calculation of aerodynamic noise of wing airfoils by

Calculation of AerodynamicNoise of Wing Airfoils by Hybrid

Methods

Rabea Matouk

Department of Aero-Thermo-MechanicsBrussels School of Engineering, Université Libre de Bruxelles

This dissertation is submitted for the degree ofDoctor of Engineering Sciences and Technology

Promotors:Prof. Gérard DegrezProf. Jean Louis Migeot Academic year: 2016-2017

Acknowledgements

First, I would like to thank a lot my supervisor Prof. Gérard Degrez (ULB) for the directionof this work, for his valuable advice and support during the realization of this thesis and forhis availability.

Next, I want to thank my co-supervisor Prof. Jean Louis Migeot (ULB,FFT) and theFFT Company for providing ACTRAN, for their permanent support and for giving theopportunity for an internship in the company.

I want to thank Dr. Christophe Julien (the von Karman Institute for Fluid Dynamics) forhis help and rich discussions about my work, for sending me his results to compare with myresults and for the help to realize my first paper.

I also want to thank a lot Dr. Yves Detandt (FFT) for his help and support to realize thesecond paper and for his advice during the thesis committee meetings.

I would also like to acknowledge my committee members and the PhD thesis commissionmembers:Prof. Gérard DegrezProf. Jean Louis MigeotProf. Herman DeconinckDr. Yves DetandtProf. Christophe SchramProf. Ghader Ghorbaniasl

My family and best friends have been encouraging, supporting and showing belief in meand my work. So thanks a lot to all you.

I acknowledge the Université Libre de Bruxelles (ULB) and in particular the School ofEngineering, Department of Aero-Thermo-Mechanics to welcome my research and all its

iv

members especially Dr. Xavier Deschamps (ULB) and Shirley Wayne for their support.

I gratefully acknowledge the University of Aleppo (Syria) for the financial support ofthis research, and in particular the mechanical engineering faculty and all its professorsparticularly Prof. Mustafa Taki, my supervisor in Syria.

Finally, I would like to thank as well the Consortium des Equipements de Calcul Intensifen Fédération Wallonie Bruxelles CECI for providing the supercomputing resources fundedby FRS-FNRS (Belgium).

Abstract

This research is situated in the field of Computational AeroAcoustics (CAA). The thesisfocuses on the computation of the aerodynamic noise generated by turbulent flows aroundwing, fan or propeller airfoils. The computation of the noise radiated from a device is thefirst step for designers to understand the acoustical characteristics and to determine the noisesources in order to modify the design toward having acoustically efficient products. As acase study, the broadband or trailing-edge noise emanating from a CD (Controlled-Diffusion)airfoil, belonging to a fan is studied. The hybrid methods of aeroacoustic are applied tosimulate and predict the radiated noise. The necessary tools were researched and developed.The hybrid methods consist in two steps simulations, where the determination of the aerody-namic field is decoupled from the computation of the acoustic waves propagation to the farfield, so the first part of this thesis is devoted to an aerodynamic study of the considered airfoil.In this part of the thesis, a complete aerodynamic study has been performed. Some aspectshave been developed in the used in-house solver SFELES, including the implementation ofa new SGS model, a new outlet boundary condition and a new transient format which isused to extract the noise sources to be exported to the acoustic solver, ACTRAN. The secondpart of this thesis is concerned with the aeroacoustic study where four methods have beenapplied, among them two are integral formulations and the two others are partial-differentialequations. The first method applied is Amiet’s theory, implemented in Matlab, based on thewall-pressure spectrum extracted in a point near the trailing edge.The second method is Curle’s formulation. It is applied proposing two approaches; thefirst approach is the implementation of the volume and surface integrals in SFELES to becalculated simultaneously with the flow in order to avoid the storage of noise sources whichrequires a huge space. In the second approach, the fluctuating aerodynamic forces, alreadyobtained during the aerodynamics simulation, are used to compute the noise consideringjust the surface sources. Finally, Lighthil and Möhring analogies have been applied via theacoustic solver ACTRAN using sources extracted via SFELES. Maps of the radiated noiseare demonstrated for several frequencies. The refraction effects of the mean flow have beenstudied.

Table of contents

List of figures xi

List of tables xix

Nomenclature xxi

1 Introduction 11.1 Fan noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Airfoil trailing-edge noise mechanisms . . . . . . . . . . . . . . . 3

1.2 Computational methods of Aeroacoustics . . . . . . . . . . . . . . . . . . 5

1.2.1 The direct numerical acoustics method: . . . . . . . . . . . . . . . 5

1.2.2 The hybrid methods of aeroacoustics: . . . . . . . . . . . . . . . . 6

1.3 Objectives of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Main contributions and original work . . . . . . . . . . . . . . . . . . . . 8

2 Review of aeroacoustics theories, sound sources definition 112.1 Lighthill’s analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Approximation of Lighthill’s stress tensor . . . . . . . . . . . . . . 13

2.1.2 Integral solution of Lighthill’s analogy . . . . . . . . . . . . . . . . 13

2.2 Curle’s formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Ffowcs Williams and Hall’s theory . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Möhring’s analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Amiet’s aeroacoustic theory . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5.1 Derivation of the generalized trailing-edge noise formulation . . . . 20

2.6 Sound sources definition: Monopole, dipole & quadrupole . . . . . . . . . 27

viii Table of contents

3 Solvers and numerical methods 313.1 The CFD solver, SFELES . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 Turbulence modeling . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 The acoustic solver, ACTRAN . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 The variational FE formulation of the acoustic analogies as imple-mented in ACTRAN . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.2 The infinite elements . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.3 Mapping methods . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Flow and acoustic characteristic scales, meshes building criteria . . . . . . 46

4 Flow regimes of Controlled-Diffusion Airfoils 494.1 Description of the configuration . . . . . . . . . . . . . . . . . . . . . . . 49

4.1.1 The computational domain and CFD meshes . . . . . . . . . . . . 504.1.2 The Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 524.1.3 Previous experimental and numerical studies of the CD airfoil . . . 53

4.2 Flow patterns around the CD airfoil according to flow Reynolds number . . 554.2.1 Attached flow (creeping) 0<Re<270 . . . . . . . . . . . . . . . . . 564.2.2 Steady, separated flow 270<Re<1300 . . . . . . . . . . . . . . . . 564.2.3 2-d unsteady laminar oscillating flow (vortex street) 1300<Re<6450 574.2.4 3-d unsteady laminar oscillating flow 6450<Re<14000 . . . . . . . 574.2.5 3-d turbulent wake, 2-d laminar boundary layer regime

14000<Re<47500 . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.6 Recirculation bubble appearance near the leading edge, laminar

boundary layer Re=47500 . . . . . . . . . . . . . . . . . . . . . . 594.2.7 Recirculation bubble explosion, 3-d laminar periodic boundary layer

and turbulent wake 50000=<Re<52000 . . . . . . . . . . . . . . . 604.2.8 Fully turbulent regime Re>=52000 . . . . . . . . . . . . . . . . . . 604.2.9 Pressure and friction coefficients distribution . . . . . . . . . . . . 614.2.10 Evolution of the lift and drag coefficients with Reynolds number and

the flow regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2.11 Evolution of the Strouhal number with Reynolds number . . . . . . 65

5 Turbulent flow over CD airfoil (Re=160 000) 695.1 Evolution of Ghorbaniasl’s model constant Cs . . . . . . . . . . . . . . . . 705.2 Flow topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3 Pressure and friction coefficients distribution on the airfoil surface . . . . . 735.4 Boundary layer velocity profiles . . . . . . . . . . . . . . . . . . . . . . . 75

Table of contents ix

5.5 Wall pressure spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.6 The average velocity profiles in the wake . . . . . . . . . . . . . . . . . . . 795.7 Stresses in the turbulent boundary layer and the law of the wall . . . . . . . 845.8 Spanwise pressure coherence function and length . . . . . . . . . . . . . . 875.9 Spatial convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.9.1 Flow topology, pressure and friction coefficients distribution . . . . 905.9.2 Boundary layer velocity profiles . . . . . . . . . . . . . . . . . . . 935.9.3 Stresses in the turbulent boundary layer . . . . . . . . . . . . . . . 935.9.4 Wall pressure spectra . . . . . . . . . . . . . . . . . . . . . . . . . 945.9.5 Spanwise pressure coherence function and length . . . . . . . . . . 94

5.10 Spanwise extension effects, z/C = 0.2 . . . . . . . . . . . . . . . . . . . . . 96

6 Aerocoustics results 1016.1 Broadband noise at Reynolds number of 160000 . . . . . . . . . . . . . . . 101

6.1.1 Amiet’s aeroacoustics theory . . . . . . . . . . . . . . . . . . . . . 1016.1.2 Curle’s integral formulation . . . . . . . . . . . . . . . . . . . . . 1066.1.3 Lighthill’s analogy . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.2 Comparison of the applied hybrid methods, conclusions . . . . . . . . . . . 131

7 Conclusions and perspectives 139

References 145

Appendix A The leading-edge noise or turbulence impact noise formulation [29] 155

Appendix B Amiet’s theory: transfer functions derivation [29, 31–33, 99] 163B.1 Leading edge case (turbulence-interaction noise): . . . . . . . . . . . . . . 163

B.1.1 The analytic solution using Schwarzschild’s technique . . . . . . . 165B.2 Trailing edge case: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

B.2.1 Supercritical gust: . . . . . . . . . . . . . . . . . . . . . . . . . . 170B.2.2 Subcritical gust: . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Appendix C Ghorbaniasl’s SGS model derivation [50] 173

Appendix D The spatial discretization of the 2D NS equations by Galerkin finiteelements methods [43] 177

Appendix E Ensight Gold format as implemented in SFELES 181

x Table of contents

Appendix F Hydrodynamics reflections at the mesh outlet, the physical boundarycondition 183F.1 The physical boundary condition . . . . . . . . . . . . . . . . . . . . . . . 186

F.1.1 The governing equations . . . . . . . . . . . . . . . . . . . . . . . 186F.1.2 Adaptation for a structured polar grid and a Cartesian grid . . . . . 189F.1.3 Implementation in SFELES for unstructured grids, generalization

for 3D flows, proposing a pressure equation for the outlet BC . . . 190F.1.4 Validation of the physical boundary condition: Velocities approach . 191F.1.5 Validation of the physical boundary condition: Pressure approach . 196

Appendix G Tonal noise corresponding to the vortex shedding at Reynolds num-ber of 12000 203G.1 The sound radiated of the CD airfoil in a 2d laminar unsteady regime, Re=12000204

G.1.1 Results for Lighthill’s analogy . . . . . . . . . . . . . . . . . . . . 204G.1.2 Results for Möhring’s analogy . . . . . . . . . . . . . . . . . . . . 208G.1.3 Results for Curle’s formulation . . . . . . . . . . . . . . . . . . . . 212

G.2 The sound radiated of the CD airfoil in a 3d laminar unsteady regime, Re=12000215

List of figures

1.1 The aircraft noise contributors. Source: Airbus . . . . . . . . . . . . . . . 2

1.2 The turbofan noise contributors. Source: Rolls-Royce plc [2] . . . . . . . . 2

1.3 Diagram illustrating the creation mechanism of the trailing-edge noise . . . 3

1.4 Production mechanisms of the trailing-edge noise, Figures reproduced fromBrooks et al. [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Regions in a hybrid aeroacoustic problem . . . . . . . . . . . . . . . . . . 12

2.2 Coordinate system for the finite-chord thin plate used in the application ofFfowcs Williams and Hall’s theory. Figure is reproduced from Wang et al [58] 17

2.3 2D problem with trailing-edge coordinates. Figure reproduced from [33] . . 20

2.4 3D problem for the trailing-edge model. Figure reproduced from [33] . . . 23

2.5 Representation of directivity pattern of sound sources: a)-Monopole, b)-Dipole, c)-Quadrupole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6 Diagram of a dipole [42] . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Diagram of a quadrupole [42] . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1 Simulation of turbulent combustion a)-DNS, b)-LES, c)-RANS [77] . . . . 36

3.2 Representation of the turbulent kinetic energy spectrum . . . . . . . . . . . 37

3.3 Helmholtz problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Mapping methods a)-the sampling method, b)-the integration method [95] . 46

4.1 The automotive cooling package, its 9-blades fan and the airfoil of the blade.Figure reproduced from [57] . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 The mesh refinement at the trailing edges and the boundary layer mesh. A)-the first (structured) mesh M1 , B)- the second mesh (unstructured) M2 . . . 51

4.3 Representation of the computational domain with the Boundary conditions . 52

xii List of figures

4.4 Inlet velocity profiles on the restricted domain, extracted from RANS com-putations: (left) longitudinal velocity and (right) transverse velocity. Figurereproduced from [69] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.5 (Left) Experimental Set-up of ECL Large Wind Tunnel (right) Representativefigure of the RANS simulation of the Test Configuration . . . . . . . . . . 55

4.6 Instantaneous contours of the longitudinal velocity U(1 : 3.25 : 14) at Re=50 564.7 Instantaneous contours of the longitudinal velocity, red: U(1 : 3.25 : 14),

blue: U(–0.7 : 0.1 : –0.1) at Re=1250 . . . . . . . . . . . . . . . . . . . . . 574.8 Contours of the spanwise vorticity at Re=2000, red (15:7:50), blue (-50:7:-15) 574.9 Contours of spanwise velocity at Re=6450. Red surfaces mark positive

values whereas the blue surfaces mark the negative values . . . . . . . . . . 584.10 Contours of vorticity at Re=7000. Red and blue surfaces mark positive

(10:1:20) and negative (-20:1:-10) values of the transverse vorticity whereasthe green and yellow mark positive and negative surfaces of the streamwisevorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.11 Contours of the vorticity magnitude (200) at Re=15000 colored by thelongitudinal velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.12 Contours of the vorticity magnitude (100:400) at Re=47500 colored by thelongitudinal velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.13 (Top): Contours of the Q criterion (100) colored by the longitudinal velocity,(bottom): Contours of vorticity at Re=50000. Red and blue surfaces markpositive and negative values of the streamwise vorticity whereas the greenand yellow surfaces mark the transverse vorticity . . . . . . . . . . . . . . 60



4.16 All flow patterns around the CD airfoil according to flow Reynolds number 624.17 Average pressure and friction coefficients distribution on the airfoil surface 634.18 Evolution of the lift and drag coefficients with Reynolds number . . . . . . 644.19 Evolution of the Strouhal number with Reynolds number (Rec) . . . . . . . 654.20 Evolution of the Strouhal number with Reynolds number Red . . . . . . . . 664.21 The recirculation region around the trailing edge with the supposed diameter D 67

5.1 Evolution of Ghorbaniasl’s model constant CS: a) x/C=-0.6 for 64 and 32Mat 9t*, b) x/C=-0.02 for 64 and 32M at 9t* , c) x/C=-0.6 for 9t* and 16t* for64M, d) x/C=-0.02 for 9t* and 16t* for 64M . . . . . . . . . . . . . . . . . 71

List of figures xiii

5.2 Flow topology of simulations performed on M1 described by the criterionQ(Q. C2

U20

= 1000) and colored by the longitudinal instantaneous velocity with

the model: a) Smagorinsky 32M, b) Smagorinsky 64M, c) Ghorbaniasl 32M,d) Ghorbaniasl 64M, e) WALE 64M . . . . . . . . . . . . . . . . . . . . . 72

5.3 (Above): Average pressure coefficient distribution (Cp) on the airfoil surface,(bottom): Average friction coefficient (Cf) distribution . . . . . . . . . . . 74

5.4 Zoom of the leading edge region on (Cf) curves characterizing the recircula-tion bubble size for all simulations . . . . . . . . . . . . . . . . . . . . . . 75

5.5 Average velocity profiles in the boundary layer on the upper surface atsections a) x/C = -0.6, b) x/C = -0.32, c) x/C = -0.14, d) x/C = -0.02 . . . . 76

5.6 Pressure fluctuations on the airfoil suction surface at the positions: x/C=-0.08,x/C=-0.02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.7 Power Spectral Density of pressure fluctuations at the positions: x/C=-0.08,x/C=-0.02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.8 The longitudinal (U) average velocity profiles in several positions in the wake 80

5.9 The normal (V) average velocity profiles in several positions in the wake . 81

5.10 The longitudinal (u′) velocity fluctuations RMS profiles in several positionsin the wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.11 The vertical (v′) velocity fluctuations RMS profiles in several positions inthe wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.12 The viscous, Reynolds and SGS stresses (τxy) in the section x/C=-0.7 . . . 85

5.13 The law of the wall in the section x/C = -0.7 . . . . . . . . . . . . . . . . . 86

5.14 Spanwise coherence function and length of the fluctuating pressure on thesuction side at x/C=-0.02 . . . . . . . . . . . . . . . . . . . . . . . . . . . 89


U20

= 1000) and colored by the longitudinal instantaneous velocity with

the model: a) Smagorinsky 32M, b) Smagorinsky 64M, c) Ghorbaniasl 64M,d) WALE 64M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.16 (Above): Average pressure coefficient distribution (Cp) on the airfoil surfaceusing mesh M2, (Bottom): Average friction coefficient (Cf) distributionusing mesh M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.17 Average velocity profiles in the boundary layer on the upper surface atsections x/C = -0.14 and x/C = -0.02 . . . . . . . . . . . . . . . . . . . . . 93

5.18 The viscous, Reynolds and SGS stresses in the section x/C=-0.7 . . . . . . 94

5.19 Power Spectral Density of pressure fluctuations at the position x/C=-0.02 . 95

xiv List of figures

5.20 Comparison of the spanwise coherence function and length between the twocomputational meshes M1 and M2 . . . . . . . . . . . . . . . . . . . . . . 95


U20

= 1000) and colored by the longitudinal instantaneous velocity for a

spanwise extension: a) Smagorinsky 128M, z/C = 0.2, b) Smagorinsky 64M,z/C = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.22 (Left): Average pressure coefficient distribution (Cp) on the airfoil surfaceusing mesh M2, (Right): Average friction coefficient (Cf) distribution usingmesh M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.23 Average velocity profiles in the boundary layer on the upper surface at sectionx/C = -0.02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.24 Power Spectral Density of pressure fluctuations at the position x/C=-0.02 . 985.25 The viscous, Reynolds and SGS stresses in the section x/C=-0.7 for the two

spanwise extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.26 Comparison of the spanwise coherence function and length between the two

spanwise extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.1 Transfer function directivity patterns for parallel and supercritical gusts forthe considered airfoil, trailing edge formulation, (left) main scattering termL1 (right) leading-edge back-scattering correction L2 . . . . . . . . . . . 102

6.2 Trailing edge sound using Amiet’s theory for the three SGS models (above)without the leading-edge correction (bottom) with the leading edge correction.The receiver is placed in the mid-span plane above the trailing edge (R=2 m,θ = 90o) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3 The noise directivity patterns [dB] for a): Ghorbaniasl’s model, b): WALEmodel, c): Smagorinsky model . . . . . . . . . . . . . . . . . . . . . . . . 105

6.4 Trailing edge sound using Amiet’s theory using coherence length extractedfrom LES simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.5 Curle: overall noise (implementation in SFELES), (CFD Mesh M2) . . . . 1096.6 Surface and volume sources contributions to the overall far-field acoustic

pressure for Smagorinsky’s model . . . . . . . . . . . . . . . . . . . . . . 1106.7 Surface and volume sources contributions to the overall far-field acoustic

pressure for WALE model . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.8 Surface and volume sources contributions to the overall far-field acoustic

pressure for Ghorbaniasl’s model . . . . . . . . . . . . . . . . . . . . . . . 1116.9 Retarded time effects on the far field SPL, for Smagorinsky’s model . . . . 1126.10 Curle: overall noise (implementation in SFELES), (CFD Mesh M1) . . . . 112

List of figures xv

6.11 Comparison of the SPL obtained using the two CFD meshes M1 and M2 forGhorbaniasl’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.12 Comparison of the SPL obtained with 0.1C and 0.2C spanwise extensionsfor Smagorinsky’s model (before scaling) . . . . . . . . . . . . . . . . . . 114

6.13 Comparison of the averaged SPL obtained for 0.1C and 0.2C spanwiseextensions for Smagorinsky’s model (before scaling) . . . . . . . . . . . . 114

6.14 Curle: dipole noise (aeroforces), (CFD Mesh M2) . . . . . . . . . . . . . . 116

6.15 Comparison of the surface contribution obtained using the two approachesfor Smagorinsky’s model, (CFD Mesh M2) . . . . . . . . . . . . . . . . . 116

6.16 Hybrid approach of aeroacoustics using SFELES CFD and ACTRAN acous-tics solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.17 The 3-D CFD geometry and mesh with a zoom on the corner, built by EnsightGold format, spanwise extension is 0.1C . . . . . . . . . . . . . . . . . . . 118

6.18 The 3-D acoustic mesh, the imposed spanwise extension is 0.3C . . . . . . 119

6.19 ACTRAN acoustic simulation setup, 1): The acoustic mesh, 2): Field mapsplane of a dimension 1.4*1.4 m, 3): A group of 25 receivers for the directivity,4): The considered receiver . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.20 (Above) Truncation phenomenon at the outlet for the frequency 300 Hz,(middle) the applied cosine filter, (bottom) the same acoustic field after theapplication of the filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.21 Lighthill sources on the acoustic mesh at frequencies, 400 Hz, 800 Hz, 1200Hz and 1600 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.22 Lighthill near field acoustic pressure at frequencies, 400 Hz, 800 Hz, 1200Hz and 1600 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.23 Lighthill, acoustic waves’ propagation to far field for the frequencies, 300Hz and 400 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.24 Lighthill, acoustic waves’ propagation to far field for the frequencies, 600Hz and 800 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.25 Comparison of the SPL obtained with ACTRAN/Lighthill’s analogy and thatmeasured from the experiments . . . . . . . . . . . . . . . . . . . . . . . . 128

6.26 Directivity patterns at frequencies, 250 Hz, 400 Hz, 800 Hz, 1200 Hz and1800 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.27 Sources truncation phenomenon effect, comparison of the SPL obtained withand without the application of the filter near the domain borders . . . . . . 129

6.28 Comparison of the SPL obtained with ACTRAN/Lighthill’s analogy, SFE-LES/Curle’s formulation and that measured from the experiments . . . . . 130

xvi List of figures

6.29 Comparison of the SPL obtained with ACTRAN/Lighthill’s analogy, SFE-LES/Curle’s formulation, Amiet’s theory and that measured from the experi-ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.30 Lighthill, acoustic waves’ propagation to far field for the frequencies, 300Hz, 400 Hz , 500 Hz and 600 Hz . . . . . . . . . . . . . . . . . . . . . . . 134

6.31 Lighthill, acoustic waves’ propagation to far field for the frequencies, 700Hz, 800 Hz , 900 Hz and 1000 Hz . . . . . . . . . . . . . . . . . . . . . . 135

6.32 Lighthill, acoustic waves’ propagation to far field for the frequencies, 1100Hz, 1200 Hz , 1300 Hz and 1400 Hz . . . . . . . . . . . . . . . . . . . . . 136

6.33 Lighthill, acoustic waves’ propagation to far field for the frequencies, 1500Hz, 1600 Hz , 1800 Hz and 2000 Hz . . . . . . . . . . . . . . . . . . . . . 137

A.1 Representation of the skewed gust impinging to the linearized airfoil . . . . 155

A.2 The airfoil with a dipole source located at X0 and a receiver located at X . . 156

B.1 Representation of the two steps procedure for Amiet’s leading edge: Inci-dent gust on a finite-chord airfoil (top), main scattering half-plane problem(bottom left) and trailing-edge correction (right) . . . . . . . . . . . . . . . 165

D.1 The global tent form basis function associated with a node j of a two-dimensional elements P1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

E.1 Building six node pentahedron elements in the geometry .geo created byEnsight gold format from the three node triangle elements used in SFELES 181

F.1 Lighthill sources: hydrodynamics reflections at the mesh M1 outlet . . . . . 184

F.2 Lighthill sources: hydrodynamics reflections are removed, the mesh M2 . . 184

F.3 Flow topology described by the longitudinal velocity field U at Re=12000:(Above) the mesh M1, (bottom) the mesh M2 . . . . . . . . . . . . . . . . 185

F.4 Contour of the vorticity at Re=12000: (Above) the mesh M1, (bottom) themesh M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

F.5 Representation of an arbitrary external surface S enclosing a rigid body Sband region Ω, illustration of the two irreducible circuits C1 and C2 enclosinga region D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

F.6 Illustrations for implementation of boundary conditions on (Left) radial grid,(right) Cartesian grid grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

F.7 Representation of the physical boundary condition implementation in SFELES191

F.8 Instantaneous vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

List of figures xvii

F.9 The average and instantaneous evolution of the pressure along the outlet . . 192F.10 The temporal evolution of the aerodynamics coefficients . . . . . . . . . . 193F.11 The mean pressure coefficient distribution Cp . . . . . . . . . . . . . . . . 193F.12 Longitudinal and normal average velocities, longitudinal and vertical velocity

fluctuations RMS in the wake at x/C=2 . . . . . . . . . . . . . . . . . . . . 194F.13 The average and instantaneous evolution of the pressure along the outlet . . 195F.14 Mean pressure and friction coefficients distribution, comparison between the

outlet phy. BC and zero-pressure BC . . . . . . . . . . . . . . . . . . . . . 196F.15 Longitudinal and normal average velocities, longitudinal and vertical velocity

fluctuations RMS in the wake at x/C=2.3 . . . . . . . . . . . . . . . . . . . 197F.16 The average and instantaneous evolution of the pressure along the outlet . . 197F.17 The mean pressure coefficient distribution Cp . . . . . . . . . . . . . . . . 198F.18 Longitudinal and normal average velocities, longitudinal and vertical velocity

fluctuations RMS in the wake at x/C=2 . . . . . . . . . . . . . . . . . . . . 199F.19 The average and instantaneous evolution of the pressure along the outlet . . 199F.20 Mean pressure distribution, comparison between the outlet phy. BC and

zero-pressure BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200F.21 Longitudinal and normal average velocities, longitudinal and vertical velocity

fluctuations RMS in the wake at x/C=2.3 . . . . . . . . . . . . . . . . . . . 201

G.1 The 3-D acoustic mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204G.2 (Above) Truncation phenomenon at the outlet for the frequency 369 Hz,

(middle) the applied Cosine Filter, (bottom) the same acoustic field after theapplication of the filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

G.3 Lighthill’s sources on the acoustic mesh for the frequencies 369 Hz, 628 Hzand 731 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

G.4 Lighthill’s acoustic pressure maps for the near field for frequencies 369 Hz,628 Hz and 731 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

G.5 Lighthill’s analogy, acoustic waves’ propagation to far field for the frequency369.656 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

G.6 Lighthill far field acoustic pressure spectra for the receiver located at mid-plane, 2 m above the trailing edge . . . . . . . . . . . . . . . . . . . . . . 209

G.7 Directivity patterns at frequencies, 369 Hz, 628 Hz, 731 Hz and 998 Hz . . 209G.8 Möhring’s sources on the acoustic mesh for the frequencies 369 Hz, 628 Hz

and 731 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210G.9 Möhring’s acoustic pressure maps for the near field for frequencies 369 Hz,

628 Hz and 731 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

xviii List of figures

G.10 Möhring’s analogy, acoustic waves’ propagation to far field for the frequency369.656 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

G.11 Comparison between Lighthill and Möhring far field acoustic pressure spectra213G.12 Far field acoustic pressure at the considered receiver via Curle’s formulation 213G.13 Curle’s formulation, surface and volume contributions to the far field acoustic

pressure compared to Lighthill’s result . . . . . . . . . . . . . . . . . . . . 214G.14 Comparison of the far field acoustic pressure at the considered receiver

obtained via the three methods: Lighthill, Möhring and Curle . . . . . . . . 214G.15 Comparison of the far field acoustic pressure obtained considering 2d un-

steady and 3d unsteady regimes corresponding to Re=12000 . . . . . . . . 215G.16 Lighthill’s acoustic pressure maps for the near field for frequency 167 Hz,

(above): 2d case, (bottom): 3d case . . . . . . . . . . . . . . . . . . . . . . 216G.17 Lighthill’s analogy, acoustic waves’ propagation to far field for the frequency

167 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217G.18 Lighthill’s sources on the acoustic mesh for the frequency 167 Hz . . . . . 217G.19 Comparison of Strouhal number for the 2d unsteady and 3d unsteady regimes

corresponding to Re=12000 . . . . . . . . . . . . . . . . . . . . . . . . . . 218G.20 Comparison of the far field acoustic pressure obtained via Lighthill and the

second approach of Curle using aerodynamics forces for the 3d unsteadyregimes corresponding to Re=12000 . . . . . . . . . . . . . . . . . . . . . 218

List of tables

4.1 Comparison of Strouhal number between the airfoil and a circular cylinder . 66

5.1 The simulations performed on the CD airfoil at Re=160000 . . . . . . . . . 705.2 Recirculation bubble size: comparsion with other simulations . . . . . . . . 735.3 Recirculation bubble size obtained on mesh M2: comparsion for the 3 SGS

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1 Comparison of the hybrid methods of aeroacoustics, Lighthill’s analogyCurle’s formulation and Amiet’s theory as applied in this research . . . . . 133

7.1 The evolution of the flow regime with Reynolds number . . . . . . . . . . 140

F.1 Comparison of the aerodynamics coefficients between the outlet phy. BCand zero-pressure BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

F.2 Comparison of the aerodynamics coefficients and Strouhal number betweenthe outlet phy. BC and zero-pressure BC . . . . . . . . . . . . . . . . . . . 198

F.3 Comparison of the aerodynamics coefficients between the outlet phy. BCand zero-pressure BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Nomenclature

Greek Symbols

δij Kronecker delta

λ Sound wave length

µ Dynamic viscosity

ν Kinematic viscosity

νt Eddy viscosity

ρ Flow Density

ρ0 Atmospheric Density

ρa Acoustic Density

τij Viscous stress tensor

τSGSij Subgrid stress tensor

t Time step (sec)

uτ Friction velocity

Other Symbols

C Airfoil chord length

c0 Sound speed

Cs Smagorinsky constant

p0 Atmospheric pressure

xxii Nomenclature

pa Acoustic pressure

Sij Strain rate tensor

t, t∗ Time, dimensionless time

Acronyms / Abbreviations

CAA Computational Aero-Acoustics

CD Controlled-Diffusion

CFD Computational Fluid Dynamics

DES Detached Eddy Simulation

DNA Direct Numerical Acoustic Simulation

DNS Direct Numerical Simulation

FEM Finite Element Method

FWH Ffowcs-Williams and Hawkings

LES Large Eddy Simulation

LHS Left Hand Side

NS Navier-Stokes Equations

RANS Reynolds-Averaged Navier-Stokes

Re Reynolds number

RHS Right Hand Side

RMS Root Mean Square

SGS Sub Grid Scale

URANS Unsteady Reynolds-Averaged Navier-Stokes

WALE Wall-Adapting Local Eddy-Viscosity

Chapter 1

Introduction

The technological progress made in the computer hardware industry with an increasedcomputational power, together with the development of the Computational Fluid Dynamics(CFD) lead new fields of research using the CFD techniques which extend to the aeroacousticsfield. Aeroacoustics is the discipline that studies the aerodynamic noise, generated by theturbulent or any flow unsteady fluctuations. Aeroacoustics also involves the analysis andmodeling of the sound propagation in complex turbulent flow as well. A large varietyof industrial problems are concerned with aeroacoustic phenomena: automotive industryfocusing on the external aerodynamic noise of electric or engine powered car; high speedtrains impacted by the turbulent noise whistling to passengers and aircraft industry concernedwith the noise certifications of new aircraft and improvement of passenger comfort. For theaircraft noise, the main contributors are engine noise and airframe noise. Engine noise comesfrom the fan/propeller, compressor, turbine, combustor and jet exhaust whereas airframenoise is produced by air flows around lifting and control surfaces, such as flaps and slats, andaround landing gears [1]. Figure 1.1 shows the contribution of these sources in the approachphase. Considering the turbofan engine noise sources, they are shown in Fig 1.2. This figureshows the difference in the main noise contributors between early and modern turbofans. Inthe early turbofan generations, the sound fields were dominated by the noise of the jet mixingwith the cold ambient air. The engine had a single flow stream with high-speed hot jets.Nowadays, turbofan designs have been developed to two-stream architectures that include ahot core flow and a cold bypass flow. The accompanying fan noise therefore becomes thedominant source [2]. Increasing the bypass ratio is always desired for commercial aircrafts todecrease fuel consumption. Consequently, the fan noise becomes more and more important.The understanding, the prediction and then the reduction of fan noise are therefore crucialchallenges. Due to the important contribution of the airframe noise, propeller noise, fan

2 Introduction

Fig. 1.1 The aircraft noise contributors. Source: Airbus

Fig. 1.2 The turbofan noise contributors. Source: Rolls-Royce plc [2]

noise and edge noise of high-lift devices, trailing-edge noise has received much attentionin recent years and becomes an intense area of research. In this thesis, the broadband noiseor trailing-edge noise of a fan airfoil is simulated and predicted using the hybrid methodsof aeroacoustics as presented in the following sections. The mechanisms of the airfoiltrailing-edge noise will be explained to better understand the physical phenomena involvedin our study.

1.1 Fan noise

Fan noise is a complex phenomenon since the aerodynamic noise is highly influenced by thepresence of other objects in the close vicinity of the rotating components, such as rotor-statorinteraction. The noise spectrum contains both tonal components (dipoles), at the bladepassing frequency and its harmonics, and broadband components (quadrupoles). Rotatingmachinery is found in confined flows (HVAC-systems, turbofan engines,. . . ), as well as in

1.1 Fan noise 3

free-field applications (wind turbines [3], helicopters [4],. . . ). The noise generated by thesedevices is referred to as fan noise [5]. The simulation of a complete fan is challenging and wefocus the analysis on some fan noise components. New quieter fan designs requires accurateand fast simulations means to assess the improvements up to the far field measurement points.

1.1.1 Airfoil trailing-edge noise mechanisms

The trailing edge noise is the major contribution of airfoil noise when the upstream turbulencelevel is small. It is the minimum noise that can be produced by rotating machines [9]. Theturbulent structures of the boundary layer created on the airfoil surface are highly modifiedas they pass to the trailing edge. The energy carried on by vortices is scattered by the trailingedge singularity in acoustics propagated to the far field. It is therefore the interaction betweenthe singularity at the trailing edge and the unsteady nature of the flow, which is usuallymanifested by turbulent structures presented in the near wall and into the near wake of thetrailing edge [6]. These mechanisms are represented in Fig. 1.3

Fig. 1.3 Diagram illustrating the creation mechanism of the trailing-edge noise

In fact, the understanding of trailing-edge noise mechanisms is largely due to the studyconducted by Brooks et al. [6, 7]. Experiments were conducted on a two-dimensionalacademic profile, the NACA 0012 at a Reynolds number of 3.106, so that the boundary layeris fully turbulent at the trailing edge. The mechanisms of trailing-edge noise production aredivided into several categories represented in Fig. 1.4. These mechanisms are:

• Diffraction of turbulent structures contained in the boundary layers by the trailing edge(especially at high Reynolds numbers).

• Detached boundary layer at the trailing edge, which is accompanied by a release ofturbulent structures.

4 Introduction

Fig. 1.4 Production mechanisms of the trailing-edge noise, Figures reproduced from Brookset al. [6]

• Generation of a vortex shedding street by a truncated trailing edge.

• The presence of a vortex shedding in the wake in case of a laminar flow.

• Deep Stall, when the incidence is very important a detachment of the boundary layerwith large vortices scales is produced.

• Tip Vortex produced by the interaction between the three-dimensional vortex forms atthe trailing edge, with the surface of the wing tip.

The singularity of the geometry of the profile at the trailing edge generates a discontinuity inthe acoustic wave created by the turbulence, and acting as quadrupoles [7]. This creates anintense acoustic field at the trailing edge. When the profile dimensions are small comparedto the wavelength of the radiated acoustic waves (C ≪ λ) where C the airfoil chord andλ the acoustic wavelength, the fluctuating flow generates pressure fluctuations transmittedinstantaneously to the near field. The radiated field follows a dipolar behavior with anintensity proportional to M6 [8]. If C ≫ λ, then the trailing edge diffracts the quadrupole

1.2 Computational methods of Aeroacoustics 5

due to turbulence; the noise has a multipolar nature and its intensity is proportional to M5.For the subsonic regime, the noise intensity is more important than the dipole in the previouscase [8]. These mechanisms present noise levels in different characteristics. For example, thediffraction of the turbulent structures by a trailing edge produces a broadband component; ifthe angle of attack is sufficiently large, the boundary layer detaches and results in a vortexstreet, fairly large, producing a low frequency noise. This vortex street forces the appearanceof a discrete mode in the pressure spectrum. If the profile is truncated, the wake is organizedas a von Karman vortex street. The produced noise is characterized by a spectral line extendedaround the frequency of the vortex shedding, which depends on the trailing-edge thickness[10]. The appearance and dominance of these mechanisms also depend on the conditionsupstream. A profile placed in a uniform upstream flow engenders only trailing-edge noise.However, when it is placed in a turbulent flow, the leading-edge noise is generated andtherefore, it is greater than the trailing-edge noise [11]. Another parameter influencing theproduced sound level is the thickness of the boundary layer developed on the profile surface.Different profiles do not generate the same noise level, even if the upstream conditions aresimilar [12].

1.2 Computational methods of Aeroacoustics

There are several approaches in computational aeroacoustics. They can be categorized intotwo groups: direct and hybrid methods.

1.2.1 The direct numerical acoustics method:

The direct numerical acoustics method (DNA) consists of resolving the compressible un-steady Navier–Stokes (NS) equations on a domain that includes the acoustic source regionand extends to the far field [13–15]. In this approach, the aerodynamic and the acousticfields are provided in a single Computational Fluid Dynamics (CFD) computation. Indeed,compressible Navier-Stokes equations describe not only the aerodynamic fluctuations butalso acoustic fluctuations. The advantage of this method is its accuracy where the computedacoustic field is exact because no acoustic model is used. However, following Tam [16]analysis, the direct computation requires to fulfill some important requirements. In particular,appropriate numerical techniques, such as non-reflecting boundary conditions and schemespreserving the non-dispersive feature of acoustic waves must be used. CFD schemes areusually dissipative and the acoustic signal could be lost if the numerical scheme has not thenecessary low-dispersive propagation characteristics. The development of low-dispersion

6 Introduction

and low-dissipation schemes is always aimed by researchers as an alternative to the classicalmethods of applied mathematics for computational fluid mechanics [18]. Furthermore thecomputational mesh has to be built so that both the flow and its sound can be well representedrespecting the CFD and acoustic criteria relying on the smallest scales to be resolved [17]. Infact, the use of these methods is unpractical for most industrial applications due to their ex-cessive computational cost. However it is noticed that the complexity of addressed problemshave gradually progressed from idealized cases (co-rotating vortices [19], vortex pairingin a two-dimensional compressible mixing layer [20]) towards the direct computation of acomplete flow-regions for supersonic jet [21] and subsonic [22] but for moderate Reynoldsnumbers.

1.2.2 The hybrid methods of aeroacoustics:

Hybrid methods of aeroacoustics consist in two-steps simulations, where the determination ofthe aerodynamic field is decoupled from the computation of the acoustic waves’ propagation.This decoupling gives the possibility to adapt the numerical techniques to the constraints ofeach computational step, such as the mesh refinement and boundary conditions. The first stepin this methodology is to determine a space-time evolution of a turbulent aerodynamic fieldfrom a solution of NS equations to obtain the near-field velocities and pressure fluctuationseither by direct calculation (DNS), Large Eddy Simulation (LES), used in this study, orunsteady Reynolds Averaged Navier-Stokes (URANS). The second step is to propagate theaerodynamic fluctuations obtained in the first step to the far field. The acoustic analogiesare used at this stage. The analogies are clever rearrangements of compressible Navier-Stokes equations leading to a partial-differential operator that represents acoustic propagationoperator in the left-hand side and an expression to the right-hand side representing anequivalent sound sources term which is computed in the CFD simulation. Among themost common hybrid approaches, we can cite Lighthill [23], Möhring [24], Curle [25],Amiet [29, 31–33], Ffowcs Williams and Hall [38] and Ffowcs-Williams and Hawkings [37]methods. The first four methods are applied in this thesis and will be detailed in the nextchapter. Therefore, pressure, density, and/or velocities are stored and then used to build theacoustic source terms which are integrated into a propagation model, leading to the radiatednoise in the far field. One advantage of the hybrid approach is that the acoustic pressurefluctuations are directly related to the instantaneous aerodynamic variables fluctuations.In contrast, these hybrid approaches still lack information on the interactions between theaerodynamic and acoustic fields. The effect of sound waves on the fluid flow and the potentialcouplings between the aerodynamic and the acoustic fields are neglected due to the largedifference in energy content where the feedback from acoustics to the flow is very small for

1.3 Objectives of the dissertation 7

subsonic flows.Moreover, there is a third approach of CAA which can be considered in the hybrid methodgroup, it is the semi-empirical models, Stochastic Noise Generation and Radiation (SNGR)method originally presented by Bechara et al. [40]. It is attractive because it requires onlysteady CFD but it makes high approximation on the acoustic source where a generatedrandom velocity field by a finite sum of discrete Fourier modes based upon averaged dataof the flow field is used to determine the source for acoustic perturbation so its results areinaccurate. As a conclusion of this paragraph, the direct and SNGR methods have challengeseither efficiency or accuracy. The hybrid methods offer a good balance between these twocriteria so it is used in this thesis.

1.3 Objectives of the dissertation

The main objective is to compute, to simulate and to determine the sources of the aero-dynamic noise generated by turbulent flows around a Controlled Diffusion (CD) airfoil,belonging to a fan, via the hybrid methods of aeroacoustics.To achieve this objective, four methods have been applied to study the effect of differentsources on the predicted noise.Lighthill’s analogy is applied to compute the acoustic pressure of the quadrupole sources.The convectional effects of the mean flow on the acoustic waves propagation has beenaccounted for via Möhring’s analogy (considering the laminar case) whereas via Curle’sintegral formulation, the dipole sources caused by the presence of the airfoil are computedseparately from the quadrupole contribution. Amiet’s theory is applicable and valid for highfrequencies.The efficiency, reliability and cost of these four methods are studied comparing to experi-mental data.The broadband noise is due to the turbulence, the effect of the Subgrid scale (SGS) model hasbeen studied considering three models among them one is implemented and validated in theframework of this study. The dissertation will focus on analyzing the physical mechanismsof aeroacoustic radiation and the different mechanisms plying a role in airfoil contribution.

1.4 Thesis organization

The thesis is divided into seven chapters and seven appendices.The first chapter is an introduction to aeroacoustics. The fan noise problem and airfoil trailingedge in particular is introduced. The different numerical approaches of CAA are compared

8 Introduction

briefly.In the second chapter, the aeroacoustics theories, theirs derivations, assumptions and restric-tions are detailed.The third chapter presents the used solvers and methods. The aerodynamic solver SFELESand the acoustic solver ACTRAN are defined. The finite elements, the infinite elements andother acoustic and CFD principles are presented.The fourth chapter is devoted to define the case study and to study the evolution of the flowregimes with Reynolds number for the CD airfoil.In the fifth chapter, results of the complete aerodynamic study of the turbulent flow are pre-sented. The effects of different parameters are studied such as SGS models, mesh refinement,span-wise extension and other parameters.The sixth chapter presents the acoustic results of the three methods applied to the turbulentflow of the CD airfoil.The conclusions are summarized in the seventh chapter.In Appendix A, the leading-edge noise or turbulence impact noise formulation proposed byAmiet is presented in details. In Appendix B, the derivation of Amiet’s theory transfer func-tions is presented whereas Appendix C is devoted for Ghorbaniasl’s SGS model derivation.The spatial discretization of the 2D NS equations by Galerkin finite elements methods ispresented in Appendix D.In Appendix E, Ensight Gold format is defined as implemented in SFELES.Hydrodynamics reflections at the mesh outlet and the physical boundary condition imple-mentation and results are presented in Appendix F.Finally, the tonal noise corresponding to the vortex shedding at Reynolds number of 12000 isstudied in Appendix G applying Lighthill, Möhring and Curle methods.

1.5 Main contributions and original work

Main results of this thesis are presented in chapters 4,5 and 6 and appendices F and G. Asoriginal works we can cite:

• The determination of flow regimes around the CD airfoil according to flow Reynoldsnumber which has never been addressed in the literature despite its importance soit is performed as an original contribution to the physics of controlled-diffusion airfoils.

• Implantation and validation of the new SGS model proposed by Ghorbaniasl which isfound more efficient than the static Smagorinsky’s model especially concerning the

1.5 Main contributions and original work 9

size of transition region to turbulent boundary layer and the corresponding results.

• Implantation of the physical boundary condition which provides an outlet pressureprofile varying with the time step and the node location. This is more physical than im-posing zero pressure outlet boundary condition because the studied flows are unsteady,it could guarantee the non-reflection of the pressure waves at the exit.

• Amiet’s theory is implemented in Matlab, using the wall-pressure spectrum and thecoherence length of SFELES.

• Curle’s integral formulation is applied proposing two original approaches; the firstapproach is the implementation of the volume and surface integrals in SFELES to becalculated simultaneously with the flow in order to avoid the storage of noise sourceswhich requires a huge space. In the second approach, the fluctuating aerodynamicforces, already obtained during the aerodynamics simulation, are used to compute thenoise considering just the surface sources. The results of the proposed approachescorrespond acceptably to the experimental results.

• Finally, Lighthill’s analogy has been applied, for the first time for the CD airfoil, in thecontext of hybrid method via the acoustic solver ACTRAN using sources computedby SFELES and extracted via the Ensight Gold interface, implemented in this work,which builds 3d pressure/velocity fields and a 3d geometry from the 2D CFD meshand Fourier modes.

Chapter 2

Review of aeroacoustics theories,sound sources definition

2.1 Lighthill’s analogy

Lighthill’s analogy is derived from the Navier-Stokes equations without any approximations.Moreover, it is an exactly valid equation, which admits an acoustic interpretation. Theacoustic domain is decomposed into a flow region containing the source region in which theturbulent fluctuations generate the sound and an acoustic quiescent and uniform region wherethe acoustic waves propagate to the far field listener’s position as shown in Fig.2.1.

Lighthill started from the continuity and momentum equations which are written, inabsence of the external forces and mass sources and for compressible flow, as:

∂ρ

∂t+∂ρui∂xi

= 0 (2.1)

∂ρui∂t

+∂ρuiuj

∂xj= –

∂p∂xi

+∂τij

∂xj(2.2)

where τij is the viscous stress tensor. For a Stokesian gas it can be expressed in terms of thevelocity gradients as:

τij = µ(∂ui∂xj

+∂uj

∂xi–

23δij

∂uk∂xk

) (2.3)

12 Review of aeroacoustics theories, sound sources definition

Fig. 2.1 Regions in a hybrid aeroacoustic problem

where µ is the dynamic viscosity of the fluid and δij is the Kronecker delta. From themomentum equation 2.2, we can write:

∂ρui∂t

= –∂

∂xj(ρuiuj +δijp –τij) (2.4)

Adding and subtracting the term c20∂ρ/∂xi to the precedent equation give:

∂ρui∂t

+ c20∂ρ

∂xi= –

∂Tij

∂xj(2.5)

Where c0 is the sound speed and Tij is the Lighthill’s stress tensor given as:

Tij = ρuiuj +δij[(p – p0) – c20(ρ–ρ0)] –τij (2.6)

With ρ0 and p0 the atmospheric density and pressure respectively. Differentiating thecontinuity equation 2.1 with respect to time, taking the divergence of the equation 2.5 andsubtracting the results lead to the following expression:

∂2ρa

∂t2– c2

0∂2ρa

∂x2i

=∂2Tij

∂xi∂xj(2.7)

2.1 Lighthill’s analogy 13

This hyperbolic partial differential equation is the Lighthill’s inhomogeneous wave equationsolved for the acoustic density fluctuation ρa [23]. The left hand side of this expression isdefined as the acoustic wave operator for a uniform medium at rest (sound propagation),while the terms appearing in the right hand side are defined as the aeroacoustic sources(sound generation), which clearly behave like a quadrupole sound source due to the presenceof the second order spatial derivative.

2.1.1 Approximation of Lighthill’s stress tensor

In general, Lighthill’s tensor’s expression can be simplified. The first assumption is that thesource term vanishes outside the turbulent region. Indeed, for a turbulent flow embedded in auniform atmosphere at rest, Lighthill’s stress tensor can be neglected outside the turbulentregion itself. Inside the turbulent region, the contribution of each term of Tij will beconsidered separately.The term δij[(p – p0) – c2

0(ρ–ρ0)] is related to entropy variations inside the source region, itvanishes exactly for isentropic flows. Moreover, the effects of viscosity and heat conductionare expected to cause only a slow damping over very large distances due to the conversion ofacoustic energy into heat. Therefore, for high Reynolds number it is possible to neglect theviscous stress tensor τij [41]. Based on these assumptions, the Lighthill’s tensor reduces to:

Tij = ρ0uiuj (2.8)

2.1.2 Integral solution of Lighthill’s analogy

Using a free-space Green’s function, the solution for the inhomogenous wave equation 2.7 is[41, 23]:

ρa =1

4πc20

∂2

∂xi∂xj

∫V

(Tij(y, t –

∣∣x–y∣∣

c0)∣∣x – y

∣∣ )dV (2.9)

where V is the volume containing the turbulent structures that generate the sound (sourceregion). The spatial differentiation can be replaced by a temporal differentiation, when theobserver is located at a distance large compared with (2π)–1 times a typical wavelength,[41, 23]:

∂

∂xi= –

1c0

∂

∂t∼ 2πf

c0(2.10)


Changing the spatial derivatives, in equation 2.9, into temporal yields in a far-field solutionof Lighthill’s acoustic analogy as:

ρa =1

4πc20

∫V

((xi – yi)(xj – yj)∣∣x – y

∣∣3 )1c2

0

∂2

∂t2Tij(y, t –

∣∣x – y∣∣

c0)dV (2.11)

For the geometrical far-field approximation, detailed in next paragraph, the receiver is locatedat large distances compared to the source region dimensions so we can write xi – yi ≈ xi,assuming that the coordinate system origin is in the source region y = 0, no solid boundariesare immersed in the fluid, and applying pa = ρa ∗ c2

0 give the final far-field integral solutionfor Lighthill’s analogy as the following [23]:

pa =xixj

4π |x|3 c20

∂2

∂t2

∫V

[Tij]tedV(y) (2.12)

This solution is only valid for exterior problems. [.]te means a term evaluated at the retardedtime defined as the propagation delay time between the sound emission (source) and reception

(observer) given as te = t –∣∣x–y

∣∣c0

= t – rc0

.

2.2 Curle’s formulation

Lighthill’s analogy has been extended by Curle [25] in order to incorporate the influenceof the presence of stationary solid boundaries on the aerodynamic sound. The surface isexpected to reflect and diffract the radiated sound, changing the wave characteristics. There-fore, the overall radiated sound field is here a contribution of three origins:

1. Quadrupole sources (Lighthill), which are present in the vicinity of the solid bound-aries and due to the turbulence such as the dispersion of turbulent boundary layer in the wakeof a body, an airfoil or a cylinder etc.2. Dipole sources, generated by fluctuating aerodynamic forces on the solid boundaries,acting on the fluid.3. Monopole sources, due to the mass flux through the surface S or the kinematics of the body.

Curle extension is reflected in a new term ∂fi∂xi

, representing the fluctuating aerodynamicforces, added to the right hand side of the equation 2.7. The most general far-field solution of

2.2 Curle’s formulation 15

the inhomogeneous wave equation 2.7 on a bounded domain is [26]

pa =1

4πc20

∂2

∂t2

[xixj

|x|3

∫V

TijdV(y)]

te–

14π

∂

∂t

[1|x|

∫SρvinidS

]te

–

14πc0

∂

∂t

[xj

|x|2

∫S

[Pij +ρvivj]nidS]

te

(2.13)

With Pij = pδij –τij is the stress tensor, ni the wall normal pointing into the flow. Assumingthat the source is acoustically compact, the variations of the retarded time te are so smallcomparing to the retarded time itself te = t – r

c0, so they can be neglected. Furthermore, the

second integral refers to a monopole-like sound field, can be omitted because the studiedairfoil is not permeable. The viscosity term τij will not be considered further. It is alsoassumed that the position of the receiver is large compared to L (chord) which meansgeometrical far-field sound. After all these simplifications, the relation is written:

pa(x, t) =1

4πc20

∂2

∂t2

[xixj

|x|3

∫V

TijdV(y)]

te–

14πc0

∂

∂t

[xj

|x|2

∫S

pnidS]

te(2.14)

This formulation is considered in this study and computed in SFELES, the CFD solver usedin the present thesis. It is important to note that it is an approximated integral solution ofLighthill’s analogy where some simplifying assumptions are applied making it applicable tospecific problems. The most important assumptions needed for the validity to this formulationare itemized as:

• Viscous effects are neglected.

• Stationary and thin airfoils, only valid for exterior noise problems.

• Geometrical far-field sound: The position of the listener is large compared to thetypical dimension of the solid boundaries, |x| ≫ C. This is fulfilled in our applicationwhere |x| = 2 m and C = 0.1356 m.

• Acoustical far-field sound: The position of the listener is large enough compared to theacoustic wavelength, |x| ≫ λ. This implies that:

|x| >c0

fmin=⇒ fmin > 170[Hz] (2.15)


which is acceptable for frequencies of interest (higher than 200 [Hz]).

• Acoustically compact source: This assumption leads for our study to:

L < λ→ chord <c0

fmax=⇒ fmax < 2507[Hz] (2.16)

which is also suitable for frequencies of interest, [200-2000] [Hz]. So, this assumptionmakes the proposed approach not really valid at very high frequencies.

Ffowcs Williams and Hawkings (FWH) [37] extended Curle’s formulation to include theinfluence of moving surfaces. When the solid boundary is in motion, it has an additionaleffect on the acoustic field similar to a mass source or monopole source due to the kinematicsof the solid boundary. It is present along the surface boundary.The details of FWH formulation are not considered in this thesis because it is not included inthe application. In contrast, Ffowcs Williams and Hall’s theory [38] is presented hereaftersince it is applied to the considered airfoil by Wang [58] and its results will be used for thecomparison with some results obtained in this work.

2.3 Ffowcs Williams and Hall’s theory

This theory is based on the solution of the Lighthill’s wave equation using a Green’s functionfor a semi-infinite flat plate whose derivative vanishes on the airfoil surface [38]. The finalacoustic pressure formulation in the far field is given in the frequency domain as:

pa(X,ω) ≈ei(k|X|–π

4 )

252π

32∣∣X∣∣ [ksin(φ)]

12 sin(

θ

2)S(ω) (2.17)

Where S(ω) is the source term that needs to be computed during the simulation. In the timedomain, it is given as:

S(t) =∫

V

ρ0

r320

(u2θ – u2

r )sin(θ02

) – 2uruθcos(θ02

)

d3y (2.18)

X = (r,θ, z) and y = (r0,θ0, z0) represent far field observer and source positions respectively,as shown in the coordinate system defined in Fig. 2.2. The half-plane formulation 2.17 isvalid for kC ≫ 1. The finite-chord effect is accounted for by a correction factor χ, followingthe derivations of Howe [39], the far-field acoustic pressure is therefore given as:

pca(X,ω) = χ.pa(X,ω) (2.19)

2.4 Möhring’s analogy 17

Fig. 2.2 Coordinate system for the finite-chord thin plate used in the application of FfowcsWilliams and Hall’s theory. Figure is reproduced from Wang et al [58]

The correction factor at mid-span φ = π2 and for a receiver located directly above the airfoil

θ = π2 , is given as:

χ = 1 + 2F

(√2kCπ

) ( eikC√

2πkC– e–iπ4

)(

e–ikC + eikC

2πikC

) (2.20)

F is the Fresnel integral auxiliary function. The reader is referred to [112] for more detailsabout approximations and limitations of this formulation.

2.4 Möhring’s analogy

Möhring’s analogy is an extension of the Lighthill’s theory for the application where themean flow convection effects on acoustic propagation are important. In Lighthill’s analogy,the derivation considers the acoustic propagation in a medium at rest. The convection andrefraction effects occurring in the mean flow are contributing by means of the right hand sidesources, which is not natural. In Möhring, the mean flow effects are directly accounted inthe acoustic propagation operator and the sources are close to real aerodynamic noise sourcecontributions. Therefore the resulted analogy is well suited to study the sound radiationfrom vortices convected in an irrotational flow [24]. Möhring’s analogy is based on thestagnation enthalpy B, which obeys a linear convected wave equation, as an acoustic variable.Starting from compressible Navier-Stokes equations (Crocco’s form) and using the energyconservation equation and other thermodynamic relations, Möhring arrived to his analogy as


the following [24, 87].The continuity equation:

∂ρ

∂t+ ∇.(ρu) = 0 (2.21)

The momentum equation in terms of the total enthalpy B:

ρ∂u∂t

+ρ∇B = ρT∇s + ρu× (∇× u) – ∇τ (2.22)

where ρ is the density, u is the flow velocity, B is defined as B = h+ 12 ∥ua∥2, ua is the velocity

fluctuations, h is the flow enthalpy and τ is the viscous stress tensor. The energy equation:

ρDBDt

–∂p∂t

= ∇.(u.τ) + ∇.(λcon∇T) (2.23)

where T is the temperature and λcon denotes the material’s conductivity. Neglecting thedissipation caused by viscous stresses and heat conduction leads to a relation relating thepressure to the enthalpy:

DBDt

=1ρ

∂p∂t

(2.24)

Combining the continuity equation 2.21 and the simplified energy equation 2.24 gives:

∂ρ

∂t= –∇.(ρu) =

1c2

0

∂p∂t

–∂ρ

∂s∂s∂t

=ρ

c20

DBDt

–∂ρ

∂s∂s∂t

(2.25)

Replacing ρ ∂u∂t with ∂

∂t (ρu) – u∂ρ∂t in the momentum equation 2.22, we get:

∂ρu∂t

–ρuc2

0

DBDt

+∂ρ

∂su∂s∂t

+ρ∇B = ρT∇s + ρu× (∇× u) – ∇τ (2.26)

This equation needs to be generalized so we introduce the parameters ρT, the total densityfield and the scaled enthalpy b defined by: Db

Dt = ρTDBDt . Combining ∇ of the equation 2.26

with ∂∂t

1ρT

of the equation 2.25 leads to the following scalar equation:

∂

∂t(

ρ

ρ2Tc2

0

DbDt

) + ∇(ρuρ2

Tc20

DbDt

–ρ

ρ2T

∇b) = R (2.27)

R = –∇[1ρT

(ρu× (∇× u) – ∇τ)] + ∇[1ρT

(∂ρ

∂su∂s∂t

–ρT∇s)]+

+∂

∂t[

1ρT

∂ρ

∂s∂s∂t

+ ρu.∇1ρT

](2.28)

2.5 Amiet’s aeroacoustic theory 19

In the previous equation, Eq. 2.27, only the dissipation of viscous stresses and heat releasedby conduction have been neglected in the energy equation. The left-hand side corresponds toan acoustic wave operator in the presence of a heterogeneous flow. The right-hand side isconsidered as acoustic sources where it contains the flow fluctuations represented by threeterms. The first term represents the turbulent noise and the other two terms represent thecombustion noise. Neglecting the viscous effects and the total density fluctuations, for anisentropic flow the source term in equation 2.28 simplifies to:

R=-∇[ 1ρT

(ρu× (∇× u)] (2.29)

2.5 Amiet’s aeroacoustic theory

Amiet’s theory is a semi-analytical approach used to compute the broadband noise radiatedby stationary, thin airfoils. Amiet started first by deriving a formulation to predict thenoise generated by the impact of a turbulent flow upstream [29, 31] or what is the so-calledleading-edge noise and then he generalized it to be applicable to compute the noise causedby the diffraction of the turbulent boundary layer developing on the profile by its trailingedge [32] or the trailing-edge noise. The source-field, defined by wall-pressure fluctuationsand spanwise coherence for the trailing-edge formulation for instance, can be determinednumerically or experimentally. The aeroacoustics transfer functions in both leading andtrailing edge noise formulations are based on an iterative process of diffraction by a half-plane following a technique of Schwarzschild [48, 49]. Amiet’s theory is widely used in theliterature [27, 28, 55], but has some limitations and assumptions. The first assumption thatthe airfoil is a flat plate extended to infinity in the upstream or downstream direction, forrespectively the trailing or leading edge noise, therefore it is applicable only for airfoil-likeshapes with small airfoil thickness, camber and angle of attack. The corresponding scatteringeffects are supposed to appear at the airfoil edges (leading and trailing edges) so we havedifferent formulations to deal with leading and trailing edges noise. The effects of airfoilshape, camber and thickness are not considered in the model directly but within the wall-pressure spectrum for the trailing-edge formulation. The receiver is assumed to be located inthe acoustic far-field. The flow impinging into the airfoil is assumed to be uniform in thespanwise direction. The leading-edge noise or turbulence impact noise formulation proposedby Amiet is presented in details in Appendix A while the trailing-edge noise formulation ispresented in the next paragraph.


2.5.1 Derivation of the generalized trailing-edge noise formulation

Amiet proposed in 1976 [32] an adaptation of the leading-edge noise model presented in1975 [29], Appendix A, to be applied for the trailing-edge noise computation for the case ofa semi-infinite plane, with a trailing edge but no leading edge. Amiet’s trailing-edge modeldescribes how the hydrodynamic waves convected within the boundary layer are scatteredby the trailing edge. Schwarzschild’s procedure [48, 49] is applied iteratively such that theturbulent boundary layer is considered as a series of gusts traveling towards the trailing edge.Amiet considered that the noise is mainly generated by the induced surface dipoles nearthe trailing edge so the main input of this model is the convecting wall pressure spectrumupstream of the trailing edge. It is basically assumed that the turbulent velocity field isunaffected by the presence of the trailing edge. This assumption allows the calculation ofthe trailing edge noise from the spectral characteristics of wall pressure which would existin the absence of the trailing edge. For finite chord airfoils, the cancellation condition ofthe potential upstream of the airfoil is not considered by Amiet and needs to be taken intoaccount. Roger & Moreau [33, 84] extended this model to include the leading-edge effect.For the derivation, let us assume C = 2b is the airfoil chord length, U is the fluid uniformvelocity upstream of the airfoil as shown in Fig 2.3.

Fig. 2.3 2D problem with trailing-edge coordinates. Figure reproduced from [33]

The starting point is a two-dimensional Fourier decomposition of the incident hydrodynamicwall pressure induced by the turbulent boundary layer developed on the airfoil surface. Theconvected wave equation, obtained from the linearized Euler equations, is written in theplane normal to the airfoil as:

∇2p′ –

1c2

0

D2p′

Dt2= 0 (2.30)


with DDt = ∂

∂t + U ∂∂x . The disturbance pressure can be expressed as p′(x,z, t) = P(x,z)eiωt,

leading to the complex equation:

β2∂2P

∂x2 +∂2P∂z2 – 2ikM

∂P∂x

+ k2P = 0 (2.31)

with k = ω/c0 the acoustic wavenumber, M = U/c0 and β =√

1 – M2.A variables change is performed as P(x,z) = p(x,z)ei(kM/β2)x, leading to the equation:

β2∂2P

∂x2 +∂2P∂z2 +

(KMβ

)2P = 0 (2.32)

where K = ω/M the convective wavenumber and k = KM. By further transforming andadimensionnalising the problem with x = x

b , z = βzb , K = Kb and µ = KM

β2 , a canonicalHelmholtz wave equation is obtained:

∂2P∂x2 +

∂2P∂z2 + µ2P = 0 (2.33)

In the non-dimensional variables, the airfoil extends over –2 ≤ x ≤ 0. The boundary con-ditions corresponding to the previous wave equation need to be determined. Upstream ofthe trailing edge, the turbulence of the boundary layer, convected with a velocity Uc, isrepresented by an incident gust of pressure as:

p′(x,0, t) = eiωte–iαKx = P0eiωt (2.34)

with α = U/Uc. Amiet extended the airfoil to infinity upstream to be as a half-plane definedby x < 0. The first boundary condition is the Kutta condition has to be satisfied at the trailingedge and in the wake, P0 must be canceled. This is done by adding a disturbance pressureP1 such that P = P0 + P1 is zero for x < 0. The half-plane is assumed perfectly rigid, whichprovides the second boundary condition which implies that the normal derivative of P1 mustbe zero for x < 0. The following system of equations is therefore obtained to be resolved, viaSchwarzschild’s solution (defined in Appendix B), to derive the main scattering term:

∂2P1∂x2 + ∂2P1

∂z2 + µ2P1 = 0 x < 0

p1 = –e–iKx[α+(M2/β2)] x ≥ 0

(2.35)


Applying Schwarzschild’s solution leads to the equation:

p1(x,0) = –1π

∫∞

0

√–xξ

e–iµ(ξ–x)

ξ– Xe–iKξ[α+(M2/β2)]dξ

= –eiµ(x)

π

∫∞

0

√–xξ

1ξ– x

e–i[αK+(1+M)µ]ξdξ

(2.36)

The integral is computed by Gradshteyn & Ryszik [106] and given as:

∫∞

0

√–xξ

1ξ– x

e–iAξdξ = πe–iAx[1 –eiπ/4√π

∫ –Ax

0

e–it√

tdt] (2.37)

Recognizing the Fresnel integrals R′

and S′, defined by:

E∗(x) =∫ x

0

e–it√

2πtdt = R

′(x) – iS

′(x) (2.38)

It is eventually arrived to the airfoil surface pressure jump formulation derived by Amiet [32]for x < 0 as

P1(x,0) = e–iαKx(1 + i)E∗(–αK + (1 + M)µ

x) – 1

(2.39)

Leading edge back-scattering correction

Amiet’s model 2.39 is restricted to high frequencies. The extension to include low frequencieswas developed by Roger & Moreau [33, 84] by taking into account the back-scatteringfrom the leading edge, accounting for the finite chord length and respecting the upstreampotential cancellation condition. The needed correction is derived, again using a two-stepSchwarzschild’s solution. The final formulation for the pressure correction term is:

P2(x,0) ≈ (1 + i)e–4iµ

2√π(α– 1)K

1 –Θ21√

A1ei(M–1)µx

.

i[K + (M – 1)µ

][F(x)]c +

[∂F(x)∂x

]c (2.40)

withF(x) = e2iµ(x+2)1 – (1 + i)E∗[2µ(x + 2)]

(2.41)

∂F(x)∂x

= e2iµ(x+2)[2iµ

1 – (1 + i)E∗ [2µ(x + 2)]

– (1 + i)√

2µ/2π(x + 2)e–2iµ(x+2)]

(2.42)


Θ1 =√

A1A , A1 = K1 + (1 + M)µ, A = K + (1 + M)µ and K1 = ω/Uc.

The notation [–]c stands for an imaginary part multiplied by the correction factor ϵ given as

ϵ =(

1 + 14µ

)–1/2. For more details about the derivation, the reader is addressed to [33].

Generalization to three-dimensional case

Fig. 2.4 3D problem for the trailing-edge model. Figure reproduced from [33]

In the case of a 3d problem, the wave equation can be written:

∂2p′

∂x2 +∂2p′

∂y2 +∂2p′

∂z2 –1c2

0

D2p′

Dt2== 0 (2.43)

The solution is in the form of:

p′(x,y,z, t) = P(x,y,z)eiωt

P(x,y,z) = p(x,y,z)ei(kM/β2)xe–iK2y

and the incident wall pressure gust is generalized as

P0 = e–iαKxe–iK2y

with y = y/b. The wave equation is read:

∂2p∂x2 +

∂2p∂z2 + κ2p = 0 (2.44)

with κ2 = µ2 – K22

β2 and K2 = αK. The mathematical nature of the problem depends on the

sign of κ2 where we can distinguish two cases:


- If κ2 > 0, the differential equation is therefore hyperbolic and the gust is called supercritical.- If κ2 < 0, the differential equation is elliptic and the gust is said subcritical.The supercritical solution is therefore an extension of the two-dimensional problem as:

P1(x,0) = e–iαKx(1 + i)E∗(–αK + κ+ Mµ

x) – 1

(2.45)

P2(x,0) ≈ (1 + i)e–4iκ

2√π(α– 1)K

1 –Θ2√αK + Mµ+ κ

ei(Mµ–κ)x

.

i[K + Mµ– κ

][F(x)]c +

[∂F(x)∂x

]c (2.46)

where Θ =√

αK+Mµ+κ′K+Mµ+κ′ .

For supercritical gusts, it is detailed by Roger & Moreau [33]. The final formulations for theinduced pressure:

P1(x,0) = e–iαKX

(1 + i)Φ0√

(–αK + Mµ– iκ′

x) – 1

(2.47)

with κ′ =

√(K2β

)2– µ2

P2(x,0) ≈ (1 + i)e–4iκ

2√π(α– 1)K

1 –Θ′2√αK + Mµ+ κ′

ei(Mµ–κ)x

.

i[K + Mµ– κ

][F′(x)

]c +[∂F′(x)∂x

]c (2.48)

with Θ′ =√

αK+Mµ–iκ′K+Mµ–iκ′ , F′(x) = 1 – erf

(√2κ′(x + 2)

)Φ0(Z) = 1√

π

∫ Z2

0e–z

z dz and Φ0(√

ix) =√

2eiπ/4E∗(x).

Far-field acoustic radiation and power spectral density

The acoustic far-field pressure corresponding to a disturbance wall pressure is given by theradiation integral, found in [29], as

p(X,ω) =–iωx34πc0σ

20

∫ 0

–2b

∫ L/2

–L/2PeiωRt/c0dydx (2.49)


X = (x1,x2,x3) the receiver location with the origin fixed at the trailing edge. P = 2(P1 +P2) = 2P stands for the induced source distribution as given by the Schwarzschild’s solution.The airfoil extends from –L/2 to L/2 in the spanwise direction. The convectional effects areaccounted for through the modified coordinates as

Rt = 1β2 (Rs – M(x1 – x))

Rs = σ0

(1 – x1x+β2x2y

σ20

)σ0 =

√x2

1 +β2(x22 + x2

3)

P can be written as: P(X,ω) = f(x)e–i(K1x+K2y) with f the complex amplitude of the sourcedistribution. Then the radiation integral becomes

P(X,ω) =–iωx34πc0σ

20

b2∫ 0

–2

∫ L/2b

–L/2bf(x)e–i(K1x+K2y)

.e–i k

β2

[σ0– x1x+β2x2y

σ0b–M(x1–bx)

]dydx

(2.50)

The integral with respect to y is given by:

b∫ L/2b

–L/2be–i(K2–kx2/σ0)ydy = Lsinc

L2b

(K2 –

kx2σ0

)(2.51)

where sinc(x) = sin(x)x . The general formulation of the acoustic pressure is obtained:

p(X,ω) =–iωx3Lb4πc0σ

20

sinc

L2b

(K2 –

kx2σ0

).e

–i kβ2 (σ0–Mx1)

∫ 0

–2f(x)e–iNxdx

(2.52)

where N = K1 – µ( x1σ0

– M). The calculation of the pressure induced on the airfoil was madefor a unit gust with wavenumbers

(K1, K2

)at the reduced frequency ω. The far-field sound

power spectral density at the same frequency results from an integration over all gusts with2D wavenumbers contributing to this frequency. The frozen turbulence hypothesis pastthe trailing edge leads to the streamwise aerodynamic wavenumber as K1 = ω/Uc. NotingA0( ωUc

) for the amplitude of the incident pressure, the corresponding disturbance pressuredistribution P over the airfoil surface is written:

P(x,y,ω) =1

Uc

∫∞

–∞

g(x,ω

Uc,K2)A0(

ω

Uc,K2)e–iK2ydK2 (2.53)


with Amiet’s function g (related to f) denoting the transfer function between the incidentpressure P0 of amplitude A0 and the disturbance pressure P.The spectral power density of the incident pressure between two points on the surface (x,y)and (x′,y′), with η = y – y′, is defined as:

SPP(x,x′,η,ω) =1

Uc

∫∞

–∞

g(x,ω

Uc,K2)g∗(x′,

ω

Uc,K2)e–iK2ηΠ0(

ω

Uc,K2)dK2 (2.54)

Π0 stands for the wavenumber spectral density of the incident gust amplitudes A0. Thecorresponding PSD of the far-field sound is therefore given as:

SPP(X,ω) =

(ωx3Lb2πc0σ

20

)21b

∫∞

–∞

Π0(ω

Uc,K2)

sinc2

L2b

(K2 – k

x2σ0

)∣∣∣∣L (ω

Uc, K2)

∣∣∣∣2 dK2

(2.55)

The spectral density Π0 is expressed by Schlinker & Amiet [34] as:

Π0(ω

Uc,K2) =

UcπΦpp(ω)ly(ω, K2) (2.56)

sinc2

L2b

(K2 – k

x2σ0

)≃ 2πb

Lδ

(K2 – k

x2σ0

)(2.57)

For very large aspect ratio airfoils, the final far-field acoustic pressure PSD formulation isread:

SPP(X,ω) =

(ωCx3

4πc0σ20

)2L2

∣∣∣∣L (ω

Uc, k

x2σ0

)∣∣∣∣2Φpp(ω)ly(ω, k

x2σ0

) (2.58)

This formulation is simplified for low Mach number around 0.1, as shown in [84], to:

Spp(X,ω) = (sinθ2πR

)2.(kC)2d.|L |2Φpp(ω)ly(ω) (2.59)

This formulation is implemented and applied in this research so its parameters are detailed.In this equation d is the assumed flat plate semi-span, where the span is assumed to equalto 40 C, with C is the airfoil chord length, k is the acoustic wavenumber. R and θ are thelistener location. L is the aeroacoustic transfer function (L = L1 +L2) with L1 the transferfunction of the trailing edge and L2 the back-scattering leading edge correction. L1 and L2formula are detailed in the Appendix B. Φpp(ω) is the wall-pressure power spectral densityand ly(ω) is the spanwise correlation length near the trailing edge. So the input data here isrepresented by the wall-pressure spectrum and the correlation length which can be obtained

2.6 Sound sources definition: Monopole, dipole & quadrupole 27

from experiments [53] or extracted from the LES computations [57] as in this study. Thisprocedure is physically justified by the fact that when the incident aerodynamic wall pressureis convected past the trailing edge, it behaves as equivalent acoustic sources. Concerning thespanwise correlation length near the trailing edge ly(ω), it is computed using the empiricalmodel of Corcos [86] given by the relation: ly(ω) = b.Uc

ω . This model is widely used in theliterature as in [55–57] with Uc = 0.7U0 the convection speed and the coefficient b = 1.5. Thecoherence length ly(ω) can be determined experimentally as in [53, 54] or it may be extractedfrom the LES computations according to the relation ly(ω) =

∫∞0√γ2(η,ω)dη where γ is

the coherence function. But it has been shown by Christophe [57] that the convergence of thespanwise coherence near the trailing edge is really poor for low frequencies. The LES resultslargely overpredict the experimental coherence and present a peak between 500 and 1000[Hz] which does not exist in the experiments. In order to improve it and to study the effectthe spanwise extent, a LES simulation using Fluent has been carried out with a spanwise of0.3C. It has been shown that a larger spanwise is necessary to correctly capture the spanwisecoherence which is overpredicted in case of small extent due to the confinement of the flowbetween the spanwise boundaries. But even if for a 0.3C span, the agreement still not goodenough to be applied in the aeroacoustics theory of Amiet taking into account the very highcost of the simulation (three times more expensive in comparison with the case of 0.1 Cspan). In this study, the Corcos’ model is used. However, the coherence length extractedfrom the present LES computations is also used. The results and more details are presentedin Chap. 6.

2.6 Sound sources definition: Monopole, dipole & quadrupole

We can identify three categories of sound sources due to flow: monopoles, dipoles andquadrupoles. A mass flow is an example of a monopole. A dipole can be created whenfluctuating forces exist in the flow, like the von Karman vortex shedding formed in the wakeof a cylinder or an airfoil. The coupling of the fluctuating forces can form quadrupoles, forinstance turbulence is a typical example. Figure 2.5 shows a representation of the directivitypatterns of these three sources.

The monopole

A monopole is a source which radiates sound equally well in all directions. A loudspeakercan be approximated as a monopole source at low frequencies. The sound pressure for a


Fig. 2.5 Representation of directivity pattern of sound sources: a)-Monopole, b)-Dipole,c)-Quadrupole

simple monopole source at a distance r is given as [42]:

p(r,ω) = iρωq(ω)e–ikr

4πr(2.60)

where q(ω) = 4πF(ω)iρω is the volumetric flow rate of a punctual source [m3/s] and F(ω) is the

source amplitude [42].

The dipole

A dipole is the superposition of two monopoles in opposite phase. Let us consider two sourceswith amplitudes A and –A located in Q1 and Q2 as shown in Fig 2.6. The sound pressure at

Fig. 2.6 Diagram of a dipole [42]

the receiver is obtained by adding the sound pressure generated by the two monopoles and

2.6 Sound sources definition: Monopole, dipole & quadrupole 29

can be expressed as:

p = A(e–ikr2

r2–

e–ikr1

r1) (2.61)

Considering the distance L small enough or its limit goes to zero and putting D = AL, thedipole moment, it is arrived to the final formulation as the following:

p = ikD.e–ikr

r.cosθ.(1 +

1ikr

) (2.62)

The acoustic field of a dipole is the product of four terms [42]:An amplitude ikD which is the product of the dipole moment D and the factor ik characteristicof a dipole.An unitary monopolar field e–ikr

r .A directivity term cosθ.A first order harmonic polynomial of ikr which goes to zero when kr goes to the infinity.

The quadrupole

The quadrupole source can be obtained by the superposition of two dipole sources of thesame strength that are in antiphase or four monopoles. Considering four sources as in Fig 2.7.The total pressure at the receiver is:

Fig. 2.7 Diagram of a quadrupole [42]

p = A(–e–ikr1

r1+

e–ikr2

r2+

e–ikr3

r3–

e–ikr4

r4) (2.63)


It is finally arrived to the formulation:

p = (ik)2Q.e–ikr

r.sinθcosθ.(1 +

2ikr

+2

(ikr)2 ) (2.64)

with Q = AL2

2 . The reader is referred to [42] for the detailed derivation.

Chapter 3

Solvers and numerical methods

As mentioned before, different hybrid aeroacoustic approaches are used to simulate the noisedue to turbulent flow over a CD airfoil. In these methods, the aerodynamic and acoustic fieldsare resolved separately. The flow data are obtained using the in-house LES solver SFELES[43, 73, 74]. The ACTRAN acoustic solver [87] is used to solve the acoustics and to providethe near and far field acoustic propagation. This chapter is devoted to describe the solversand to present the numerical methods. In addition, some basic backgrounds in aerodynamicsand acoustics are presented.

3.1 The CFD solver, SFELES

SFELES stands for ’Spectral Finite Elements Large Eddy Simulation’ which indicate theprinciples of the solver. SFELES is a hybrid spectral/finite elements code for 3D unsteadyincompressible viscous flows over axisymmetric or planar geometries. It is a large eddysimulation (LES) solver based on a combination of a Fourier expansion in the periodic (az-imuthal or transverse) direction and a P1 finite element (FE) formulation in the perpendicular(meridian or longitudinal) plane. This solver has been developed by group of researchersof Brigham Young University (USA), Université Libre de Bruxelles and the Von KarmanInstitute for Fluid Dynamics (Belgium). The main advantage of this code is its ability toexploit the existence of a direction of periodicity in the geometry to improve the computa-tional efficiency. Another advantage is that a single 2D (structured or unstructured) meshneeds to be stored (coordinates, connectivity, etc.) reducing the memory requirements incomparison with traditional 3D solvers. The Navier-Stokes equations are discretized andresolved numerically in SFELES. For an incompressible viscous flow and in the absence of

32 Solvers and numerical methods

volume forces, these equations are written in Cartesian coordinates in the form:

∂ux∂x + ∂uy

∂y + ∂uz∂z = 0

∂ux∂t + ux

∂ux∂x + uy

∂ux∂y + uz

∂ux∂z = –∂p

∂x +ν(∂2ux∂x2 + ∂2ux

∂y2 + ∂2ux∂z2 )

∂uy∂t + ux

∂uy∂x + uy

∂uy∂y + uz

∂uy∂z = –∂p

∂y +ν(∂2uy∂x2 + ∂2uy

∂y2 + ∂2uy∂z2 )

∂uz∂t + ux

∂uz∂x + uy

∂uz∂y + uz

∂uz∂z = –∂p

∂z +ν(∂2uz∂x2 + ∂2uz

∂y2 + ∂2uz∂z2 )

(3.1)

Since the in-plane discretization (FE) is different from the periodic direction (spectralmethods), let us re-formulate the NS equations introducing an in-plane operator ∇ and anin-plane velocity defined by u = uxex + uyey. That gives:

∇.u + ∂uz∂z = 0

∂u∂t + (u.∇)u + uz

∂u∂z = ˜–∇p +ν(∇2u + ∂2u

∂z2 )∂uz∂t + (u.∇)uz + uz

∂uz∂z = –∂p

∂z +ν(∇2uz + ∂2uz∂z2 )

(3.2)

For planar geometries, the flow variables are approximated as

qh(x,y,z, t) =1

Nm

Nm/2

∑k=–Nm/2+1

[Nn

∑j=0

qkj (t)φj(x,y)]exp

(2πIk

zL

)(3.3)

where I =√

–1, Nm is the number of Fourier modes, Nn is the number of nodes in each finiteelement plane, qk

j (t) is the k-th mode of the variable q at node j, L is the spanwise dimensionof the domain, k is the Fourier mode number and φj(x,y) stands for the P1 piecewise linearbasis function associated with node j.The governing equations are discretized using a stabilized finite element approach with theclassical streamline upwind (SUPG) and pressure (PSPG) stabilizations [71, 72], i.e.∀w,q weight functions in the approximation space, the variational formulation of the NSequations 3.2 leads to the following weak form as:

∫Ω

[w.(∂u∂t

+ (u.∇)u + uz∂u∂z

)– ∇.wp +ν

(∇w.∇u – w.

∂2u∂z2

)+

wz

(∂uz∂t

+ (u.∇)uz + uz∂u∂z

+ p)

+ν

(∇wz.∇uz – wz

∂2uz

∂z2

)+

q(

∇.u +∂uz∂z

)]dΩ+

∫Γn

[w.((pI –ν∇u).n

)–νwz∇uz).n]dΓ + ST = 0

(3.4)

3.1 The CFD solver, SFELES 33

where ST is the stabilization terms given as:

ST = ∑e

∫Ωe

[(τSUPG(u.∇)w +τPSPG∇q

).R +τSUPG(u.∇wz)Rz

]dΩ (3.5)

with R and Rz are the residuals given by the following relations:R = ∂u∂t + (u.∇)u + uz

∂u∂z + ∇p –ν(∇2u + ∂2u

∂z2 )

Rz = ∂uz∂t + (u.∇)uz + uz

∂uz∂z + ∂p

∂z –ν(∇2uz + ∂2uz∂z2 )

(3.6)

Note that the stabilization terms involve only gradients in the finite element plane, identifiedby the symbol ∇. Concerning the stabilization terms, τSUPG and τPSPG, different proposi-tions exist in the literature. In our code SFELES, those proposed in [71, 72] are used as thefollowing:

τPSPG =1√

1τ2

c+ 1τ2

t+ 1τ2ν

(3.7)

where: τc = he2U , τt = ∆t

2 , τν = h2e

4ν and he =√

4Ωeπ is the hydraulic diameter of the element, U

is a reference velocity, Ωe is the triangle finite element area.

τSUPG =1√

1τ2

c+ 1τ2

t+ 1τ2ν

(3.8)

where: τc = he2∥ue

SUPG∥, τt = ∆t

2 , τν = h2e

4ν . he =2∥ue

SUPG∥3∑j∈e

ueSUPG.∇wj

is the maximum length of the

considered element according to its average velocity direction. ueSUPG = 1

33∑

j∈euj.

Stabilization terms: SUPG and PSPGIn fact, when the Galerkin FE formulation is applied for an incompressible flow, we mayface two problems of stability:Oscillations appear in the approximate solution of the velocity field: They appear for highflow Reynolds number because the convection becomes dominant so convective instabilitiesappear. In order to remove them, it is necessary that the Peclet number remains (Pe = ρuhe

ν 6 2)on all the elements of mesh (for a one-dimensional mesh) or SFELES uses the method ofSUPG (Streamline - Upwind Petrov - Galerkin) via which an artificial diffusive term is addedto the discretized equations of motion to eliminate these oscillations.Parasite pressure oscillations: They appear because of the combination of the shape functions


for speed and pressure, in our case we used the element (P1/P1). We can eliminate theseoscillations using a lower order shape function for the pressure than for the velocity (P2/P1or P2/P0) or SFELES uses the PSPG (Pressure Stabilised Petrov - Galerkin) approach viawhich a least squares term is added to the discretized equation of continuity.

Full discretization and the matrix form

A DFT in the spectral direction is performed on the finite element semi-discretization systemgiven in 3.4 to obtain the full space discretized system as the following:

∫Ω

[w.∂

Ûk

∂t– ∇.wPk +ν

(∇w : ∇

Ûk + (

2πkL

)2w.ˆ

Uk)

+

wz

(∂Uk

z∂t

+2πIk

LPk

)+ν

(∇wz.∇Uk

z + (2πkL

)2wzUkz

)+

q(

∇.ˆ

Uk +2πIk

LUk

z

)]dΩ+

ˆhk + hk

z+∫Γn

[w.(

(PkI –ν∇ˆ

Uk).n)

–νwz∇Ukz).n]dΓ + STk

SUPG + STkPSPG+

∑e

∫Ωe

τPSPG∇q.[∂

Ûk

∂t+ ∇

ˆPk +ν(

2πkL

)2 Ûk]dΩe = 0

(3.9)

All non-linear terms (convective terms, SUPG stabilization terms, convective contributions toPSPG stabilization terms) are computed using a pseudo-spectral approach as the following:

ˆhk = DFT

[∫Ω w.

((u.∇)u + uz

∂u∂z

)dΩ]

hkz = DFT

[∫Ω w.

((u.∇)uz + uz

∂uz∂z

)dΩ]

STkSUPG = DFT

[∑eτSUPG

∫Ωe

((u.∇)w.R + u.∇wzRz

)dΩe

]STk

PSPG = DFT[∑eτPSPG

∫Ωe

∇q.(

(u.∇)u + uz∂u∂z

)dΩe

] (3.10)

These terms are computed in the physical space and transferred back to the spectral spaceapplying a DFT transformation. The 2/3 dealiasing technique is used to avoid aliasing errorsin the pseudo-spectral treatment [73]. That means, if N modes are imposed for the continuousrepresentation of a variable in the periodic direction, only 2/3 N active modes are kept duringthe transformation from spectral to physical space and the other modes being set to zero.


The space-discretized equations can finally be written in a matrix form as:

MijdΦk

j

dt+ Lk

ijΦkj = –Ck

i (φ– N

2 +1j , ...,φ

N2j ) (3.11)

Φkj is the eight components vector in Fourier space, which contains the k-th mode for the

real and imaginary parts of the velocity vector and the pressure at node j.φ

pj is the four components vector in physical space associated with node j in the p-th finite

element plane.Where Mij is the mass matrix and it is linked to the time derivatives dependent terms, Lijcontains the linear terms of the NS equations, Ci contains the non-linear terms of the NSequations and the SUPG, SPSG terms. The reader is referred to [43, 73] for more detailsabout the hydrodynamic components of these matrices. As far as the temporal discretizationis concerned, the implicit Crank-Nicolson method is used for the pressure and diffusiveterms. To enhance the computational efficiency, all non-linear terms (convective terms, SUPGstabilization terms, convective contributions to PSPG stabilization terms) are treated explicitlyusing an Adams-Bashforth method. Because of the explicit treatment of all non-linear (modecoupling) terms, the resulting algebraic system to solve at each time step decomposes into aset of decoupled (2D) systems for each Fourier mode, which can be very efficiently solved inparallel.

MijQk,n+1

j – Qk,nj

∆t+ Lij

Qk,n+1j + Qk,n

j

2= –

32

Ck,ni +

12

Ck,n–1i (3.12)

or, in ∆-form [Mij

∆t+

Lij

2

]∆Qj = –LijQ

k,nj –

32

Ck,ni +

12

Ck,n–1i (3.13)

Since the physical variables are real, we can benefit from the following Fourier symmetry:q–k = [q]∗k where the * indicates to the complex conjugate, therefore we need to solve thefirst N/2 modes.

3.1.1 Turbulence modeling

Although there is no universal definition of turbulence, its main properties are the following[76]: it is an unsteady, unstable and unpredictable state of the flow. It is characterized bya wide range of spatial and temporal scales and large fluctuations of flow variables. Theinteraction between these scales causes irregularities and asymmetries. Turbulence is stronglyrelated to the non-linearities of the governing equations [78] and strong chaotic vorticity field


are observed.Richardson [79] proposed the energy cascade idea. The turbulence is considered as a set ofeddies of varying sizes. It states that kinetic energy enters the turbulence only at its largestscales of motion by the production mechanism. These large eddies are unstable, they breakup, transferring their energy to slightly smaller eddies. The smaller eddies undergo a similarbreak-up and transfer their energy to up to an eddy size where the viscous effects dominateand are converted into heat. Despite the evolution of the power of supercomputers, it is stillvery difficult and expensive to resolve all the spatial and temporal scales of turbulence. Thereare three aspects between which we should make a compromise according to our needs:

• The level of description: it corresponds to the quantity of information that we desire toaccess. It concerns large structures, small ones, or the whole structures in the flow.

• The accuracy: it depends on the numerical technique for the simulation and which cangive more or less accurate results.

• The computational cost: the time and resources required to perform the calculation.

There are three main approaches for turbulence modeling: DNS, RANS and LES whichis used in this study by SFELES. Figure 3.1 presents an example of turbulent combustionsimulations performed using the three methods. In DNS, no turbulence model is used, the

Fig. 3.1 Simulation of turbulent combustion a)-DNS, b)-LES, c)-RANS [77]

Navier Stokes equations are resolved directly therefore, all turbulence scales are calculatedexplicitly. The principle of RANS is to solve the Navier-Stokes equations for mean quantities,such that the only the mean motion is calculated. All the turbulence structures are modeled.The LES approach is a kind of compromise between DNS and RANS. The LES approach


is based on the concept of scales separation in turbulent flows by application of a filteringoperation such that the large scales of turbulence, which are anisotropic and affected by theboundary conditions, are separated from the small scales that are more isotropic, universaland affected by the viscosity [78]. Then the large scales are computed directly as in DNSwhile the small ones are modeled by a subgrid-scale stress model. The turbulent kineticenergy spectrum is represented in Fig 3.2 with comparison between the resolved scales in thethree approaches.

Fig. 3.2 Representation of the turbulent kinetic energy spectrum

Subgrid scale models

The main function of the Subgrid scale (SGS) model is to take into account the energytransfer between large and small scales. The eddy viscosity models are most commonlyused in the literature. In this study three SGS models have been used. These are the staticSmagorinsky’s model [75] and the WALE model [80] which are well known in the literature.The third model is a recent model proposed by Ghorbaniasl et al. [50], it is implemented inSFELES and validated in this study.

Smagorinsky’s model [75] is based on the Boussinesq assumption which uses the con-cept of eddy viscosity νt with the strain rate tensor of the filtered velocity field Sij to give theSGS Reynolds stress according to the relation:

τSGSij = 2νtSij +

13τSGS

kk δij (3.14)


with Sij = 12

(∂ui∂xj

+∂uj∂xi

). The Smagorinsky model assumes that νt is proportional to the filter

width ∆ and to the resolved shear according to the following relation:

νt = C2s .∆2.

∣∣S∣∣ (3.15)

∣∣S∣∣ =√

2SijSij is the resolved local strain rate, Cs ∼ 0.2 Smagorinsky constant. In the presentstudy, it is imposed to 0.17. This value is commonly used in the literature for exterior flowsas in [120–122]. The imposed constant value can be justified on a theoretical basis proposedby Lilly [118] supposing isotropic turbulence as the following:The resolved dissipation rate can be estimated as

ε = νt|S|2 = (Cs)2|S|3 (3.16)

Assuming that |S| can be estimated from the energy spectrum, we get

|S|2 = 2∫ kc

0k2E(k)dk ≈ 2

∫ kc

0k2K ε2/3k–5/3dk ≈ 3

2K ε2/3k4/3

c (3.17)

Lilly assumed that kc = π/ is the cut-off wavenumber in Fourier space, from Eq. 3.16 andEq. 3.17 we get:

Cs ≈1π

(2

3K

)3/4(3.18)

Using K = 1.5 as a value for the Kolmogorov constant, one gets the considered value of theconstant Cs ≈ 0.17. It is important to mention that near the wall, the eddy viscosity musttend to zero. This is accomplished in SFELES by applying the exponential damping functionof van Driest [81, 43].

WALE model [80] is based on the square of the velocity gradient tensor and accountsfor both the strain and the rotation rates of the smallest resolved turbulent fluctuations. Thefinal formulation of this model for the eddy viscosity is:

νt = (Cw.∆)2(Sd

ijSdij)

3/2

¯(SijSij)5/2 + (SdijS

dij)

5/4(3.19)

Where Cw is a constant whose value is in the range 0.55 ≤ Cw ≤ 0.60. Nicoud and Ducros[80] proposed that C2

w ≈ 10.6C2s which links WALE model constant to Smagorinsky constant

in the same flow conditions. Using the imposed value of Smagorinsky constant used inthis study, Cs ≈ 0.17, leads to Cw ≈ 0.553. So in the performed simulations, it is imposed


equal to 0.55. SdijS

dij = 1

6(S2S2 +Ω2Ω2) + 23S2Ω2 + 2IVSΩ with the notations S2 = SijSij,

Ω2 = ΩijΩij and Ωij = 12

(∂ui∂xj

–∂uj∂xi

)is the rotation rate tensor.

Comparing to the static Smagorinsky model, WALE model has the following advantages:

• The eddy-viscosity goes naturally to zero near walls so no damping function is needed.

• Both the local strain and rotation rates are included in the eddy viscosity formulation,thus all the turbulent structures are supposed to be detected by the proposed model.

Ghorbaniasl’s model [50] is a new formula for estimating the Smagorinsky constant Cs

which is supposed to be a function of the flow variables including the rotational and transla-tional velocities. The final formulation of this model is:

Cs = min(Gx,Gy,Gz) (3.20)

where Gx = |Ωxux|2D , Gy = |Ωyuy|

2D , Gz = |Ωzuz|2D and D =

√¯(Ωxux)2 + ¯(Ωyuy)2 + ¯(Ωzuz)2. Using

these relations, the constant Cs is computed dynamically in the sense that its values changewith time and space but without the need of a dynamic procedure, like the one used in thedynamic Smagorinsky model [82], which provides significant CPU time savings. Further-more, the eddy viscosity vanishes at walls and in irrotational flow regions through the modelcoefficient vanishing without the use of any damping function near walls. Another advantageof this model is the ease of incorporation to any code with structured or unstructured mesh incomparison with the dynamic Smagorinsky model, presented in the next paragraph.

Dynamic Smagorinsky model [82]The dynamic Smagorsinky model is presented here for sake of completeness, but has notbeen implemented in SFELES in the framework of this thesis. The dynamic Smagorinskymodel, which is proposed by Germano et al. [82] and developed by Lilly [83], is obtained byapplying a second filter, referred to as the test LES filter (denoted with an over-hat .), to theonce-filtered NS equations, by a grid LES filter, (denoted with an over-bar .). The applicationof the test filter generates another unknown residual stress tensor defined as:

Tij = uiuj – ˆui ˆuj (3.21)

Germano’s identity between the grid and test filtered fields, Lij = Tij – τij, with τij = uiuj –uiuj,is used to dynamically determine (Cs)2 in the Smagorinsky model. The tensor Lij can beexpressed in terms of the filtered or resolved velocity as well as in terms of the Smagorinskymodel.


In terms of the resolved velocity, Lij reads:

Lij = uiuj – ûi ûj (3.22)

In terms of the Smagorinsky model, the deviatoric portion of Lij can be expressed by applyinga test filtering as

τdij = –2(Cs)2 |S|Sij (3.23)

AndTd

ij = –2(Csˆ)2| ˆS| ˆSij (3.24)

In the previous equation, the strain-rate tensor ˆSij and its norm ˆ|S| are based on the double-filtered velocity ûi. The model coefficient (Cs)2 is computed by minimizing the square ofthe difference between the modeled Ld

ij and the resolved Ldij with respect to Cs [83] where

the difference is given asDij = Ld

ij – 2(Cs)2Mij (3.25)

whereMij = |S|Sij –β| ˆS| ˆSij (3.26)

β = (ˆ)2 is the square of the filter width ratio. The minimization procedure, leads to

(Cs)2 =12

[LdijMij]avg

[MijMij]avg=

12

[LijMij]avg

[MijMij]avg(3.27)

[]avg indicates to an averaging over homogeneous directions to prevent eddy viscosity frombecoming negative.

3.2 The acoustic solver, ACTRAN

ACTRAN, acronym of ACoustic TRANsmission, is a finite element program for modelingsound propagation, transmission and absorption in an acoustic, vibro-acoustic or aero-acoustic problems [87]. ACTRAN is being developed by Free Field Technologies. In thisthesis, it has been used to solve the acoustics and to provide the near and far fields propagationin a hybrid aeroacoustic context via Lighthill and Möhring analogies. These analogies havebeen detailed in the second chapter. Hereafter, the variational formulations of these two

3.2 The acoustic solver, ACTRAN 41

analogies are presented in the frequency domain and then some acoustic background ispresented.

3.2.1 The variational FE formulation of the acoustic analogies as im-plemented in ACTRAN

Lighthill’s analogy

Lighthill’s equation in frequency domain is obtained by applying a DFT transformation tothe Lighthill’s equation Eq. 2.7:

–ω2ρa – c20ρa =

∂2Tij

∂xixj(3.28)

Taking Na as a test function, the weighted residual formulation is then given as:

∫V

Na

[–ω2ρa – c2

0∂2ρa

∂x2i

]dV =

∫V

Na∂2Tij

∂xi∂xjdV (3.29)

Integrating the spatial derivatives by parts and using Gauss theorem, the weak variationalformulation is therefore written:

–ω2∫

VNaρadV –

∮sNac2

0∂ρa∂xi

.nidS +∫

V

∂Na∂xi

c20∂ρa∂xi

dV =∮sNa

∂Tij

∂xj.nidS –

∫V

∂Na∂xi

∂Tij

∂xjdV

(3.30)

Let us regroup the two surface integrals.

∮sNac2

0∂ρa∂xi

.nidS +∮

sNa

∂Tij

∂xi.nidS =

∮sNa

∂

∂xj[c2

0ρaδij + Tij]nidS (3.31)

Substituting Lighthill tensor Tij by its relation, Tij = ρuiuj +δij[(p – p0) – c20(ρ–ρ0)] –τij in

the precedent equation and using the momentum equation, Eq. 2.2, the surface integral istherefore given as: ∮

sNa

∂ρavi∂t

nidS (3.32)

And in the frequency domain:

iω∮

sNaρavinidS (3.33)


Replacing the precedent relation in the equation 3.30, it is arrived to the final finite elementformulation as implemented in ACTRAN as the following:

ω2∫

VNaρadV –

∫V

∂Na∂xi

c20∂ρa∂xi

dV = iω∮

sNaρavinidS +

∫V

∂Na∂xi

∂Tij

∂xidV (3.34)

The left-hand side (LHS) corresponds to an acoustic wave operator and the RHS is the sourceterm consisting of surface and volume contribution. The surface contribution representsa mass flow across the surface. It is not needed if the surface is fixed or vibrating in itsown plane so it is not considered in our study since the airfoil is fixed and not permeable.Neglecting the entropic sources and the viscous effects in the Lighthill stress tensor Tij asexplained in Chap. 2, the volume sources related to turbulent noise is reduced to:

∫V

∂Na∂xi

∂Tij

∂xjdV =

∫V

∂Na∂xi

∂ρvivj

∂xjdV (3.35)

Möhring’s analogy

Following the same steps as in the precedent paragraph and applying the simplifications ofthe sources terms as explained in chapter 2 (considering just the turbulent noise source termand neglecting the combustion noise, the viscous effects and the total density fluctuations)the final finite elements formulation of Möhring’s analogy in the frequency domain for anisentropic flow is given by:

–ω2∫

VNa

ρbρ2

Tc2dV + iω

∫V

Naρviρ2

Tc2∂b∂xi

dV –∫

V

∂Na∂xi

[iω

ρvibρ2

Tc2+ρvivj

ρ2Tc2

∂b∂xj

–ρ

ρ2T

∂b∂xi

]dV =

∫V

∂Na∂xi

DFT[ρ

ρT(−→v ×−→ω)i

](3.36)

3.2.2 The infinite elements

ACTRAN uses a combination of finite and infinite elements discretization to solve theacoustic problem on the whole domain. The acoustic near field (source region) is modeledwith the finite elements whereas the infinite elements are used to model the unboundedacoustic domain. Another objective of them is to prohibit the reflection of the acousticwaves into the finite element domain; they act as a non-reflective boundary condition and tocompute the sound pressure levels (SPL) in far field.An infinite element is defined by Burnett and Holford [92] as a finite element in which the


field variable, typically the acoustic pressure, is represented by a trial function over an entiresemi-infinite sector of space. Other researches and works on the infinite elements are done in[88–91]. The infinite elements are based on the multipole expansion of the wave equationsolution. The expansion order governs the accuracy and the convergence of the solution. Theexpansion order is a way to define a number of virtual nodes on the infinite edges of theinfinite elements. Increasing the expansion order allows modeling a more complex radiationpattern but it increases the cost of the problem. Higher order is required for high frequenciesand for turbulent flow in comparison with laminar flow case as we will see in the studiedcases in next chapters. Let us consider the Helmholtz problem as shown in Fig 3.3 where

Fig. 3.3 Helmholtz problem

Ω denotes the unbounded domain, Γ is the radiating surface, Γb a truncated computationalsurface at a radial radius r = b and Ωb denotes the computational domain between Γ and Γbor the finite elements domain. As mentioned, we are looking for the acoustic pressure in theexterior domain Ω whose complex amplitude p(x,ω) satisfies the Helmholtz equation in theabsence of mean flow such that:

∇2p(x,ω) + k2p(x,ω) = 0 (3.37)


where k is the wave number. The natural boundary condition is satisfied on Γ such as:∇p.n = –ρan with an is the normal surface acceleration and n is the outward unit normal. Asboundary condition in the far field, the Sommerfeld radiation condition has to be satisfied as:

limr→∞

r(∂p∂r

+ ikp)

= 0 (3.38)

This equation can be approximated for large distance r = R to give an impedance-typecondition as:

∂p∂r

= –ikp (3.39)

To obtain the variational statement of Helmholtz equation, it is multiplied by a test functionw(r,ω), integrated over ΩR and then integrated by parts. Taking into account the boundarycondition of Eq. 3.39 and the natural boundary condition it is arrived to the followingformulation: ∫

ΩR

(∇p.∇w – k2pw

)dΩR +

∫ΓR

(ikpw)dΓR –∫ΓρanwdΓ = 0 (3.40)

As stated by Holford [93], the complex pressure amplitude can be expanded in the regionexterior to the ellipsoid Ωb as a series of multipole, in the spherical polar system it takes theform:

p(r,θ,φ;k) = e–ikr∞

∑v=1

Hv(θ,φ)(r/b)v (3.41)

Hv(θ,φ) is a directivity function. e–ikr

r gives the expected far field pressure whereas the otherterms contribute to the near and intermediate fields. The sum in the precedent equation istruncated according to the infinite element order.

The infinite elements shape and test functions

One of the principal features of the infinite elements is that the trial functions in each elementare chosen so that they include asymptotic, far field solutions of the continuous problem.Generally, two approaches are used to construct the shape functions for the acoustic infiniteelements: the separable method and the mapped approach [94]. The separable formulationrelies on the edges or faces of the element aligned with iso-surfaces in an orthogonalcurvilinear coordinate system. In the mapped approach, the infinite element is transformedto a unit square or block. For a node γ, the associated shape function is defined as:

φγ(r,ω) =

br

e–k(r–b) (3.42)


The oscillatory nature of the shape function is clear. This function is able to represent exactlythe outward propagating component of the solution in the far field.Concerning the test function wγ, there are three formulations most commonly used:

• The unconjugated formulation: the test function is chosen to be the same as the shapefunction as:

wγ(r,ω) = φγ(r,ω) =

br

e–k(r–b) (3.43)

• The conjugated formulation: the test function is chosen to be the complex conjugate ofthe shape function as:

wγ(r,ω) = φ∗γ(r,ω) =

br

e+k(r–b) (3.44)

This formulation is used in ACTRAN.

• The Astely-Leis formulation: this formulation also uses the complex conjugate of theshape function but multiplied by a geometric weighting as the following:

wγ(r,ω) =

br

2φ∗γ(r,ω) =

br

3e+k(r–b) (3.45)

3.2.3 Mapping methods

In order to transform the sources from the CFD mesh to the acoustic mesh, a mappingprocess is needed. This is available in the tool ICFD integrated with ACTRAN. There aretwo methods of mapping: the sampling method and the conservative integration method.Figure 3.4 shows the difference between these two methods.In the sampling method, all nodal acoustic values are sampled into the CFD mesh. Then alinear interpolation from the CFD mesh to the acoustic nodes is performed. The tool iCFDidentifies the closest CFD cell to each acoustic node. Since the acoustic mesh is coarser thanthe CFD mesh, some information may be lost.In the integration method, the sources are integrated over the CFD nodes using the shapefunctions of the acoustic mesh nodes so all information is conserved in the process. Thismethod has been used in this study to avoid losing information about the noise sources. [96]


Fig. 3.4 Mapping methods a)-the sampling method, b)-the integration method [95]

3.3 Flow and acoustic characteristic scales, meshes build-ing criteria

The smallest turbulent scales to be resolved or modeled are characterized by the Kolmogorovlength scale η which is characteristic of the smallest eddies size dissipating the kinetic energy.Bailly et al [97] proposed a dimensional law linking the Kolmogorov scale η to the flowintegral scale L supposing a homogeneous and isotropic turbulence as the following:

Lη ∼ Re3/4

L or η≈ L.Re–3/4L with ReL = UL

ν

where U is a characteristic velocity scale of the flow, ν is the fluid kinematic viscosity andReL is the flow Reynolds number. In three dimensions simulation, the number of nodesnecessary to describe all scales characteristic of turbulence varies as:

N ∝ Re9/4L (3.46)

3.3 Flow and acoustic characteristic scales, meshes building criteria 47

In order to estimate the total cost we need to add the time discretization cost. The characteris-tic time scale associated to the Kolmogorov scale: tη = η

U . Defining tL = LU as a characteristic

time of the flow, the number of time steps n is then the ratio of the two characteristic times asthe following:

n =tLtη

=L

U.tη(3.47)

Considering CFL = c0∆t∆x ∼ c0tη

η ∼ O1 for a stable simulation, the total computation cost forthe turbulent flow is then proportional to:

N∗n ∝Re3

LM

(3.48)

M is the characteristic flow Mach number. This is the simulation cost for a DNS simulation,it is clear that it is not affordable for high Reynolds number. In the case of LES simulation,

used in this research, the cost is reduced to Re2L

M according to Bailly et al [97] where the effectof the smallest scales are modeled.The characteristic acoustic length scale is the acoustic wave length λ, it is linked to theintegral turbulence scale L as:

λ =c0f

=L

St.M(3.49)

The corresponding time scale:

T =1f

=L

St.U(3.50)

The cost thus is estimated as λL ∼ 1

M . Due to the difference in scales to be modeled betweenthe flow and the acoustics as it has been demonstrated (λ >> η), different criteria for buildingmeshes exist. In this study, concerning the CFD meshes, the criterion y+ < 1 is taken intoaccount, where y+ = y∗uτ

ν , with uτ =√

τwρ the friction velocity and τw the shear stress on the

wall. It means the refinement was done such that the first grid points of the surface lie withinone wall unit of the wall. The acoustic mesh is coarser than the CFD mesh, the criterion 6quadratic elements by an acoustic wave is taken in consideration basing on the maximumrequired frequency to be solved (2500 Hz).

Chapter 4

Flow regimes ofControlled-Diffusion Airfoils

The controlled-diffusion (CD) airfoil is a benchmark for preliminary assessments of aeroa-coustic numerical methods. The flow developing over the airfoil includes complex flowfeatures such as boundary layer detachment close to the leading edge, recirculation and tran-sition to turbulent boundary layer. To better understand the mechanics of noise generation,the different flow regimes with the Reynolds number are presented in the following sections,as an original contribution to the physics of controlled-diffusion airfoils.Before presenting the aerodynamics results, it is important to define the studied case, thecomputational domain, the CFD meshes, the imposed boundary conditions and the previousaerodynamics and experimental studies already performed for the CD airfoil. This is done innext paragraphs.

4.1 Description of the configuration

As mentioned earlier, the considered configuration is a controlled-diffusion airfoil from anautomotive cooling fan. This airfoil family can control effectively the deceleration of thefluid from the maximum speed point to the trailing edge thanks to its camber and thicknessdesign [98, 100]. The aim here being to minimize the separation of the boundary layer on thesuction side. This type of airfoil also reduces the losses related to the viscosity and reducesthe trailing edge noise whose intensity is proportional to the thickness of the boundary layer[99]. It has been chosen as a case study in this thesis on the one hand, because the turbulentflow around this airfoil includes complicated phenomena such as the leading-edge boundarylayer separation, shear layer transition to turbulent behavior, reattachment and trailing-edge

50 Flow regimes of Controlled-Diffusion Airfoils

Fig. 4.1 The automotive cooling package, its 9-blades fan and the airfoil of the blade. Figurereproduced from [57]

vortex shedding and on the other hand we have reference experimental aerodynamics andaeroacoustics data for it. Figure 4.1 shows the automotive cooling package, its 9-blades rotorand the profile of the blade, cut at mid-span. The airfoil thickness is 4%C, where C = 0.1356m is the airfoil chord and its camber angle is 12. The airfoil is set at 8 angle of attack.

4.1.1 The computational domain and CFD meshes

To check the convergence of the solution, two different meshes have been used in the CFDcomputations. The first mesh is structured with a similar level of refinement as the oneused by Wang et al. [69, 58] whereas the second mesh is unstructured. The size of thecomputational domain, illustrated in Fig 4.3, is 4C in the stream-wise direction (x) and 2.5Cin the transverse direction (y). In the first mesh, the domain was divided into triangularelements with approximately 160,000 triangular elements (80,000 nodes). This mesh isrefined around the airfoil and in the wake to capture small perturbations in the flow. Therefinement ensures that the first grid points of the surface lie within one wall unit of thewall (y+ < 1). The second mesh is an unstructured and has approximately 210,000 triangularelements (105,000 nodes). The refinement was concentrated around the airfoil, particularlyin the area around the leading and trailing edges because it was observed in the simulationsperformed on the first mesh that this region is very sensitive and there is a strong pressuregradient near the leading edge recirculation bubble and a small vortex shedding near the

4.1 Description of the configuration 51

trailing edge. The Lighthill and Möhring solutions are sensitive to these regions where thesources are computed. In any case, the effect of the mesh refinement on the aerodynamicresults will be presented in Chap. 5. Moreover, since the boundary layer is the most delicate

Fig. 4.2 The mesh refinement at the trailing edges and the boundary layer mesh. A)- the first(structured) mesh M1 , B)- the second mesh (unstructured) M2

part of the mesh, a boundary layer mesh is generated around the airfoil in the second meshusing Gambit. It consists of 20 layers with the rate (1.1), where the thickness of the first layeris (3e–4 m) corresponding to maximum values (y+ < 1) on the airfoil surface . Concerningthe coordinate system, it is fixed at the trailing edge of the airfoil such that the trailing edgecorresponds to x/C = 0 and the leading edge corresponds to x/C = –1, x is aligned withthe chord and pointing in the streamwise direction, y is crosswise. Figure 4.2 shows therefinement done around the trailing edge in the two meshes and the boundary layer mesh. Inthe remaining part of this dissertation, M1 will refer to the first (structured) mesh, and M2 tothe second (unstructured) mesh.


4.1.2 The Boundary conditions

Velocity was prescribed at the inlet (along the outer C boundary), zero pressure conditionwas imposed at the outlet and no-slip conditions were imposed on the airfoil as representedin Fig 4.3. Inlet velocity profiles on the restricted LES domain were extracted from RANS

Fig. 4.3 Representation of the computational domain with the Boundary conditions

computations. The first LES numerical study has been performed by Wang et al. [69, 58],using a LES hybrid finite differences/spectral code. Another recent study has been performedby Christophe [57], using LES OpenFoam finite volumes simulations. The two studies havebeen achieved on a sub-domain around the airfoil using velocities extracted from a RANSsimulation performed by Wang et al. [69] on a larger domain as boundary conditions on theLES domain inlet. This procedure was applied in order to reproduce closely experimentalconditions, performing a two-dimensional RANS simulation on a large computational domainrepresenting the main features of the open-jet wind tunnel configuration illustrated in Fig. 4.5(left). As shown in Fig. 4.5 (right), it includes the airfoil, the nozzle exit geometry and the jet.It has been shown by Moreau et al. [70] that there are significant differences between theresults obtained from a simulation performed on an isolated airfoil in a uniform stream andthat performed on an airfoil in an open-jet wind tunnel facility. It was assumed here that theairfoil mock-up spanwise extent was large enough such that the 3D effects were negligible.The same procedure is applied in the present work and the velocities, extracted from theRANS simulation are used as inlet boundary condition. The longitudinal and transversevelocities are interpolated on the meshes inlet nodes and plotted on Fig. 4.4. Furthermore, aperiodicity condition in the spanwise direction (z) is assumed in all 3D simulations performedin this study. In order to trigger 3D instabilities, some random perturbations are added tothe solution during some simulations for a period of 0.4 t* for all mesh nodes. This allows

4.1 Description of the configuration 53

Fig. 4.4 Inlet velocity profiles on the restricted domain, extracted from RANS computations:(left) longitudinal velocity and (right) transverse velocity. Figure reproduced from [69]

starting the turbulent motions where the intensity of the applied fluctuations is 0.1% for allsimulations.

4.1.3 Previous experimental and numerical studies of the CD airfoil

The first experimental and numerical study of the CD profile was performed by Moreau etal. in 2001 [70]. RANS is used to reproduce the mean pressure coefficient Cp measuredexperimentally. The simulation and measurements showed significant discrepancies becausein the simulation, the profile is considered isolated while in the tests, it is placed at the outletof a jet confined between two plates to adjust the angle of the attack. The interaction of thejet with the profile modifies the lift distribution [101]. As a result of these observations, thegeometry of the convergent was included in subsequent RANS simulations to capture thatinstallation effect. Roger and Moreau applied Amiet’s theory with its extensions based on thewall pressure spectra close the trailing edge to predict the noise radiated in free field [102, 53].Their results were in good agreement with experimental noise measurements performed bythe Turbulent Shear Flow Laboratory (TSFL) from the University of Michigan. The use ofLES to solve the flow around the CD airfoil is first done by Wang via a LES hybrid finitedifferences/spectral code [58]. These simulations have been achieved on a sub-domain aroundthe airfoil using velocities extracted from a RANS simulation as mentioned before. Withthis methodology, Wang managed to achieve a very good agreement with measurements interms of mean and fluctuating velocity and pressure. In 2005, Moreau performed other DES


and LES with the Star-CD solver [85]. LES has produced results comparable to experimentsbut the calculation was sensitive to the mesh refinement and quality. DES however, failed tocapture the transition from the laminar to turbulent regime near the leading edge where arecirculation bubble must exist as in the LES and experiments. Addad [65] then showed in2008 that improving the quality and refinement of the mesh with the same solver improvesthe results comparing with the experiments in terms of wall pressure spectrum. Usingcommercial finite volume solvers (Fluent and OpenFoam), Christophe studied the effectsof the mesh refinement, the boundary conditions and the solver [57, 66, 103]. He managedto reproduce correctly the mean pressure distribution over the blade with overestimationof the transition region where the recirculation bubble size is found equal to 5.3%C. Thewall pressure spectrum is underestimated at high frequencies which has direct influenceson the radiated noise estimation via Amiet’s model as it will be shown in Chap. 6. Anotherinteresting result in this research is that it has been proved that the convergence of thespanwise coherence near the trailing edge is really poor for low frequencies. The LES resultslargely overpredict the experimental coherence and present a peak between 500 and 1000[Hz] which does not exist in the experiments. In order to improve it and to study the effect ofthe spanwise extent, a LES simulation using Fluent has been carried out with a spanwise of0.3C. It has been shown that a larger spanwise is necessary to correctly capture the spanwisecoherence which is overpredicted in case of small extent due to the confinement of the flowbetween the spanwise boundaries. But even for a 0.3C span, the agreement is still not goodenough to be applied in the aeroacoustics theory of Amiet taking into account the very highcost of the simulation (three times more expensive in comparison with the case of 0.1 C span).Finally, Tannoury performed LES simulations in 2013 on the CD airfoil using the SC/Tetra[104]. The comparisons with experimental measurements are considered satisfactory. Aslight deterioration was observed by going from a hexahedral mesh to a tetrahedral mesh.This is due to a coarser resolution for the tetrahedral mesh at the wall. The experimentsperformed at ECL and TSFL on the CD airfoil are presented in the next paragraph.

Experimental database

The experiments were performed at Ecole Centrale de Lyon (ECL) by Moreau and Roger[51] and Moreau et al. [52] to provide the reference data for comparison with numericalsimulations results. After having highlighted the influence of the confinement of the jet onthe lift distribution of the profile, all subsequent experiments were made in the large anechoicwind tunnel of ECL. The airfoil mock-up chord length (C) is 0.134 m and its span is 0.3 m. Itis placed at the exit of the open-jet anechoic wind tunnel nozzle (as shown in Fig. 4.5 (left))and embedded in a 50 cm wide jet (≈ 3.7C). The tests were run with a speed U0 = 16 m/s,

4.2 Flow patterns around the CD airfoil according to flow Reynolds number 55

Fig. 4.5 (Left) Experimental Set-up of ECL Large Wind Tunnel (right) Representative figureof the RANS simulation of the Test Configuration

which corresponds to a Reynolds number based on the airfoil chord length Rec = 1.6∗105,and for an angle of attack of 8. The mean and fluctuating pressure are measured from 20Hz to 25 kHz via 21 flush-mounted remote Electret microphone probes (RMP) placed atthe mock-up midspan. Spanwise coherence lengths are measured by three additional RMPslocated in the spanwise direction near the trailing edge. The far field noise is obtained using asingle B&K 1/2′′ 1.27 cm Type-4181 microphone located in the midspan plane at a distanceof 2 m from the airfoil trailing edge.The experiment was reproduced on TSFL wind tunnel at the University of Michigan withan outlet section of 0.61 m2. Hot-wire measurements upstream the CD airfoil were made toensure that identical conditions to those of the ECL tunnel are applied [98]. The flow speedis U0 = 16 m/s, which corresponds to a Reynolds number based on the airfoil chord lengthRec = 1.6∗105, and the angle of attack is kept 8. Hot-wire measurements were acquired inthe airfoil wake, the boundary layer and at the LES domain boundaries. Simple sensors orX probes were used for these measurements based on information provided by the RANScalculations on the two or three dimensional nature of the flow at a given position.

4.2 Flow patterns around the CD airfoil according to flowReynolds number

This paragraph is devoted to study in details the flow patterns around the CD airfoil forReynolds number range 0<Re<60000 and hereafter the results. Concerning the performedsimulations set-up and the obtained results, it is important to mention the following details:All simulations in this chapter have been carried out on the second computational mesh


(M2). 16 Fourier modes have been imposed in the spanwise, periodic direction, for Reynoldsnumber higher than 3000. The static Smagorinsky turbulence model is used with Cs = 0.17.In order to trigger 3-d instabilities, some random perturbations are added to the solutionduring the simulation for a period of 3t∗ for all mesh nodes where t∗ = t.U0

C , U0 = 16m/s.This triggers the turbulent fluctuations, also in the spanwise direction. The intensity of theapplied fluctuations is 0.1% for all simulations. Once the turbulent perturbations are switchoff, a longer signal (more than 10t∗) was recorded and analyzed to determine the flow regime.For this and the following chapters, average or mean aerodynamic quantities mean that theyare obtained by averaging in spanwise direction and in time for more than 6t∗.

4.2.1 Attached flow (creeping) 0<Re<270

In the Reynolds number range of 0<Re<270, the flow is very slow and completely attached,no flow separation occurs, because of the very high viscosity. Figure 4.6 shows the contoursof the longitudinal velocity at Reynolds number of 50.

Fig. 4.6 Instantaneous contours of the longitudinal velocity U(1 : 3.25 : 14) at Re=50

4.2.2 Steady, separated flow 270<Re<1300

At Reynolds number of 270, the adverse pressure gradient becomes high enough to make theflow detached from the airfoil surface. A very small flow separation on the airfoil suctionside near the trailing edge has been noticed. At higher Reynolds numbers, until 1300, theflow separation becomes bigger modifying the outside potential flow and pressure field. Theflowfield remains steady in this Reynolds numbers range where no oscillations appear in thewake. The flowfield, at Reynolds number of 1250, is shown via the instantaneous contoursof the longitudinal velocity in Fig 4.7. Red color indicates positive values whereas the bluecolor indicates negative values.


Fig. 4.7 Instantaneous contours of the longitudinal velocity, red: U(1 : 3.25 : 14), blue:U(–0.7 : 0.1 : –0.1) at Re=1250

4.2.3 2-d unsteady laminar oscillating flow (vortex street) 1300<Re<6450

In the Reynolds number range of 1300<Re<6450, the downstream eddies and vortices becomeunstable, separate from the airfoil and are alternately shedding downstream. The alternateshedding is called the Karman vortex street and it occurs at a certain frequency (Strouhalnumber will be determined). This type of flow is periodic, it is 2-d unsteady but repeats itselfat some time interval. The pressure variation associated with the velocity changes produces asound called tonal noise as we will see later. This vortex shedding represents the major noisesource for this case. Figure 4.8 shows the contours of the vorticity in the spanwise direction(Ωz) at Reynolds of 2000. Blue lines indicate negative values and they shed from the suctionsurface of the airfoil whereas the positive values are colored by red and they shed from thepressure surface.

Fig. 4.8 Contours of the spanwise vorticity at Re=2000, red (15:7:50), blue (-50:7:-15)

4.2.4 3-d unsteady laminar oscillating flow 6450<Re<14000

Three dimensional structures start to appear at Reynolds number of 6450. Contours ofspanwise velocity at this Reynolds number are depicted in Fig 4.9. The 3-d structures becomemore and more important and the flow continue developing in the spanwise direction forhigher Reynolds numbers until 14000. These 3-d structures disrupt the regular 2-d shedding


Fig. 4.9 Contours of spanwise velocity at Re=6450. Red surfaces mark positive valueswhereas the blue surfaces mark the negative values

process and the spectrum of shedding frequencies is broadened as it will be shown whenthe lift coefficient and Strouhal number analysis are addressed. Figure 4.8 presents surfacecontours of the vorticity at Re=7000. Red and blue surfaces mark positive and negativevalues of the transverse vorticity whereas the green and yellow surfaces indicate positiveand negative surfaces of the streamwise vorticity. The periodicity in the spanwise directionis clear here. 3-d vortex shedding occurs, so the flow regime is therefore 3-d laminar andunsteady.

Fig. 4.10 Contours of vorticity at Re=7000. Red and blue surfaces mark positive (10:1:20)and negative (-20:1:-10) values of the transverse vorticity whereas the green and yellow markpositive and negative surfaces of the streamwise vorticity

4.2.5 3-d turbulent wake, 2-d laminar boundary layer regime14000<Re<47500

At a Reynolds number of 14000 the flow separation at the airfoil trailing edge becomessignificant such that the resulting downstream eddies and vortices become highly unstablecausing more important three-dimensional chaotic structures. The periodic flow startsbreaking down into a chaotic wake generating turbulent structures. The wake is fullyturbulent whereas the boundary layer is still laminar and 2D. These flow phenomena are clearon Fig 4.11 which displays the flow topology at a Reynolds number of 15000 representedby the surface contours of the vorticity magnitude colored by the longitudinal velocity. The


same flow regime applies up to Reynolds 47500 where the laminar flow separation appearsin the leading edge region.

Fig. 4.11 Contours of the vorticity magnitude (200) at Re=15000 colored by the longitudinalvelocity

4.2.6 Recirculation bubble appearance near the leading edge, laminarboundary layer Re=47500

At Reynolds number of 47500, a laminar separation region called recirculation bubble isgenerated near the airfoil leading edge. Its size is very small and its pressure level is low atthis Reynolds number but they become more important as the Reynolds number is furtherincreased from 47500 up to the Reynolds number of 50000 where it generates 3-d laminarstructures and very few turbulent structures on the airfoil surface as it will be explained in thenext paragraph. The regime in the Reynolds numbers range 47500<Re<50000 still features a3-d turbulent wake and a 2-d laminar boundary layer. Figure 4.12 shows the flow topology

Fig. 4.12 Contours of the vorticity magnitude (100:400) at Re=47500 colored by the longitu-dinal velocity

at Reynolds number 47500 represented by the surface contours of the vorticity magnitudecolored by the longitudinal velocity. The wake is fully turbulent with a small recirculationbubble which is visible on friction coefficient distribution, it is characterized by negative Cfvalues.


4.2.7 Recirculation bubble explosion, 3-d laminar periodic boundarylayer and turbulent wake 50000=<Re<52000

At a Reynolds number around 50000, the recirculation bubble created near the leading edgeexplodes, triggering 3-d laminar structures and very few turbulent structures in the boundarylayer which, for higher Reynolds number, becomes completely 3-d turbulent (Re>52000).Figure 4.13 shows the developed turbulent structures (above) described by the contours of Qcriterion defined as Q = 1

4(ω2 + 2SijSij). This figure also shows the 3-d laminar spanwise-periodic structures (bottom). So we can say the regime in this Reynolds numbers range is3-d laminar spanwise-periodic boundary layer and turbulent wake.

Fig. 4.13 (Top): Contours of the Q criterion (100) colored by the longitudinal velocity,(bottom): Contours of vorticity at Re=50000. Red and blue surfaces mark positive andnegative values of the streamwise vorticity whereas the green and yellow surfaces mark thetransverse vorticity

4.2.8 Fully turbulent regime Re>=52000

At Reynolds number 52000, the recirculation bubble created near the leading edge explodes,triggering the turbulent structures in the boundary layer which becomes for higher Reynoldsnumber completely 3-d turbulent. The boundary layer transition to chaotic turbulent flowwith vortices of many different scales being shed in the turbulent wake weakening the period-icity of the wake progressively with increasing Reynolds number in this range. The wakeis not quite as wide as for the previous cases because of the reattachment of the flow nearthe trailing edge resulting from the breaking of the shear layer coming from the laminarboundary layer on the upper surface.We can summarize here that the transition from laminar to full turbulent boundary layer forthe CD airfoil starts at Reynolds number of 52000 or higher. The initiation of 3-d turbulentstructures shed from the recirculation bubble in the boundary layer at Reynolds number


52000 is shown on Fig 4.14 which shows the flow topology represented by the surfacecontours of the vorticity magnitude. At Reynolds number of 60000, the boundary layer


becomes fully 3-d turbulent as shown in Fig 4.15.


All possible flow patterns around the CD airfoil according to flow Reynolds number aresummarized in Fig 4.16.

4.2.9 Pressure and friction coefficients distribution

As mentioned in previous sections, at a Reynolds number of 47500, a laminar separationstarts to appear in the presence of an adverse pressure gradient disrupting the aerodynamics ofthe profile. This bubble is characterized by negative friction values as it is shown in Fig 4.17.In this figure, a comparison of the mean pressure Cp and friction Cf coefficients are presentedat Reynolds numbers of 45000, 47500, 50000 and 52000 in order to better understand themechanisms responsible for the recirculation bubble appearance and then the transitionfrom the laminar to turbulent boundary layer regime. As it is clear at Reynolds number of45000 no negative Cf values are noticed. At Reynolds number of 47500 a small laminarrecirculation bubble occurs. Its size is almost 2%C. At a higher Reynolds number of 50000,the adverse pressure increases making the recirculation bubble explode triggering 3-d laminarstructures and very few turbulent structures in the boundary layer, the friction decreases atthe leading edge region and increases in the region near the trailing edge because the shear


Fig. 4.16 All flow patterns around the CD airfoil according to flow Reynolds number

layer produced on the upper surface becomes less strong, consequently, the separation region


Fig. 4.17 Average pressure and friction coefficients distribution on the airfoil surface

becomes less important. By increasing further the Reynolds number, the flow reattaches tothe airfoil when the turbulent structures are more intensive. At Reynolds number of 52000 theadverse pressure arrives to high level sufficient to make the bubble explode giving vorticesand triggering turbulent structures in the boundary layer. The recirculation bubble extension


is very big at this Reynolds number, it is almost 14%C as it is seen in the friction coefficientcurve. The transition to turbulent boundary layer occurs in the separated flow zone near theleading edge. The flow reattaches to the airfoil when it wins the kinetic energy required tooffset the effect of the adverse pressure gradient, creating the recirculation bubble. In Chap.5, more information about this particular phenomenon is presented, the effect of the meshrefinement and the SGS models on its size and on the corresponding pressure and frictioncoefficients in that region are studied and compared to other solvers and numerical methods.

4.2.10 Evolution of the lift and drag coefficients with Reynolds numberand the flow regime

The evolution of the lift and drag coefficients with the Reynolds number is studied andpresented in Fig 4.18. It is noticed that generally the lift coefficient increases with theReynolds number up to the Reynolds number of 50000 where it reaches the maximum value(1.02). For Reynolds numbers corresponding to the turbulent boundary layer regime, thelift coefficient almost takes a fixed value (50000-70000). The drag coefficient decreases byincreasing the Reynolds number.

Fig. 4.18 Evolution of the lift and drag coefficients with Reynolds number


4.2.11 Evolution of the Strouhal number with Reynolds number

The Strouhal number is a dimensionless number describing the oscillating flow mechanisms.It has the formula St = f∗d

U0where f is the frequency of the vortex shedding, d is a characteristic

length (it is taken as the airfoil trailing edge diameter in our case d/C = 0.0111078) and U0 isthe free-stream flow velocity. The evolution of the Strouhal number with respect to Reynoldsnumber is studied and plotted on Fig 4.19. As the figure shows generally, it increases withthe Reynolds number. However, when reaching the Reynolds number of 6450 where the flowregime starts to be 3-d as explained above, this causes a decrease of the Strouhal numberwhich means that the main shedding frequency is affected by the 3-d effects. Then, Strouhalnumber continues to increase and it takes its highest value (0.2695) at Reynolds number14000. By further increasing the Reynolds number, the Strouhal number and the sheddingfrequency decrease because, as shown before, from this Reynolds number the turbulentstructures grow up in the wake breaking the periodicity of the shedding and weakening itgradually.

Fig. 4.19 Evolution of the Strouhal number with Reynolds number (Rec)

The vortex shedding is responsible for generating a tonal component of the noise radiatedfrom the airfoil. The tones are associated to the dominant frequency (Strouhal of shedding)and its harmonics. In contrast, the turbulent structures generate broadband noise. For eachflow regime, should correspond a characteristic noise regime. In appendix G, the noisecomputed with Curle’s, Lighthill and Möhring approaches is presented for the Re=12000corresponding to a 3D laminar configuration.


In the Strouhal analysis, it could be interesting to compare the case of the airfoil with thecase of a circular cylinder since the trailing edge of the airfoil is circular and we havereference results for the cylinder [43]. To perform this comparison, it is based on thediameter of the trailing edge as characteristic length for Reynolds and Strouhal numberswhere d/C = 0.0111078. To achieve that, the evolution of the Strouhal number with Reynoldsnumber is reproduced basing on d (Red) and shown on Fig 4.20.Now the comparison is done and the results are shown in table 4.1. It is noticed that there are

Fig. 4.20 Evolution of the Strouhal number with Reynolds number Red

ReD The cylinder StD The airfoil Std140 0.180 0.263100 0.165 0.24260 0.135 0.210

Table 4.1 Comparison of Strouhal number between the airfoil and a circular cylinder

differences between the two cases. This can justified by the flow detachment occurring towardthe trailing edge of the airfoil which causes the formation of a recirculation region. Thisregion behaves as a rigid mass of fluid forcing the flow to turn around it and start shedding.Therefore, the flow turn around this region and not around the trailing edge with a velocity,around this area, different from the undisturbed free stream velocity as in the case of thecylinder. So, the diameter which may taken into account is the diameter D shown in Fig 4.21


which is the diameter of the trailing edge plus the diameter of the recirculation region and thevelocity can be computed approximately in the region in front of the recirculation region.

Fig. 4.21 The recirculation region around the trailing edge with the supposed diameter D

Chapter 5

Turbulent flow over CD airfoil(Re=160 000)

The first step in the hybrid aeroacoustic approaches is to solve the flow field in order todetermine the noise sources. The accurate prediction of the broadband noise using thesemethods is directly linked to the accurate determination and computation of the aerodynamicsources. The aerodynamic results need to be validated before being used to build the aeroa-coustic sources terms. A comprehensive and detailed aerodynamic analysis is presented hereusing the LES solver SFELES. The averaged pressure and friction coefficients distribution,boundary layer profiles, wall-pressure spectra, wake mean and turbulent velocities are com-puted and compared with experimental results and LES OpenFoam computational results.The experimental results were obtained by Moreau and Roger [51] and Moreau et al. [52]whereas OpenFoam simulations results have been obtained by Christophe [57]. In somecases, the results are also compared to the results obtained by Wang et al. [58] using a hybridfinite differences/spectral solver. Three sub grid-scale models have been used to study theinfluence of the LES SGS models on the results. Those are the static Smagorinsky modelwith Cs = 0.17, the WALE model with Cw = 0.55 and Ghorbaniasl’s model already describedin Chap. 3 while its derivation is presented in Appendix C. In order to study the influenceof some flow parameters on the results, more than ten simulations of the CD-profile wereperformed by changing the number of spanwise Fourier modes, the computational mesh, thespanwise extension and the SGS model, according to table 5.1.

70 Turbulent flow over CD airfoil (Re=160 000)

The computational meshes M1 and M2, mentioned in the table 5.1, are the same pre-

Simulation Fourier modes CFD mesh Time step (sec) SGS model Span z/C

1 64M M1 ∆t = 5.10–6 Smagorinsky 0.12 32M M1 ∆t = 5.10–6 Smagorinsky 0.13 64M M1 ∆t = 10–6 WALE 0.14 64M M1 ∆t = 5.10–6 Ghorbaniasl 0.15 32M M1 ∆t = 5.10–6 Ghorbaniasl 0.16 64M M2 ∆t = 10–5 Smagorinsky 0.17 64M M2 ∆t = 10–5 WALE 0.18 64M M2 ∆t = 10–5 Ghorbaniasl 0.19 32M M2 ∆t = 10–5 Smagorinsky 0.1

10 128M M2 ∆t = 10–5 Smagorinsky 0.2

Table 5.1 The simulations performed on the CD airfoil at Re=160000

sented and used in Chapter 4 (Fig 4.2).

5.1 Evolution of Ghorbaniasl’s model constant Cs

The spatial and temporal evolution of Ghorbaniasl’s constant is studied in two positions in thecomputational domain. The first is in the first half of the airfoil (x/C=-0.6) and the second isnear the trailing edge (x/C=-0.02) and for two given instants 9t* and 16t* as shown in Fig. 5.1.We can observe that the Cs amplitude is always positive and the maximum value is 0.14.This is another advantage of this model while, with the dynamic Smagorinsky model, theamplitude can be negative and result in numerical instabilities [50]. Ghorbaniasl’s constantdecreases away from the surface and falls to zero which means that this model enforces thezero contribution from the sub-grid scales in irrotational flow regions and in the viscoussub-layer. In the following sections, the aerodynamic results of the simulations performed onthe first computational mesh M1 will be presented followed by those obtained on the secondmesh M2 to study the mesh refinement effects.

5.2 Flow topology 71

Fig. 5.1 Evolution of Ghorbaniasl’s model constant CS: a) x/C=-0.6 for 64 and 32M at 9t*,b) x/C=-0.02 for 64 and 32M at 9t* , c) x/C=-0.6 for 9t* and 16t* for 64M, d) x/C=-0.02 for9t* and 16t* for 64M

5.2 Flow topology

Figure 5.2 shows the flow topology described by the contours of the Q criterion withQ = 1

4(ω2 + 2SijSij). The figure depicts the level of vorticity and the size of turbulentstructures in the flow at a given instant for the simulations performed on the first structuredmesh M1. The turbulent flow around the profile at Re=160000 is characterized by a laminarboundary layer on the lower surface and a turbulent and transitional boundary layer on theupper surface. It is also noticed that there is a recirculation bubble near the airfoil leadingedge as expected from Chap. 4 and from the literature. This recirculation bubble triggersa fast transition towards a fully turbulent boundary layer. It is found that the bubble size


Fig. 5.2 Flow topology of simulations performed on M1 described by the criterion Q(Q. C2

U20

=

1000) and colored by the longitudinal instantaneous velocity with the model: a) Smagorinsky32M, b) Smagorinsky 64M, c) Ghorbaniasl 32M, d) Ghorbaniasl 64M, e) WALE 64M

highly depends on the solver and the SGS model as shown in the table 5.2. In this table, therecirculation bubble size is compared to the present simulations and for several simulationsalready performed on the CD airfoil including different solvers, numerical methods andSGS models. It is clear that LES with the static Smagorinsky model overpredicts the bubblesize regardless of the numerical method used in the solver whereas the results with thedynamic Smagorinsky and Ghorbaniasl models, have a more reasonable recirculation bubblesize. The recirculation is predicted by RANS simulation, but not by DES technique. Sincewe do not have experimental data for the friction coefficient to determine precisely therecirculation bubble size, it has been defined using the size of the plateau region on thepressure coefficient distribution whose size is found to be (2-3)% C. The vorticity is moreimportant for 64 Fourier modes than for 32 modes and there are many more structures in thiscase, because more information is resolved about the flow. The figure shows that the natureof the developed turbulent structures is largely influenced by the size and the structure of therecirculation bubble. In the simulation with Ghorbaniasl’s model, the turbulent structuresare finer and more intense than for the simulation with the Smagorinsky model but thetotal energy convected at the trailing edge is almost the same as it will be shown when the

5.3 Pressure and friction coefficients distribution on the airfoil surface 73

Author Method/Solver SGS model Bubble sizeStatic Smagorinsky 32M 9.8%CStatic Smagorinsky 64M 8.4%C

Present LES FEM spectral/SFELES Ghorbaniasl 32M 3.55%CGhorbaniasl 64M 3.5%C

WALE 64M 7%CChristophe LES FV/OpenFoam Dynamic Smagorinsky 5.26%C

[57] LES FV/Fluent Dynamic Smagorinsky 5.04%CWang et al. LES FD/spectral (Re=150000) Dynamic Smagorinsky 3.7%C

[69, 58] RANS SST k –ω 3.1%CMoreau et al. LES FVM/STAR-CD (Re=120000) Static Smagorinsky 11.2%C

[85] DES FVM/STAR-CD (Re=120000) k –ε 0

Table 5.2 Recirculation bubble size: comparsion with other simulations

wall-pressure spectrum is computed (see Fig. 5.7). Furthermore, a small vortex sheddingfrom the pressure side is noticed. It is more important with Ghorbaniasl’s model, especiallyfor 64M. This shedding depends largely on the angle of attack of the profile and the freestream velocity [84]. The shedding is responsible of the most important noise mechanism. Anarrow-band or tonal contribution is added to the overall sound [84]. This component is notconsidered in the sound predictions in this study. In the following paragraphs, the results ofthe simulations with 64M are just considered where the effect of the number of spanwiseFourier modes will be addressed for the second mesh M2.

5.3 Pressure and friction coefficients distribution on theairfoil surface

The mean pressure and friction coefficients distributions on the airfoil are shown in Fig. 5.3.As far as the pressure distribution is concerned, first considering the Smagorinsky and WALEresults, one observes a very good agreement with experimental results as well as OpenFoamand Wang simulations except near the leading edge where the results are influenced bythe size of the recirculation bubble which is larger than in OpenFoam and Wang results asmentioned in table 5.2. In contrast, for Ghorbaniasl’s model, we have a very good agreementwith OpenFoam and Wang results in this region. As far as skin friction on the upper surfaceis concerned, there is a region toward the leading edge in which the friction coefficient isnegative, this region corresponds to the recirculation bubble. A zoom in this region is donein Fig. 5.4 to show clearly the flow detachment and reattachment points locations (defined


Fig. 5.3 (Above): Average pressure coefficient distribution (Cp) on the airfoil surface,(bottom): Average friction coefficient (Cf) distribution

5.4 Boundary layer velocity profiles 75

by Cf = 0) for all cases. SFELES results with Gorbaniasl’s model predict almost the samereattachment point as Wang’s and OpenFoam simulations whereas there is a small differencein the detachment point location. Downstream of the reattachment, the present results areseen to be in good agreement with the OpenFoam and Wang results, in particular for thesimulation with 64 M using Ghorbaniasl’s model. The skin friction on the lower surface isuniform and less important because the boundary layer is laminar.

Fig. 5.4 Zoom of the leading edge region on (Cf) curves characterizing the recirculationbubble size for all simulations

5.4 Boundary layer velocity profiles

The average longitudinal velocity profiles in the boundary layer have been extracted inseveral positions along the airfoil (x/C = –0.6, x/C = –0.32, x/C = –0.14 and x/C = –0.02)as illustrated in Fig. 5.5. It is noticed that the thickness of the boundary layer developedon the upper surface increases from the leading edge to the trailing edge. One observeson Fig. 5.5 that the boundary layer evolution along the suction side is well predicted. Agood agreement with the OpenFoam simulations is obtained, it is noticed that the results arealmost identical in terms of boundary layer thickness and overall profiles especially in thesections near the trailing edge and for Smagorinsky and WALE models. In contrast, whenthe comparison with the experimental data is considered, SFELES results as well as the LES


Fig. 5.5 Average velocity profiles in the boundary layer on the upper surface at sections a)x/C = -0.6, b) x/C = -0.32, c) x/C = -0.14, d) x/C = -0.02

OpenFoam simulations tend to show a poor agreement in terms of boundary layer thicknessand overall profiles. To match correctly the experimental boundary-layer profiles, it hasbeen demonstrated by Christophe [57] that a larger spanwise extent (0.3C) is required, butleads to more computationally expensive simulations. In the present analysis, two spanwiseextensions have been simulated, demonstrating that 0.2C improves the velocity profile.

5.5 Wall pressure spectra 77

5.5 Wall pressure spectra

The pressure time signal is recorded and averaged in spanwise direction in two positions nearthe trailing edge on the suction side: x/C=-0.08 and x/C=-0.02 during a period of 6t* for allsimulations. Figure 5.6 shows the pressure fluctuations for the Smagorinsky and GhorbaniaslSGS models. The wall-pressure spectra are computed and expressed as the Power Spectral

Fig. 5.6 Pressure fluctuations on the airfoil suction surface at the positions: x/C=-0.08,x/C=-0.02

Density (PSD), of the pressure fluctuations, which is a function of a random sequence P(t),


Fig. 5.7 Power Spectral Density of pressure fluctuations at the positions: x/C=-0.08, x/C=-0.02

named Spp(f) describing how its mean squared value σ2 is distributed in frequencies:

σ2p =

+∞∫–∞

Spp(f)df =+∞∫0

Gpp(f)df (5.1)

5.6 The average velocity profiles in the wake 79

Spp(f) and Gpp(f) are respectively two-sided and one-sided Power Spectral Density. It is

expressed in[

pa2

Hz

][108]. The results are then plotted and compared with experimental,

OpenFoam and Wang computations results in Fig. 5.7. In order to express the PSD in [dB],

the following transformation is applied: SPL = 20log10(√

Gpp(f)P0

) where P0 = 2 ∗10–5[pa]

is the reference pressure. Welch method is applied using an averaging over segments of 29

points and with a multiplication by a Hanning window and 50% overlap between the segments.At the two positions, the results provide an excellent agreement with the experiments, asOpenFoam computations, especially for the low and mid frequency ranges, for the three SGSmodels with a difference up to 2dB. In contrast, Wang’s results show an overestimation upto 6dB for the first position. At high frequencies, SFELES results have similar trends andare closer to experimental results than the other simulations results where Wang’s resultsunderestimate the pressure spectra with a difference up to more than 10dB. There are smalldifferences between the SGS models, Ghorbaniasl and Smagorinsky curves are closer to theexperiments. In contrast, WALE model curve has higher values for high frequencies. It isnoticed that the recirculation bubble size has no important influence on the trailing-edge wallpressure spectra. The wall pressure spectrum is one of the primary inputs of Amiet’s theoryfor predicting the trailing-edge noise.

5.6 The average velocity profiles in the wake

Longitudinal (U) and normal (V) average velocity profiles, and the longitudinal (u′) andvertical (v′) velocity fluctuations RMS profiles were extracted in different positions in thewake of the blade from x/C = 0.0574 to x/C = 1.0133 as Figures 5.8, 5.9, 5.10 and 5.11illustrate respectively. Figure 5.8 illustrates the longitudinal velocity U profiles. It is noticedthat in the first Figure which corresponds to the position (x/C = 0.0574), there is a sharp peakthat can be explained by the fact that the disturbances are concentrated in a small area at thetrailing edge. Moving away from the trailing edge, the peaks become smaller and smallerand the wake thickness increases, due to the turbulent diffusion. The velocity V illustrated inFig. 5.9 can be explained in the same way. As far as the velocity fluctuations u′ and v′, shownin Figures 5.10 and 5.11 are considered, there are two peaks one corresponds to the pressureside and the other to the suction side. The two peaks are higher in the sections closest to thetrailing edge and they become smaller and smaller and their thickness become larger andlarger away from the trailing edge. There is a good agreement with OpenFoam results atmost sections and it is remarked that the results of SFELES are closer to the experimentalresults except for the two first sections of Fig. 5.10.


Fig. 5.8 The longitudinal (U) average velocity profiles in several positions in the wake


Fig. 5.9 The normal (V) average velocity profiles in several positions in the wake


Fig. 5.10 The longitudinal (u′) velocity fluctuations RMS profiles in several positions in thewake


Fig. 5.11 The vertical (v′) velocity fluctuations RMS profiles in several positions in the wake


5.7 Stresses in the turbulent boundary layer and the lawof the wall

The stresses in turbulent boundary layer decomposes into 3 contributions: τxx τxy τxz

τyx τyy τyz

τzx τzy τzz

= 2ν

Sxx Sxy Sxz

Syx Syy Syz

Szx Szy Szz

+

+ 2νt

Sxx Sxy Sxz

Syx Syy Syz

Szx Szy Szz

–ρ

u′u′ u′v′ u′w′

v′u′ v′v′ v′w′

w′u′ w′v′ w′w′

+ 2ρkSGSδij

(5.2)The first contribution is the viscous stress, the second is the sub-grid scale part correspondingto the effect of unresolved scales and the third represents the resolved Reynolds stress. νt

is the SGS kinematic viscosity, ν is the fluid kinematic viscosity, Sij is the rate of averagestrain tensor. u′, v′ and w′ are the turbulent velocity components, δij is the Kronecker symboland 2kSGS = u′u′ + v′v′ + w′w′ . Since the dominant component is τxy, only this part willbe considered in the following. The stresses are calculated and presented in Fig. 5.12 forthe three SGS models in the position x/C=-0.7. It is noticed that viscous stress has the samecontribution for the three SGS models whereas there are differences for the other two stresscomponents. Th subgrid scale stress term has a very small contribution except in the closevicinity to the wall. Its contribution is more important for WALE model than the two othermodels. Its maximum contribution is almost 7% whereas it is found equal to 4% for theSmagorinsky model and 2.5% for the Ghorbaniasl model, which means that the contributionof LES is more important for WALE model in this case. Considering the resolved Reynoldsstress, it is more important for Ghorbaniasl model, the peak found equal to 1.08 whereas it isfound equal to 0.97 for the Smagorinsky model and 0.87 for the WALE model. However itdrops to zero sharper than the two other cases, this is linked to the difference in recirculationbubble size which influences the boundary layer thickness and stresses in the region near theleading edge (bigger recirculation bubble size causes higher stresses in the boundary layeraround the leading edge, this will be clearer when stresses are computed for the mesh M2).The stress τtn is given as:

τtn = τxycos(2θ) + (τyy –τxx)sin2(θ

2) (5.3)

5.7 Stresses in the turbulent boundary layer and the law of the wall 85

Fig. 5.12 The viscous, Reynolds and SGS stresses (τxy) in the section x/C=-0.7

θ is the angle between the normal and y axis in the section x/C = –0.7. Since θ is very smallat this position, τtn and τxy are found the same so just τxy is shown here.As far as the law of the wall is considered, according to Tennekes and Lumley [109], theboundary layer region can be divided into three different regions:- Viscous or laminar sub-layer: y+ < 5In this zone the governing law is u+ = y+. At the surface all the stress is viscous stress andthe velocity fluctuations do not contribute much to the total stress because of the viscosity.- Buffer layer: 5 < y+ < 30


The region where neither one of the stresses can be neglected is called the buffer layer. Thisis the region where the linear velocity in the viscous layer is linked to logarithmic velocityprofile in the inertial sub-layer.- Inertial sub-layer or the logarithmic region: 30 < y+ < 300The profile of the mean velocity in this zone is logarithmic. It is represented by the followinganalytic relationship:

u+ = κ∗ log10(y+) + B (5.4)

where u+ = Uavguτ is the dimensionless velocity with uτ =

√τwρ representing the friction

velocity and τw stands for the wall shear stress. κ ≈ 0.41 is the von Karman’s constant,B≈ 5.2 is constant. y+ = yn∗uτ

ν is the dimensionless wall distance with yn the normal distanceon the wall.The relationship between u+ and y+ has been extracted from the performed simulationsand shown in Fig. 5.13 for the three SGS models. It is noticed that there is a very goodagreement with the analytic relation in the laminar sub-layer zone for the three models. In thelogarithmic zone only the Ghorbaniasl model reproduces the linear behavior of the velocityu+ as the analytic relation whereas its value is slightly overestimated for the other two models.

Fig. 5.13 The law of the wall in the section x/C = -0.7

5.8 Spanwise pressure coherence function and length 87

5.8 Spanwise pressure coherence function and length

The coherence function is essentially the two-point correlation coefficient in the frequencydomain, defined as:

γ2(ω,z) =

∣∣φ(ω, z1,z2)∣∣2∣∣φ(ω, z1,z1)

∣∣ ∣∣φ(ω, z2,z2)∣∣ (5.5)

where φ(ω, z1, z2) and φ(ω, z1, z1) denote the cross-power spectral density and the auto-power spectral density of pressure fluctuations respectively. The coherence length is theintegral of the coherence function over the span:

lz(ω) = limL→∞

∫ L

0γ(ω,z)dz (5.6)

This relation is not used in the present study due to convergence issues already demonstratedin several LES studies as in [57, 110, 111] because of the very small spanwise extensionimposed in the simulations, so coherence length is calculated via an approach based onexponential function:

γ(ω,z) = e– |z|lz(ω) (5.7)

which is found to be a robust alternative of the integral relation [110, 111].We need to compute the coherence function and length for two reasons:- To verify whether the imposed airfoil spanwise in the simulations is sufficient to ensure thatthe resolved turbulent structures are decorrelated within the span (the coherence should reachzero when reaching the edges of the domain and the spanwise extension need to be largerthan the coherence length).- The acoustic numerical results are compared to experimental data obtained by Moreau andRoger [51] and Moreau et al. [52]. In the experiments, the spanwise was 0.3C whereas in thepresent LES it is equal to 0.1C. Therefore the ratio Lz–exp

Lz–LESequal to 3 in our case. In order to

predict the sound pressure radiated by the full span as experiments, the spanwise coherencefunction and length of the wall pressure fluctuations needs to be evaluated.According to Wang and Moin [112], the sound pressure spectrum radiated by the full spanLz–exp can be rescaled by one of the following cases:

• If Lz–LES > lz, the spanwise extension of the computational domain is larger than thecoherence length of the acoustical source therefore the acoustic sources in each span-wise domain are considered to radiate in a statistically independent manner. Therefore,the total sound spectrum is the linear summation of the contributions of each spanwisedomain, resulting in Lz–exp

Lz–LESindependent sources. That means Stot

pp = [ Lz–expLz–LES

].Spp.


• If Lz–LES < lz, the acoustic source is coherent along the entire span of the forward stepsurface and the source could be considered 2-D so it is rescaled by Stot


]2.Spp.

In order to identify the wall pressure coherence function and length, the pressure in spanwisedirection is sampled along a line in the position x/C = –0.02. The results are presented inFig. 5.14 for the three SGS models. The coherence length is compared to Corcos model[86]. As one can notice, the coherence function drops considerably and converges to zero athigh frequencies. No significant difference is noticed between the three SGS models at highfrequencies, while at low frequencies, the coherence function with WALE model drops fasterthan the other two models. The spanwise coherence length matches acceptably with thedecay and values of the empirical model of Corcos especially at the high frequencies. Whencomparing the two spanwise separations for ∆z/C = 0.02 and ∆z/C = 0.04, it is observed thatthe coherence decays for high separations. In our LES simulation, the spanwise extension is0.1 C. The results show that the imposed spanwise is sufficient to ensure that the resolvedturbulent structures are correlated within a fraction of the span (coherence length) which islower than the spanwise dimension for most frequencies of interest (200-2000 Hz). ThereforeStot


].Spp is expected to be suitable to rescale the radiated noise in our study. Thistheory is proved and confirmed before its use in our research by computing and comparingthe noise of two spanwise extensions for 0.1 C and 0.2 C via Curle’s formulation as it will beshown in Chap. 6.

5.8 Spanwise pressure coherence function and length 89

Fig. 5.14 Spanwise coherence function and length of the fluctuating pressure on the suctionside at x/C=-0.02


5.9 Spatial convergence

In this section the mesh refinement effects on the aerodynamics results are studied using thesecond computational mesh M2 which is unstructured, finer than mesh M1 and contains aboundary layer mesh as presented in Chap. 4. The spanwise discretization is also analyzedby comparing simulations with 64 and 32 transverse Fourier modes. The effects of theSGS models have been already addressed using the mesh M1, the present comparisons areperformed only with the static Smagorinsky model.

5.9.1 Flow topology, pressure and friction coefficients distribution

Figure 5.15 shows the flow topology described by the contours of the Q criterion at a giveninstant for simulations performed on the second un-structured mesh M2. Comparing to the


U20

=

1000) and colored by the longitudinal instantaneous velocity with the model: a) Smagorinsky32M, b) Smagorinsky 64M, c) Ghorbaniasl 64M, d) WALE 64M

flow topology obtained using the first structured mesh M1, we can notice that they are almostsimilar except in the region near the leading edge where the laminar separation occurs. Thesimulations based on mesh M2 are less sensitive to the SGS model than the simulationsperformed on mesh M1, close transition region sizes are found for the three SGS as shown in

5.9 Spatial convergence 91

the table 5.3 but with the Smagorinsky model, as on the first mesh M1 and in the literature,the overestimation always exists. This can be justified by the high refinement done in theboundary layer when imposing the boundary layer mesh and the region around it which leadsto less LES contribution and therefore less differences between the SGS models concerningthe transition phenomenon. Considering the comparison between the two simulations with64M and 32M using Smagorinsky model, it is clear that the intensity of vortex structures inthe turbulent boundary layer is much more important for 64M than with 32M. The transitionsize is bigger for the simulation with 32M than with 64M. The mean pressure and frictioncoefficients distribution on the airfoil using the mesh M2 are computed for the three SGSmodels and plotted on Fig 5.16 which presents a comparison with that obtained using themesh M1 with the Smagorinsky SGS model, and between the simulations with 64M and 32Mperformed on the mesh M2 with the Smagorinsky SGS model. The Cp and Cf coefficients

Simulation’s model Smag/32M Smag/64M WALE/64M Ghorbaniasl/64MBubble size as %C 6.2 4.3 3.5 3.5

Table 5.3 Recirculation bubble size obtained on mesh M2: comparsion for the 3 SGS models

are very similar for the three SGS models, the simulation with WALE model exhibits lessfriction on the suction surface. Comparing to mesh M1, for the Smagorinsky model, it isclear that the recirculation bubble size is much smaller in this case (4.2%C vs 8.4%C) andthis leads to pressure and friction differences in this region. Apart from the leading edgeregion, we have almost identical curves. The same thing is noticed when the simulations with64M and 32M are compared, the transition region size is found equal to 4.2%C and 6.2%Crespectively for 64M and 32M simulations. It is also noticed that we have less friction in theregion around the airfoil center on the suction surface, this can be linked to the lack of smallvortex structures in this region which confirms that 32M are insufficient to model enoughsmall vortices comparing with the simulation with 64M.


Fig. 5.16 (Above): Average pressure coefficient distribution (Cp) on the airfoil surface usingmesh M2, (Bottom): Average friction coefficient (Cf) distribution using mesh M2


5.9.2 Boundary layer velocity profiles

The average longitudinal velocity profiles have been extracted in two positions around theairfoil trailing edge (x/C = –0.14 and x/C = –0.02) and plotted on Fig. 5.17. As a comparisonbetween the two meshes, the velocity profiles using the unstructured mesh still show a pooragreement in terms of boundary layer thickness and overall profiles in comparison withexperimental results despite the little improvement obtained in the upper part of the boundarylayer (y/C > 0.028).

Fig. 5.17 Average velocity profiles in the boundary layer on the upper surface at sections x/C= -0.14 and x/C = -0.02

5.9.3 Stresses in the turbulent boundary layer

Stresses in the turbulent boundary layer are compared between the two meshes for thesimulations with Smagorinsky model and 64M. Considering the SGS and viscous stresses,they are very similar. In contrast, a significant difference is noticed considering the Reynoldsstresses, they are less important for the unstructured mesh. They do not act in the upper layerof the boundary layer (y/C > 0.02). They drop sharply to zero at (y/C = 0.01) on this meshwhereas on the structured mesh they act until the thickness (y/C = 0.04) where they becomenull. We can link these differences to the corresponding difference in the recirculation bubblesize formed near this position towards the leading edge because bigger transition size causesmore important stresses acting for bigger distance in the boundary layer.


Fig. 5.18 The viscous, Reynolds and SGS stresses in the section x/C=-0.7

5.9.4 Wall pressure spectra

The wall pressure spectrum is compared for the two computational meshes at the positionx/C = –0.02 and depicted on Fig. 5.19. The results of the unstructured mesh present anoverestimation up to 5 dB for low frequencies and an underestimation up to 5 dB for highfrequencies in comparison with the results obtained on the structured mesh. These resultsare noticed to be identical to those obtained by Wang for frequencies higher than 700 Hzwhereas for lower frequencies there are some differences. The number of transverse Fouriermodes does not play a significant role below 4000 Hz, in contrast at higher frequencies thepressure spectra for 32M are similar to those obtained using the first mesh and closer toexperimental data.

5.9.5 Spanwise pressure coherence function and length

The coherence function and length are compared for the two meshes on Fig. 5.20. As for thefirst mesh, the coherence function drops towards zero for high frequencies and shows lower


Fig. 5.19 Power Spectral Density of pressure fluctuations at the position x/C=-0.02

Fig. 5.20 Comparison of the spanwise coherence function and length between the twocomputational meshes M1 and M2

values for bigger spanwise separation ∆z/C = 0.04 in comparison with ∆z/C = 0.02. As acomparison between the two meshes, the coherence function computed on the second mesh


M2 presents almost similar results for the spanwise separation ∆z/C = 0.04, whereas forthe separation ∆z/C = 0.02 the results show higher values with a peak at a frequency range[1000-1500] Hz which does not exist using the first mesh M1. Considering the coherencelength for the spanwise separation ∆z/C = 0.02, the results are very similar and fluctuatingclose to the Corcos’s model curve except over the frequency range [1000-1500] Hz where apeak exists.

5.10 Spanwise extension effects, z/C = 0.2


U20

=

1000) and colored by the longitudinal instantaneous velocity for a spanwise extension: a)Smagorinsky 128M, z/C = 0.2, b) Smagorinsky 64M, z/C = 0.1

In this section the influence of the spanwise extension on the results is studied. A simu-lation has been carried out on the unstructured mesh M2 using the Smagorinsky model fora spanwise extension z/C = 0.2 with 128 Fourier modes in the direction of periodicity andcompared to the simulation performed as well on the mesh M2, spanwise extension z/C = 0.1with 64M and the Smagorinsky model. Figure 5.21 shows the flow topology for the twocases. The same vortices form, size and intensity are obtained. The recirculation bubbleformed near the leading edge has also the same structure and size as it is clear on Fig. 5.22which presents the mean pressure and friction coefficients curves. As it is shown, the curvesare identical so the spanwise extension has no effect on the average pressure and frictiondistribution on the airfoil surface nor on the leading edge transition region size.

The average longitudinal velocity profile in the boundary layer has been extracted inthe position x/C = –0.02 near the trailing edge and plotted on Fig. 5.23. One observes thatthe velocity profile has been improved in general and it is closer to the experimental data for

5.10 Spanwise extension effects, z/C = 0.2 97

Fig. 5.22 (Left): Average pressure coefficient distribution (Cp) on the airfoil surface usingmesh M2, (Right): Average friction coefficient (Cf) distribution using mesh M2

Fig. 5.23 Average velocity profiles in the boundary layer on the upper surface at section x/C= -0.02


the spanwise extension z/C = 0.2 than the extension z/C = 0.1. The boundary layer thicknessis very well predicted and it is identical to the experimental value.

The wall pressure spectrum is compared for the two spanwise extensions at the posi-tion x/C = –0.02 on Fig. 5.24. The results show that the pressure spectrum has been improvedat low and mid frequencies until 2000 Hz to be closer to the experimental spectrum than withthe extension z/C = 0.1. In contrast, at higher frequencies the results are similar.

Fig. 5.24 Power Spectral Density of pressure fluctuations at the position x/C=-0.02

Stresses in the turbulent boundary layer have been extracted in the section x/C = –0.7 andcompared for the two meshes on Fig. 5.25. The average stresses curves are almost identicaland no significant difference is noticed.

Spanwise pressure coherence function and length are finally compared for the two span-wise extensions as shown in Fig. 5.26. The coherence function curves show lower valuesand faster convergence to zero for the extension z/C = 0.2 comparing to the case z/C = 0.1subsequently, closer coherence length curve to the Corcos’s model is found.

5.10 Spanwise extension effects, z/C = 0.2 99

Fig. 5.25 The viscous, Reynolds and SGS stresses in the section x/C=-0.7 for the twospanwise extensions

Fig. 5.26 Comparison of the spanwise coherence function and length between the twospanwise extensions

Chapter 6

Aerocoustics results

In this chapter, the results of the aeroacoustics methods applied on the CD airfoil are presentedand compared to the experimental data. The results of the broadband noise computed viaAmiet’s theory, Curle’s formulation and Lighthill’s analogy at a Reynolds number of 160000are presented in details.

6.1 Broadband noise at Reynolds number of 160000

6.1.1 Amiet’s aeroacoustics theory

The following far-field acoustic pressure PSD formulation is applied:

Spp(X,ω) = (sinθ2πR

)2.(kC)2d.|L |2Φpp(ω)ly(ω) (6.1)

The parameters are defined in Chap. 2 in details. L is the global aeroacoustic transferfunction given as

L = L1 +L2 (6.2)

where L1 is the transfer function of the trailing edge and L2 is the back-scattering leadingedge correction. L1 and L2 formula are presented in the Appendix B. Φpp(ω) is the wall-pressure power spectral density in a point located near the airfoil trailing edge. It is computedat the position x/C = –0.02 already presented in Chap. 5. ly(ω) is the spanwise correlationlength near the trailing edge. The input data are represented by the wall-pressure spectrumand the spanwise correlation length. For convergence reasons of the correlation length, thesemi-empirical Corcos’ model [86] is used as in the literature [55–57]. The correlation length

102 Aerocoustics results

obtained from the LES simulations, using an approach based on an exponential function Eq.5.7, is used. The results are presented and compared for the two cases.

Implementation of Amiet’s theory in Matlab

Amiet’s formulation, Eq. 6.1, is implemented in Matlab as a part of this research. Theparameters of this equation are defined in Chap. 2. The aeroacoustic transfer functionsformulations, shown in Appendix B, B.43 and B.44, are obviously the most important termsin the implementation. So, first, they are implemented, their directivity patterns are studiedand compared to those obtained in the literature by Moreau & Roger [84] to validate theimplementation. Figure 6.1 shows the transfer function directivity patterns, considering

Fig. 6.1 Transfer function directivity patterns for parallel and supercritical gusts for theconsidered airfoil, trailing edge formulation, (left) main scattering term L1 (right) leading-edge back-scattering correction L2

parallel and supercritical gusts, given as∣∣∣kC.L . z

σ0

∣∣∣ for the dimensionless frequencieskC = 1,5,10,20,40 for mid-span plane. In the left figure, the main scattering term L1 isconsidered while the right figure shows the leading-edge back-scattering correction L2(presented on smaller scale). It can be noticed that the sound directivity has two symetriclobes at low frequencies. In contrast, at high frequencies, the lobes multiply and bendtoward the leading edge for the main scattering term and toward the trailing edge for theback-scattering term. However, a noticeable effect on the radiation is just produced, by

6.1 Broadband noise at Reynolds number of 160000 103

the correction term, at low frequencies whereas, as expected, the leading-edge correctioncontribution vanishes at high frequencies. These results and observations agree very wellwith similar calculations by Moreau & Roger [84].

Trailing edge sound using coherence length computed by Corcos’ model

Figure 6.2 shows the PSD of the sound pressure level at the considered observer. Consideringthe above figure without the leading-edge back-scattering correction, we have a very goodagreement with the experimental results with a discrepancies up to 2 dB at low frequencies,which is expected because the actual chord length is not yet considered. On the bottom figurewhere the leading-edge correction is applied, the results are improved for low frequenciesbetween 100 and 300 Hz as expected. The back-scattering correction has no importantinfluence on the results for high frequencies. It is clear that SFELES results provide verygood agreement with experimental results, being actually closer to experimental data thanOpenFoam in particular for frequencies between 1500 and 2000 Hz. Comparing with Open-Foam results, the results are in good agreement in the range 100-1500 Hz and then SFELEScurves have higher values but closer to experimental results (1500-2000 Hz), this is linkeddirectly to the differences in the wall-pressure spectra estimation, (see Fig. 5.7). The noisecomputed by Wang [58], using the Ffowcs Williams and Hall [38] theory, is compared onthe same figure (Fig. 6.2). At frequencies in the range 500-2000 Hz, the present resultsare very similar to Wang’s results whereas for frequencies smaller than 500 Hz, SFELEScurves have higher values but closer to experiments. At high frequencies, above 2000 Hz,the difference is larger but experimental results are not available. As far as the comparisonbetween the three SGS models is concerned, one can notice that there are no importantdifferences because the wall-pressure spectra are very similar, the difference does not exceed2 dB for the frequencies of interest (100-2000 Hz). The directivity patterns of the acousticpressure are presented in Fig. 6.3 for the three SGS models. The dipolar behavior is clearwhere there are two big dominant lobes, they multiply and orient towards the leading edge forhigh frequencies. As a comparison between the considered SGS models, similar directivitiesare logically found since directivity is imposed by the sin(θ) term in Eq. 6.1, corrected bythe analytical aeroacoustic transfer function.


Fig. 6.2 Trailing edge sound using Amiet’s theory for the three SGS models (above) withoutthe leading-edge correction (bottom) with the leading edge correction. The receiver is placedin the mid-span plane above the trailing edge (R=2 m, θ = 90o)


Fig. 6.3 The noise directivity patterns [dB] for a): Ghorbaniasl’s model, b): WALE model,c): Smagorinsky model


Trailing edge sound using Amiet’s theory with coherence length extracted from LESsimulation

In the literature, the coherence length required for the Amiet’s formula is computed usingthe Corcos’ model. In the present section, the LES results are used to compute the pressurecoherence length to improve the results and bypass the use of Corcos’ model. The coherencelength is computed assuming the exponential decay for better convergence over a small span.The results are depicted on Fig. 6.4. It can be observed that the proposed approach providevery good agreement with the experimental SPL for the three SGS models without significanteffect of the SGS model at frequencies higher than 400 Hz. In contrast, at lower frequencies,the spanwise extension effect appears to cause significant discrepancy in comparison withthe experimental data. This is directly related to the differences observed in the coherencelength for the different turbulence models.

Fig. 6.4 Trailing edge sound using Amiet’s theory using coherence length extracted fromLES simulations

6.1.2 Curle’s integral formulation

Curle’s formulation (Eq. 2.14) has been applied in this study proposing two approaches;the first approach is the implementation of the volume (quadrupole) and surface (dipole)integrals of the relation 2.14 in the CFD solver SFELES to be calculated simultaneously with


the flow in order to avoid the storage of noise sources which requires a huge space. The timederivatives are computed using the second order central difference method via Matlab as apost-processing then the DFT transformation is performed. Instead of evaluating the integralsin the retarded time (t – r/c0), it is accounted for approximately in the post-processing via theaccumulation time method for the observer [123]. In this approach, the source time becomesthe reference time and the observer time is approximated by:

tobserver =[(

tsource + r/c0)

/t]

.t (6.3)

where [] denotes the integer value function and t denotes the time step used for timeadvancement of the flow. This method is used in the literature as in [124, 125].In the second approach, the fluctuating aerodynamic forces, already obtained during theaerodynamics simulation, are used to compute the noise considering just the surface sourcesof the equation 2.14 as follows in the far field:

pa(x, t) u –xj

4π |x|2 c0

∂

∂t

[∫S

pnidS]

te(6.4)

The integral in the precedent relation represents the aerodynamics loads, for thin airfoils:

∂

∂t

[∫S

pnidS]

te=∂Fi∂t

(Y, te) (6.5)

Therefore the relation 6.4 is written:

pa(x, t) u –xj

4π |x|2 c0

∂Fi∂t

(Y, te) (6.6)

where Fi(Y, te) the aerodynamic forces on the airfoil evaluated at the retarded time. Thisrelation is known as Gutin’s principle for compact and rigid body [126]. It can be written as:

pa(x, t) u –1

4πc0r2

[x.∂D(Y, te)

∂t+ y.

∂L(Y, te)∂t

](6.7)

where x = r.cos(θ) and y = r.sin(θ). r = 2m and θ = 90 are the receiver coordinates. In thisstudy the receiver is placed in the mid-span plane above the trailing edge as mentioned before.The aerodynamic forces are written:

D(t) = 0.5∗ρ∗U2∞ ∗S∗Cd(t) (6.8)

L(t) = 0.5∗ρ∗U2∞ ∗S∗CL(t) (6.9)


For the drag and lift forces respectively. The time history of the aerodynamics coefficientsCd and CL is already stored during the aerodynamic simulation so this relation is applied bya simple post-processing procedure by Matlab. The aerodynamic coefficients consist of twoparts, one related to the pressure and the other to the viscosity, which is neglected here.

Cd(t) = Cdp(t) + Cdν(t) (6.10)

CL(t) = CLp(t) + CLν(t) (6.11)

The influence of the SGS model is studied by comparing the obtained SPL for the threeSGS models: Smagorinsky, WALE and Ghorbaniasl. The results are compared with theexperimental data and with those obtained by Christophe [57] using the acoustical solverVirtual.Lab [114] which considers only the dipole surface sources of Curle’s formulation.It is important to mention that using this solver, the storage of unsteady pressure evolutionon the airfoil is needed during the CFD simulation carried out using LES OpenFoam solverwith the dynamic Smagorinsky model.In next paragraphs and on corresponding figures, it will be referred to the proposed approachesfor Curle’s formulation as the following:

• First approach: Curle: overall noise (implementation in SFELES).

• Second approach: Curle: dipole noise (aeroforces).

The far-field acoustic pressure obtained using the approach, Curle: overall noise (im-plementation in SFELES):

The SPL perceived at the considered receiver using the first approach (Curle: overall noise)is shown on Fig. 6.5. A very good agreement is obtained with the experimental data for thefrequencies of interest [200-2000] Hz. No significant influence of the SGS model on the radi-ated noise spectra is noticed for frequencies up to 2500 Hz. In contrast, at higher frequencies,Ghorbaniasl’s model presents higher values than WALE and Smagorinsky models but closerto Christophe Virtual.Lab results. The two curves have the same trends which is justified bythe effect of the SGS model constant since the two models are dynamic models (Cs evolveswith time and space). Because of the source compactness supposed as an approximationwhen this method has been derived in its integral formulation, it is subsequently restrictedto low frequency application. For the CD airfoil case, it is valid for frequencies less than2507 Hz, as presented in Chap. 2. The SPL results have been presented for frequencies up10000 Hz to show the effect of the SGS model since the difference is noticed to occur at highfrequencies and as a kind of validation as a similar tendency is obtained using the Virtual.Lab


Fig. 6.5 Curle: overall noise (implementation in SFELES), (CFD Mesh M2)

solver. The rise of the spectrum at frequencies above 2500 Hz, for Gohrbaniasl’s model, maybe also justified by the fact that the compactness assumption is not valid anymore so theresults are questionable.

The contribution of surface (dipole) and volume (quadrupole) sources to the overallSPL

At low Mach number configurations and for acoustically compact solid boundaries, it is welldocumented that the scattered (dipole) field dominates the direct (quadrupole) field as shownby Gloerfelt et al. [127] and by Hardin & Lamkin [128].For the considered case, the contribution of the surface (dipole) and volume (quadrupole)sources to the overall noise are studied for the three SGS models and shown on Figures6.6, 6.7 and 6.8 for Smagorinsky, WALE and Ghorbaniasl models respectively. It can benoticed that the surface source contribution is dominant, as expected, however, the volumecontribution is important as well and not negligible. The volume source contribution issignificant at high frequencies (from 1000 Hz) for Smagorinsky and WALE models whereasfor Ghorbaniasl’s model, it is important for almost all frequencies of interest [200-2000] Hz.This is somewhat unexpected based on the literature but it has never been addressed, for


Fig. 6.6 Surface and volume sources contributions to the overall far-field acoustic pressurefor Smagorinsky’s model

Fig. 6.7 Surface and volume sources contributions to the overall far-field acoustic pressurefor WALE model


Fig. 6.8 Surface and volume sources contributions to the overall far-field acoustic pressurefor Ghorbaniasl’s model

the case of an airfoil, because the storage of the volume sources is very costly, contrary tothe proposed approach. In any case, considering the quadrupole contribution improves theoverall noise, especially at high frequencies, and makes it closer to the experimental SPL, asshown in Fig. 6.5.

Quantification of retarded time effects

In order to quantify the retarded time effects for the considered application, a comparisonof SPL is performed between the case where the retarded time is included and that whereit is neglected considering the simulation with Smagorinsky’s model. The result is shownin Fig 6.9. It can be noticed that the retarded time effect is not significant and limited tolow frequencies, therefore it can be neglected. So, it will not be taken into account in thefollowing sections (results of computational mesh M1, chord 0.2C and the aerodynamicsforces approach).


Fig. 6.9 Retarded time effects on the far field SPL, for Smagorinsky’s model

CFD mesh influence/sensitivity on the acoustic pressure

Fig. 6.10 Curle: overall noise (implementation in SFELES), (CFD Mesh M1)


Since the CFD mesh is used to compute the acoustics and since the mesh criteria aredifferent, the mesh sensitivity is important to be studied by applying the proposed approach,Curle: overall noise (implementation in SFELES), to the two CFD meshes used in this thesis.The results presented above are obtained using the second unstructured mesh M2. Hereafter,the far-field acoustic pressure using the structured CFD mesh M1 is plotted on Fig. 6.10for the three SGS models. The results are very similar to those obtained using the meshM2 with some differences at high frequencies over than 2500 Hz (Fig. 6.5). This is linkedto the mesh refinement where the mesh M2 is finer than M1 as explained before in Chap.4. This confirms that the proposed approach is not very sensitive to the CFD mesh for thefrequency domain of application. This is reasonable since the acoustic mesh is coarser thanthe CFD mesh generally. Figure 6.11 shows a comparison of the SPL using the two meshesfor Ghorbaniasl’s model.

Fig. 6.11 Comparison of the SPL obtained using the two CFD meshes M1 and M2 forGhorbaniasl’s model

Spanwise extension influence, validation of the scaling theory of Wang and Moin [112]

The far-field acoustic pressure is computed for a 0.2C spanwise extension and comparedwith the 0.1C spanwise extension. This is done to verify whether the scaling theory proposedby Wang and Moin [112] is applicable in the present case. According to this theory andto aerodynamics results concerning the spanwise coherence length, it is expected that the


Fig. 6.12 Comparison of the SPL obtained with 0.1C and 0.2C spanwise extensions forSmagorinsky’s model (before scaling)

Fig. 6.13 Comparison of the averaged SPL obtained for 0.1C and 0.2C spanwise extensionsfor Smagorinsky’s model (before scaling)


airfoil with 0.2C span will radiate 3 dB more than the 0.1C span case which means that thespan 0.2C can be divided into two sources that radiate the noise in an independent way froma statistics point of view. The SPL results are depicted on Fig. 6.12 for the two spanwiseextensions and then averaged on Fig. 6.13 to get the comparison easier. We can see that themean difference is around 3 dB for most frequencies which are higher than 300 Hz. Forlower frequencies, the discrepancy is less but the theory still applicable with an acceptableapproximation. These results confirm the consistency of the applied scaling method and thecomputation of the coherence length using the logarithmic approach.

The far-field acoustic pressure obtained using the approach, Curle: dipole noise (aero-forces):

The second approach of Curle’s formulation, which is based on the aerodynamics forcesand takes into account only the surface source contribution, is applied here. The result ispresented in Fig. 6.14 for the three SGS models. It is clear that this approach providesacceptable approximation for frequencies up to 1500 Hz. At higher frequencies up to 2000Hz, the acoustic pressure spectrum is underestimated with a value up to 4 dB. This is becausethe volume contribution, which is shown to contribute significantly at high frequencies, isnot taken into account in this approach. The acoustic pressure spectrum obtained via thisapproach is compared with that obtained using the first approach considering just the dipolecontribution on Fig. 6.15 for the Smagorinsky’s model. We can see that the two curves arefully matching each other. This confirms that the proposed approach can provide acceptableresults for frequencies up to 1500 Hz and without any additional computational efforts.Finally, the results obtained applying Curle’s formulation via the two proposed approachesmatch in an acceptable approximation the experimental and the Virtual.lab acoustic solverresults. The non-sensitivity of the CFD mesh and the SGS models at the frequencies ofinterest is demonstrated. The application of this formulation shows that it is a simple, notcostly (in terms of CPU time and memory or storage), reliable formulation and can give afast idea about the far field radiated noise from airfoils with the limitations and restrictionsmentioned in Chap. 2 such as compactness source region (valid up to moderate frequencies),usable only for exterior noise problems in free field, etc. . . .


Fig. 6.14 Curle: dipole noise (aeroforces), (CFD Mesh M2)

Fig. 6.15 Comparison of the surface contribution obtained using the two approaches forSmagorinsky’s model, (CFD Mesh M2)


6.1.3 Lighthill’s analogy

Lighthill and Möhring (for laminar flow case) analogies have been applied in the context ofthe hybrid methods of aeroacoustics in which the the aerodynamic and acoustic fields aredecoupled and simulated separately via SFELES and ACTRAN solvers respectively. Thefinal formulation of Lighthill’s analogy used via ACTRAN in this research is the following:

∂2ρa

∂t2– c2

0∂2ρa

∂x2i

=∂2 [ρ0uiuj

]∂xi∂xj

(6.12)

The Right Hand Side (RHS) represents the sources, they are computed and extracted duringthe aerodynamic simulation. In order to extract and to store the primary variables (velocityand/or pressure) to be used for building the Lighthill and Möhring sources via ACTRAN, thetransient format Ensight Gold, supported by ACTRAN, is implemented in SFELES. Thisformat builds 3-D fields and geometry from the 2-D modes solutions and the original CFDmesh. The reader is referred to [107] and Appendix E for more details about this format. Theprocess applied for the complete hybrid approach is shown in Fig. 6.16.

Fig. 6.16 Hybrid approach of aeroacoustics using SFELES CFD and ACTRAN acousticssolvers


Fig. 6.17 The 3-D CFD geometry and mesh with a zoom on the corner, built by Ensight Goldformat, spanwise extension is 0.1C

We can summarize the process by the following steps:First, the aerodynamic simulations have been carried out, on the well refined CFD meshM2, using the SFELES solver, the imposed span is 0.1C. The reader is referred to Chap. 5and to [117] where some of the aerodynamic results have been recently published by theInternational Journal of Aeroacoustics (IJA). When the simulation reaches to a statisticallyconverged solution, the velocity components are extracted in the Ensight Gold format on


Fig. 6.18 The 3-D acoustic mesh, the imposed spanwise extension is 0.3C

every point of the reconstructed 3-D CFD mesh, shown in Fig. 6.17, for more than 5 airfoilflow-through times (almost 0.1 sec) to cover frequencies up to 2500 [Hz].Second, the sources are projected from the reconstructed 3-D CFD mesh on the acousticmesh, shown in Fig. 6.18, using the integration method which conserves the sources byintegrating them on the acoustic mesh elements [116]. This is performed by ICFD toolavailable in the ACTRAN suite. Then the Lighthill and Möhring volume source terms arecomputed in the time domain and a Fourier transform (DFT) is applied to convert them tofrequency domain. As explained in Chap. 5, the spanwise extension in the experimentsis 3 times larger than that imposed in the aerodynamic simulations, this is done to avoidthe very high cost of the CFD simulation. It has been shown that the imposed spanwiseextension in the simulation (0.1C) is sufficient to obtain acceptable results and to ensurethat the resolved turbulent structures are well correlated within a fraction of the span in areasonable and affordable cost. Then, the theory proposed by Wang and Moin [112], canbe used to predict the sound pressure radiated by the full span (0.3C) as it has bean alreadyapplied to rescale the SPL results of Curle’s formulation. Here, it is proposed to have anacoustic mesh with the same spanwise extension as in the experiments. It means to build amesh of 0.3C spanwise extension composed of three slices of 0.1C thickness. Since the flowis periodic in the spanwise direction and the turbulent structures are decorrelated within the0.1C spanwise extension, this implies that the sources are repeated in the same way on thethree slices. Therefore, it is possible to project the same aerodynamic sources on the three


slices of the acoustic mesh. This procedure is applied at this stage. Furthermore, the acousticmesh is smaller in dimensions in the plane (x,y), this is done in a way to enclose the sourcesregion, and its elements are coarser than in the CFD mesh, the criterion 6 quadratic elementsby acoustic wavelength is taken in consideration based on the maximum required frequencyto be solved (2500 Hz).Third, the acoustic simulation is performed via ACTRAN to compute the radiated noise,represented by the sound pressure level (SPL) in the near and the far fields. Figure 6.19presents the simulation set-up. As it can be seen, the configuration includes the acousticmesh, a Field maps plane of a dimension 1.4*1.4 m in order to show the acoustic wavespropagation to the far field, outside the finite elements domain. A group of 25 microphones(including the considered receiver) are placed around the trailing edge on a semi-circle of 2m radius in order to obtain the directiviy patterns. The SPL in far field is computed using

Fig. 6.19 ACTRAN acoustic simulation setup, 1): The acoustic mesh, 2): Field maps planeof a dimension 1.4*1.4 m, 3): A group of 25 receivers for the directivity, 4): The consideredreceiver

the Infinite Elements which provide the solution at any point of the outer domain. Infiniteelements are used to model non-reflecting boundary conditions and the solution outside thefinite element region. The Infinite Elements are applied on the small 4 surfaces of the acousticmesh (located in the (x,z) and (y,z) planes) whereas on the other two surfaces parallel to


the airfoil section (located in the (x,y) plane), a rigid plate condition is imposed, as in theexperiments. The proposed acoustic approach with the boundary conditions, rigid plates onparallel surfaces, is not a propagation of channeled or guided acoustic since there is no ductproperly defined: it lacks two transverse surfaces to have a duct. Thus, it only reveals modesbeyond the so-called cutoff frequency. The time signal (0.1 sec) is subdivided in 9 intervalsof 0.02 sec with 50% overlap between segments during the Fourier transform analysis. Thisis similar to the process applied on turbulent experimental signals, except that the number ofsamples is generally much higher for real signal processing.

Lighthilll sources on the acoustic mesh, near and far field acoustic pressure maps

Truncation of the Lighthill source terms at the domain boundaries can create spurious soundespecially at low frequencies. This is called truncation phenomenon which pollutes theacoustic field with artificial sources and causes an overprediction of the acoustic pressurefield. Figure 6.20 (above) presents this phenomenon obtained in our study for the turbulentcase flow where an artificial dipole is visible close to the domain outlet for the frequency300 Hz. In order to prevent this phenomenon, a spatial filter of cosine type is applied nearthe borders. The source terms are damped out slowly toward the sources boundaries by theapplied filter as shown in Fig. 6.20 (middle). In this Figure an amplitude value of 1 (redcolor) means that the sources in that area are 100% taken into account whereas 0 value (bluecolor) means that the area is not taken into account at all. The result of the application ofthis filter is shown in Fig. 6.20, it is clear that the artificial dipole has been removed fromthe acoustic field. The spatial filtering is applied in all acoustic simulations performed viaACTRAN in this study. The filter distance needed to ensure the perfect elimination of thisphenomenon is 0.3 C for the turbulent flow cases. The truncation phenomenon effects arequantified and shown in Fig. 6.27. This Figure presents a comparison of the SPL obtainedwith and without the application of the filter. Without spatial filtering, the noise level isoverestimated due to spurious sources present at the end of the computational and related tostrong wake’s structures leaving the domain.The sources, near and far field maps are presented hereafter considering just the contributionof one loadcase. The second loadcase, located in the middle of the acoustic mesh is cho-sen. The maps are presented on a cut-plane located at mid distance between the surfacessupporting the airfoil. Figure 6.21 shows Lighthill sources computed on the acoustic meshfor frequencies 400 Hz, 800 Hz, 1200 Hz and 1600 Hz respectively. It is clear that mainnoise sources come from the turbulent boundary layer and wake. For high frequencies, thesources are less efficient. The corresponding near field acoustic pressure maps are shownin Fig. 6.22. These maps show the mechanism of the trailing-edge noise generation by the


turbulence. At low and mid frequencies, the airfoil radiates more important noise than athigh frequencies which is in accordance with the experimental observations. It is noticed,as expected, that the center of acoustic radiation is the trailing edge. The acoustic wavespropagation from the sources region to the far field is shown in Figures 6.23 and 6.24 via thefield maps for frequencies 300 Hz, 400 Hz and 600 Hz, 800 Hz respectively. The acousticwaves leave the computational domain boundaries smoothly and without any reflections. Ahigher Infinite Elements order is generally requested to ensure it compared with the laminarcase. Field maps for frequencies from 300 Hz to 2000 Hz are shown in the end of this chapter(Figures 6.30, 6.31, 6.32 and 6.33) with more saturated acoustic pressure scale in order tobetter show the directivity patterns. At high frequencies the quadripolar type character of thesound pattern is clear on the acoustic pressure maps.


Fig. 6.20 (Above) Truncation phenomenon at the outlet for the frequency 300 Hz, (middle)the applied cosine filter, (bottom) the same acoustic field after the application of the filter


Fig. 6.21 Lighthill sources on the acoustic mesh at frequencies, 400 Hz, 800 Hz, 1200 Hzand 1600 Hz


Fig. 6.22 Lighthill near field acoustic pressure at frequencies, 400 Hz, 800 Hz, 1200 Hz and1600 Hz


Fig. 6.23 Lighthill, acoustic waves’ propagation to far field for the frequencies, 300 Hz and400 Hz


Fig. 6.24 Lighthill, acoustic waves’ propagation to far field for the frequencies, 600 Hz and800 Hz


The SPL perceived at receiver, directivity patterns

The far field acoustic pressure spectra are predicted at the considered receiver and comparedwith the experimental data. As mentioned about the methodology applied for Lighthill’s

Fig. 6.25 Comparison of the SPL obtained with ACTRAN/Lighthill’s analogy and thatmeasured from the experiments

analogy, a single ACTRAN run is performed to investigate the nine acoustic signals (load-cases) due to the different correlated time series of a period of time 0.02 sec obtainedwhen the DFT is applied. Then an average of the nine acoustic signals is performed. Byapplying this procedure, an envelope of the SPL variation characterizing the maximumand minimum resulting values at each frequency is obtained. The results are depicted onFig. 6.25. The comparison of the numerical results with the experimental measurementsshows an overall excellent agreement confirming the capability of SFELES (LES sources)combined with ACTRAN (Lighthill’s analogy) to predict correctly the noise generated byturbulent flows around airfoils. The acoustic spectrum presents an overprediction up to 5 dBat the frequencies 300 Hz and 550 Hz and an underprediction about 5 dB at the frequencies1100 Hz and 1750 Hz. These discrepancies of the spectrum may be reduced if the sources areextracted for longer periods to improve the frequency step precision. The directivity patternsare extracted from the 25 microphones placed at a distance of 2 m around the trailing edge tocover a half-circle (180 degrees). Five frequencies have been chosen, 250 Hz, 400 Hz, 800 Hz,1200 Hz and 1800 Hz. Figure 6.26 shows the result. It can be noticed that, at low frequencies,the SPL is more important and the number of lobes, which indicates the complexity of themodel, is lower. By increasing the frequency, the pattern becomes more complex where the


Fig. 6.26 Directivity patterns at frequencies, 250 Hz, 400 Hz, 800 Hz, 1200 Hz and 1800 Hz

Fig. 6.27 Sources truncation phenomenon effect, comparison of the SPL obtained with andwithout the application of the filter near the domain borders


lobes multiply denoting more efficient contribution of the quadrupole sources due to theturbulent eddies. The global pressure amplitude reduces with the increase of the frequency.The acoustic field pattern also becomes more complex with the frequency, showing multiplelobes at higher frequencies. The directivity differs from the Amiet’s solution. The reasonsare probably related to the hypothesis used to derive the analytical expression for Amiet’stheory. The Lighthill’s solution is not restricted on the same hypothesis and is more general.

Fig. 6.28 Comparison of the SPL obtained with ACTRAN/Lighthill’s analogy, SFE-LES/Curle’s formulation and that measured from the experiments

The SPL is compared with the case where no filter is applied near the domain boundaries inFig. 6.27. The overestimation of the SPL for all frequencies is clearly due to the artificialdipole characteristics of the truncation phenomenon.Finally, the SPL is compared with those obtained by the two proposed approaches of Curle’sformulation in Fig. 6.28. The SPL derived from the Curle’s formulation is inside the envelopeof the SPL provided by Lighthill’s solution as expected. The SPL obtained via the firstapproach of Curle’s formulation, overall noise (implementation in SFELES), matches verywell that obtained via Lighthill with a little overestimation at frequencies between 600-700Hz. The second approach, dipole (aeroforces), presents lower values of the SPL at highfrequencies in comparison with Lighthill’s analogy, this is due to not considering the volumesources in this approach. These results represent another confirmation of the validity of theproposed approaches, simplifications and the supposed compactness of the sources in the

6.2 Comparison of the applied hybrid methods, conclusions 131

application of Curle’s formulation. Möhring’s analogy has been applied considering just thelaminar flow case for Reynolds number 12000 (Appendix G). No significant differences arenoticed in comparison with Lighthill’s analogy. This means that the convectional effects arenot so important for the considered Mach number, 0.047. Since the same Mach number isconsidered for the turbulent flow case, this analogy is not applied here.

6.2 Comparison of the applied hybrid methods, conclusions

A comparison of the far field acoustic pressure spectra perceived at the considered receiver,radiated from the studied CD airfoil for the turbulent flow at a Reynolds number of 160000and a Mach number of 0.047, and obtained via the application of three hybrid methods,ACTRAN/Lighthill’s analogy, SFELES/Curle’s formulation (overall noise) and Amiet’stheory, is presented in Fig. 6.29 with the experimental measurements. Generally, a very good

Fig. 6.29 Comparison of the SPL obtained with ACTRAN/Lighthill’s analogy, SFE-LES/Curle’s formulation, Amiet’s theory and that measured from the experiments

agreement with the experiments is obtained using the three analogies. This means that thehybrid methods of aeroacoustics are applicable and reliable for industrial applications. Theintegral approaches allow to calculate the far fields during the CFD calculation whereas thevariational approach allows to take into account interactions with other objects in the vicinity


and take into account some acoustic phenomena in stage of post-treatment, and cost more. Adetailed comparison of the three hybrid methods including the required time, computationalcost, frequency domain of application, sources and main accessible results is organized inthe following table. 6.1.


Lig

hthi

ll’s

anal

ogy

Cur

le’s

form

ulat

ion

Am

iet’s

theo

ryPr

ogra

m/c

ode

AC

TR

AN

SFE

LE

S/M

atla

bM

atla

b

CFD

sour

ces

∂2[ T

ij]

∂x i∂

x j(T

ij,P

i)or

∂F i ∂t

(Y,t

)Φ

(Kx,

k y)|,

γ(ω

,z)

Form

ulat

ion

Var

iatio

nalf

orm

Inte

gral

form

(Gre

en’s

func

tion)

Inte

gral

form

(Sch

war

zsch

ild’s

proc

edur

e)A

pplic

atio

nsA

llki

ndof

inte

rior

and

exte

rior

flow

sA

irfo

ils(t

onal

and

boad

band

nosi

e)V

ery

thin

airf

oils

(boa

dban

dno

sie)

exce

ptin

case

ofco

nvec

tion

effe

cts

are

high

Lim

itaio

nsV

alid

fora

llfr

eque

ncie

sV

alid

forf

requ

enci

esup

to25

07H

zV

alid

fora

llfr

eque

ncie

sM

ain

assu

mpt

ions

The

acou

stic

field

does

notm

odif

yth

eflo

w,

Com

pact

ness

sour

ces,

airf

oil∼

flate

plat

eSo

lidbo

unda

ryL

(C)<

λ

Acc

essi

ble

resu

ltsFa

rfiel

dSP

L,q

uadr

upol

eso

urce

s,Fa

rfiel

dSP

L,d

ipol

ean

dqu

adru

pole

Farfi

eld

SPL

map

sof

near

and

farfi

elds

cont

ribu

tions

sepa

rate

dC

ost:

Sour

ces

volu

me

(MB

)ve

loci

ties:∼

1000

00In

tegr

als:∼

100

pres

sure

:∼50

0E

xtra

ctio

nso

urce

stim

e(H

ours

)∼

250

H(u

sing

256

Proc

s)∼

0∼

0A

cous

ticsi

mul

atio

nse

tup

∼10

hour

s(B

uild

ing

mes

h+so

urce

sm

appi

ng)

∼0

(pos

t-pr

oces

sing

)∼

0(p

ost-

proc

essi

ng)

time

and

sim

ulat

ion

time

5ho

urs

(AC

TR

AN

sim

ulat

ion)

Tabl

e6.

1C

ompa

riso

nof

the

hybr

idm

etho

dsof

aero

acou

stic

s,L

ight

hill’

san

alog

yC

urle

’sfo

rmul

atio

nan

dA

mie

t’sth

eory

asap

plie

din

this

rese

arch


Fig. 6.30 Lighthill, acoustic waves’ propagation to far field for the frequencies, 300 Hz, 400Hz , 500 Hz and 600 Hz


Fig. 6.31 Lighthill, acoustic waves’ propagation to far field for the frequencies, 700 Hz, 800Hz , 900 Hz and 1000 Hz


Fig. 6.32 Lighthill, acoustic waves’ propagation to far field for the frequencies, 1100 Hz,1200 Hz , 1300 Hz and 1400 Hz


Fig. 6.33 Lighthill, acoustic waves’ propagation to far field for the frequencies, 1500 Hz,1600 Hz , 1800 Hz and 2000 Hz

Chapter 7

Conclusions and perspectives

The present thesis focuses on the computation of the aerodynamic noise generated by laminarand turbulent flows around wings or fan blades. As a case study, the tonal and broadbandor trailing-edge noise emanating from a CD (Controlled-Diffusion) airfoil, belonging to afan are studied. The hybrid methods of aeroacoustics are applied to simulate and predictthe radiated noise. The necessary tools were researched and developed. Since the hybridmethods consist in two steps simulations, where the determination of the aerodynamic fieldis decoupled from the computation of the acoustic waves’ propagation to the far field, thefirst part of this thesis is devoted to a detailed aerodynamic study of the considered flowincluding the determination of all expected flow patterns for the Reynolds number range0<Re<60000. Some aspects have been developed in the in-house solver SFELES, includingthe implementation of a new SGS model (Ghorbaniasl’s model), a new outlet boundarycondition (the physical BC), a new transient format (Ensight gold format) which is used toextract the noise sources to be exported to the acoustic solver, ACTRAN and other neededsubroutines. Three sub grid-scale models have been used to study the influence of the LESSGS models on the results. Those are the static Smagorinsky model with Cs = 0.17, theWALE model with Cw = 0.55 and Ghorbaniasl’s model. In order to study the influence ofsome flow parameters on the results, for the turbulent flow (Re = 160000), more than tensimulations of the CD-profile were performed by changing the number of spanwise Fouriermodes, the computational mesh, the spanwise extension and the SGS model.The second part of this thesis is concerned with the aeroacoustic study where four meth-ods have been applied, among them two are integral formulations and the two others arepartial-differential equations. The first method applied is Amiet’s theory, implemented inMatlab, based on the wall-pressure spectrum extracted in a point near the trailing edge andthe spanwise pressure coherence length.

140 Conclusions and perspectives

The second method is Curle’s integral formulation. It is applied proposing two approaches;the first approach is the implementation of the volume and surface integrals in SFELES to becalculated simultaneously with the flow in order to avoid the storage of noise sources whichrequires a huge space. In the second approach, the fluctuating aerodynamic forces, alreadyobtained during the aerodynamics simulation, are used to compute the noise considering justthe surface sources. Finally, Lighthill (for tonal and broadband noise) and Möhring (just fortonal noise) analogies have been applied via the acoustic solver ACTRAN using sourcesextracted via SFELES. As main results in this research we can cite:All expected flow patterns of the CD airfoil are well determined with respect to Reynoldsnumber as shown in the following table:

Reynolds number Corresponding flow regime of the CD airfoil0<Re<270 Attached flow (creeping)

270<Re<1300 Steady, separated flow1300<Re<6450 2-d unsteady laminar oscillating flow (vortex street)

6450<Re<14000 3-d unsteady laminar oscillating flow14000<Re<47500 3-d turbulent wake, 2-d laminar boundary layer regime

Re=47500 Recirculation bubble appearance near the leading edge,laminar boundary layer

50000=<Re<52000 Recirculation bubble explosion,3-d laminar periodic boundary layer and turbulent wake

Re>=52000 Recirculation bubble explosion, turbulent boundary layer and wake

Table 7.1 The evolution of the flow regime with Reynolds number

The average pressure and friction coefficients distribution, boundary layer profiles, wall-pressure spectra, wake mean and turbulent velocities, coherence function and length, bound-ary layer stresses and other quantities are computed and compared with experimental andLES OpenFoam computational results. All these results are found in very good agreementwith the experimental data.

It is found that the new SGS formula enforces the zero contribution from the sub-gridscales in irrotational flow regions and in the viscous sub-layer where its value goes to zeroon the surface of the airfoil without the need of any damping function. The use of thisSGS model improves the aerodynamic results in general, especially near the leading edge,comparing with OpenFoam, Wang and experimental results. The recirculation bubble size isfound equal to 3.5%C for 64 Fourier modes in the spanwise direction.

The high variability of the leading edge recirculation bubble size with the solver and theSGS model is found as shown in table 5.2 and already observed in literature. This leads to

141

differences of the results in the region towards the leading edge.The recirculation bubble size has no important effect on the wall-pressure spectra near the

trailing edge, whereas it influences largely the pressure and friction coefficients in the regionnear the leading edge. The wall-pressure spectra computed by SFELES are in excellentagreement with experiments and in closer agreement to experimental data than OpenFoamfor frequencies between 1500 Hz and 2000 Hz and than Wang’s spectra for all frequencies.The wall-pressure spectra are insensitive to the choice of the SGS model.

It is noticed that the vorticity is much clearer and more intense for 64 Fourier modes thanfor 32 modes and there are many more structures in this case. So, 64 modes are necessaryand sufficient for acceptable convergence of the results with acceptable cost.

It is observed that the velocity profiles have been improved in general and are closer to theexperimental data for the spanwise extension z/C = 0.2 than for the extension z/C = 0.1. Theboundary layer thickness is very well predicted and it is identical to the experimental value.

The preliminary results of the physical boundary condition show acceptable results incomparison with the zero-pressure outlet with some improvements. The obtained pressureprofile on the outlet nodes varies with the time step and the node location. This is morephysical since the studied flows are unsteady, it could guarantee the non reflection of thepressure waves on the outlet border.

Tonal noise is simulated via Lighthill’s analogy, Curle’s formulation and Möhring’sanalogy. It is found that the von Karman vortex shedding formed in the wake by the laminarboundary layer is the main acoustic source. A dominant peak is found at the frequency of thevortex shedding (369 Hz). The convectional effects are found not significant at the consid-ered Mach number, as expected. Finally, the results of the proposed approaches for Curle’sformulation match well the results obtained by Lighthill and Möhring analogies. The vortexshedding frequency, its harmonics and the corresponding acoustic pressure are well capturedfor the laminar case. So we can say that Curle’s formulation is a simple, approximated andreliable formulation which can give a quick idea of the radiated noise of the studied airfoilwith the limitations and restrictions already mentioned such as compactness source region,applicable only for exterior noise problem in free field, etc. . . .

As far as the broadband noise is concerned, the predicted noise spectra obtained, viaAmiet’s theory, using the wall-pressure spectrum and the coherence length of SFELESare in very good agreement with experimental results and closer to experimental data thanOpenFoam, in particular for frequencies between 1500 and 2000 Hz regardless of the SGSmodel. The radiated noise computed using Amiet’s theory is compared as well with Wang’scomputations using FW-H and it is found that at frequencies between (500-2000 Hz), the

142 Conclusions and perspectives

present results are very similar to Wang’s results whereas for frequencies smaller than 500Hz, our curves have higher values but closer to experimental results.

The dipolar behaviour of the noise sources, computed via Amiet’s theory, is found andthe lobes multiply and orient towards the leading edge for high frequencies. The predictednoise spectrum is found insensitive to the choice of the SGS model, the difference does notexceed 2 dB for frequencies of interest (200-2000 Hz), this is linked to the insensitivity ofthe wall-pressure spectra to the choice of the SGS model.

With respect to broadband noise computation using Lighthill’s analogy, sources and nearfield maps show that the turbulent boundary layer and wake are the more efficient sourcesfor the turbulent flow around the airfoil and the center of radiation is the trailing edge. Thecomparison of the numerical SPL results with the experimental measurements shows anoverall excellent agreement. The acoustic spectrum presents an overprediction up to 5 dB atthe frequencies 300 Hz and 550 Hz and an underprediction about 5 dB at the frequencies1100 Hz and 1750 Hz. The directivity patterns are extracted using Lighthill’s analogy. It isnoticed that, at low frequencies, the SPL is more important and the number of lobes, whichindicates the complexity of the model, is lower. By increasing the frequency, the patternbecome more complex where the lobes multiply denoting more efficient contribution of thequadrupole sources due to the turbulent eddies.

Turning to Curle’s formulation, the SPL obtained using the two proposed variants ofCurle’s formulation matches very well the experimental results. The dipole sources aredominant whereas the quadrupole contribution is found important, in particular for highfrequencies, despite of the low Mach number considered in this study. The compactnessassumption is found valid for the studied airfoil for the frequencies of interest (lower than2500 Hz). So, this formulation can be used to obtain fast, approximate and acceptable resultsabout the noise radiation avoiding the storage of noise sources which requires a huge space.The obtained results are found very similar to those obtained in [57] using Curle’s acousticsolver Virtual.Lab.

Finally, the hybrid methods of aeroacoustics are found very efficient and capable tosimulate and compute the radiated noise for laminar and turbulent flows and we can say theyare applicable and reliable for industrial applications.

As perspectives:

Möhring’s analogy could be applied for the turbulent flow of the CD airfoil for higherMach number to quantify the acoustic waves refraction caused by the convectional effects ofthe mean flow.

Further validation cases and developments could be performed on the outlet physical

143

boundary condition which is candidate to be a non-reflective boundary condition for theSFELES solver. Lighthill’s analogy should be applied using aerodynamic sources with theapplication of the physical boundary condition to see the implied improvements of the sourcesand therefore the resulting estimated noise. Extracting sources for Lighthill’s analogy forlonger periods (0.2 sec, 0.4 sec ...) could carry some improvements of the computed SPL.

Imposing higher number of Fourier Modes in the spanwise direction as 128 M or 256 Mcould also improve the aerodynamic and aeroacoustic results.

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List of journal publications, conferences contributions andpresentations performed during this research

List of journal publications (peer reviewed):Matouk, R., Christophe, J., and Degrez, G., Aerodynamics and aeroacoustics study of theflow around an automotive fan airfoil, International journal of aeroacoustics, volume 14,number (78), Pages: 1049-1070. 2015

Matouk, R., Detandt, Y., and Degrez, G., The Hybrid Methods of Aeroacoustics Applied for aCD Airfoil in a Turbulent Regime, International journal of aeroacoustics, 2016 (submitted).

List of conferences contributions and presentations:Matouk, R., Degrez, G. and Christophe, J., , Aerodynamics and aeroacoustics study of theflow around an automotive fan airfoil, Proceedings of WCCM 2014, ECCM 2014 and ECFD2014, Pages: 5677-5688, Barcelona, Spain, July 2014.

Matouk, R., and Degrez, G., Calculation of aerodynamic noise of wing profiles by hybridmethods, International Graduate Summer School in Aeronautics and Astronautics, Beijing,China, July 2015.

Matouk, R., and Degrez, G., COMPUTATION OF THE RADIATED NOISE OF A CONTROLLED-DIFFUSION AIRFOIL VIA THE HYBRID METHODS OF AEROACOUSTICS, Proceed-ings of ISUAAAT 14, Stockholm, Sweden, 8-11 of September 2015.

Appendix A

The leading-edge noise orturbulence impact noiseformulation [29]

Fig. A.1 Representation of the skewed gust impinging to the linearized airfoil

This problem is solved in two stages. First the distribution of unsteady lift on the profileis determined. Then, using the formula for the radiation of a dipole in a medium in motionand by means of a statistical analysis, the spectrum of the acoustic field radiated by theprofile is obtained. In order to derive the acoustic field formulation in this case, let usconsider a stationary airfoil of span 2d and chord C = 2b subjected to an upstream turbulentflow as shown in Fig. A.1. A single skewed gust is supposed to sweep the airfoil with amean flow velocity U. The origin of the coordinates system is fixed at the airfoil center.The x-axis is in the streamwise direction. The y and z-axes are aligned with the spanwiseand the crosswise directions, respectively. The spectral wavenumbers kx and ky represent

156 The leading-edge noise or turbulence impact noise formulation [29]

chordwise and spanwise components of the frozen turbulence. As mentioned above, theairfoil is assumed to be a flat plate with zero thickness, within the scope of a linearizedtheory, hence the crosswise component of the turbulence kz is not considered. The acoustic

Fig. A.2 The airfoil with a dipole source located at X0 and a receiver located at X

radiation of the airfoil subjected to incoming turbulence involves the radiation of spanwiseand chordwise distributed dipoles on the airfoil surface. The airfoil and the coordinatesystem are represented on Fig. A.2. The observer is defined as X = (x,y,z) and the source,represented by a dipole, is located on the airfoil surface and defined as X0 = (x0,y0). Theairfoil is placed in a uniform flow defined by a Mach number M = U/c0 and the correspondingparameter β =

√1 – M2, with c0 the speed of sound. The position of the receiver must take

into account the convection effect so it is given by the vector :

Xt = [((x – x0) – Mσs)/β2,y – y0, z] (A.1)

with σs =√

(x – x0)2 +β2[(y – y0)2 + z2]. The propagation distance between the source andthe observer is not the linear distance but a corrected distance including the convectioneffects, it is therefore written as σt = [σs – M(x – x0)]/β2.The acoustic pressure generated by a single dipole located on the airfoil for a frequency ω isgiven as [44, 25]:

p(X,ω;X0) =ikXt.F(X0,ω)

4πσ2s

e–ikσt(1 +1

ikσs) (A.2)

with k = ω/c0 and F is the force vector in the Fourier domain corresponding to the forcestrength of the dipole. The factor 1/ikσs corresponds to the acoustical near-field effect.Considering only the lift component of the dipole l which is perpendicular to the airfoil, theformulation A.2 is therefore reduced to:

p(X,ω;X0) =ikz.l(X0,ω)

4πσ2s

e–ikσt(1 +1

ikσs) (A.3)

157

The total acoustic pressure field of the airfoil is obtained by integrating the dipole sources onthe complete airfoil surface Sy:

p(X,ω) =∫ d

–d

∫ b

–b

ikz.l(X0,ω)4πσ2

se–ikσt(1 +

1ikσs

)dx0dy0 (A.4)

=∫

Sy

ikz.l(X0,ω)4πσ2

se–ikσt(1 +

1ikσs

)dSy (A.5)

The power spectral density (PSD) of the sound pressure perceived at the observer positiondue to the dipoles distribution on the airfoil is therefore expressed as:

Spp(X,ω) =∫

Sy

∫S′

y(

kz4π

)2 1

σ2sσ

′2s

(1 +1

ikσs)(1 +

1

ikσ′s)∗...

Sll(X0,X′0,ω)e–ik[σt–σ

′t ]dS

′ydSy

(A.6)

()∗ denotes the complex conjugate operator and Sll is the cross-power spectral density ofthe local lift function. The cross-PSD of the local lift function is directly connected to theincoming gust and the corresponding pressure jump induced on the airfoil. Considering aconvective sinusoidal gust of the form

w(X0, t) = ˆw(kx,ky)ei(kx(Ut–x0)–kyy0) (A.7)

ˆw is the double spatial Fourier transform of the incident perturbation. The resulting pressurejump on the airfoil can be written as

l(X0, t) = P(X0, t) = 2πρ0U ˆw(kx,ky)g(x0,kx,ky)ei(kx(Ut–x0)–kyy0) (A.8)

Where g(x0,kx,ky) Amiet’s function denoting the transfer function linking the disturbancepressure jump to the incoming gust. For all the incoming gusts, the local lift density is thusexpressed

l(X0, t) =∫

∞∫

–∞

l(X0, t)dk′xdky (A.9)

Applying the Fourier transform of the local lift density gives:

l(X0, t) = 2πρ0U∫

∞∫

–∞

ˆw(kx,ky)g(x0,kx,ky)e–ikyy0(U2π

∫∞

–∞

ei(kxU–ω)tdt)dk′xdky (A.10)

This relation is simplified recognizing the Fourier transform of the Dirac function, the locallift density is therefore written

l(X0, t) = 2πρ0

∫∞

–∞

ˆw(Kx,ky)g(x0,Kx,ky)e–ikyy0dky (A.11)


where Kx = ω/U is a particular value of the chordwise wavenumber. The cross-PSD for thelocal lift density function in Eq. A.6 is written:

Sll(X0,X′0,ω) = (2πρ0)2

∫∞∫

–∞

ˆw(Kx,ky) ˆw∗(Kx,k′y)...

g(x0,Kx,ky)g∗(x′0,Kx,k

′y)e–i(kyy0–k

′yy

′0)dkydk

′y

(A.12)

However, due to the statistical orthogonality of the turbulence wavevectors [45] it can beshown that

ˆw(Kx,ky) ˆw∗(Kx,k′y) = Uδ(ky – k

′y)Φww(Kx,ky) (A.13)

where Φww is the two-dimensional turbulence energy spectrum. The cross-PSD of the liftfunction becomes

Sll(X0,X′0,ω) = (2πρ0)2U

∫∞

–∞

Φww(Kx,ky)e–iky(y0–y′0)

g(x0,Kx,ky)g∗(x′0,Kx,ky)dky

(A.14)

The final expression of the PSD for the acoustic field of an airfoil in a turbulent stream istherefore obtained by replacing the equation A.14 into the equation A.6

Spp(X,ω) =∫

Sy

∫S′

y(ρ0kz

2)2U

1

σ2sσ

′2s

(1 +1

ikσs)(1 +

1

ikσ′s)∗e–ik[σt–σ

′t ]...∫

∞

–∞

Φww(Kx,ky)e–iky(y0–y′0)g(x0,Kx,ky)g∗(x

′0,Kx,ky)dkydS

′ydSy

(A.15)

No assumptions on the acoustical or geometrical far-field are applied in the development ofthis formulation so it can be used in a general context. In order to make this formulationapplicable with less numerical efforts, assumptions and simplifications will be done asmentioned in the introduction. First, the observer is assumed to be located in the acousticalfar-field

∣∣Xt∣∣≫ λ [46]. Thus, the acoustical near-field contribution 1/ikσ

′s can be neglected

leading to the far field formulation:

Spp(X,ω) =∫

Sy

∫S′

y(ρ0kz

2)2U

1

σ2sσ

′2s

e–ik[σt–σ′t ]...∫

∞

–∞

Φww(Kx,ky)e–iky(y0–y′0)g(x0,Kx,ky)g∗(x

′0,Kx,ky)dkydS

′ydSy

(A.16)

Considering a large aspect ratio airfoil, 2d ≫ C located far away from the airfoil comparedto its span-length, σ ≫ d. Consequently, (x – x0)2 ≈ x2 can be supposed. Therefore, thepropagation distance term can be simplified to:

σs = σk =√

x2 +β2[(y – y0)2 + z2] (A.17)

159

Taking the first order development in Taylor series around the center of the airfoil (x0,y0) = 0for the observer distance σs gives the approximation:

σs =√

(x – x0)2 +β2[(y – y0)2 + z2]

≈ σ0(1 –xx0 +β2yy0

σ20)

≈ σ0 + x0∂σs∂x0

|x0=0 + y0∂σs∂y0

|y0=0

(A.18)

where σ20 = x2 +β2(y2 + z2). Using the equation A.1 we can write:

σt –σ′t =

1β2σ0

[(x – Mσ0)(x′0 – x0) +β2y(y

′0 – y0)] (A.19)

Returning to the equation A.16, it is now expressed:

Spp(X,ω) = (ρ0kz

2)2U

∞∫–∞

Φww(Kx,ky)∫∫ d

–d

1σk2σk′2

e–i(

kyσ0

–ky)(y′0–y0)

dy0dy′0 ..

∫ b

–bg(x0,Kx,ky)e

i k(x–Mσ0)β2σ0

x0dx0

∫ b

–bg∗(x0,Kx,ky)e

–i k(x–Mσ0)β2σ0

x′0dx

′0

(A.20)

The spanwise and chordwise integrals are separated and can be solved independently. Thechordwise integral results as:

∫ b

–bg(x0,Kx,ky)e

i k(x–Mσ0)β2σ0

x0dx0 = bL (x,Kx,ky) (A.21)

where L (x,Kx,ky) is the aeroacoustic transfer function linking the impinging gust to theacoustic field of the airfoil [32, 29]. The reader is addressed to Appendix A for furtherderivations of the transfer functions. The spanwise integral can be computed independentlyand the formulation A.20 is finally expressed:

Spp(X,ω) = (ρ0kzb

2)2U

∞∫–∞

|κ(x,Kx,ky)|2Φww(Kx,ky)|L (x,Kx,ky)|2dky (A.22)


where the function κ(x,Kx,ky) has the following formulation according to [30]:

κ(x,y,z,Kx,ky) =iei(Ky+ky)y

2β2√

x2/β2 + z2...

e–(Ky+ky)√

x2/β2+z2E1

[–(Ky + ky)

√x2/β2 + z2 – i(Ky + ky)(d – y)

]–e–(Ky+ky)

√x2/β2+z2

E1

[–(Ky + ky)

√x2/β2 + z2 – i(Ky + ky)(–d – y)

]–e(Ky+ky)

√x2/β2+z2

E1

[(Ky + ky)

√x2/β2 + z2 – i(Ky + ky)(d – y)

]+e(Ky+ky)

√x2/β2+z2

E1

[(Ky + ky)

√x2/β2 + z2 – i(Ky + ky)(–d – y)

]

(A.23)

with Ky = ky/σ0 and E1 is defined as E1(x) =∫

∞x

e–t

t dt [35]. As a last simplification, (y–y0) ≈y can be supposed in the case that the receiver is placed in the far field with a large distancefrom the airfoil compared to the airfoil span. Therefore, the scaling factor of formulationA.16 can be approximated as:

1σ2

s≈ 1

σ20

(A.24)

And the spanwise integral reads:∫∫ d

–d

1σ4

0e

–i(kyσ0

–ky)(y′0–y0)

dy0dy′0 =

4σ4

0

sin2[(Ky – ky)d](Ky – ky)2 (A.25)

The equation A.20 is now expressed as:

Spp(X,ω) = (ρ0kzbσ2

0)2πUd

∞∫–∞

sin2[(Ky – ky)d](Ky – ky)2 Φww(Kx,ky)|L (x,Kx,ky)|2dky (A.26)

When the semi-span d increases, the square of the sine cardinal function in Eq. A.26 tends toa delta function as the following:

limd→∞

(sin2[(Ky – ky)d]πd(Ky – ky)2 ) = δ(Ky – ky) (A.27)

This means that for high spanwise extensions, the spanwise wave number ky is replaced bythe particular one, Ky, including the ones in the turbulence spectrum and in the acoustictransfer function. This is physically explained by the fact that the observer only hears thegust producing acoustic wave-fronts perpendicular to the line joining the source and theobserver [47]. Using the similarity rules defined by Graham [36], the equation A.26 canbe simplified. Φww and L become nearly independent of ky, allowing them to be takenoutside the integral, if MKx ≫ ky because the airfoil response function becomes independentof ky. Replacing the spanwise integral with the particular value of the Ky, the PSD for the

161

large-span airfoils is then given by the formulation:

Spp(X,ω) = (ρ0kzbσ2

0)2πUdΦ(Kx,ky)|L (x,Kx,ky)|2 (A.28)

This is the final formulation to predict far-field noise of thin airfoils due to the interaction ofturbulent flow with the airfoil leading-edge. The complete derivation of the transfer functionL (x,Kx,ky) is presented in Appendix B.

Appendix B

Amiet’s theory: transfer functionsderivation [29, 31–33, 99]

B.1 Leading edge case (turbulence-interaction noise):

Amiet’s model is based on the linearized theory of thin airfoils and the linear acoustics. Theprofile is therefore assumed a flat plate, with zero thickness and no incidence, defined bythe plane z = 0 and 0 ≤ x ≤ 2b (Fig. B.1). The disturbance is introduced upstream of theprofile and convected by the flow to sweep the leading edge. Let us denote φ the potential ofthe velocity perturbation induced by the presence of the profile, M = U/c0 is the flow Machnumber. The first step is to determine the potential φ satisfying the convected linearizedwave equation, the convected Helmholtz equation, as the following:[

∇2 –

1c2

0

D2

Dt2

]φ(x,y,z, t) = 0 (B.1)

with the derivative D expressed as

DDt

=∂

∂t+ U

∂

∂x(B.2)

To solve the boundary value problem represented by Eq. B.1, three boundary conditions needto be imposed:

φ(x,y,0, t) = 0 x ≤ 0

∂φ∂z (x,y,0, t) = –w(x) 0 < x ≤ 2b

DφDt (x,y,0, t) = 0 x > 2b

(B.3)

These three equations mean respectively:- Zero velocity potential upstream of the profile.- Zero normal velocity on the profile.- Zero pressure difference between the pressure and suction sides at the trailing edge and in

164 Amiet’s theory: transfer functions derivation [29, 31–33, 99]

the wake, in accordance with the Kutta-Joukowski condition.The wave equation B.1 is written in the three dimensional coordinates:

∂2φ

∂x2 +∂2φ

∂y2 +∂2φ

∂z2 –1c2

0

[∂2φ

∂t2+ 2U

∂2φ

∂t∂x+ U2∂

2φ

∂x2

]= 0 (B.4)

Or

β2∂2φ

∂x2 +∂2φ

∂y2 +∂2φ

∂z2 –1c2

0

∂2φ

∂t2–

2Uc2

0

∂2φ

∂t∂x= 0 (B.5)

where β2 = 1 – M2. The incident disturbance velocity is written in the form:

w = w0ei(ωt–kxx–kyy) (B.6)

kx and ky are the chordwise and spanwise aerodynamic wavenumbers respectively. A vari-ables change has been performed to get dimensionless variables as the following:

ki = kic2 , x = 2x

c , y = 2βyc , z = 2βz

c , σ = kxβ2

The flow potential can be defined as a Fourier-type function:

Φ(x, z) = φ(x,y,z, t)e–iωtei[–σM2x+kyy/β] (B.7)

Then we obtain the equation:∂2Φ

∂x2 +∂2Φ

∂z2 +κ2Φ = 0 (B.8)

where k = ω/c0 = kxM, κ2 = µ2 –k2

yβ2 , µ = kxM

β2

The mathematical nature of the problem depends on the sign of κ2 where we can distinguishtwo cases:- If κ2 > 0, the differential equation is therefore hyperbolic and the gust is called supercritical.In this kind of problem, an initial perturbation is not seen at the same instant on all positionsof the flow, but it is wave-likely propagated with constant and finite speed along characteristiclines.- If κ2 < 0, the differential equation is elliptic and the gust is said subcritical. Subcriticalgusts contribute to the far-field for finite-span airfoils only, as they produce evanescent wavesotherwise. The boundary conditions are expressed with the new variables:

Φ(x,0) = 0 x ≤ 0

∂Φ∂z = c0

2βe–iσx 0 < x ≤ 2

∂Φ∂x + iσΦ = 0 2 < x

(B.9)

In fact, there is no exact analytic solution for the equation system consisting of the waveequation B.8 with the three boundary conditions B.9. The problem is linear, it is typically

B.1 Leading edge case (turbulence-interaction noise): 165

resolved by an iterative process described by Schwarzschild procedure [48, 49] presented inthe next paragraph.

B.1.1 The analytic solution using Schwarzschild’s techniqueFirst, we need to determine a solution of the wave equation, satisfying just the rigid wallcondition expressed by the second equation of B.9, denoted Φ(0) on an infinite plane. Theflow potential is supposed to be given by the relation:

Φ(0)(x, z) = R.e(sx–i√κ2+s2z) (B.10)

As in [105]. R and s and are constants determined from the rigid wall condition as:

R = –c02k , s = –iσ, k =

√k2

x + k2y

Consequently, the uncorrected potential velocity is given by:

Φ(0)(x, z) =–c02k

e–iσx–kz/β (B.11)

The problem is then resolved by successive half-planes for the leading and trailing edgescorrections respectively as show in Fig. B.1.Second, In this step, the airfoil is considered as a semi-infinite plane extending downstream

Fig. B.1 Representation of the two steps procedure for Amiet’s leading edge: Incident gust ona finite-chord airfoil (top), main scattering half-plane problem (bottom left) and trailing-edgecorrection (right)


to compute the leading edge correction Ψ1 such that Φ(1) = Φ(0) +Ψ1. The potential iscanceled upstream satisfying the first boundary equation of B.9. The solution Φ(1) satisfiestherefore the two first boundary conditions of B.9 so Φ(1) is the solution of the system:

∂2Φ(1)

∂x2 + ∂2Φ(1)

∂z2 +κ2Φ(1) = 0

Φ(1)(x,0) = 0 x ≤ 0

∂Φ(1)

∂z (x,0) = –c02β e–iσx x > 0

(B.12)

And Ψ1 is therefore the solution of the following system:

∂2Ψ1∂x2 + ∂2Ψ1

∂z2 +κ2Ψ1 = 0

Ψ1(x,0) = –Φ(0)(x,0) x ≤ 0

∂Ψ1∂z (x,0) = 0 x > 0

(B.13)

This system is resolved using the Schwarzschild’s procedure presented in the next section.

Schwarzschild’s solution

Schwarzschild’s problem states that if a function ϕ is governed by an equation system of theform:

∂2ϕ∂x2 + ∂2ϕ

∂z2 +κ2ϕ = 0

ϕ(x,0) = f(x) x ≤ 0

∂ϕ∂z (x,0) = 0 x > 0

(B.14)

The solution for z = 0 and x < 0 is therefore given by the following relation:

ϕ(x,0) =1π

∞∫0

G(x,ξ,0)f(ξ)dξ (B.15)

where G is the Green’s function solution of the boundary value problem given as:

G(x,ξ,0) =√

–xξ

e–κ(ξ–x)

ξ– x(B.16)


And now, applying Schwarzschild’s solution to the equation system B.13 gives the solution:

Ψ1(x,0) = –1π

∞∫0

√xξ

e–iκ(ξ+x)

ξ+ xΦ(0)(–ξ,0)dξ (B.17)

By replacing Φ(0), Eq. B.11, in the precedent equation we obtained:

Ψ1(x,0) =1π

e–iκx –cw02k

∞∫0

√xξ

e–iξ(κ–σ)

ξ+ xdξ (B.18)

The integral is computed by Gradshteyn & Ryszik [106] and given as:

∞∫0

√xξ

e–iAξ

ξ+ xdξ = πeiAx

[1 –

eiπ/4√π

∫ Ax

0

e–it√

tdt

](B.19)

Returning to Eq. B.18, Ψ1 is now written:

Ψ1(x,0) =cw02k

e–iσx

1 – (1 + i)

–(σ–κ)x∫0

e–it√

2πtdt

(B.20)

Recognizing the Fresnel integrals, defined by:

E(x) =x∫

0

eit√

2πtdt (B.21)

And it is known as well that: (1 + i)E∗(–x) = (1 – i)E(x)The potential Φ(1) is eventually written:

Φ(1) = –cw02k

(1 – i)E[(σ–κ)x] (B.22)

Returning to the dimensional parameters, we obtain:

φ1(x,y,0, t) = –w0(1 – i)√

k2x + k2

y

E[

2(σ–κ)xc

]ei(Ukxt–kxx–kyy) (B.23)

The link between the potential velocity and pressure is used to express the load fluctuationsinduced by the impact of the incident gust:

p1(x,y,0, t) = –ρ0Dφ1Dt

= –ρ0

(∂φ1∂t

+ U∂φ1∂x

)(B.24)


And this leads to the final expression of the unsteady pressure generated by a disturbed flowon a semi-infinite downstream plane.

p1(x,y,0, t) = ρ0w0Ue–iπ/4√

πx(kx +β2κ)e

i[(M2σ–κ) 2x

c +Ukxt–kyy]

(B.25)

Third, It is time to add a corrective term taking into account the finite character of the chordand the Kutta condition B.9 at the trailing edge. A semi-infinite upstream plane is consideredin this step. We will seek directly for the correction term P2 which is the solution of thefollowing equation system:

∂2P2∂x2 + ∂2P2

∂z2 +κ2P2 = 0

P2(x,0) = –P1(x,0) x ≥ 2

∂P2∂z (x,0) = 0 x < 2

(B.26)

with P1 expressed as:

P1(x,0) = p1(x, y,0, t)e–iσM2xei(kyy/β–ωt) (B.27)

It is also a Schwarzschild’s problem, so its solution is:

P2(x,0) =1π

∞∫0

G(x – 2,ξ,0)P1(2 +ξ,0)dξ (B.28)

That gives:

P2(x,0) = –ρ0w0Ue–iπ/4

π√π(kx +β2κ)

e–4iκeiκx∞∫

0

√2 – x

ξ(ξ+ 2)e–2iκξ

ξ+ 2 – xdξ (B.29)

The integral is simplified by Amiet supposing that ξ+ 2 ≈ 2 near the trailing edge where thecontribution of small values of ξ is dominant.

P2(x,0) ≈ –ρ0w0Ue–iκx–iπ/4√2π(kx +β2κ)

1 – (1 + i)E∗ [2κ(2 – x)]

(B.30)

Thus, the trailing edge correction to satisfy the Kutta condition is:

p2(x,y,0, t) ≈ ρ0w0U√2π(kx +β2κ)

ei[(M2σ–κ) 2x

c –π/4+ωt–kyy]

1 – (1 + i)E∗[

2κ(2 –2xc

)](B.31)


The final relation of the resulting unsteady pressure is the sum of the results obtained fromthe two iterations in the process.

p(x,y,0, t) = p1(x,y,0, t) + p2(x,y,0, t) (B.32)

Additional iterations can be performed but on the one side that will lead to complex integralsthat can not be computed analytically and on the other side, it has been already shown byAmiet in his paper [31] that the first two iterations are sufficient.

The aeroacoustics transfer functions:

Now we can find the aeroacoustics transfer functions formulations for the leading edge theory.

A. For a supercritical gust:

The aeroacoustic transfer function for a supercritical gust is defined as:

L (x,y,z,kx,ky) =+1∫

–1

g(ξ,kx,ky)e–iµ(M–x/σ0)dξ (B.33)

with σ0 =√

x2 +β2(y2 + z2) and g is Amiet’s function (the reduced lift function) linking thedisturbance pressure jump to the incoming gust as:

p(x,0, t) = 2πρ0Uw0g(x,kx,ky)eiωt (B.34)

As mentioned in Chap. 2, the airfoil thickness and camber are considered negligible so thelocal lift can be approximated to equal twice of the pressure fluctuation:

p(x,0, t) = 2p(x,y,0, t)eikyy (B.35)

Therefore Amiet’s function g(x,kx,ky) is obtained by the equality of the two precedentequations as:

g(x,kx,ky) =p(x,y,0, t)eikyye–iωt

πρ0Uw0(B.36)

Replacing the computed pressure with the leading and trailing edges corrections equations(B.25 and B.31) gives:

g1(x,kx,ky) =e–π/4

π√

π(kx +β2κ)(x + 1)e–i(κ–M2σ0)(x+1) (B.37)

g2(x,kx,ky) = –e–π/4

π√

2π(kx +β2κ)

1 – (1 + i)E∗ [2κ(1 – x)]

e–i(κ–M2σ0)(x+1) (B.38)


Replacing g1 and g2 in the aeroacoustic function relation Eq. B.33:

L1(x,y,z,kx,ky) =1π

√2

(kx +β2κ)θ1E∗ [2θ1

]eiθ2 (B.39)

L2(x,y,z,kx,ky) =eiθ2

θ1π√

2π(kx +β2κ)

i(1 – e–i2θ1) + (1 – i)E∗ [4κ] –

√2κθ3

e–i2θ1E∗ [2θ3]

(B.40)where θ1 = κ–µx/σ0, θ2 = µ(M – x/σ0) –π/4, θ3 = κ+µx/σ0.

A. For a subcritical gust:

Applying the same methodology and development steps as for the supercritical gust case, wecan arrive to the following aeroacoustic transfer functions formula for a subcritical gust:

L ′1(x,y,z,kx,ky) =

1π

√2

(kx + iβ2κ)θ′1E∗ [2θ′1]eiθ2 (B.41)

L ′2(x,y,z,kx,ky) =

eiθ2

θ′1π√

2π(kx + iβ2κ′)....

i(1 – e–i2θ′1 – erf[√

–4κ′]) –2e–i2θ′1

1 + i

√2κ′

θ′3Φ0[√

2iθ′3] (B.42)

where θ′1 = iκ′ –µx/σ0, θ′3 = iκ′ +µx/σ0 and κ′ = –κ = kyβ2 –µ2

B.2 Trailing edge case:

The aeroacoustic transfer functions for the trailing edge theory are obtained following thesame procedures and steps performed for the leading edge case. A Schwarzschild problemis obtained twice first for the trailing edge and then for the leading edge back-scatteringcorrection. The same assumptions and hypothesis supposed for the leading edge case areconsidered. In this theory, the convected wave is supposed to arrive and sweep the trailingedge where the Kutta condition is imposed whereas in the leading edge theory, the convectedwave sweeping the leading edge.

B.2.1 Supercritical gust:

L1(x,y,z,K,ky) = –ei2C

iC

(1 + i)e–i2C

√B

B – CE∗[2(B – C)] – (1 + i)E∗[2B] + 1 – e–i2C

(B.43)

B.2 Trailing edge case: 171

And the leading edge back-scattering correction is found as:

L2(x,y,z,K,ky) ≈ H

[ei4κ(1 – (1 + i)E∗[4κ])]c – ei2D + i[D + K +µM –κ]G

(B.44)

where:α = U/Uc, A = K + (1 + M)µ,A1 = Kc + (1 + M)µ, B = K +µM +κ,C = Kc –µ(x/σ0 – M), D = κ–µx/σ0,ϵ = (1 + 1

4µ )–1/2

G = (1 +ϵ)ei(2κ+D) sin(D – 2κ)(D – 2κ)

+ (1 –ϵ)ei(–2κ+D) sin(D + 2κ)sin(D + 2κ)

...

+(1 +ϵ)(1 – i)

2(D – 2κ)ei4κE∗[4κ] –

(1 –ϵ)(1 + i)2(D + 2κ)

e–i4κE[4κ]...

+ei2D

2

√2κD

E∗[2D][(1 –ϵ)(1 + i)

D + 2κ–

(1 +ϵ)(1 – i)D – 2κ

]

(B.45)

H = (1+i)e–i4κ(1–Θ21)

2√π(α–1)K

√B

, Kc = ω/Uc, Θ1 =√

A1A .

The bar (.) indicates a non-dimensionalisation by the semi-chord and .c correspondsto a multiplication of the imaginary part by the factor ϵ.

B.2.2 Subcritical gust:

L ′1(x,y,z,K,ky) ≈ –

ei2C

iC

e–i2C

√A′

1D′ Φ

0[√

2iD′] –Φ0[√

2iA′1] + 1

(B.46)

L ′2(x,y,z,K,ky)≈

e–i2D′

iD′

A′(ei2D′

[1 – erf(√

4κ′)]) +√

2κ′(K +µ(M – x/σ0))Φ0[

√–2iD′∗]

[√

–iD′∗]

(B.47)

where:

A′ = K + Mµ– iκ′, A′1 = Kc + Mµ– iκ′, D′ = µx/σ0 –κ′,

Θ′1 =

√A′

1A′ ,

H′ = (1+i)e–i4κ(1–Θ′21 )

2√π(α–1)K

√A′

1

,

Φ0(Z) = 2√π

∫ Z0 (e – Z2)dZ, erf(x) = 2√

π

∫ x0 (e – t2)dt,

E(x) =∫ x

0eit

√2πt

dt

More details about the transfer functions derivation exit in [33, 57].

Appendix C

Ghorbaniasl’s SGS modelderivation [50]

This model proposes a relation to estimate the Smagorinsky SGS constant Cs dynamicallysuch that it takes its value according to the transitional and rotational velocities fluctuationsthrough the mesh elements or regions. That is more efficient than fixing the constant valuebecause the eddy viscosity should vanish in some regions where the flow is irrational or nearthe body surface, this is guaranteed by this model. So it has the same idea as Smagorinskydynamic model but it is simpler and easier to be implemented to CFD codes and moreefficient in terms of computational efforts and cost. Moreover, the constant value is stillpositive avoiding some instabilities. It is well known that the SGS model is used to close thefiltered NS equations by modeling the anisotropic part of the subgrid stress tensor. The eddyviscosity concept is the most common used as:

τDij = τSGS

ij –13τSGS

ij δij = –2νtSij (C.1)

νt = UL (C.2)

U is a characteristic velocity scale and L is a characteristic length scale. Smagorinsky treatedthese two terms according to the following hypothesis:

L ∝ = Cs (C.3)

νt = L2 ∣∣S∣∣ = C2s .∆2.

∣∣S∣∣ (C.4)

where ∆ is the filter width and Cs ∼ 0.2 for external flows.Ghorbaniasel started from the characteristic length scale equation, Eq. C.3, to derive hismodel for the constant Cs. It is hypothesized that:

L = Urc.Tt

c (C.5)

Where Urc and Tt

c are characteristic velocity and time scales respectively. The superscripts.r and .t denote the rotational and transitional velocities since the philosophy of the SGSmodels in LES, implying that the characteristic length scale can be assumed proportional tothe product of cross velocity terms, is used. So, the characteristic velocity in each element

174 Ghorbaniasl’s SGS model derivation [50]

center is obtained basing on the resolved rotation rate as the following:

Urc =∣∣Ω∣∣ .

2(C.6)

The time scale is obtained basing on the translational velocity and using a normalization termDn as:

Ttc = |u|

Dn(C.7)

Dn dimension is m/s2, it is given by the following relation:

Dn =√

(Ωxux)2 + (Ωyuy)2 + (Ωzuz)2 (C.8)

Substituting these relations in L equation gives:

L =

∣∣Ωu∣∣

2Dn(C.9)

For three dimensional case one can write:

Lx =

∣∣Ωxux∣∣

2Dn(C.10)

Ly =

∣∣Ωyuy∣∣

2Dn(C.11)

Lz =

∣∣Ωzuz∣∣

2Dn(C.12)

Finally, the length scale characterizing the unresolved motion is taken as the minimum ofLx, Ly and Lz. This is justified by the fact that the scale of the the unresolved scales isdetermined basing on the smallest resolved scales

L = min(Lx,Ly,Lz) (C.13)

The eddy viscosity relation C.4 is rewritten basing on the new definition of the length scalederived here, Eq. C.13, that gives:

νt =[min(Lx,Ly,Lz)

]2 .∣∣S∣∣ (C.14)

The precedent equation can be written as:

νt =[min(Rx,Ry,Rz).

]2 .∣∣S∣∣ =

[min(Rx,Ry,Rz)

]2 .2.∣∣S∣∣ (C.15)

where:

Rx =|Ωxux|

2Dn(C.16)

Ry =|Ωyuy|

2Dn(C.17)

175

Rz =|Ωzuz|2Dn

(C.18)

Comparing the two relations of the eddy viscosity, Eq. C.15 and Eq. C.4, we arrive finally tothe Smagorinsky constant Cs relation as:

Cs = min(Rx,Ry,Rz) (C.19)

This model has been incorporated to our solver SFELES, evaluated in Chap. 5 and 6 for theaerodynamics and aeroacoustics study and its results are found reasonable and more efficientthan the static Smagorinsky model.

Appendix D

The spatial discretization of the2D NS equations by Galerkinfinite elements methods [43]

In this method the solution is approximated by a linear combination the so-called basisfunctions:

fn(x) =n

∑j=1

fjϕj(x) (D.1)

where the representation parameters fj need to be determined. The basis or shape functionsϕj are defined locally as a function of polynomial interpolation functions. They must belinearly independent and respect the condition:

ϕxj = δij (D.2)

with δij is Kronecker’s function. Figure D.1 shows the global basis function associated witha node j of a two-dimensional elements P1, it takes the tent shape. In Galerkin approach, a

Fig. D.1 The global tent form basis function associated with a node j of a two-dimensionalelements P1

178The spatial discretization of the 2D NS equations by Galerkin finite elements methods [43]

single set of functions is used. This means that the basis and weight functions are chosen thesame, wj = ϕj. Consequently, the problem becomes simpler in comparison with the oppositecase. Let’us now apply this method to discretize the 2-d steady Navier Stokes equationswithout the convection terms given by the following system:

ν

(∂2u∂x2 +

∂2u∂y2

)–∂p∂x

= 0

ν

(∂2v∂x2 +

∂2v∂y2

)–∂p∂y

= 0

∂u∂x

+∂v∂y

= 0

(D.3)

Denoting un, vn and pn for the approximate solution of the velocity components and thepressure, we can write:

un(x,y) =n

∑j=1

ujϕj(x,y) (D.4)

vn(x,y) =n

∑j=1

vjϕj(x,y) (D.5)

pn(x,y) =n

∑j=1

pjϕj(x,y) (D.6)

Using ξuxi ,ξ

uyi and ξ

pi as weight functions for the momentum the continuity equations

respectively, we obtain the formulation of weighted residuals as the following:

∫Ωξux

i

[ν

(∂2un

∂x2 +∂2un

∂y2

)–∂pn

∂x

]dΩ = 0 (D.7)

∫Ωξ

uyi

[ν

(∂2vn

∂x2 +∂2vn

∂y2

)–∂pn

∂y

]dΩ = 0 (D.8)

∫Ωξ

pi

[∂un

∂x+∂vn

∂y

]dΩ = 0 (D.9)

When applying the Galerkin formulation, the weighting functions are taken the same as theshape function as the following:

ξuxi = [ϕi,0]

ξuyi = [0,ϕi]

ξpi = ϕi

(D.10)

179

Substituting these relationships and the approximate solution relationships to the equationsD.7, D.8 and D.9, we obtain:∫

Ωϕi

[ν

(∂2

∂x2

n

∑j=1

ujϕj +∂2

∂y2

n

∑j=1

ujϕj

)–∂

∂x

n

∑j=1

pjϕj

]dΩ = 0 (D.11)

∫Ωϕi

[ν

(∂2

∂x2

n

∑j=1

vjϕj +∂2

∂y2

n

∑j=1

vjϕj

)–∂

∂y

n

∑j=1

pjϕj

]dΩ = 0 (D.12)

∫Ωϕi

[∂

∂x

n

∑j=1

ujϕj +∂

∂y

n

∑j=1

vjϕj

]dΩ = 0 (D.13)

Since the shape functions are defined locally on each element, we can write the integrals inthe previous equations as the sum of the elements Ωe constituting the domain Ω:

∑e

∫Ωe

ϕi

[ν

(∂2

∂x2

n

∑j=1

ujϕj +∂2

∂y2

n

∑j=1

ujϕj

)–∂

∂x

n

∑j=1

pjϕj

]dΩe = 0 (D.14)

∑e

∫Ωe

ϕi

[ν

(∂2

∂x2

n

∑j=1

vjϕj +∂2

∂y2

n

∑j=1

vjϕj

)–∂

∂y

n

∑j=1

pjϕj

]dΩe = 0 (D.15)

∑e

∫Ωe

ϕi

[∂

∂x

n

∑j=1

ujϕj +∂

∂y

n

∑j=1

vjϕj

]dΩe = 0 (D.16)

The nodal values are constant therefore the summations can be taken out of the integrals as:

∑e

n

∑j=1

[(ν

∫Ωe

ϕi

(∂2ϕj

∂x2 +∂2ϕj

∂y2

)dΩe

)uj –(∫

Ωeϕi

∂ϕj

∂xdΩe

)pj

]= 0 (D.17)

∑e

n

∑j=1

[(ν

∫Ωe

ϕi

(∂2ϕj

∂x2 +∂2ϕj

∂y2

)dΩe

)vj –(∫

Ωeϕi

∂ϕj

∂ydΩe

)pj

]= 0 (D.18)

∑e

n

∑j=1

[(∫Ωe

ϕi∂ϕj

∂xdΩe

)uj +

(∫Ωe

ϕi∂ϕj

∂ydΩe

)vj

]= 0 (D.19)

But these equations contain second order derivatives of the linear basis functions. Thisproblem is handled using the integration by parts and applying the divergence theorem.Neglecting the integrations on the borders, it is arrived to the final discretized formulation ofthe 2-d Navier-Stokes equations by the Galerkin finite elements approach as:

∑e

n

∑j=1

[(–ν∫Ωe

(∂ϕi∂x

∂ϕj

∂x+∂ϕi∂y

∂ϕj

∂y

)dΩe

)uj +

(∫Ωe

∂ϕi∂x

ϕjdΩe

)pj

]= 0 (D.20)

180The spatial discretization of the 2D NS equations by Galerkin finite elements methods [43]

∑e

n

∑j=1

[(–ν∫Ωe

(∂ϕi∂x

∂ϕj

∂x+∂ϕi∂y

∂ϕj

∂y

)dΩe

)vj +

(∫Ωe

∂ϕi∂y

ϕjdΩe

)pj

]= 0 (D.21)

∑e

n

∑j=1

[(∫Ωe

ϕi∂ϕj

∂xdΩe

)uj +

(∫Ωe

ϕi∂ϕj

∂ydΩe

)vj

]= 0 (D.22)

Appendix E

Ensight Gold format asimplemented in SFELES

The transient format Ensight Gold [107], is implemented in SFELES. This format builds 3-Dfields and geometry from the 2-d modes solutions and the original CFD mesh. As it is clearnow, in SFELES a 2-d CFD mesh with triangular elements is used.The Ensight Gold format has been implemented such that it builds a 3-d six node pentahedronelements from two three node triangular elements of two successive modes as shown inFig. E.1

Fig. E.1 Building six node pentahedron elements in the geometry .geo created by Ensightgold format from the three node triangle elements used in SFELES

The EnSight Gold format supports structured and unstructured meshes. Both ASCII andbinary formats are implemented in SFELES.The EnSight Gold format output consists of three types of files:* .case file* .geo

182 Ensight Gold format as implemented in SFELES

* .variable (.Velocity, .Pressure, .....)

The .case file is an ASCII free format file that contains all the file and name informationfor accessing model (and measured) geometry, variable and time information. It comprisesof five sections (FORMAT, GEOMETRY, VARIABLE, TIME, FILE).The .geo file is the geometry with the 3-d mesh.The .variable files are written such that for every requested time step and for every variable afile is written. To activate this format in SFELES, it is needed to add the following lines tothe command file.

————————————————–!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!————————————————–Extract the solution (pressure velocities) in the Ensight gold format (ACTRAN)(yes:1,no:0)INT 1 IENSIGHTFORMAT0How many time steps to extract the Ensight format for?INT 1 IENSTIMESTEPS30000Interval between two stepsINT 1 IENSINTERVAL1Extraction of the solution must start after a number of time steps(it should be more than 100 because the first pressure values fluctuate badly)Writing of the solution in this format will start after this time stepINT 1 IENSSTARTTIMESTEP200The format of ensight glod files (ascii:2,binary:1)INT 1 IENSFILESFORMAT1————————————————–!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!————————————————–

This format is supported by Ensight gold and ACTRAN softwares. It is also adapted tobe read by Paraview (we can make videos to show some transient flows phenomena orinteractions which need to be demonestrated in videos to be understood).

Appendix F

Hydrodynamics reflections at themesh outlet, the physicalboundary condition

In steady flows, boundary conditions like pressure being constant or a Neumann boundarycondition are often used. The computations involving these boundary conditions, however,require a large distance placement of the outflow boundary which increase the computationalcost otherwise, hydrodynamics distortions may be created near the outlet.When the Lighthill and Möhring analogies are applied, the volume sources are needed tobe extracted on each node of the mesh. Some hydrodynamics distortions, perturbationsand/or reflections are noticed near the outlet. They contaminate the flow-field and causean overestimation of the radiated noise because they represent parasite sources. In orderto resolve this problem and to improve the quality of the sources, two solutions have beenproposed.The first is build another CFD mesh, more refined in the wake and near the outlet to makethe flow leave the computational domain without reflections.The second is to generalize and implement a suitable non-reflection boundary condition inthe CFD solver SFELES such as the physical boundary condition proposed by N. Hasan[113]. The first solution was done by building the unstructured mesh M2 and this gave verygood results. The flow field contamination has been well removed. Figure F.1 and Figure F.2show a comparison of Lighthill sources computed using the two meshes at Reynolds number160000. It is clear that the problem is resolved. So in SFELES when the mesh is refined wellin the wake and near the outlet, the flow exits without significant distortion or reflections.Figure F.3 and F.4 present another example of the reflections observed at Reynolds numberof 12000. They show how the Karman vortex street suffers of distortions and reflections nearthe domain outlet on the first mesh M1 whereas the mesh refinement done in the wake onthe second mesh M2 ensures the smooth exit of the flow at the domain outlet. Extendingthe domain far downstream of the body and applying an artificial damping by increasing theviscosity near the outlet is a traditional solution to minimize the effects of the outlet boundarycondition but it increases the simulation cost and this may weaken and kill some sources foraeroacoustics applications where the wake is an important noise source.The implementation of the non-reflection boundary condition and its results are presented inthe next paragraph.

184 Hydrodynamics reflections at the mesh outlet, the physical boundary condition

Fig. F.1 Lighthill sources: hydrodynamics reflections at the mesh M1 outlet

Fig. F.2 Lighthill sources: hydrodynamics reflections are removed, the mesh M2

185

Fig. F.3 Flow topology described by the longitudinal velocity field U at Re=12000: (Above)the mesh M1, (bottom) the mesh M2

Fig. F.4 Contour of the vorticity at Re=12000: (Above) the mesh M1, (bottom) the mesh M2


F.1 The physical boundary condition

The implementation of a non-reflection boundary condition could improve more the sourcesand prohibit even the small flow reflections resulting from the imposition of a constant zeropressure at all outlet nodes. This is not quite accurate in the case of unsteady flows and usingsmall meshes where the distance between the outlet and the wall (the airfoil) is not largeenough. It means instead of building huge meshes to respect that we can just apply a suitablenon-reflection boundary condition at the outlet which will ensure the non reflections andenhance the computational efficiency (time and memory).The physical boundary condition is a new approach proposed in [113]. It consists in theextrapolation of the velocities at the outflow boundary in the simulations of external incom-pressible flows around rigid bodies. This procedure is based on the radial behavior of thevelocity field at large distances from the body. It is called the physical boundary conditionbecause the velocities at the outflow boundary are extrapolated from the closest interiornodes (adjacent to the outlet) basing on the expected radial variation of velocity field at largedistances from the rigid body, which is inferred from physical laws such as mass conservation(continuity equation) and vorticity considerations far from the rigid body. So, the proposedboundary condition is considered to be physically consistent.

F.1.1 The governing equations

The theoretical basis of this approach has been known for a long time [115], however it hasnot been applied in CFD. In this section, we will derive the formulations of the velocity fieldin the radial direction at very large distances from a body. Figure F.5 shows a representationof a rigid body Sb enclosed by a surface S of unit depth. Denoting V(r, t) for the velocity

field, V′(r, t) for the perturbed velocity field (caused by the presence of the body) and U∞ for

the uniform stream velocity, we can write:

lim|r|→∞

V(r, t) = U∞ (F.1)

V′(r, t) = V(r, t) – U∞ (F.2)

From the mas conservation law, it is deduced:

∇.V′= ∇.V = 0 (F.3)

Applying Gauss theorem to the perturbed velocity field,∫Ω

(∇.V′)dΩ =

∫S

V′.ndS –

∫Sb

V′.ndS = 0 (F.4)

Therefore ∫S

V′.ndS =

∫Sb

V′.ndS (F.5)

F.1 The physical boundary condition 187

Fig. F.5 Representation of an arbitrary external surface S enclosing a rigid body Sb andregion Ω, illustration of the two irreducible circuits C1 and C2 enclosing a region D

The perturbed velocity field at the body surface is given as

V′= Ub(r, t) – U∞ (F.6)

As a general case, the body is assumed deformable with a velocity at its surface denotedUb(r, t). ∫

S

V′.ndS =

∫Sb

[Ub(r, t) – U∞].ndS (F.7)

However, ∫Sb

U∞.ndS = U∞

∫Sb

ndS = 0 (F.8)

Consequently ∫S

V′.ndS =

∫Sb

Ub(r, t).ndS (F.9)


For a non-deforming rigid body as in our study, the integral on the right-hand side of theprecedent relation equal to zero, thus ∫

S

V′.ndS = 0 (F.10)

This integral must be independent of the shape and the location of the surface S from the body.To achieve that, a cylinder C with a surface Scyl and a radius r and unit depth is consideredto analyze the local behavior of the velocity field at any point on the surface S, (Fig. F.5).In the following relations, Vr∞ and Vθ∞ represent the radial and azimuthal components ofinfinity velocity field U∞. ∮

Scyl(vr – Vr∞)rdθ = 0 (F.11)

The integral in the last relation must vanish when r → ∞, to realize that the term (vr – Vr∞)must behave at least as

(vr – Vr∞) ∼ 1r2 (F.12)

This relation gives the radial velocity at large distances. We need also to deduce a relationfor the azimuthal component vθ. Considering two irreducible circuits C1 and C2 enclosingthe body as shown in Fig. F.5. Denoting D for the region between C1 and C2. Taking thecurl of F.2 gives the following relation for the vorticity field

ω′= ω (F.13)

Supposing that Γ1 and Γ2 are circulations around C1 and C2, respectively, then using Stokestheorem leads to ∫

Dω

′dA = Γ1 – Γ2 (F.14)

Far away from the body, the vorticity is quite small in viscous flows. Since C1 and C2 are farfrom the body, therefore the left-hand side of the previous equation is generally quite smalland negligible. The circulation around a circuit becomes nearly constant and independentof the shape and distance of the circuit from the body. To analyze the radial behavior of vθat a point on the contour C1, consider a circular contour C of radius r passing through thepoint under consideration on C1 at a large distance from the body. The circulation around Cis given as:

Γ =∮

C(vθ – Vθ∞)rdθ (F.15)

There are only two possibilities for the precedent relation, either Γ is non-zero or zero∮(vθ – Vθ∞)rdθ = 0, ifΓ = 0∮(vθ – Vθ∞)rdθ = 0, ifΓ = 0

(F.16)

Therefore, as r → ∞, vθ → Vθ∞ must behave as(vθ – Vθ∞) ∼ 1

r2 , ifΓ = 0(vθ – Vθ∞) ∼ 1

r , ifΓ = 0(F.17)


F.1.2 Adaptation for a structured polar grid and a Cartesian grid

The relations F.12 and F.17 give the velocity field at large distances from the body. Theserelations can be utilized to extrapolate velocities from the interior to the outflow boundaryof the computational domain. The extrapolation procedure is straightforward to apply on astructured polar grid and with the aid of Fig. F.6 (left). It shows two constant curves withradius r = r1 and r = r2 and a constant θ line in a structured polar grid. The outlet boundaryof the domain is represented by r = r2 and r = r1 is the constant radius curve adjacent to r =r2. The extrapolation involves obtaining the velocity field at a typical boundary grid point

Fig. F.6 Illustrations for implementation of boundary conditions on (Left) radial grid, (right)Cartesian grid grid

2 on r2. The grid point 1 lies on r = r1 and also on the constant θ line connecting point 2to point 1. Since point 1 is the nearest radial neighbor of point 2 in the structured grid, thevelocity components vr and vθ at point 2 can be readily computed using F.12 and F.17 as:

(vr – Vr∞)1r21 = (vr – Vr∞)2r2

2 (F.18)

Since the extrapolation is radial, (Vr∞)1 = (Vr∞)2.

(vr)2 =

r1r2

2(vr)1 + (Vr∞)2

1 –(

r1r2

)2

(F.19)

Similarly for the azimuthal velocity(vθ)2 =

r1r2

2(vθ)1 + (Vθ∞)2

1 –(

r1r2

)2

, ifΓr=r1 = 0

(vθ)2 =

r1r2

(vθ)1 + (Vθ∞)2

1 –(

r1r2

), ifΓr=r1 = 0

(F.20)


Fig. F.6 (right) illustrates the procedure for implementation of boundary conditions on aCartesian structured grid. Let x = x2 be the outlet boundary grid line and x = x1 be theadjacent grid line. Let φ be the slope of the radial line drawn from a typical boundary gridpoint 2. This radial line intersects x = x1 at y1 which can be written as

y1 = y2 – (x2 – x1)tanφ (F.21)

This relation gives the location of point 1 on grid line x = x1. Let the nearest neighbor ofpoint 1 on the grid line x = x1 be point A. The value of any flow variable can be obtained atpoint 1 employing Taylor expansion around point A

φ1 = φA + (∂φ

∂y)A(y1 – yA) + (

∂2φ

∂y2 )A(y1 – yA)2

2+O

[(y1 – yA)3

](F.22)

(vr)1 and (vθ)1 can thus be obtained at point 1 from F.22. Finally, the boundary values (vr)2and (vθ)2 are obtained from F.19 and F.20.

F.1.3 Implementation in SFELES for unstructured grids, generaliza-tion for 3D flows, proposing a pressure equation for the outletBC

The extrapolation of radial and azimuthal velocities on the outlet nodes is given by therelations F.19 and F.20. So we need to determine a near neighbor node or the nearest nodein the interior domain (node 1) to use its velocities values in the extrapolation procedure toget the velocities in the outlet node (node 2). Since SFELES is a finite element solver and ituses a structured or an unstructured triangular-elements mesh, it is tricky to determine thenearest node. Figure F.7 shows a representation of the procedure applied to adapt the physicalboundary condition approximately for the implementation in SFELES for an unstructuredmesh. We proposed to plot an imaginary vertical line near the outlet boundary at a distance∆x controlled by the user in the command file (red line (1)). This line intersects the radialline (plotted from the body center to the infinity passing the considered outlet node j) ina node k which is considered for the extrapolation node. We need now to determine thelocation and velocities at the node k. The x and y coordinates of the node k are determinedby the relations.

xk = xj –∆x (F.23)

yk = yj – (xj – xk)tanφ (F.24)

Then we determine the triangular element to which the node k belongs. The node k velocitiesare computed by interpolating the velocities of the three nodes of the triangular element.Then the velocities in node j are extrapolated using the relations F.19 and F.20.In order to use this boundary condition for 3D turbulent flows, we need an equation forthe velocity in the spanwise direction. The following equation has been proposed andimplemented in SFELES:(w)2 =

r1r2

2(w)1 + (W∞)2

1 –(

r1r2

)2

, ifΓr=r1 = 0

(w)2 =

r1r2

(w)1 + (W∞)2

1 –(

r1r2

), ifΓr=r1 = 0

(F.25)


Like that we have equations for u,v and w that have been incorporated to the solver to be asan outlet boundary condition. Another equation is proposed for the pressure as the following:

(p)2 =

r1r2

2(p)1 + (P∞)2

1 –(

r1r2

)2

(F.26)

The idea here is instead of imposing the velocities on the outlet we can use this equationto impose pressure extrapolated and implemented in the same way as the first approachand similarly to imposing zero-pressure outlet but here the pressure evolves with time andlocation of the node on the outlet boundary as we will see. The validation of the proposedprocedure of the boundary condition and the sensitivity of the location of the imaginary line(1) is studied in the next paragraph.

Fig. F.7 Representation of the physical boundary condition implementation in SFELES

F.1.4 Validation of the physical boundary condition: Velocities approachLaminar unsteady flow case at Re=2000

This Reynolds number has been chosen as a validation case because of the production of ahuge vortex shedding street with strong vortices in the wake which will exit the outlet of thedomain causing high variations of the pressure and velocities values. When these vorticesreach the outlet boundary, they may lead to large spurious reflections which can perturb theflow as it has been already shown in this appendix when the Lighthill sources are computed.This flow is therefore a good prototype to test the physical outlet boundary condition.The instantaneous vorticity obtained using the physical boundary condition is shown onFig. F.8. The flow exits without significant distortion or reflections at the outlet. The averageand instantaneous pressure variation on the outlet are plotted on Fig. F.9. This figure showsthat the pressure on the outlet nodes varies with the time step and the node location which


Fig. F.8 Instantaneous vorticity

ensures the non reflection of the pressure waves on the outlet border. It can be seen that, atthe nodes located where the vortices exit, the pressure takes negative values could arrive to-12 pa. In contrast, away from this region where the influence of the vortex street is low, thepressure takes small values around zero. These observations are physically reasonable. In

Fig. F.9 The average and instantaneous evolution of the pressure along the outlet

order to study the effect of the physical boundary condition on the aerodynamics coefficients,a comparison of the temporal evolution of the aerodynamics coefficients is done with thezero fix pressure boundary condition. This is depicted on Fig. F.10. The same evolution andconvergence of the solution is obtained. The same lift coefficient Cl, drag coefficient Cd andStrouhal number St are obtained using the physical boundary condition.


Fig. F.10 The temporal evolution of the aerodynamics coefficients

Fig. F.11 The mean pressure coefficient distribution Cp


The mean pressure coefficient distribution Cp is also found identical for the two boundaryconditions as shown on Fig. F.11. Longitudinal (U) and normal (V) average velocity profiles,and the longitudinal (u′) and vertical (v′) velocity fluctuations RMS profiles are extractedin the wake in the position x/C = 2 which is near the outlet. As shown on Fig. F.12, almostidentical results are obtained using the two boundary conditions for (U), (u′) and (v′) whereasa significant difference is noticed for the normal velocity. The values corresponding to thesuction surface decay faster which means smaller wake width and that the vortices exit thedomain more smoothly.

Fig. F.12 Longitudinal and normal average velocities, longitudinal and vertical velocityfluctuations RMS in the wake at x/C=2

Turbulent flow case at Re=160000

The physical boundary condition has been applied for a turbulent flow case, the CD airfoil, atRe=160000 and the results are compared to the zero boundary condition outlet. The averageand instantaneous pressure variation on the outlet are plotted on Fig. F.13. It could be seenthat the pressure varies with the time step and the node location on the outlet boundary. Thepressure fluctuates almost around the value -5 pa. The mean pressure and friction coefficientsdistribution on the airfoil are shown in Fig. F.14. The results are almost identical. Thecomparison of the aerodynamics coefficients is also considered, the results are shown in tableF.1. A very slight increasing of the lift coefficient is noticed whereas the drag coefficient hasthe same value for the two boundary conditions. Longitudinal (U) and normal (V) averagevelocity profiles, and the longitudinal (u′) and vertical (v′) velocity fluctuations RMS profilesare extracted in the wake in the position x/C = 2.3 which is adjacent to the outlet. Figure F.15shows very similar behavior of the obtained profiles.These results confirm the validity of the proposed boundary condition for the laminar andturbulent flows. It is important to mention that for some cases, where the pressure may havevery important fluctuations, it is needed to fix the pressure to zero on a node of the meshto prevent the divergence if the pressure arrives to very high values (infinity). Concerning



Aerodynamics coefficients The physical BC The zero pressure BCCl 1.0031 0.9989Cd 0.0859 0.0858

Table F.1 Comparison of the aerodynamics coefficients between the outlet phy. BC andzero-pressure BC

the location of the imaginary line which determine the nearest nodes of the outlet, it hasto be located near the outlet with a distance less than the smallest element of the mesh atthe boundary. For instance, in the previous validation cases, it is chosen to be located at adistance of 0.001 m. No significant difference is noticed if it is located at a distance of 0.005m . More developments and validations cases need to be performed to ensure the validity ofthis boundary condition and to show the benefits and advantages in comparison with the zeropressure coefficients. This BC may improve the aeroacoustics results in the hybrid methodssince it may prohibit spurious sources accumulated with time and caused by some reflectionsof the flow.


Fig. F.14 Mean pressure and friction coefficients distribution, comparison between the outletphy. BC and zero-pressure BC

F.1.5 Validation of the physical boundary condition: Pressure approachLaminar unsteady flow case at Re=2000

The validation of the pressure approach is made following the same steps as in the previousparagraph. The average and instantaneous pressure variation on the outlet are plotted onFig. F.16. It can be noticed that the pressure takes negative values in the region correspondingto the wake on the outlet border while it takes zero value apart from this region. As acomparison with the velocities approach, it is noticed the maximum negative pressure value


Fig. F.15 Longitudinal and normal average velocities, longitudinal and vertical velocityfluctuations RMS in the wake at x/C=2.3



Aerodynamics coefficients/Strouhal number The physical BC The zero pressure BCCl 0.5696 0.5687Cd 0.1325 0.1324St 0.1536 0.1536

Table F.2 Comparison of the aerodynamics coefficients and Strouhal number between theoutlet phy. BC and zero-pressure BC

arrives to -1.2 pa whereas using the velocities approach it can arrive to -12 pa as it has beenshown. Almost identical aerodynamics coefficients and Strouhal number are obtained asshown in table F.2 with a very slight increase of the lift coefficient.

Fig. F.17 The mean pressure coefficient distribution Cp

Identical mean pressure coefficient and average wake velocities are obtained and shownin Fig. F.17 and Fig. F.18 respectively.

Turbulent flow case at Re=160000

The average and instantaneous pressure variation on the outlet are shown on Fig. F.19. Itcould be seen that the pressure fluctuates around the zero value taking negative and positivevalues between -0.7 to 0.7 pa. Almost identical aerodynamics coefficients values are obtainedas shown in table F.3 with a very slight increase of the lift coefficient. Mean pressurecoefficient and average wake velocities are obtained and shown in Fig. F.20 and Fig. F.21respectively. The results are identical to those obtained using the zero pressure boundary


Fig. F.18 Longitudinal and normal average velocities, longitudinal and vertical velocityfluctuations RMS in the wake at x/C=2


condition. A very slight increase of the lift coefficient is noticed whereas the drag coefficienthas almost the same value for the two boundary conditions.

Figure F.21 shows very similar behavior of the obtained profiles.


Fig. F.20 Mean pressure distribution, comparison between the outlet phy. BC and zero-pressure BC

Aerodynamics coefficients The physical BC The zero pressure BCCl 1.0027 0.9989Cd 0.0860 0.0858

Table F.3 Comparison of the aerodynamics coefficients between the outlet phy. BC andzero-pressure BC

Another motivation to develop this boundary condition and need to be validated later comesfrom the fact that if the size of the computational mesh around the studied body can belimited to relatively smaller dimensions, the computational efficiency can be enhancedwithout affecting the results.


Fig. F.21 Longitudinal and normal average velocities, longitudinal and vertical velocityfluctuations RMS in the wake at x/C=2.3

Appendix G

Tonal noise corresponding to thevortex shedding at Reynoldsnumber of 12000

Lighthill and Möhring analogies and Curle’s formulation have been applied to study andpredict the tonal noise produced by the 2d unsteady and 3d unsteady laminar regimes corre-sponding to Reynolds number of 12000. The obtained results are presented and comparedhereafter. These analogies have been applied using the same methodology explained in Chap.6.The same parameters are used in the aerodynamic simulation as performed in Chap. 4with one difference which is adding perturbations in order to push the flow in the spanwisedirection in one case, this leads to a 3d unsteady laminar regime, periodic in the spanwisedirection. In the other case, no perturbations are applied, so the regime is still 2d laminar.The idea behind this study is to predict the radiated noise caused by the vortex shedding,then to study the 3d effects caused by the periodic flow in the spanwise direction tryingto find the dominant frequency in this direction. So, in the 3d case, it is expected to havetwo series of tones. The first corresponds to the frequency of the vortex shedding and thesecond corresponds to the periodic flow in the spanwise direction in comparison with the 2dcase. Another goal is trying to justify some sub-harmonics noticed in the far field acousticpressure for the 2d case. First the 2d case is completely studied then the 3d case is studiedand compared. The CFD mesh used in this chapter is the computational mesh M2. Theacoustic mesh is smaller in dimensions and coarser than the CFD mesh, the criterion 6quadratic elements by an acoustic wave. Concerning the CFD sources for Lighthill andMöhring analogies, the velocity components are extracted in the Ensight Gold format onevery point of the reconstructed 3-D CFD mesh for more than 5 airfoil flow-through times(almost 0.1 sec). The acoustic mesh is shown in Fig. G.1. This Figure shows also the orderand the interface of the infinite elements used to predict the SPL in the far field and prohibitthe acoustics waves reflections.

204 Tonal noise corresponding to the vortex shedding at Reynolds number of 12000

Fig. G.1 The 3-D acoustic mesh

G.1 The sound radiated of the CD airfoil in a 2d laminarunsteady regime, Re=12000

G.1.1 Results for Lighthill’s analogy

As it is already shown in Chap. 6, when Lighthill and/or Möhring analogies are applied,a truncation phenomenon is occurred at the domain boundaries and can create spurioussound especially at low frequencies for laminar flows. Figure G.2, (above) presents thisphenomenon, an artificial dipole has appeared near the domain outlet for the frequency 369Hz. In order to prevent this phenomenon, a spatial filter of type cosine is applied near theborders. The source terms are damped out slowly toward the sources boundaries by theapplied filter as shown in Fig. G.2, (middle). The result of the application of this filter isshown in Fig. G.2, (bottom), it can be seen that the artificial dipole has been removed fromthe acoustic field. A filter distance of 0.5 C is needed to ensure the elimination of the parasitedipole. Figure G.3 shows the Lighthill’s sources for the frequencies 369 Hz, 628 Hz and 731Hz respectively. These frequencies have been chosen to show that the sources correspondingto the dominant frequency 369 Hz, which is the von Karman vortex shedding frequency andits harmonics, are strong and important whereas for other frequencies the sources are weak.Lighthill’s acoustic pressure maps for the near field are presented in Fig. G.4 for frequencies369 Hz, 628 Hz and 731 Hz respectively. These results show that the acoustic pressure isimportant for the frequency 369 Hz which corresponds to the vortex shedding which is themain source of the obtained tonal noise. For the frequency 628 Hz, the acoustic pressure isvery small whereas for the frequency 731 Hz, the acoustic pressure is important because thisfrequency corresponds to the first harmonic of the dominant frequency. In order to visualize

G.1 The sound radiated of the CD airfoil in a 2d laminar unsteady regime, Re=12000 205

Fig. G.2 (Above) Truncation phenomenon at the outlet for the frequency 369 Hz, (middle)the applied Cosine Filter, (bottom) the same acoustic field after the application of the filter

the acoustic pressure outside the finite elements domain where the infinite elements are usedto model the unbounded domain, a Field maps plane of a dimension 1.4*1.4 m is added inthe mid-span to show the acoustic waves’ propagation from the sources (near filed) to the farfield where the receiver is located. Results are shown in Fig. G.5 for the frequency 369 Hz. It


Fig. G.3 Lighthill’s sources on the acoustic mesh for the frequencies 369 Hz, 628 Hz and 731Hz

is clear that the acoustic waves leave the computational domain boundaries smoothly withoutany reflections; this is thanks to the use of the infinite elements. Far field acoustic pressurespectra for the considered receiver are depicted on Fig. G.6. The result presents a main


Fig. G.4 Lighthill’s acoustic pressure maps for the near field for frequencies 369 Hz, 628 Hzand 731 Hz


Fig. G.5 Lighthill’s analogy, acoustic waves’ propagation to far field for the frequency369.656 Hz

peak corresponding to the von Karman vortex shedding frequency which is 369 Hz. Othertones exist and they correspond to harmonics of the dominant frequency. Apart from thesefrequencies, the acoustic pressure is really negligible. Some sub-harmonics are noticed inthe far field acoustic pressure spectra. In order to obtain the directivty patterns, 25 receiversare mounted on a semi-circle of 2 m radius around the trailing edge. The result is shown inFig. G.8 for 4 frequencies including the dominant frequency and its harmonics. It can beseen that, at the frequency 369 Hz, the dipole behavior is clear which means the dominanceof the dipolar sources (surface contribution). However, at high frequencies, the multipolarbehaviour is present indicating the efficient contribution of the volume or quadripolar sources.

G.1.2 Results for Möhring’s analogyMöhring’s analogy has been applied using the same steps as Lighthill’s analogy according tothe methodology shown in Chap. 6. The importance of this analogy that it allows to studythe convectional effects of the mean field on the acoustic waves’ propagation. The finalformulation of Möhring’s analogy used via Actran in this research is the following:

∂

∂t(

ρ

ρ2Tc2

DbDt

) + ∇(ρvρ2

Tc2DbDt

–ρ

ρ2T

∇b) = –∇[1ρT

(ρv× (∇×v)] (G.1)

Figure G.8 shows Möhring sources mapped on the acoustic mesh at frequencies 369 Hz,628 Hz and 731 Hz. Möhring acoustic pressure maps for the near field corresponding to thementioned frequencies are shown in Fig. G.9.


Fig. G.6 Lighthill far field acoustic pressure spectra for the receiver located at mid-plane, 2m above the trailing edge

Fig. G.7 Directivity patterns at frequencies, 369 Hz, 628 Hz, 731 Hz and 998 Hz


Fig. G.8 Möhring’s sources on the acoustic mesh for the frequencies 369 Hz, 628 Hz and 731Hz

The acoustic waves’ propagation from the sources to the far field is shown on a plane inFig. G.10 for the frequency 369 Hz. The far field pressure is predicted and compared withthat obtained using Lighthill’s analogy in Fig. G.11. The tonal noise is very similar; the


Fig. G.9 Möhring’s acoustic pressure maps for the near field for frequencies 369 Hz, 628 Hzand 731 Hz


Fig. G.10 Möhring’s analogy, acoustic waves’ propagation to far field for the frequency369.656 Hz

acoustic pressure corresponding to the main frequency (369 Hz) is almost the same. However,for mid and high frequencies the acoustic pressure is a little bit more important for Möhring’sanalogy than for Lighthill. This can be justified by the diffraction of the acoustic wavescaused by the mean flow. However, these effects are almost negligible for the consideredMach number, 0.047.

G.1.3 Results for Curle’s formulation

The effect of solid boundary (airfoil) is taken into account as well as the volume sources inthis formulation applying the first proposed approach which is the implementation of thesetwo integrals in SFELES. Figure G.12 presents the far field acoustic pressure considering thesurface and volume sources contributions where these terms are implemented in SFELES. Itcan be seen that the surface source contribution is dominant however, the volume contributionis important as well and not negligible especially for mid and high frequencies as shown aswell in Fig. G.13 which presents a comparison of these two contributions with Lighthill’sanalogy.

A comparison of the far field acoustic pressure obtained by the three methods: Lighthill,Möhring and Curle is made and shown in Fig. G.14. This Figure shows that the dominantfrequency, its harmonics and the corresponding acoustic pressure are well captured by theproposed approach for Curle’s formulation. However, for mid and high frequencies peaks,Curle’s acoustic pressure has higher values. Some sub-harmonics are not captured by thisapproach. These results confirms that Curle’s formulation is a reliable formulation for thelaminar flow noise as well as for the turbulent flows already confirmed and shown in Chap.


Fig. G.11 Comparison between Lighthill and Möhring far field acoustic pressure spectra

Fig. G.12 Far field acoustic pressure at the considered receiver via Curle’s formulation

6. It can give a fast idea about the far field radiated noise from airfoils without the need ofsources extraction which need sometimes a lot amount of space (for the turbulent flows).


Fig. G.13 Curle’s formulation, surface and volume contributions to the far field acousticpressure compared to Lighthill’s result

Fig. G.14 Comparison of the far field acoustic pressure at the considered receiver obtainedvia the three methods: Lighthill, Möhring and Curle


G.2 The sound radiated of the CD airfoil in a 3d laminarunsteady regime, Re=12000

Since Möhring and Lighthill presented very similar results, just Lighthill’s analogy is con-sidered here. Curle’s formulation is considered applying the second approach proposed inChap. 6 basing on the temporal history of the aerodynamic forces. Far field acoustic pressurespectra are depicted on Fig. G.15 and compared with the 2d case. The result presents a mainseries of tones, as expected, corresponding to the von Karman vortex shedding frequency atalmost the same frequencies for the 2d case with a delay of almost 20 Hz and with almostthe same SPL values. The main differences noticed are:First, a second series of tones is produced with a dominant frequency around 167 Hz andharmonics around 533 Hz and 898 Hz. This could be justified as the tones corresponding tothe periodic flow in the spanwise direction.Second, the acoustic pressure spectra are more important for all frequencies with fewersub-harmonics. Since the same behavior is obtained considering the noise produced by the

Fig. G.15 Comparison of the far field acoustic pressure obtained considering 2d unsteadyand 3d unsteady regimes corresponding to Re=12000

vortex shedding, just the near and far field maps and the source of the dominant frequencycharacterizing the spanwise periodic flow will be demonstrated. These results are shownin Figures G.16, G.17 and G.18 respectively. It can be noticed that, for the 3d flow at thefrequency 167 Hz, we have an important source which produces an important peak of SPL,while at this frequency, the SPL is not important for the 2d case.

In order to confirm these observations, the Strouhal analysis of the lift coefficient isconsidered for the 2d and 3d flows. The results are depicted on Fig. G.19. It can be noticedthat for the 3d unsteady spanwise-periodic flow, the figure exhibits a second peak at the


Fig. G.16 Lighthill’s acoustic pressure maps for the near field for frequency 167 Hz, (above):2d case, (bottom): 3d case

frequency 167 Hz, which does not exist for the 2d unsteady case. The delay is also clearconsidering the peak corresponding to the vortex shedding. Finally, the aerodynamic forcesapproach of Curle is applied and compared to Lighthill’s result. It can be seen that thisapproach can reproduce in acceptable approximation the far field SPL. The two series oftones are captured with some differences at some frequencies.As a conclusion of this appendix, the tonal noise case is studied for a laminar flow at Reynoldsnumber of 12000. Tonal noise that arises from vortex shedding generated by laminar boundarylayer is the main acoustic source for the laminar flow where a dominant peak is found atthe frequency of the vortex shedding (369 Hz). The dipole sources are dominant whereasthe quadrupole contribution is found important despite of the low Mach number consideredin this study. The convectional effects are found negligible at the considered Mach number.Finally, the results of the proposed approaches for Curle’s formulation match well the resultsobtained by Lighthill and Möhring analogies. The vortex shedding frequency, its harmonicsand the corresponding acoustic pressure are well captured for the laminar case. A secondseries of tones is produced with a dominant frequency around 167 Hz characterizing the


Fig. G.17 Lighthill’s analogy, acoustic waves’ propagation to far field for the frequency 167Hz

Fig. G.18 Lighthill’s sources on the acoustic mesh for the frequency 167 Hz

sound produced by the spanwise periodic flow. The acoustic pressure spectra are found moreimportant for all frequencies for the 3d case with fewer sub-harmonics in comparison withthe 2d case.


Fig. G.19 Comparison of Strouhal number for the 2d unsteady and 3d unsteady regimescorresponding to Re=12000

Fig. G.20 Comparison of the far field acoustic pressure obtained via Lighthill and the secondapproach of Curle using aerodynamics forces for the 3d unsteady regimes corresponding toRe=12000

calculation of aerodynamic noise of wing airfoils by

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