Improvement of wall treatment in Large Eddy Simulation for aeroacoustic applications

Faculté de génie

Département de génie mécanique

Amélioration de Loi de Paroi de Simulation

aux Grands Échelles pour des Applications

Aéroacoustiques

Improvement of wall treatment in Large Eddy Simulation for

Aeroacoustic applications

Thèse de doctorat

Specialité: génie mécanique

Chaofan ZHANG

Sherbrooke (Québec) Canada

Stéphane MOREAU

Marlène SANJOSÉ

Karine TRUFFIN

Aloïs SENGISSEN

Sébastien PONCET

Le bruit de train d’atterrissage, généré par l’interaction de l’écoulement turbulent avec des corps solides et le décollement de la couche limite, sont les sources principales de bruit d’un avion en phase d’atterrissage. Les données expérimentales existantes ne sont pas suffisantes pour fournir les informations détaillées sur ces mécanismes de génération du bruit, et, depuis des années, les simulations numériques ont prouvé être un moyen efficace pour la prévision du bruit de ce type. Comparée à la Simulation Numérique Directe et aux modèles à moyenne de Reynolds, la Simulation aux Grandes Échelles (SGE), est un compromis efficace entre la précision des résultats et le coût de calcul. Cependant, la prévision de l’écoulement dans la couche limite turbulente reste un défi en SGE. En effet, les simulations existantes résolvent souvent les plus petites échelles aux parois, nécessitant alors un maillage très raffiné proche des surfaces, augmentant considérablement le coût de calcul.

Par conséquent, un modèle de paroi qui est capable de reconstituer la contrainte de ci-saillement à la paroi sur la base de données extraites à une certaine distance au-dessus du paroi est nécessaire pour réduire les coûts. La revue de litérature met en évidence le modèle analytique proposé par Afzal [6] qui considère les effets de gradient de pression défavorables avec un surcoût négligeable. Outre les effets de gradient de pression, la couche limite lami-naire dans la partie amont du cylindre avant la transition vers la turbulence pose un autre problème. L’utilisation des lois de paroi pour la couche limite turbulente peut être imprécise et même changer complètement le régime d’écoulement. Pour surmonter cet obstacle, un modèle a été proposé dans ce travail pour estimer la contrainte de cisaillement de la paroi dans la couche limite laminaire lorsque le gradient de pression est important. Un capteur de transition basé sur le modèle de sous-maille a été utilisé pour déclencher l’utilisation de la loi de paroi turbulente.

L’écoulement d’un cylindre circulaire dans le régime critique a été considérée comme une première validation de la loi d’Afzal et son extension. La valeur du nombre de Reynolds choisi correspond à la configuration de l’écoulement qui se trouve sur la jambe principale du train d’atterrissage LAGOON. L’écoulement complexe du cylindre est examiné par une SGE résolue, qui a ensuite été utilisée extensivement comme base de données de validation intense pour la loi d’Afzal et son extension. Tous les modèles de paroi sont capables de prédire correctement la moyenne et le RMS de la pression pariétale de la simulation de référence. L’utilisation des lois turbulentes sur toute la surface du cylindre entraîne une contrainte de cisaillement de la paroi inférieure dans la région laminaire et supérieure dans la région turbulente par rapport à la simulation résolue. L’extension de la loi d’Afzal fournit une prédiction améliorée dans les régions laminaires et turbulentes. Comme dans les systèmes du train d’atterrissage réels, il existe des interactions entre ses composants cylindriques, tels que la barre de traction avec la jambe principale. L’expérience canonique de barreau-profil pour une telle interaction, est donc sélectionnée comme deuxième cas de validation. Les simulations avec loi de paroi montrent des résultats acoustiques en champ lointain en bon accord avec les messures.

GOON#1. En général, toutes les simulations prédisent précisément la pression moyenne pariétale. Cependant, l’application d’un modèle pour la couche limit turbulente partout prévoient des valeurs RMS et des spectres de pression plus élevés sur le périmètre de la roue depuis la première position de mesure expérimentale. Une transition plus précoce se produit systématiquement. L’extension de la loi d’Afzal retarde la transition et permet de mieux prédire le spectre de pression des parois, à la fois sur la surface de la roue et sur la jambe principale. Toutes les simulations sont capables de récupérer les spectres de pres-sion des parois dans la région séparée. Malgré ces divergences sur le développement de la couche limite, toutes les simulations prédisent une valeur OASPL acceptable dans le champ lointain, avec une amélioration notable de l’extension de la loi d’Afzal.

Mots-clés : Aéroacoustiques, Bruit de célllule, Simulation aux Grands Échelles, loi de parois, gradient de pression adverse, train atterrissage

Airframe noise, generated through the interaction of turbulent flow with solid bodies such as landing gears becomes the main contributor to the airplane noise during approach and landing phases, since significant progress has been made on the noise reduction of turbo-jet engines. The existing experimental data haven’t been able to provide sufficiently detailed information on airframe noise mechanism and numerical simulations have been considered as an effective method in understanding both aerodynamic and noise generation mechanisms. Among different numerical methods, Large Eddy Simulation (LES) is considered as the best trade-off between predictive accuracy and computational cost. However, wall-bounded flows at high Reynolds number remain the most crucial challenge for LES since the resolution of the boundary layer dominates the computational cost which is close to Direct Numerical Simulations.

One solution to overcome this difficulty is the use of wall models to provide boundary con-ditions for the LES simulation. The classical logarithmic-law is not suitable in simulations of landing gear flows in which the longitudinal adverse pressure gradient have significant effects. A new analytical wall model (proposed by Afzal [6]) which accounts for the adverse pressure gradient effect has been considered to tackle the noise prediction of a realistic landing gear. Another challenge of such flows is the presence of the laminar state boundary layer. The use of wall models for the turbulent boundary layer can be inaccurate and even change completely the flow regime. To overcome this obstacle, a model has been proposed in this work to approximate the wall-shear stress in the laminar boundary layer when im-portant pressure gradient effects are present. A transition sensor based on the subgrid-scale model has been used to trigger the use of wall law for the turbulent boundary layer. The benchmark of the circular cylinder flow in the critical regime has been considered as a first validation for the above wall models. The flow at such a critical Reynolds number combines complex features: large favorable and adverse pressure gradient, separation and turbulence transition and flow reattachment. This flow regime is also the most relevant for landing gear flow applications because of the Reynolds number range involved on its components. The complex cylinder flow has been investigated by a wall-resolved LES which has then been used extensively as validation database for Afzal’s law and its extension. All the wall-models are able to predict the mean and the RMS wall pressure distributions of the reference simulation. The use of a turbulent wall model on the entire surface results in lower wall-shear stress in the laminar region and higher in the turbulent region compared with the resolved simulation. The extended model shows improved prediction of the shear stress in both laminar and turbulent regions. All of the models recover the dipole pattern with similar OASPL levels as in the wall-resolved simulation. Since in actual landing gear systems, there are actually interaction between various cylindrical components such as the tow bar with the main strut for instance. The canonical experiment for such an interaction, the rod-airfoil interaction is therefore selected as a second validation case. These models show reasonable aerodynamic and acoustic results compared with the experimental references.

In general, the mean wall pressure profiles are accurately predicted by all the simulations. However, turbulent wall models predict higher rms and spectra of pressure on the wheel perimeter since the first experimental measurement position. Earlier transition systemati-cally occurs. The extended Afzal’s law delays the transition and shows improved prediction of the wall pressure spectra both on the wheel surface and on the main leg. All the models are able to recover the wall-pressure spectra in the separated region. Despite these discrep-ancies on the boundary layer development, all the simulations predict satisfactory OASPL in the far-field with a significant improvement from the extended Afzal’s law.

Keywords: Aeroacoustic, Airframe noise prediction, Large Eddy Simulation, Wall treat-ment, Adverse pressure gradient, Landing Gear

Tout à bord, je remercie sincèrement mes encadrants, Pr. Stéphane Moreau, Dr. Marlène Sanjosé et Dr. Aloïs Sengissen pour leur soutiens solids, conseils pertinents et patience du début jusqu’à la fin. Je remercie particulièrement la chaire industrielle d’aéroacoustique de l’Université de Sherbrooke et Airbus pour avoir financé cette thèse.

Je voudrais évidemment remercier les membres de jury, Pr. Sébastien Poncet et Dr. Karine Tuffin pour le suivi de ma thèse depuis mon prédoc à deuxième année jusqu’à la fin. Merci pour la relecture de ma thèse et les conseils perspicaces. Je voudrais aussi remercier Pr. Thierry Poinsot et Pr. Laurent Gicquel du côté CERAFCS et Pr. Franck Nicoud de l’Université de Montpellier pour les discussions productives.

Je voudrais aussi remercier mes collègues au sein du groupe d’aéroacoustique de l’UdeS, dont nombreux sont devenus des amis: informaticient Léonard Thomas, Carlos, Manqi, Gyuzel, Aurelian et Yann qui étaient à l’autre côté de la table durant les réunion de discus-sion scientifique. Sans être oubliés bien sûr les supports incontournables de Hao, Prateek, Dominic, Dipali, Safouane, Miguel, Régis, Pavel, Aurellian et Bruno.

I definitely would like thank the computational resources and technical supports made avail-able on the NIAGARA and Mammouth-Parallèle-II of Computecanada and CalculQuebec. Je voudrais aussi remercier mes amis: Bo, Céline, Wei Qi, Jianyu, Yu and Jiaping. Peu important la dispersion dans le monde, les supports s’amortissent pas. Je dois aussi de remerciement à Stéphanie et Yanfei, ma famille d’adoption locale au Québec.

Ce travail n’aurait pas été possible sans les supports infinis de ma famille. Un grand merci aussi pour la présence assurante de Sanae du fond de mon coeur durant les périodes difficile d’écriture.

I must say thank you, dear reader, to have the interest for this research work. The work is less meaningful to me without being shared.

1 Introduction 1

1.1 Industrial context . . . 1

1.2 Landing gear noise . . . 2

1.3 Methodology and the LAGOON project . . . 4

1.3.1 Numerical Method for Landing gear noise prediction . . . 5

1.3.2 The LAGOON project . . . 7

1.4 Objectives, simulation cases and organization of the thesis . . . 9

1.4.1 Objectives and computational cases . . . 9

1.4.2 Organization of the thesis . . . 10

2 Large Eddy Simulation 13 2.1 Introduction . . . 13

2.2 Governing equations . . . 13

2.2.1 Navier-Stokes equations . . . 13

2.2.2 Equation of state and fluid thermodynamics . . . 14

2.2.3 The LES approach . . . 15

2.2.4 Equations of LES . . . 17

2.3 Turbulence closure . . . 19

2.4 Numerics in AVBP . . . 20

2.4.1 Cell-vertex approach . . . 20

2.4.2 Numerical schemes . . . 23

2.5 Hybrid grid topology . . . 24

2.5.1 Test case configuration . . . 24

2.5.2 Results . . . 26

3 Turbulent boundary layer 33 3.1 Turbulent boundary layer without pressure gradient . . . 33

3.1.1 Introduction and definitions . . . 33

3.1.2 Mean flow properties and the law of wall . . . 35

3.1.3 Turbulent structures . . . 38

3.2 Effect of adverse pressure gradient . . . 40

3.2.1 Introduction and definitions . . . 40

3.2.2 Boundary layer profiles . . . 41

3.2.3 Wall-pressure fluctuations . . . 46

4 Wall models for Large Eddy Simulation 49 4.1 Wall-stress models . . . 50

4.1.1 Analytical models . . . 50

4.1.2 Afzal’s law . . . 53

4.1.3 Models based on TBL equations . . . 54

4.1.4 Integral models . . . 57 xi

4.2 RANS/LES hybrid methods . . . 59

4.3 Summary . . . 60

5 Verification and implementation of Afzal’s law 63 5.1 A priori validation with NACA0012 setup . . . 63

5.1.1 Numerical setup of the verification case . . . 63

5.1.2 Results and conclusion . . . 64

5.2 Implementation of Afzal’s law . . . 68

5.2.1 Interaction of the WMLES approach with the flow solver . . . 68

5.2.2 The pressure gradient term . . . 70

5.3 An extended model to the laminar layer with transition sensor . . . 72

5.3.1 Momentum balance in a laminar boundary layer subjected to pressure gradient . . . 72

5.3.2 Transition sensor . . . 73

6 Results on benchmark cases 75 6.1 Motivations . . . 75

6.2 Flow around a circular cylinder . . . 75

6.2.1 Introduction . . . 75

6.2.2 Numerical Parameters . . . 76

6.2.3 Results of the wall resolved LES . . . 79

6.2.4 Wall modeled LES . . . 96

6.3 Rod-Airfoil configuration . . . 107

6.3.1 Benchmark and physical phenomenon . . . 107

6.3.2 Numerical Setup . . . 107

6.3.3 Results and discussions . . . 109

6.4 Conclusion . . . 115

7 Landing gear noise prediction 121 7.1 Introduction . . . 121

7.1.1 Experimental setup and description of the database . . . 121

7.1.2 Previous numerical simulations . . . 123

7.2 Numerical setup . . . 125

7.3 Aerodynamic results . . . 127

7.3.1 Phenomenology of the flow . . . 127

7.3.2 Mean coefficients at the wall . . . 131

7.3.3 Mean and rms velocity contours and profiles . . . 136

7.3.4 Wall-pressure spectra . . . 139

7.4 Far-field acoustic results . . . 147

7.4.1 PSD of the far-field acoustic pressure . . . 148

7.4.2 Overall sound pressure level . . . 149

7.4.3 Noise source separation . . . 152

7.5 Conclusion . . . 155

9 Conclusion et perspectives 161

A Vortex convected along the interface 165

1.1 A B747 taking-off at Sydney . . . 1 1.2 Typical noise distribution of different elements at approaching for an Airbus

long haul airplane, from [142]. . . 2 1.3 Nose landing gear of A320. . . 3 1.4 LAGOON mock-ups, from Manoha et al. [95] . . . 7 1.5 Pressure taps, kulites and microphones position in the measurement, from Manoha

et al. [95] . . . 8 2.1 Plot of the conceptual representation of RANS, LES and DNS applided to a

homogeneous isotropic turbulent field: turbulent kinetic energy spectrum as a function of the inverse length scale. . . 16 2.2 Primary cells: white and dual cells: shaded. Solid and empty dots are nodes

of primary and dual cells respectively. . . 21 2.3 Cell nodal normal. . . 22 2.4 Dissipation and dispersion properties as a function of dimensionless

wavenum-ber of LW, TTGC and TTG4A schemes for CFL=0.7, from Lamarque [81]. . 24 2.5 Sketch of the mid-plane of the simulation domain. . . 25 2.6 A zoom view of different grids tested in Table 2.1: from top to bottom: PT,

PT-Perturbation and PT-centaur. . . 26 2.7 Absolute values of the pressure difference between numerical resultspnumand

theoretical valuespthin the simulation domain mid-plane at different instants

using Pri/Tet-regular grid: left, LW; center, TTGC and right, TTG4A. . . . 28 2.8 Absolute values of the pressure difference between numerical resultspnumand

theoretical values pth at the simulation domain mi-plane at different instants

using Pri/Tet-perturbation grid: left, LW; center, TTGC and right, TTG4A. 30 2.9 Absolute values of the pressure difference between numerical resultspnumand

theoretical values pth at the simulation domain mi-plane at different instants

using Pri/Tet-centaur grid: left, LW; center, TTGC and right, TTG4A. . . . 31 2.10 Temporal evolution of theL1 error on the pressure for different hybrid grids. 32

3.1 The development of boundary layer over flat plate. . . 34 3.2 Decomposition of shear stress in a turbulent boundary layer, using several

DNS data of Lee and Moser [86] with different Reynolds numbers. Blue dash-dotted lines: viscous shear stress; red dashed lines: Reynolds stresses; black solid lines: total shear stress. . . 36 3.3 Mean velocity profiles in wall unit for fully developed turbulent channel flow

measured, using the same data as in Fig. 3.2. . . 37 3.4 Schematic of streamwise vortex and hairpin structures: bottom, streaks; top,

hairpin structures; middle, streak and hairpin structures. From Robinson [128]. 39 xv

3.5 Sketch of the organization of hairpin and packets in a boundary layer. Hair-pins are mostly found in the lower half of the boundary layer, especially in the logarithmic layer, creating uniform momentum zones in layers. They

sometimes extend to the edge of the boundary layer, from Adrian [5]. . . 39

3.6 Smoked visualization of the hairpin structures and large scale motions in a turbulent boundary layer, from Falco [53]. . . 40

3.7 Mean velocity profile of TBL subjected to pressure gradient. _{•: dp/dx = 0,} Reθ = 13, 000, H = 1.1334; ◦: dp/dx > 0, Reθ = 67, 400, H = 2.324. From Cousteix [34]. . . 42

3.8 Mean velocity profiles of non-equilibrium TBL subjected to APG over flat plate. The mean velocity defect and shape factor H increases with β, which increases in the streamwise direction. The velocity profile with H = 3.69 is near separation. Figure adapted from Maciel et al. [92]. . . 42

3.9 Turbulence intensity profiles (urms) in wall units for: APG TBL, β = 0.35 with Reθ = 1460, DNS results of Araya and Castillo [7]; ZPG TBL, β = 0, Reθ = 1550, DNS of Simens et al. [147]. . . 43

3.10 Comparison of the production of TKE in ZPG and APG TBL, using outer scales, after Simens et al. [146]. . . 44

4.1 Application of a wall-law in LES. . . 51

4.2 Sketch of the TLM procedure of Wang and Moin [166]. . . 56

5.1 A priori validation procedure of Afzal’s law. . . 63

5.2 Hybrid mesh for the test case configuration. . . 64

5.3 Comparison of mean pressure and skin-friction coefficients along the airfoil of the present RANS, panel method with the experimental measurements of similar flow around a NACA12 from Garcia-Sagrado and Hynes [58]. . . 65

5.4 Distribution of pressure gradient dp/ds, and dimensionless pressure gradient βi computed from RANS and panel method results. . . 66

5.5 Comparison of velocity profiles. (a): RANS results against experimental results from Garcia-Sagrado and Hynes [58]. (b): Plot of wall law using Eq. (4.19) against experimental results. Three pressure gradient values are used βi = 0, βi = 0.01 and βi = 0.02, where the last two values are taken from Fig. 5.4(b) at x = 0.8 C and x = 0.9 C respectively. . . 67

5.6 Comparison of skin-friction predicted by the generalized log-law and Afzal’s law (Eq. (4.19)) with RANS results; right y-axis shows the dimensionless pressure gradient along the airfoil. Data are extracted for y+ _{around 50. . . . 67}

5.7 Diagram of the different processes of the wall-modeled LES for an adiabatic wall. . . 69

5.8 Schematic view of the application of the wall model. . . 70

5.9 Illustration of the time average. . . 71

5.10 Blending function used in the transitional region. . . 74

6.1 Simulation domain of the cylinder flow: dashed line is the porous FWH-surface. θ is the angle from the front stagnation point and equals to 180◦ at the rear point in the wake. . . 77

6.2 Mesh of cylinder flow in the mid-span plane: Top: mesh of the coarse and resolved case around the cylinder; Center: zoom views for 85◦ _{< θ < 95}◦_;

Bottom: zoom of the resolved case mesh around the tripping near θ = 72◦. . 78

6.3 Wall resolution of three computational cases. . . 79

6.4 Drag coefficient as a function of Reynolds number: comparison of present simulation with measurements from Fage and Falkner [52], Spitzer [158], Achen-bach and Heinecke [4], Schewe [136] and Vaz et al. [163]; LES results Lehmkuhl et al. [87] and Cheng et al. [25]. . . 79

6.5 Mean pressure coefficient and wall-pressure fluctuation at the circumference of the cylinder. TheCp of current simulation is compared with measurements from Fage and Falkner [52], Achenbach [1], LES results from Cheng et al. [25] and Lehmkuhl et al. [87]. . . 81

6.6 Time and spanwise averaged pressure and pressure fluctuation around the cylinder of the present LES. . . 81

6.7 Skin-friction coefficient at the circumference of the cylinder: comparison of current simulation with measurements from Achenbach [1] and LES results from Cheng et al. [25]. . . 82

6.8 Flow field near the laminar separation bubble (LSB). Black-dashed line in (f) shows the location of probes in the shear layer from 90◦ _to120◦ _{for every} 2.5◦_{. . . 83}

6.9 Tangential velocity uθ at different locations: 90◦, 105◦ and 115◦ are respec-tively the beginning, the center and the end of the separation bubble; 125◦, the beginning of the final separation. . . 84

6.10 Edge velocity and boundary layer thickness. . . 85

6.11 Velocity profiles normalized by ue. Dashed and dotted lines in (a) are the Polhausen’s polynomial velocity profiles, using the upper (Λ = 12) and the lower (Λ = _{−12) limit of Λ which is a local dimensionless quantity defined} as Λ = δ2 ν dUE dx . . . 86

6.12 Global boundary layer parameters. . . 86

6.13 Velocity fluctuations and turbulent kinetic energy profiles. . . 88

6.14 Dilation and magnitude of the vorticity field views. . . 89

6.15 PSD of wall-pressure fluctuations. . . 90

6.16 PSD of streamwise (a) and crosswise (b) velocity in the shear layer. . . 90

6.17 PSD of acoustic pressure at different far-field positions obtained with the porous and solid surfaces . . . 91

6.18 Directivity of far-field acoustic pressure using solid-FWH surface (a): OASPL integrated between St = 0.1 and 10; (b): Directivity at the shedding fre-quency: 3 points at the maximum are used for the integration and values are referred to the one at 90◦. . . 92

6.19 Illustration of the extraction position of the unsteady pressure signal for the correlation: wall surfaces, shear-layer probes, near-field probes and far-field microphones. The flow is displayed by an instantaneous magnitude of the vorticity. Only results for the near/far-field probe/microphone at θobs = 90◦ are shown. . . 93

6.21 Correlation between shear-layer and near/far-field pressure fluctuations. . . . 94

6.22 Drag coefficient as a function of Reynolds number: comparison of present simulation with measurements from Fage and Falkner [52], Spitzer [158], Achen-bach and Heinecke [4], Schewe [136] and Vaz et al. [163]. . . 96

6.23 Mean pressure coefficient and wall-pressure fluctuation at the circumference of the cylinder. The Cp of the wall modeled simulations are compared with measurements from Fage and Falkner [52], Achenbach [1] and the wall re-solved LES of the present work. . . 97

6.24 Comparison of the mean skin-friction coefficient of the wall modeled simu-lations with the wall resolved LES and with the experiments of Achenbach [1]. . . 98

6.25 (a) Distribution ofµt/µ and TKE around the cylinder at the sensor detection position (first off-wall grid points); (b) the skin friction and the contribution of the pressure gradient term from Eq. (5.9) of the results of the simulation with the LAF model. . . 99

6.26 Mean tangential velocity profiles at θ = 60◦_{, 90}◦ _{and 120}◦_{. . . 99}

6.27 Zoom view of the mean tangential velocity profiles. Dashed-dotted lines are the position of the first grid points of the simulations with wall models. . . . 101

6.28 Edge velocity and boundary layer thickness. . . 102

6.29 TKE profiles at different positions. . . 102

6.30 PSD of wall-pressure fluctuations. . . 103

6.31 PSD of crosswise velocity at (a) P1 and (b) P3 in the shear layer. . . 103

6.32 PSD of the far-field acoustic pressure obtained by solid-FWH surface. . . 104

6.33 Directivity of far-field acoustic pressure using solid-FWH surface (a): OASPL integrated between St = 0.1 and 10; (b): Directivity at the shedding fre-quency: 3 points at the maximum are used for the integration and values are referred to the one at 90◦. . . 105

6.34 Vorticity magnitude, extracted from wall-resolved LES results of Giret et al. [61]. . . 107

6.35 (a) Mesh used for the AF-XC and LG-XC and zoom on the cylinder and 20%C leading edge; (b) Comparison of leading edge resolution: top to bot-tom are meshes for AF-XC/LG-XC, NS-C and NS-F cases; left: view of 0.5%C and right: view of 20%C. . . 108

6.36 Contours of the mean streamwise velocity in the mid-span plane, extracted from NS-F case. . . 109

6.37 Comparison of mean velocity and RMS velocity at section x/C = −0.225 before the leading edge of the airfoil with HW measurements from Jacob et al. [73] on the rod-airfoil configuration and cases in [61]. . . 110

6.38 Mean velocity profiles (a) and RMS velocity (b) around the airfoil: from fine-mesh case of rod-airfoil configuration in [61] and experimental results of the isolated airfoil configuration from Garcia-Sagrado and Hynes [58]. . . 111

6.39 Distributions of mean pressure coefficient (a) and (b) pressure fluctuations along the airfoil. The Cp is compared with the experimental results of the isolated airfoil configuration from Garcia-Sagrado and Hynes [58]. . . 112

6.40 Skin friction coefficient (a) and its streamwise gradient (b). The experimental results of the isolated airfoil configuration from Garcia-Sagrado and Hynes [58] are compared with the present Rod-Airfoil setup. . . 113 6.41 Mean wall tangential velocity profiles on the airfoil upper surface at different

streamwise positions. . . 113 6.42 Root mean square velocity profiles at different streamwise positions on the

airfoil. . . 116 6.43 Wall-pressure PSD around the airfoil. . . 117 6.44 PSD of acoustic pressure at different far-field positions obtained with the

porous and solid surfaces. . . 118 6.45 Solid FWH PSD of the far-field acoustic pressure of separated elements. . . . 119

7.1 LAGOON#1 model in F2 aerodynamic wind tunnel (left) and CEPRA19

anechoic chamber (right). . . 122

7.2 LAGOON#1 mock up (left), sketch (center) and F2 coordinates systems

used in the present simulations (right). . . 122 7.3 Simulation domain. . . 125 7.4 Crinkle cut at z = 0 plane: global view (top) and zoom views of the mesh

around the landing gear wheels (center and bottom). The right-bottom figure shows the cell volume in this region. . . 126 7.5 y+ _{distribution of the present simulations. . . 127}

7.6 Iso-surface of Q criterion of LAF case. . . 128 7.7 Streamlines near the wheel colored by the velocity magnitude. The front,

bottom and wake views are given in Fig. 7.6(a). . . 129 7.8 Streamlines at the inboard and outboard sides of the wheel colored by the

velocity magnitude. . . 130 7.9 Streamlines in the near region of the wheel. . . 130 7.10 Mean wall-pressure and fluctuations at the perimeter of the left wheel: (c)

RMS computed from the temporal pressure signal; (d) Symbols are RMS computed from the PSD spectra (integration from 200 Hz to 10 kHz) and curves are the same as in (c). . . 132 7.11 Bottom view of the iso-surface of Q criterion near the wheels colored by the

Mach number. The color map is the same as in Fig. 7.6. The dashed lines indicate the positions of -60◦ and -90◦ as defined in Fig. 7.10(a). . . 133 7.12 Mean wall-pressure coefficient on the main leg (b) and the axle (c). . . 134 7.13 Mean wall-pressure coefficient, pressure fluctuations and skin friction on the

center part of the upper leg. . . 135 7.14 Flow passing near the upper leg of the present simulations. . . 135 7.15 Comparison of velocity profile at 90◦ on the wheel. . . 136 7.16 Mean streamwise velocity in the wake of the wheels in the plane z = 0. The

black dashed lines denote from the top to the bottom the cutting lines at x =−0.16 m, −0.18 m and −0.22 m respectively, where velocity profiles are compared. . . 137

7.17 Mean and rms axial velocity in the cutting lines x = −0.16 m, −0.18 m and _{−0.22 m. Experimental results are obtained using the Laser Doppler} Velocimetry (LDV). . . 138 7.18 Mean and rms vertical velocity in the wake of the wheels in the plane z = 0

for the cutting lines atx = 0.16 m, 0.18 m and 0.22 m. Experimental results are obtained using the Laser Doppler Velocimetry (LDV). . . 139 7.19 Wall-pressure PSD on the perimeter of the right wheel. . . 141 7.20 Wall-pressure PSD on the mid-perimeter of the right wheel (continued). . . . 142 7.21 Flow on the inboard side of the right wheel. . . 142 7.22 TKE near Kulites K13 to K15, on the inboard side of the right wheel. . . 143 7.23 Wall-pressure PSD on the inboard side of the right wheel. . . 143 7.24 Kulites on the axle and leg. . . 144 7.25 Velocity contours in the plane crossing the kulites 24 to 27 of LAF case. . . . 145 7.26 TKE distribution around the main leg. The TKE is normalized by the square

of the inlet velocity. The zoom views around K25 are shown in the center of each figure. . . 145 7.27 Flow field near K23. . . 146 7.28 Distribution of the TKE normalized by u2

∞ in the cross plane containing K23.146

7.29 Wall-pressure PSD on the main leg and axle. . . 147 7.30 Microphones positions for the far-field acoustic measurements, adapted from Manoha

et al. [95]. . . 148 7.31 Dilatation field of the LAF case in different planes. . . 149 7.32 Power spectral density of the far-field acoustic pressure on the flyover arc. . . 150 7.33 Power spectral density of the far-field acoustic pressure on the sideline arc. . 151 7.34 OASPL of the flyover microphones. . . 151 7.35 OASPL of the sideline microphones. . . 152 7.36 Acoustic source separation in the flyover arc at 90◦_{. . . 153}

7.37 Acoustic source separation in the sideline arc at 90◦. . . 154 7.38 Elements contribution to the OASPL in the flyover arc. . . 154 7.39 Elements contribution to the OASPL in the sideline arc. . . 155 A.1 Absolute values of the pressure difference between numerical results pnum and

theoretical valuespth at the simulation domain mi-plane at different instants

using Pri/Tet-regular grid: left, LW; center, TTGC and right, TTG4A. . . . 166 A.2 Absolute values of the pressure difference between numerical results pnum and

theoretical valuespth at the simulation domain mi-plane at different instants

using Pri/Tet-perturbation grid: left, LW; center, TTGC and right, TTG4A. 167 A.3 Absolute values of the pressure difference between numerical results pnum and

theoretical valuespth at the simulation domain mi-plane at different instants

using Pri/Tet-centaur grid: left, LW; center, TTGC and right, TTG4A. . . . 168 A.4 Temporal evolution of the L1 error on the pressure for different hybrid grids. 169

1.1 Summary computational cases in this thesis . . . 10 2.1 Summary of the vortex convection cases. . . 26 6.1 Summary of cylinder flow simulations . . . 77 6.2 Pressure and skin distribution parameters . . . 97 6.3 Summary of the simulation cases. . . 109 7.1 Summary of LAGOON simulations . . . 127

NOMENCLATURE

βi Dimensionless streamwise pressure gradient, [-]

u1 Tangential velocity at first off-wall grids, [m/s]

y1 Wall normal distance of first off-wall grids, [m]

ν Kinetic iscosity, [m2/s] ντ Turbulent viscosity, [m2/s] uτ Friction velocity, [m/s] up Pressure velocity, [m/s] u_∞ Freestream velocity, [m/s] ue Edge velocity, [m/s]

τw Wall-shear stress, [Pa]

δ Boundary layer thickness, [m]

θ Momentum thickness, [m]

δ∗ _{Displacement thickness, [m]}

H Shape factor, [-]

κ von Karman constant, [-]

Re Reynolds number, [-]

Reθ Momentum Reynolds number, [-]

Reτ Friction Reynolds number, [-]

APG, FPG, ZPG Adverse, Favorable, Zero Pressure Gradient

CAA Computational AeroAcoustics

CFD Computational Fluid Dynamics

CFL Courant-Friedrichs-Lewy number

DES Detached Eddy Simulation

DNS Direct Numerical Simulation

LAGOON LAnding Gear nOise database for CAA validatiON

LBM Lattice Boltzmann Method

LES Large Eddy Simulation

OASPL OverAll Sound Pressure Level

PSD Power Spectral Density

(U)RANS (Unsteady) Reynolds Averaged Navier-Stokes

SGS Sub-Grid Scale

Introduction

1.1

Industrial context

World air traffic has been growing rapidly over the last few decades. Indeed, the number of air passengers has grown more than eight-fold between 1970 and 2010 [50]. International Air Transport Association1 released a passenger growth forecast, projecting that passenger numbers are expected to reach 7 billion by 2034 with a 3.8% average annual growth in demand (2014 baseline year). That is more than double the 3.3 billion who flew in 2014. Airbus2_{gave a similar growth forecast. Besides, this strong growth of air traffic is combined}

with increasing urbanization of peripheries of meca-cities. The original remote airport areas are surrounding with more and more populations. Fig (1.1) shows an airplane taking-off at Sydney.

Figure 1.1 A B747 taking-off at Sydney

Different approaches have been considered to air traffic noise management. They consist of identifying the noise problem at an airport and then analyzing the various measures available to reduce the noise. These approaches include the reduction of noise at the source, land-use planning, noise abatement operational procedures and operating restrictions. For

1. http://www.iata.org/pressroom/pr/Pages/2015-11-26-01.aspx 2. http://www.airbus.com/company/market/forecast/

instance, many countries have taken action on the withdrawal of operations of most noisy aircraft at their airports3.

At the same time, aircraft manufacturers are making efforts to reduce noise at the source. Recent measurements allow them to accurately quantify the contributions of the main sound sources at both taking-off and landing phases, as shown in Fig. 1.2 for an Airbus long-haul aircraft. These measurements indicate that the main noise source at taking-off is the jet noise. During the approach phase, where engines are in idle mode, the airframe noise is the most significant. The extremely non-aerodynamic landing-gear is the first airframe noise contributor. The flow features and mechanism of landing gear noise are described in following section.

Cavities are also common sources, which in general generate tonal noise. Even the latter has no significant impact on current certification level, but can emerge from the background noise and will be eventually a problem in the near future with the progress of the noise reduction of other sources.

Figure 1.2 Typical noise distribution of different elements at approaching for an Airbus long haul airplane, from [142].

1.2

Landing gear noise

The main function of a landing gear is to support the airplane for taxiing, taking-off and landing. During the taxiing operation, the nose landing gear controls the direction. While during landing, they are constructed to absorb the impact energy when the airplane touches the runway and ensure the airplanes stop at a short distance with the brake systems. To

ensure these objectives, both nose and main landing gears consist of diverse components such as main leg, wheels, axles, support struts, wheel-wells, actuators and hydraulic sys-tems. Most of them can be approximated by circular cylinders with various diameters. And landing gears can be considered as a cluster of bluff-body elements: a vast number of small component attached to the main structures. Fig. 1.3 shows the nose landing gear of an Airbus A320.

Figure 1.3 Nose landing gear of A320.

One of the first experimental studies of landing gear noise was performed by Heller and Dobrzynski [67]. In their works, simplified scaled model nose gears of Douglas DC-10 and of a main gear of Boeing 747 are studied. They concluded that the noise generated by the main component of the landing gear is low frequency nature. The experimental work of Dobrzynski and Buchholz [43] in anechoic wind tunnel with two realistic landing gears emphasizes the contribution on higher frequency noise between 1 and 2 kHz. These higher frequency noise components are induced by the small components of the landing gear. Ravetta et al. [124] confirmed this conclusion experimentally.

Dobrzynski et al. [44] performed tests with different full-scale landing gears in the Reduc-tion of Airframe and InstallaReduc-tion Noise (RAIN) project. They proved that the noise level increases with the number of axles and the leg length. They showed that the noise was produced by the impact of wakes of the upstream turbulent flow onto the downstream elements.

Following Dobrzynski [42], the main characteristics of landing gear noise can be summarized as:

– The noise is essentially broadband in nature. The broadband noise is generated by: the turbulent flow separation from blunt structures; the interaction between turbulent wake flows and downstream components, which causes a small fraction of the turbulence energy to be transferred into propagating acoustic waves.

– Occasionally narrow band noise appears, which is generated by coherent periodic vor-tex shedding from small struts and hydraulic lines with laminar flow separation since the flow is in the sub-critical regime at approach and landing conditions. However, there is very little experimental evidence that these vortex shedding tones are a major noise problem for current landing gear architectures.

– Large structures contribute preferentially to low frequencies; middle size elements contribute to the medium frequencies while small items will contribute to the high frequencies.

– The directivity of almost all the tested landing gears is almost omnidirectional in the flight axis, maximum in the front and at the rear and minimum at 90◦_{. For clean}

configuration without any small-scale element, a significant reduction of the high-frequency noise component is observed and the directivity becomes omnidirectional. Interaction noise is primarily governed by the velocity of the impinging flow and its tur-bulence characteristics, such as intensity and mean eddy length. Sound intensity increases with flow velocity as compact dipole source. Therefore, a reduction of local velocity has a beneficial effect. For example, a reduced inflow velocity is observed for the main landing gears due to the wing lift induced circulation which leads to a decrease of noise level. The increase of turbulence intensity has adverse effect. In a realistic landing gear configuration, both the inflow velocity and turbulence characteristics are different at each component, resulting in different source intensities.

1.3

Methodology and the LAGOON project

Although the rich experimental data have shed light on the mechanisms and characteris-tics of the landing gear noise, they are not able to provide sufficiently detailed information on the flow field. Theoretically, accurate numerical simulation can provide data that had never been measured previously: unobtrusive measurements of velocity, velocity gradients, pressure, passive scalars, etc. Databases obtained from the simulations can provide com-plete flow field views and new insights on the physics of the flow. Moreover, since noise reduction is to be considered as a design criterion for the overall aircraft configuration as well as for those noise generating components, comprehensive and predictive landing gear noise simulation methods are required to evaluate and optimize new designs to meet the noise reduction criterion. However, all the numerical methods (Computational Fluid

Dy-namics (CFD) or Computational Aero-Acoustic (CAA)) must be first of all validated for this complex unsteady flow with massive separation and interaction problem. The lack of extensive experimental data prevents this validation.

1.3.1

Numerical Method for Landing gear noise prediction

Reynolds Averaged Navier-Stokes simulation (RANS)

RANS equations are solved to obtain a statistically averaged flow field, while all the tur-bulent scales and kinetic energy are modeled. Hence the number of degrees of freedom is largely reduced and the number of grid points depends only weakly on ReL. This method

is widely used in practical engineering such as flow around landing gears.

The mean solution from RANS can help to select the designs with less interactions between the flow and the structures. For instance, Dobrzynski et al. [45] used RANS solutions in a qualitative way to optimize the main landing gear of the A340 by reducing the strong flow-surface interactions illustrated by the mean solution. However, these simulations are not able to provide quantitative information of the unsteady properties of flow such as transition and separation on the blunt surfaces, which directly affects the acoustic sources. Complementary investigations with wind tunnel experiments or more accurate simulations are needed.

Large Eddy Simulation (LES)

An alternative method called Large Eddy Simulation has been investigated intensively dur-ing the last decades [131]. In LES, only the energy-carrydur-ing large structures are computed exactly, while the more universal small scales of turbulence are modelled by a subgrid-scale (SGS) model. The LES method is detailed in Chapter 2. The grid point requirement of LES is thus significantly reduced for solid-wall free flows compared with the Direct Numer-ical Simulation (DNS) of the Navier-Stokes equations in which all the turbulent scales are calculated. Nevertheless, moving towards the wall, turbulent structures become progres-sively smaller and more anisotropic in wall bounded flows and to resolve these near wall structures, the grid point requirement in LES shows a dependence of the Reynolds number asRe1.9

L , which is close to the cost of a DNS [27, 115]. The wheel diameter based Reynolds

number of the nose landing gear of A320 in approach condition is about ReD = 5× 106,

which is far beyond the current capacity for a Wall-Resolved LES.

Giret [59] was the first one to use LES approach to study the flow and noise of a generic 0.4 scaled and simplified mock-up of the nose landing gear of an A320, coupled with the Ffowcs-Williams and Hawkings analogy [168] for the propagation of the acoustic sources in

the far field. Although the best resolved mesh has a mean y+ _{≈ 30, this is the only LES}

simulation in the scope of landing gear noise prediction.

Significant efforts have been dedicated to the development of wall-models which allow the use of a coarse resolution in the near wall-region, resulting in a reduction in computational costs [115]. However, these approaches lack the maturity for turbulent boundary layers with adverse pressure gradients and eventually massive separation as in the landing gear configuration.

Another direction to overcome the cost obstacle in wall-bounded LES is the Detached Eddy Simulation, which is discussed in the following section.

Detached Eddy Simulation (DES)

Spalart et al. [152] proposed the use of Detached Eddy Simulation (DES) in order to reduce the near wall cost of LES. DES is a hybrid method of RANS and LES, computing the near wall region with RANS equations and beyond the wall with LES on a single mesh. Recently, a lot of research work was based on this method, applied to simplified landing gear configurations, [89, 133, 134, 155]. Results vary as a function of mesh and turbulence model used in the RANS zone. Fröhlich and von Terzi [57] underlined also that DES is sensitive to the initial conditions. Besides, the wall-shear stress is consistently under-predicted by about 10− 15% which is caused by the mismatch around the interface between RANS and LES [109, 116]. Detailed discussion can be found in Section 4.2.

Lattice Boltzmann Method (LBM)

Instead of solving the Navier–Stokes equations, the discrete Boltzmann equation is solved to simulate the flow of a Newtonian fluid with collision models. Chen et al. [24] shows the possibility of recovering the Navier-Stokes equations from the Boltzmann’s equations. The function f (x, u, t) represents the probability of a Lattice gas particular with velocity u, at position x at time t in the space _{{(x, x + dx), (u, u + du), (t, t + dt)}. The velocity} space u is then projected into a base of elementary vectors representing all the possible motions of the particle. It is shown that the use of 19 elementary vectors (collision scheme D3Q19) is sufficient to obtain a resolution equivalent to the Navier-Stokes equations for an isentropic gas [35] with a Mach number less than 0.5. The macroscopic variables ρ and u can be found by calculating the expectation of the function f (x, u, t):

ρ(x, t) = X i fi(x, t), and ρu(x, t) = X i fi(x, t)ui (1.1)

The discrete Boltzmann equation is solved on cubic elements, called voxels. There is a factor 2 between the cubic size in two adjacent zones. The mesh generation for complex geometries is easier than the traditional unstructured meshes. For high Reynolds number turbulent flows, a turbulence model can be used to account for the unresolved subgrid-scale model.

This method has been widely applied for the simulation of flow around landing gears [8, 19, 21, 77, 88, 104, 143, 144]. Two advantages have been reported by these studies compared with classical Navier-Stokes solvers: reduced computational cost and easier to generate meshes for complex geometries.

Nevertheless, the resolution of the inner layers of the turbulent boundary layers by LBM for a high Reynolds number proves to be significantly more expensive than for a Navier-Stokes solvers for high Reynolds numbers. Finally, the use of the D3Q19 collision model restricts the use of the method to low Mach number flows (M <0.5), which nevertheless is not restrictive for simulations of landing gear at approach condition.

1.3.2

The LAGOON project

The LAGOON (LAnding Gear nOise datebase for computational aeroacoustics validatiON) project [95], funded by Airbus, aims at providing an accurate and extensive experimental database for the validation of computational aeroacoustics methods for the landing gear applications. Three different mock-up configurations are tested, incorporating different level of complexities. They are representative of a 0.4 scaled Airbus A320 nose landing gears, see Fig 1.4.

The wheel diameter is D = 0.3 m with realistic cross section: two cavities are present on each wheel. The diameter of the main leg is Dl = 55 mm in the lower part (close to the

axle) and Du = 71 mm in the upper part. The spanwise separation distance S between

the median plane of two wheels is S = 198 mm. The Reynolds number based on the wheel diameter at M = 0.23 is ReD = 1.5× 106. They are simplified but yet remain generic:

the typical bluff-body and massive flow-surface interaction features are retained. In the present study, the configuration in Fig. 1.4a is considered. The mock-up is equipped with on-board pressure measurement devices, microphones, on wheels, the main leg and the axle. Fig 1.5 shows the locations of these measurement devices on the mock-up. Planar Particular Image Velocimetry (PIV) and Laser Doppler Velocimetry (LDV) are used for the mean flow measurement and LDV/hotwire-X-probes are used for unsteady flow measurement. Far-field and near-field microphone measurements are available for the acoustic pressure comparison. More details about the LAGOON project can be found in [95, 96].

Figure 1.5 Pressure taps, kulites and microphones position in the measurement, from Manoha et al. [95]

Several numerical studies have been performed on the LAGOON#1 configuration. The challenges on this configuration are to correctly predict the laminar to turbulent transition and the cavity modes. Most of them used numerical methods based on the DES (Detached Eddy Simulation) [37, 89, 133, 134] and LBM (Lattice-Boltzman Method) method [21, 143]. Only one simulation using LES [59] is found. Restricted by cost, the LES case by Giret [59] reached a mean y+_{≈ 30, which is not fully wall-resolved. Giret [59] attempted to improve}

the wall boundary condition by using a logarithmic-law and concluded that the standard equilibrium log-law model is not suitable for the flow around bluff-body geometry. The solution is either to increase the mesh refinement at the wall which is not practical with current computational capacity or to use a more advanced wall-model including adverse pressure gradient effects. The latter is denoted as wall-modeled LES (WMLES) which is absent in the prediction of landing-gear noise within the LAGOON project. Following this conclusion, the objectives of the present thesis are defined in the following section.

1.4

Objectives, simulation cases and organization of the

thesis

1.4.1

Objectives and computational cases

The objectives of this thesis are to improve the classical wall-model based on the logarithmic-law in LES for the application towards the prediction of both aerodynamic and acoustic prediction of bluff-body flows, such as landing-gears. Towards this goal, a wall-model which accounts for the adverse pressure gradient effects is considered. An a priori validation of this model is performed using flow data extracted from a separate simulation of NACA12 flow at zero angle of attack. The model is then implemented in the LES solver AVBP [23, 140], developed by Cerfacs and IFP-Energies-Nouvelles. It has been shown in previous section that the two key influential features of the landing gear noise is the separation around the bluff-body elements and the wake-surface interaction. Thus, two benchmark configurations are considered for the validation of the implementation of the new wall-model before the application to the LAGOON#1 configuration:

– Cylinder: it is considered as a simplification of the landing gear and the benefits of using this novel wall modeling approach for bluff bodies can be better estimated. De-spite the simplification, the challenging features such as: transition, strong pressure gradient, separation, shear layer and large structures in the oscillating wake are re-tained. Both wall resolved and wall modeled Large Eddy Simulation (WR-LES and WM-LES) have then been carried out for the flow prediction at critical regime. The former is validated against experimental and numerical results from literature and then served as a reference for the WM-LES.

– Rod-Airfoil: this experimental benchmark configuration [73] has been widely used for both CFD and Computational AeroAcoustic (CAA) validation for the flow-surface interaction. A wall-resolved simulation of this set-up from Giret et al. [61] is also available and used as an extensive database to validate the implementation of the model for both aerodynamic and aeroacoustic predictions.

Another objective is to assess the use of hybrid mesh. In the wall bounded flows with complex realistic geometries, the generation of structured grid are often prohibitively ex-pensive even impossible and unstructured grid can be the only solution. Hybrid grids with several layers of prismatic or hexahedral elements above the wall for body fitted grids and tetrahedral elements elsewhere are often preferred for the good resolution at wall. However, the use of this kind of mesh topology might introduce numerical instabilities and special care must be taken. An investigation of the performance of different numerical schemes on hybrid grids in AVBP is performed by convecting a theoretically known vortex on hybrid grid topology on a regular domain. In the present work, all the simulations are performed using the AVBP v7.0.1.

Table 1.1 Summary computational cases in this thesis

Case Objective Method

NACA12 a priori validation RANS/Panel method

Cylinder reference case WR-LES

validation WM-LES

Rod-Airfoil validation WM-LES

LAGOON#1 application WM-LES

1.4.2

Organization of the thesis

The work is decomposed into the following parts: Chapter 2

This chapter is attached to the description of Large Eddy Simulation. The main concepts of the LES approach and the governing equations behind are described. Common subgrid-scale models for the turbulence closure and the numerics in the AVBP solver are presented. The assessment of the hybrid mesh approach is discussed in this chapter as well.

Chapter 3

The main properties of turbulent boundary layer (TBL) are summarized in this chapter. The main properties including the mean flow properties and the important structures of the turbulent wall layer with zero-pressure gradient and subjected to the adverse pressure gradient.

Chapter 4

In this chapter, common wall modeling strategies are described: different wall-stress models (based on how the wall-stress is estimated); LES/RANS hybrid methods resolving the near

wall region using RANS solver to reduce the near wall computation cost. From another point of view, these models can be divided into equilibrium and non-equilibrium models which include non-equilibrium terms such as pressure gradient in the boundary layer momentum equation. The wall function proposed by Afzal [6] which accounts for the adverse pressure gradient effects is presented in the last part of the session in a generalized form.

Chapter 5:

A priori validation and implementation of Afzal’s law and the implementation of Afzal’s law are detailed. A model to extend Afzal’s law to laminar regions is proposed by the combined use of an approximation for the shear stress in the laminar region and a transition sensor to trigger the turbulent wall model based on Afzal’s law.

Chapter 6: Benchmark cases

In this chapter, two benchmark cases are considered for the validation of Afzal’s law and the extended Afzal’s law. They are the cylinder at the critical regime and the rod-airfoil flow from Jacob et al. [73]. The flow around an circular cylinder is considered to validate the wall-model LES approach using Afzal’s law and its extended version for the prediction of both aerodynamic and aeroacoustic results for the bluff bodies which are the fundamental component of a landing gear. Besides, in realistic landing gear systems, there are actually interaction between various cylindrical components such as the tow bar with the main strut. Therefore, the canonical experiment for such an interaction, the rod-airfoil interaction is considered here.

Chapter 7: LAGOON simulation

In this chapter, the wall-modeled LES approach is applied to the LAGOON benchmark case of Airbus. The complex flow features are shown by both the iso-surface of Q criterion and the streamlines from the mean flow. Detailed flow fields at each component are shown by different flow parameters. The mean wall pressure, wall-pressure fluctuations on the surfaces of different components are compared with the experimental results. The effects of different wall-models on the flow transition are discussed as well. The far-field acoustic results are also compared with the microphone measurements. The contribution of each component to the far-field acoustic pressure are also investigated.

Large Eddy Simulation

2.1

Introduction

This chapter presents the equations that are implemented in the Large Eddy Simulation (LES) solver AVBP. AVBP handles multi-phase, reacting and non reacting compressible flows using unstructured grids. Complete equations can be found in Poinsot and Veynante [118] and the AVBP handbook [23]. In the scope of the present study, only equations of mono-species and non reacting flows are described.

2.2

Governing equations

2.2.1

Navier-Stokes equations

The standard Navier-Stokes equations consist of mass and momentum conservation equa-tions. An equation of conservation of total energy is added to account for heat transfer. The index notation (Einstein’s rule of summation) is adopted for the description of the governing equations. ∂ρ ∂t + ∂(ρui) ∂xi = 0 (2.1) ∂(ρuj) ∂t + ∂(ρuiuj) ∂xi + ∂P ∂xj = ∂τij ∂xi (2.2) ∂(ρE) ∂t + ∂(ρEui) ∂xi +∂(P ui) ∂xi = ∂(τijui) ∂xj − ∂qi ∂xi (2.3)

In Eqs. (2.1, 2.2, 2.3), t, ρ, P , ui, xi denote time, density, pressure, velocity and position

(i = 1, 2, 3) respectively; E, τij, qi represent total energy, viscous stress tensor and heat

flux (i = 1, 2, 3).

It is possible to write the above equations in matrix form: ∂w

∂t +∇.F(w) = s (2.4)

where w= (ρ, ρui, ρE)T is the vector of conservative variables and F denotes the flux tensor

and s is the source term. It is common to decompose the flux tensor F into an inviscid and a viscous component.

F(w) = F(w)I _{+ F(w)}V _(2.5)

They are respectively noted as: Inviscid terms: F(w)I ₌    ρui ρuiuj + P δij (ρE + P δij)ui    (2.6)

where the static pressure P is defined by an equation of state for a perfect gas. Viscous terms: F(w)V ₌    0 −τij −(uijτij) + qi    (2.7)

The viscous stress tensor τij is defined by:

τij = 2µ(Sij − 1 3δijSkk) (2.8) and Sij = 1 2( ∂ui ∂xj + ∂uj ∂xi ) (2.9)

where µ is dynamic viscosity of the fluid and the heat flux qij as modeled by the Fourier

conduction law:

qi =−λ

∂T ∂xi

(2.10) with λ the thermal conductivity of the fluid.

2.2.2

Equation of state and fluid thermodynamics

The equation of state for an ideal gas writes:

P = ρrT (2.11)

where r = R/Mair. R = 8.3143 J/(mol.K) is the universal gas constant and Mair =

28.9 g/mol is the molecular weight. The adiabatic exponent is given by γ = Cp/Cv.

computation of the Courant number (CFL) is given by:

c2 = γrT (2.12)

Sutherland’s law [160] is used to determine the dynamic viscosity which is a function of temperature: µ(T ) = µref  T Tref 3/2 Tref + S T + S (2.13)

where µref = 1.71× 10−5 kg/(m.s), Tref = 273 K and S = 110.4 K. The heat conduction

coefficient is computed by introducing the molecular Prandtl number which is considered to be constant in time and space in the present study as:

λ = µCp

P r (2.14)

2.2.3

The LES approach

The Direct Numercial Simulation (DNS) approach directly solves the Navier-Stokes equa-tions without any modeling and therefore is the most accurate method. In a DNS, the discretization should capture the largest (L) and smallest eddies (η). The ratio L/η in a homogeneous isotropic turbulent flow is directly related to the Reynolds number:

L

η ∝ Re

3/4 _(2.15)

Therefore, the grid requirement of a three-dimensional problem can be estimated by

N _≈ L

η 3

∝ Re9/4 (2.16)

For a turbulent boundary layer flow, this number is N ∝ Re11/4δ [27], where Reδ is the

Reynolds number based on the boundary layer thickness δ. Besides, the time resolution depends also on the size of the smallest element which limits the DNS to relatively low Reynolds number flows. In the scope of this project, landing gears are considered as a collection of bluff bodies with various sizes. Circular cylinder represents the most common components such as the leg, axle, hydraulic wires and other cylindrical coverings. The Reynolds number ReD which is based on the diameter of these cylindrical components at

the approch condition can vary from 104 _to106_{. At the time of writing this work, no DNS}

has been performed on a circular cylinder for Reynolds number larger than ReD = 104.

Alternative turbulence modeling approaches exist and the choice depends on the compro-mise between the accuracy and the computational cost. The most common approaches are Large Eddy Simulation (LES) and Reynolds Averaged Navier-Stokes (RANS). The gov-erning equations of LES are obtained by applying the filter operators to the compressible Navier-Stokes equations. Generally the filtering is linked to the mesh cell size. Large en-ergy containing structures which correspond to low frequencies are resolved and only the homogeneous subgrid-scales need to be modeled to simulate the turbulent transport and dissipation effects. For a turbulent boundary layer flow, the grid number required for a wall-resolved LES is N ∝ Re7.3/4δ [27]. In RANS, unclosed terms are modeling the physics

taking place over the entire frequency range. Fig. 2.1 illustrates the different concepts be-tween the RANS, LES and DNS approaches. The large modeling simplification of RANS makes it much cheaper than the LES and DNS approaches therefore has been widely used for industrial problems for decades. However, the cylinder flow especially at high Reynolds numbers such as in the present work, remains challenging for RANS with its complex and unsteady features.

LES, recognized as an intermediate approach compared to RANS and to DNS, seems to be the appropriate approach to provide accurate physical insights for the present problem. Indeed, the largest Reynolds number of circular cylinder flow using LES to tackle the critical to super-critical flow regimes [25, 87]. Therefore, the LES approach and its implementation in the AVBP solver are briefly described in this section.

Figure 2.1 Plot of the conceptual representation of RANS, LES and DNS ap-plided to a homogeneous isotropic turbulent field: turbulent kinetic energy spec-trum as a function of the inverse length scale.

2.2.4

Equations of LES

A physical quantity f in LES is decomposed into the sum of the filtered and resolved component f and the unresolved subgrid-scale part f0 _{= f − f. For compressible flows, a}

mass-weighted Favre filtering is introduced as: ˜ f = ρf

ρ (2.17)

The conservative equations for LES by filtering the Navier-Stokes equations are: ∂ρ ∂t + ∂(ρ˜ui) ∂xi = 0 (2.18) ∂(ρ˜uj) ∂t + ∂(ρ˜uiu˜j) ∂xi + ∂P ∂xj = ∂ ∂xj [τij − τtij] (2.19) ∂(ρ ˜E) ∂t + ∂(ρ ˜E ˜ui) ∂xi +∂(P ˜ui) ∂xi = ∂ ∂xj [τijui− qj − qjt] (2.20)

Similarly to the previous section, the filtered NS equation can be rewritten in a matrix form:

∂ ˜w

∂t +∇.˜F(w) = ˜s (2.21)

where w˜ = (ρ, ρ˜ui, ρ ˜E)T is the vector of conservative variables and s is the filtered source˜

term. ˜F is the flux tensor that can be divided in three parts: the inviscid part ˜FI, the viscous part ˜FV and the subgrid scale turbulent part ˜Ft_:

˜

F= ˜FI + ˜FV + ˜Ft (2.22)

Inviscid flux terms have the similar components as in DNS but based on the filtered quantities: ˜ FI =    ρ˜uj ρ˜uiu˜j + P δij ρ ˜E ˜uj + P ujδij    (2.23)

Viscous flux terms take the form: ˜ FV =    0 τij −uiτij+ qj    (2.24)

Following Poinsot and Veynante [118], the viscous filtered stress tensor is given by:

τij = 2µ(Sij − 1 3§kkδij)≈ 2µ ˜Sij − 1 3S˜kkδij (2.25) with ˜ Sij = 1 2( ∂ ˜ui ∂xj + ∂ ˜uj ∂xi ) (2.26)

The filtered heat flux reads:

qi =−λ ∂T ∂xi ≈ −λ ∂ ˜T ∂xi (2.27)

The additional subgrid scale terms read:

˜ Ft=    0 τt ij qt j    (2.28)

with the subgrid scale stress tensor:

τtij =−ρ(ugiuj − ˜uiu˜j)≈ 2ρνtS˜ij + 1 3τkkδij (2.29) and ˜ Sij = 1 2( ∂ ˜ui ∂xj + ∂ ˜uj ∂xi )₋ 1 3 ∂ ˜uk ∂xk δij (2.30)

The subgrid scale heat flux reads:

qti = ρ(guiE− ˜uiE)˜ (2.31) which is modeled by qt i =− ρνtCp P rt ∂ ˜T ∂xi (2.32)

The closure of these terms involves the subgrid scale (SGS) turbulent viscosity νt and the

turbulent Prandtl number P rt _{which is assumed to be constant in this study.}

2.3

Turbulence closure

The influence of the SGS on the resolved motion is taken into account by a SGS model based on the introduction of a turbulent viscosity, νt. Such an approach assumes the effect

of the SGS field on the resolved field to be purely dissipative. In this section, two subgrid scale (SGS) models are described.

Smagorinsky model

The Smagorinsky model was developed by Smagorinsky [150] and is the first SGS model for LES. The turbulent viscosity is assumed to be proportional to the subgrid characteristic width scale ∆ and to a characteristic turbulent velocity taken as the local strain rate | ˜Sij|

through the following equation:

νt= (CS∆)2

q

2 ˜SijS˜ij (2.33)

where CS denotes the model constant. The filter width is usually set to be the cube-root

of the cell volume. CS takes a value of 0.18 for homogeneous isotropic turbulence (HIT)

and can vary from 0.1 to 0.18 depending on the type of flow.

This model is simple and shows the right dissipation of kinetic energy in a HIT flows. Nevertheless, it is shown that this model is too dissipative near the walls and its use for transition regimes to turbulence is not recommended [131].

Wall-Adapting Local-Eddy model

The Van Driest damping function [162] is often used to obtain the right scaling of νt in

the vicinity of a solid wall boundary. However, this correction is difficult in a complex geometry configuration. A more elegant way to handle the near wall behavior and scaling is the Wall-Adapting Local Eddy-viscosity model (WALE) as proposed by Nicoud and Ducros [108].

The characteristic filtered strain rate ˜Sij in the Smagorinsky model is replaced by terms

that detect both the local strain and rotation rates: νt= (Cw∆)2

(sd ijsdij)3/2

( ˜SijS˜ij)5/2+ (sdijsdij)5/4

with sd ij = 1 2 (˜g 2 ij + ˜gji2)− 1 3 g˜ 2 kkδij (2.35) ˜ gkk = ∂ ˜ui ∂xj (2.36) The characteristic width∆ is the same as in Eq. (2.33). Cw = 0.4929 is the model constant

and g˜ij denotes the resolved velocity gradient.

As both the local strain and rotation rates are considered by the model, all the turbulence structures relevant for the kinetic energy dissipation are detected by the model. In the vicinity of a wall the eddy-viscosity scales as y3 _{with no need of either (dynamic) constant}

adjustment or damping function in case of wall bounded flows. The model produces zero eddy viscosity in case of a pure shear as in a laminar boundary layer and has been reported to be able to reproduce the laminar to turbulent transition process through the growth of linear unstable modes by Nicoud and Ducros [108].

2.4

Numerics in AVBP

The numerical implementations of the LES equations in the unstructured compressible AVBP LES solver are briefly described in this section. The cell-vertex approach introduces several particularities compared to the standard FV/FE methods. The unstructured meshes with different types of elements (hexahedral, prismatic, pyramidal and tetrahedral) make the meshing of complex geometries easier than for structured or block-structured solvers, but make the development of high-order accurate schemes difficult. In AVBP, the convective schemes are3rd _{order accurate based on a Taylor expansion for such meshes. More detailed}

descriptions can be found in the AVBP handbook and the PhD thesis of Lamarque [81].

2.4.1

Cell-vertex approach

The cell-vertex method is a residual method that consists of solving the conservation equa-tions on the grid cells and storing solution at the grid nodes. Basic notaequa-tions and metrics are presented firstly using a 2-D unstructured mesh as shown in Fig. 2.2.

Metrics

– Ω is the computational domain.

– C represents a primary cell of the mesh which is defined by the edges that connect its nodes.

– Cj represents the dual cell which is defined by the centroids of the primary cells.

j

∂C_j

C

C_j

k

Figure 2.2 Primary cells: white and dual cells: shaded. Solid and empty dots are nodes of primary and dual cells respectively.

– k is the local numbering of the vertices of a cell C.

– n represents the normal vector, defined at each face and each vertex, as shown in Fig. 2.3

Residual computation

The matrix form of the NS equation is considered and the source term is neglected for simplicity.

∂w

∂t +∇.F(w) = 0 (2.37)

As shown above, the fluxes can be divided into an inviscid, convective part Fi and a viscous part Fv and one should note that the viscous fluxes not only depends onw, but also on its gradients _∇w.

The first step is to mesh the simulation domain Ω into small cells C:

Ω = [

C∈Th

C (2.38)

Integrating Eq. (2.37) on the cell C: ∂ ∂t Z C wdv + Z C∇.Fdv = 0 (2.39) The use of the Green-Gauss theorem converts the above volume integration into a surface flux: ∂ ∂t Z C wdv + I ∂C F.nds = 0 (2.40)

The cell residual rC is defined as: rC := 1 |Vc| Z C ∇.Fhdv = 1 |Vc| I ∂C Fh.nds (2.41)

with Vc the volume of cell C and Fh is the numerical approximation of the fluxes F.

Trapezoidal rule is used to integrate the fluxes along the cell edges. This means the fluxes is considered to vary linearly along the edge. In AVBP the nodal normal associated with vertex k is a linear combination of the face normal of the adjacent faces as in Eq. (2.42).

nk = X f3k −nd nfv nf (2.42)

where nd is the number of dimensions. In the example of a triangle cell, the nodal normal

equals to its opposite edge (face) normal as shown in Fig. 2.3.

C

n_f1

n_f2 n_f3

nk1 = nf1

Figure 2.3 Cell nodal normal.

Therefore, the cell residual in Eq. (2.41) can be numerically calculated by: rC =− 1 nd|Vc| X k∈C Fk.nk (2.43)

Once the cell residual is calculated, it has to be distributed to the mesh nodes to obtain as many equations as there are degrees of freedom and update the solution vector. The nodal residual rj is defined as a volume-weighted average of residuals of the adjacent cells of node

j: rj = 1 |VC,j| X C∈Dj Dj,CrC|VC| (2.44) with |VC,j| = X C∈Dj |VC| nC v (2.45)