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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
2.4 The cross section of the grid in the y − z plane at x = 15ro. (Everyother grid point is shown) . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 The cross section of the grid in the y − z plane at x = 25ro. (Everyother grid point is shown) . . . . . . . . . . . . . . . . . . . . . . . . 22
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
3.16 One-dimensional sprectrum of the streamwise velocity fluctuations atthe x = 15ro, y = ro, z = 0 location for hAMP . . . . . . . . . . . . . 39
3.17 One-dimensional sprectrum of the streamwise velocity fluctuations atthe x = 20ro location on the jet centerline for hAMP . . . . . . . . . . 40
3.18 One-dimensional sprectrum of the streamwise velocity fluctuations atthe x = 20ro, y = ro, z = 0 location for hAMP . . . . . . . . . . . . . 40
3.19 One-dimensional sprectrum of the streamwise velocity fluctuations atthe x = 25ro location on the jet centerline for hAMP . . . . . . . . . . 41
3.20 One-dimensional sprectrum of the streamwise velocity fluctuations atthe x = 25ro, y = ro, z = 0 location for hAMP . . . . . . . . . . . . . 41
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
4.11 Acoustic pressure spectra at R = 60ro, θ = 25o in the far-field for eachtest compared against the Baseline case for a closed control surface. 56
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
4.13 Acoustic pressure spectra at R = 60ro, θ = 90o in the far-field for eachtest compared against the Baseline case for a closed control surface. 57
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.8 Instantaneous streamwise velocity contours after 2,000 time steps with∆t = 0.015. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
A.9 Instantaneous streamwise velocity contours after 4,000 time steps with∆t = 0.015. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
A.10 Instantaneous streamwise velocity contours after 8,000 time steps with∆t = 0.015. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
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
Figure 2.4. The cross section of the grid in the y−z plane at x = 15ro.(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 2.5. The cross section of the grid in the y−z plane at x = 25ro.(Every other grid point is shown)
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
0.045 Baselinerf4 modesrf6 modesrf8 modeshAMP
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
0.05 Baselinerf4 modesrf6 modesrf8 modeshAMP
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
Baselinerf4 modesrf6 modesrf8 modeshAMP
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
Baselinerf4 modesrf6 modesrf8 modeshAMP
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
Baselinerf4 modesrf6 modesrf8 modeshAMP
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
Baselinerf4 modesrf6 modesrf8 modeshAMP
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
Figure 3.16. One-dimensional sprectrum of the streamwise velocityfluctuations at the x = 15ro, y = ro, z = 0 location for hAMP .
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
Figure 3.17. One-dimensional sprectrum of the streamwise velocityfluctuations at the x = 20ro 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
Figure 3.18. One-dimensional sprectrum of the streamwise velocityfluctuations at the x = 20ro, y = ro, z = 0 location for hAMP .
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
Figure 3.19. One-dimensional sprectrum of the streamwise velocityfluctuations at the x = 25ro location on the jet centerline for hAMP .
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
Figure 3.20. One-dimensional sprectrum of the streamwise velocityfluctuations at the x = 25ro, y = ro, z = 0 location for hAMP .
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
Baseline to Baseline Open CSrf4 to Baseline Open CSrf6 to Baseline Open CSrf8 to Baseline Open CShAMP to Baseline Open CS
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
Baseline to Baseline Closed CSrf4 to Baseline Closed CSrf6 to Baseline Closed CSrf8 to Baseline Closed CShAMP to Baseline Closed CS
Figure 4.11. Acoustic pressure spectra at R = 60ro, θ = 25o 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 1.1-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Baseline to Baseline Open CSrf4 to Baseline Open CSrf6 to Baseline Open CSrf8 to Baseline Open CShAMP to Baseline Open CS
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
Baseline to Baseline Closed CSrf4 to Baseline Closed CSrf6 to Baseline Closed CSrf8 to Baseline Closed CShAMP to Baseline Closed CS
Figure 4.13. Acoustic pressure spectra at R = 60ro, θ = 90o in thefar-field for each test compared against the Baseline case for a closedcontrol surface.
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.
LIST OF REFERENCES
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
Figure A.8. Instantaneous streamwise velocity contours after 2,000time steps with ∆t = 0.015.
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
Figure A.9. Instantaneous streamwise velocity contours after 4,000time steps with ∆t = 0.015.
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
Figure A.10. Instantaneous streamwise velocity contours after 8,000time steps with ∆t = 0.015.
83
z
y
-1 0 1
-1
-0.5
0
0.5
1
u1.51.351.21.050.90.750.60.450.30.150
Figure A.11. Instantaneous streamwise velocity contours after 10,000time steps with ∆t = 0.015.
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 -
6th-Order Tri-diagonal αf = 0.34926 -
8th-Order Tri-diagonal αf = 0.30677 -
10th-Order Tri-diagonal αf = 0.24080 -
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
5,333 0.240 0.292 0.115 0.173 2-D LES 6th-Order Tri-diagonal Filter
5,333 0.231 0.292 0.116 0.161 2-D LES 8th-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
x = 70 δω(0)x = 100δω(0)x = 130δω(0)x = 160δω(0)x = 200δω(0)
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
x = 70 δω(0)x = 100δω(0)x = 130δω(0)x = 160δω(0)x = 200δω(0)
Figure B.9. Normalized Reynolds shear stress σxy profiles for 6th-Order penta-diagonal filter.