Calculation of Aerodynamic Noise of Wing Airfoils by Hybrid Methods

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Calculation of Aerodynamic

Noise of Wing Airfoils by Hybrid

Methods

Rabea Matouk

Department of Aero-Thermo-Mechanics

Brussels School of Engineering, Université Libre de Bruxelles

This dissertation is submitted for the degree of

Doctor of Engineering Sciences and Technology

Promotors:

Prof. Gérard Degrez

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Acknowledgements

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

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

I want to thank Dr. Christophe Julien (the von Karman Institute for Fluid Dynamics) for his help and rich discussions about my work, for sending me his results to compare with my results 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 the second paper and for his advice during the thesis committee meetings.

I would also like to acknowledge my committee members and the PhD thesis commission members:

Prof. Gérard Degrez Prof. Jean Louis Migeot Prof. Herman Deconinck Dr. Yves Detandt

Prof. Christophe Schram Prof. Ghader Ghorbaniasl

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

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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 of this research, and in particular the mechanical engineering faculty and all its professors particularly Prof. Mustafa Taki, my supervisor in Syria.

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Abstract

This research is situated in the field of Computational AeroAcoustics (CAA). The thesis focuses on the computation of the aerodynamic noise generated by turbulent flows around wing, fan or propeller airfoils. The computation of the noise radiated from a device is the first step for designers to understand the acoustical characteristics and to determine the noise sources in order to modify the design toward having acoustically efficient products. As a case 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 to simulate 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 far field, 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 aspects have been developed in the used in-house solver SFELES, including the implementation of a new SGS model, a new outlet boundary condition and a new transient format which is used to extract the noise sources to be exported to the acoustic solver, ACTRAN. The second part of this thesis is concerned with the aeroacoustic study where four methods have been applied, among them two are integral formulations and the two others are partial-differential equations. The first method applied is Amiet’s theory, implemented in Matlab, based on the wall-pressure spectrum extracted in a point near the trailing edge.

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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 from

Brooks 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 of

Ffowcs 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.

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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. Figure reproduced from [69] . . . 53 4.5 (Left) Experimental Set-up of ECL Large Wind Tunnel (right) Representative

figure 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 56 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 . . . 57 4.8 Contours of the spanwise vorticity at Re=2000, red (15:7:50), blue (-50:7:-15) 57 4.9 Contours of spanwise velocity at Re=6450. Red surfaces mark positive

values whereas the blue surfaces mark the negative values . . . 58 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 mark positive and negative surfaces of the streamwise vorticity . . . 58 4.11 Contours of the vorticity magnitude (200) at Re=15000 colored by the

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

longitudinal 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 mark positive and negative values of the streamwise vorticity whereas the green and yellow surfaces mark the transverse vorticity . . . 60 4.14 Contours of the vorticity magnitude (100:400) at Re=52000 colored by the

longitudinal velocity . . . 61 4.15 Contours of the vorticity magnitude (100:400) at Re=60000 colored by the

longitudinal velocity . . . 61 4.16 All flow patterns around the CD airfoil according to flow Reynolds number 62 4.17 Average pressure and friction coefficients distribution on the airfoil surface 63 4.18 Evolution of the lift and drag coefficients with Reynolds number . . . 64 4.19 Evolution of the Strouhal number with Reynolds number (Rec) . . . 65

4.20 Evolution of the Strouhal number with Reynolds number Re_d . . . 66 4.21 The recirculation region around the trailing edge with the supposed diameter D 67 5.1 Evolution of Ghorbaniasl’s model constant C_S: a) x/C=-0.6 for 64 and 32M

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List of figures xiii

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) 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 simularecircula-tions . . . 75 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 . . . . 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 positions

in the wake . . . 82 5.11 The vertical (v′) velocity fluctuations RMS profiles in several positions in

the 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 the

suction side at x/C=-0.02 . . . 89 5.15 Flow topology of simulations performed on M2 described by the criterion

Q(Q.C2

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 surface

using mesh M2, (Bottom): Average friction coefficient (Cf) distribution using mesh M2 . . . 92 5.17 Average velocity profiles in the boundary layer on the upper surface at

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xiv List of figures

5.20 Comparison of the spanwise coherence function and length between the two computational meshes M1 and M2 . . . 95 5.21 Flow topology of simulations performed on M2 described by the criterion

Q(Q.C2

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 surface

using mesh M2, (Right): Average friction coefficient (Cf) distribution using mesh M2 . . . 97 5.23 Average velocity profiles in the boundary layer on the upper surface at section

x/C = -0.02 . . . 97 5.24 Power Spectral Density of pressure fluctuations at the position x/C=-0.02 . 98 5.25 The viscous, Reynolds and SGS stresses in the section x/C=-0.7 for the two

spanwise extensions . . . 99 5.26 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 for

the considered airfoil, trailing edge formulation, (left) main scattering term L1(right) leading-edge back-scattering correctionL2 . . . 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): WALE

model, c): Smagorinsky model . . . 105 6.4 Trailing edge sound using Amiet’s theory using coherence length extracted

from LES simulations . . . 106 6.5 Curle: overall noise (implementation in SFELES), (CFD Mesh M2) . . . . 109 6.6 Surface and volume sources contributions to the overall far-field acoustic

pressure for Smagorinsky’s model . . . 110 6.7 Surface and volume sources contributions to the overall far-field acoustic

pressure for WALE model . . . 110 6.8 Surface and volume sources contributions to the overall far-field acoustic

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List of figures xv

6.11 Comparison of the SPL obtained using the two CFD meshes M1 and M2 for Ghorbaniasl’s model . . . 113 6.12 Comparison of the SPL obtained with 0.1C and 0.2C spanwise extensions

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

extensions 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 approaches

for 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 Ensight

Gold 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 maps

plane 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 the application of the filter . . . 123 6.21 Lighthill sources on the acoustic mesh at frequencies, 400 Hz, 800 Hz, 1200

Hz and 1600 Hz . . . 124 6.22 Lighthill near field acoustic pressure at frequencies, 400 Hz, 800 Hz, 1200

Hz and 1600 Hz . . . 125 6.23 Lighthill, acoustic waves’ propagation to far field for the frequencies, 300

Hz and 400 Hz . . . 126 6.24 Lighthill, acoustic waves’ propagation to far field for the frequencies, 600

Hz and 800 Hz . . . 127 6.25 Comparison of the SPL obtained with ACTRAN/Lighthill’s analogy and that

measured from the experiments . . . 128 6.26 Directivity patterns at frequencies, 250 Hz, 400 Hz, 800 Hz, 1200 Hz and

1800 Hz . . . 129 6.27 Sources truncation phenomenon effect, comparison of the SPL obtained with

and without the application of the filter near the domain borders . . . 129 6.28 Comparison of the SPL obtained with ACTRAN/Lighthill’s analogy,

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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, 300

Hz, 400 Hz , 500 Hz and 600 Hz . . . 134 6.31 Lighthill, acoustic waves’ propagation to far field for the frequencies, 700

Hz, 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 by

Ensight 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) the

mesh M2 . . . 185 F.5 Representation of an arbitrary external surface S enclosing a rigid body Sb

and region Ω, illustration of the two irreducible circuits C1 and C2 enclosing a region D . . . 187 F.6 Illustrations for implementation of boundary conditions on (Left) radial grid,

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List of figures xvii

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

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

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

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

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

zero-pressure BC . . . 200 F.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 . . . 204 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 . . . 205 G.3 Lighthill’s sources on the acoustic mesh for the frequencies 369 Hz, 628 Hz

and 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 frequency

369.656 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 . . 209 G.8 Möhring’s sources on the acoustic mesh for the frequencies 369 Hz, 628 Hz

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

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xviii List of figures

G.10 Möhring’s analogy, acoustic waves’ propagation to far field for the frequency 369.656 Hz . . . 212 G.11 Comparison between Lighthill and Möhring far field acoustic pressure spectra213 G.12 Far field acoustic pressure at the considered receiver via Curle’s formulation 213 G.13 Curle’s formulation, surface and volume contributions to the far field acoustic

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

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

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

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

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

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

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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 . . . 70 5.2 Recirculation bubble size: comparsion with other simulations . . . 73 5.3 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 analogy

Curle’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. BC

and zero-pressure BC . . . 195 F.2 Comparison of the aerodynamics coefficients and Strouhal number between

the outlet phy. BC and zero-pressure BC . . . 198 F.3 Comparison of the aerodynamics coefficients between the outlet phy. BC

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Nomenclature

Greek Symbols δ_ij Kronecker delta λ Sound wave length µ Dynamic viscosity ν Kinematic viscosity νt Eddy viscosity ρ Flow Density ρ₀ Atmospheric Density ρa Acoustic Density

τ_ij Viscous stress tensor τSGS_ij Subgrid stress tensor △t Time step (sec) uτ Friction velocity

Other Symbols

C Airfoil chord length c₀ Sound speed

Cs Smagorinsky constant

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xxii Nomenclature

p_a Acoustic pressure S_ij 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

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Chapter 1

Introduction

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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 attention in recent years and becomes an intense area of research. In this thesis, the broadband noise or trailing-edge noise of a fan airfoil is simulated and predicted using the hybrid methods of aeroacoustics as presented in the following sections. The mechanisms of the airfoil trailing-edge noise will be explained to better understand the physical phenomena involved in our study.

1.1 Fan noise

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1.1 Fan noise 3

free-field applications (wind turbines [3], helicopters [4],. . . ). The noise generated by these devices is referred to as fan noise [5]. The simulation of a complete fan is challenging and we focus the analysis on some fan noise components. New quieter fan designs requires accurate and 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 turbulence level is small. It is the minimum noise that can be produced by rotating machines [9]. The turbulent structures of the boundary layer created on the airfoil surface are highly modified as they pass to the trailing edge. The energy carried on by vortices is scattered by the trailing edge singularity in acoustics propagated to the far field. It is therefore the interaction between the singularity at the trailing edge and the unsteady nature of the flow, which is usually manifested by turbulent structures presented in the near wall and into the near wake of the trailing 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 study conducted by Brooks et al. [6, 7]. Experiments were conducted on a two-dimensional academic profile, the NACA 0012 at a Reynolds number of 3.106, so that the boundary layer is fully turbulent at the trailing edge. The mechanisms of trailing-edge noise production are divided 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).

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4 Introduction

Fig. 1.4 Production mechanisms of the trailing-edge noise, Figures reproduced from Brooks et 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 layer with large vortices scales is produced.

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

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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 previous case [8]. These mechanisms present noise levels in different characteristics. For example, the diffraction of the turbulent structures by a trailing edge produces a broadband component; if the angle of attack is sufficiently large, the boundary layer detaches and results in a vortex street, fairly large, producing a low frequency noise. This vortex street forces the appearance of a discrete mode in the pressure spectrum. If the profile is truncated, the wake is organized as a von Karman vortex street. The produced noise is characterized by a spectral line extended around 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 conditions upstream. 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 and therefore, it is greater than the trailing-edge noise [11]. Another parameter influencing the produced 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 are similar [12].

1.2 Computational methods of Aeroacoustics

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

1.2.1 The direct numerical acoustics method:

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6 Introduction

and low-dissipation schemes is always aimed by researchers as an alternative to the classical methods of applied mathematics for computational fluid mechanics [18]. Furthermore the computational mesh has to be built so that both the flow and its sound can be well represented respecting the CFD and acoustic criteria relying on the smallest scales to be resolved [17]. In fact, 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 problems have gradually progressed from idealized cases (co-rotating vortices [19], vortex pairing in a two-dimensional compressible mixing layer [20]) towards the direct computation of a complete flow-regions for supersonic jet [21] and subsonic [22] but for moderate Reynolds numbers.

1.2.2 The hybrid methods of aeroacoustics:

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1.3 Objectives of the dissertation 7

subsonic flows.

Moreover, there is a third approach of CAA which can be considered in the hybrid method group, 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 only steady CFD but it makes high approximation on the acoustic source where a generated random velocity field by a finite sum of discrete Fourier modes based upon averaged data of the flow field is used to determine the source for acoustic perturbation so its results are inaccurate. As a conclusion of this paragraph, the direct and SNGR methods have challenges either efficiency or accuracy. The hybrid methods offer a good balance between these two criteria 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 different sources 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 been accounted for via Möhring’s analogy (considering the laminar case) whereas via Curle’s integral formulation, the dipole sources caused by the presence of the airfoil are computed separately from the quadrupole contribution. Amiet’s theory is applicable and valid for high frequencies.

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 has been studied considering three models among them one is implemented and validated in the framework of this study. The dissertation will focus on analyzing the physical mechanisms of 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.

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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 SFELES and the acoustic solver ACTRAN are defined. The finite elements, the infinite elements and other acoustic and CFD principles are presented.

The fourth chapter is devoted to define the case study and to study the evolution of the flow regimes 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 turbulent flow 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 by Amiet 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 is presented 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 is studied 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. As original works we can cite:

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

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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 pressure profile 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 the coherence length of SFELES.

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

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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. The acoustic domain is decomposed into a flow region containing the source region in which the turbulent fluctuations generate the sound and an acoustic quiescent and uniform region where the 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, in absence of the external forces and mass sources and for compressible flow, as:

∂ρ ∂t + ∂ρu_i ∂x_i = 0 (2.1) ∂ρu_i ∂t + ∂ρu_iu_j ∂x_j = – ∂p ∂x_i+ ∂τ_ij ∂x_j (2.2)

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

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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 δ_ij is the Kronecker delta. From the momentum equation 2.2, we can write:

∂ρu_i ∂t = –

∂

∂x_j(ρuiuj+ δijp – τij) (2.4) Adding and subtracting the term c2₀∂ρ/∂x_ito the precedent equation give:

∂ρu_i ∂t + c 2 0 ∂ρ ∂x_i = – ∂T_ij ∂x_j (2.5)

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

T_ij= ρu_iu_j+ δ_ij[(p – p₀) – c2₀(ρ – ρ₀)] – τ_ij (2.6) With ρ₀ and p₀ the atmospheric density and pressure respectively. Differentiating the continuity equation 2.1 with respect to time, taking the divergence of the equation 2.5 and subtracting the results lead to the following expression:

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2.1 Lighthill’s analogy 13

This hyperbolic partial differential equation is the Lighthill’s inhomogeneous wave equation solved for the acoustic density fluctuation ρa[23]. The left hand side of this expression is

defined 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 presence of 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 the source term vanishes outside the turbulent region. Indeed, for a turbulent flow embedded in a uniform atmosphere at rest, Lighthill’s stress tensor can be neglected outside the turbulent region itself. Inside the turbulent region, the contribution of each term of T_ij will be considered separately.

The term δ_ij[(p – p₀) – c2₀(ρ – ρ₀)] is related to entropy variations inside the source region, it vanishes exactly for isentropic flows. Moreover, the effects of viscosity and heat conduction are expected to cause only a slow damping over very large distances due to the conversion of acoustic energy into heat. Therefore, for high Reynolds number it is possible to neglect the viscous stress tensor τ_ij[41]. Based on these assumptions, the Lighthill’s tensor reduces to: T_ij= ρ₀u_iu_j (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 ∂x_i∂x_j Z V( T_ij(y, t – x–y c₀ ) x – y )dV (2.9)

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

ρa= 1 4πc2₀ Z V( (x_i– y_i)(x_j– y_j) x – y 3 ) 1 c2₀ ∂2 ∂t2Tij(y, t – x – y c₀ )dV (2.11) For the geometrical far-field approximation, detailed in next paragraph, the receiver is located at large distances compared to the source region dimensions so we can write x_i– y_i≈ x_i, assuming that the coordinate system origin is in the source region y = 0, no solid boundaries are immersed in the fluid, and applying p_a= ρa∗ c2₀ give the final far-field integral solution

for Lighthill’s analogy as the following [23]:

p_a= xixj 4π |x|3c2₀ ∂2 ∂t2 Z V[Tij]tedV(y) (2.12)

This solution is only valid for exterior problems. [.]te means a term evaluated at the retarded

time defined as the propagation delay time between the sound emission (source) and reception (observer) given as te= t –

x–y

c₀ = t –cr₀.

2.2 Curle’s formulation

Lighthill’s analogy has been extended by Curle [25] in order to incorporate the influence of the presence of stationary solid boundaries on the aerodynamic sound. The surface is expected 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 wake of 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 aerodynamic

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2.2 Curle’s formulation 15

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

p_a= 1 4πc2₀ ∂2 ∂t2 _x ixj |x|3 Z VTijdV(y) te – 1 4π ∂ ∂t 1 |x| Z S ρv_in_idS te – 1 4πc₀ ∂ ∂t _x j |x|2 Z S [P_ij+ ρv_iv_j]n_idS te (2.13)

With P_ij= pδ_ij– τ_ijis the stress tensor, n_i the wall normal pointing into the flow. Assuming that the source is acoustically compact, the variations of the retarded time te are so small

comparing to the retarded time itself te= t –_cr₀, so they can be neglected. Furthermore, the

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

p_a(x, t) = 1 4πc2₀ ∂2 ∂t2 _x ixj |x|3 Z VTijdV(y) t_e – 1 4πc₀ ∂ ∂t _x j |x|2 Z SpnidS t_e (2.14)

This formulation is considered in this study and computed in SFELES, the CFD solver used in the present thesis. It is important to note that it is an approximated integral solution of Lighthill’s analogy where some simplifying assumptions are applied making it applicable to specific problems. The most important assumptions needed for the validity to this formulation are 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 the typical dimension of the solid boundaries, |x| ≫ C. This is fulfilled in our application where |x| = 2 m and C = 0.1356 m.

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

|x| > c0

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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 assumption makes the proposed approach not really valid at very high frequencies.

Ffowcs Williams and Hawkings (FWH) [37] extended Curle’s formulation to include the influence of moving surfaces. When the solid boundary is in motion, it has an additional effect on the acoustic field similar to a mass source or monopole source due to the kinematics of 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 in the application. In contrast, Ffowcs Williams and Hall’s theory [38] is presented hereafter since it is applied to the considered airfoil by Wang [58] and its results will be used for the comparison 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 function for a semi-infinite flat plate whose derivative vanishes on the airfoil surface [38]. The final acoustic pressure formulation in the far field is given in the frequency domain as:

ˆp_a(X, ω) ≈ e

i(k_|X_|–π₄)

252_π32_X

[ksin(φ)]12_sin(θ

2) ˆS(ω) (2.17)

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

S(t) = Z V ρ₀ r 3 2 0 (u2_θ– u2_r)sin(θ0 2 ) – 2uruθcos( θ₀ 2 ) 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 is valid for kC ≫ 1. The finite-chord effect is accounted for by a correction factor χ, following the derivations of Howe [39], the far-field acoustic pressure is therefore given as:

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2.4 Möhring’s analogy 17

Fig. 2.2 Coordinate system for the finite-chord thin plate used in the application of Ffowcs Williams 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 r 2kC π ! eikC √ 2πkC– e –iπ₄

e–ikC+_2πikCeikC

(2.20)

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

2.4 Möhring’s analogy

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

∂ρ

∂t + ∇.(ρ⃗u) = 0 (2.21) The momentum equation in terms of the total enthalpy B:

ρ∂⃗u

∂t + ρ∇B = ρT∇s + ρ⃗u × (∇ ×⃗u) – ∇τ (2.22) where ρ is the density,⃗u is the flow velocity, B is defined as B = h +1₂∥ua∥2, uais the velocity

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

ρDB Dt –

∂p

∂t = ∇.(⃗u.τ) + ∇.(λcon∇T) (2.23) where T is the temperature and λcon denotes the material’s conductivity. Neglecting the

dissipation caused by viscous stresses and heat conduction leads to a relation relating the pressure to the enthalpy:

DB Dt = 1 ρ ∂p ∂t (2.24)

Combining the continuity equation 2.21 and the simplified energy equation 2.24 gives: ∂ρ ∂t = –∇.(ρ⃗u) = 1 c2₀ ∂p ∂t – ∂ρ ∂s ∂s ∂t = ρ c2₀ DB Dt – ∂ρ ∂s ∂s ∂t (2.25)

Replacing ρ∂⃗u_∂t with _∂t∂(ρ⃗u) –⃗u∂ρ_∂t in the momentum equation 2.22, we get: ∂ρ⃗u ∂t – ρ⃗u c2₀ DB Dt + ∂ρ ∂s⃗u ∂s ∂t+ ρ∇B = ρT∇s + ρ⃗u × (∇ ×⃗u) – ∇τ (2.26) This equation needs to be generalized so we introduce the parameters ρ_T, the total density field and the scaled enthalpy b defined by: Db_Dt = ρ_TDB_Dt. Combining ∇ of the equation 2.26 with _∂t∂ _ρ1

T of the equation 2.25 leads to the following scalar equation:

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2.5 Amiet’s aeroacoustic theory 19

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

R=-∇[_ρ1

T(ρ⃗u × (∇ ×⃗u)] (2.29)

2.5 Amiet’s aeroacoustic theory

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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 in 1975 [29], Appendix A, to be applied for the trailing-edge noise computation for the case of a semi-infinite plane, with a trailing edge but no leading edge. Amiet’s trailing-edge model describes how the hydrodynamic waves convected within the boundary layer are scattered by the trailing edge. Schwarzschild’s procedure [48, 49] is applied iteratively such that the turbulent 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 near the trailing edge so the main input of this model is the convecting wall pressure spectrum upstream of the trailing edge. It is basically assumed that the turbulent velocity field is unaffected by the presence of the trailing edge. This assumption allows the calculation of the trailing edge noise from the spectral characteristics of wall pressure which would exist in the absence of the trailing edge. For finite chord airfoils, the cancellation condition of the potential upstream of the airfoil is not considered by Amiet and needs to be taken into account. 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 uniform velocity 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 hydrodynamic wall pressure induced by the turbulent boundary layer developed on the airfoil surface. The convected wave equation, obtained from the linearized Euler equations, is written in the plane normal to the airfoil as:

∇2p′– 1 c2₀

D2p′

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with _DtD = _∂t∂ + U_∂x∂ . The disturbance pressure can be expressed as p′(x, z, t) = P(x, z)eiωt, leading to the complex equation:

β2∂ 2_P ∂x2 + ∂2P ∂z2 – 2ikM ∂P ∂x+ k 2_{P = 0} _(2.31)

with k = ω/c₀ the acoustic wavenumber, M = U/c₀and β = √

1 – M2.

A variables change is performed as P(x, z) = p(x, z)ei(kM/β2)x, leading to the equation: β2∂ 2_P ∂x2 + ∂2P ∂z2 + KM β 2 P = 0 (2.32)

where K = ω/M the convective wavenumber and k = KM. By further transforming and adimensionnalising the problem with ¯x = x_b, ¯z = βz_b , ¯K = Kb and ¯µ = KM¯

β2 , a canonical

Helmholtz wave equation is obtained: ∂2P ∂¯x2 +

∂2P ∂¯z2 + ¯µ

2_{P = 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 of the trailing edge, the turbulence of the boundary layer, convected with a velocity Uc, is

represented 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 defined

by ¯x < 0. The first boundary condition is the Kutta condition has to be satisfied at the trailing edge and in the wake, P₀must be canceled. This is done by adding a disturbance pressure P₁ such that P = P₀+ P₁ is zero for ¯x < 0. The half-plane is assumed perfectly rigid, which provides the second boundary condition which implies that the normal derivative of P₁ must be zero for ¯x < 0. The following system of equations is therefore obtained to be resolved, via Schwarzschild’s solution (defined in Appendix B), to derive the main scattering term:

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Applying Schwarzschild’s solution leads to the equation:

p₁(¯x, 0) = –1 π Z _∞ 0 r –¯x ξ e–i ¯µ(ξ–¯x) ξ – X e –i ¯Kξ[α+(M2/β2)]_dξ = –e i ¯µ(¯x) π Z _∞ 0 r –¯x ξ 1 ξ – ¯xe –i[α ¯K+(1+M) ¯µ]ξ_dξ (2.36)

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

Z _∞ 0 r –¯x ξ 1 ξ – ¯xe

–iAξ_{dξ = πe}–iA¯x_{[1 –}e_√iπ/4

π Z –A¯x 0 e–it √ tdt] (2.37) Recognizing the Fresnel integrals R′ and S′, defined by:

E∗(x) = Z 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–iα ¯K¯x(1 + i)E∗(–α ¯K + (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 frequencies was developed by Roger & Moreau [33, 84] by taking into account the back-scattering from the leading edge, accounting for the finite chord length and respecting the upstream potential cancellation condition. The needed correction is derived, again using a two-step Schwarzschild’s solution. The final formulation for the pressure correction term is:

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Θ₁= q

A₁

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

The notation [–]cstands for an imaginary part multiplied by the correction factor ϵ given as ϵ =1 +_{4 ¯}1_µ–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 – 1 c2₀ 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–iα ¯K¯xe–i ¯K2¯y

with ¯y = y/b. The wave equation is read: ∂2p ∂¯x2+ ∂2p ∂¯z2 + ¯κ 2 p = 0 (2.44) with ¯κ2 = ¯µ2–K¯ 2 2

β2 and ¯K2 = α ¯K. The mathematical nature of the problem depends on the

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- 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–iα ¯K¯x(1 + i)E∗(–α ¯K + ¯κ + M ¯µ ¯x) – 1 (2.45) P₂(¯x, 0) ≈ (1 + i)e –4i ¯κ 2√π(α – 1) ¯K 1 – Θ2 p α ¯K + Mµ + ¯κe i(M ¯µ– ¯κ)¯x . iK + M ¯¯ µ – ¯κ [F(¯x)]c+ ∂F(¯x) ∂¯x c (2.46) where Θ = r α ¯K+M ¯µ+ ¯κ′ ¯ K+M ¯µ+ ¯κ′ .

For supercritical gusts, it is detailed by Roger & Moreau [33]. The final formulations for the induced pressure: P₁(¯x, 0) = e–iα ¯KX (1 + i)Φ0 q (–α ¯K + M ¯µ – i ¯κ′ ¯x) – 1 (2.47) with ¯κ′= r _K_¯ 2 β 2 – ¯µ2 P₂(¯x, 0) ≈ (1 + i)e –4i ¯κ 2√π(α – 1) ¯K 1 – Θ′2 p α ¯K + Mµ + ¯κ′e i(M ¯µ– ¯κ)¯x . iK + M ¯¯ µ – ¯κ F′(¯x)c+ ∂F ′_(¯x) ∂¯x c (2.48) with Θ′= r α ¯K+M ¯µ–i ¯κ′ ¯ K+M ¯µ–i ¯κ′ , F ′_{(¯x) = 1 – erf}p 2κ′(¯x + 2) Φ0(Z) = √1 π RZ2 0 e –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 the radiation integral, found in [29], as

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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 are accounted for through the modified coordinates as

Rt= _β12(Rs– M(x1– x)) Rs= σ0 1 –x1x+β2x2y σ2₀ σ₀ = q x2₁+ β2(x2₂+ x2₃)

P can be written as: P(X, ω) = f(¯x)e–i( ¯K1¯x+ ¯K2¯y) _{with f the complex amplitude of the source}

distribution. Then the radiation integral becomes

P(X, ω) = –iωx3 4πc₀σ2₀b 2Z 0 –2 Z L/2b –L/2bf(¯x)e –i( ¯K₁¯x+ ¯K₂¯y) .e –i k β2 σ₀–x1 ¯x+β2x2 ¯y σ0 b–M(x1–b¯x) d¯yd¯x (2.50)

The integral with respect to ¯y is given by:

b

Z L/2b

–L/2b

e–i( ¯K2–¯kx2/σ0)¯y_{d¯y = Lsinc} L

2b ¯ K₂– ¯kx2 σ₀ (2.51)

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

p(X, ω) =–iωx3Lb 4πc₀σ2₀ sinc L 2b ¯ K₂– ¯kx2 σ₀ .e–i k β2(σ0–Mx1) Z 0 –2f(¯x)e –iN¯x_d¯x (2.52) where N = ¯K₁– ¯µ(x1

σ₀– M). The calculation of the pressure induced on the airfoil was made

for a unit gust with wavenumbers K¯₁, ¯K₂ at the reduced frequency ω. The far-field sound power spectral density at the same frequency results from an integration over all gusts with 2D wavenumbers contributing to this frequency. The frozen turbulence hypothesis past the trailing edge leads to the streamwise aerodynamic wavenumber as K₁= ω/Uc. Noting

A₀(_Uω

c) for the amplitude of the incident pressure, the corresponding disturbance pressure

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with Amiet’s function g (related to f) denoting the transfer function between the incident pressure 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 Z _∞ –∞g(x, ω Uc , K₂)g∗(x′, ω Uc , K₂)e–iK2η_Π 0( ω Uc , K₂)dK₂ (2.54)

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

SPP(X, ω) = ωx₃Lb 2πc₀σ2₀ !2 1 b Z _∞ –∞Π0( ω Uc , K₂) sinc2 L 2b ¯ K₂– ¯kx2 σ₀ L ( ¯ ω Uc , ¯K₂) 2 d ¯K₂ (2.55)

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

Π₀(ω Uc , K₂) = Uc π Φpp(ω)ly(ω, ¯K2) (2.56) sinc2 L 2b ¯ K₂– ¯kx2 σ₀ ≃ 2πb L δ ¯ K₂– ¯kx2 σ₀ (2.57) For very large aspect ratio airfoils, the final far-field acoustic pressure PSD formulation is read: SPP(X, ω) = ωCx₃ 4πc₀σ2₀ !2 L 2 L ( ω Uc , ¯kx2 σ₀) 2 Φpp(ω)ly(ω, ¯kx2 σ₀) (2.58) This formulation is simplified for low Mach number around 0.1, as shown in [84], to:

Spp(X, ω) = (

sinθ 2πR)

2_.(kC)2_d.|_{L |}2_Φ

pp(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 equal to 40 C, with C is the airfoil chord length, k is the acoustic wavenumber. R and θ are the listener location.L is the aeroacoustic transfer function (L = L₁+L₂) withL₁the transfer function of the trailing edge andL₂ the back-scattering leading edge correction. L₁ andL₂ formula are detailed in the Appendix B. Φpp(ω) is the wall-pressure power spectral density

and ly(ω) is the spanwise correlation length near the trailing edge. So the input data here is

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2.6 Sound sources definition: Monopole, dipole & quadrupole 27

from experiments [53] or extracted from the LES computations [57] as in this study. This procedure is physically justified by the fact that when the incident aerodynamic wall pressure is convected past the trailing edge, it behaves as equivalent acoustic sources. Concerning the spanwise correlation length near the trailing edge ly(ω), it is computed using the empirical

model of Corcos [86] given by the relation: ly(ω) = b.U_ωc. This model is widely used in the

literature as in [55–57] with Uc= 0.7U0the convection speed and the coefficient b = 1.5. The

coherence length ly(ω) can be determined experimentally as in [53, 54] or it may be extracted

from the LES computations according to the relation ly(ω) =R₀∞

p

γ2(η, ω)dη where γ is the coherence function. But it has been shown by Christophe [57] that the convergence of the spanwise coherence near the trailing edge is really poor for low frequencies. The LES results largely 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 the spanwise extent, a LES simulation using Fluent has been carried out with a spanwise of 0.3C. It has been shown that a larger spanwise is necessary to correctly capture the spanwise coherence which is overpredicted in case of small extent due to the confinement of the flow between the spanwise boundaries. But even if for a 0.3C span, the agreement still not good enough to be applied in the aeroacoustics theory of Amiet taking into account the very high cost of the simulation (three times more expensive in comparison with the case of 0.1 C span). In this study, the Corcos’ model is used. However, the coherence length extracted from the present LES computations is also used. The results and more details are presented in Chap. 6.

2.6 Sound sources definition: Monopole, dipole & quadrupole

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

The monopole

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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 sources with 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]

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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, the dipole moment, it is arrived to the final formulation as the following:

p = ikD.e

–ikr

r .cosθ.(1 + 1

ikr) (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 characteristic of 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 the same 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]

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It is finally arrived to the formulation:

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Chapter 3

Solvers and numerical methods

As mentioned before, different hybrid aeroacoustic approaches are used to simulate the noise due to turbulent flow over a CD airfoil. In these methods, the aerodynamic and acoustic fields are 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 provide the near and far field acoustic propagation. This chapter is devoted to describe the solvers and to present the numerical methods. In addition, some basic backgrounds in aerodynamics and acoustics are presented.

3.1 The CFD solver, SFELES

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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( ∂2u_x ∂x2 + ∂2u_x ∂y2 + ∂2u_x ∂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 (spectral methods), let us re-formulate the NS equations introducing an in-plane operator ˜∇ and an in-plane velocity defined by ˜u = uxex+ uyey. That gives:

         ˜ ∇. ˜u +∂u_∂zz = 0 ∂˜u

∂t+ (˜u. ˜∇) ˜u + uz∂˜u∂z =–∇p + ν( ˜˜ ∇2˜u + ∂2˜u ∂z2)

∂uz

∂t + (˜u. ˜∇)uz+ uz∂u∂zz = – ∂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

ˆqk_j(t)φ_j(x, y)] exp2πIkz L

(3.3)

where I =√–1, Nmis the number of Fourier modes, Nn is the number of nodes in each finite

element plane, ˆqk_j(t) is the k-th mode of the variable q at node j, L is the spanwise dimension of the domain, k is the Fourier mode number and φ_j(x, y) stands for the P1 piecewise linear basis function associated with node j.

The governing equations are discretized using a stabilized finite element approach with the classical streamline upwind (SUPG) and pressure (PSPG) stabilizations [71, 72], i.e. ∀w, q weight functions in the approximation space, the variational formulation of the NS equations 3.2 leads to the following weak form as:

Calculation of Aerodynamic Noise of Wing Airfoils by Hybrid Methods