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Analytical and Numerical Methods in Vortex-Body Aeroacoustics

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Analytical and Numerical Methods in Vortex-Body Aeroacoustics Numéro d'ordre :2002-13 ANNÉE 2002 THÈSE présentée devant le POLITECNICO DI TORINO etl'ÉCOLE CENTRALE DE LYON pour obtenir le double titreItalo-Français et le titre Européen de DOCTEUR spécialité MÉCANIQUE DES FLUIDES et ACOUSTIQUE par Damiano CASALINO ANALYTICAL AND NUMERICAL METHODS IN VORTEX-BODY AEROACOUSTICS Soutenue le 8 avril 2002 devatit la Commissiond'Examen JURY Président: Prof.A. COGHE Examinateurs : Prof.M. ROGER Dr. M. JACOB Prof.G. P. ROMANO Prof.E. CARRERA Rapporteurs: Prof.M. HIRSCHBERG Prof.W. SCHRODER Dipartimento di Ingegneria Aeronautica e Spaziale Politecnico di Torino et Laboratoire de Mécanique des Fluides et d'Acoustique, UMR CNRS 5509 Ecole Centrale de Lyon Numéro d'ordre : 2002-13 ANNÉE 2002 THÈSE présentée devant le POLITECNICO DI TORINO et l'ÉCOLE CENTRALE DE LYON pour obtenir le double titre Italo-Français et le titre Européen de DOCTEUR spécialité MÉCANIQUE DES FLUIDES et ACOUSTIQUE par Damiano CASALINO ANALYTICAL AND NUMERICAL METHODS IN VORTEX-BODY AEROACOUSTICS Soutenue le 8 april 2002 devant la Commission d'Examen JURY Président: Prof.A. COGHE Examinateurs: Prof.M. ROGER Dr. M. JACOB Prof.G. P. ROMANO Prof.E. CARRERA Rapporteurs: Prof.M. HIRSCHBERG Prof.W. SCHRÖDER Dipartimento di Ingegneria Aeronautica e Spaziale Politecnico di Torino et Laboratoire de Mécanique des Fluides et d'Acoustique, UMR CNRS 5509 Ecole Centrale de Lyon -r Ecole Centralede Lyon BtBLIOTHEQUE 36, avenue GuydeCoUOflgUe F - 69134ECULLY CEDEX Preface This work deals with that branch of Aeroacoustics concerning the noise generated by the interaction between vortical flows and rigid surfaces.It is the outcome of a PhD research shared among the Dipartimento di Ingegneria Aeronautica e Spaziale at Politecnico di Torino and the Laboratoire de Mécanique des Fluides et d'Acoustique at Ecole Centrale de Lyon. Results concerning a nominal two-dimensional flow, the rod-airfoil configuration,are brought to- gether with a description of the noise generation mechanisms in fluid-body interactions. The rod-airfoil configuration is the object of part I, where I summarizedmy analytical contribu- tions to the vortex-airfoil interaction problem and to the development of numerical methodologies of aeroacoustic prediction. The description of the sound generation mechanisms in fluid-body interactions is the object of part II. This constitutes the theoretical basis on which I founded my PhD education. Therefore, part I constitutes my PhD Thesis and part II should be assumed as formally separated by part I. However, because of the great engagement required by writing part H, my opinion and feeling are to include it in the present work and to consider part I and part II as substantially joined. Of course, part I is in its definitive form, because it represents the outcome of a time constrained research. On the contrary, thanks to its formal autonomy, part II will be reviewed in the next future. Damiano Casalino i Contents IInteraction Noise from an Airfoil in the Wake ofa Cylinder 9 iIntroduction 13 1.1A Brief Description of the Vortex Dynamics in a Rod-Airfoil Configuration 14 1.2Theoretical and Practical Relevance of the Rod-Airfoil Configuration 15 1.3Part I Overview 16 2 Vortex-Airfoil Interaction: Aerodynamic Modeling 19 2.1Introduction 20 2.2The Aerodynamic Problem 22 2.2.1Flow Model 23 2.2.2The Kutta Condition and the Physical Role of Vortex Shedding 26 2.2.3A Fixed-Wake Formulation of the Vortex-Airfoil Interaction Problem 27 2.2.4The Oncoming Vortex Trajectory 30 2.2.5 A Free-Wake Formulation of the Vortex-Airfoil Interaction Problem 30 2.2.6The Cloud of Oncoming Vortices 33 2.2.7The Double Row of Counter-Rotating Vortices 36 2.2.8The Aerodynamic Force on the Airfoil 38 2.3Conclusions 39 3Vortex-Airfoil Interaction: Acoustic Modeling 53 3.1Acoustic Analogy Approach 53 3.2Aeroacoustic Sources 54 3.3A Linear Model for the Vortex-Airfoil Interaction Noise 55 3.4A Matched Asymptotic Expansion Model of the Vortex-Airfoil Interaction Noise 59 3.4.1 Inner Problem 60 3.4.2 Outer Problem 61 3.4.3Solution and Matching 62 3.4.4Discussion 66 3.5Conclusions 67 4 Vortex-Airfoil Interaction: Results and Discussion 69 4.1Effects of the Vortex Convection Velocity 69 4.1.1 Aerodynamic Results 70 4.1.2 Acoustic Results 72 4.1.3 Comparisons with Howe's Analytical Model 72 4.2Effects of the Vortex Distortion 75 4.2.1 Aerodynamic Results 75 4.2.2Acoustic Results 76 4.2.3An Example of Vortex Splitting 78 4.3Effects of the Airfoil Camber 80 3 4 CONTENTS 4.3.1 Aerodynamic Results 80 4.3.2 Acoustic Results 81 4.4The Unsteady Pressure Field on the Airfoil Surface 84 4.4.1Trailing Edge Behaviour 84 4.4.2 Aeroacoustic Sources Characterization 85 4.5Effects of the Free-Stream Velocity 90 4.5.1 Aerodynamic Results 90 4.5.2Acoustic Results 91 4.6Effects of the Airfoil Angle of Attack 93 4.6.1Aerodynamic Results 93 4.6.2Acoustic Results 93 4.7Comparison with Experimental Results 97 4.7.1Aerodynamic Results 97 4.7.2Acoustic Results 99 4.8Conclusions 102 5Rod-Airfoil Experiment 103 5.1Experimental Set-Up 103 5.1.1 Acoustic Measurements 103 5.1.2Surface Pressure Measurements 104 5.2Experimental Results 104 5.2.1 Acoustic Measurements 105 5.2.1.1 Isolated rod noise 105 5.2.1.2 Rod-airfoil configuration noise 107 5.2.1.3 Airfoil noise 107 5.2.2Spatial Coherence and Correlation Measurements 113 5.2.2.1 Rod configuration 113 5.2.2.2 Rod-airfoil configuration 118 5.3A Hydrogen Bubble Visualization Experiment 127 5.4Conclusions 131 6 Acoustic Analogy Formulation 139 6.1 Introduction 139 6.2Aeroacoustic Formulation 141 6.2.1 The FW-H Equation 141 6.2.2The FW-H Equation versus the Kirchhoff Equation 143 6.2.3The Retarded Time Formulation of the FW-H Equation 144 6.2.3.1 Non-dimensionalized FW-H Integral Equation 148 6.2.4The Advanced Time Fornmlation 149 6.3Numerical Assessment of Advanlia 152 6.3.1 Two-dimensional Tests 152 6.3.1.1 Test 1 152 6.3.1.2 Test 2 155 6.3.2Three-dimensional Tests 156 6.3.2.1 Test 1 156 6.3.2.2 Test 2 162 6.3.2.3 Test 3 166 6.3.3Discussion 169 6.4On the Feasibility of a Hybrid CFD/FW-H Aeroacoustic Prediction 169 6.5Conclusions 169 CONTENTS 5 7 Spanwise Statistical Modeling of a Circular Cylinder Flow 177 7.1Introduction 177 7.2Vortex Dynamics in the a Wake of a Circular Cylinder 178 7.2.1Three-Dimensional Effects 180 7.2.1.1 Three-Dimensionality at Low Reynolds Numbers 180 7.2.1.2 Three-Dimensionality at High Reynolds Numbers 181 7.3A Statistical Method for Aeroacoustic Predictions 182 7.3.1Phillips' Model 182 7.3.2The Method of the Phase Variance Distribution 183 7.3.3Random Amplitude Modulation versus Spanwise Phase Dispersion 187 7.3.4Aeroacoustic Implementation of the Statistical Model 188 7.4Aeroacoustic Prediction of a Circular Cylinder Flow 188 7.4.1 Aerodynamic Computation 188 7.4.2Acoustic Computation 189 7.4.3Aerodynamic Results 189 7.4.4Acoustic Results 192 7.5Conclusions 198 8 RANS/FW-H Rod-Airfoil Aeroacoustic Prediction 201 8.1Introduction 201 8.2On the Adequacy of a Hybrid RANS/FW-H Aeroacoustic Prediction 204 8.3The Rod-Airfoil Aerodynamic Simulation 205 8.3.1The Aerodynamic Solver 205 8.3.2Computational Parameters 205 8.3.3Flow Parameters and Initial Conditions 206 8.3.4Geometrical Parameters 206 8.3.5 Computational Mesh 207 8.4The Rod-Airfoil Acoustic Computation 207 8.5Results and Discussion 210 8.5.1 Aerodynamic Results 210 8.5.1.1 Unsteady force on the airfoil and wall pressure field 210 8.5.1.2 Airfoil results versus rod results 214 8.5.1.3 Mean and fluctuating flow near the airfoil 216 8.5.1.4 Mean and fluctuating flow past the cylinder 217 8.5.1.5 Snapshots of the Rod-Airfoil Aerodynamic Field 225 8.5.2 Acoustic Results 234 8.5.2.1 Influence of the integration surface 234 8.5.2.2 Comparison with acoustic measurements 235 8.5.2.3 Effects of the airfoil angle of attack 238 8.6Conclusions 243 9 Epilogue of part I 245 IIAerodynamic Noise in Fluid-Body Interactions 247 1Basic Equations of Fluid Mechanics 251 1.1 Introduction 251 1.2Reynolds' Transport Theorem 251 1.3Governing Equations of Fluid Motion 253 6 CONTENTS 1.3.1 The Continuity Equation 253 1.3.2The Momentum Equation 254 1.3.3The Energy Equation 258 1.3.4Convective Form of the Flow Governing Equations 259 1.4Potential Flows 260 1.4.1 Green's Functions of Wave Equations 264 1.5The Helmholtz Decomposition 265 2Nonlinearity and Modes of Fluctuation 267 2.1Introduction 267 2.2 A Perturbative Expansion of the Navier-Stokes Equations 267 2.3Physics of Modal Bilateral Interaction 274 3 The Pressure Field at the Wall of a Turbulent Boundary-Layer 277 3.1Introduction 277 3.2Wall Pressure Wavenumber-Frequency Spectrum 278 3.3Coreos' Similarity Model 280 3.3.1 Wavenumber/Phase-Velocity Spectrum 287 3.3.2Discussion on the Linearizing Assumption 288 3.4Landahi's Wave-Guide Model 288 3.5Shubert & Coreos' Linear Model 292 3.6Ffowcs Williams' Extension of Coreos' Model 297 3.7Chase's Wall Pressure Spectrum Model 307 4Gust-Response Aerodynamic Theories 317 4.1Introduction 317 4.2Possio's Integral Equation 318 4.3Sears' Gust-Response Solution 319 4.4Filotas' Model of Oblique Gust-Airfoil Interaction 323 4.5Amiet's Theory of Low- and High-Frequency Unsteady Flow Pasta Thin Airfoil .
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