Dissertation Alireza

8/18/2019 Dissertation Alireza

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Large Eddy Simulation of Sound Generation by Turbulent Reacting andNonreacting Shear Flows

Alireza Najafi-Yazdi

Doctor of Philosophy

Department of Mechanical Engineering

McGill University

Montreal,Quebec

December 2011

A dissertation submitted to McGill University in partial fullfilment of therequirements for the degree of Doctor of Philosophy

Copyright c

2011 by Alireza Najafi-Yazdi


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To my mother, Sharhzad

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ABSTRACT

The objective of the present study was to investigate the mechanisms of sound

generation by subsonic jets. Large eddy simulations were performed along with

bandpass filtering of the flow and sound in order to gain further insight into the role

of coherent structures in subsonic jet noise generation.

A sixth-order compact scheme was used for spatial discretization of the fully

compressible Navier-Stokes equations. Time integration was performed through

the use of the standard fourth-order, explicit Runge-Kutta scheme. An implicit

low dispersion, low dissipation Runge-Kutta (ILDDRK) method was developed and

implemented for simulations involving sources of stiffness such as flows near solid

boundaries, or combustion. A surface integral acoustic analogy formulation, called

Formulation 1C, was developed for farfield sound pressure calculations. Formulation

1C was derived based on the convective wave equation in order to take into account

the presence of a mean flow. The formulation was derived to be easy to implement

as a numerical post-processing tool for CFD codes.

Sound radiation from an unheated, Mach 0.9 jet at ReD = 400, 000 was consid-

ered. The effect of mesh size on the accuracy of the nearfield flow and farfield sound

results was studied. It was observed that insufficient grid resolution in the shear layer

results in unphysical laminar vortex pairing, and increased sound pressure levels inthe farfield. Careful examination of the bandpass filtered pressure field suggested

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that there are two mechanisms of sound radiation in unheated subsonic jets that

can occur in all scales of turbulence. The first mechanism is the stretching and the

distortion of coherent vortical structures, especially close to the termination of the

potential core. As eddies are bent or stretched, a portion of their kinetic energy is

radiated. This mechanism is quadrupolar in nature, and is responsible for strong

sound radiation at aft angles. The second sound generation mechanism appears to

be associated with the transverse vibration of the shear-layer interface within the

ambient quiescent flow, and has dipolar characteristics. This mechanism is believed

to be responsible for sound radiation along the sideline directions.

Jet noise suppression through the use of microjets was studied. The microjet

injection induced secondary instabilities in the shear layer which triggered the tran-

sition to turbulence, and suppressed laminar vortex pairing. This in turn resulted

in a reduction of OASPL at almost all observer locations. In all cases, the bandpass

filtering of the nearfield flow and the associated sound provides revealing details of

the sound radiation process. The results suggest that circumferential modes are sig-

nificant and need to be included in future wavepacket models for jet noise prediction.

Numerical simulations of sound radiation from nonpremixed flames were also

performed. The simulations featured the solution of the fully compressible Navier-

Stokes equations. Therefore, sound generation and radiation were directly captured

in the simulations. A thickened flamelet model was proposed for nonpremixed flames.

The model yields artificially thickened flames which can be better resolved on the

computational grid, while retaining the physically currect values of the total heat


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released into the flow. Combustion noise has monopolar characteristics for low fre-

quencies. For high frequencies, the sound field is no longer omni-directional. Major

sources of sound appear to be located in the jet shear layer within one potential core

length from the jet nozzle.


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Résumé

L’objectif de cette étude est d’obtenir la meilleure compréhension des mécanismes

de géneration de bruit par des jet subsoniques. Cette étude est basée sur simulations

aux grandes échelles de jets réactifs et sans réactifs.

Des calculs numériques employant des schéme compacts de sixiéme ordre. L’integration

temporelle fut éxéciteé à l’aide de schéme Runge-Kutta de de quatrième ordre. Des

schéme à faible dispersion et dissipation numérique. Un formulation intégrale basée

sur les analogies acoustiques fut développées pour la prédiction du champ acous-

tique lointain pour les sources et observateure en mouvement dans un fluide avec

vitesse uniforme. La formulation fut implémentée à l’aide d’algorithmes facilitant

l’implémentation pour le traitement de données d’écoulement en haute performance

utilisant des outils de simiulation á grande échelle.

Les champs sonore produit par un jet turbulent non-réactif avec nombre de Mach

de 0.9, et un nombre de Reynolds ReD = 400, 000 fut étudié. L’effect de la taille

du maillage sur la précision de l’écoulement en champs proche et e champs sonore

loin de source fut analysé. La sous-résolution de la couche decisaillement à la sortie

du jet méne à l’apparition de structures cohérentes et forte radiation qui no sort pas

physiquement réalistes. Deux mécanismes principaux de génération sonore par jets

subsoniques furent identifiés.

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Le premier mécanisme est l’étirement et la distorsion de structures tourbillon-

naires cohérentes, en particulier prés de la fin du coere potentiel. Ce mécanisme

est quadripolaire, et émet principalement vers l’arriére du jet dans la direction de

l’écoulement. Le seconde mécanisme semble être constitué de vibration transversale

de la couche de cisaillement en réponse á la présemce de structures cohérentes dans

la jet. Semblable à la radiation d’une plaque à bonds finis, la contribution de ce

méchanisme est dipolaire et domine la champs sonore dans la direction transversale,

perpendiculaire au jet.

L’utilisation de plusieurs microjet fut investiguée pour la réduction du bruit.

L’injection à l’aide de microjets précipite la transition à la turbulence, favorisent le

mélange et la destrcutction de structures cohérentes de grande échelle.

Un filtrage en bandes de étroites fut effectué. Ce traitement des données numérique

permet de visualiser les relations complexes entre l’écoulement et les onds sonores

émises. Les résultats démontrent l’importance de modes circumférenciele, ce qui a

des conśequenecs pour les modiles dits de paquets d’onde pour la preédiction du

bruit du jet.

Des simulation numériques d’écoulement et champs sonore d’une flame sans pré-

mélange furent aussi éxécutées. Les simulations incluent encore une fois l’écoulement

et le champ sonore associé, obtenus directement des équations de Navier Stokes com-

pressibles. Un modèle flammelette épaissie fut proposé que donne flammes épaissies

artificiellement qui peuvent être mieux résolus sur le maillage. Le bruit de combus-

tion a des caractéristiques monopolaires aux basses fréquences. Principales sources

de bruit semblent être situé dans la couche de cisaillement.


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Acknowledgements

The undertaking of the work presented in this thesis would have not been pos-

sible without the support, guidance, and encouragement of my advisor, Prof. Luc

Mongeau. His patience, diligence, willingness, and enthusiasm to explore new ideas

made my doctoral studies a wonderful experience. His being a mentor passes way

beyond daily research matters.

I am also thankful to Prof. Stephane Moreau and Dr. Marlene Sanjose for not

only our academic discussions, but also for their friendship. Sincere thanks also go

to Prof. Jeffrey Berghtorson. A great teacher and a true mentor, he has always been

generous with his time for me. I am also thankful to Prof. Siva Nadarajah who

kindly served on my thesis advisory committee.

Special thanks go to Prof. Thierry Poinsot, and Dr. Benedicte Cuenot who

made my visit to CERFACS possible. They were very generous with their time for

my various questions on turbulent combustion modeling. Their expertises and kind-ness to share their knowledge helped me to develop a better understanding in the

field of turbulent combustion.

I am also grateful to Dr. Ali Uzun, Dr. Christophe Bogey, and Prof. Christophe

Bailly for sharing their jet results and for their useful comments.

I would like to thank my friends and colleagues, Dr. Phoi-Tack (Charlie) Lew,

Kaveh Habibi, Hani Bakhshaei, and others who provided a fun and stimulating envi-

ronment during the course of my studies at McGill. I am also grateful to my family

for their unconditional love and support.

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Financial support from McGill University, through the McGill Engineering Doc-

toral Award (MEDA) and Lorne Trottier Fellowship, is gratefully acknowledged.

My stay at CERFACS was made possible through the financial support of the EC-

COMET program and the Marie Curie Fellowship. Financial support from the Exa

Corporation, Green Aviation Research & Development Network (GARDN), Pratt

& Whitney Canada, and the National Science and Engineering Research Council

(NSERC) of Canada is also gratefully acknowledged.

The computational resources were provided by Compute/Calcul Canada through

the CLUMEQ and the RQCHP High Performance Computing Consortia.

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TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Background on Unheated Jet Noise . . . . . . . . . . . . . . . . . 41.3 Background on Reacting Jet Noise . . . . . . . . . . . . . . . . . 11

1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . 151.6 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

I Nonreacting Flows 20

2 Governing Equations and Numerical Methods . . . . . . . . . . . . . . . 21

2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Approximate Deconvolution Model . . . . . . . . . . . . . . . . . 252.3 Spatial Discretization Scheme . . . . . . . . . . . . . . . . . . . . 272.4 Spatial Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Temporal Integration Scheme . . . . . . . . . . . . . . . . . . . . 302.6 Nonreflective Boundary Conditions . . . . . . . . . . . . . . . . . 312.7 Sponge Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.8 Inflow Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.9 LES Code Parallelization . . . . . . . . . . . . . . . . . . . . . . . 34

3 Convective Ffowcs Williams-Hawkings Equation: Formulation 1C . . . . 36

3.1 Originial Ffowcs Williams - Hawkings Acoustic Analogy . . . . . . 363.2 Convective FW-H Equation . . . . . . . . . . . . . . . . . . . . . 39

3.3 Formulation 1C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.1 Thickness Noise . . . . . . . . . . . . . . . . . . . . . . . . 453.3.2 Loading Noise . . . . . . . . . . . . . . . . . . . . . . . . . 49

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3.3.3 The Special Case of “Wind-tunnel” . . . . . . . . . . . . . 513.4 Numerical Implementation and Verification . . . . . . . . . . . . . 52

3.4.1 Stationary monopole in a moving medium . . . . . . . . . . 52

3.4.2 Stationary dipole in a moving medium . . . . . . . . . . . . 543.4.3 Rotating monopole in a moving medium . . . . . . . . . . . 55

4 Large Eddy Simulations of an Isothermal High Speed Subsonic Jet . . . . 60

4.1 Computational Setup . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Computational Grid Setup . . . . . . . . . . . . . . . . . . . . . . 62

4.2.1 Grid-30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2.2 Grid-88 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2.3 Grid-380 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2.4 Grid Resolution and Subgrid Scales . . . . . . . . . . . . . 63

4.3 Nearfield Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4 Farfield Sound Predictions . . . . . . . . . . . . . . . . . . . . . . 68

5 Sound Generation by Subsonic Jets: A Band-Pass Filtering Study . . . . 82

5.1 Band-Pass Filtering Procedure . . . . . . . . . . . . . . . . . . . . 825.2 Band-Pass Filtering of Pressure Field . . . . . . . . . . . . . . . . 835.3 Sound Generation Mechanism in Subsonic Jets . . . . . . . . . . . 86

6 Large Eddy Simulation of Jet Noise Suppression by Impinging Microjets 94

6.1 Computational Setup . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Nearfield Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.3 Farfield Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.4 Bandpass Filter Visualization of Acoustic Nearfield . . . . . . . . 100

II Reacting Flows 114

7 Reacting Flow Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.1 Governing Equations for Reacting Flows . . . . . . . . . . . . . . 1167.2 Simplifying assumptions . . . . . . . . . . . . . . . . . . . . . . . 118

7.2.1 Diffusion Fluxes . . . . . . . . . . . . . . . . . . . . . . . . 118

7.2.2 Unit Lewis Number . . . . . . . . . . . . . . . . . . . . . . 1197.2.3 Buoyancy effects . . . . . . . . . . . . . . . . . . . . . . . . 119

7.3 Mixture Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

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7.4 Flamelet Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 1207.5 Flamelet/Progress Variable Modeling . . . . . . . . . . . . . . . . 1227.6 The flamelet code . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.7 Flamelet Modeling for Compressible Flows . . . . . . . . . . . . . 1237.8 Thickened Flamelet Model . . . . . . . . . . . . . . . . . . . . . . 1247.9 Thickened Flamelet vs. PDF modeling . . . . . . . . . . . . . . . 1267.10 Sound Generation by Nonpremixed Flames . . . . . . . . . . . . . 1287.11 Sound Generation by a Reacting Mixing Layer . . . . . . . . . . . 130

7.11.1 Computational Setup . . . . . . . . . . . . . . . . . . . . . 1307.11.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 133

7.12 Sound Generation by a Nonpremixed Jet Flame . . . . . . . . . . 1357.12.1 Computational setup . . . . . . . . . . . . . . . . . . . . . 1357.12.2 Nearfield results . . . . . . . . . . . . . . . . . . . . . . . . 1377.12.3 Farfield Sound . . . . . . . . . . . . . . . . . . . . . . . . . 137

8 Conclusions & Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1548.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.2.1 Source location identification . . . . . . . . . . . . . . . . . 1578.2.2 Heated jets . . . . . . . . . . . . . . . . . . . . . . . . . . . 1588.2.3 Effect of chevrons and lobed mixers . . . . . . . . . . . . . 1588.2.4 Wavepacket models . . . . . . . . . . . . . . . . . . . . . . 1588.2.5 Thickened flamelet model . . . . . . . . . . . . . . . . . . . 1588.2.6 Combustion noise models . . . . . . . . . . . . . . . . . . . 159

8.2.7 Extension of Formulation 1C for diffraction effects . . . . . 1 5 9

III Appendices 160

A A Low-Dispersion and Low-Dissipation Implicit Runge-Kutta Scheme . . 161

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161A.2 Dispersion and dissipation of RK schemes . . . . . . . . . . . . . . 162A.3 ILDDRK scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 165A.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . 168

A.4.1 Linear Wave Equation . . . . . . . . . . . . . . . . . . . . . 169A.4.2 Nonlinear Euler equations: One-Dimensional Case . . . . . 1 7 1

A.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

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B Systematic definition of progress variables and Intrinsically Low-Dimensional,Flamelet Generated Manifolds for chemistry tabulation . . . . . . . . 177

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177B.2 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . 179

B.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . 179B.2.2 Principal direction . . . . . . . . . . . . . . . . . . . . . . . 180B.2.3 Singular Value Decomposition . . . . . . . . . . . . . . . . 181

B.3 IL-FGM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183B.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 186

B.4.1 Flamelets with a single-progress variable . . . . . . . . . . 186B.4.2 Flamelets with multi-progress variables . . . . . . . . . . . 187

B.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

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LIST OF TABLES

4–1 Reported LES of sound radiation from SP07 jet. . . . . . . . . . . . . 61

4–2 Normalized cut-off wavenumbers for the grid spacing in the jet shear

layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4–3 Simulation time steps and runtime. The term AFTT denotes acoustic

flow-through time and is equal to 32 D j/U j. . . . . . . . . . . . . . . 66

5–1 Parameters used in the design of the bandpass filters. The frequencies

are cast in nondimensional form as the Strouhal numbers, f D j/U j. . 84

A–1 Optimal coefficients for the fourth-order, low-dispersion, low-dissipation,

implicit Rung-Kutta scheme. . . . . . . . . . . . . . . . . . . . . . . 168

A–2 The L2 norm of the error between the numerical results and the exact

solution at t = 300. . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

A–3 Error between the numerical results and the exact solution at t = 300

for different CFL numbers. . . . . . . . . . . . . . . . . . . . . . . 171

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LIST OF TABLES xv

B–1 The weight coefficients of each species, as obtained from PCA, to define

the progress variable for a premixed C H 4-air flame at φ = 0.85. The

results are shown to the fourth decimal. The entries for species with

significant weight are in bold. . . . . . . . . . . . . . . . . . . . . . 192

B–2 The weight coefficients of each species, as obtained from PCA, to define

the progress variable for a premixed CH 4-air flame at φ = 1.9. The

results are shown to the fourth decimal. The entries for species with

significant weight are in bold. . . . . . . . . . . . . . . . . . . . . . 193

B–3 The weight coefficients of each species, as obtained from PCA, to

define the first progress variable, c1, for a premixed CH 4-air flame

at φ ∈ [0.6, 1.5]. The results are shown to the fourth decimal. Theentries for species with significant weight are in bold. . . . . . . . . 198

B–4 The weight coefficients of each species, as obtained from PCA, to

define the second progress variable, c2, for a premixed CH 4-air flame

at φ ∈ [0.6, 1.5]. The results are shown to the fourth decimal.Theentries for species with significant weight are in bold. . . . . . . . . 199


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LIST OF FIGURES

1–1 Schematic of jet flow field. The jet plume is illustrated with contours of

vorticity (hot color scheme) superimposed on the acoustic pressure

(gray-scale color scheme). Microphone locations are commonly

specified by their distance, R, from the jet nozzle, and polar angle,

Θ, with respect to the jet centerline. . . . . . . . . . . . . . . . . . 17

1–2 Wavepacket concept for jet flows. This figure shows how the existence

of coherent structures results in a wavepacket pressure distribution

in the shear layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1–3 A one-dimensional wavepacket illustration in (a) the physical and (b)

the frequency domain. The gray area corresponds to the range of

radiating wavenumbers. The wavenumber corresponding to sonic

propagation is denoted by ka. . . . . . . . . . . . . . . . . . . . . . 18

1–4 Similarity spectra suggested by Tam et al. (1996) for turbulent

mixing noise: solid line: large-scale similarity spectrum ; dashed

line: fine-scale similarity spectrum. . . . . . . . . . . . . . . . . . . 19

1–5 Sound pressure spectra measured by Bogey et al. (2007) at R = 100D

of a subsonic jet (M j = 0.9); solid line: measurements at Θ = 30◦;

dashed line: measurements at Θ = 90

◦

. . . . . . . . . . . . . . . . . 192–1 The seven-point overlap at the interface between two adjacent blocks. 35

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LIST OF FIGURES xvii

2–2 The computation time needed to simulate 30 time steps in a jet flow

simulation vs. increasing number of processors. . . . . . . . . . . . 35

3–1 Flow over a rigid body whose motion is defined by f (x, t). . . . . . . 57

3–2 Farfield directivity pattern of a point monopole, measured at r = 340l,

radiating in (a) a flow at M 0 = 0.5 , and (b) a flow at M 0 = 0.85

measured at r = 340l; solid line: exact solution; symbols: FW-H

code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3–3 Nearfield directivity pattern of a monopole, measured at r = 5l,

radiating in (a) a flow at M 0 = 0.5 , and (b) a flow at M 0 = 0.85

(test case 1); solid line: exact solution; symbols: FW-H code. . . . 58

3–4 The directivity pattern of a point dipole, measured at r = 30l,

radiating in (a) a medium at rest, and (b) a flow at M 0 = 0.5

moving in the positive x1-direction (θ = 0 [deg]); solid line: exact

solution; symbols: FW-H code. . . . . . . . . . . . . . . . . . . . . 58

3–5 The schematic of a rotating monopole in a moving medium. . . . . . 59

3–6 Time history of the sound pressure generated by a rotating monopole

in a moving medium (test case); (a) at (2l, 0, 0) where solid line

corresponds to the exact solution and symbols represnt the results

obtained from the FW-H code; (b) comparison between the results

at (2l, 0, 0), θ = 0[deg], and (−2l, 0, 0), θ = 180[deg]. . . . . . . . . 594–1 Grid stretching for Grid-30; left: axial direction; right: transverse

direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4–2 Grid-30 shown in x-y plane; every 4th node is shown. . . . . . . . . . 72


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LIST OF FIGURES xviii

4–3 Grid stretching for Grid-380; left: axial direction; right: transverse

direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4–4 Grid-380 shown in x-y plane; every 6th node is shown. . . . . . . . . 73

4–5 Variation of the attenuation due to molecular viscosity and LES filter

with wavenumber, k. In (a), solid line: numerical filter; dashed

line: molecular viscosity (Grid-30); dashed-dotted line: molecular

viscosity (Grid-88); dotted line: molecular viscosity (Grid-380); the

symbol ∆ corresponds to the lateral grid spacing at r = D j/2. In

(b), solid line: molecular viscosity; dashed line: LES filter (Grid-

30); dashed-dotted line: LES filter (Grid-88); dotted line: LES

filter (Grid-380). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4–6 Snapshots of vorticity field superimposed on acoustic pressure; (a):

Grid-30; (b): Grid-88; (c): Grid-380. . . . . . . . . . . . . . . . . . 75

4–7 Contours of normalized mean axial velocity, < U > /U j, in the x-y

plane; (a): Grid-30; (b): Grid-88; (c): Grid-380. . . . . . . . . . . . 76

4–8 Contours of normalized mean axial Reynolds stress, < σxx >=< uu >

/U 2 j , in the x-y plane; (a): Grid-30; (b): Grid-88; (c): Grid-380. . . 77

4–9 Variations of (a) centerline mean axial velocity, and (b) axial tur-

bulence intensity. Solid line: LES (Grid-380); dashed line: LES

(Grid-88); dashed-dotted line (Grid-30); : LES of Bogey et al.

(2011); : Arakeri et al. (2003); : Lau et al. (1979). . . . . . . . . 78


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LIST OF FIGURES xix

4–10 Inverse of mean streamwise velocity along the centerline normalized

by the inflow jet velocity. Legend: solid line, eq. (4.7); , Arakeri

et al. (2003); , LES (Grid-380). . . . . . . . . . . . . . . . . . . . 79

4–11 The Ffowcs Williams-Hawkings surface setup for farfield sound pre-

diction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4–12 Sound pressure spectra obtained at 52D j from the jet nozzle: (a) at

Θ = 30◦ ; (b) at Θ = 90◦; solid line: LES (Grid-380); dashed-line:

LES (Grid-30); :Bogey et al. (2007); :Tanna (1977). . . . . . . . 80

4–13 Directivity of overall sound pressure level (OASPL) in [dB] at 52D j

from the jet nozzle; (a): LES (Grid-30); (b): LES (Grid-380).

Legend: •, LES with Grid-30; , LES (unfiltered) with Grid-380; , LES (filtered) with Grid-380; ,Bogey et al. (2007); ,

Mollo-Christensen et al. (1964); , Lush (1971). . . . . . . . . . . . 81

5–1 The magnitude response, in dB, of the one-third-octave bandpass

filter with center frequency equivalent to Stc = f cD j/U j = 0.4.

This frequency band corresponds to S t = f D j/U j ∈ [0.36, 0.45]. . . 885–2 The magnitude response, in dB, of the one-third-octave bandpass

filter with center frequency equivalent to Stc = f cD j/U j = 4. This

frequency band corresponds to S t = f D j/U j ∈ [3.56, 4.48]. . . . . 885–3 Snapshots of the vorticity field superimposed on the acoustic pressure;

(a): Unfitered field; (b): bandpass filtered around Stc = 0.4; (c):

bandpass filtered around Stc = 4. . . . . . . . . . . . . . . . . . . . 89


20/248

LIST OF FIGURES xx

5–4 Snapshots of the vorticity field superimposed on the bandpass filtered

acoustic pressure; The band-center frequency was S tc = f D j/U j =

0.4; snapshots are shown in sequence with a time difference of

∆t = 0.84D j/U j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5–5 Wavepackets and radiated sound obtained from the low-frequency

bandpass filtering; (a): pressure and vorticity field shown together;

(b): pressure field only. The potential core length, L, and dominant

radiation directions are also shown with arrows for reference. . . . . 91

5–6 Wavepackets obtained from the bandpass filtering of vorticity and

pressure fields; the color scheme corrosponds to bandpass filtered

vorticity field while the gray scale color scheme corrosponds to the

bandpass-filtered pressure contours; (a): low-frequency passband;

(b): high-frequency passband. . . . . . . . . . . . . . . . . . . . . 92

5–7 Directivity of passband overall sound pressure level (OASPL) in

[dB] at R = 52D j ; Legend: , low-frequency radiated sound; ,

high-frequency radiated sound. . . . . . . . . . . . . . . . . . . . . 93

6–1 Contours of instantaneous vorticity superimposed on the pressure field

(gray-scale colors); (a) Base round jet; (b) with Microjet setup.

The nozzle is shown only schematically here for comparison, and

was not included in the computational domain. . . . . . . . . . . . 101

6–2 Centerline distribution of (a) mean axial velocity and (b) axial

turbulence intensity for the base round jet; solid line: present LES;

: Zaman (1986), : Lau et al. (1979). : Arakeri et al. (2003). . 102


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LIST OF FIGURES xxi

6–3 Inverse of mean streamwise velocity along the centerline normalized

by the inflow jet velocity; solid line: LES of the base round jet;

dashed dotted line: linear regression. . . . . . . . . . . . . . . . . . 103

6–4 Centerline distribution of mean axial velocity; solid line: base round

jet; dashed dotted line: with microjets. . . . . . . . . . . . . . . . . 103

6–5 Contours of normalized mean axial velocity; (a) Base round jet; (b)

with microjets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6–6 The three-dimensional spatial evolution of the mean flow velocity

with streamwise distance in the presence of microjets. . . . . . . . 105

6–7 The effect of microjets on the mean streamwise velocity at 1D from

the nozzle exit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6–8 Contours of normalized axial Reynolds stress, σxx = ux u

x

U 2j; (a) base

round jet, (b) with microjets. . . . . . . . . . . . . . . . . . . . . . 106

6–9 The streamwise distribution of peak (a) axial turbulence intensity

and (b) radial turbulence intensity; solid line: base round jet ;

dashed-dotted line: with microjets . . . . . . . . . . . . . . . . . . 107

6–10 Measured and predicted far-field noise directivity; 1: base round jet

(present LES); 2: with microjets (present LES); 3: base round jet

(Bogey et al., 2007); 4: base round jet (Alkislar et al., 2007); 5:

with microjets (Alkislar et al., 2007). The data are scaled for a

common distance of R = 100D j. . . . . . . . . . . . . . . . . . . . 108


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LIST OF FIGURES xxii

6–11 The Power Spectral Density (PSD) of the farfield acoustic sound

pressure; (a) radiation angle Θ = 30◦; (b) Θ = 90◦. The data are

reported for a common distance of R = 100D j. . . . . . . . . . . . 109

6–12 The band-pass filtered acoustic nearfield of the base setup; the

frequency band corresponds to St = 0.225 to St = 0.375; (a)

tU jDj

= 438; (b) tU jDj

= 443; (c) tU jDj

= 448; (d) tU jDj

= 453. The arrows

correspond to dominant radiation directions. . . . . . . . . . . . . . 110

6–13 The band-pass filtered acoustic nearfield of the base setup; the

frequency band corresponds to St = 0.925 to St = 1.075; (a)

tU jDj

= 438; (b) tU jDj

= 443; (c) tU jDj

= 448; (d) tU jDj

= 453. . . . . . . . . 111

6–14 The band-pass filtered acoustic nearfield of the microjet setup; the

frequency band corresponds to St = 0.225 to St = 0.375 (a)

tU jDj

= 272; (b) tU jDj

= 302; (c) tU jDj

= 332; (d) tU jDj

= 362. . . . . . . . . 112

6–15 The band-pass filtered acoustic nearfield of the microjet setup; the

frequency band corresponds to St = 0.925 to St = 1.075 (a)

tU jDj

= 272; (b) tU jDj

= 302; (c) tU jDj

= 332; (d) tU jDj

= 362. . . . . . . . . 113

7–1 Solution of the steady flamelet equations for partially premixed

CH 4/air combustion; the flame condition corresponds to the ex-

periment of Cabra et al. (2005). (a): S-shaped curve showing

the variation of the temperature at stoichiometry as a function of

stoichiometric scalar dissipation rate, χst; (b): the flamelet solution

in Z space at χst = 100s−1

. . . . . . . . . . . . . . . . . . . . . . . 140


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LIST OF FIGURES xxiii

7–2 A schematic of the effect of flamelet thickening; left: the original

flamelet; right: the thickened flamelet. . . . . . . . . . . . . . . . . 141

7–3 The heat-release source term, ω̇T obtained from solving the flamelet

equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7–4 The heat-release source term, ω̇T at χst = 0.6; solid line: original

flamelet solution as shown in Fig. 7–3; dashed line: thickened flamelet. 142

7–5 Vorticity contours of a nonreactive, naturally developing mixing layer

at Reδω,0 = 1200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7–6 Vorticity contours of the reacting mixing layer. . . . . . . . . . . . . . 143

7–7 Mixture fraction contours for the reacting mixing layer.Pure fuel

corresponds to Z = 1, while pure oxidizer corresponds to Z = 0. . . 144

7–8 Mixture fraction contours for the reacting mixing layer.Pure fuel

corresponds to Z = 1, while pure oxidizer corresponds to Z = 0. . . 144

7–9 Temperature contours and dilatation field for the reacting mixing layer.145

7–10 Locations of the virtual probes for acoustic pressure measurements. . 146

7–11 The Fourier transform of the pressure fluctuation, pac = p/p∞ − 1,measured at R = 70δ ω,0, from x = 92δ ω,0, and y = 0; solid line:

Θ = 10 [deg]; dashed line: Θ = −10 [deg]. . . . . . . . . . . . . . . 1477–12 The Fourier transform of the pressure fluctuation, pac = p/p∞ − 1,

measured at R = 70δ ω,0, from x = 92δ ω,0, and y = 0; solid line:

Θ = 50 [deg]; dashed line: Θ = −50 [deg]. . . . . . . . . . . . . . . 147


24/248

LIST OF FIGURES xxiv

7–13 The Fourier transform of the pressure fluctuation, pac = p/p∞ − 1,measured at R = 70δ ω,0, from x = 92δ ω,0, and y = 0; solid line:

Θ = 90 [deg]; dashed line: Θ = −90 [deg]. . . . . . . . . . . . . . . 1487–14 The Fourier transform of the pressure fluctuation, pac = p/p∞ − 1,

measured at R = 70δ ω,0, from x = 92δ ω,0, and y = 0; solid line:

Θ = 140 [deg]; dashed line: Θ = −140 [deg]. . . . . . . . . . . . . . 1487–15 The directivity of radiated sound pressure level measured at R =

70δ ω,0, from x = 92δ ω,0, and y = 0; (a): at frequency f 0; (b): at

frequency 4f 0. The ambient pressure was assumed to be atmospheric.149

7–16 Instantaneous snapshots of jet flame flow field; (a): contours of

temperature (hot color scheme) superimposed on the acoustic

pressure (gray-scale color scheme); (b): mixture fraction. . . . . . . 150

7–17 Mean and rms statistics of (a) axial velocity, (b) mixture fraction,

and (c) temperature along the jet flame centerline. . . . . . . . . . 151

7–18 Sound pressure level spectra at a distance of R = 100D j from the jet

flame nozzle; : radiation angle Θ = 30◦; : radiation angle Θ = 90◦.152

7–19 Directivity of overall sound pressure level (OASPL) in [dB] at 100D j

from the jet nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7–20 Temperature field superimposed on the bandpass-filtered pressure field

of the jet flame; (a): center frequency corresponds to StD = 0.4;

(b): center frequency corresponds to StD = 1; (c): center frequency

corresponds to StD = 4.0 . . . . . . . . . . . . . . . . . . . . . . . 153


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LIST OF FIGURES xxv

A–1 Amplification factor (a) and difference in phase of the Runge-Kutta

schemes: •: the new ILDDRK scheme, , standard explicit RK4;

, SDIRK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

A–2 (a) Dissipation and (b) dispersion error of the Runge-Kutta schemes

in logarithmic scales: •: the new ILDDRK scheme, , standardexplicit RK4; , SDIRK . . . . . . . . . . . . . . . . . . . . . . . . 174

A–3 Numerical solution of eq. (A.29) at t = 300; (a) CFL=0.5; (b)

CFL=1.0; (c) CFL=1.5; (d) CFL=2.0. . . . . . . . . . . . . . . . . 175

A–4 Optional caption for list of figures . . . . . . . . . . . . . . . . . . . . 176

B–1 A schematic profile of the mass fraction of a species in a one-

dimensional freely propagating flame. The flame is sampled at n

locations in the physical domain to construct the data set. . . . . . 1 9 1

B–2 The structure of a freely propagating flame in a CH 4-air mixture for

φ = 0.85, T 0 = 473 [K], and P = 1 [atm] in the physical space; solid

line: physical space solution; symbols: solution retrieved from the

previously generated flamelet table. (a): temperature; (b): CO2

mass fraction; (c): CO mass fraction; (d): OH mass fraction. . . . 194

B–3 The solution of the flame, shown in Fig. B–2 versus the progress

variable; solid line: physical space solution; symbols: flamelet

table. (a): temperature; (b): CO2 mass fracion; (c): CO mass

fracion; (d): OH mass fraction. . . . . . . . . . . . . . . . . . . . . 195


26/248

LIST OF FIGURES xxvi


φ = 1.9, T 0 = 473 [K], and P = 1 [atm] in the physical space;

solid line: physical space solution; circles: solution retrieved from

the previously generated flamelet table using the new definition of

progress variable; triangles: solution retrieved from the previously

generated flamelet table using c = Y CO2 + Y CO + Y H 2O + Y H 2. (a):

temperature; (b): CO2 mass fraction; (c): CO mass fraction; (d):

OH mass fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

B–5 The temperature distribution in the progress variable space; solid

lines: physical space solution for various equivalence ratios ; sym-

bols: interpolated values for equivalence ratio φ = 1.15. . . . . . . . 197


φ = 1.15, T 0 = 473 [K], and P = 1 [atm] in the physical space; solid

line: physical space solution; symbols: solution retrieved from the

previously generated flamelet table. (a): temperature; (b): CO2

mass fraction; (c): CO mass fraction; (d): OH mass fraction. . . . 200


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NOMENCLATURE

Roman Symbols

B Jet decay rate parameter; see eq. (4.7) on page 68

D j Jet diameter

Dk Diffusivity of species k

DT Thermal diffusivity

Dµ Microjet diamater

Da Damköhler number

F, G, H Inviscid flux vectors in the Navier-Stokes equations

Fv, Gv, Hv Viscous flux vectors in the Navier-Stokes equations

F r Froud number

G(x, y; ∆) LES filter function; see eq. (2.1) on page 21

g Body force vector

H (f ) Heaviside function

I a Acoustic intensity

J Jacobian of coordinate transformation

k Waveknumber

Le Lewis number

M Mach number

xxvii


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NOMENCLATURE xxviii

P r Prandtl number

Q Vector of flow conserved variables

q Heat flux vector

R Distance from the jet nozzle exit; see Fig. 1–1 on page 17

R Universal gas constant

R Ideal Gas constant

Re Reynolds number

S Entropy

S ij Strain rate tensor

St Strouhal number

T Temperature

T ij Lighthill’s stress tensor

t Time

U j Jet velocity

u = [u v w]T , Velocity vector

(u,v,w) Velocity components physical coordinates

V k,j j-component of diffusion velocity of species k ; see eq. (7.2) on page 117

x = [x y z ]T , spatial coordinate vector

Y Species mass fraction

Z Mixture fractionZ 2 Residual scalar variance of mixture fraction


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NOMENCLATURE xxix

Greek Symbols

αf Filtering parameter; see eq. (2.36) on page 29

∆ Grid spacing

∆0 Minimum grid spacing

δ ij Kronecker delta

δ θ Shear layer momentum thickness

δ ω Shear layer vorticity thickness

εn Random parameter in forcing procedure; see eq. (2.48) on page 33

γ Specific heat ratio

Π Phillips’ Acoustic parameter; see eq. (1.6) on page 6

µ Molecular viscosity

ω Angular frequency

ω̇k Chemical source term per unit mass; see eq. (7.2) on page 117

ω̇T Chemical heat release term per unit mass; see eq. (7.2) on page 117

ρ Density

σij Shear stress tensor

σ Sponge zone damping parameter

τ e Emission time; see eq. (3.44) on page 48

Θ Emission angle; see Fig. 1–1 on page 17

χ Scalar dissipation rate; see eq. (7.15) on page 121

(ξ , η , ζ ) Generalized curvilinear coordinates


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NOMENCLATURE xxx

(ξ x, ξ y, ξ z) Partial derivatives of ξ along physical coordinates

(ηx, ηy, ηz) Partial derivatives of η along physical coordinates

(ζ x, ζ y, ζ z) Partial derivatives of ζ along physical coordinates

Superscripts, Subscripts, and Accents

(·)0 Ambient condition

(·)∞ Freestream condition

(·) j Jet

(·)r Reference condition(·)rms Root mean square

(·)sgs Subgrid scale

(·)st Stoichiometric condition

(·) LES filtered variable, see eq. (2.1) on page (2.1)

(·) Favre-averaged quantity, see eq. (2.3) on page (2.3)(·)µ Microjet properties

< · > Reynolds averaged property

Abbreviations

CFD Computational Fluid Dynamics

dB Decibel

DNS Direct Numerical Simulation

FW-H Ffowcs-Williams & Hawkings


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NOMENCLATURE 1

LES Large Eddy Simulation

LHS Left hand side

MCAAP McGill Computational Acoustic Analogy Package

NSCBC Navier-Stokes Characteristic Boundary Conditions

OASPL Overall sound pressure level

RHS Right hand side

RMS Root mean squared

SPL Sound pressure level


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“The sensation of sound is a thing sui generis , not comparable with any of our

sensations.”

– Lord Rayleigh in The Theory of Sound

2


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CHAPTER 1Introduction

1.1 Motivation

More than a century after the first powered flight by the Wright brothers, avia-

tion is a dominant industry and a major job creator, specially in the North American

economy. With increases in air traffic, aircraft noise has increased and become a nui-

sance for communities living close to active airports. This has led governments to

create increasingly stringent regulations for the maximum sound levels allowable dur-

ing aircraft landing and takeoff. Despite significant improvements over the past few

decades, aircraft noise remains a significant challenge. Governments and aerospace

industries have laid out plans to reduce the current aircraft noise level by 20 EPNdB1

by the year 2020 (Weasoky, 1998).

One of the most significant contributor to aircraft sound emission is the engine.The sound radiated from the propulsion system consists of fan noise, combustion/core

noise, and jet exhaust noise. The jet exhaust noise and combustion core noise are

especially significant during takeoff phase. After more than half a century, sound

generation by high speed turbulent jet flows remains one of the most recalcitrant and

challenging problems in aeroacoustics. It is widely accepted that coherent structures

1 Effective Perceived Noise level dB

3


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1.2. BACKGROUND ON UNHEATED JET NOISE 4

play a significant role in sound generation by turbulent jets. Despite numerous

experimental and numerical studies, the details of sound generation mechanisms by

turbulent jet flows are still not well understood, and the nature of the sound sources,

especially for subsonic jets, is still the subject of debates among researchers. In the

present study, the role of coherent structures in sound radiation from jet exhaust

flows is studied. It is expected that the findings of this study be useful to describe

mechanisms of sound generation by subsonic jets, and develop new phenomenological

models.

With the introduction of high-by-pass ratio engines to reduce jet exhaust noise,

other sources such as combustion and core noise have become more prominent. Sound

radiation from nonpremixed flames is also studied in the present work.

1.2 Background and Literature Review: Nonreacting Jet Noise

The first attempt to develop a theory for jet noise was made by Sir James

Lighthill (Lighthill, 1952, 1954) who introduced the concept of acoustic analogy .

Lighthill’s idea was to recast the exact equations of motion, i.e., the conservation

of mass and momentum, as an inhomogeneous wave equation where the nonlinear

terms are moved to the right hand side (RHS) and are treated as sources of sound.

The problem of calculating the turbulence-generated sound is then equivalent to

solving the radiation of a distribution of sources into an ideal fluid at rest. Lighthill’s

inhomogeneous wave equation is

1

c

2

0

∂ 2

∂t2

− ∇2

p =

∂ 2T ij

∂xi ∂x j

, (1.1)


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where T ij is called the Lighthill’s stress tensor and is given by

T ij = ρ ui u j + ( p − p0) − c20(ρ − ρ0) δ ij − σij . (1.2)In eq. (1.2), σij is the shear stress tensor, ui and u j are the flow velocity components,

and p0, ρ0, and c0 are the ambient pressure, density, and speed of sound, respectively.

The first term in Lighthill’s stress tensor, eq. (1.2), is commonly referred to as the

Reynolds stress, and is a significant source of sound in turbulent flows. The second

term is due to wave amplitude nonlinearity and entropy variations in the source

region. The third term represents the attenuation of sound waves due to viscous

stresses, and is usually neglected. Due to the appearance of the double divergence

operator ∂ 2

∂xi∂xj, the source of eq. (1.1) is referred to as a quadrupole with strength

T ij (Howe, 2003).

Lighthill’s analogy, eq. (1.1) implies that the problem of turbulence-generated

sound is equivalent to the radiation of a distribution of quadrupole sources with

strength T ij into a stationary, ideal fluid (Howe, 2003). Using his acoustic analogy,

Lighthill (1954) showed that the acoustic intensity, I a, of the sound radiated from a

low Mach number jet is given by

I a = K ρ2 jD

2 j U

8 j

ρ0c50R2

, (1.3)

where K is a constant of the order of 10−5.

The effect of source convection was also discussed by Lighthill (1952), and then

investigated in more details by Ffowcs Williams (1960, 1963). They showed that

the Doppler shift due to the convection of quadrupole sources results in stronger


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radiation in the aft direction, such that the acoustic intensity is equal to

I a = K ρ2 jD

2 j U

8 j

ρ0c50 (1 − M c cos Θ)2 + lrτ rc025/2 R2, (1.4)

where M c = U c/c0 is the convective Mach number, lr is the turbulence characteristic

length scale, and τ r is the turbulence characteristic time scale. The parameters R

and Θ specify the location of the microphone, as shown in Fig. 1–1. The term

(1 − M c cos Θ) captures the effects of the Doppler shift, whereas the term lr/τ rc0takes into account the spatial extent of eddies. The latter term is of significance in

directions normal to the Mach waves, where M c cos Θ = 1.

In Lighthill’s analogy, the convection and refraction of the emitted sound waves

are neglected, and are lumped into the source term which can lead to inaccurate

predictions for high speed jet noise. Therefore, Lighthill’s analogy was subsequently

modified (Phillips, 1960; Goldstein & Howe, 1973; Lilley, 1974; Goldstein, 2003) to

take propagation effects into account by including the corresponding terms in the

wave operator.

Phillips (1960) derived a convected wave equation,

D2

D t2Π − ∂

∂xi

c2

∂ Π

∂xi

=

∂ui∂x j

∂u j∂xi

+ D

Dt

1

c p

dS

dT

− ∂

∂x j

1

ρ

∂σij∂x j

, (1.5)

where the parameter Π is given by

Π = 1

γ ln

p

p0. (1.6)


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The first term on the RHS of eq. (1.5), ∂ui∂xj

∂uj∂xi

, represents the aerodynamic sources

due to velocity fluctuations, while the second and third terms correspond to the

effects of entropy sources and viscosity fluctuations, respectively.

In comparison to Lighthill’s analogy, eq. (1.1), a convective term has been moved

to the left-hand side of eq. (1.5). This leads to the appearance of a second-order total

time derivative in the wave operator. The spatial dependence of the speed of sound

has also been included in the wave operator to account for refraction effects.

Lilley (1974) argued that the first term on the RHS of Phillips’ equation contains

propagation effects for shear flows that should be included in the convective wave

operator on the LHS. He then derived a third-order wave equation2 ,

D

Dt

D2Π

Dt − ∂

∂xi

c2

∂ Π

∂xi

+ 2

∂u j∂xi

∂

∂x j

c2

∂ Π

∂xi

= −∂u j

∂xi

∂uk∂x j

∂ui∂xk

+ Ψ , (1.7)

where Ψ represents the effects of entropy generation and viscosity fluctuations which

are generally neglected. Some industrial jet noise prediction tools, such as General

Electric’s MGB approach (Balsa et al., 1978), are based on Lilley’s analogy.

More recently, Goldstein (2003) has proposed a generalized acoustic analogy in

which the Navier-Stokes equations are recast as a set of linearized inhomogeneous

2 Lilley’s equation, as published in the original paper (Lilley, 1974), reads slightlydifferent from eq. (1.7). In the original paper, it was assumed that the mean flow isparallel to x1 axis.


38/248


Euler equations with source terms representing shear-stress and energy-flux pertur-

bations. This approach requires the proper choice of a base flow around which the

Navier-Stokes equations are linearized.

Acoustic analogies have some intrinsic drawbacks. The linearization of the gov-

erning equations and the dissociation of the propagation effects (i.e. refraction of

sound waves) from the source terms are usually somewhat arbitrary without any a

priori knowledge of the sound field. This means that the source terms in acoustic

analogy theories are not necessarily true sources of sound. Moreover, the source

terms are assumed to be known, making acoustic analogies dependent on separate

experimental measurements or theoretical or numerical calculations. With the ex-

ception of Lighthill’s equation, almost all acoustic analogies involve nonlinear dif-

ferential operators which generally cannot be integrated analytically. This implies

that a quantitative prediction of the farfield sound requires the numerical solution

of the acoustic analogy partial differential equations, which can be computationally

intensive.

As an alternative to acoustic analogy theories, some researchers have investi-

gated phenomenological models based on the observation of coherent structures in

turbulent jets. Mollo-Christensen (1967) suggested a wavepacket concept to model

sound generation by coherent structures. Figure 1–2 illustrates how the existence of

coherent structures in the shear layer results in a pressure distribution that can be

modeled as a wave packet.

In supersonic jets, such coherent structures generate sound through Mach wave

radiation to the farfield, a process analogous to the sound radiation by a supersonic


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wavy wall (Tam, 1995). The introduction of the wavepacket concept led researchers

to explore the flow stability theory as a theoretical framework to analyze the nearfield

dynamics and its relation to farfield sound radiation (Michalke, 1970, 1972; Mankbadi

& Liu, 1984). Models based on linear convecting instability waves were developed

by Tam & Morris (1980) and Tam & Burton (1984a,b). More recently, Wu (2005)

investigated the sound radiation from nonlinearly evolving instabilities.

The relevance of models based on instability wave radiation for subsonic jet noise

is debatable. This is because the energy in wave-numbers with supersonic phase

speeds appears only with the growth and decay of subsonically convected instability

waves. In other words, it can be argued that instability waves do not propagate

supersonically to radiate Mach waves. The appearance of supersonic phase speeds

may be explained as an artifact of the Fourier transform (cf. Fig. 1–3). Despite this

fact, several authors have investigated the sound emissions from wave packet pressure

fields (Crighton & Huerre, 1990; Avital & Sandham, 1997; Le Dizès & Millet, 2007;

Obrist, 2009, 2011; Cavalieri et al., 2011).

Following an exhaustive review of supersonic jet noise data, Tam et al. (1996)

showed that far-field sound pressure spectral densities can be fit using two similarity

spectra. The spectrum shape that dominated the aft angles was associated with

Mach wave radiation by large scale structures. Hence, this spectrum is commonly

referred to as the Large Scale Similarity (LSS) spectrum (Morris, 2009). The second

spectrum was used to fit the sound pressure spectra radiated toward sidelines. Tam

et al. (1996) argued that the sound radiated toward sidelines is generated by small-

scale structures. Hence, the second spectrum is commonly designated as the Fine


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Scale Similarity (FSS) spectrum. At intermediate angles, a combination of the two

spectra is needed to match the experimental data.

The two similarity spectra were used by Viswanathan (2002) and Tam et al.

(2008) to fit further experimental results, including the data for subsonic jets. Tam

et al. (2008) argued that these results reinforce the idea that the two similarity spectra

correspond to two separate noise generation mechanisms in jets for all operating

conditions. This model is similar to the one initially proposed by Schlinker (1975),

and Laufer et al. (1975) for turbulence mixing noise in supersonic jets.

The large-scale/small-scale description of jet noise radiation is primarily based

on the fact that a combination of the two similarity spectra can be used to fit a given

farfield sound pressure spectrum provided that the peak frequencies and spectral

magnitudes are correctly chosen. However, the decomposition of a given spectrum

into two or more spectra does not necessarily yield a unique solution. In other words,

one can choose another set of, say, three ad-hoc similarity spectra and combine them

such that their sum reproduce a given spectrum. Another important observation

which seems to be inconsistent with Tam & Morris (1980)’s hypothesis is that the

peak of the supposedly fine scale radiated sound spectrum, measured for example at

90◦ from the jet axis, is observed at relatively low Strouhal numbers (St ≈ 0.3). Thisvalue of Strouhal number is close to that of the spectrum measured at 30◦ which is

supposed to be the large scale radiated sound spectrum (c.f. Fig. 1–5). But, one

would expect that the peak frequency of small-scale radiated sound be at least an

order of magnitude larger than that of large-scale radiated sound.


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1.3. BACKGROUND ON REACTING JET NOISE 11

A more rigorous approach to identify the sound generation mechanism by jets is

to look for direct relationships between the nearfield flow properties and the radiated

sound field. For example, Panda & Seasholtz (2002) and Panda et al. (2005) used

a Rayleigh-scattering technique to measure correlations between the radiated sound

pressure and nearfield quantities, such as density and velocity fluctuations, for jets at

Mach numbers 0.8, 0.95, 1.4 and 1.8. They reported significant correlations between

nearfield density fluctuations and farfield pressure at aft angles where the convective

velocity was supersonic, while such correlations were negligible for subsonic jets. Bo-

gey & Bailly (2007) used a similar causality method to investigate sound generation

from isothermal jets at Mach numbers 0.6 and 0.9. The nearfield results were ob-

tained from the large eddy simulations of the jets. The simulations were performed

at two Reynolds numbers, ReD = 1.7 × 103 and ReD = 4 × 105, with the aim of studying the effects of both Mach and Reynolds number on the correlations between

the radiated sound pressure and nearfield flow quantities. They reported significant

levels of correlation between the centerline turbulence and sound radiated to Θ = 40 ◦

for all four jets. Maximum correlations were observed at the end of the potential

core. Based on the analysis, Bogey & Bailly (2007) suggested the presence of a noise

generation mechanism near the end of the potential core.

1.3 Background and Literature Review: Reacting Jet Noise

Combustion can be a significant contributor to the aero-engine core noise (Smith,

2004). The interaction of acoustic sound waves with the flame can also lead to

combustion instability in gas turbine engines and industrial furnaces. Depending

on the generation mechanism, combustion noise can be characterized as direct or


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indirect (Strahle, 1971, 1978). Direct noise is caused by volumetric expansion due to

heat release, and constitutes a monopole source. Indirect sound is generated by the

differential acceleration of entropy non-uniformities and their interaction with solid

boundaries. This mechanism constitutes a dipole source (Howe, 1998).

A review of literature shows that the majority of the combustion noise studies

are either theoretical or experimental. Numerical simulations of sound radiation by

open flames can help improve our understanding of broadband combustion noise. In

the absence of a retro-active feedback leading to instability, the flame can be excited

in a controlled manner to measure the flame transfer function, a quantity useful

for combustion instability studies. This approach is called system identification in

numerical studies of combustion instability (Poinsot & Veynante, 2005).

Zhao & Frankel (2001) preformed the DNS of sound radiation from an axisym-

metric premixed reacting jet. They used a 6th-order compact scheme (Lele, 1992) for

spatial discretization, and a 4th-order Runge-Kutta algorithm for temporal integra-

tion. A generic one-step global reaction was considered to model the chemistry. Their

computational domain included both the nearfield and the farfield. They observed

that combustion heat release had a significant effect on the vortical structure of the

jet, as well as the radiated sound pressure level and directivity. Lighthill’s acoustic

analogy was used to identify apparent sound source locations. It was concluded that

heat release stabilized the jet, enhanced sound radiation levels, and altered the fre-

quency of the most unstable modes to lower values, which led to a broader sound

spectrum.


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Flemming et al. (2007) performed an LES of the H3 flame (Tacke et al., 1998).

The simulation was based on the low-Mach-number assumption in which the pres-

sure work is neglected in the energy equation (Poinsot & Veynante, 2005). This

assumption reduces the computational cost by relaxing the CFL restriction. But,

the results can no longer be used to directly predict the generation and propagation

of sound. A linear wave equation with a monopole source term was used to estimate

the radiated sound pressure level.

Bui et al. (2007) used a hybrid LES/CAA approach, in which a low-Mach num-

ber LES was combined with an acoustic perturbation equation modified for reacting

flows. Mühlbauer et al. (2008) used a RANS/statistical model to simulate broad-

band combustion noise of the open non-premixed DLR-A jet flame. Ihme et al.

(2009) also used the low-Mach-number assumption to perform an LES of an N 2-

diluted C H 4 − H 2/air flame. They developed a modified form of Lighthill’s acousticanalogy to predict combustion generated sound. Their acoustic analogy utilized the

flamelet/progress-variable model to formulate the excess density.

So far numerical studies of nonpremixed flames have used the low-Mach-number

assumption to simulate the hydrodynamic field. The radiated sound has been cal-

culated based on simplified models. It is interesting to directly simulate the sound

production and radiation by diffusion flames. Such studies can enhance our cur-

rent understanding of turbulence/acoustic/flame interactions, and can be used as

benchmarks to further validate combustion noise models or develop new ones.


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1.4. OBJECTIVES 14

1.4 Objectives

The literature review presented in Sec. 1.2 suggests that further studies on the

nature of jet noise sources and their dynamics are required. While acoustic analogies

seem to be incapable of providing more insight into the physics of sound sources,

phenomenological models, such as wavepacket theory, seem promising as models

of generation by shear flows. Developing such models, and calibrating them for

practical noise predictions require a better understanding, both qualitatively and

quantitatively, of sound radiation by coherent structures at different scales.

Although two-point space-time correlations provide some information about the

relation between the nearfield flow and the sound perceived in the farfield, they

do not provide a comprehensive picture of sound radiation patterns by turbulent

structures at different scales. The objective of the present study was to investigate

a new approach, bandpass filtering of the flow field, to provide further insight into

the dynamics of sound radiation by coherent structures of subsonic jet flows. This

approach shows how different scales contribute to farfield sound radiation. In the

present study, Large-Eddy Simulations (LES) of high-speed subsonic jet flows were

performed to obtain both the nearfield sources and the farfield radiated sound. Band-

pass filtering was then used as a post-processing tool to visualize and characterize

the radiated acoustic field in different frequency bands. The result of this study were

used to identify source mechanisms in subsonic jets. The results suggest that Tam

et al. (2008) description of jet noise generation may be oversimplified.

Similar analysis was also used to perform a preliminary study of the sound

radiation by a nonpremixed reacting mixing layer, and a diffusion jet flame. The


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1.5. ORGANIZATION OF THE THESIS 15

results provide a better understanding of combustion/core noise problem, which has

not received as much attention as the jet noise problem.

1.5 Organization of the Thesis

The thesis is divided into two parts. The problem of sound radiation from

nonreacting, high speed, subsonic jets is studied in Part I, while combustion noise is

the subject of Part II. Chapter 2 presents the governing equations, and the numerical

schemes used in the LES of nearfield flow. The farfield sound prediction method is

presented in Chapter 3. The LES results for a Mach 0.9 jet at ReD = 4 × 105 arepresented in Chapter 4. Chapter 5 describes the bandpass filtering procedure, and the

major findings obtained from this approach. Using the LES and bandpass filtering,

the use of mircojets for jet noise suppression is studied in Chapter 6. Numerical

simulations of sound radiation from a reacting mixing layer, and a jet diffusion flame

are presented in Chapter 7. Some concluding remarks, and suggestions for future

work are presented and in Chapter 8.

1.6 Contributions

The following list summarizes the major contributions of the present work:

• High-fidelity, direct computations of sound radiation from reacting and nonre-acting jet flows.

• Investigation of grid resolution effects on the accuracy of jet noise simulations.• A band-pass filter visualization and analysis of the source region and the acous-

tic field of subsonic jets.

• Development of an Implicit, Low-Dispersion, Low-Dissipation Runge-Kutta(ILDDRK) scheme of fourth order accuracy.


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1.6. CONTRIBUTIONS 16

• Development of a new surface-integral acoustic analogy formulation, Formula-tion 1C , to predict the sound radiated by moving sources in uniformly moving

media.

• Large Eddy Simulation of jet noise suppression by impinging microjets.• Development of a thickened flamelet model for simulations of turbulent non-

premixed flames.

• Development of a systematic method to define progress variables and the Intrin-sically Low-Dimensional, Flamelet Generated Manifold (IL-FGM) modeling for

chemistry tabulation in LES of turbulent flames

• Direct noise computation of a reacting mixing layer, and a jet diffusion flame.


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FIGURES OF CHAPTER 1 17

x

Θ

R

Figure 1–1: Schematic of jet flow field. The jet plume is illustrated with contours of vorticity (hot color scheme) superimposed on the acoustic pressure (gray-scale colorscheme). Microphone locations are commonly specified by their distance, R, fromthe jet nozzle, and polar angle, Θ, with respect to the jet centerline.

p(x)

x

Figure 1–2: Wavepacket concept for jet flows. This figure shows how the existence of coherent structures results in a wavepacket pressure distribution in the shear layer.


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x

|q (x)|q 0

2l(a)

k

k0

ka−ka

downstream upstream

q̂ (k)

(b)

Figure 1–3: A one-dimensional wavepacket illustration in (a) the physical and (b) thefrequency domain. The gray area corresponds to the range of radiating wavenumbers.The wavenumber corresponding to sonic propagation is denoted by ka.


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-20

-15

-10

-5

0

5

10

0.1 1 10

S P L

[ d B ]

f / f p

Figure 1–4: Similarity spectra suggested by Tam et al. (1996) for turbulent mixingnoise: solid line: large-scale similarity spectrum ; dashed line: fine-scale similarityspectrum.

80

85

90

95

100

105

110

115

120

0.01 0.1 1

S P L [ d B ] / S t

St D = f D / U j

Figure 1–5: Sound pressure spectra measured by Bogey et al. (2007) at R = 100Dof a subsonic jet (M j = 0.9); solid line: measurements at Θ = 30

◦; dashed line:measurements at Θ = 90◦.


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Part I

Nonreacting Flows

20


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CHAPTER 2Governing Equations and Numerical Methods

In the present study, the Navier-Stokes equations are numerically solved in the

context of Large-Eddy Simulation (LES).The basic idea behind LES is to decompose

the flow properties into a large-scale or resolved component, ψ , and a small-scale or

subgrid component, ψsg. This decomposition is achieved by applying a spatial-filter

(Pope, 2000),

ψ(t, x) =

ψ(t, y)G(x, y;∆)dy , (2.1)

where ∆ is the filter size, and G is the filter function satisfying the normalization

condition G(x, y; ∆) = 1 . (2.2)

For compressible and reacting flows where the density changes significantly, Favre

filtering is employed. A Favre-filtered quantity, ψ is defined asψ = ρψ

ρ =

1

ρ

ρ(t, y)ψ(t, y)G(x, y;∆)dy . (2.3)

It is customary to filter the Navier-Stokes equations first, and then solve the

resulting equations for the filtered quantities. This procedure, also known as implicit

filtering , always results in the so-called closure problem, i.e. some terms remain

unclosed and need modeling. These terms correspond to the effects of subgrid-scale

dynamics. Commonly used subgrid scale models include the classic Smagorinsky

21


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2.1. GOVERNING EQUATIONS 22

model (Smagorinsky, 1963), and the dynamic Smagorinsky model (Germano et al.,

1991; Moin et al., 1991).

In the implicit filtering, the filter function is not necessarily known as it is defined

by the numerical grid (Kravchenko & Moin, 1997; Meyers & Sagaut, 2007; Lund,

2003). Hence, the filter effect on energy dissipation cannot be quantified. This can

be alleviated by explicitly filtering the flow properties at each step as an integral part

of the numerical simulation (Lund, 2003; Bose et al., 2010). In the present study, the

explicit filtering technique is adopted in order to quantify the energy dissipation by

the LES filter and assess the effects of filtering on the quality of the LES results. The

filtering procedure adopted in this work is based on the approximate deconvolution

model (ADM) (Stolz & Adams, 1999; Mathew et al., 2003; Bogey & Bailly, 2006) of

subgrid scales.

In this chapter, the governing equations, the LES subgrid modeling through

ADM, and some details on the numerical schemes and boundary conditions are pre-

sented.

2.1 Governing Equations

The unsteady, non-dimensional, compressible form of the Navier-Stokes equa-

tions were solved on curvilinear grids. The governing equations, in the generalized

coordinates, (ξ , η , ζ ), are given by

1

J

∂ Q

∂t +

∂

∂ξ

F − Fv

J

+

∂

∂η

G − Gv

J

+

∂

∂ζ

H − Hv

J

=

S

J , (2.4)

where, Q is the the vector of conserved variables,

Q = 1

J [ρ ρu ρv ρw E ]T , (2.5)


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and total energy is defined as

E = p

γ − 1 +

1

2

ρuiui . (2.6)

The inviscid flux vectors are

F =

ρU

ρuU + ξ x p

ρvU + ξ y p

ρwU + ξ z p

(E + p)U

, G =

ρV

ρuV + ηx p

ρvV + ηy p

ρwV + ηz p

(E + p)V

, (2.7)

and

H =

ρW

ρuW + ζ x p

ρvW + ζ y p

ρwW + ζ z p

(E + p)W

, (2.8)

where the variables U , V , and W are the transformed velocity vectors in the com-

putational space given by

U = uξ x + vξ y + wξ z , (2.9)

V = uηx + vηy + wηz , (2.10)

and

W = uζ x + vζ y + wζ z . (2.11)


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The viscous flux vectors are defined as follows:

Fv =

F v1

F v2

F v3

F v4

F v5

=

ξ xΨxx + ξ yΨxy + ξ zΨxz

ξ xΨyx + ξ yΨyy + ξ zΨyz

ξ xΨzx + ξ yΨzy + ξ zΨzz

uF v2 + vF v3 + wF v4 − ξ xq x − ξ yq y − ξ zq z

, (2.12)

Gv =

Gv1

Gv2

Gv3

Gv4

Gv5

=

ηxΨxx + ηyΨxy + ηzΨxz

ηxΨyx + ηyΨyy + ηzΨyz

ηxΨzx + ηyΨzy + ηzΨzz

uGv2 + vGv3 + wGv4 − ηxq x − ηyq y − ηzq z

, (2.13)

and

Hv =

H v1

H v2

H v3

H v4

H v5

=

ζ xΨxx + ζ yΨxy + ζ zΨxz

ζ xΨyx + ζ yΨyy + ζ zΨyz

ζ xΨzx + ζ yΨzy + ζ zΨzz

uH v2 + vH v3 + wH v4 − ζ xq x − ζ yq y − ζ zq z

. (2.14)

The shear stress tensor, Ψij is given by

Ψij = 2µ

Re

S ij − 1

3S kkδ ij

, (2.15)


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2.2. APPROXIMATE DECONVOLUTION MODEL 25

where S ij is the strain rate tensor defines as

S ij = 1

2 ∂ui

∂x j+

∂ u j

∂xi . (2.16)The heat flux vector, q i, is defined as

q i =

µ

(γ − 1)M 2r RePr

∂ T

∂xi, (2.17)

where M r is the reference Mach number,

M r = U r√

γRT r. (2.18)

The Jacobian of coordinate transformation, J , and transformation metrics are

calculated in a conservative form as outlined by Visbal & Gaitonde (2002). If present,

the source terms are represented by the vector S on the right hand side of eq. (2.4).

2.2 Explicit Filtering and Approximate Deconvolution Model

Consider a one-dimensional transport equation of the form

∂u

∂t +

∂ f (u)

∂x = 0 , (2.19)

where f (u) is a nonlinear flux function. Applying a low pass filter, G, to eq. (2.19)

yields

∂u

∂t + G ∗ ∂ f (u)

∂x = 0 , (2.20)

where G ∗ (.) denotes low pass filtering by convolution as outlined in eq. (2.1).Eq. (2.20) can be recast as

∂u

∂t +

∂ f (u)

∂x = Rsgs (2.21)


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2.2. APPROXIMATE DECONVOLUTION MODEL 26

where the subgrid scale residual,

Rsgs =

∂f (u)

∂x −G

∗ ∂ f (u)

∂x

, (2.22)

needs to be modeled. The subgrid model should describe the subgrid scale resid-

ual, Rsgs as a function of the filtered solution u. The approximation deconvolutionmethod models Rsgs with the following relation

Rsgs = ∂f (u)∂x

− G ∗ ∂ f (u∗)

∂x , (2.23)

where u∗(x, t) is an approximation of u(x, t) obtained through a deconvolution (Mathew

et al., 2003),

u u∗ = Q ∗ u . (2.24)

The deconvolution function, Q, is supposed to be the exact inverse of G; however, it

is usually an approximation to the exact inverse of G such that QG is unity for low

wavenumbers.

Substitution of eq. (2.24) into eq. (2.20) yields

∂u

∂t + G ∗ ∂ f (u

∗)

∂x = 0 , (2.25)

which can be recast to give

G ∗

∂u∗

∂t +

∂ f (u∗)

∂x

= G ∗ ∂ u

∗

∂t − ∂ u

∂t . (2.26)

Since G ∗ u∗ ≈ G ∗ u, the RHS of eq. (2.26) may be set equal to zero, i.e.

G ∗∂u∗

∂t +

∂ f (u∗)

∂x

= 0. (2.27)


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2.3. SPATIAL DISCRETIZATION SCHEME 27

which implies to

∂u∗

∂t +

∂ f (u∗)

∂x = 0 . (2.28)

Thus eq. (2.28) is solved instead of eq.’s (2.20) or (2.21). The numerical imple-

mentation is as follows.

Consider the filtered solution at time step n denoted by u(n). The unfiltered

solution is obtained from

u∗ (n) = Q ∗ u(n) , (2.29)

which is used to numerically integrate eq. (2.28) and obtain u∗ (n+1). The filtered

solution is then obtained from

u(n+1) = G ∗ u∗ (n+1) . (2.30)

When executed sequentially, the filtering step, eq. (2.30), and the deconvolution

step, eq. (2.29), can be combined to form a single filtering step, Q ∗ G ∗ u∗ (n+1).Since Q is not the exact inverse of G, the operator Q ∗ G removes high wavenumber

components of the solution. In effect, the filter H = Q ∗ G is a low pass filtersimilar to G, but with a higher cut-off frequency. In summary, the approximate

deconvolution model is implemented through explicitly applying a low-pass filter

during time integration.

2.3 Spatial Discretization Scheme

A sixth-order, non-dissipative, central difference compact scheme (Lele, 1992),

α∂f ∂ξ i−1 + ∂f ∂ξ i + α∂f ∂ξ i−1 = a f i+1 − f i−12∆ξ + b f i+2 − f i−24∆ξ , (2.31)


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2.4. SPATIAL FILTERING 28

was used for spatial discretization where α = 1/3, a = 14/9, and b = 1/9. For

the boundary points i = 1 and i = N , the following third-order one-sided compact

scheme equations were used, respectively:∂f

∂ξ

1

+ 2

∂f

∂ξ

2

= 1

2∆ξ (−5f 1 + 4f 2 + f 3) , (2.32)

and ∂f

∂ξ

N

+ 2

∂f

∂ξ

N −1

= 1

2∆ξ (5f N − 4f N −1 − f N −2) . (2.33)

For points i = 2 and i = N −1, the following forth-order, central difference, compact

scheme equations were used, respectively:

1

4

∂f

∂ξ

1

+

∂f

∂ξ

2

+ 1

4

∂f

∂ξ

3

= 3

4∆ξ (f 3 − f 1) , (2.34)

and

1

4

∂f

∂ξ

N −2

+

∂f

∂ξ

N −1

+ 1

4

∂f

∂ξ

N

= 3

4∆ξ (f N − f N −2) . (2.35)

The above equations form a tri-diagonal system of linear equations which can

be efficiently solved using the Tridiagonal Matrix Algorithm (TDMA) (Conte &Boor, 1980). The combination of the sixth-order scheme for interior points and the

third-order, one-sided scheme for boundary points results in a fourth-order global

accuracy.

2.4 Spatial Filtering

Since the central difference compact scheme, eq. (2.31), is non-dissipative, spatial

filtering of the solution is required at each time step to remove high wavenumber

components and keep the solution stable. The filtering also serves as the ADM

subgrid modeling for LES as described in Sec. 2.2.

Dissertation Alireza

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