to concentrate the vorticity around the rotor tip,

73 3. Actuator Line validation: comparison with blade design parameters and an experimental work7 8 6 d F I I <V Vz>θ z /<V >θ z Figure 3.2: Axial distance behind the rotor plane as a function of Vz F h/r=., h/r=. V I z θ / V I z θ :. Figure 3.2: Sketch of the behavior of the axial velocity, Vz F, induced by a finite helical vortexfilament of length L/R = 2d = 6. Near the central part, L/2, the velocity Vz F is equal to that induced by an infinite helical vortex-filament Vz I. Bolnot has considered rotor blades designed such that the circulation and the angle of attack are constant along the blade. These two constraints fix the geometry of the blade (chord and twist) for considered control parameters (λ, Re). The A8 airfoil profile was used for the section of the tested blade (from the UIUC applied aerodynamics group, htt p :// database.html). The chord and the twist angle along the span are given in Figure In our numerical simulations we used, as depicted in Figure 3.26, the geometric and aerodynamic characteristics of the blade developed by Bolnot 22 (8) and the computed wake characteristics are compared with experimental results. A special attention is given to the computed circulation and angle of attack along the blade that, contrarily to what is done in the blade de-

74 72 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS.5 at z= at z=8r <V z F >θ r/r Figure 3.22: Axial induced velocity: V F z θ d= = /2 V F z θ d=8 for h/r=. Figure 3.23: A sketch of the tip and hub vortices used to calculate velocities for comparison with those of the actuator line method. sign, are not fixed in our simulations. Therefore, A good validation for our numerical procedure is to recover the circulation and the angle of attack of the blade design with the NS solver. For the simulations, the circulation is obtained using the lifting-line theory: Γ(r) = 2 c(r)c L(r)U rel (r), (3.) U rel (r) = Uz 2 +(Ωr U θ ) 2, (3.) where c is the chord length, C L is the lift coefficient and Ω is the angular velocity of the blade. As shown in Figure 3.27, a good agreement is found between theoretical blade design, experimental data and numerical simulations for the circulation along the blade as well as for the angle of attack (Figure 3.28). A small difference is shown around the tip of the blade, it could be the result to the three dimensional behavior of the vortex at the tip and the presence of a radial flow

75 3.2 Conclusion 73 Figure 3.24: Azimuth averaged velocities at the rotor plane. Green color: infinite vortex; blue color: finite vortex with a length L 6R; red color: numerical simulation using the actuator line method in a computational box with an axial length L 6R. along the span of the blade for both numerical and experimental case, contrary to the design case. Therefore, our numerical procedure, can reproduce theoretical values of the circulation and the angle of attack especially for the tip speed ratio of design λ D = ΩR b /W. 3.2 Conclusion In this part, the numerical method, we developed, was presented, it consists on the coupling between a 3D, incompressible, Navier-Stokes equation solver and a (local) 2D aerodynamic approach to calculate the body forces exerted by the blade on the fluid. The numerical method was validated: (i) comparing the velocity fields induced by numerical simulations and analytical formulations (for finite-length and infinite-length helical vortex), and (ii) comparing the aerodynamic properties along the blade obtained by numerical simulations and those described in a previous experimental work for the same considered blade (design and experimental data).

76 74 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS chord/r b r/r b.2 Twist angle r/r b Figure 3.25: The geometry of the blade used in the present simulations: (a) chord; (b) twist angle; (c) blade section. Theoretical Design Numerical approach Γ(r)=Γ, α(r)=α, λ D Blade Geometry: C(r), γ(r) Design Processes Actuator Line Method, λ D Blade Geometry: C(r), γ(r) Γ(r)== Γ, α(r)== α Figure 3.26: Main steps of the theoretical blade design and of the present numerical approach. In the design phase the circulation (Γ) and the angle of attack (α) are fixed and the blade geometry (chord, C(r), and twist angle γ(r)) is determined, for a given tip speed ratio λ D. In the numerical computations, this blade geometry is used and the circulation, as well as the angle of attack, of the design are retrieved (for the same λ D ) as a validation of the numerical procedure. Therefore, the Actuator Line Method can be used to study in details the helical vortex generated in rotor wake.

77 3.2 Conclusion 75 = D =8.34 =9.6 /(W R b ) /(W R b ) r/r b r/r b.5 =2.3.4 /(W R b ) r/r b Figure 3.27: Circulation along the blade: comparison between numerical simulations and experimental results for different tip speed ratios λ: numerical, experimental, design. α (deg) r/r b Figure 3.28: Angle of attack along the blade for λ = λ D : numerical, design.

78 76 CHAP. 3: NUMERICAL METHOD AND VALIDATIONS

79 Chapter 4 Characterization of the wake behind a one-bladed rotor In this part, the wake system behind the one-bladed rotor (used in the experimental work of Bolnot) is characterized. In a first time, global properties are studied (pitch, wake expansion). In a second time, the tip vortex is studied in details: the core shape and size, vorticity and velocity profiles inside the core and circulation. Finally we present a general vortex core model for helical tip vortex in rotor wake. Many simulations are done for different Reynolds numbers and Tip Speed Ratios. The computational domain is about 3 blade lengths (3R b ) in the radial direction and 2 blade lengths in the axial direction. The blade is placed at 5 blade lengths from the inflow boundary. Neumann conditions are used in lateral and outflow boundaries. As previously mentioned, the blade is subject to a uniform axial flow W during time units (R b /W ) corresponding to time steps. The Reynolds number is defined as Re= W R b /ν. The Tip Speed Ratio is defined as λ = ΩR b /W, with Ω the angular velocity of the blade. The resolution used is(n r = 3,N θ = 64,N z = 2) in the radial, azimuthal and axial directions respectively (about 23 million grid points). The CPU time needed to simulate unit of time (R b /W ), at this resolution, is about days for the sequential computing that is reduced to about.5 day for the parallel computing. Simulations were done using a desktop computer (2 processors) and the JADE cluster of the Centre Informatique National de l enseignement supèrieur-france (CINES) equipped with SGI machines (intel Xeon X556 processor). Figure 4.: Helical wake developed behind a rotating blade. 77

80 78 CHAP. 4: CHARACTERIZATION OF THE WAKE BEHIND A ONE-BLADED ROTOR 4. Vortex wake geometry As shown in figure 4.2, the helical tip vortex is well established using the actuator line method. An interesting result is the strong influence of λ on the shape of the wake as it is illustrated Figure 4.2: The Helical vortex (in red) generated behind a one bladed rotor, obtained as a solution of Navier-Stokes equations. The norm of the vorticity is shown. The hub vortex is shown in blue in figure 4.3. For low λ, the wake extends continuously away from the rotating blade and the helical structure is well identified. Contrarily, for high λ, the wake contracts and remains close to the rotor plane. So, the near wake of a rotor (here a wind turbine blade is used), where the blade characteristics influence the wake properties, can extend several rotor diameters downstream contrarily to many previous studies that consider the near wake about or 2 rotor diameters from the blade plane. It is true that the turbulence plays an important role in the wake dynamics, but many experimental studies showed that we can obtain, also, a well established helical-vortex for a distance relatively far from the rotor plane (Felli et al. 2 (5)). An important parameter characterizing the wake geometry is the tip vortex pitch that can be defined as the axial distance travelled by a particle of the tip vortex during a one blade revolution. For a particle having a helical trajectory and moving with an angular velocity, Ω, and an axial velocity, W, the pitch, h (in unit of R b ), of the trajectory is given by (hr b )/W = 2π/Ω leading to h=2π/λ for this motion with no-induction (kinematic case). Figure 4.4 shows the tip vortex pitch for both numerical results and the kinematic case. Results show a trend of decreasing pitch with increasing λ. The numerically obtained pitch is lower than the value of the kinematic case. This can be explained by the existence of the vortex core due to the viscous effects and the mutual induction which is not considered in the kinematic model. It is found that the Reynolds number, for a given λ, does not affect the pitch value although it greatly affects the core size (discussed in the following sections). The vortex wake geometry can be characterized by the spatial location of the vortex center during its evolution. As shown in Figure 4.5, the rotor wake has a radial extent greater than the blade radius for different tip speed ratios λ, in agreement with what is observed experimentally for wind turbine blades. The measured extension is about 5% for low λ and about 2% for the highest one. So the radial convection increases when λ increases. The Reynolds number has a week influence on this extension. It is important to note that there is no analytical model

81 4. Vortex wake geometry r r z z (a) (b) r r z 7 z (c) (d) Figure 4.3: The influence of λ on the shape of the wake. Contours of the vorticity in the (r, z) plane are presented: (a) λ = 4, Ωt/(2π) = 5 (b) λ = 6, Ωt/(2π) = 8 (c) λ =, Ωt/(2π) = 3 (d) λ = 5, Ωt/(2π) =. For figures (a) and (b) the wake conserves its form for longer times and it extends up to the outlet boundary. All the figures are obtained for Re =. Numerical 2π/λ.2 Pitch λ Figure 4.4: Pitch of the helical wake as a function of the tip speed ratio, λ, for numerical simulations and for the no-induction case. describing the extension of the wake, until today.

82 8 CHAP. 4: CHARACTERIZATION OF THE WAKE BEHIND A ONE-BLADED ROTOR.. Radial position.5.95 Radial position.5.95 (a) Axial position (b) Axial position Radial posistion Radial position (c) Axial position (d) Axial position Figure 4.5: Radial and axial positions of the vortex center for different Reynolds numbers: Re=5, Re=, + Re=5, Re=2 for (a) λ = 5, (b) λ = 6, (c) λ = 9, (d) λ = Vortex core properties The 3D numerical simulations allow us to avoid many difficulties and errors encountered in experimental studies due to the low PIV resolution and to the averaging of the measured quantities. Thus core details, obtained by direct numerical simulations, are shown and studied in this section. For all cases, the parameter of body forces regularization, ε, is taken equal to 6δz where δz = L z /N z is the grid spacing in the axial direction z. In fact we find that ε has an influence on the core radius of the starting vortex (vortex age less than the time of one turn, i. e., Ωt < 2π) as shown in figure Characterization procedure The characterization of the wake generated behind the blade, and the vortex system created, is based on the moments of vorticity. It is done in axial (r,z) planes, that are not orthogonal to the local vortex axis. As shown in figure 4.7 a (r,z) plane and the plane of the vortex core form an angle χ given by: χ = arctan(h/2π) = arctan(/λ), (4.)

83 4.2 Vortex core properties 8 a c ǫ/ǫ m in Figure 4.6: Influence of the smoothing force parameter, ε, on the initial (Ωt < 2π) vortex core radius. The results presented in this figure are obtained for λ = 6, Re=2 and ε min = 6δz. where h=2π/λ is the tip vortex pitch. For the tip speed ratios used in this study (5<λ < 2), 6 5 θ (rad) z Figure 4.7: Presentation of a developed helical vortex in a plane(θ z) for λ = 5 and Re=2: trace of the vortex core plane, trace of a(r z) plane. cos(χ) is practically constant (.98<cos(χ)<.9965). Thus, the error arising from the fact that the characterization of the vortex core is done in a(r,z) plane is less than 2%. To characterize the vortex, first, its center (R c,z c ) is determined. It corresponds to the spatial point with a local maximum vorticity (ω θ (R c,z c )=maxω θ ). We define the vortex core using a rectangular domain, D, such that z Z c [ h/2,h/2] and r R c [ R b /2,R b /2]. Then, to determine the vortex core size, a c, ellipticity (aspect ratio e = σ z/σ r, figure 4.8), the angle of ellipticity (Θ e ) and the vortex strength (Γ), a calculation based on the vorticity moments is used. The vortex core geometry is illustrated in figure 4.8. The circulation Γ in the vortex core is given by Γ= ω θ dd. (4.2) D

84 82 CHAP. 4: CHARACTERIZATION OF THE WAKE BEHIND A ONE-BLADED ROTOR Figure 4.8: Vortex core shape. We also considered the following moments of vorticity: I = ω θ r z dd, (4.3) D I 2 = ω θ z 2 dd, (4.4) D I 3 = ω θ r 2 dd. (4.5) D The characteristics of the vortex core are obtained using the following relations (the definition of Θ e is shown in figure 4.8): The ellipticity angle: tan(2θ e )= 2I I 2 I 3 (4.6) The vortex core size along the first axis σ 2 z = Γ [(2sinΘ e cosθ e )I +(cos 2 Θ e )I 2 +(sin 2 Θ e )I 3 ], (4.7) The vortex core size along the second axis σ 2 r = Γ [( 2sinΘ e cosθ e )I +(sin 2 Θ e )I 2 +(cos 2 Θ e )I 3 ]. (4.8) Therefore, the core size ( radius ) is given by: a 2 c = σ 2 z + σ 2 r = I 2+ I 3 Γ. (4.9)

85 4.2 Vortex core properties 83 Finally, the ellipticity, e, of the vortex core is given by: e= σ z. (4.) σ r Core size One of the main results of the present work, is the determination of the core size evolution during the vortex life. Most previous studies (no one has used NS equations and they are essentially experimental) have determined the core for just one blade revolution as a vortex age (Thompson et al. 988 (48) and references therein). We determine the tip vortex core properties during a greater period (up to 6 blade revolutions) before the appearance of turbulent features. Let us recall some known results about vorticity moments for two dimensional flows that, as it will be shown, are useful for the vorticity evolution, of the helical vortex, in an axial plane. Poincaré 89 (38) has shown, for plane (2D) motions of a homogeneous viscous and incompressible fluid that Γ (equation 4.2) remains constant and that: d(i 2 + I 3 ) dt = 4Γ Re, (4.) where I 2 and I 3 are given in equations 4.5. Since Γ is constant and, according to our definition, I 2 + I 3 = a 2 cγ, one should obtain for a two dimensional flow: (a c (t) 2 a c (t ) 2 )= 4(t t ). (4.2) Re The computed vortex core lengths, in an axial plane, for the helical (3D) vortex wake are shown in figure 4.9. Core size evolution (for different tip speed ratios) is shown as a function of the wake age, expressed in n=ω(t t )/(2π) revolutions. The linear 2D-diffusion law is plotted, in the same figure, using a discontinuous dotted line. Clearly, although the helical vortex flow is three dimensional, in the axial plane the dynamics of the vortex core is ultimately asymptotic to a two dimensional diffusion one. Note however that for some tip-speed ratios and Reynolds numbers the importance of nonlinear interactions between consecutive turns is such that the vortex core does not follow a two dimensional evolution. These nonlinearities give an overshoot (for λ > 5) to the core size compared to a 2D diffusive evolution. The vortex core size increases due to the viscous effects and adjacent vortices interactions (explaining transients). Knowing that the helical structure is hard to break, the axial interaction is promoted until the axial merging of the vortices as shown in figure 4. (however, no rotation of one vortex above the other, like in the 2D vortex merging, is observed). These results are important because most analytical models, used in rotor analysis, consider that the vortex core size remains constant in the wake and has a constant circular shape. Finally, it is also found that the initial core size of the helical vortex, a c (t ), does not depend on either the Reynolds number or the tip speed ratio. The initial core size a c (t ) is about 4 or 5% of the blade length R b.

86 84 CHAP. 4: CHARACTERIZATION OF THE WAKE BEHIND A ONE-BLADED ROTOR 4 4 (a c (t) 2 a c (t ) 2 )Re 3 2 (a c (t) 2 a c (t ) 2 )Re 3 2 (a) Ω(t t )/(2π) 4 (b) Ω(t t )/(2π) 4 (a c (t) 2 a c (t ) 2 )Re 3 2 (a c (t) 2 a c (t ) 2 )Re 3 2 (c) Ω(t t )/(2π) (d) Ω(t t )/(2π) Figure 4.9: Core size evolution as a function of the vortex age: influence of Reynolds number: Re=5, Re=, Re=5, Re=2 for (a) λ = 5, (b) λ = 6, (c) λ = 9, (d) λ = 2. The linear, two dimensional, diffusion law is plotted using a discontinuous dotted line Core shape As we have mentioned above, the vortex core has an elliptical shape which can be characterized with two parameters: the aspect ratio e=σ r /σ z and the angle of ellipticity (angle between the major axis of the ellipse and the flow axis). As depicted in figures 4., the aspect ratio has two evolution phases. The first phase is the creation phase of the helical vortex (roll up process) where the core undergoes a large deformation and so, a large variation of the aspect ratio, during practically the first rotor revolution. After being generated, the shape of the core is nearly preserved. The aspect ratio is greater than one and less than two proving that the vortex core is far from being circular. Regarding the ellipticity angle, it has the same behavior evolution as the aspect ratio. As illustrated in Figures 4.2, during the creation phase, Θ e varies rapidly until reaching an equilibrium value Θ e. The equilibrium elliptical angle Θ e is around for law λ (the principal axis of the vortex core is parallel to the axial direction of the cylindrical domain) and it increases with λ (it is around 45 for λ = 2).

87 4.2 Vortex core properties r r.9.9 (a) z (b) z r r.9.9 (c) z (d) z Figure 4.: Time evolution of two adjacent vortices for λ=2: (a) Ω(t t )/2π =, (b) Ω(t t )/2π = 2, (c) Ω(t t )/2π = 4, (d) Ω(t t )/2π = Maximum vorticity in the core For a two dimensional circular vortex with radius a(t), the conservation of angular momentum reads: πa(t) 2 ω m (t) = πa(t ) 2 ω m (t ) where the maximum of the vorticity is noted ω m. Using equation 4.2 one obtains (for a circular vortex a=a c ): ω m (t) ω m (t ) = +X, X = 4(t t ) Rea(t ) 2, (4.3) demonstrating that, for a circular vortex, X is a diffusion similarity variable for the evolution of the maximum of vorticity. For the helical vortex, the vortex core is not circular, as shown previously. However, the maximum of the vorticity is still self-similar, in a diffusion time scale, but now according to: ω m (t) ω m (t ) = α α+ X β, (4.4) as shown in figure 4.3. Ranges (obtained for different Re) of α and β are given in table 4.. It can be seen that the maximum of the vorticity, of the helical vortex, behaves like that of a 2D circular vortex for low tip speed ratios. It differs from the 2D diffusion when λ increases.

88 86 CHAP. 4: CHARACTERIZATION OF THE WAKE BEHIND A ONE-BLADED ROTOR (a) (b) (c) (d) Figure 4.: Evolution of the aspect ratio e during the vortex age. Re = 5, Re =, Re=5, Re=2 for (a) λ = 5, (b) λ = 6, (c) λ = 9, (d) λ = Vorticity and velocity profiles In this section we present vorticity and (azimuthal) velocity profiles in the vortex core of the helical wake. Profiles in an axial plane (rotating with the blade) are considered. The origin of the coordinate system(ρ,θ,ζ) is now placed at the center of the first formed vortex core (in the axial plane) and we follow its evolution in a frame rotating with the blade. Let u ρ, u Θ and u ζ denote, respectively, the radial, azimuthal and axial velocity in the local coordinate system and ω denotes the local axial vorticity (it corresponds to ω θ in the cylindrical coordinate system of the computational domain). Results for different vortex ages will be presented. Precisely, the ages n = Ω(t t )/2π = [,3,5,7] are considered. Profiles of vorticity (Normal to the plane Table 4.: Ranges (obtained for different Re) of α and β in equation 4.4 λ D α β

89 4.3 Vorticity and velocity profiles 87 Figure 4.2: The evolution of the ellipticity as a function of the vortex age: influence of Reynolds number: Re = 5, Re =, Re = 5, Re = 2 for (a) λ = 5, (b) λ = 6, (c) λ = 9, (d) λ = 2. containing axis Oz) and azimuthal velocity (in the plane) are shown. Profiles are defined along the line Θ=π/2 (Θ is the local angle when the origin is placed in the vortex core center). We have tried other lines and found that main conclusions are the same. These profiles should give, for example, an insight of those to be used for the study of short wavelength (compared to the vortex core) instability of a helical vortex. To the best knowledge of the author this instability has not yet been studied in the helical symmetry framework Vorticity in the vortex core It is found that vorticity profiles, in the vortex core, for different vortex ages are self-similar. To simplify notations, the vorticity (Normal to the plane containing axis Oz) in the core is noted ω. An example of these profiles is shown in figure 4.4 (a) (where ρ is the distance from the vortex core center). Let the vorticity scale be defined as: the radius scale, ρ /2, defined as: ω s (t)=ω(ρ =,t), (4.5) ω(ρ /2 /a c,t)= 2 ω s(t), (4.6)

90 88 CHAP. 4: CHARACTERIZATION OF THE WAKE BEHIND A ONE-BLADED ROTOR.8.8 ω m /ω m ().6.4 ω m /ω m () (a) X.8 (b) X.8 ω m /ω m ().6.4 ω m /ω m () (c) X (d) X Figure 4.3: The decay in time of the the maximum vorticity inside the vortex core for (a) λ = 5, (b) λ = 6, (c) λ = 9, (d) λ = 2. Similarity for different Reynolds number: Re = 5, Re =, Re=5, Re=2, 2D diffusion (α = and β = ) in equation 4.4. and the radial similarity variable given by: Then, it is found that: ω(ξ,t) ω s (t) ξ = ρ ρ /2. (4.7) = ω(ξ), (4.8) that is the scaled ω as a function of ξ is independent of t. Consequently, all vorticity profiles (for different ages) collapse onto a single curve as shown in figure 4.4 (b). The function ω is given by: ω(ξ)=exp( αξ 2 ). (4.9) Knowing that ω() = exp( α) and ω() = /2 lead to α = ln(2). Note that the variation in time of the vorticity scaling, ω s (t), is given by equation 4.4 and it is found, for instance, that (a c /ρ /2 ) t 2.34, for Re = 2. It is also interesting to note that for a Lamb-Oseen vortex (axisymmetric, 2D circular vortex) the similarity scaling and variable are ω s (t) and ρ/a c,

91 4.3 Vorticity and velocity profiles 89 respectively (Saffman 992 (4)). 5.8 ω 5 ω (a) ρ/a c (b) 2 3 ξ Figure 4.4: (a) Profiles of the vorticity in the vortex core for different vortex ages (expressed in n rotor revolutions) obtained for λ = 2 and Re = 2. Dashed n =, n = 3, n = 5, n = 7. (b) Profiles of the scaled vorticity as functions of the similarity variable ξ (equation 4.7). The equation 4.9 is plotted using a continuous line. The collapse of all profiles shows the self-similarity of vorticity in the core Azimuthal velocity in the vortex core Let u Θ denotes the azimuthal velocity in the core. Figure 4.5 (a) shows an example of profiles of this velocity. As done in the previous paragraph, let now the azimuthal velocity scale be defined as: u s Θ (t)=maxu Θ(ρ,t). (4.2) ρ For the azimuthal velocity there are two radii, ρ/2 a and ρb /2, corresponding to the half of maximum velocity. That is: Let, then, the radial similarity variable, η, be defined as: It is found that: u Θ (ρ a,b /2 /a c,t)= 2 max ρ u Θ(ρ,t). (4.2) 2ρ η = ρ/2 a +. (4.22) ρb /2 u Θ (η,t) u s Θ (t) = ũ Θ (η). (4.23) So, the scaled u Θ as a function of η is independent of t and all profiles collapse onto a single curve as depicted in figure 4.5 (b). It is interesting to note that we tried the location of the maximum azimuthal velocity as a similarity variable and found no similarity. The function ũ Θ (η) is given by: ũ Θ (η)=a( exp( (η/b) 2 ))/η. (4.24)

92 9 CHAP. 4: CHARACTERIZATION OF THE WAKE BEHIND A ONE-BLADED ROTOR Table 4.2: Ranges (obtained for different Re) of a and b in equation The 2D case corresponds to the azimuthal velocity of a Lamb-Oseen vortex to which the same procedure of section is applied. λ D a.94,.97.86,.9.87,.89.84,.88.9 b.58,.63.54,.55.55,.57.52, The ranges of a and b (obtained for different Re) are given in table 4.2. The values of a and b for the azimuthal velocity of a Lamb-Oseen vortex, to which the same procedure is applied, are also given. As an example, it is found that u s Θ (t) t.6 and that a c /(ρ a /2 + ρb /2 ) t.48 for Re=2. We tried other values of the local angle Θ. The self-similar behavior of vorticity and azimuthal velocity profiles is preserved. However, ρ /2 and (ρ/2 a + ρb /2 )/2 depend slightly on Θ since the vortex core is elliptical and we find its ellipticity (aspect ratio: major-axis/minor-axis) less than 2. ũθ (a) (b) Figure 4.5: (a) Profiles of the azimuthal velocity in the vortex core for different vortex ages (expressed in n rotor revolutions) obtained for λ = 2 and Re=2. Dashed n=, n=3, n=5, n=7. (b) Profiles of the scaled azimuthal velocity as functions of the scaling variable η (equation 4.22). The equation 4.24 is plotted using a continuous line. 4.4 A vortex core model for a helical vortex In this section, the azimuthally averaged profiles of vorticity and velocities in an axial plane (rotating with the blade) are considered.

93 4.4 A vortex core model for a helical vortex Vorticity profile in the vortex core It is found that averaged vorticity profiles, in the vortex core, for different vortex ages are selfsimilar. To simplify notations, the azimuthally averaged vorticity (normal to the axial plane) in the core is noted ω = (2π) 2π ω dθ. An example of these profiles is shown in figure 4.6- (left) (where ρ is the distance from the vortex core center). Let the velocity scale be defined as: the radius scale, ρ /e, defined as: ω s (t)= ω(ρ =,t), (4.25) and the radial similarity variable given by: Then it is found that: ω(ρ /e,t)= e ω s(t), (4.26) ω(η, t) ω s (t) η = ρ ρ /e. (4.27) = ω(η), (4.28) that is the scaled ω as a function of η is independent of t. Consequently, all vorticity profiles (for different ages) collapse onto a single curve as shown in figure 4.6-(right). The function ω is given by: ω(η)=exp( η 2 ), (4.29) Note that the variation with time of the vorticity scaling, ω s (t), is given by equation 4.4. It is important to note that for a Lamb-Oseen vortex (axisymmetric, 2D circular vortex) the similarity scaling and variable are also ω s (t) and η. Using Parseval s identity: due to the periodicity along the azimuthal direction, we can write: ω 2 Θ 2π ω 2 dθ= ω 2 + δ, 2π ( ω is as defined above), so the relative error introduced by the azimuthal averaging is ε = ω 2 Θ ω2 ω 2. Θ We have found that this error is small for ρ h/2 and n 7: ε 2% for λ = 5, δ 3% for λ = 8.4 and δ 7% for λ = 2 (for all the Reynolds numbers tried). This shows that the Lamb-Oseen vortex is a good approximation for the vorticity profile in the core of a rotor wake

94 92 CHAP. 4: CHARACTERIZATION OF THE WAKE BEHIND A ONE-BLADED ROTOR ω ω (a) ρ η ω 5 ω (b) ρ η.8 ω 5 ω (c) ρ η Figure 4.6: (left) Profiles of the averaged vorticity in the vortex core for different vortex ages (expressed in rotor revolutions n 7) obtained for Re = 2: (a) λ = 5, (b) λ = 8.4, (c) λ = 2. (right) Profiles of the scaled vorticity as functions of the scaling variable η (equation 4.27) for Re=2: (a) λ = 5, (b) λ = 8.4, (c) λ = 2. The equation 4.29 is plotted using a discontinuous line. The collapse of all profiles shows the self-similarity of vorticity in the core. helical-vortex. Moreover, figure 4.7 shows that the time evolution of the radius scale follows the linear 2D diffusion law except for the first blade revolution where the vortex is forming (the roll-up).

95 4.4 A vortex core model for a helical vortex (ρ /e (t) 2 ρ /e (t ) 2 )/ν (a) Ω(t t )/(2π) 25 2 (ρ /e (t) 2 ρ /e (t ) 2 )/ν 5 5 (b) Ω(t t )/(2π) 6 4 (ρ /e (t) 2 ρ /e (t ) 2 )/ν (c) Ω(t t )/(2π) Figure 4.7: Time evolution of ρ /e (equation 4.26) for Re=2: (a) λ = 5, (b) λ = 8.4 and (c) λ = 2. The 2D diffusion law is plotted using a discontinuous line (8πn/Ω) Azimuthal velocity profile in the vortex core Let Û Θ = (2π) 2π u Θ dθ denotes the averaged azimuthal velocity in the core. Figure 4.8- (left) shows an example of profiles of this velocity. As done in the previous paragraph, let now the the azimuthal velocity scale be defined as: ÛΘ s (t)=max Û Θ (ρ,t). (4.3) ρ

96 94 CHAP. 4: CHARACTERIZATION OF THE WAKE BEHIND A ONE-BLADED ROTOR The variable η given by equation 4.27 remains a similarity variable for the azimuthal velocity. It is found that: Û Θ (η,t) Û s Θ (t) = Ũ Θ (η). (4.3) So, the scaled U Θ as a function of η is independent of t and all profiles collapse onto a single curve as depicted in figure 4.8-(right). The function Ũ Θ (η) is given by: Ũ Θ (η)=a( exp( (η) 2 ))/η, (4.32) with a=η /( exp( η 2 )), η corresponds to the radial position such that Ũ Θ (η )= and is equal to.2 (so a =.57). This behavior is the same for a Lamb-Oseen vortex to which the same scaling procedure is applied Axial velocity profile in the vortex core Let Û ζ = (2π) π u ζ dθ denotes the averaged axial velocity in the core. Figure 4.9-(left) shows an example of profiles of this velocity. Let now that the axial velocity scale can be defined as: A scaled U ζ is given by: Ûζ s (t)=max Û ζ (ρ,t). (4.33) ρ Ũ ζ (η)= Ûζ(η,t). (4.34) Ûζ s (t) Contrarily to the vorticity and azimuthal velocity (in the core) it is found that the axial velocity does not present a self-similarity feature, at least in the time range considered here (of the order of the time needed for 7 rotor revolutions). This can be checked in figures 4.9- (right) obtained for different tip speed ratios, λ, and Re=2. Note that the maximum of this velocity could be either in the vortex core (λ 8.4) or outside it. Note also that it is found that the axial velocity decreases when λ increases. With these profiles a swirl number, S, measuring the importance of rotation compared to the axial flow in the core could be defined as follows: S(λ)= max ρ,t(û Θ ) max ρ,t (Û ζ ), ρ a c = ρ /e. (4.35) The variation of S as a function of λ could be summarized with the following: for λ < 8.4, S 2; for λ 8.4, S 6; for λ > 8.4, S,

97 4.4 A vortex core model for a helical vortex 95.4 (a) Ûθ ρ.4 UΘ η.5 Ûθ.3.2. UΘ.5 (b) ρ η.5 Ûθ.3.2. UΘ.5 (c) ρ η Figure 4.8: (left) Profiles of the azimuthal velocity in the vortex core for different vortex ages (expressed in rotor revolutions n 7) obtained for Re = 2: (a) λ = 5, (b) λ = 8.4, (c) λ = 2. (right) Profiles of the scaled azimuthal velocity as functions of the scaling variable η (equation 4.27) for Re=2: (a) λ = 5, (b) λ = 8.4, (c) λ = 2. The equation 4.32 is plotted using a discontinuous line. The collapse of all profiles shows the self-similarity of vorticity in the core. and the general tendency is an increase of the swirl number with the increase of the tip speed ratio.

98 96 CHAP. 4: CHARACTERIZATION OF THE WAKE BEHIND A ONE-BLADED ROTOR Ûz..5 Ũζ.5 (a) ρ η.5.4 Ûz.2 Ũζ.5 (b) ρ η Ûz.5 Ũζ 2.2 (c) ρ η Figure 4.9: (left) Profiles of the axial velocity in the vortex core for different vortex ages (expressed in rotor revolutions n 7) obtained for Re=2: (a) λ = 5, (b) λ = 8.4, (c) λ = 2. (right) Profiles of the scaled axial velocity as functions of the scaling variable η (equation 4.27) for Re = 2: (a) λ = 5, (b) λ = 8.4, (c) λ = 2. A Gaussian function is plotted using a discontinuous line. 4.5 Conclusions In this part, helical vortex generated behind a one-bladed rotor, placed in a uniform axial flow, was studied. Numerical computations at different Tip Speed Ratios, λ, and different Reynolds numbers, Re, were carried out using our numerical method. In addition to characterizing the global wake, particular attention was given to the Tip Vortex. The computations demonstrated that the global flow field behind the rotating blade is well re-

99 4.5 Conclusions q (a) Ω(t t )/(2π) 3 2 q (b) Ω(t t )/(2π) 3 log( q ) 2 (c) Ω(t t )/(2π) Figure 4.2: Swirl for different vortex ages (expressed in rotor revolutions n 7) obtained for Re=2: (a) λ = 5, (b) λ = 8.4, (c) λ = 2 produced and the helical shape of the vortex is well established. The results indicate the strong influence of the Tip Speed Ratio on the shape of the wake and on the near wake length. For the tip helical vortex, the core size is found to have two different evolution regimes during the vortex age. Its square increases linearly as for the 2D diffusion of vortices. Then, it increases more rapidly until the destruction of the helical structure by wake contraction or axial vortex-merging. In particular it is shown, for some tip-speed ratios and Reynolds numbers, that the importance of transients (due to nonlinear interaction between consecutive turns) is such that the vortex core does not follow a two dimensional evolution. These transients give an overshoot (for λ > 5) to the core size compared to a 2D diffusive evolution. It is also shown that inside the vortex core the vorticity and azimuthal velocity profiles are self-similar if they are considered

100 98 CHAP. 4: CHARACTERIZATION OF THE WAKE BEHIND A ONE-BLADED ROTOR for different ages of the helical vortex. This is not the case for the axial flow in the vortex core. These profiles should give an insight of those to be used in the study of short wavelength (compared to the vortex core) instability of a helical vortex. To the best knowledge of the author this instability has not yet been studied in the helical symmetry framework. Comparing the Lamb- Oseen like vortex (obtained by averaging along the local azimuthal direction in an axial plane) and the (full) vorticity, we found that the relative error, in considering the vortex core as axisymmetric, is at most 7%. Therefore, the Lamb-Oseen vortex is a good approximation of the core vorticity of the helical vortex wake (for the parameters (λ, Re) used here).

101 Chapter 5 Instability of a helical vortex In this part, the stability of the helical tip-vortex, generated by a single-blade rotor, is studied. As done in previous analytical works (Widnall 972 (54) for a helical filament case and Gupta 974 (9) for the multi-helical filaments case), in the numerical investigations done by Ivanell et al 2 (25) (LES simulations for the Tjaereborg wind turbine) and in the experimental work of Bolnot 22 (8), the perturbation is introduced as a deformation on the helical form of the vortex. It is done through a small variation of the pitch h over the time. The perturbation is characterized by its frequency /T, normalized by the angular velocity Ω, and its amplitude d normalized by the pitch h of the unperturbed helical vortex. We calculate the growth rate of the perturbation for different wave numbers. We use Γ/(2h 2 ) as the normalized scale for the growth rate, with the core vortex circulation Γ and the pitch of the helix h are those of the unperturbed one. A comparison with previous research work is done. An important case to note is the 2D pairing instability of a street of straight vortices (Lamb 932 (28)). Its normalized growth rate is equal to σ 2D = π/2. This case is considered as a reference for many vortex stability studies (as in the aircraft wake instability). In his work, Bolnot 22 (8) showed that the pairing instability is the most unstable mode in the case of an infinite street of vortex rings perturbed with a white noise, and the growth rate is found to be close to the 2D case for heigh ring radius R b /a 25 and aspect ratio h/r b /2 (here h is the distance between rings). A similar result was found by Levy & Forsdyke 928 (29) in their filamentary approach. Its important to note that according to the characterization of the helical tip-vortex done in the previous chapter, the case studied here is well within the range of validity of the 2D approach. Therefore, comparing the growth rates obtained by our numerical simulations to that of the 2D pairing instability remains a very interested challenge. In this part, the angular velocity considered is that of the design Ω D R b /W = 8.4, the same spatial resolution is used (N r N θ N z = ), the Reynolds number is Re=2, the time step is dt =. for a periode of simulation T f = unit time R b /W (except for a few cases where T f = 2). 99

102 CHAP. 5: INSTABILITY OF A HELICAL VORTEX 5. Perturbation Considering an inviscid flow, for a constant axial velocity W and angular velocity Ω D (the parameters of the studied flow), the rotating blade generates at its tip a helical vortex with a constant pitch h (as described in previous chapters) where the axial position and the angular position are related with the following expression: z= h 2π θ = hω Dt, (5.) where Ω D is expressed in a number of turns per second (tr/s). To generate a perturbed wake and force a local pairing, an angular velocity Ω(t) dependent on time is used, as done in the work of Hadrien Bolnot 22 (8). This variation produces a sinusoidal deformation in the helical form of the tip-vortex as shown in figure 5.. Figure 5.: Presentation of the perturbed helical vortex: (left) a front view of the 3D configuration and (right) a perspective view: in red color the perturbed vortex and in blue color the unperturbed vortex (stationary). The parameters used are: Ω D = tr/s, h =, T = 2 and d p /h=5%. The form of the perturbed vortex is controlled by the amplitude of the perturbation d p, which corresponds to the axial displacement done by the perturbed vortex relatively to the original unperturbed one and is less than the half of the pitch (d p h/2), and the period of the perturbation T (tr). So, for the perturbed vortex, the relation between axial and angular position can be expressed as follows: z= h 2π θ + d p cos( θ T )=hω Dt, (5.2) assuming that the pitch h is still constant for the unperturbed vortex. Taking the time derivative of this expression, an expression for the angular velocity, as shown in figure 5.2, is obtained and

103 5. Perturbation is given by: 2πΩ D θ(t)= ( ) Ω(t) (5.3) 2π d p T h sin θ(t) T Ω(t) n (rotor revolutions) Rbθ z Figure 5.2: (Left) The temporal evolution of the angular velocity (as a function of the rotor revolutions): in blue color the constant angular velocity and in red color the angular velocity used to perturb the helical wake. (Right) Presentation of the developed helical vortices (about their axis): in blue color the stationary vortex and in red the perturbed vortex. The parameters used are: Ω D = tr/s, h/r b =, T = 2 and d p /h=5%. It is important to note that this perturbation induces a deformation in the helical form only along the z-axis (no deformation along the radial direction as shown in figure 5.3), unlike the work of Widnall where the perturbations are imposed in both directions (radial and axial). Figure 5.3: An r θ presentation of the unperturbed vortex (in blue color) and the perturbed one (in red color): there is no radial displacement of the perturbed vortex In the present study, two ways of perturbations are used, regarding on the trigger time T p at which the perturbation is introduced. In the first case, the blade was rotated using the perturbed

104 2 CHAP. 5: INSTABILITY OF A HELICAL VORTEX angular velocity Ω(t) from the beginning of the simulation T p =. And in the second case, the blade was rotated with a constant angular velocity Ω D, a stationary flow is established, and from a time T p >, it was rotated with the perturbed angular velocity Ω(t). It is found that the two cases give same results. This implies that the evolution of the perturba Ω(t)/ΩD Ω(t)/ΩD time time Figure 5.4: Perturbation for T = 2/3, d p /h=3% and T p =. tion introduced in the flow is independent of the trigger time T p. The first type of perturbation is used in the following simulations. 5.2 Growth rate In his thesis, Bolnot showed that the growth rate of the perturbation can be measured by considering the full dynamics of three adjacent vortices (in the fully generated wake). A numerical study of the pairing instability in the case of an infinite vortex-ring street was also done in the thesis of Bolnot. Considering three adjacent vortices, as shown in the figure 5.5, the distance between the first vortex and the second is noted d 2 and the distance between the second vortex and the third one is noted d 23. To consider the dynamics of the pairing as a function of only one Figure 5.5: Positions of three adjacent vortex for two different instants.

105 5.2 Growth rate 3 variable, a distance d is considered and is defined as: d = d 23 d 2. (5.4) Bolnot found that the growth rate of the distance d is in good agreement with the growth rate of the linear mode of the 2D pairing instability (figure 5.6). Therefore, this original method of growth rate measuring was used in his experimental study and used in the present numerical study of the helical wake instability. The method, we use to calculate the temporal evolution of the distance d, consists as shown in Figure 5.6: Comparison of the temporal evolution of the distance d/h ( d = d) with the growth of the linear mode in the pairing instability case (red dotted line). The parameters of the infinite vortex-ring street considered are: R/a c = 4, h/a c =2 and Re=5 where R is the ring radius, Γ is the vortex circulation, h is the pitch and a c is the vortex radius. figure 5.7 on following the positions of the considered vortex in a frame rotating, with an angular velocity equal to Ω D, along the helical wake established at a given instant (at the end of the simulation). The temporal growth of the instability is given using the relation between the axial position and the time z=hω D t. Figure 5.7: The spatial evolution of the positions of the three vortices in the helical wake.

106 4 CHAP. 5: INSTABILITY OF A HELICAL VORTEX 5.3 Results The growth of the perturbation introduced in the flow was studied for different (selected) frequencies (/T ) and a perturbation amplitude chosen equal to d p /h=3%. The circulation of the vortex center is γ/(r b W ) =.25 and the pitch of the helix is h/r =.6 (for the unperturbed vortex wake). As shown in figure 5.8, two principal families of modes can be distinguished: the in-phase modes which are stable, and the out-of-phase modes which are the (locally) most unstable ones. It is important to note that the growth rate values are in the same order as that of the 2D pairing instability (σ 2D = π/2), especially for the frequency /T = / Growth rate σ frequency /T Figure 5.8: Normalized growth rate (by Γ/(2h 2 )) as a function of the frequency of the perturbation /T. The growth rate of the 2D pairing is plotted in discontinuous red line, σ 2D = π/ The in-phase modes They correspond to the frequencies equals to integers: /T = n, with n=,,2,... Their growth rates are almost zero for the considered time of the study. As shown in figures 5.9-(right), the considered vortices move in the axial direction keeping their distance constant and equal to the helical pitch h. This means that the three vortices have identical corrugated trajectories as shown in figure 5.. The used perturbations introduce, normally, only axial displacement and so the helical vortex must be deformed only along the axial direction z as shown in figure 5.2 and figure 5.3. But, for this stable mode, the radial positions of the vortices in the perturbed wake are also affected by the perturbation. This radial displacements are caused by the expansion of the wake due to

107 5.3 Results Radial position R c.2. Axial position Z c t t Figure 5.9: Mode /T = 2. Temporal evolution of the radial and axial positions of the considered adjacent vortices: vortex in blue continuous line, vortex 2 in red dashed line and vortex 3 in green dashed-dotted line..8 Rbθ/(2π) z Figure 5.: Presentation of the developed (simplified) mode shape of the in-phase mode /T = 2: vortex in blue continuous line, vortex 2 in red dashed line and vortex 3 in green dasheddotted line. the induced velocities (a wind blade rotor is considered) and then the vortices move radially in phase as shown in the simplified configuration of figure 5.. Therefore the distances d 2 and d 23 oscillate significantly in time as shown in figure 5.2. Despite that, the distance d= d 23 d 2 remains equal to until the occurrence of the features of turbulence. A corrugated helical tip-vortex is established (figure 5.3) in the wake. The frequency of the perturbation controls the corrugated form. It is shown that increasing the frequency increases the (local) radius of curvature of the vortex. Thus, the vortex is destructed leading to the appearance of a turbulent zone which length is greater when as the frequency is greater as shown in figure 5.4 for /T =, figure 5.3 for /T = 2 and figure 5.5 for /T = 3. A spatio-temporal evolution of the perturbed vortices for /T = 2 is illustrated in figure 5.6, where it can be seen that the disturbance is not amplified.

108 6 CHAP. 5: INSTABILITY OF A HELICAL VORTEX (a) (b) (c) 3 Figure 5.: A simple presentation of the perturbed vortex-motion, in a cross section perpendicular to the streamwise direction during two periods 2τ (rotations) for different frequencies: (a) /T =, (b) /T = 2 and (c) /T = 3. The continuous circle represents the motion of the center of an unperturbed vortex, the direction of the motion is given by point numbers (t() < t() <.. < t(n)). The red dashed line represents the trajectory Ψ of the vortex center during the first period (t τ), and the green dashed-dotted line is its trajectory Ψ 2 during the second period (τ t 2τ). It is shown that for all frequencies the two trajectories are superimposed Ψ 2 Ψ, its means that the adjacent vortices are in phase.

109 5.3 Results 7.68 d 2,d d/h t t Figure 5.2: (Left) The temporal evolution of the distances between the vortices (/T = 2): d 2 in blue continuous line, d 23 in red dashed line. (Right) The temporal evolution of the distance d used to calculate the growth rate. Figure 5.3: The 3D presentation of the in-phase mode shape /T = 2: the corrugated vortex remains stable until the appearance of a turbulent zone caused by the destruction of the wake under the fusion effect as in the unperturbed case (studied in the previous Chapter). Vorticity is shown. Figure 5.4: The 3D presentation of the in-phase mode shape /T = : the corrugated vortex remains stable until the appearance of a turbulent zone caused by the destruction of the wake under the fusion effect as in the unperturbed case (studied in the previous Chapter).

110 8 C HAP. 5: I NSTABILITY OF A HELICAL VORTEX Figure 5.5: The 3D presentation of the in-phase mode shape /T = 3: the corrugated vortex remains stable until the appearance of a turbulent zone caused by the destruction of the wake under the fusion effect as in the unperturbed case (studied in the previous Chapter). r.2 r z (t ) z r.2 r z (t3 ) z (t4 ) 8.5 r.2 r z (t5 ) z (t6 ).2 r r (t7 ) 7.5 (t2 ) z 9 8 (t8 ) 8.5 z Figure 5.6: Spatio-temporal evolution of the perturbed wake for mode /T = 2 with d/h = 3%. We note τ = 2π/ΩD the time needed for one blade rotation (in unit of Rb /W ): the snapshots are separated with τ/4 and the first one is for t = 2τ.

111 5.3 Results The out-of-phase modes The out-of-phase modes are the most unstable ones and correspond to frequencies given by /T = n+/2, with n =,,2,... For this kind of perturbation, adjacent vortices are out-ofphase as shown in figures 5.7. Therefore, their corrugated trajectories, presented in figure 5.8 can not be superimposed, but they intersect at different points. Radial position R c t Axial position Z c t Figure 5.7: Mode /T = 3/2. Temporal evolution of the radial and axial positions of the considered adjacent vortices: vortex in blue continuous line, vortex 2 in red dashed line and vortex 3 in green dashed-dotted line..8 Rbθ/(2π) z Figure 5.8: Presentation of the developed (simplified) mode shape of the in-phase mode /T = 3/2: vortex in blue continuous line, vortex 2 in red dashed line and vortex 3 in green dasheddotted line. It is important to note that the number of the intersection points and their position in the space is weakly related to the frequency. It is found that the number of these fixed points is N f = 2 (/T) and their positions in the r θ plane (perpendicular to the streamwise direction) are defined by the angular positions given by θ f (i)=i (2π)/(N f ). It is possible that these positions depend on the position of the blade at T p. In our case the angular position of the blade is θ b ()= for T p =. A simple presentation illustrates the deformation of a vortex trajectory and the formation of the

112 CHAP. 5: INSTABILITY OF A HELICAL VORTEX fixed points, and is shown in figure 5.9. The (π) out-of-phase is well illustrated. For this modes, adjacent vortices tend to roll-up one around another, leading to a kind of local pairing instability, and the wake is destructed there (figure 5.25 and 5.23). Therefore, the oscillatory radial displacements are imposed not only because of the expansion as the in the case of in-phase modes, but also by the interaction between vortices caused by the pairing instability. Then the perturbation is amplified, as shown in figure 5.2 and a non-zero growth rate σ can be calculated for the linear part of the dynamics (figure 5.2). A spatio-temporal evolution of the perturbed flow for frequency /T = 3/2 is illustrated in figure 5.22, on which the motion in space of three adjacent vortices is noticed. The local pairing leads to the fusion of the vortices two by two until the appearance of turbulence features. An important mode of this type is that for a frequency equal to /T = /2. It is the most unstable one for the considered perturbations. The spatio-temporal evolution of the pairing is illustrated in figure 5.24 and its 3D presentation is shown in figure 5.25.

113 5.3 Results 2 (a) (b-) 5 5 (b-2) (c-) 3 (c-2) Figure 5.9: A simple presentation of the perturbed vortex-motion, in a cross section perpendicular to the streamwise direction during two periods 2τ (rotations) for different frequencies: (a) /T = /2, (b) /T = 3/2 and (c) /T = 5/2. The continuous circle represents the motion of the center of an unperturbed vortex, the direction of the motion is given by point numbers (t()< t()<..<t(n)). The red dashed line represents the trajectory Ψ of the vortex center during the first period (t τ), and the green dashed-dotted line is its trajectory Ψ 2 during the second period (τ t 2τ). It is shown that for all frequencies the two trajectories are out-ofphase (their phase shift is Φ=π. Same results are found in the numerical simulations: (c-) and (c-2) represent the r θ plot of the established 3D wake.

114 2 CHAP. 5: INSTABILITY OF A HELICAL VORTEX d 2,d t Figure 5.2: Mode /T = 3/2. (Left) The temporal evolution of the distances between the vortices: d 2 in blue continuous line, d 23 in red dashed line. d/h t log(d/h) t Figure 5.2: Mode /T = 3/2. The temporal evolution of the distance d used to calculate the growth rate. The envelope of the function d/h is used to calculate the growth rate during the linear regime.

115 5.3 Results 3 r.2 r z (t ) z r.2 r z (t3 ) z (t4 ) 8.5 r.2 r z (t5 ) z (t6 ).2 r r (t7 ) 7.5 (t2 ) z 9 8 (t8 ) 8.5 z Figure 5.22: Spatio-temporal evolution of the perturbed wake for mode /T = 3/2 with d/h = 3%, we note τ = 2π/Ω the time needed for one blade rotation (in unit time): the snapshots are separated with τ/4 and the first one is for t = 2τ. Figure 5.23: The 3D presentation of the out-of-phase mode shape /T = 3/2: the amplification of the disturbance leads to a local pairing. The wake is destroyed and a turbulent zone is established.

116 4 CHAP. 5: INSTABILITY OF A HELICAL VORTEX Figure 5.24: Mode /T = /2: snapshots illustrate the spatio-temporal evolution of pairing instability. Instants are separated by one unit of R b /W (t n t n = R b /W ). Figure 5.25: The 3D presentation of the out-of-phase mode shape /T = /2: the amplification of the disturbance leads to a local pairing. The wake is destroyed and a turbulent zone is established.

117 5.3 Results Other modes For other frequencies (/T n and /T n+/2, n is integer), modes are found to be unstable. Their growth rates are less than those of the (π) out-of-phase modes. The phase shift Φ between adjacent vortices is given by: n</t < n+/2 <Φ<π; n+/2</t < n+ π < Φ<. (5.5) The mode corresponding to the frequency /T = 5/4 is illustrated in figures Radial position R c t Axial position Z c t Figure 5.26: Mode /T = 5/4. Temporal evolution of the radial and axial positions of the considered adjacent vortices: vortex in blue continuous line, vortex 2 in red dashed line and vortex 3 in green dashed-dotted line..8 Rbθ/(2π) z Figure 5.27: Presentation of the developed (simplified) mode shape of the mode /T = 5/4: vortex in blue continuous line, vortex 2 in red dashed line and vortex 3 in green dashed-dotted line.

118 6 CHAP. 5: INSTABILITY OF A HELICAL VORTEX.9.8 d 2,d t Figure 5.28: Mode /T = 5/4: (left) the temporal evolution of the distances between the vortices: d 2 in blue continuous line, d 23 in red dashed line. d/h t log(d/h) t Figure 5.29: Mode /T = 5/4: The temporal evolution of the distance d used to calculate the growth rate.

119 5.3 Results 7.2 r r z (t ) r.2 r z (t3 ) z (t4 ) r.2 r z (t5 ) 8 z (t6 ) r.2 r (t7 ) 7.5 z (t2 ) z (t8 ) z Figure 5.3: Spatio-temporal evolution of the perturbed wake for mode /T = 5/4 with d/h = 3%, we note τ = 2π/Ω the time needed for one blade rotation (in unit time): the snapshots are separated with τ/4 and the first one is for t = 2τ. Figure 5.3: Mode /T = 5/4: 3D presentation of the mode shape.

120 8 CHAP. 5: INSTABILITY OF A HELICAL VORTEX 5.4 Conclusion In this chapter, stability of a helical tip-vortex generated by a single-blade rotor is studied. Using an angular velocity, Ω(t), dependent in time, some dynamic modes are excited. The deformation imposed to the helical form is, normally, only axial. The perturbations are controlled by two parameters: their frequencies /T (normalized by the angular rotation Ω D ), and their amplitudes d (normalized by the pitch of the unperturbed helix h). The growth rates are found in the order of that of the pairing instability. Three different type of modes are identified depending on the frequency: (a) The in-phase modes, correspond to the frequencies that are integers. These modes are found to be stable for the considered time and amplitude. For these modes, adjacent vortices move in the space (in phase) keeping their distances constant. It is found that the perturbation induce not only axial corrugations but also radial ones. They are caused by the expansion of the wake. Despite that, the modes remain stable. These modes correspond well to the stable analytical modes identified by Widnall in his, inviscid, stability analysis of an infinite helical vortex-filament of finite core. The corrugated wake still stable until the appearance of a turbulent zone caused either by the vortex fusion as in the unperturbed case (as shown in previous Chapters), or by the destruction of the helical form due to high local deformation (high curvature radius) in the case of high frequencies. (b) The out-of-phase modes, correspond to frequencies given by /T = n+/2 with n an integer. As in the 2D pairing instability, adjacent vortices attempt to roll-up one around another leading to local pairing instability. The vortices merge two by two forming larger ones until features of turbulence appear and the helical form of the wake is destroyed. As predicted by previous works (Widnall 972 (54)and Bolnot 22 (8)), these modes are the most unstable ones for this kind of disturbance. The perturbations grow linearly until saturation (for some case only linear growth phase is observed for the considered time) and a growth rate σ (in the linear part of the dynamics) is measured based on the relative displacement of three adjacent vortices. The growth rates, normalized by Γ/(2h 2 ), are found to be in good agreement with that of the 2D paring instability, where the core vortex circulation Γ and the helical pitch h are those of the unperturbed helical vortex (for the same angular rotation Ω D ). It is important to note that the motion (out-of-phase) of adjacent coils (vortices) leads to their intersection and the vortex is destroyed in the intersection points, that are found to be related to frequencies such that their number is given by N f = 2 (/T) and their angular locations are separated by(2π)/(n f ). (c) For the other frequencies, modes are found to be also unstable. Their shapes similar to those of the out-of-phase modes. The phase shift Φ between adjacent vortices (Φ ±pi and Φ ) allows the growth and the amplification of the perturbation until the destruction of the helical form of the tip vortex. Their growth rates are found to be less than those of the out-ofphase modes (at least locally). It is important to note that this study was done for selected frequencies and it is found that the

121 5.4 Conclusion 9 computing of the growth rate is critical due to the quick merging of the wake (diffusive effects), and so the linear growth phase of the instability is short. Up to now, it is not possible to have a wake life, without turbulence features, longer than six rotor revolutions (its is the first time to reach 6 rotor revolutions as seen in the previous chapter). The low Reynolds number, Re, used in the present study is limited by the heigh resolution (about 23 millions grid points) needed to have smooth solutions of Navier-Stokes equations.

122 2 CHAP. 5: INSTABILITY OF A HELICAL VORTEX

123 Chapter 6 Conclusions & Perspectives In this thesis, numerical study of a helical tip vortex, generated behind a single-blade rotor, is done. The numerical program is developed from scratch. The numerical method combines a 3D Navier-Stokes (NS) equation solver with a technique in which body forces, exerted by the blade, are distributed radially along each blade (considered, ideally, as a line). Thus, the dynamics of the wake is determined by a full 3D NS simulations whereas the influence of the rotating blade on the surrounding flow field is included using tabulated aerofoil data to represent the loading on each blade. The aerofoil data subsequent loading are determined by computing local angles of attack from the movement of the blade and the local flow field. This concept developed by Sørensen & Shen (22) (47), enables us to study in details the dynamics of the wake and the tip vortices. The NS equations are solved using a second-order, centred, finite-difference scheme with a fractional time step method combined with an approximate-factorization technique. The 3D Poisson equation, to enforce incompressibility, is solved by a fast inversion method, based on fast Fourier transforms along the θ-direction and a resolution of a two-dimensional (2D) Helmholtz problem (for each azimuthal wavenumber) using the cyclic reduction technique. To compute the non-solenoidal velocity field, a third-order, low-storage, Runge-Kutta method combined with an implicit Crank-Nicholson scheme, for viscous terms, are used. This method is second-order accurate in time for the viscous terms and third-order accurate for the convective terms, the overall accuracy being second-order in time. The implementation of the method was parallelized using the Open-MP libraries. The body forces, distributed radially along a line, are smoothed using a Gaussian kernel controlled by a parameter ε taken as small as possible to not influence the wake structure. The NS solver was validated by comparing numerical growth rates of an unstable jet flow with those predicted by linear instability theory. The good agreement found comparing the induced velocity fields in numerical simulations and analytical formulations (for finite-length and infinite-length helical vortices) allows as to validate the implementation of the Actuator Line concept. It is found that the method predicts well the aerodynamic properties along the blade by 2

124 22 CHAP. 6: CONCLUSIONS & PERSPECTIVES comparing the properties obtained numerically and those described in the experimental work of Bolnot (8), for the same considered blade. The method was, then, used to study in details the helical tip vortex generated by a singleblade rotor placed in a uniform axial inflow. Numerical computations at different tip speed ratios, λ, and different Reynolds numbers, Re, are carried out. The vortex core size is found to have two different evolution regimes during the vortex age. Its square increases linearly as for the 2D diffusion of vortices. Then, it increases more rapidly until the destruction of the helical structure by wake contraction or axial vortex-merging. In particular it is shown, for some tip-speed ratios and Reynolds numbers, that the importance of transients (due to nonlinear interaction between consecutive turns) is such that the vortex core does not follow a two dimensional evolution. These transients give an overshoot (for λ > 5) to the core size compared to a 2D diffusive evolution. It is also shown that inside the vortex core the vorticity and azimuthal velocity profiles are selfsimilar if they are considered for different ages of the helical vortex. This is not the case for the axial flow in the vortex core which decreases when λ increases. A general model for the vortex core was given by taken the azimuthally averaged profiles. It is found that the (well known) Lamb-Oseen vortex is a good approximation for the vorticity in the core of a helical vortex, in the wake of a rotor. These profiles should be used to ameliorate instability studies like those done in previous works (Okulov 24 (35), Fukumoto & Okulov 25 (6), Okulov & Sørensen 27 (34) and Widnall 972(54)) where ad-hoc vorticity and velocity profiles in the core are used, and should be used especially as next order basic-profiles in the study of short wavelength (compared to the vortex core length) instabilities (Hattori 24 (23)). In the final part of this work, the stability of the helical tip-vortex was studied. The perturbations were introduced using an angular velocity Ω(t) dependent in time. They are controlled by two parameters: their frequencies /T normalized by the angular rotation Ω D, and their amplitudes d normalized by the pitch of the helix h. Three different type of modes are identified depending on the frequency: (a) The in-phase modes, correspond to the frequencies that are integers. These modes are found to be stable in the considered time and amplitude. In this case, adjacent vortices move in the space (in phase) keeping their distances constant. It is found that the perturbation induce not only axial corrugation but also radial ones. They are caused by the expansion of the wake. Despite that, the modes remain stable. These modes correspond well to the stable analytical modes identified by Widnall 972 (54) in his, inviscid, stability analysis of an infinite helical vortexfilament of finite core. The corrugated wake is stable until appearance of a turbulent zone caused either by the vortex fusion as in the unperturbed case, or by the destruction of the helical form due to high local deformation (high curvature radius) in the case of high frequencies. (b) The out-of-phase modes, correspond to the frequencies given by /T = n+/2 with n an integer. As in the 2D pairing instability, adjacent vortices attempt to roll-up one around

125 23 another leading to local pairing instability. The vortices merge two by two forming larger ones until features of turbulence appear and the helical form of the wake is destroyed. As predicted by previous works (Widnall 972 (54) and Bolnot 22 (8)), the modes are the most unstable for this disturbance. The perturbations grow linearly until a saturation (for some cases only linear growth phase is observed for the considered time) and a growth rate σ (in the linear part of the dynamics) is measured based on the relative displacement of three adjacent vortices. The numerical growth rates, are found to be in good agreement with that of the 2D paring instability. The positions of the pairing zone are related to the perturbation frequencies such that their number is given by N f = 2 (/T) and their angular locations are separated by(2π)/(n f ). (c) For the other frequencies, the modes are found to be also unstable. Their shapes resemble to those of the out-of-phase modes. The phase shift Φ between adjacent vortices (Φ ±pi and Φ ) allows the growth and the amplification of the perturbation until the destruction of the helical form of the tip vortex. Their growth rates are found to be less than those of the out-ofphase modes (at least locally). It is important to note that the stability study was done for selected frequencies and it is found that the computing of the growth rate is critical due to the quick merging of the wake, and so the linear growth phase of the instability is short. Up to now, it is not possible to have a wake life, without turbulence features, longer than six rotor revolutions due to the low Reynolds number Re used. It is limited by the heigh resolution used in the simulations (about 23 millions grid points) needed for the obtaining of smooth solutions of Navier-Stokes equations. Therefore, an important future possible direction is to improve the computational capacity of the developed method by parallelizing the code using a more efficient methods such that MPI language. This allows the use of the code as a tool for (direct numerical) simulations and the study of more complicated configurations encountered in rotor wakes: non-uniform inflow, severalblade rotors, wake interaction...

126 24 CHAP. 6: CONCLUSIONS & PERSPECTIVES

127 Bibliography [] M. Abid. Nonlinear mode selection in a model of trailing line vortices. Journal of Fluid Mechanics, 65:9 45, [2] M. Abid and M. E. Brachet. Numerical characterization of the dynamics of vortex filaments in round jets. Physics of Fluids A-Fluid Dynamics, 5(): , , 8, 38, 62 [3] M. Abid and M. E. Brachet. Direct numerical simulations of the batchelor trailing vortex by a spectral method. Physics of Fluids, (2): , , 8, 38, 62 [4] M. Ali and M. Abid. Helical vortex wake: How far is the infinity? In The 23 International Conference on Aerodynamics of Offshore Wind Energy Systems and Wakes, Copenhagen, Denmark, 23. 5, 39 [5] Mohamed Ali and Malek Abid. Self-similar behaviour of a rotor wake vortex core. Journal of Fluid Mechanics, 74:R, 24. 5, 39 [6] R.M. Beam and R.F. Warming. Journal of Computational Physics, 8, [7] Mahendra J. Bhagwat and J. Gordon. Stability analysis of helicopter rotor wakes in axial flight. Journal of the American Helicopter Society, 45(3):65 78, [8] H. Bolnot. Instabilités des tourbillons hélicoïdaux: application au sillage des rotors. PhD thesis, Aix-Marseille Université, 22. 9, 28, 29, 37, 38, 4, 45, 46, 7, 7, 99,, 8, 22, 23 [9] H Bolnot, T Leweke, and S Le Dizes. Spatio-temporal development of the pairing instability in helical vortices. AIAA Paper, 3927, [] Ivan Delbende, Jean-Marc Chomaz, and Patrick Huerre. Absolute/ convective instabilities in the batchelor vortex: a numerical study of the linear impulse response. Journal of Fluid Mechanics, 355: , [] Ivan Delbende, Maurice Rossi, and Olivier Daube. Dns of flows with helical symmetry. Theoretical and Computational Fluid Dynamics, 26(-4):4 6,

128 26 BIBLIOGRAPHY [2] I. Dobrev. Modele hybride de surface active pour l analyse du comportement aerodynamique des rotors eoliens a pales rigides ou deformables. PhD thesis, Ecole Nationale Superieure d Arts et Metiers, [3] Ivan Dobrev, Bassem Maalouf, Niels Troldborg, and Fawaz Massouh. Investigation of the wind turbine vortex structure. In 4th international symposium on applications of laser techniques to fluid mechanics, Lisbon, Portugal, pages 7, [4] Ir J Meijer Drees and Ir WP Hendal. Airflow patterns in the neighbourhood of helicopter rotors: A description of some smoke tests carried out in a wind-tunnel at amsterdam. Aircraft Engineering and Aerospace Technology, 23(4):7, [5] M Felli, R Camussi, and F Di Felice. Mechanisms of evolution of the propeller wake in the transition and far fields. Journal of Fluid Mechanics, 682:5 53, 2. 37, 78 [6] Y. Fukumoto and V. L. Okulov. The velocity field induced by a helical vortex tube. Physics of Fluids (994-present), 7():, 25. 5, 9, 33, 38, 39, 64, 66, 22 [7] Yasuhide Fukumoto and Yuji Hattori. Curvature instability of a vortex ring. Journal of Fluid Mechanics, 526:77 5, [8] Hermann Glauert. Airplane propellers. In Aerodynamic theory, pages Springer, , 54 [9] BP Gupta and RG Loewy. Theoretical analysis of the aerodynamic stability of multiple, interdigitated helical vortices. AIAA Journal, 2():38 387, , 37, 99 [2] M.O.L. Hansen, J.N. Sørensen, S. Voutsinas, N. Sørensen, and H.Aa. Madsen. State of the art in wind turbine aerodynamics and aeroelasticity. Progress in Aerospace Sciences, 42(4):285 33, 26. 6, 3 [2] Jay C Hardin. The velocity field induced by a helical vortex filament. Physics of Fluids ( ), 25(): , , 38 [22] F.H. Harlow and J.E. Welch. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Physics of Fluids, 8(2): , [23] Yuji Hattori and Yasuhide Fukumoto. Modal stability analysis of a helical vortex tube with axial flow. Journal of Fluid Mechanics, 738: , 24. 5, 39, 22 [24] S.S.A. Ivanell. Numerical Computations of Wind Turbine Wakes. PhD thesis, Royal Institute of Technology, Linné Flow Center, Department of Mechanics, Stockholm, Sweden, , 34 [25] Stefan Ivanell, Robert Mikkelsen, Jens N. Sørensen, and Dan Henningson. Stability analysis of the tip vortices of a wind turbine. Wind Energy, 3(8):75 75, 2. 36, 99 [26] Stefan Ivanell, Jens N. Sørensen, Robert Mikkelsen, and Dan Henningson. Analysis of numerically generated wake structures. Wind Energy, 2():63 8,

129 BIBLIOGRAPHY 27 [27] J. Kim and P. Moin. Application of a fractional-step method to incompressible navierstokes interaction in two and three dimensions. Journal of Computational Physics, 59:38 323, , 49 [28] Horace Lamb. Hydrodynamics. Cambridge university press, , 99 [29] H Levy and AG Forsdyke. The steady motion and stability of a helical vortex. Proceedings of the Royal Society of London. Series A, 2(786):67 69, , 33, 35, 45, 99 [3] F Massouh and I Dobrev. Exploration of the vortex wake behind of wind turbine rotor. Journal of Physics: Conference Series, 75():236, [3] Kenneth W McAlister. Rotor wake development during the first revolution. Journal of the American Helicopter Society, 49(4):37 39, [32] Vyacheslav V Meleshko and Hassan Aref. A bibliography of vortex dynamics Advances in applied mechanics, 4(97):97 292, [33] R. Mikkelson. Actuator Disc Method Applied to Wind Turbines. PhD thesis, Technical University of Denmark, DTU, Department of Mechnical Engineering, Denmark, , 56 [34] Valery L Okulov and Jens Nørkær Sørensen. Stability of helical tip vortices in a rotor far wake. Journal of Fluid Mechanics, 576: 25, 27. 5, 33, 34, 36, 39, 22 [35] VL Okulov. On the stability of multiple helical vortices. Journal of Fluid Mechanics, 52:39 342, 24. 5, 33, 36, 39, 22 [36] Benjamin Piton. Simulations de tourbillons à symétrie hélicoïdale. PhD thesis, Paris 6, [37] Benjamin PITON, Ivan DELBENDE, and Maurice ROSSI. Simulations numériques de vortex à symétrie hélicoïdale. 2ème Congrès Français de Mécanique, 28 août/2 sept Besançon, France (FR), [38] Henri Poincaré. Théorie des tourbillons, chap.. Jacques Gabay, , 83 [39] M.M. Rai and P. Moin. Direct simulations of turbulent flow using finite difference schemes. Journal of Computational Physics, 96:5 53, , 5 [4] Philip G Saffman. Vortex dynamics. Cambridge university press, , 38, 4, 64, 65, 89 [4] B Sanderse. Aerodynamics of wind turbine wakes. Energy Research Center of the Netherlands (ECN), ECN-E 9-6, Petten, The Netherlands, Tech. Rep, [42] U. Schumann and Roland A. Sweet. A direct method for the solution of poisson s equation with neumann boundary conditions on a staggered grid of arbitrary size. Journal of Computational Physics, 2:7 82,

130 28 BIBLIOGRAPHY [43] Wen Zhong Shen, Jian Hui Zhang, and Jens Nørkær Sørensen. The actuator surface model: a new navier stokes based model for rotor computations. Journal of Solar Energy Engineering, 3():2, [44] Michael Sherry, John Sheridan, and DavidLo Jacono. Characterisation of a horizontal axis wind turbine Äôs tip and root vortices. Experiments in Fluids, 54(3): 9, [45] Jens Nørkær Sørensen. Aerodynamic aspects of wind energy conversion. Annual Review of Fluid Mechanics, 43: , [46] Jens Nørkær Sørensen. Instability of helical tip vortices in rotor wakes. Journal of Fluid Mechanics, 682(): 4, 2. 32, 37 [47] Jens Nørkær Sørensen and Wen Zhong Shen. Numerical modeling of wind turbine wakes. Journal of fluids engineering, 24(2): , 22. 4, 6, 33, 37, 38, 4, 2 [48] TL Thompson, NM Komerath, and RB Gray. Visualization and measurement of the tip vortex core of a rotor blade in hover. Journal of aircraft, 25(2):3 2, , 35, 83 [49] N. Troldborg. Actuator Disc Method Applied to Wind Turbines. PhD thesis, Technical University of Denmark, DTU, Department of Mechnical Engineering, Denmark, , 34, 35, 56 [5] L.J. Vermeer, J.N. Sørensen, and A. Crespo. Wind turbine wake aerodynamics. Progress in Aerospace Sciences, 39(6 Äì7):467 5, 23. 6, 3 [5] R Verzicco and P Orlandi. A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. Journal of Computational Physics, 23(2):42 44, , 42 [52] J H Walther, M Guénot, E Machefaux, J T Rasmussen, P Chatelain, V L Okulov, J N Sørensen, M Bergdorf, and P Koumoutsakos. A numerical study of the stabilitiy of helical vortices using vortex methods. Journal of Physics: Conference Series, 75():234, [53] J Whale, C.G Anderson, R Bareiss, and S Wagner. An experimental and numerical study of the vortex structure in the wake of a wind turbine. Journal of Wind Engineering and Industrial Aerodynamics, 84(): 2, 2. 34, 35 [54] Sheila E Widnall. The stability of a helical vortex filament. Journal of Fluid Mechanics, 54(4):64 663, , 29, 33, 35, 36, 37, 39, 4, 45, 99, 8, 22, 23 [55] Zifeng Yang, Partha Sarkar, and Hui Hu. An experimental investigation on the wake characteristics of a wind turbine in an atmospheric boundary layer wind. In 29th AIAA Applied Aerodynamics Conference, pages 8, [56] Larry A Young. Vortex core size in the rotor near-wake. NASA Technical Memorandum, 22275, , 35

131 Chapter 7 Appendix 29

132 3 CHAP. 7: APPENDIX 7. Paper presented in the International Conference on Aerodynamics of Offshore Wind Energy Systems and Wakes, Denmark, 23

133 ICOWES23 Conference 7-9 June 23, Lyngby FULL PAPER Helical vortex wake: How far is the infinity? Mohamed Ali, Malek Abid IRPHE, U M R CNRS et Université d Aix-Marseille 49 rue Joliot Curie - BP 46 Technopôle de Château Gombert 3384 MARSEILLE Cedex 3 ali@irphe.univ-mrs.fr, abid@irphe.univ-mrs.fr ABSTRACT In this paper, we discuss the assumption that the helical vortex wake, such as the wind turbine wake, can be considered as an infinite vortex filament or tube. In the real life, vortices in helical wakes are of finite length. We compare the velocity field induced by an infinite helical vortexfilament and the velocity field induced by a finite length one. The analytical solution given by Fukumoto & Okulov is used for the infinite case and a Biot-Savart formulation is used for the finite case. A basic difference is found for the radial induced velocity due to the asymmetry of the finite helical vortex (near its boundaries). For the axial velocity field, the distance behind the rotor plane required to superimpose the two models is given and it is found weakly related to the pitch-radius ratio (h/r). Keywords: Helical vortex tube/filament, Biot-Savart Formulation, rotor wake INTRODUCTION Helical vortices are observed in various types of flows. They are often created as tip vortices in rotating devices such as propellers, wind turbines, etc. This type of flow has been the subject of many theoretical, experimental and numerical studies. Many proposed analytical models, especially the Joukowsky model, assumed that the helical vortex wake can be modeled as an infinite vortex filament or tube (or N vortices in the case of a N-bladed rotor). This idealistic model was used for the stability analysis of helical vortex systems [2,6,8,9] and the design of rotor blade [,7,]. In this context, Hardin [3] developed in 982 an exact solution for the velocity field, both interior and exterior, induced by an infinite helical vortex filament using Kepteyn s series. Recently, Fukumoto & Okulov [5] succeeded to represent this analytical solution in closed form using a singularity-separation technique. However, in the real life, vortices are of finite length (Figure ).

134 ICOWES23 Conference 7-9 June 23, Lyngby In the present work, we study the influence of the geometric parameters of the vortex (pitchradius ratio h/r and the number of coils N c ) on the validity of the above assumption. We calculate, for a finite-length vortex filament, the number of coils needed to consider that the induced velocity field coincides with that of an infinite filament (N c = ) with the same circulation Γ, radius R and pitch h. So, we compare the induced velocities of an infinite filament V I, given by Fukumoto and Okulov solution [5], to those of a finite-length one V F, given by a Biot-Savart formulation [4]. Particular attention is given to the axial induced velocity knowing that it is the important component in wake modeling and blade design. Figure : Helical vortex system: (left) an infinite helical vortex-tube, (right) a semi-infinite helical vortex-tube FORMULATION Velocity field induced by a finite helical vortex-filament We consider a right-handed helical vortex-filament (figure 2) of circulation Γ and constant pitch h, with a finite length L = 2πhN c, the axis of which is that of a cylinder with radius R. The position of the vortex filament is given, in a cylindrical coordinates system, (r,θ,z), z being the axial direction, by the following equation: r (θ )=R e r (θ )+hθ e z. () The velocity induced by the finite filament on a point M ( r (θ)=r e r (θ)+z e z ) is given by the Biot-Savart relation as follows: V F ( r) = Γ t (s ) ( r r (s )) 4π Helix r r (s ) 3 ds, r / Helix (2)

135 ICOWES23 Conference 7-9 June 23, Lyngby L=2hN c z 2 e θ e θ e r M(r,θ,z) R y Figure 2: Geometry of the helical vortex filament e r R where t is the unit tangent vector to the helical vortex and s is the (curvilinear) abscissa along t and is given by : t d r dθ = d r dθ = d r ds (3) ds dθ = d r dθ (4) r (θ ) = R e r(θ )+kθ e z (5) r(r,θ,z) = r e r (θ)+z e z (6) (7) with e r = cos(θ θ ) e r + sin(θ θ ) e θ (8) e θ = sin(θ θ ) e r + cos(θ θ ) e θ (9) This allows us to write : ds = t = R 2 + k 2 dθ () R 2 + k 2(R e θ + k e z) ()

136 ICOWES23 Conference 7-9 June 23, Lyngby and so, t ( r r )= R 2 + k 2 Rsin(θ θ ) Rcos(θ θ ) k r Rcos(θ θ ) Rsin(θ θ ) z kθ (2) Finally, the velocity induced by the finite filament is given as follows: V F ( r) = Γ 4π [θ ] ( (r Rcos(θ θ )) 2 + R 2 sin 2 (θ θ )+(z kθ ) 2) 3 /2 Rcos(θ θ )(z kθ ) Rk sin(θ θ ) k(r Rcos(θ θ )) (z kθ )Rsin(θ θ ) R 2 rrcos(θ θ ) Velocity field induced by an infinite helical vortex-filament dθ (3) For the velocity induced by an infinite filament, V I, we consider the solution given by Fukumuto & Okulov [3]: u r (r, χ)= Γ [ e iχ 4 (k 2πrk 2 + r 2 )(k 2 + R 2 )Im e ξ e iχ ± k ( 2k 2 + 9R 2 24 (k 2 + R 2 ) 3/2 2k2 + 9r 2 ] )ln( e ±ξ+iχ (k 2 + r 2 ) 3/2 (4) ( u θ (r, χ)= Γ ( 2πr N b ) ( u z (r, χ)= Γ ( Nb 2πr where ) + 4 k 2 + R 2 4 k 2 + r 2 4 k 2 + R 2 4 k 2 + r 2 N b [ ±e iχ n Re n= e ξ e iχ + k ( 3r 2 2k 2 n 24 (k 2 + r 2 ) 3/2 + 9R2 + 2k 2 ) (k 2 + R 2 ) 3/2 ln( e )] ) ±ξ+iχ (5) N b [ ±e iχ n Re n= e ξ e iχ + k ( 3r 2 2k 2 n 24 (k 2 + r 2 ) 3/2 + 9R2 + 2k 2 ) (k 2 + R 2 ) 3/2 ln( e )] ) ±ξ+iχ (6) χ = θ z/k (7) = χ+ 2π(n ) N b (8) χ n e ξ = r(+ +R 2 /kl 2 )exp( +r 2 /k 2 ) R(+ +r 2 /k 2 )exp(+ +R 2 /k 2 ) where Re[.] and Im[.] indicate the real and imaginary parts of the complex expression, respectively. For the notations ± and (:), the upper sign or symbol in parenthesis corresponds to r<r and the lower to r R. (9) RESULTS AND DISCUSSION

137 ICOWES23 Conference 7-9 June 23, Lyngby Radial induced velocity at the rotor plane In what follows the induced axial flow-velocity of the infinite helical vortex is chosen as a velocity scale. One effect of breaking the symmetry of the wake is the creation of a radial velocity in the rotor plane. Figure 3 shows that the azimuth-averaged radial velocity in the rotor plane has a magnitude with same order as the axial induced velocity (only the h/r= case is presented here, but the same behavior is found for different h/r). The presence of the radial velocity brings into question many theoretical approaches which neglect the radial velocity effect on the wind turbine wakes or does not consider it in the blade design process. The non-zero radial velocity at the rotor plane is the most important difference between the two approaches considered here (infinite and finite vortex filament)..2 <V r > θ Finite case Inifinite case r/r Figure 3: Radial induced velocity, averaged in the azimuthal direction, V r θ : comparison between velocity induced by the finite helical vortex and the infinite helical vortex at the rotor plane and for h/r=. Axial induced velocity As the axial induced velocity is the important component for the wake modeling and blade design, we discuss here its behavior for the two considered configurations. As shown in Figure 4, the azimuth-averaged axial velocity induced by a finite helical vortexfilament depends on the pitch-radius ratio (h/r) and the number of coils which form the filament considered. It s found that, for small h/r values, a high number of coils is needed to consider that V F is equivalent to V I (i.e., ( V F V I )/ V I =ε, ε ), contrarily to the cases with high h/r where few coils are sufficient. In what follows, unless otherwise stated, V F and V I are evaluated at r= and z=l/2, and only results for their axial components, Vz F, and Vz I are shown. From an experimental point of view, the helical tip vortex generated during few rotor revolutions is equivalent to an infinite helical vortex with the same parameters (Γ, R, h) when h/r O()

138 ICOWES23 Conference 7-9 June 23, Lyngby <V z F Vz I >θ /<V z I >θ h/r=. h/r=.5 h/r= Number of coils N c <V z F Vz I >θ /<V z I >θ h/r=. h/r=.2 h/r=.5 h/r= h/r= Number of coils N c Figure 4: Comparison between a finite vortex filament (Biot-Savart formulation) and an infinite one (Fukumoto and Okulov analytical solution) for different h/r: the azimuthal average of the axial-velocity relative-difference is presented as a function of the number of coils N c. which is the case of wind turbine rotors. But for small h/r which is the case of helicopter rotors, the helical vortex generated can not be considered equivalent to an infinite helical vortex until it is formed by a large number of coils. Based on our calculations, the relative difference between the two velocities Vz I and Vz F, for a given h/r, follows the following law: V F z Vz I Vz I N c = = exp( N c /N c), α β + h/r. (2) We find α 2 and β 2 3, two coefficients approximating the characteristic number

139 ICOWES23 Conference 7-9 June 23, Lyngby of coils N c. Note that this law is the same (data not chown) for r if V F z θ and V I z θ are considered with. θ the average in the azimuthal direction θ. It is interesting to note that, the number of coils can be transformed to a non-dimensional axial distance d, behind the rotor plane, given by: d = N c h/r (2) As shown in Figure 5, the difference between the two (azimuth-averaged) axial induced-velocities has a similarity behavior, i.e., all the curves collapse when the variable d is used, and V F z Vz I Vz I = exp(d/d ), d = N c h/r= α +βr/h, (22) It is shown in Figure 6 that d converges quickly to a constant value equal to α (i.e.,d = 2), it is weakly related to h/r, unlike N c, ans so d is weakly related to the pitch-radius ratio. It is shown in the Figure 7 that the axial velocity field induced by the finite helical filament is equivalent to that given by an infinite helical filament for a nondimensional axial position d 8 from the rotor plane (with less than % relative difference). So the length, L, of the finite helical vortex-filament, that must be considered to ensure that the axial induced velocity coincides (near L/2) with that of an infinite vortex-filament prediction, is at least L/R=2d 6 as sketched in Figure 8. θ )/<V I >θ z (<V z F >θ V z I d Figure 5: The comparison between the averaged axial velocity induced by the finite filament and that induced by the infinite filament at r = as a function of the axial distance d, for different h/r : h/r=., h/r=., h/r=.5, h/r=.8, h/r=, h/r=2,+h/r=,. exp( d/.65), exp( d/2). Furthermore, this study justifies that the axial velocity induced by a finite helical vortex at

140 ICOWES23 Conference 7-9 June 23, Lyngby 2.9 d * Figure 6: The characteristic distance d converge rapidly to α = 2 for increasing pitch-radius ratio h/r h/r 8 6 d F I I <V Vz>θ /<V >θ z z Figure 7: Axial distance behind the rotor plane as a function of Vz F., h/r=. V I z θ / V I z θ :. h/r= the rotor plane is the half of the velocity induced by an infinite helical vortex if L 6R. Figure 9 shows the difference between the averaged axial velocities at the rotor plane (d = ) and at axial position d = 8. Comparison with a numerical simulation of a rotor wake In this paragraph we present a comparison of azimuth averaged velocities, of infinite and finite helical (filament) vortices, with those of a finite helical vortex (with a core) in the wake of a rotating blade, obtained in a numerical simulation using the actuator line method []. A helical

141 ICOWES23 Conference 7-9 June 23, Lyngby Figure 8: Sketch of the behavior of the axial velocity, Vz F, induced by a finite helical vortexfilament of length L/R = 2d = 6. Near the central part, L/2, the velocity Vz F is equal to that induced by an infinite helical vortex-filament Vz I..5 at z= at z=8r <V z F >θ r/r Figure 9: Axial induced velocity: V F z θ d= = /2 V F z θ d=8 for h/r=. vortex filament, having the circulation and radius of the hub vortex, is also used in the finite and infinite computations of vortex filament velocities (see figure ). The azimuth averaged velocities, in the rotor plane, are presented in figure. In the finite cases, the lengths of vortices are L 6R. It is shown that average velocities of finite length vortices are in good agreement, however they are different from those of the infinite case. Note, however, the factor two in magnitude between the finite and infinite cases for azimuthal and axial velocities. CONCLUSION

142 ICOWES23 Conference 7-9 June 23, Lyngby Figure : A sketch of the tip and hub vortices used to calculate velocities for comparison with those of the actuator line method. Figure : Azimuth averaged velocities at the rotor plane. Green color: infinite vortex; blue color: finite vortex with a length L 6R; red color: numerical simulation using the actuator line method in a computational box with an axial length L 6R.

143 ICOWES23 Conference 7-9 June 23, Lyngby A comparison between an infinite helical vortex-filament and a (realistic) finite helical vortexfilament has been done. The study allows to verify the validity of some approximations used to model wind turbine wakes and/or to design new blades. The main results of this study can be summarized as follows:. The symmetry of the infinite helical vortex-filament is broken when we consider a finite helical vortex-filament. Therefore, a not-negligible radial velocity appears at the rotor plan. 2. For a helical vortex-filament with a length L 6R (Figure 8), the axial induced velocity could be considered equivalent to that induced by an infinite helical vortex-filament near L/2, with a relative difference less than %. This length has been found to be independent of the pitch-radius ratio h/r. 3. The axial induced velocity at the rotor plane is found equal to the half of the velocity induced by the infinite vortex when L 6R. REFERENCES [] H. Glauert, Airplane propellers, In: Durand, WF. (ed) Division in Aerodynamic Theory, Reprinted edition 943, vol IV, pp [2] S.E. Widnall, The stability of helical vortex filament, J. Fluid Mech, vol 54 (972) pp [3] J.C. Hardin, The velocity field induced by a helical vortex filament, Phys. Fluid, vol.25 (982) pp [4] P.G. Saffman, Vortex Dynamics, Cambridge University Press 992. [5] Y. Fukumoto, V.L. Okulov, The velocity field induced by a helical vortex tube, Phys. Fluid, vol.7 (25) pp. -9. [6] V.L. Okulov, J.N. Sørensen, Stability of helical tip vortices in a rotor far wake, J. Fluid Mech, vol 576 (27) pp [7] V.L. Okulov, J.N. Sørensen, Maximum efficiency of wind turbine rotors using Joukowsky and Betz approaches, J. Fluid Mech, vol 649 (2) pp [8] V.L. Okulov, J.N. Sørensen, Applications of 2D helical vortex dynamics, Theor. Comput. Fluid Dyn., vol 24 (2) pp [9] I. Delbende, M. Rossi, O. Daube, DNS of flows with helical symmetry, Theor. Comput. Fluid Dyn., (2) [] H. Bolnot, Instabilitées des tourbillons hélicoïdaux :application au sillage des rotors, PhD thesis, Aix-Marseille University, IRPHE, France 22. [] Sorensen, J. N. and Shen, W. Z., Numerical modeling of Wind Turbine Wakes, J. Fluids Eng., Vol. 24, 22, pp

144 42 CHAP. 7: APPENDIX 7.2 Paper 2 published on Journal of Fluid Mechanics, 24

145 J. Fluid Mech. (24), vol. 74, R, doi:.7/jfm Self-similar behaviour of a rotor wake vortex core Mohamed Ali and Malek Abid Institut de Recherche sur les Phénomènes Hors Équilibre (IRPHE), CNRS, UMR 7342, Centrale Marseille, and Aix-Marseille Université, 49, rue F. Joliot Curie, B.P. 46, 3384 Marseille CEDEX 3, France (Received 7 October 23; revised 2 November 23; accepted 2 December 23; first published online 8 January 24) We report a self-similar behaviour of solutions (obtained numerically) of the Navier Stokes equations behind a single-blade rotor. That is, the helical vortex core in the wake of a rotating blade is self-similar as a function of its age. Profiles of vorticity and azimuthal velocity in the vortex core are characterized, their similarity variables are identified and scaling laws of these variables are given. Solutions of incompressible three-dimensional Navier Stokes equations for Reynolds numbers up to Re = 2 are considered. Key words: Navier Stokes equations, wakes, vortex dynamics. Introduction Helical tip vortices in rotor wakes are important for many applications, especially in the study of wind turbine and propeller flow dynamics. For example Okulov & Sørensen (27) have noted the consequences of strong tip vortices interacting with other turbines in a wind farm. Therefore, many studies have considered the stability problem of a helical vortex system (Widnall 972; Okulov 24; Fukumoto & Okulov 25; Okulov & Sørensen 27). However, in these studies basic flows are not solutions of Navier Stokes equations and ad hoc circular vortex cores are used. Furthermore, core radius is fixed in time and vorticity and velocity profiles are not known in the core. Recently, considering uniform vorticity, Lucas & Dritschel (29) have found the form of the vortex core, for different (infinite) helical vortices, using Euler equations. In the present work we consider solutions of Navier Stokes equations (without restrictions on the core vorticity) in the wake of a single-blade rotor, obtained numerically using the actuator line method (Sørensen & Shen 22). The rotor, with an angular velocity Ω, is placed in a uniform axial inflow with a characteristic address for correspondence: abid@irphe.univ-mrs.fr c Cambridge University Press R-

146 M. Ali and M. Abid velocity W and a kinematic viscosity ν. The blade has a characteristic radius R b. A numerical investigation is conducted to characterize (for many blade revolutions) the wake structure and the evolution of the helical tip vortex. The paper is organized as follows. In the 2 the mathematical model and the computational algorithm are described. Section 3 describes the methods used to characterize the helical vortex properties. The similarity behaviour is presented in 4. Section 5 is devoted to our conclusions. 2. Numerical method and validation The most frequently used models for studying rotor wakes are based on axisymmetric assumptions, as in the actuator disc model (Vermeer, Sørensen & Crespo 23; Hansen et al. 26). The main limitation of these models is that the forces are distributed evenly along the actuator disc, hence the influence of the blades is taken as an integral quantity in the azimuthal direction. To overcome this limitation, an extended three-dimensional (3D) actuator disc model has been developed. The model combines a 3D Navier Stokes (NS) equation solver with a technique in which body forces are distributed radially along each blade (considered, ideally, as a line). Thus, the dynamics of the wake is determined by a fully 3D NS simulation whereas the influence of the rotating blades on the flow field is included using tabulated aerofoil data to represent the loading on each blade. The aerofoil data and subsequent loading are determined by computing local angles of attack from the movement of the blades and the local flow field. This concept, developed by Sørensen & Shen (22), enables us to study in detail the dynamics of the wake and the tip vortices and their influence on the induced velocities in the rotor plane. Globally, the wake computation is based on two principal steps: determine body forces; inject the body force distribution into the NS equations and solve them. We present briefly the numerical scheme used to solve NS equations and the technique used to calculate the aerodynamic forces. 2.. Governing equations and Navier Stokes solver The flow is governed by the incompressible 3D NS equations: U t + (U )U = p ρ + Re 2 U + f, U =, (2.) where U is the velocity vector, p is the pressure, ρ is the constant fluid density (taken here equal to unit of density), Re W R b /ν is the Reynolds number and f represents body forces. These equations are solved in a cylindrical coordinate system with a basis (e r, e θ, e z ). The singularities introduced by the first and the second derivatives on the axis (r = ) are bypassed using a staggered grid and primitive variables as defined by Verzicco & Orlandi (996) (q r = rv r, q θ = v θ, q z = v z ), where v r, v θ and v z are the velocity components in the radial, azimuthal and axial directions, respectively. The axial inflow velocity, W, is used as a velocity scale and the rotor radius, R b, as a length scale. Unless otherwise stated, time is made non-dimensional using the time scale R b /W. The computational domain is cylindrical and defined by a longitudinal length L z and a maximum radius R max. In the present work, we have considered a uniform staggered grid using (N r, N θ, N z ) points in the radial, azimuthal and axial directions respectively. The primitive variables q i are calculated at different points located at the centre of the surfaces of each cell while the pressure is calculated at cell centres. The advantage of 74 R-2

147 Self-similar behaviour of a rotor wake vortex core using the staggered grid is that only the component q r is evaluated at the axis r =, and there q r = by definition. As boundary conditions, we have considered: (i) a periodic condition along the azimuthal direction θ; (ii) at the axis r =, q r = (by definition), q θ = and q z / r = ; (iii) for the lateral boundary r = R max, Neumann conditions for the different variables, q i / r =, are used; (iv) uniform axial inflow velocity is used at the inlet boundary z =, q z = W, q r = q θ = ; (v) homogeneous Neumann boundary conditions are used at the outlet boundary z = L z, q i / z =. The computations are done for different combinations among four Reynolds numbers, Re = (5,, 5, 2), and four different tip speed ratios, λ ΩR b /W = (5, 6, 9, 2), which represent a large range of operational conditions of the rotor Numerical implementation The mathematical equations described above are solved using a second-order, centred, finite-difference scheme with a fractional time step method combined with an approximate-factorization technique. The 3D Poisson equation, to enforce incompressibility, is solved by a fast inversion method, based on fast Fourier transforms along the θ-direction and a resolution of a two-dimensional (2D) Helmholtz problem (for each azimuthal wavenumber) using the cyclic reduction technique. To compute the non-solenoidal velocity field, a third-order, low-storage, Runge Kutta method combined with an implicit Crank Nicholson scheme, for viscous terms, are used. This method is second-order accurate in time for the viscous terms and third-order accurate for the convective terms, the overall accuracy being second-order in time. The implementation of the method was parallelized using the Open-MP libraries Body force computation To determine body forces, we use the blade-element method (Sørensen & Shen 22) which allows us to calculate the local aerodynamic load, for each radial position, knowing the angle of attack (from the NS solver) and the corresponding aerodynamic coefficients (tabulated lift and drag for a given blade) at that position. The aerodynamic coefficients of the A8 aerofoil profile (from the UIUC applied aerodynamics group, database.html) are used here. To avoid the singular behaviour of the aerodynamic forces, distributed radially along a line, the forces are smoothed using a Gaussian kernel. The smoothing is obtained by taking the convolution of the computed load f (designating any component of the aerodynamic force) and a regularization kernel η ǫ, We used a regularization kernel defined in 2D as f ǫ = f η ǫ. (2.2) η ǫ (p) 2D = ǫ 2 π exp( (p/ǫ) 2), (2.3) 74 R-3

148 M. Ali and M. Abid (a) Pitch Numerical (b) FIGURE. (a) Pitch, h, of the helical wake as a function of the tip speed ratio, λ, for numerical simulations and for the theoretical case h = 2π/λ. (b) Presentation of a developed helical vortex in a plane (θ z) for λ = 5 and Re = 2 (the norm of the vorticity is shown with red colour corresponding to the maximum value and blue colour to. of the maximum):, trace of the vortex core plane;, trace of an (r z) plane. where p is the distance between cell-centred grid points and points at the actuator line. The parameter ǫ is a constant introduced as a smearing parameter that serves to adjust the concentration of load regularization, and its size is related to the local grid size. The forces are therefore distributed in planes normal to the blade (infinite planes orthogonal to the actuator line). The smoothing parameter, ǫ, is chosen to be few cell lengths and should be as small as possible to not influence the wake structure Validation The NS solver is validated by reproducing growth rates of an unstable jet flow as done by Abid & Brachet (993, 998) (data available, not shown). Principally, different time and space resolutions are used and we compare (approximate) growth rates obtained using the NS solver with those predicted by linear instability theory (considered as exact values of growth rates). In particular, we check that growth rate errors go to zero like the grid spacing to the power 2, and like the time step to the power 2 also, when space and time resolutions become finer, respectively. This is in agreement with the finite-difference discretization (of order two) and the Runge Kutta/Crank Nicholson method (of order two) that we used for space and time discretizations, respectively. The implementation of the actuator line method is validated using an important parameter characterizing the wake geometry, namely the tip-vortex pitch that can be defined as the axial distance travelled by a particle of the tip vortex during one blade revolution. For a particle having a helical trajectory and moving with an angular velocity, Ω, and an axial velocity, W, the pitch, h (in units of R b ), of the trajectory is given by (hr b )/W = 2π/Ω leading to a theoretical pitch h = 2π/λ for this motion. Figure (a) shows the tip-vortex pitch for both numerical results and the theoretical case. Results show a trend of decreasing pitch with increasing λ in good agreement with theoretical predictions. The numerically obtained pitch is slightly lower than the theoretical value. This can be explained by viscous effects, that promote the interactions between adjacent vortices, and the existence of the vortex core itself. It is found that the Reynolds number does not affect the pitch value although it greatly affects the core size (discussed in the following sections). 74 R-4

149 Self-similar behaviour of a rotor wake vortex core (a) (b) FIGURE 2. (a) Tip helical vortex (in red) generated behind a one-bladed rotor, obtained as a solution of the Navier Stokes equations. (The norm of the vorticity is shown. The hub vortex, which is not the subject of the present study is shown in blue.) (b) Tip-vortex core shape (contour lines of the vorticity ω θ are shown from the maximum, in red, to. of the maximum, in blue, with a regular interval). 3. Characterization of the helical vortex If t = is the starting time of the blade rotation, we wait a time t for the starting vortex to move away from the rotor plane. Precisely, the time t is defined as the time needed for the blade to make seven revolutions. We check that for a given z the vortex core radius is independent of time for z7h (where h is the pitch of the helical vortex) and t t. The origin of the coordinate system is then placed at the centre of the first formed vortex core (in an axial plane containing (O, z)) and we follow its evolution in a frame rotating with the blade. Since, for a helix, z = hθ/(2π), a correspondence between z and time t exists: z = hω(t t )/(2π). In the following we will use vortex age n = Ω(t t )/(2π). Results for different vortex ages will be presented. Specifically, the ages n7 are considered. The characterization of the wake generated behind the blade, and the vortex created, is based on the moments of vorticity. It is done in axial (r, z) planes, which are not orthogonal to the local vortex axis. As shown in figure (b) an (r, z) plane and the plane of the vortex core form an angle χ given by χ = arctan(h/(2π)) = arctan(/λ), where h = 2π/λ is the tip-vortex pitch. For the tip speed ratios used in this study (5 < λ < 2), cos(χ) is practically constant (.98 < cos(χ) <.9965). Thus, the error arising from the fact that the characterization of the vortex core is done in an (r, z) plane is less than 2 %. To characterize the vortex, first, its centre (R c, Z c ) is determined. It corresponds to the spatial point with a local maximum vorticity (ω θ (R c, Z c ) = max ω θ ). We define the vortex core using a rectangular domain, D, such that z Z c [ h/2, h/2] and r R c [ R b /2, R b /2]. Then, to determine the vortex core size, a c, a calculation based on the vorticity moments is used. We find that the vortex core shape is elliptical and its corresponding geometry is illustrated in figure 2(b). We have checked that the vortex core shape is also elliptical in a plane orthogonal to the local vortex axis. For all Reynolds numbers considered we find that the ellipticity aspect ratio (major/minor axis lengths) is less than R-5