EFFECTS OF INFLOW FORCING ON JET NOISE USING 3-D …lyrintzi/Charlie-thesis.pdfEFFECTS OF INFLOW...

EFFECTS OF INFLOW FORCING ON JET NOISE USING 3-D LARGE EDDY

SIMULATION

A Thesis

Submitted to the Faculty

of

Purdue University

by

Phoi-Tack Lew

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science in Aeronautics and Astronautics

May 2004

ii

To my family...

iii

ACKNOWLEDGMENTS

I would like to thank my advisors Professor Anastasios S. Lyrintzis and Professor

Gregory A. Blaisdell who have been a constant source of guidance, wisdom and

leadership during this project. In addition, I would to thank Professor Steve H.

Frankel for serving as the third member of my committee. I would also like to thank

my colleague Dr. Ali Uzun who provided both his 3-D LES and aeroacoustic post-

processing codes. His assistance in understanding the inner workings of his code is

also greatly appreciated. Further thanks also goes out to Charles W. Wright for all

his help and patience for running the RANS computations using WIND. I would also

like to acknowledge the partial support from a Computational Science & Engineering

(CS&E) Fellowship during this project. The work summarized in this thesis is part

of a joint project with Rolls-Royce, Indianapolis and was sponsored by the Indiana

21st Century Research & Technology Fund. It was also partially supported by the

National Computational Science Alliance under the grant CT0100032N, and utilized

both the SGI Origin 2000 and IBM-SP4 supercomputer systems at the University

of Illinois at Urbana-Champaign. Some of the simulations were also carried out on

Purdue University’s 320 processor and Indiana University’s 508 processor IBM-SP3

supercomputers.

iv

TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Main Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 3-D LARGE EDDY SIMULATION METHODOLOGY . . . . . . . . . . . 6

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Vortex Ring Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Setup and Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 FORCING EFFECTS ON TURBULENT FLOW DEVELOPMENT . . . . 23

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Growth Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Reynolds Stresses and Turbulence Intensities . . . . . . . . . . . . . . 24

3.4 Potential Core Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5 Energy Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 FORCING EFFECTS ON JET AEROACOUSTICS . . . . . . . . . . . . . 42

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Ffowcs Williams-Hawkings Surface Integral Acoustic Method . . . . . 42

4.3 Far-Field Aeroacoustics . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK . 58

v

Page

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 59

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

A AN ATTEMPT AT COUPLING RANS AND LES FOR JET AEROA-COUSTICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.3 Brief Description of RANS and LES Methodology . . . . . . . . . . . 71

A.4 Setup and Coupling Methodology . . . . . . . . . . . . . . . . . . . . 71

A.5 Interpolation Methodology and Setup . . . . . . . . . . . . . . . . . . 73

A.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 74

A.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

B EFFECT OF TRI-DIAGONAL FILTERS FOR A PLANE MIXING LAYERUSING 2-D LARGE EDDY SIMULATION . . . . . . . . . . . . . . . . . . 84

B.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

B.2 Numerical Methods and Setup . . . . . . . . . . . . . . . . . . . . . . 85

B.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 88

B.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

vi

LIST OF TABLES

Table Page

2.1 Test case legend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 Initial jet growth rates. . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Peak turbulent Reynolds stresses for each test case at x = 25ro. . . . 30

3.3 Axial, radial and azimuthal peak root mean square velocity fluctu-ations and peak locations along the shear layer r = ro for all testcases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Axial, radial and azimuthal peak root mean square velocity fluctua-tions and peak locations along the jet centerline for all test cases. . . 31

3.5 Location of potential core break up. . . . . . . . . . . . . . . . . . . . 31

4.1 Peak SPL difference, frequency location and crossover frequency lo-cation for each test case with respect to Baseline for an open controlsurface at R = 60ro, θ = 60o in the far-field. . . . . . . . . . . . . . . 50

4.2 Peak SPL difference, frequency location and crossover frequency loca-tion for each test case with respect to Baseline for an closed controlsurface at R = 60ro, θ = 60o in the far-field. . . . . . . . . . . . . . . 50

B.1 Test case filter coefficients . . . . . . . . . . . . . . . . . . . . . . . . 92

B.2 Comparison of the normalized peak Reynolds stresses and growthrates with available experimental and computational data. . . . . . . 92

vii

LIST OF FIGURES

Figure Page

2.1 Boundary conditions used in the 3-D LES code. . . . . . . . . . . . . 20

2.2 The cross section of the computational grid on the z = 0 plane. (Everyother grid point is shown) . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 The cross section of the grid in the y − z plane at x = 5ro. (Everyother grid point is shown) . . . . . . . . . . . . . . . . . . . . . . . . 21



3.1 Normalized mean streamwise velocity profiles for Baseline case. . . . 32

3.2 Streamwise variation of the half-velocity radius normalized by the jetradius. Baseline case. . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Normalized Reynolds Stress σxx for each forcing case at x = 25ro . . . 33

3.4 Normalized Reynolds Stress σrr for each forcing case at x = 25ro . . . 33

3.5 Normalized Reynolds Stress σrx for each forcing case at x = 25ro . . . 34

3.6 Normalized Reynolds Stress σθθ for each forcing case at x = 25ro. . . 34

3.7 Axial profile of the root mean square of the axial fluctuating velocityalong the shear layer r = ro for all test cases . . . . . . . . . . . . . . 35

3.8 Axial profile of the root mean square of the radial fluctuating velocityalong the shear layer r = ro for all test cases . . . . . . . . . . . . . . 35

3.9 Axial profile of the root mean square of the azimuthal fluctuatingvelocity along the shear layer r = ro for all test cases . . . . . . . . . 36

3.10 Axial profile of the root mean square of the axial fluctuating velocityalong the jet centerline for all test cases . . . . . . . . . . . . . . . . . 36

3.11 Axial profile of the root mean square of the radial fluctuating velocityalong the jet centerline for all test cases . . . . . . . . . . . . . . . . . 37

3.12 Axial profile of the root mean square of the azimuthal fluctuatingvelocity along the jet centerline for all test cases . . . . . . . . . . . . 37

viii

Figure Page

3.13 One-dimensional spectrum of the streamwise velocity fluctuations atthe x = 10ro location on the jet centerline for hAMP . . . . . . . . . . 38

3.14 One-dimensional sprectrum of the streamwise velocity fluctuations atthe x = 10ro, y = ro, z = 0 location for hAMP . . . . . . . . . . . . . 38

3.15 One-dimensional sprectrum of the streamwise velocity fluctuations atthe x = 15ro location on the jet centerline for hAMP . . . . . . . . . . 39






4.1 The open FW-H stationary control surface around the turbulent jet. . 51

4.2 Schematic showing four of the sixteen partitioned blocks of the com-putational domain. Boundaries are numbered in (). . . . . . . . . . . 51

4.3 Schematic showing the center of the arc and how the angle θ is mea-sured from the jet axis. . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Overall sound pressure levels at R = 60ro from the nozzle exit for allcases with the open control surface. . . . . . . . . . . . . . . . . . . . 52

4.5 Overall sound pressure levels at R = 60ro from the nozzle exit for allcases with the closed control surface. . . . . . . . . . . . . . . . . . . 53

4.6 Acoustic pressure spectra at R = 60ro, θ = 60o in the far-field foreach test case with an open control surface. . . . . . . . . . . . . . . . 53

4.7 Acoustic pressure spectra at R = 60ro, θ = 60o in the far-field foreach test case with a closed control surface. . . . . . . . . . . . . . . 54

4.8 Acoustic pressure spectra at R = 60ro, θ = 60o in the far-field for eachtest compared against the Baseline case for an open control surface. 54

4.9 Acoustic pressure spectra at R = 60ro, θ = 60o in the far-field for eachtest compared against the Baseline case for a closed control surface. 55

ix

Figure Page

4.10 Acoustic pressure spectra at R = 60ro, θ = 25o in the far-field for eachtest compared against the Baseline case for a open control surface. . 55


4.12 Acoustic pressure spectra at R = 60ro, θ = 90o in the far-field for eachtest compared against the Baseline case for a open control surface. . 56


A.1 The cross centerline section of the turbofan considered. . . . . . . . . 78

A.2 Lobed mixer geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 78

A.3 Converged density contours (sectional) from WIND at the exit nozzleplane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

A.4 Sectional LES grid on RANS grid for interpolation . . . . . . . . . . 79

A.5 Converged streamwise velocity contours from WIND at the exit nozzleplane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

A.6 Interpolated RANS solution for the streamwise velocity on LES gridand Streamwise velocity contours at t = 0. . . . . . . . . . . . . . . . 80

A.7 Instantaneous streamwise velocity contours after 1,000 time steps with∆t = 0.015. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81




A.11 Instantaneous streamwise velocity contours after 10,000 time stepswith ∆t = 0.015. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

B.1 Transfer functions of various filters used in this study. . . . . . . . . . 93

B.2 Computational grid used in this LES. (Every 5th node is shown) . . . 93

B.3 Instantaneous streamwise vorticity contours in a naturally developingmixing layer. (Mc = 0.074, Reω = 5333) . . . . . . . . . . . . . . . . 94

B.4 Scaled velocity profiles along with the error-function profile for 6th-Order penta-diagonal filter. . . . . . . . . . . . . . . . . . . . . . . . 94

x

Figure Page

B.5 Vorticity thickness growth in the mixing layer for 6th-Order penta-diagonal filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

B.6 Transfer functions of the tri-diagonal filters used for the point next tothe boundary, i.e. i = 2. . . . . . . . . . . . . . . . . . . . . . . . . . 95

B.7 Normalized Reynolds normal stress σxx profiles for 6th-Order penta-diagonal filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

B.8 Normalized Reynolds normal stress σyy profiles for 6th-Order penta-diagonal filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

B.9 Normalized Reynolds shear stress σxy profiles for 6th-Order penta-diagonal filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

xi

NOMENCLATURE

Roman Symbols

Csgs Subgrid-scale model constant in the original Smagorinsky model

CI Compressibility correction constant in the subgrid-scale model

c Speed of sound

Dj Jet nozzle diameter

E(k) Turbulent kinetic energy spectrum

Ef (f) Power spectral density of velocity fluctuations

eφ Unit vector in the φ direction of the spherical coordinate system

er Unit vector in the r direction of the spherical coordinate system

eθ Unit vector in the θ direction of the spherical coordinate system

et Total energy

f Arbitrary variable; frequency

f Large scale component of variable f

fsg Subgrid-scale component of variable f

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

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

G(~x, ~x′

, ∆) Filter function

xii

J Jacobian of the coordinate transformation from physical to com-

putational domain

k Wavenumber

kx Axial wavenumber

M Mach number

Mj Mach number at jet nozzle exit

Mr Reference Mach number

N Number of grid points along a given spatial direction

p Pressure

Pr Prandtl number

Prt Turbulent Prandtl number

q,Q Vector of conservative flow variables

qi Resolved heat flux vector

q′

,RHS(q; t) Right-hand side of the governing equations

Qi Subgrid-scale heat flux vector

Qtarget Target solution in the sponge zone

Re Reynolds number

ReD Reynolds number based on jet diameter

Reω Reynolds number based on vorticity thickness

ro Jet nozzle radius

r1/2 Half velocity radius

xiii

r Radial direction in cylindrical coordinates

S Control surface

Sij Favre-filtered strain rate tensor

St Strouhal number

t Time

tn Time step n

T Temperature

Tr Reference temperature

U , V , W Contravariant velocity components

Uc Jet centerline velocity as a function of streamwise distance

Uo Jet centerline velocity at nozzle exit

Ur Reference velocity

u = (u, v, w) Mean velocity vector

u Velocity component in the x direction of Cartesian coordinates

(u, v, w) Velocity vector in Cartesian coordinates

ui Alternate notation for (u, v, w)

v Velocity component in the y direction of Cartesian coordinates

vr Velocity component in the radial (r) direction of cylindrical coor-

dinates

vθ Velocity component in the azimuthal (θ) direction of cylindrical

coordinates

xiv

vx Velocity component in the axial (x) direction of cylindrical coor-

dinates

V Integration volume

Vg Acoustic group velocity

w Velocity component in the z direction of Cartesian coordinates

(x, y, z) Cartesian coordinates

x Streamwise direction in both Cartesian and cylindrical coordi-

nates

y Transverse direction in Cartesian coordinates

z Transverse direction in Cartesian coordinates

xi or ~x Alternate notation for (x, y, z)

Greek Symbols

αf Filtering parameter of the tri-diagonal filter

α Parameter that controls the strength of the vortex ring forcing

φi Spatially filtered variable at grid point i

χ(x) Parameter that controls the strength of the sponge zone damping

term

∆ Local grid spacing or eddy viscosity length scale

δij Kronecker delta

δω Vorticity thickness of shear layer

∆t Time increment

xv

∆ξ Uniform grid spacing along the ξ direction in the computational

domain

γ Ratio of the specific heats of air

µ Molecular viscosity

µr Reference viscosity

ν Kinematic viscosity

ρ Density

ρr Reference density

Ψij Resolved shear stress tensor

σij Normalized Reynolds stress components

σxx = 〈vx′

vx′

〉U2

cNormalized Reynolds normal stress in the axial (x) direction of

cylindrical coordinates

σrr = 〈vr′

vr′

〉U2

cNormalized Reynolds normal stress in the radial (r) direction of

cylindrical coordinates

σθθ = 〈vθ′

vθ′

〉U2

cNormalized Reynolds normal stress in the azimuthal (θ) direction

of cylindrical coordinates

σrx = 〈vr′

vx′

〉U2

cNormalized Reynolds shear stress in cylindrical coordinates

τ Retarded time

Tij Subgrid-scale stress tensor

θ Azimuthal direction in cylindrical coordinates; angle from down-

stream jet axis

(ξ, η, ζ) Generalized curvilinear coordinates

xvi

Other Symbols

( )i′ Spatial or time derivative at grid point i

( )η Spatial derivative along the η direction

( )ξ Spatial derivative along the ξ direction

( )ζ Spatial derivative along the ζ direction

( ) Mean quantity

( ) Spatially filtered quantity

( ) Favre averaged quantity

( )′ Perturbation from mean value; acoustic variable

( )∞ Ambient flow value

( )o Flow value at jet centerline on the nozzle exit

〈〉 Time averaging operator

∂∂x

, ∂∂y

, ∂∂z

Partial spatial derivative operators in Cartesian coordinates

∂∂ξ

, ∂∂η

, ∂∂ζ

Partial spatial derivative operators in computational domain

∂∂t

Partial time derivative operator

Abbreviations

CAA Computational Aeroacoustics

DES Detached Eddy Simulation

DNS Direct Numerical Simulation

FWH Ffowcs Williams - Hawkings

xvii

LES Large Eddy Simulation

OASPL Overall Sound Pressure Level

RANS Reynolds Averaged Navier-Stokes

RHS Right-Hand Side of the Navier-Stokes Equations

rms Root Mean Square

SGS Subgrid-Scale

SPL Sound Pressure Level

xviii

ABSTRACT

Lew, Phoi-Tack. M.S.A.A.E, Purdue University, May, 2004. Effects of Inflow Forc-ing on Jet Noise using 3-D Large Eddy Simulation. Major Professors: AnastasiosS. Lyrintzis and Gregory A. Blaisdell.

This study uses 3-D Large Eddy Simulation (LES) to predict the noise emitted

from a Mach 0.9, Reynolds number ReD = 100, 000 jet. Recent discoveries have

shown that by adjusting selected inflow forcing parameters, the jet turbulent flow

development, and most importantly jet noise, could be greatly influenced. To im-

plement fully a nozzle structure in a high-end simulation like LES would require a

prohibitive number of grid points (approximately 150-200 million grid points) to re-

solve the nozzle boundary layer for realistic Reynolds numbers (ReD = 1, 000, 000).

Thus, inflow forcing currently seems to be a reasonable substitute for a nozzle geome-

try. However, the drawback of this approach is that the flow field results are sensitive

to the inflow forcing parameters used. With LES as an investigative tool, this the-

sis studies the effects of inflow forcing with particular emphasis on the number of

azimuthal modes and forcing amplitude. It is observed that halving the forcing am-

plitude and removing the first few modes results in the jet developing more slowly,

i.e. having a longer potential core. Furthermore, the peak turbulence intensities

increase when we reduce the forcing amplitude by half or remove the first 6 and 8

azimuthal modes of forcing. Due to this, high peak turbulence intensities we found,

using the Ffowcs Williams-Hawkings method, that the overall sound pressure level

(OASPL) also increases at all observation angles.

1

1. INTRODUCTION

1.1 Motivation

During the past several years, airports locally and abroad have implemented strict

regulations on aircraft with high jet noise emissions including imposing penalty fees

and restricting hours of operation. This not only causes a burden to airlines but also

to the communities surrounding the airport, which have to bear these high noise

levels. Hence, jet engine manufacturers have invested millions of dollars in theoreti-

cal, experimental, and computational research in the hopes of reducing jet noise and

thus remaining competitive in the aircraft industry. The underlying mechanisms

that cause jet noise are still not well understood and, therefore, hitherto cannot be

fully controlled or optimized. Thus, the jet noise problem still remains one of the

most elusive problems in aeroacoustics.

With the advent of fast supercomputers, the application of advanced compu-

tational techniques to jet noise prediction is becoming more feasible. The most

advanced approach is Direct Numerical Simulation (DNS). DNS solves for the dy-

namics of all the relevant length scales of turbulence and thus no form of turbulence

modeling is used. Freund et al. [1] were the first to study noise from a turbulent

jet using DNS. They simulated a Reynolds number 2000, Mach 1.92 supersonic tur-

bulent jet. The computed overall sound pressure levels (OASPL) were compared to

experimental data and found to be in good agreement with that of similar Mach

number jets. Later, Freund et al. [2] also used DNS to simulate a Reynolds number

3600, subsonic turbulent jet with a Mach number of 0.9. The computed mean flow

field and radiated sound field were in excellent agreement to a similar laboratory

experiment performed by Stromberg et al. [3]. Unfortunately, due to the wide range

of time and length scales present in turbulent flows and because of the limitations

2

of current computational resources, DNS is still restricted to low Reynolds number

flows as shown in the examples above. As a rule of thumb for DNS, the number

of grid points scales as Re9

4 with the Reynolds number and the computational cost

scale as Re3. Consider a Reynolds number ReD = 100, 000 turbulent jet which is

the Reynolds number used in this study. If the rule of thumb is applied here, the

number of grid points points required to solve all the relevant length scales would

approximately be 187 billion! Hence, one can imagine the enormous magnitude in

terms computational resources required to solve such a DNS problem.

In contrast to DNS, Large Eddy Simulation (LES), which computes the large

scales directly and models the small scales or the subgrid scales, yields a cheaper

alternative to DNS. It is assumed that the large scales in turbulence are generally

more energetic compared to the small scales and are affected by the boundary con-

ditions directly. In contrast, the small scales are more dissipative, weaker, and tend

to be more universal in nature. Furthermore, most turbulent jet flows that occur

in an experimental or an industrial setting are at high Reynolds numbers, usually

greater than 100,000. With this idea in mind it is more appropriate to use LES as a

tool for jet noise prediction, since it is capable of simulating high Reynolds number

flows but at a fraction of the cost of DNS. Furthermore, LES plays two important

roles in the simulation of complex turbulent flows of engineering interest. First,

LES can be used to test and validate lower order models such as Reynolds Average

Navier-Stokes (RANS) κ− ǫ, algebraic stress, and full Reynolds stress models. LES

can provide important additional data that would be otherwise impossible to obtain

experimentally, and which is at much higher Reynolds numbers than can be reached

by DNS [4]. Secondly, with computers today getting more powerful, LES can also be

used as an engineering tool rather than a research tool. Although LES still remains

an expensive alternative, it will likely be the only means of computing complex flows

for which lower order turbulence models fall short [4].

In the context of Computational Aeroacoustics (CAA), the first use of LES as

an investigative tool for jet noise prediction was carried out by Mankbadi, et al. [5].

3

They performed a simulation of a low Reynolds number supersonic jet and applied

Lighthill’s analogy [6] to calculate the far-field noise. Lyrintzis and Mankbadi [7]

were the first to use Kirchhoff’s method with LES to compute the far-field noise. A

string of other numerical experiments [8–14] were then carried out by investigators

at higher Reynolds numbers and were also found to be in good agreement with

experimental results.

In essence, all of the above numerical simulations using LES have one common

feature, they do not include the jet nozzle in the computations. The reason is that

if one were to include part of the jet nozzle, the number of grid points needed

to accurately resolve the boundary layer would be prohibitive unless the Reynolds

numbers are kept very low [15]. Thus, most of today’s jet CFD calculations which

include part of or the entire jet engine geometry are mainly restricted to using the

Reynolds Average Navier-Stokes (RANS) approach as an investigative tool [16–20].

Hence for LES, in place of a nozzle geometry and the turbulent boundary layers on

the nozzle walls, some form of forcing is needed to artificially simulate turbulence

in the jet flow field. These forcing functions are ideally divergence free so as not to

cause too much artificial noise in the simulation. Recently, Glaze and Frankel [21]

studied the behavior of two different stochastic inlet conditions which are intended

to simulate a turbulent inflow for a round jet. They tested a Gaussian random

forcing technique as the baseline case, and a version of the weighted amplitude

wave superposition spectral representation method as an improved technique. They

found that the Gaussian random inlet fluctuations model turbulent inflow poorly

and dissipate almost immediately, whereas the spectral inlet fluctuations reproduce

the jet near field much more accurately and allow the flow to transition rapidly to

self-sustaining turbulence. Another example of inflow forcing, which is used here in

this thesis, was developed by Bogey, et al. [22]. It takes the form of a vortex ring

placed close to the inflow boundary. Random perturbations are added to the flow

in order to “naturally” break up the potential core of the jet as the simulation is

time advanced. However, Bogey and Bailly [23] found that by carefully manipulating

4

selected forcing parameters, such as the amplitude and the number of modes, not

only were the turbulent flow properties changed but the far-field noise was altered

as well. The parameter that had the greatest impact was the number of azimuthal

forcing modes. By removing the first four modes out of a total of sixteen forcing

modes, they found that the simulation resulted in a quieter jet compared to a baseline

case with all modes turned on. Similar observations were reported by Bodony and

Lele [24].

Perhaps this effect is not very surprising when one considers that we are actually

attempting to simulate real turbulence numerically by artificial means, i.e. not with

a nozzle structure in this case. Bodony and Lele [24] explain that this behavior is

believed to be linked to the lack of three dimensionality inherent in most forcing

functions. These perturbations being fed into the inlet are highly coherent in the

azimuthal direction. They later suggest that these observations are also supported

by experimental data. Hence, the flow results and noise levels are sensitive to the

parameters used. At this point in time forcing is the only means of getting reasonable

turn-around time for a high-end simulation such as LES or DNS and needs to be

investigated further.

1.2 Main Objectives

The aim in this thesis is to investigate trends in our 3-D LES methodology devel-

oped by Uzun [25] on turbulent flow development and most importantly sensitivities

in jet noise by changing the forcing amplitude and number of modes present in the

vortex ring inflow forcing. Hence, the organization of this thesis is as follows. Chap-

ter 2 will discuss the governing equations and the numerical methodology used for

the 3-D Large Eddy Simulation code. Chapter 3 will then discuss several results

pertaining to the effect of the inflow forcing on turbulent flow development such as

jet growth rates and Reynolds stresses. Chapter 4 will first discuss the aeroacoustic

surface integral methods used and also what effect the vortex ring has on the far-field

5

jet noise. Finally, Chapter 5 will cover some concluding remarks and suggestions for

further work. Parts of this work were published as a conference paper listed as

Reference [26].

In addition, the Appendix section gives two brief reports on some previous work

done using LES. An attempt to couple the 3-D LES methodology used here with

RANS for the eventual study of jet noise is covered in Appendix A. Appendix B,

however, studies the effect of a tridiagonal spatial filter on a spatially developing

plane mixing layer using 2-D LES.

6

2. 3-D LARGE EDDY SIMULATION METHODOLOGY

2.1 Introduction

The 3-D LES methodology briefly described in this chapter was developed by

Uzun [14,25] as an initial platform to study the noise radiated from a high Reynolds

number, subsonic turbulent jet. Hence, for a more detailed description of this 3-D

LES methodology, please refer to Reference [25]. In addition, this chapter will also

present the setup and test cases that were considered.

2.2 Governing Equations

As mentioned in the previous chapter, Direct Numerical Simulation (DNS) solves

all the relevant temporal and spatial length scales present in a turbulent flow field.

Thus, due its cost, DNS at the present time is mainly restricted to being a research

tool. In the Reynolds Averaged Navier-Stokes approach, the entire turbulent flow

field is decomposed into a time averaged and fluctuating component and one only

solves for the time averaged flow field. Hence when compared to DNS, RANS is an

attractive tool due to its low computational cost. However, the use of RANS has its

limitations when applied to problems, like the one in this present study, where the

unsteady information is of great importance in accurately predicting the far-field jet

noise. Large Eddy Simulation (LES) can be seen as a compromise between the two

methods above, i.e. DNS and RANS. For LES, the turbulent field is decomposed

into a large-scale or resolved-scale component (f) and a small-scale or subgrid-scale

component (fsg). Hence, for an arbitrary variable f ,

f = f + fsg. (2.1)

7

A filtering operation is applied to f so that it only maintains the large-scale infor-

mation, f . This filtering operation is defined as a convolution integral operated on

f as follows

f(~x) =

∫

V

G(~x, ~x′

, ∆)f(~x′

) d~x′

(2.2)

where G(~x, ~x′

, ∆) is some spatial filter. Thus, the filtering operation removes the

information of the small-scale structures and the resulting governing equations con-

tain only the large-scale turbulent motions while the effect of the small-scales on the

resolved scales can be modeled by using a subgrid-scale (SGS) model such as the

classical Smagorinsky model [27] or the more sophisticated but expensive dynamic

Smagorinsky model proposed by Germano et al. [28].

The 3-D LES methodology incorporates the compressible form of the Navier-

Stokes equations. Hence, the large-scale component is written in terms of a Favre-

filtered variable

f =ρf

ρ. (2.3)

The Favre-filtered, compressible, non-dimensionalized continuity, momentum, and

energy equations are written in conservative form and are expressed as follows

∂ρ

∂t+

∂ρui

∂xi

= 0, (2.4)

∂ρui

∂t+

∂ρuiuj

∂xj

+∂p

∂xi

−∂

∂xj

(Ψij − Tij) = 0, (2.5)

∂et

∂t+

∂ui(et + p)

∂xi

−∂

∂xi

uj(Ψij − Tij) +∂

∂xi

(qi + Qi) = 0. (2.6)

In the momentum equation, the resolved shear stress tensor is given by the expression

Ψij =2µ

Re

(Sij −

1

3Skkδij

), (2.7)

whereas the Favre-filtered strain rate tensor is given by

Sij =1

2

(∂uj

∂xi

+∂ui

∂xj

). (2.8)

In the energy equation, the total energy is defined as

et =1

2ρuiui +

p

γ − 1, (2.9)

8

and the resolved heat flux is

qi = −

[µ

(γ − 1)Mr2RePr

]∂T

∂xi

. (2.10)

The temperature T is obtained from using the filtered pressure and density via the

ideal gas relation

p =ρT

γMr2 , (2.11)

Sutherland’s law is used for the molecular viscosity

µ

µr

=

(T

Tr

)3/2Tr + S

T + S. (2.12)

The Sutherland constant, S, is set to 110oK, while the reference temperature at the

centerline is Tr = 286oK, and the molecular viscosity, µr, is set as the jet centerline

temperature viscosity.

Due to the filtering operation, extra terms appear in the momentum and energy

equations, i.e. the subgrid-scale stress tensor and subgrid-scale heat flux expressed

as

Tij = ρ(uiuj − uiuj), (2.13)

Qi = ρ(uiT − uiT ). (2.14)

The 3-D LES code uses either the classical [27] or a localized dynamic [29] Smagorin-

sky (DSM) subgrid-scale model together with the compressibility correction proposed

by Yoshizawa [30] to model the subgrid-scale stress tensor. However, the modelling

of the subgrid-scale stress tensor has been debated for sometime [31–34]. Since the

small-scales are energy-dissipating and are not resolved, it is agreed that artificial

damping is required [35]. This is done through the use of an eddy-viscosity hypoth-

esis based on the physical interpretation of Tij, just like the classical Smagorinsky

model. However, Uzun et al. [36] reported that for their turbulent jet simulations us-

ing LES, the peak Reynolds stresses were highly sensitive to the chosen Smagorinsky

constant, Csgs. One reason why a particular flow property might be sensitive to the

chosen Smagorinsky constant is that the eddy-viscosity has the same functional form

9

as the molecular viscosity. Thus, it is difficult to define the effective Reynolds number

for the simulated flow [37]. Hence, to alleviate the uncertainty of the Smagorinsky

constant, a localized dynamic subgrid-scale model was implemented [14]. The big

drawback of using this sophisticated model is the simulation runtime. Simply put,

it takes 50% longer to run our LES code with the DSM compared to without any

SGS model at all. A comparison of the results with a DSM and without an SGS

model can be found in Reference [25]. Since our main objective is to identify trends

of the effects of inflow forcing, we decided to run the 3-D LES code without the

SGS model, i.e. setting Tij = 0 and Qi = 0. In place of an explicit SGS model, a

spatial filter [38] (See Section 2.3) will be used as an implicit SGS model to damp

the turbulent energy.

The 3-D LES methodology solves the governing equations for generalized curvi-

linear grids. This is useful for problems that include complex geometries. The

transformed governing equations can be written as

1

J

∂Q

∂t+

∂

∂ξ

(F − Fv

J

)+

∂

∂η

(G − Gv

J

)+

∂

∂ζ

(H − Hv

J

)= 0. (2.15)

Here t is the time, ξ, η, and ζ are the corresponding generalized coordinates in

computational space, and J is the Jacobian of the coordinate transformation from

the physical space to computational space, which can be expressed as

J =1

xξ

(yηzζ − yζzη

)− xη

(yξzζ − yζzξ

)+ xζ

(yξzη − yηzξ

) . (2.16)

In Equation (2.15) the bold face variables are the vector quantities and are expressed

as

Q =

ρ

ρu

ρv

ρw

et

F =

ρU

ρuU + ξxp

ρvU + ξyp

ρwU + ξzp

(et + p)U

G =

ρV

ρuV + ηxp

ρvV + ηyp

ρwV + ηzp

(et + p)V

H =

ρW

ρuW + ζxp

ρvW + ζyp

ρwW + ζzp

(et + p)W

,

(2.17)

10

Fv =

Fv1

Fv2

Fv3

Fv4

Fv5

Gv =

Gv1

Gv2

Gv3

Gv4

Gv5

Hv =

Hv1

Hv2

Hv3

Hv4

Hv5

, (2.18)

Fv1

Fv2

Fv3

Fv4

Fv5

=

0

ξx(Ψxx − Txx) + ξy(Ψxy − Txy) + ξz(Ψxz − Txz)

ξx(Ψxy − Txy) + ξy(Ψyy − Tyy) + ξz(Ψyz − Tyz)

ξx(Ψxz − Txz) + ξy(Ψyz − Tyz) + ξz(Ψzz − Tzz)

uFv2 + vFv3 + wFv4 − ξx(qx + Qx) − ξy(qy + Qy) − ξz(qz + Qz)

,

(2.19)

Gv1

Gv2

Gv3

Gv4

Gv5

=

0

ηx(Ψxx − Txx) + ηy(Ψxy − Txy) + ηz(Ψxz − Txz)

ηx(Ψxy − Txy) + ηy(Ψyy − Tyy) + ηz(Ψyz − Tyz)

ηx(Ψxz − Txz) + ηy(Ψyz − Tyz) + ηz(Ψzz − Tzz)

uGv2 + vGv3 + wGv4 − ηx(qx + Qx) − ηy(qy + Qy) − ηz(qz + Qz)

,

(2.20)

Hv1

Hv2

Hv3

Hv4

Hv5

=

0

ζx(Ψxx − Txx) + ζy(Ψxy − Txy) + ζz(Ψxz − Txz)

ζx(Ψxy − Txy) + ζy(Ψyy − Tyy) + ζz(Ψyz − Tyz)

ζx(Ψxz − Txz) + ζy(Ψyz − Tyz) + ζz(Ψzz − Tzz)

uHv2 + vHv3 + wHv4 − ζx(qx + Qx) − ζy(qy + Qy) − ζz(qz + Qz)

,

(2.21)

where Q is the vector of conservative flow variables, F, G, and H are the inviscid

flux vectors, Fv, Gv, and Hv are the viscous flux vectors. U , V , W are given by

U = ξxu + ξyv + ξzw, (2.22)

11

V = ηxu + ηyv + ηzw, (2.23)

W = ζxu + ζyv + ζzw. (2.24)

Please note that Tij and Qi are set to zero. Furthermore, ξx, ξy, ξz, ηx, ηy, ηz, ζx, ζy,

ζz are the grid transformation metrics. To ensure metric cancellation for a general

3-D curvilinear grids when high-order spatial descretization schemes are used, the

code uses the following “conservative” form of evaluating the metric expressions [38]

ξx/J =(yηz

)ζ−

(yζz

)η,

ηx/J =(yζz

)ξ−

(yξz

)ζ, (2.25)

ζx/J =(yξz

)η−

(yηz

)ξ,

ξy/J =(zηx

)ζ−

(zζx

)η,

ηy/J =(zζx

)ξ−

(zξx

)ζ, (2.26)

ζy/J =(zξx

)η−

(zηx

)ξ,

ξz/J =(xηy

)ζ−

(xζy

)η,

ηz/J =(xζy

)ξ−

(xξy

)ζ, (2.27)

ζz/J =(xξy

)η−

(xηy

)ξ.

The grid filter width, ∆ is given as

∆ =

(1

J

)1/3

. (2.28)

2.3 Numerical Methods

As mentioned in the previous section, the 3-D LES code solves the governing

equations in computational space where the grid spacing is uniform. The spatial

12

derivatives at the interior grid points away from the boundaries are computed using

a non-dissipative sixth-order compact scheme proposed by Lele [39]

1

3f ′

i−1 + f ′i +

1

3f ′

i+1 =7

9∆ξ

(fi+1 − fi−1

)+

1

36∆ξ

(fi+2 − fi−2

). (2.29)

Here, f′

i is the approximation of the first derivative of f at point i in the ξ direction,

and ∆ξ is the grid spacing in the ξ direction which is uniform. For the points next

to the boundaries, i = 2 and i = N − 1, the following fourth-order central compact

scheme used

1

4f ′

1 + f ′2 +

1

4f ′

3 =3

4∆ξ

(f3 − f1

), (2.30)

1

4f ′

N−2 + f ′N−1 +

1

4f ′

N =3

4∆ξ

(fN − fN−2

). (2.31)

Finally, for the points on the left and right boundary, i.e. i = 1 and i = N , the

following one-sided third-order compact scheme is used

f ′1 + 2f ′

2 =1

2∆ξ

(−5f1 + 4f2 + f3

), (2.32)

f ′N + 2f ′

N−1 =1

2∆ξ

(5fN − 4fN−1 − fN−2

). (2.33)

In order to eliminate numerical instabilities that can arise from the boundary

conditions, unresolved scales, and mesh nonuniformities, the sixth-order tri-diagonal

spatial filter proposed by Visbal and Gaitonde [38] is employed for the interior grid

points

αff i−1 + f i + αff i+1 =3∑

n=0

an

2(fi+n + fi−n) , (2.34)

where the an coefficients are defined as

a0 =11

16+

5αf

8a1 =

15

32+

17αf

16a2 =

−3

16+

3αf

8a3 =

1

32−

αf

16. (2.35)

The parameter αf satisfies the inequality given by −0.5 < αf < 0.5. A higher value

value of αf implies a less dissipative filter. Setting αf = 0.5 implies no filtering

effect. In the 3-D LES code, the filter coefficient is set to αf = 0.47. Now, for the

13

points next to the left-hand side boundary, i.e. i = 2, 3, the following sixth-order,

one-sided right-hand side stencil is used [38]

αff i−1 + f i + αff i+1 =7∑

n=1

an,ifn i = 2, 3, (2.36)

where

a1,2 =1

64+

31αf

32a2,2 =

29

32+

3αf

16a3,2 =

15

64+

17αf

32,

a4,2 =−5

16+

5αf

8a5,2 =

15

64−

15αf

32a6,2 =

−3

32+

3αf

16, (2.37)

a7,2 =1

64−

αf

32,

and

a1,3 =−1

64+

αf

32a2,3 =

3

32+

13αf

16a3,3 =

49

64+

15αf

32,

a4,3 =5

16+

3αf

8a5,3 =

−15

64+

15αf

32a6,3 =

3

32−

3αf

16, (2.38)

a7,3 =−1

64+

αf

32.

A similar procedure is applied for the points near the right boundary point, i = N

αff i−1 + f i + αff i+1 =6∑

n=0

aN−n,ifN−n i = N − 2, N − 1, (2.39)

where

aN−n,i = an+1,N−i+1 i = N − 2, N − 1 n = 0, 6. (2.40)

The boundary points, i = 1 and i = N are left unfiltered. Keep in mind that we

also use this spatial filter as an implicit SGS model since we have turned-off both

the classical Smagorinsky and localized Dynamic Smagorinsky models.

For time advancement, the 3-D LES code utilizes the standard fourth-order ex-

plicit Runge-Kutta sheme. See References [25] and [40] for details regarding the

formulation.

The state-of-the-art Tam and Dong’s [41] radiation and outflow boundary condi-

tions are implemented. This boundary condition was originally developed in 2-D and

14

was recently extended to 3-D by Bogey and Bailley [42]. The radiation boundary

conditions in spherical coordinates and are given by

1

Vg

∂

∂t

ρ

u

v

w

p

+

(∂

∂r+

1

r

)

ρ − ρ

u − u

v − v

w − w

p − p

= 0, (2.41)

and are applied to the lateral boundaries of the computational domain shown in

Figure 2.1. Here, ρ, u, v, w, p are the local primitive flow variables on the boundary,

ρ, u, v, w, p are the local mean flow properties, Vg is the acoustic group velocity

expressed as

Vg = (u + c) · er = u · er +

√|c|2 − (u · eθ)2 − (u · eφ)2. (2.42)

From the above equation, c is the local mean sound velocity vector and is defined

as follows: Draw a vector from the acoustic source location, (xsource, ysource, zsource),

to the boundary point at which Vg is being computed. This vector is the local mean

sound speed velocity vector. er, eθ, eφ denote the unit vectors in r, θ and φ directions

of the spherical coordinate system. These unit vectors can be expressed in terms of

Cartesian coordinates as

er = (sin θ cos φ, sin θ sin φ, cos θ),

eθ = (cos θ cos φ, cos θ sin φ,− sin θ), (2.43)

eφ = (− sin φ, cos φ, 0).

The acoustic group velocity, Vg, is the same as the wave propagation speed, and is

equal to the projection of the vector sum of the local mean sound velocity and local

mean flow velocity onto the sound propagation direcition. It is assumed that in the

far-field, the outgoing acoustics disturbances are propagating in the radial direction

relative to the acoustic source [25].

15

The position vector, r is obtained by

r =√

(x − xsource)2 + (y − ysource)2 + (z − zsource)2, (2.44)

where x, y, z are the coordinates of the boundary point, and xsource, ysource, zsource

are the coordinates of the acoustic source location. The source location is usu-

ally chosen as the end of the potential core of the jet and in this case it is set to

(xsource, ysource, zsource) = (10ro, 0, 0) in the 3-D LES code (where ro is one jet radii).

The derivative along the r direction is expressed in terms of the derivatives in the

Cartesian coordinate system as follows

∂

∂r= ∇ · er = sin θ cos φ

∂

∂x+ sin θ sin φ

∂

∂y+ cos θ

∂

∂z, (2.45)

and ∇ is the gradient operator in the Cartesian coordinate system. On the outflow

boundary, however, where entropy and vorticity waves in addition to the acoustic

waves cross, the above formulation of radiation Tam and Dong’s is not suitable. On

the outflow, the following formulation is used [42]

∂ρ

∂t+ u · ∇(ρ − ρ) =

1

c2

(∂p

∂t+ u · ∇(p − p)

),

∂u

∂t+ u · ∇(u − u) = −

1

ρ

∂(p − p)

∂x,

∂v

∂t+ u · ∇(v − v) = −

1

ρ

∂(p − p)

∂y, (2.46)

∂w

∂t+ u · ∇(w − w) = −

1

ρ

∂(p − p)

∂z,

1

Vg

∂p

∂t+

∂(p − p)

∂r+

(p − p)

r= 0.

For a more in-depth discussion on the numerical implementation methodology of this

boundary condition in the 3-D LES code, refer to Uzun [25].

In addition, a sponge zone [43] is attached to the end of the computational

domain to dissipate the vortices present in the flow field before they hit the outflow

boundary. This is done so that unwanted reflections from the outflow boundary

are suppressed. Grid strecthing as well as explicit filtering are applied along the

16

streamwise direction in the sponge zone to dissipate the vortices before they exit the

outflow boundary. Uzun [25] reports that the combination of explicit filtering and

Tam and Dong’s outflow boundary conditions were found to be stable and did not

cause any problems. In the sponge zone, the turbulent flow field is forced towards a

target solution with the use of a damping term added to the right-hand side of the

governing equations

∂Q

∂t= RHS −X (x)(Q − Qtarget), (2.47)

where the damping term X (x) is expressed as

X (x) = Xmax

(x − xphy

xend − xphy

)3

. (2.48)

Here, RHS is the right hand side of the governing equations, x is the streamwise

coordinate in the sponge zone, xphy is the streamwise coordinate of the end of the

physical domain and xend is the streamwise coordinate of the of the end of the sponge

zone. Q as before, is the vector containing the conservative variables, Qtarget is the

target solution in the sponge zone, and X (x) is function that determines the strength

of the damping term. In the 3-D LES code, the damping amplitude, Xmax is set to

1.0 and the target solution is specified as the self-similar solution of an isothermal

incompressible round jet.

2.4 Vortex Ring Forcing

To excite the mean flow, randomized perturbations in the form of induced ve-

locities from a vortex ring [22] are added to the velocity profile at a short distance

(approximately one jet radius) downstream from the inflow boundary. This is done

to ensure the break up of the potential core. The length of the potential core here

is determined by the location where the jet centerline velocity reduces to 95% of the

inflow jet velocity, Uc(xc) = 0.95Uj. The streamwise and radial velocity components

17

of the vortex ring (vx, vr) are added to the local velocity components (vxo, vro) as

shown by the formulation below

vx = vxo+ αUxring

Uo

nmodes∑

n=0

ǫn cos(nΘ + ϕn)

︸︷︷︸v′

x

(2.49)

vr = vro+ αUrring

Uo

nmodes∑

n=0

ǫn cos(nΘ + ϕn)

︸︷︷︸v′

r

(2.50)

where Θ = tan−1(y/z), ǫn and ϕn are randomly generated numbers that satisfy

−1 < ǫn < 1 and 0 < ϕn < 2π. Uo is the mean jet centerline velocity at the inflow

boundary. The parameter that determines the amplitude of the forcing is α and it

is set to α = 0.007. Finally, the parameter of interest is the number of modes given

by nmodes. Velocity perturbations in the azimuthal direction are not added. Uxring

and Urringare the mean nondimensional streamwise and radial velocity components

induced by the vortex ring and are given by

Uxring= 2

ro

r

r − ro

∆o

exp

(−ln(2)

(∆(x, y)

∆o

)2)

(2.51)

Urring= −2

ro

r

x − xo

∆o

exp

(−ln(2)

(∆(x, y)

∆o

)2)

(2.52)

where r =√

y2 + z2 6= 0, ∆o is the minimum grid spacing in the shear layer, and

∆(x, y)2 = (x− xo)2 + (r − ro)

2. The location where the center of the vortex ring is

located is xo and for our case it is set at xo = ro. An approximate location is shown

in Figure 2.1. The radius of the vortex ring is ro and is set equal to the initial jet

radius.

2.5 Setup and Test Cases

Since we are trying to establish trends, a computational domain with a reason-

able number of grid points will be considered so that several runs can be made. The

18

physical part of the domain extends to approximately 25ro in the streamwise direc-

tion and −15ro to 15ro in the transverse y and z directions. The total number of grid

points used here is 287× 128× 128 in the x-y-z directions, respectively. This gives a

total of approximately 4.7 million grid points. Figure 2.2 shows the x − y cross sec-

tional plane of the computational domain. Notice that there are more points packed

near the shear layer in order to resolve the relatively high velocity gradients there.

Figures 2.3 through 2.5 show the y−z cross section for several stations downstream.

We consider a hyperbolic tangent velocity mean profile on the inflow boundary given

by

u(r) =1

2Uo

[1 − tanh

[b

(r

ro

−ro

r

)]](2.53)

where r, ro, and Uo are defined in the previous section. The parameter that controls

the thickness of the shear layer is b. In our code we have set this parameter to

b = 3.125. A higher value of b implies a thinner shear layer. For comparison,

Freund [2] used a value of 12.5 since his grid was fine enough to resolve thin shear

layers. Hence, our b parameter corresponds to that of a relatively thick shear layer.

In addition, the following Crocco-Buseman relation for an isothermal jet is specified

for the density profile on the inflow boundary

ρ(r) = ρo

(1 +

γ − 1

2Mr

2 u(r)

Uo

(1 −

u(r)

Uo

))−1

, (2.54)

where Mr = 0.9.

We study a subsonic jet with a Mach number of 0.9 and Reynolds number

ReD =ρjUjDj

µj

= 100, 000. (2.55)

Here, ρj, Uj (Uj = Uo) and µj are the jet centerline density, velocity and viscosity

at the inflow. Dj is simply the jet diameter. Since this is an isothermal jet, the

centerline temperature is the same as the ambient temperature. The vortex ring

used here contains a total of 16 azimuthal jet modes of forcing, i.e. nmodes + 1 =

16. Bogey and Bailly [23] performed a simulation with all modes present and later

removed the first four modes and found that the jet was quieter with the latter case.

19

Furthermore, they also reduced the forcing amplitude α by half and found that this

procedure resulted in a noisier jet. For our study we investigate five test cases. The

first four will involve the consistent removal of the number of modes present. And

in the last, we will keep all modes present and instead reduce the forcing amplitude

by half, i.e. α = 0.0035. Table 2.1 gives the test case legend and the corresponding

number of azimuthal forcing modes removed as well as the forcing strength. The

time step is calculated by

∆t =min(∆x, ∆y, ∆z)

c∞ + Uj

, (2.56)

and the minimum grid spacing here is ∆ymin = ∆zmin = 0.06ro. Also, c∞ is the

ambient speed of sound based on the centerline Mach number, i.e. Mr = Uj/c∞, and

Uj is the centerline velocity which is unity.

The 3-D LES methodology is fully parallel, incorporating the Message Passing

Interface (MPI) libraries and written in the Fortran 90 progamming language. Due

to the nature of the compact scheme and the implicit spatial filter, a transposition

strategy is used to compute these schemes in parallel. Furthermore, the code also has

a restarting capability where a simulation can run in many stages. A more in-depth

discussion on the parallelization of this 3-D LES methodology again can be found

in Reference [25]. A total runtime of 17 days is required for each test case using 16

processors on the IBM-SP3 or IBM-SP4 machines.

20

Table 2.1 Test case legend

Test Case No. Modes Removed (Remaining Modes) α

Baseline None (16) 0.007

rf4 First Four Modes (12) 0.007

rf6 First Six Modes (10) 0.007

rf8 First Eight Modes (8) 0.007

hAMP None (16) 0.0035

Tam & Dong’s radiation boundary conditions

Sponge Zone

Tam & Dong’s radiation boundary conditions

Tam &Dong’sradiationbcs

Tam &Dong’soutflowboundarycondition

Vortex ring forcing

Figure 2.1. Boundary conditions used in the 3-D LES code.

21

x/ro

y/r o

0 10 20 30-15

-10

-5

0

5

10

15

Figure 2.2. The cross section of the computational grid on the z = 0plane. (Every other grid point is shown)

z/ro

y/r o

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

Figure 2.3. The cross section of the grid in the y−z plane at x = 5ro.(Every other grid point is shown)

22

z/ro

y/r o

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15


z/ro

y/r o

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15


23

3. FORCING EFFECTS ON TURBULENT FLOW

DEVELOPMENT

3.1 Introduction

This chapter presents results pertaining to the effect of forcing on turbulent flow

development. Turbulent flow properties such as jet growth rates, Reynolds stresses,

turbulence intensities and one-dimensional energy spectra of the fluctuating velocity

are presented and discussed.

3.2 Growth Rates

Figure 3.1 shows the normalized mean streamwise velocity profiles along different

downstream locations normalized by r1/2 for the Baseline case. The half-velocity

radius, r1/2, at a particular downstream location is defined as the radial location

where the mean streamwise velocity is one-half the jet mean centerline velocity. From

Figure 3.1, it can be seen that the three downstream locations collapse fairly well and

exhibit self-similarity. Although not shown here, the same self-similarity behavior is

observed for the remaining test cases at the same three downstream locations. The

streamwise variation of the half-velocity radius normalized by the initial jet radius

is an indicator of the jet spreading rate and is shown in Figure 3.2. From Figure 3.2

we can see that the jet spreading rate for the baseline case is A = 0.076, which is

less than the values obtained from experiments. This slow growth can be explained

by the relative shortness or our streamwise domain. By x = 25ro the jet has not yet

reached its full growth rate. Uzun, et al. [14] used a streamwise domain length of

x = 60ro and obtained a value of A = 0.092 which is well within the experimental

range. They measured the growth rate from x = 30ro until the end of the physical

24

domain. Hence, we should expect to obtain the experimental growth rate values if

we were to use a longer domain. However, since our main goal is to look at trends of

the effects of the inflow parameters, the shorter domain was chosen in order to save

time. Table 3.1 gives the growth rate values for each test case. It can be seen from

this table that forcing has only a minor effect on the initial growth rates of our jet.

3.3 Reynolds Stresses and Turbulence Intensities

Next we look at the Reynolds stresses. Due to the shortness of the domain, we

found that the Reynolds stresses do not achieve their true asymptotic self-similar

state. To obtain the self-similar state, a streamwise domain of at least x = 45ro

is required [14] for a jet Reynolds number of ReD = 100, 000. It should be noted

that in experiments a distance of approximately x = 100ro is typically used for the

measurement of asymptotic rates. However, we can make some observations on the

initial peak Reynolds stresses. Figures 3.3 to 3.6 show the variation of normalized

Reynolds stresses at the end of the physical domain, x = 25ro. The normalized

Reynolds stresses are defined in cylindrical coordinates as follows:

σxx =v′

xv′x

U2c (x)

σrr =v′

rv′r

U2c (x)

σθθ =v′

θv′θ

U2c (x)

σrx =v′

rv′x

U2c (x)

, (3.1)

where v′x, v′

r, v′θ are the axial, radial and azimuthal components of the fluctuating

velocity, respectively, Uc(x) is the mean jet centerline velocity at a particular axial

location, and the overbar denotes time-averaging. Table 3.2 in turn gives the values

for the peak Reynolds stresses for all test cases. The peak values for σxx, σrr, and

σθθ show an increase when the first 4 modes are removed but there is a slight drop

in the peak value for shear stress, σrx. When the first 6 modes are removed, the

normal stresses drop slightly but now the peak shear stress increases. However, if

further modes are removed, i.e. the rf8 case, we see that this case shows the highest

peak Reynolds stresses for both the normal and shear directions. For the case of

hAMP , the peak normal Reynolds stresses, σxx, σrr, and σθθ, increases relative to

the Baseline case. The peak shear stress however, shows a slight decrease when

25

compared to the Baseline case. The somewhat inconsistent behavior between the

normal and shear stresses as the number of modes are removed could be due to the

fact that the Reynolds stresses are normalized by the local centerline velocity Uc(x)

and this value changes for each test case at x = 25ro. But generally speaking, we

see that as modes are removed from the forcing or the amplitude is reduced the

turbulence levels are increased.

Figures 3.7 through 3.9, show the axial, radial and azimuthal root mean square

(rms) velocity fluctuations, respectively, at r = ro normalized by Uo, i.e. the exit

jet centerline velocity rather than Uc(x). The location r = ro was chosen in order

to focus on the shear layer. Table 3.3 gives the peak fluctuation values along with

their locations. From Table 3.3, we see that the peak of the rms fluctuations shifts

downstream as the number of modes removed is increased. Notice also that the peak

locations occur well before the end of the potential core (approximately 2.5ro to 3ro

before xc). (The potential core length is discussed in Section 3.4.) Thus, the increase

in peak Reynolds stresses at x = 25ro is a consequence of the streamwise shift in

peak turbulence intensities downstream. Now, if we compare Figures 3.7 through 3.9

carefully, a consistent behavior persists, i.e. the variation of intensities before they

reach their peak values. If we look at x = 5ro for example, as we remove the first

few modes of forcing, the intensities for each test case decrease and consequently

their peak intensities shift downstream. By removing the first few modes of forcing,

we are reducing the azimuthal correlation between the perturbations. When Bogey

and Bailly [23] removed the first four modes of forcing using their LES methodology,

they reported a drop in their axial and radial peak turbulence intensity on the order

of 1.5% and 10%, respectively. Based on Table 3.3, we see a drop in the peak inten-

sities for rf4 on the order of 1% and 2.5% for the peak axial and radial intensities,

respectively. Hence, we do not observe the same significant drop when compared to

Bogey and Bailly. One of the reasons we believe is that their jet Reynolds number

was ReD = 400,000. Bogey and Bailly [23] did not report turbulence intensities in

the azimuthal direction, i.e. v′

θ/Uo. Hence,

26

Bogey and Bailly, however, did not perform simulations where the first six or

eight modes of forcing were removed as is done here. An interesting behavior is

observed when we look carefully at the peak values of intensities for the axial, radial

and azimuthal components. From Table 3.3, there is not much of a change between

peak intensities of the axial component for each test case but this behavior is not

seen for the radial and azimuthal component. If we consider the first three test

cases for the radial and azimuthal part, the difference in peak intensities is relatively

small but this is not so when we compare it to rf8. One possible explanation could

be the original definition of total number of modes used. In reports by Bogey, et

al. [22, 23], although they do not explicitly state the reason, the total number of

azimuthal modes chosen was at least nmodes + 1 = 10. The first three test cases

contain a total number of 10 azimuthal modes or more except for the last case, i.e.

rf8. Thus, the reduced number of azimuthal modes might possibly be a factor as to

why the radial and azimuthal peak intensities for rf8 is high compared to the other

cases, although further investigation is needed.

In addition, Bogey and Bailly also performed a simulation where they reduced the

forcing amplitude, α, by half and kept the total number of modes fixed at 16. They

found that the intensities for the this case were also reduced before they reached their

peak values. However, the peak values were higher compared to the baseline case

(both Baseline and hAMP have 16 modes). A similar observation is reported here.

Refering to Figures 3.7 through 3.9 once more, we see that hAMP has the highest

peak intensity when compared to Baseline and reduced mode cases rf4 through rf8

along the jet shear layer. However, the peak location occurs somewhere in between

Baseline and rf6. The peak intensity values and their corresponding locations are

also tabulated in Table 3.3. We can also make some comparisons to experiments.

Hussain and Zedan [44] reported a peak intensity of ((v′

x)rms/Uo)p ≃ 0.19 in their

axisymmetric free shear layer experiment. Hence, our values in Table 3.3 agree well

with their experiment. A peak radial intensity of ((v′

r)rms/Uo)p ≃ 0.13 was obtained

from Hussain and Husain’s experiment [45]. From Table 3.3 we can see that we are

27

over-predicting their peak values for all test cases. Bogey and Bailly [23] reported a

similar behavior with their LES methodology.

Figures 3.10 through 3.12 show the variation of the axial, radial and azimuthal

rms velocity fluctuation along the jet centerline. Table 3.4 gives the corresponding

peak intensity values and peak locations. From Table 3.4 we can see that the same

behavior persists along the centerline as in the the shear layer, i.e. as more modes

are removed, the peak turbulence intensities increase and again hAMP registers the

highest peak value. The differences here are that the magnitude of the peak intensi-

ties are lower and the peak locations are located further downstream (approximately

4ro to 7ro after the potential core) when compared to the intensities along the shear

layer (see Table 3.3). Again, Bogey and Bailly [23] reported a similar behavior along

the centerline with their LES methodology when they removed the first four modes

of forcing and reduced the forcing amplitude by half. Comparing the computed peak

intensities at the centerline to experiments, Arakeri et al. [46] reported peak intensi-

ties of ((v′

x)rms/Uo)p ≃ 0.12 for untripped jets whereas Lau et al. [47] reported a peak

value of ((v′

r)rms/Uo)p ≃ 0.14 and ((v′

x)rms/Uo)p ≃ 0.11 for a ReD = 106 turbulent

jet. Hence, the computed axial and radial turbulence intensities are still slightly

greater than the peak values from experiments.

3.4 Potential Core Lengths

Table 3.5 gives the location where the jet potential core region breaks up for each

case. The length of the potential core here is determined by the location where the

jet centerline velocity reduces to 95% of the inflow jet velocity, Uc(xc) = 0.95Uj. We

can see that as the number of modes is removed, the potential core length increases

consistently. Hence, the jet develops more slowly. The jump is more pronounced

when we compare Baseline to rf4. Reducing the forcing amplitude by half increases

the potential core length by about one jet radii. Hence, we could assume that the

number of forcing modes in this case has a more dominant effect on the jet core

28

length than the forcing strength. A potential core length of about 10ro with an

initially transient shear layer was reported by Raman, et al. [48]. For jets with high

Reynolds numbers and where the shear layer is initially turbulent, a potential core

length of about 14ro has been measured by Raman, et al. [48] and Lau, et al. [47]. An

interesting behavior here is that by removing more modes, based on the comparison

of core lengths, we resemble closely an initially turbulent jet. Bogey and Bailly [23]

also reported a similar behavior when they removed the first four modes, but their

potential core length was 11.9ro with ReD = 400, 000. They suggest that a jet

excited with higher modes of forcing behaves more like a turbulent jet. This is also

probably due to the modes having less coherence in the azimuthal direction.

3.5 Energy Spectra

Figures 3.13 through 3.20 show the one-dimensional spectrum of the stream-

wise velocity fluctuations at several locations downstream of the jet for hAMP .

Data needed to compute spectra were not collected during the other simulations.

The computed one-dimensional spectrum comes from the temporal spectrum of the

streamwise velocity fluctuations at a given location by making use of Taylor’s hy-

pothesis of frozen turbulence. In this hypothesis, the one-dimensional spectrum of

the streamwise velocity fluctuations is given as

E(kx) = Ef (f)u

2π, (3.2)

where kx = f(2π/u) is the axial wavenumber, f is the frequency and Ef (f) is

the power spectral density of the streamwise velocity fluctuations. From Figures

3.13 through 3.20, the one-dimensional spectrum becomes almost flat as the axial

wavenumber approaches zero. The plots also show the grid cut-off wavenumber

corresponding to our grid resolution at the given axial location. Due to the grid

resolution being coarser as we progress downstream, the wavenumber cut-off becomes

smaller. With the exception of Figure 3.13, the spectrum exhibits a decay rate which

is almost similar to Kolmogorov’s -5/3 law decay rate prediction in the inertial range

29

of turbulence before the grid cut-off wavenumber. Hence, our grid resolution is fine

enough to resolve a portion of the inertial range of the wavenumbers at a given axial

location. Figure 3.13 does not show this trend since this point (x = 10ro on the jet

axis) is located inside the potential core region. In fact, we see a build-up of energy

in the large scales and no presence of an inertial range.

30

Table 3.1 Initial jet growth rates.

Test Case Growth rate, A

Baseline 0.076

rf4 0.071

rf6 0.074

rf8 0.078

hAMP 0.081

Table 3.2 Peak turbulent Reynolds stresses for each test case at x = 25ro.

Test Case (σxx)p (σrr)p (σθθ)p (σrx)p

Baseline 0.0552 0.0334 0.0358 0.0187

rf4 0.0574 0.0353 0.0374 0.0179

rf6 0.0571 0.0347 0.0386 0.0188

rf8 0.0592 0.0361 0.0392 0.0196

hAMP 0.0569 0.0351 0.0361 0.0185

Table 3.3 Axial, radial and azimuthal peak root mean square velocityfluctuations and peak locations along the shear layer r = ro for alltest cases.

Test Case ((v′

x)rms)p

Uoxp

((v′

r)rms)p

Uoxp

((v′

θ)rms)p

Uoxp

Baseline 0.192 8.22ro 0.159 9.13ro 0.171 9.50ro

rf4 0.190 9.77ro 0.155 10.41ro 0.168 10.92ro

rf6 0.191 11.07ro 0.162 11.09ro 0.170 11.83ro

rf8 0.194 11.70ro 0.170 11.29ro 0.174 12.43ro

hAMP 0.197 10.78ro 0.174 10.04ro 0.182 10.73ro

31

Table 3.4 Axial, radial and azimuthal peak root mean square velocityfluctuations and peak locations along the jet centerline for all testcases.

Test Case ((v′

x)rms)p

Uoxp

((v′

r)rms)p

Uoxp

((v′

θ)rms)p

Uoxp

Baseline 0.148 17.10ro 0.123 15.20ro 0.119 16.00ro

rf4 0.147 19.02ro 0.120 18.62ro 0.117 16.34ro

rf6 0.152 17.48ro 0.125 17.86ro 0.124 17.87ro

rf8 0.153 17.10ro 0.126 19.00ro 0.127 16.73ro

hAMP 0.158 16.72ro 0.128 15.95ro 0.129 16.33ro

Table 3.5 Location of potential core break up.

Test Case Location

Baseline xc = 11.54ro

rf4 xc = 13.07ro

rf6 xc = 13.43ro

rf8 xc = 13.45ro

hAMP xc = 12.49ro

32

r/r 1/2

u/U

c

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 x = 15ro

x = 20ro

x = 25ro

Figure 3.1. Normalized mean streamwise velocity profiles for Baseline case.

x/ro

r 1/2

/ro

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Slope = A = 0.076

Experimental Valuesof A: 0.086 - 0.096

Figure 3.2. Streamwise variation of the half-velocity radius normal-ized by the jet radius. Baseline case.

33

r/r 1/2

σ xx

0 0.5 1 1.5 2 2.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07 Baselinerf4 modesrf6 modesrf8 modeshAMP

Figure 3.3. Normalized Reynolds Stress σxx for each forcing case at x = 25ro

r/r 1/2

σ rr

0 0.5 1 1.5 2 2.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04


Figure 3.4. Normalized Reynolds Stress σrr for each forcing case at x = 25ro

34

r/r 1/2

σ rx

0 0.5 1 1.5 2 2.50

0.005

0.01

0.015

0.02Baselinerf4 modesrf6 modesrf8 modeshAMP

Figure 3.5. Normalized Reynolds Stress σrx for each forcing case at x = 25ro

r/r 1/2

σ θθ

0 0.5 1 1.5 2 2.50

0.01

0.02

0.03

0.04


Figure 3.6. Normalized Reynolds Stress σθθ for each forcing case at x = 25ro.

35

x/ro

(v’ x)

rms/U

o

0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22Baselinerf4 modesrf6 modesrf8 modeshAMP

Figure 3.7. Axial profile of the root mean square of the axial fluctu-ating velocity along the shear layer r = ro for all test cases

x/ro

(v’ r)

rms/U

o

0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Baselinerf4 modesrf6 modesrf8 modeshAMP

Figure 3.8. Axial profile of the root mean square of the radial fluc-tuating velocity along the shear layer r = ro for all test cases

36

x/ro

(v’ θ)

rms/U

o

0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2


Figure 3.9. Axial profile of the root mean square of the azimuthalfluctuating velocity along the shear layer r = ro for all test cases

x/ro

(v’ x)

rms/U

o

0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2


Figure 3.10. Axial profile of the root mean square of the axial fluc-tuating velocity along the jet centerline for all test cases

37

x/ro

(v’ r)

rms/U

o

0 5 10 15 20 250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15


Figure 3.11. Axial profile of the root mean square of the radialfluctuating velocity along the jet centerline for all test cases

x/ro

(v’ θ)

rms/U

o

0 5 10 15 20 250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15


Figure 3.12. Axial profile of the root mean square of the azimuthalfluctuating velocity along the jet centerline for all test cases

38

kxro

E(k

xro)

/U2 or

o

5 10 1510-7

10-6

10-5

10-4

10-3

10-2

10-1

(kxro)-5/3

Grid cutoff

Figure 3.13. One-dimensional spectrum of the streamwise velocityfluctuations at the x = 10ro location on the jet centerline for hAMP .

kxro

E(k

xro)

/U2 or

o

5 10 1510-7

10-6

10-5

10-4

10-3

10-2

10-1

(kxro)-5/3

Grid cutoff

Figure 3.14. One-dimensional sprectrum of the streamwise velocityfluctuations at the x = 10ro, y = ro, z = 0 location for hAMP .

39

kxro

E(k

xro)

/U2 or

o

2 4 6 8 1010-7

10-6

10-5

10-4

10-3

10-2

10-1

(kxro)-5/3

Grid cutoff

Figure 3.15. One-dimensional sprectrum of the streamwise velocityfluctuations at the x = 15ro location on the jet centerline for hAMP .

kxro

E(k

xro)

/U2 or

o

2 4 6 8 1010-7

10-6

10-5

10-4

10-3

10-2

10-1

(kxro)-5/3

Grid cutoff


40

kxro

E(k

xro)

/U2 or

o

2 4 6 81010-7

10-6

10-5

10-4

10-3

10-2

10-1

(kxro)-5/3

Grid cutoff


kxro

E(k

xro)

/U2 or

o

2 4 6 8 1010-7

10-6

10-5

10-4

10-3

10-2

10-1

(kxro)-5/3

Grid cutoff


41

kxro

E(k

xro)

/U2 or

o

2 4 6 8 1010-7

10-6

10-5

10-4

10-3

10-2

10-1

Grid cutoff

(kxro)-5/3


kxro

E(k

xro)

/U2 or

o

2 4 6 81010-7

10-6

10-5

10-4

10-3

10-2

10-1

(kxro)-5/3

Grid cutoff


42

4. FORCING EFFECTS ON JET AEROACOUSTICS

4.1 Introduction

In this chapter, sensitivities of the far-field aeroacoustics to forcing effects is

discussed. A brief section on the aeroacoustic methodology utilized is presented first,

followed by aeroacoustic results such as the overall sound pressure level (OASPL)

and pressure spectra for a fixed point in the far-field.

4.2 Ffowcs Williams-Hawkings Surface Integral Acoustic Method

The aeroacoustic analysis is done after the LES computations are finalized. To

couple the two, the porous Ffowcs Williams-Hawkings [49,50] formulation is utilized

to study the far-field jet noise as suggested by Lyrintzis [51] and Lyrintzis and Uzun

[52]. For simplicity, a continuous stationary control surface around the turbulent jet

is used. The formulation for the disturbance pressure is as follows

p′(~x, t) = p′T (~x, t) + p′L(~x, t) + p′Q(~x, t), (4.1)

where

4πp′T (~x, t) =

∫

S

[ρ∞Un

r

]

ret

dS, (4.2)

4πp′L(~x, t) =1

c∞

∫

S

[Lr

r

]

ret

dS +

∫

S

[Lr

r2

]

ret

dS, (4.3)

and

Ui =ρui

ρ∞

, (4.4)

Li = p′δijnj + ρuiun. (4.5)

Here, p′T (~x, t) is known as the thickness noise, p′L(~x, t) is the loading noise and

p′Q(~x, t) is the quadrupole noise pressure term that includes all sources outside the

43

control surface. The quadrupole noise pressure term is neglected in this methodology

[25]. (~x, t) are the observer coordinates and time, r is the distance from the source on

the surface to the observer, c∞ and ρ∞ are the ambient speed of sound and ambient

density, respectively. A time derivative is indicated with a dot over a variable and

the subscript r and n implies a dot product of the vector with the unit vector in the

radiation direction r and in the surface normal direction n, respectively. dS is an

elemental surface to be integrated over, and the subscript ret implies the evaluation

of the integrand at the retarded time, τ = t − r/c∞. For details regarding the

numerical implementation of the Ffowcs Williams-Hawkings method, the reader is

referred to Uzun’s thesis [25].

4.3 Far-Field Aeroacoustics

Due to the nature of the grid which is curvilinear, the control surface is shaped

as in Figure 4.1. We show results for both a closed and open control surface. A

closed control surface here is defined as one for which there is also a surface at the

end of the physical domain, i.e. x = 25ro, whereas for an open control surface there

is no surface there. Note that in both cases there is no surface at x = 0 as we are not

interested in the upstream propagation. Uzun, et al. [53] performed simulations with

three control surfaces which were located at 3.9ro, 5.9ro and 8.1ro above the jet at the

inflow plane. They found that the only difference in the noise spectra is the higher

resolved Strouhal number for the inner surfaces, because the grid is finer. In any

case, since we are simulating four different test cases, only one control surface will be

considered for our study. As mentioned in Section 2.5, a total of 16 processors were

used in parallel for the 3-D LES simulation. Due to the formulation of the compact

and implicit spatial filter which are non-local (see Section 2.3 and Reference [25] for

more details), the computational grid is initially partitioned into 16 non-overlapping

blocks (since 16 processors are used) along the z direction. Now, the 3-D LES code

was written such that aeroacoustic data is collected on the boundaries of each block.

44

Figure 4.2 show four blocks of the sixteen using our grid where the boundaries of each

non-overlaping block fall near to the control surfaces studied by Uzun et al. [53] at

the inflow, i.e 3.9ro, 5.9ro and 8.1ro in the x−z plane. From Figure 4.2 the beginning

of boundaries (1), (2), (3) and (4) are approximately located at 3.7ro, 5.5ro, 5.8ro

and 8.9ro on the z-axis, respectively. The boundaries (5), (6), (7) and (8) are located

at the negative z-axis but have the same distances from the origin as boundaries (1),

(2), (3) and (4). Boundaries (1), (4), (5) and (8) which belong to Block 1 and Block

4, respectively, fall outside the ranges studied by Uzun et al. [25] and thus were

not considered. The remaining boundaries, i.e. (2), (3), (6) and (7) are possible

candidates for a control surface. Keep in mind that the main goal here is to observe

trends in the sensitivity of far-field aeroacoustics due to inflow forcing changes, and

not to validate our results with available experimental or numerical values available

in the literature. Boundaries (3) and (6) were picked as the stationary control surface

to study the far-field aeroacoustics since these boundaries fall in between the ones

studied by Uzun et al. [53]. Boundaries (2) and (7) could have been chosen, though

we do not expect the aeroacoustic results to differ much between the two, since they

are located very close to each other.

As mentioned in Chapter 2, Tam & Dong’s [41,42] radiation boundary conditions

are applied at the inflow boundary. This boundary condition at the inflow is only

an approximation and therefore the solution near the boundary may not be very

accurate. Because of this, the selected control surface starts about one jet radii

downstream and is situated at approximately 5.5r0 above and below the jet at the

inflow boundary in the y and z directions and extends streamwise until the end of

the physical domain at which point the cross stream extent of the control surface

is approximately 21.1ro. Thus, the total streamwise length of the control surface is

24ro.

Flow field data is gathered on the control surface at every 5 time steps over

a period of 25,000 time steps. The total acoustic sampling period corresponds to

a time scale in which the ambient sound wave travels about 10 times the domain

45

length in the streamwise direction. Based on the grid resolution around the control

surface and assuming that 6 points per wavelength are needed to accurately resolve

an acoustic wave [25], the maximum resolved frequency corresponds to a Strouhal

number of St = 1.1. The Strouhal number here is defined as St = fDj/Uo where f

is the frequency, Dj and Uo are the jet nozzle diameter and the jet centerline velocity

on the inflow boundary, respectively. In addition, based on the data sampling rate,

there are about 14.5 temporal points per period in the highest resolved frequency.

The overall sound pressure level (OASPL) is computed along an arc of radius of

60ro from the jet nozzle. The angle θ is measured relative to the centerline jet axis

(see Figure 4.3). Using the Ffowcs Williams-Hawkings method, the acoustic pressure

signal is computed at 8 equally spaced azimuthal points on a full circle at a particular

θ location. There are total of 14 observer angles (values of θ) on the far-field arc.

Further details on the computation of OASPL and SPL can be found in Uzun, et

al. [14].

Figure 4.4 shows the OASPL for each test case (open control surface) computed

along the arc and compared to the SAE ARP 876C [54] database prediction for

an isothermal jet operating at similar conditions to ours. This database prediction

consists of actual engine jet noise measurements and can be used to predict overall

sound pressure levels within a few dB at different jet operating conditions. As

we see, when compared to the prediction database at similar conditions, our jet is

noisier (approximately 1-2 dB) for angles 45o to 90o, but then our OASPL drops for

angles lower than 45o. This is due to the fact that our control surface extends to a

streamwise distance of only 25ro which is a relatively short domain and also the fact

that we have an open control surface at the end. Freund et al. [55] reported that

the major noise contribution comes from a point on the surface that intersects a line

between the observation and the source point. Uzun, et al. [14] showed that with an

open control surface with a streamwise length long enough (approximately 60ro or

more) the noise shape for the lower angles is better than ours but still about 2-3 dB

too high.

46

Now if we compare the OASPL for all test cases, we see for angles 45o and above

that removing the first 6 and 8 modes of forcing, results in a noisier jet. If we

compare the baseline case to rf4 we do not see much of a difference at all. Bogey

and Bailly [23] reported a quieter jet when they removed the first 4 modes of forcing.

One reason why we are not seeing this marked behavior here is because we are using

a lower Reynolds number jet with a relatively thick shear layer compared to Bogey

and Bailly. Nevertheless, what is interesting here is the behavior of rf6 and rf8

whereby removing these number of modes results in a noisier jet. Uzun also [25]

reported the same behavior, i.e. a noisier jet, when he removed the first 6 modes of

the vortex ring forcing for a ReD = 400, 000, Mach 0.9 jet. According to Bogey and

Bailly’s findings we would assume that the jet would be quieter if more modes are

removed. But the opposite trend is found here. This might not be so surprising since

it was observed before (Section 3.3) that the peak intensities of the radial velocities

increase for rf6 and rf8 when compared to the Baseline case. Thus, removing the

first six and eight of modes of forcing contributes to the higher overall noise levels

that we are seeing for the upstream angles. From Section 3.3, the hAMP test case

showed the highest peak turbulence intensity for all fluctuating velocity components

compared to the mode cases. Hence, from Figure 4.4, we see that hAMP has the

highest overall sound pressure levels for almost all observation angles between 45o to

90o when compared to the reduced mode cases except at 60o where hAMP is very

close to rf8. Bogey and Bailly [23] also reported higher OASPL for their jet when

they reduced the vortex ring forcing amplitude by half. At this point, it should be

emphasized that the sound levels do change and depend on these two different inflow

parameters, i.e. the number of modes and forcing amplitude.

Noticing that we were not capturing the noise levels well in the downstream

region, we now use a control surface closed at the end. Figure 4.5 shows the OASPL

for the same test cases but this time with a closed control surface. Again we notice the

same behavior for all test cases when compared to the SAE ARP 876C prediction, i.e.

our jet is still noisier. However, the curve shape for the downstream region now looks

47

more like the database prediction (albeit more noisy). We can also make another

observation from Figure 4.5. The shape for the upstream angles does not show the

same exact behavior as in Figure 4.4. At the upstream angles we get noise levels

that become constant instead of continuing to decrease as the angle is increased. We

believe that this is due to a spurious line of dipoles created as the quadrupole sources

move through the surface at x = 25ro. The possible effect of dipoles is discussed in

more detail in Reference [53]. Furthermore, the effect of halving the forcing strength

is less felt this time for all observation angles as compared to an open control surface.

For angles greater than 40o, we see that the OASPL curve for the hAMP case nearly

collapses with that for rf8. However, for angles less than 40o, it is observed that the

overall sound pressure levels for hAMP are now less than the reduced mode cases

but slightly higher than the baseline case.

Finally, we compare acoustic pressure spectra results for each test case. Figures

4.6 and 4.7 show the acoustic pressure spectra at an observation angle of θ = 60o

at the far-field arc of 60ro for each test case for an open and closed control surface,

respectively. It should be noted here that since the computed spectra are noisy, the

spectra that are shown are polynomial fits to the actual computed spectra. At this

observation angle, we see the difference in the spectral behavior between the closed

and open control surface is minimal. Also, there are only small differences between

rf4 and the Baseline cases. One noticeable difference is that rf4 has higher low

frequencies and lower high frequencies than the baseline in for both the open and

closed control surfaces. Figure 4.8 show the differences in SPL for each test case with

respect to the Baseline case for an open control surface. Test cases rf4 and rf6

show crossovers from low frequency to the high frequency range with the baseline

case at St ≃ 0.46 and St ≃ 0.8, respectively for an open control surface (Figure

4.8). No crossovers are observed for rf8 and hAMP , where for both cases, the

sound pressure levels per Strouhal number are higher than Baseline throughout the

frequency spectrum for an open control surface. Again from Figure 4.8 we see that

the peak SPL comes from the hAMP case with a difference of about 1.9 dB/St

48

compared to the baseline case at St ≃ 0.15. Table 4.1 presents some the above

mentioned values for each test case against Baseline.

Figure 4.9 on the other hand, shows the sound pressure level difference for each

test case when compared to the Baseline case for a closed control surface. This time

the peak difference SPL is lower in the case of hAMP when compared to Figure 4.8,

i.e. open control surface. Also note that there are two crossover points for rf4.

One at the very low frequency range and another at approximately St ≃ 0.65.

Likewise, Table 4.2 presents peak SPL difference values, peak difference frequency

and crossover frequency from Figure 4.9. In general, Figures 4.8 and 4.9 show that by

removing modes or reducing the forcing amplitude, the low frequencies are enhanced

and the high frequencies are reduced. Furthermore, Figures 4.8 and 4.9 and Tables

4.1 and 4.2 clearly point to the sensitivity of the jet noise levels to the inlet forcing

conditions.

It also interesting to look at downstream and upstream pressure spectra. Figures

4.10 and 4.11 show the sound pressure level in the far-field at θ = 25o for both an

open and closed control surface, respectively. The pressure levels here are compared

to the Baseline test case. At θ = 25o, which is located in the highly turbulent

region (large scales), we would expect the low-frequencies to dominate. However,

from Figure 4.10, we cannot draw any conclusions since, we are not capturing the

noise levels correctly due to the control surface absent at that angle (θ = 25o).

Figures 4.12 and 4.13 on the other hand, show the pressure spectra in the far-field

at θ = 90o for both an open and closed control surface, respectively. In this region,

we expect the shear noise (high frequencies) to dominate. From Figure 4.12, we see

that the high frequency region follows the trend of the peak turbulence intensities

shown in Figures 3.8 and 3.9 in Chapter 3. Here, we see that the SPL/dB for the

rf4 test case decreases when compared to the Baseline test case but increases again

for the rf6, rf8 and hAMP test case. However, if we observe Figure 4.13, i.e. for

a closed control surface, we again find that we cannot draw any conclusions since

49

the SPL/dB behaviors in Figure 4.13 may be “contaminated” by the effects of the

spurious line of dipoles.

50

Table 4.1 Peak SPL difference, frequency location and crossover fre-quency location for each test case with respect to Baseline for anopen control surface at R = 60ro, θ = 60o in the far-field.

Test Case SPLpeak Stpeak,SPL Stcrossover

Baseline - - -

rf4 0.9 0.05 0.46

rf6 0.5 0.22 0.80

rf8 1.5 0.50 -

hAMP 1.9 0.16 -

Table 4.2 Peak SPL difference, frequency location and crossover fre-quency location for each test case with respect to Baseline for anclosed control surface at R = 60ro, θ = 60o in the far-field.

Test Case SPLpeak Stpeak,SPL Stcrossover

Baseline - - -

rf4 0.5 0.32 0.15, 0.65

rf6 0.9 0.26 0.74

rf8 1.5 0.52 -

hAMP 1.6 0.15 -

51

Figure 4.1. The open FW-H stationary control surface around the turbulent jet.

x/ro

z/r o

0 10 20 30-15

-10

-5

0

5

10

15

Chosen Bondaries forControl Surfaces

Block 4

Block 1

Block 2

Block 3(8)

(7)

(6)

(5)

(4)(3)

(2)

(1)

Figure 4.2. Schematic showing four of the sixteen partitioned blocksof the computational domain. Boundaries are numbered in ().

52

x/ro

y/r o

0 5 10 15 20 25-5

0

5

10

15

R

θ

Figure 4.3. Schematic showing the center of the arc and how theangle θ is measured from the jet axis.

θ (deg)

OA

SP

L(d

B)

20 30 40 50 60 70 80 90100

102

104

106

108

110

112

114

116

118

Baseline Open CSrf4 Open CSrf6 Open CSrf8 Open CShAMP Open CSSAE ARP 876C

Figure 4.4. Overall sound pressure levels at R = 60ro from the nozzleexit for all cases with the open control surface.

53

θ (deg)

OA

SP

L(d

B)

20 30 40 50 60 70 80 90104

106

108

110

112

114

116

118

120

122

124Baseline Closed CSrf4 Closed CSrf6 Closed CSrf8 Closed CShAMP Closed CSSAE ARP 876

Figure 4.5. Overall sound pressure levels at R = 60ro from the nozzleexit for all cases with the closed control surface.

St

SP

L(d

B/S

t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

Baseline Open CSrf4 Open CSrf6 Open CSrf8 Open CShAMP Open CS

Figure 4.6. Acoustic pressure spectra at R = 60ro, θ = 60o in thefar-field for each test case with an open control surface.

54

St

SP

L(d

B/S

t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1104

105

106

107

108

109

110

111

112

113

114

115

116

Baseline Closed CSrf4 Closed CSrf6 Closed CSrf8 Closed CShAMP Closed CS

Figure 4.7. Acoustic pressure spectra at R = 60ro, θ = 60o in thefar-field for each test case with a closed control surface.

St

SP

L(d

B/S

t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-1

-0.5

0

0.5

1

1.5

2

2.5

Baseline to Baseline Open CSrf4 to Baseline Open CSrf6 to Baseline Open CSrf8 to Baseline Open CShAMP to Baseline Open CS

Figure 4.8. Acoustic pressure spectra at R = 60ro, θ = 60o in thefar-field for each test compared against the Baseline case for an opencontrol surface.

55

St

SP

L(d

B/S

t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-1

-0.5

0

0.5

1

1.5

2

2.5

Baseline to Baseline Closed CSrf4 to Baseline Closed CSrf6 to Baseline Closed CSrf8 to Baseline Closed CShAMP to Baseline Closed CS

Figure 4.9. Acoustic pressure spectra at R = 60ro, θ = 60o in thefar-field for each test compared against the Baseline case for a closedcontrol surface.

St

SP

L(d

B/S

t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-2

-1.5

-1

-0.5

0

0.5

1

1.5


Figure 4.10. Acoustic pressure spectra at R = 60ro, θ = 25o in thefar-field for each test compared against the Baseline case for a opencontrol surface.

56

St

SP

L(d

B/S

t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3



St

SP

L(d

B/S

t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3


Figure 4.12. Acoustic pressure spectra at R = 60ro, θ = 90o in thefar-field for each test compared against the Baseline case for a opencontrol surface.

57

St

SP

L(d

B/S

t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-1.5

-1

-0.5

0

0.5

1

1.5

2



58

5. CONCLUSIONS AND RECOMMENDATIONS FOR

FUTURE WORK

5.1 Conclusions

Using a 3-D Large Eddy Simulation (LES) methodology developed by Uzun [25],

we have looked at the effects of the vortex ring inflow forcing on both turbulent flow

development and, most importantly, the sensitivity of the far-field noise of a Mach

0.9, Reynolds number ReD = 100, 000, turbulent jet. The LES code utilizes high-

order accurate compact finite differencing as well as implicit spatial filtering schemes

together with the state-of-the-art Tam & Dong’s boundary conditions on the LES

domain. The implicit spatial filter was also used as an implicit subgrid-scale (SGS)

model to damp the turbulent energy. For time advancement, the explicit 4th-order,

4-stage Runge-Kutta method was used. Finally, for analyzing far-field aeroacoustics,

the Ffowcs Williams - Hawkings surface integral acoustic method was utilized.

In terms of turbulent flow development, we see that by consistently removing

low order azimuthal modes or halving the forcing amplitude in the inflow forcing,

the Reynolds stresses generally increase and the length of the potential core also

increases. Hence, the development of the jet approaches that of an initially turbulent

jet as we remove more modes. This behavior compares well with the findings of Bogey

and Bailly [23] and with experimental results. Due to the increase in core length, we

also observe that the axial, radial and azimuthal peak turbulence intensities along the

shear layer and jet axis shift downstream. However, not only do the radial intensities

shift downstream but they also increase for test cases rf6, rf8 and hAMP , where

the hAMP test case (hAMP is the test case where the forcing strength is halved)

registered the highest peak turbulence intensity.

59

As for far-field aeroacoustics, we see that due to the increase in peak radial and

azimuthal turbulence intensities, the OASPL increases by about 1-2 dB between the

Baseline and hAMP cases. Due to the impact of the above findings, we see that flow

development and overall noise levels are sensitive to the chosen number of azimuthal

forcing modes and forcing strength. Furthermore, when the first 4 modes of forcing

were removed, the overall sound pressure levels were found to not differ much with

the Baseline case. This trend was not observed by Bogey and Bailly when in fact

they reported a significant reduction the OASPL for their jet when they removed the

first four modes of forcing. This maybe due to the fact that they were running their

jet at a higher Reynolds number and using a thinner shear layer compared to ours.

In addition, when Uzun [25] removed the first 6 modes of the vortex ring forcing, he

reported higher overall noise levels but with a ReD = 400, 000 jet when compared

to no forcing modes removed at all. Hence, we see the same behavior also but for a

ReD = 100, 000 jet. Also, when we compare the noise levels to the SAE ARP 876C

prediction from actual jet engine noise data, we see that we over predict the overall

noise levels. We believe this is due to the vortex ring forcing generating excessive

energy in the large scale structures which then contribute to the noise levels being

high compared to experimental results.

5.2 Recommendations for Future Work

As observed in the previous section, changing certain inflow forcing parameters

results in changes in turbulent flow development and far-field predicted noise. Also,

the noise levels predicted are greater than those measured in experiments. Hence,

improved methods of modeling the the inflow conditions are needed. Section 1.1

discussed a method developed by Glaze and Frankel [21] that could be implemented

in the current 3-D LES code. This method involves a version of the weighted am-

plitude wave superposition spectral representation, where the fluctuations created

60

reproduce the jet near field much more accurately and allow the flow to transition

rapidly to self-sustaining turbulence.

One way of allieviating the uncertainty of any artificial inlet forcing is to include

part of the nozzle geometry altogether. However, the cost of implementing such a

feat is certainly challenging but not far out of reach for today’s supercomputers. The

reason why it is computationally expensive is due to the required grid resolution to

accurately resolve the wall shear layer using LES. Freund and Lele [56] estimate that

approximately 150-200 million grid points are needed in a high Reynolds number

(ReD = 1, 000, 000) jet flow LES to accurately resolve the acoustically significant

region near the nozzle. To perform such a simulation, our single-block LES/DNS

code would need to be modified to a multi-block version. The idea behind the

multi-block implementation is that complex domains can be broken up into smaller

more manageable domains and the already existing high-order accurate single-block

code applied in each of the simpler domains. Grid point overlaps will be required

to exchange data between each block during the computations. In fact, Xiangang

et al. [57] are currently developing such a methodology using the single-block code

developed by Uzun [25]. Yao et al. [15] performed a similar multi-block approach

when they studied turbulent flow over a rectangular trailing edge using DNS. One of

the reasons why it is also important to include the jet nozzle in the computations is

to accurately resolve the acoustically significant region near the nozzle. It is believed

that the vortex-solid body interaction process is one of the primary sources of near-

nozzle high-frequency noise generation [25]. Uzun [25] suggests that as intermediate

step simulations could be performed that include the jet nozzle lips using a multi-

block approach with a total of 50 - 80 million grid points.

Performing an LES/DNS simulation that includes part of the jet nozzle as briefly

outlined above is certainly costly. An alternative that would reduce the compu-

tational cost and yet yield accurate results, is to implement a coupled RANS-LES

approach. Schluter et al. [58–62], for example, have successfully implemented a solver

that integrates RANS and LES procedures towards a very complex problem which

61

simulates an aero-thermal flow in a entire aircraft gas turbine engine. Due to the

high flow complexity inside the engine, it was suggested that several different flow

solvers be used such as using a RANS solver in the compressor, LES in the combustor

and RANS again in the turbine. The big challenge was implementing an interface

between the two solvers as the simulation was time advanced.

Another form of coupling two types of solvers for an engineering like flow is

Detached Eddy Simulation (DES) proposed by Spalart et al. [63]. Specifically aimed

at tackling aerospace problems, Spalart et al. proposed to solve the boundary layer on

a wing using RANS and use LES in the separated flow region. Squires et al. [64] used

DES to study massively separated flows over several bluff and streamlined bodies.

A specific flow simulation that Squires et al. [64] performed is the computation of

separated flow for an entire aircraft at a high angle-of-attack. However, a single

flow solver is used, and the Spalart-Allmaras [65] model is formulated to smoothly

transition from a RANS mode to an LES approach. Hence, the above methods could

be applied to study jet noise, whereby RANS is used to solve for the region near the

wall of the nozzle lip and the flow solver transitions to LES where the flow exhibits

high flow complexity.

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62

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APPENDIX

68

A. AN ATTEMPT AT COUPLING RANS AND LES FOR

JET AEROACOUSTICS

A.1 Introduction

It has been acknowledged that the shape of lobe mixers (see Figure A.1 and

Figure A.2) has a strong effect on far-field noise, but this phenomenon is still not

well understood. Our goal is to advance the science of jet noise prediction for tur-

bofan aircraft engines. The proposed method here was to investigate the feasibility

of coupling two turbulence methodologies such as Reynolds Average Navier-Stokes

(RANS) and Large Eddy Simulation (LES) for the eventual prediction of jet aeroa-

coustics. Here, a converged RANS solution of the flow-field from the nozzle exit and

beyond will be used as an initial condition for the LES solver. Hence, RANS will

be used to solve the flow through the lobe mixer, i.e. the internal flow and LES

will be used to solve the external flow-field. However, due to the inherent nature

of the non-radiative boundary conditions of the LES code, this methodology proved

to be infeasible. The next several sections give the details as to how the coupling

methodology was implemented and how the aforementioned conclusion was reached.

A.2 Motivation

In most turbulent jet simulations the mean velocity profile specified has a smooth

idealized monotonic behavior. However, today’s high bypass engines on regional jets

are fitted with lobe mixers that enhance the mixing process and thus increases the

spread rate of the jet. This in turn reduces the far-field noise generated (albeit it

is not exactly known how). With that in mind the velocity profile is not smoothly

monotonic, i.e. ‘peaks’ and ‘valleys’ appear in the velocity profile at the exit nozzle

69

plane due to embedded streamwise vortices. Simulating the entire problem with LES

from the internal nozzle where the mixer is located to the exit plane and beyond is

not feasible. This is due to the fine near-wall resolution required to accurately

compute the boundary layers. An appropriate method for solving problems where

walls are present is RANS, where it captures the behavior of boundary layers quite

accurately. Then LES is applied where the flow exhibits high flow complexity such as

the turbulence at the jet exit and beyond. Furthermore, since LES is more accurate

than RANS, for our case it will resolve the noise generating part of the flow resulting

in a more accurate noise prediction. Hence, from this perspective it is desirable to

couple the two methodologies.

Several studies have been implemented in coupling RANS and LES to study a

variety of problems. Schluter et al. [58–62] for example, have successfully imple-

mented a solver that integrates RANS and LES procedures towards a very complex

problem which simulates an aero-thermal flow in a entire aircraft gas turbine engine.

Due to the high flow complexity inside the engine, it was suggested that several

different flow solvers be used such as using a RANS solver in the compressor, LES

in the combustor and RANS again in the turbine. The big challenge was imple-

menting an interface between the two solvers as the simulation was time advanced.

Another form of coupling two types of solvers for an engineering like flow is De-

tached Eddy Simulation (DES) proposed by Spalart et al. [63]. Specifically aimed

at tackling aerospace problems, Spalart et al. proposed to solve the boundary layer

on a wing using RANS and use LES in the separated flow region. Squires et al. [64]

used DES to the fullest to study massively separated flows over several bluff and

streamlined bodies. A specific flow simulation that Squires et al. [64] performed is

the computation of separated flow for an entire aircraft at a high angle-of-attack.

However, a single flow solver is used, and the Spalart-Allmaras [65] model is formu-

lated to smoothly transition from a RANS mode to an LES approach. Several other

RANS/LES methods have been performed to date. Recently, Chang and Park [66]

used a hybrid RANS/LES approach to study 3-D deep cavity flows. Subbareddy

70

and Candler [67] used DES to accurately compute base drag for supersonic base flow

with bleed and Xing et al. [68] used DES to study unsteady free-surface wave induced

separation. The above mentioned RANS/LES studies offer attractive methodologies

to study jet noise for lobed mixer jets. However, the difficulty here is that we do

not have an “in-house” combined RANS-LES real-time flow solver to perform such a

complex simulation. Hence, we want to attempt a different type of approach outlined

below:

1. Select a fully converged RANS solution for a high bypass jet engine which has a

lobed mixer geometry. Interpolate the external flow-field RANS solution onto an

LES grid and set the RANS mean flow result as the initial condition for the time

dependent LES solver. The 3-D LES jet solver mentioned here was developed

by Uzun et al. [69] as an initial platform to study jet aeroacoustics with velocity

profiles that are idealized. However, the 3-D LES code is only setup to solve the

external flow-field of a round jet. Hence, the 3-D LES methodology here does

not incorporate solid boundaries and thus it is necessary to couple with the

internal RANS flow-field solution. As mentioned, the 3-D LES methodology is

initialized with an idealized mean flow solution. Now, instead of using the initial

meanflow of the LES, we could use the converged mean flow-field of the RANS

solution which contain embedded vortices from the lobed mixer as an initial

solution. Hence, the code here has to be modified to reflect as best as possible

the initial conditions of the industrial jet and also to read in the interpolated

initial conditions from a set database. The same disturbance source, i.e. the

vortex ring forcing (see Chapter 2) used by Uzun et al. [25] will be implemented

to excite the jet.

2. Once the first hurdle is successful, we will examine the effect of the lobe mixer

on several turbulent flow properties such as jet growth rates, Reynolds stresses,

turbulence intensities, turbulent energy spectra and most importantly far-field

noise. The prediction methodology of the far-field noise is described in Chapter

4.

71

A.3 Brief Description of RANS and LES Methodology

The converged RANS solution was obtained using a commercial code called

WIND [70] which was developed by the NPARC Alliance, a partnership between

the NASA Glenn Research Center (GRC) and the USAF Arnold Engineering Devel-

opment Center (AEDC) in close partnership with the Boeing Company. It is based

on a second order upwind scheme. The choice of the turbulence model used to solve

the mean flow field is the one-equation Spalart-Allmaras [65] model.

The 3-D LES methodology developed by Uzun [69] et. al. is used here. Please,

refer to Chapter 2 for a more extensive description of the LES code. The original

setup of the 3-D LES code that will be used here will have the classical Smagorinsky

[27] subgrid-scale model turned-on. The subgrid-scale stress tensor is modeled as

follows:

Tij = −2Csgsρ∆2SM

(Sij −

1

3Skkδij

)+

2

3CIρ∆2S2

Mδij, (A.1)

where SM =(2SijSij

)1/2. The terms Csgs and CI are the model coefficients, and ∆

is the filter width or the eddy-viscosity length scale (Chapter 2 has the definitions of

the remaining terms). On the right hand side of Equation A.1, the first term is the

original incompressible term in Smagorinsky’s subgrid-scale model [27], whereas the

second term is the compressibility correction expression proposed by Yoshizawa [30].

For all runs, the model coefficients are set to Csgs = 0.018 and CI = 0.0066.

A.4 Setup and Coupling Methodology

As mentioned, a converged RANS solution for a jet with the lobe mixer geometry

was obtained using WIND. Only the portion containing the flow-field beyond the jet

exit was used as input to the 3-D LES methodology.

Figure A.1 shows an axial cross section near the exit of an engine that has a lobed

mixer. If we look closely we notice the lobed-mixer geometry strategically located

where it mixes the hot core flow from the turbine and the cooler fan flow. Figure A.2

gives a three dimensional representation of the mixer with the nozzle section. Figure

72

A.3 shows the density contours from a fully converged RANS solution obtained from

WIND. Only a section of the flow is considered because of the symmetry of lobe

mixer. Notice the strong vortex structure in the flow due to the lobe mixer located

about three-quarters above the centerline. Please, note that Figure A.3 is the result

obtained from a quarter scale version of the full model fitted on regional jets in order

to match experiments.

Several inflow and outflow parameters must be specified to initialize a run for

the WIND code. The most important ones are the pressure and temperature con-

ditions inside and outside the lobed mixer jet. The ambient pressure is specified at

P = 14.48 psi and the ambient temperature at T = 498.9 ◦R. For the core region

of the jet the total pressure and temperature are Pcore = 20.14 psi and Tcore = 1177

◦R, respectively. For the bypass region however, the total pressure and temperature

are Pbypass = 20.84 psi and Tbypass = 504 ◦R. These values are held steady during

the WIND computations. Hence, the Reynolds number based on jet centerline ve-

locity, density, and molecular viscosity after the RANS solution has converged is

approximately Rej,RANS = 4.2 million, whereas the computed Mach number is ap-

proximately Mj,RANS = 0.68. Now, given the current computational resources, we

will still be unable to resolve all the relevant length scales at such a high Reynolds

number using LES. We tried several matching ideas such as matching the jet thrust

and Mach number hoping that this will reduce the required Reynolds number set in

the LES code. However, this method yielded a Reynolds number that was in the or-

der of a million and thus for LES was still too high. After several attempts in trying

to match the Reynolds number we decided that it was best to just downscale the

RANS computational model based on jet diameter to a more feasible LES Reynolds

number of 100,000. This unfortunately, had an adverse impact on the vortex struc-

ture, i.e. the once strong outgoing vortex seemed to be ‘smeared’ tremendously and

not as distinct as in the previous RANS solution due to a thick boundary layer in

the small nozzle. Thus, it was decided at the moment not to match the solution

exactly and instead use the Reynolds number 4 million RANS solution as it is and

73

run the LES solver at a lower Reynolds number, i.e. there will be a Reynolds number

mismatch between the RANS solution and LES solver but the Mach numbers are

equal. It is important to note that until this point, no runs have been made with

the LES solver. The next section briefly describes the interpolation strategy used to

couple a selected RANS solution as an initial condition for the LES solver.

A.5 Interpolation Methodology and Setup

For the RANS computations, only a 15◦ pie-shaped section was considered since

it takes advantage of the symmetry present due to the periodic geometry. Figure

A.4 shows the 2-D representation of the RANS grid superimposed upon the LES

grid for interpolation at the jet exit. Firstly, notice the amount of grid points packed

near the edge of nozzle lip region. This is done to resolve the thin shear layer at the

wall of the nozzle. Furthermore, notice the amount of resolution on the RANS grid

compared to the LES grid. The RANS case in Figure A.4 has 41 × 271 points in

the azimuthal and radial directions, respectively. The original setup of the RANS

grid follows a cylindrical pattern, whereas for the LES case, a Cartesian setup is

used. Also keep in mind that Figure A.4 only shows a 2-D representation of the

solution and the RANS solution will have to be mirrored circumferentially to obtain

the full circular jet setup which will be used for the 3-D LES solver. It is also worth

mentioning that the RANS solution extends past the edge of the shear layer but this

is not shown in Figure A.4. Once the RANS grid is mirrored, a single RANS grid

plane has a resolution of 329 × 961 grid points.

If the LES method were to match the resolution used for the RANS case, a pro-

hibitive computational cost will arise. Hence, some form of interpolation is needed

to interpolate the RANS solution onto the LES grid. For this purpose, a commercial

visualization tool called Tecplot 9.2 was used to perform a full 3-D linear interpola-

tion from the RANS solution, on to the LES grid. Figure A.5 shows the streamwise

velocity contours for the converged RANS solution, whereas Figure A.6 shows the

74

interpolated streamwise velocity contours on the LES grid. Notice the once strong

outgoing vorticity being slightly smeared and also the originally thin shear layer for

the RANS now thicker due to the relative coarseness in grid resolution for the LES

case.

In terms of domain setup, the physical portion of the LES domain stretches to

approximately 25ro in the streamwise x direction and ±15ro in both the y and z

directions with a resolution of 375 × 128 × 128 which is approximately 6.1 million

grid points. The converged RANS solution is used for the initialization of the LES

solver. However, the streamwise distance of the RANS domain only extends to about

17ro. Hence, the remaining portion of the LES initial mean solution was copied from

17ro until 25ro.

A.6 Results and Discussion

The initial Reynolds number used for the LES code was 100,000 based on jet

centerline properties using the classical Smagorinsky subgrid-scale model. Based on

the Mach number and minimum grid resolution, the maximum non-dimensional time

step was found to be ∆t = 0.015. The method in determining the maximum ∆t is

shown below

∆t =min(∆x, ∆y, ∆z)

c∞ + Uj

(A.2)

where min(∆x, ∆y, ∆z) is the minimum LES grid spacing which for this case is

∆zmin = 0.045ro, c∞ is the ambient speed of sound based on the centerline Mach

number, i.e. Mj,LES = Uj/c∞, and Uj is the centerline velocity which is unity. Using

the above equation yields ∆t = 0.018. However, that is the maximum allowable

time step in order for the 3-D LES code not to blow-up. Hence, the time step was

lowered to a conservative value of ∆t = 0.015. Also, keep in mind that the initial

mean LES solution here is still based on a Rej,RANS = 4.3 million and therefore

we have the mismatch for the Reynolds number but not for the Mach number, i.e.

Mj,LES = Mj,RANS = 0.68. Using this setup, the LES code blew-up after only several

75

hundred time steps. After some investigation, we found out that the cause of the

code blowing-up was at the region at the jet exit. The instantaneous streamwise

velocity right before when the code blew-up was reaching values close to Uj = 2.

Recalculating the time step using this centerline value, we have ∆t = 0.013 as the

maximum allowable time step and this will clearly render the 3-D solution to blow-

up. Suspecting that the size of the time step might be the problem, we reduced the

time step to ∆t = 0.010. However, the code blew-up again after several hundred time

steps although the simulation progressed a little longer. As another test, the classical

Smagorinsky subgrid-scale model was turned-off and only filtering was used. This

technique also yielded the same conclusion. Thus, the Smagorinsky model in essence

had no effect. Another technique was to turn-off the vortex ring forcing. This route

also rendered the LES solver blowing-up albeit after several more hundred time steps

compared to the previous trials.

Since the Reynolds number here was based on jet centerline properties, another

route was to run the LES solver based on averaged RANS quantities, i.e. average

streamwise velocity, density, molecular viscosity at the jet exit with the diameter

unchanged. Taking average quantities made more sense as the exit profile for the

primitive variables were not smooth and monotonic. With these modifications, the

resulting average Mach number is now M j,LES = M j,RANS = 0.63 and the average

Reynolds number is approximately Rej,RANS = 4 million. Using the same method

of determining the minimum time stepping, we calculated ∆t = 0.015. Likewise,

there is still the Reynolds number mismatch here. Unfortunately, this method also

made the solver blow-up. However, an encouraging result was that this time the

code progressed slightly above a thousand time steps. Noticing this, we reduced the

time step to ∆t = 0.010 but the solver blew-up this time around two thousand time

steps. Again, we went through the remaining procedures as the paragraph before

but to no avail.

Since the above trials were unsuccessful, it was decided that the LES Reynolds

number be increased to a higher or intermediate value of Rej,LES = 400, 000 based

76

on centerline jet exit properties. A new WIND calculation was done but this time

to adhere to the changes based on the new Reyonlds number, i.e. by resizing the

geometry of the RANS grid. Performing this approximately yielded the same the

Mach number of Mj,RANS = 0.68. With this new RANS initial condition, the LES

solver this time progressed to about 4000 time steps with ∆t = 0.015. Lowering the

time stepping to ∆t = 0.010 yielded the same conclusion as before but this time

the simulation progressed to approximately 6200 time steps before solution blew-

up. As before, the region where this was happening was at the jet exit. Recalling

that we had a marked improvement when we took average values as opposed to

jet centerline values, we again used RANS to compute a solution for Rej,RANS =

Rej,LES = 400, 000. Likewise, the computed Mach number based on average values

M j,RANS = M j,LES = 0.62. Using this setup, the LES solver progressed even further

but unfortunately blew-up a little after 10,000 time steps with ∆t = 0.015.

Figures A.6 through A.11 show the behavior of the streamwise velocity contour

from the initial condition until 10,000 time steps. From these figures it is clear that

the initial outgoing vortex structure that needs to be maintained and captured is

slowly dissipating as the LES solver is advanced in time. Also notice the peak level

in the grayscale contours as the solution is progressed in time. Since we are using

average jet properties, the maximum non-dimensional initial streamwise velocity

is approximately Uj = 1.2. As the solution progresses and reaches close to the

code blowing up, i.e. Figure A.11, the centerline peak streamwise velocity level has

increased to about Uj = 1.5. As the solver progresses a little beyond 10,000 time

steps the instantaneous centerline streamwise velocity rapidly reaches values close to

Uj = 2 before the solver blows-up. In addition, notice that in Figure A.10 we have

a dipole like structure emanating from the jet exit and disappears in Figure A.11.

One possible explanation as to why we are seeing the behaviors that appear

in Figures A.7 through A.11 is the inherent nature of the Tam and Dong’s non-

reflecting boundary conditions. The nature of this type of boundary condition does

not impose or hold the solution profile that we want to capture. Mainly because the

77

profile used is strongly inhomogeneous due to the counter rotating vortices exiting

the jet exit. The boundary condition here works well for Uzun [25] et al. and

Bogey and Bailly’s [22] work due to the fact that the inlet velocity profile specified

is only slightly inhomogeneous, i.e. it has a smooth hyperbolic tangent profile which

is monotonic. Another source as to why this methodology did not work could be

due to the coarseness of the LES grid used. However, the computational cost of

increasing the number of grid points for such a high level simulation like LES is just

not justified at this time.

A.7 Summary

We have seen that the coupling of RANS and LES methodology that has been

proposed here was infeasible. This is mainly due to the inherent nature the non-

reflecting boundary condition of Tam and Dong where it does not conserve the

velocity profile structure at the inlet plane. Another reason could be due to the

coarseness of the current LES grid. Hence, it would be desirable if future work could

look into ways of devising a non-reflective boundary condition which in principle

allows outgoing acoustic waves to leave the domain but at the same time also holds

any type of solution profile specified.

78

Figure A.1. The cross centerline section of the turbofan considered.

Figure A.2. Lobed mixer geometry.

79

Region of high vorticity

Figure A.3. Converged density contours (sectional) from WIND atthe exit nozzle plane.

z

y

0 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Section of LES gridat jet exit

RANS grid at jet exit

Figure A.4. Sectional LES grid on RANS grid for interpolation

80

Figure A.5. Converged streamwise velocity contours from WIND atthe exit nozzle plane.

z

y

-1 0 1

-1

-0.5

0

0.5

1

u1.21.080.960.840.720.60.480.360.240.120

Figure A.6. Interpolated RANS solution for the streamwise velocityon LES grid and Streamwise velocity contours at t = 0.

81

z

y

-1 0 1

-1

-0.5

0

0.5

1

u1.41.261.120.980.840.70.560.420.280.140

Figure A.7. Instantaneous streamwise velocity contours after 1,000time steps with ∆t = 0.015.

z

y

-1 0 1

-1

-0.5

0

0.5

1

u1.41.261.120.980.840.70.560.420.280.140


82

z

y

-1 0 1-1.5

-1

-0.5

0

0.5

1

u1.41.261.120.980.840.70.560.420.280.140


z

y

-2 -1 0 1

-1

-0.5

0

0.5

1

1.5

2

u1.41.261.120.980.840.70.560.420.280.140


83

z

y

-1 0 1

-1

-0.5

0

0.5

1

u1.51.351.21.050.90.750.60.450.30.150


84

B. EFFECT OF TRI-DIAGONAL FILTERS FOR A

PLANE MIXING LAYER USING 2-D LARGE EDDY

SIMULATION

B.1 Motivation

The main aim of this section was to investigate the effects of a particular type of

low-pass filter on a spatially developing mixing layer using 2-D Large Eddy Simula-

tion (LES). Flow properties that we want to investigate are the mixing layer growth

rates and peak Reynolds stresses. The compact spatial filter in question was pro-

posed by Visbal and Gaitonde [38] as a tool for studying aeroacoustic phenomena on

curvilinear grids. Furthermore, this type of spatial filter can also be used for single

or multi-block applications. As an initial example, Chapter 2 in the main part of

this thesis gives a 6th-order version of this spatial filter (Equation 2.34). As can be

seen from Chapter 2, the discrete form of the this spatial filter has an implicit tri-

diagonal form that can be solved using a linear algebra package. To study the effects

of this class of filters we use a 2-D LES code developed by Uzun et al. [69]. The

2-D LES methodology was developed to test their LES simulation techniques before

extending to a full 3-D turbulent jet simulation. Mixing layers are a class of free-

shear flows that are commonly studied both experimentally and numerically. Several

experimental examples are by Wygnanski and Fiedler [71], Spencer and Jones [72]

and Bell and Mehta [73]. Numerical studies have been carried out by Rogers and

Moser [74], Stanley and Sarkar [75], and Bogey [76]. The results gathered here are

compared to the above references.

85

B.2 Numerical Methods and Setup

The numerical methods implemented in the 2-D LES code are similar to the

ones discussed in Chapter 2 with the exception of the spatial filter and boundary

conditions. Thompson’s non-reflecting boundary conditions [77,78] are applied on all

boundaries except at the inflow boundary. At the inflow boundary, a procedure based

on method of characteristics [11] is used. The spatial filter proposed by Lele [39] is

investigated. In principal, this filter when operated on the governing equations has

a discrete form which is penta-diagonal. Hence, in order to solve a penta-diagonal

system of equations, one has to perform an LU decomposition twice, i.e. one for each

direction. This procedure is computationally expensive but still manageable in a 2-

D simulation. However, the computational cost would be significant when applied

to a 3-D simulation like a turbulent jet. Hence, a tri-diagonal filter like the one

used here is computationally cheap when compared its penta-diagonal counterpart.

Nevertheless, in this study, we will make comparisons between the two by comparing

the growth rates and peak stresses. The 4th-order penta-diagonal compact filter

proposed by Lele [39] is given by

α2f i−2 + α1f i−1 + f i + α1f i+1 + α2f i+2 = a1fi + a2(fi+1 + fi−1)

+a3(fi+2 + fi−2) + a4(fi+3 + fi−3), (B.1)

where fi and f i represent the solution variable and the spatially filtered solution

variable at point i, respectively and the coefficients are given by

α1 = 0.652247 α2 = 0.170293,

a1 =2 + 3α1

4a2 =

9 + 16α1 + 10α2

32, (B.2)

a3 =α1 + 4α2

8a4 =

6α2 − 1

32.

Zhao [11] determined that filtering at and near the boundaries is not necessary and

therefore, this filter is used on grid points i = 5 through i = N − 4 where N is

the total number of grid points along the grid line. Figure B.1 shows the transfer

86

function for this type of filter. This study will also include the 6th-order version of

this filter. The 4th-order penta-diagonal filter transfer function is constrained to go

through wavenumber k/π = 0.7958 where the value of the filter transfer function

is 0.5. In order to make a justifiable comparison between the two, the 6th-order

compact penta-diagonal filter will also be constrained to pass through the above

mentioned points. Thus, the coefficients of the 6th-order compact penta-diagonal

filter are given by

α1 = 0.619399 α2 = 0.169612,

a1 =2 + 3α1

4a2 =

6 + 7α1

8, (B.3)

a3 =6 + α1

20a4 =

2 − 3α1

40.

Figure B.1 also plots the 6th-order version of this filter and when compared its

6th-order counterpart, the plots nearly collapse on one another. Hence, we should

expect the results, i.e. growth rates and peak stresses to be almost similar. Section

2.3 (Equation 2.34) in this thesis gives the 6th-order formulation of the compact

tri-diagonal filter used for the 3-D LES turbulent jet computations. This study will

consider 4 different orders in terms of accuracy of the tri-diagonal filter, i.e. 4th, 6th,

8th and 10th-Order. The parameter that governs the shape of this low pass filter is αf

and satisfies the inequality −0.5 < αf < 0.5. Applying the same constraints used for

the penta-diagonal filter, Figure B.1 shows the tri-diagonal filter transfer function

along with the corresponding values of αf for each filter of different order accuracy.

In contrast to the penta-diagonal filter, the tri-diagonal filter filters the points next

to the boundaries. Section 2.3 gives the filtering formulation for the points next to

the boundaries for the 6th-order tri-diagonal filter (Equations 2.36 through 2.40).

The 4th, 8th and 10th-order filtering expressions for points in the interior and points

next to the boundaries, refer to Gaitonde and Visbal [79]. Table B.1 summarizes the

test cases considered for this study as well as its corresponding coefficients.

87

We study a 2-D spatially developing mixing layer with the following hyperbolic

tangent inflow profile for the mean streamwise velocity

u(y) =U1 + U2

2+

U2 − U1

2tanh

(2y

δω(0)

), (B.4)

and the mean transverse velocity is given by

v(y) = 0. (B.5)

Here, U1 and U2 are the velocities of the low-speed and high-speed streams, respec-

tively, and δω(0) is the initial vorticity thickness, which is defined as

δω(0) =U2 − U1

|∂U∂y|max

. (B.6)

The Reynolds number based on the initial vorticity thickness and the velocity dif-

ference across the layer is

Re =(U2 − U1)δω(0)

ν= 5, 333. (B.7)

The relative convective Mach number is

Mc =U2 − U1

2c∞= 0.074, (B.8)

and the convection velocity is

Uc =U1 + U2

2= 0.222c∞, (B.9)

with

η =U2 − U1

U2 + U1

=1

3. (B.10)

Figure B.2 shows the computational grid used for this simulation. The physical

portion of the grid extends to about 200δω(0) in the streamwise x-direction and from

−100δω(0) to 100δω(0) in the transverse y-direction. From 200δω(0) to 400δω(0) grid

stretching is applied where the sponge zone resides. The grid has 720 points in the

streamwise direction and 384 points along the transverse direction. The minimum

grid spacing in the y-direction is about 0.16δω(0) around the centerline. The 2-D

88

LES code uses the Smagorinsky model with Csgs = 0.182 = 0.0324, CI = 0.00575

and the turbulent Prandtl number, Prt = 0.9. To simulate a naturally developing

mixing layer, random perturbations are applied on the transverse velocity on the

inflow boundary

v(y) = ǫαUc exp

(−

y2

∆y02

), (B.11)

where ǫ is a random number between −1 and 1, α = 0.0045, and ∆y0 is the minimum

grid spacing in the y-direction [25]. No perturbations are applied in the streamwise

direction. The instantaneous vorticity contours are shown in Figure B.3. Notice the

occurrence of vortex pairing at about x = 100δω(0). For a randomly perturbated

mixing layer, the location of vortex pairing does occur at a fixed downstream location.

After the initial transients have exited the domain, the code was run for 55,000

time steps which corresponds to an acoustic wave traveling a distance of approxi-

mately 17 times the domain length at the ambient speed of sound.

B.3 Results and Discussion

Figure B.4 shows the scaled mean streamwise velocity profiles at five stations.

The scaled velocity is given by

f(ξ) =U − Uc

U2 − U1

, (B.12)

where U is the time averaged streamwise velocity component, and

ξ =y − y(x)

δ(x), (B.13)

δ(x) = y0.9(x) − y0.1(x), (B.14)

y(x) =1

2[y0.9(x) + y0.1(x)]. (B.15)

y0.1(x) is the y location where

U = U1 + 0.1(U2 − U1), (B.16)

89

and, similarly, y0.9(x) is the y location where

U = U1 + 0.9(U2 − U1). (B.17)

Figure B.4 shows the scaled velocity profiles at several different stations downstream

of the mixing layer. Pope [80] shows that the following error-function profile

f(ξ) =1

2erf

(ξ

0.5518

), (B.18)

is a good fit to the experimental data of Champagne et al. [81]. Hence, from Figure

B.4 the scaled velocity profiles show a good degree of self-similarity for the 6th-order

penta-diagonal filter case. The agreement with the error-function is also very good.

Although not shown in this report, the remaining test cases show similar behavior

of self-similarity as in Figure B.4.

Figure B.5 shows how the vorticity thickness grows for a naturally developing

mixing layer for the 6th-order penta-diagonal case. From this plot it can be seen

that there is slow growth till about x = 60δω(0) and from here there is linear growth

downstream. Stanley and Sarkar [75] reported a similar observation with their 2-D

DNS numerical experiment. From the plot the slope of line is 0.051. Using this value

the growth rate can be calculated by using

1

η

∂δω(x)

∂x= 0.152, (B.19)

where η is defined in Equation B.10. This value agrees well with the experiments of

Spencer and Jones [72] where they reported a growth rate of 0.16 and the numerical

experiments of Stanley and Sarkar [75] with 0.15. The growth rate values for each

test case as well as growth rates from other references are tabulated in Table B.2.

As can be seen from Table B.2, the 2-D LES growth rates compare well with other

exception of the 6th-order tri-diagonal case which is slightly high. The 4th and 6th-

order penta-diagonal cases show little difference in terms of growth rates between one

another. This was expected since the filter transfer function used looked very similar

(see Figure B.1). The 10th-order tri-diagonal case was not run until completion

90

because the 2-D LES code blew-up midway through the simulation. Investigation as

to why this happened revealed that instantaneous values of the primitive variables

ρ, u, v, p near the boundaries were registering huge values in the order of hundreds

and thousands before the code blew-up. Upon further investigation, we found out

that the primitive solution values were being ‘amplified’ due to the treatment of the

filters on the boundaries. Figure B.6 shows the variation of the transfer function for

the point next to the boundary. As can be seen, with the exception of the 4th-order

case, all the other cases ‘amplify’ the signal/solution near the spurious wavenumber

region. However, the code did not stop for the 6th and 8th-order case. The inherent

viscosity of the code probably kept the spurious amplifications from contaminating

the solution during the simulation. However, for the 10th-order case these signals

were too strong and thus resulted in the code blowing-up.

Figures B.7, B.8 and B.9 show the normalized Reynolds stress profiles at var-

ious downstream locations for the 6th-order penta-diagonal case. The normalized

Reynolds stresses are defined as

σxx =

√u′

2

(U2 − U1)σyy =

√v′

2

(U2 − U1)σxy = sign(u′v′)

√|u′v′ |

(U2 − U1). (B.20)

The σxx and σyy profiles are observed to collapse well in the far downstream region

and hence exhibit self-similarity. The σxy profile also shows a similar trend but

the peak shear stress value at x = 130δω(0) is higher when compared to profile at

x = 160δω(0). This discrepancy could be due to the statistical sample size taken.

Hence, a bigger statistical sample size would probably eliminate this behavior. Table

B.2 also shows the normal and shear Reynolds stress peak values for all the test

cases. The peak normal Reynolds stresses, σxx and σyy, register higher overall values

when compared to the 3-D DNS done by Rogers and Mosers [74] and laboratory

experiments [71–73]. The same situation of higher peak stresses is also reported with

the 2-D simulation results of Stanley and Sarkar [75], Bogey [76] and Uzun [69]. This

is probably due to the fact that there is no velocity fluctuation in the third direction,

and the 3-D breakdown of the large scale structures into fine scale turbulence is

91

not present in the current two-dimensional calculations [69]. Hence, the energy

dissipation mechanism in a 2-D simulation is not expected to be the same as the

case in a 3-D simulation [69]. Another possible reason could come from the grid

resolution used for our LES. Since our LES probably does not have a fine enough

grid resolution, the Reynolds stresses in the streamwise direction will over-predict

the values given in experiments and high resolution DNS results [82].

B.4 Conclusions

For this report, the effect of two different families of low-pass filters have been

studied on a spatially developing planar mixing layer using 2-D Large Eddy Sim-

ulation (LES). The low-pass filters studied here has a discrete form where one is

penta-diagonal and the other is tri-diagonal. Overall, the 2-D LES peak Reynolds

stresses are higher when compared to available experimental and 3-D computational

data. This is probably due to the fact that there is no velocity present in the third

direction, and the 3-D breakdown of large scale structures into fine-scale turbulence

is not present in this 2-D simulation. Furthermore, the grid resolution used for our

LES is probably not fine enough, and thus the Reynolds stresses in the streamwise

direction will overpredict the values given from experiments and high fidelity 3-D

DNS results.

92

Table B.1 Test case filter coefficients

Type of Filter Filter Coefficients

4th-Order Penta-diagonal α1 = 0.652247 α2 = 0.170293

6th-Order Penta-diagonal α1 = 0.619399 α2 = 0.169612

4th-Order Tri-diagonal αf = 0.37888 -




Table B.2 Comparison of the normalized peak Reynolds stresses andgrowth rates with available experimental and computational data.

Reω σxx σyy |σxy|1η

∂δω(x)∂x

Reference

- 0.176 0.138 0.097 0.19 Wygnanski and Fiedler’s experiment [71]

- 0.19 0.012 0.114 0.16 Spencer and Jones’ experiment [72]

1,800 0.18 0.14 0.10 0.163 Bell and Mehta’s experiment [73]

3,200 0.16 0.13 0.10 0.13 Rogers and Moser’s 3-D DNS [74]

720 0.20 0.29 0.15 0.15 Stanley and Sarkar’s 2-D DNS [75]

5,333 0.20 0.26 0.14 0.18 Bogey’s 2-D LES [76]

720 0.22 0.28 0.11 0.15 Uzun’s [25] 2-D DNS

5,333 0.229 0.287 0.124 0.155 2-D LES 4th-Order Penta-diagonal Filter

5,333 0.228 0.287 0.122 0.152 2-D LES 6th-Order Penta-diagonal Filter

5,333 0.244 0.278 0.122 0.158 2-D LES 4th-Order Tri-diagonal Filter



93

k/π

Filt

er

Tra

nsf

erF

un

ctio

n

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.54th-Order tri-diagonal filter withαf = 0.378886th-Order tri-diagonal filter withαf = 0.349268th-Order tri-diagonal filter withαf = 0.3067710th-Order tri-diagonal filter withαf = 0.240804th-Order penta-diagonal filter6th-Order penta-diagonal filter

Figure B.1. Transfer functions of various filters used in this study.

x/δω(0)

y/δ ω

(0)

0 100 200 300 400-100

-50

0

50

100

150

200

250

Figure B.2. Computational grid used in this LES. (Every 5th node is shown)

94

x/δω(0)

y/δ ω

(0)

0 50 100 150 200-100

-75

-50

-25

0

25

50

75

100

Figure B.3. Instantaneous streamwise vorticity contours in a natu-rally developing mixing layer. (Mc = 0.074, Reω = 5333)

ξ

f(ξ)

-2 -1 0 1 2-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100δω(0)x = 130δω(0)x = 160δω(0)x = 200δω(0)error function

Figure B.4. Scaled velocity profiles along with the error-functionprofile for 6th-Order penta-diagonal filter.

95

x/δω(0)

δ ω(x

)/δω(0

)

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10 ComputationLinear Fit

Figure B.5. Vorticity thickness growth in the mixing layer for 6th-Order penta-diagonal filter.

k/π

Bo

und

ary

Filt

erT

ran

sfer

Fu

nct

ion

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

4th-Order tri-diagonal filter withαf = 0.378886th-Order tri-diagonal filter withαf = 0.349268th-Order tri-diagonal filter withαf = 0.3067710th-Order tri-diagonal filter withαf = 0.24080

Figure B.6. Transfer functions of the tri-diagonal filters used for thepoint next to the boundary, i.e. i = 2.

96

ξ

σ xx

-3 -2 -1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

x = 70 δω(0)x = 100δω(0)x = 130δω(0)x = 160δω(0)x = 200δω(0)

Figure B.7. Normalized Reynolds normal stress σxx profiles for 6th-Order penta-diagonal filter.

ξ

σ yy

-3 -2 -1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3


Figure B.8. Normalized Reynolds normal stress σyy profiles for 6th-Order penta-diagonal filter.

97

ξ

σ xy

-3 -2 -1 0 1 2 3-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04


Figure B.9. Normalized Reynolds shear stress σxy profiles for 6th-Order penta-diagonal filter.

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