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The Pennsylvania State University
The Graduate School
THE PREDICTION OF BROADBAND SHOCK-ASSOCIATED
NOISE USING REYNOLDS-AVERAGED NAVIER-STOKES
SOLUTIONS
A Dissertation in
Aerospace Engineering
by
Steven Arthur Eric Miller
c 2009 Steven Arthur Eric Miller
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
December 2009
UMI Number: 3399683
All rights reserved
INFORMATION TO ALL USERS
The quality of this reproduction is dependent upon the quality of the copy submitted.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.
UMI 3399683
Copyright 2010 by ProQuest LLC.
All rights reserved. This edition of the work is protected against
unauthorized copying under Title 17, United States Code.
ProQuest LLC
789 East Eisenhower Parkway
P.O. Box 1346
Ann Arbor, MI 48106-1346
The dissertation of Steven Arthur Eric Miller was reviewed and approved? by the
following:
Philip J. Morris
Boeing/A.D. Welliver Professor of Aerospace Engineering
Dissertation Advisor, Chair of Committee
Dennis K. McLaughlin
Professor of Aerospace Engineering
Kenneth S. Brentner
Professor of Aerospace Engineering
Daniel C. Haworth
Professor of Mechanical Engineering
George A. Lesieutre
Professor of Aerospace Engineering
Head of the Department of Aerospace Engineering
?
Signatures are on file in the Graduate School.
Abstract
Broadband shock-associated noise is a component of jet noise generated by supersonic jets operating off-design. It is characterized by multiple broadband peaks and
dominates the total noise at large angles to the jet downstream axis. A new model
is introduced for the prediction of broadband shock-associated noise that uses the
solution of the Reynolds averaged Navier-Stokes equations. The noise model is an
acoustic analogy based on the linearized Euler equations. The equivalent source
terms depend on the product of the fluctuations associated with the shock cell
structure and the turbulent velocity fluctuations in the jet shear layer. The former are deterministic and are obtained from the Reynolds averaged Navier-Stokes
solution. A statistical model is introduced to describe the properties of the turbulence. Only the geometry and operating conditions of the nozzle need to be known
to make noise predictions. Unlike other models, the developed broadband shockassociated noise model is a true prediction scheme and not calibrated for a finite
range of operating conditions. The broadband shock-associated noise model developed represents the only prediction method in existence that has no restrictions on
nozzle geometry or jet operating conditions. This overcomes the limitations and
empiricism present in previous broadband shock-associated noise models. Extensive validation of the Reynolds averaged Navier-Stokes solution is performed using
experimental data. Validation efforts of the Reynolds averaged Navier-Stokes solutions include comparisons of Pitot and static probe measurements and schlieren
visualization. These validations show both the strengths and deficiencies for modeling strategies of supersonic jets operating off-design using Reynolds averaged
Navier-Stokes equations and associated turbulence closure schemes. Predictions
for various nozzle and operating conditions are compared with experimental noise
measurements of the associated jets to validate the broadband shock-associated
noise model. The operating conditions include under-expanded and over-expanded
jets with and without heating.
iii
Table of Contents
List of Figures
vi
List of Tables
xiii
List of Symbols
xiv
Acknowledgments
xviii
Chapter 1
Introduction
1.1 Supersonic Jets . . . . . . . . . . . . . . . . . . .
1.1.1 Structure of the Flow Field . . . . . . . .
1.1.2 Three Components of Supersonic Jet Noise
1.1.2.1 Mixing Noise . . . . . . . . . . .
1.1.2.2 Screech . . . . . . . . . . . . . .
1.1.2.3 BBSAN . . . . . . . . . . . . . .
1.2 Current BBSAN Prediction Methodology . . . . .
1.2.1 Harper-Bourne and Fisher?s Model . . . .
1.2.2 Tam?s Model . . . . . . . . . . . . . . . .
1.2.3 Current Model Limitations . . . . . . . . .
Chapter 2
Computational Fluid Dynamics
2.1 Introduction to CFD . . . . . .
2.1.1 Governing Equations . .
2.1.2 Grid Generation . . . . .
2.1.3 Solver . . . . . . . . . .
2.1.4 Post Processing . . . . .
iv
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1
5
8
11
14
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24
26
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31
32
32
36
40
42
2.2
CFD Results . . . . . . . . . . . . . . . . . . .
2.2.1 Computational Grids . . . . . . . . . . .
2.2.2 General CFD Results . . . . . . . . . . .
2.2.3 Menter SST and Chien K- Comparisons
2.2.4 Pitot Probe Comparisons . . . . . . . . .
2.2.5 Schlieren Comparisons . . . . . . . . . .
2.2.6 Off-Design Study . . . . . . . . . . . . .
2.2.7 Laminar and Turbulent Flow Nozzle . .
2.2.8 Helium and Hot Jet Comparisons . . . .
Chapter 3
Broadband Shock-Associated Noise
3.1 Model Development . . . . . . . . . . . . .
3.2 Implementation . . . . . . . . . . . . . . .
3.3 Parametric Studies . . . . . . . . . . . . .
3.4 Single Stream Axisymmetric Jets . . . . .
3.5 Dual Stream Axisymmetric Jets . . . . . .
3.6 Three Dimensional Jets . . . . . . . . . . .
3.7 The Effect of Laminar Flow in the Nozzle
3.8 Helium / Air Mixture and Hot Air . . . .
3.9 Turbulent Scale Coefficients . . . . . . . .
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43
45
52
56
59
76
84
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93
93
104
115
118
140
145
156
158
160
Chapter 4
Conclusion
163
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Appendix A
Correlation and Sound Pressure Level
172
Appendix B
Model Integrations
177
B.1 Integration of I?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
B.2 Integration of I?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Appendix C
Green?s Function
180
Bibliography
184
v
List of Figures
1.1
1.2
1.3
1.4
1.5
2.1
2.2
2.3
Possible operating conditions of converging and converging-diverging
nozzles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The three possible types of operating conditions in a supersonic
on- and off-design supersonic jet. Numerical contours of ??/?y are
gradients of density in the cross-stream direction representative of
a schlieren image. a) An over-expanded jet. b) An on-design jet.
c) An under-expanded jet. . . . . . . . . . . . . . . . . . . . . . . .
SP L per unit St of a Md = 1.00, Mj = 1.50, N P R = 3.67,
T T R = 1.00, D = 0.0127 m, jet at R/D = 100 and various observer locations, ?. The experimental data is courtesy of NASA. . .
The Harper-Bourne and Fisher model prediction compared with
experimental results of a Md = 1.00, Mj = 1.42, D = 0.025m jet
at 118 D, and ? = 135 degrees. The experimental data is digitized
from Harper-Bourne and Fisher [1]. . . . . . . . . . . . . . . . . . .
Tam?s model prediction compared with experimental results of a
Md = 1.00, Mj = 1.50, N P R = 3.67, T T R = 1.00, D = 0.0127 m
jet at R/D = 100 and ? = 130. The experimental data is courtesy
of NASA. Note that the prediction has been translated by -5 dB to
better match experiment. . . . . . . . . . . . . . . . . . . . . . . . .
Example placement of boundary conditions for CFD of an axisymmetric jet simulation. . . . . . . . . . . . . . . . . . . . . . . . . . .
Nozzle profiles and closeup of the computational grid near the nozzle
exit plane. a) Contour of the converging nozzle. b) Closeup of the
nozzle exit of the converging nozzle. c) Contour of the convergingdiverging nozzle. d) Closeup of the nozzle exit for the convergingdiverging nozzle. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dual stream nozzle profile and closeup view of computational grid
near the jet exit planes. . . . . . . . . . . . . . . . . . . . . . . . .
vi
7
10
13
23
27
39
46
48
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
The computational domain of the rectangular nozzle with Md =
1.50. The domain extends 75 equivalent diameters downstream
and 50 equivalent diameters in the cross-stream directions from the
nozzle exit center plane. The width of the nozzle exit is w = 0.0208
m and the height is h = 0.0119 m. . . . . . . . . . . . . . . . . . . .
An enlarged view of the rectangular nozzle computational grid. . . .
Contours of the velocity component, u m/s, for a Md = 1.50,
Mj = 1.30, T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contours of the Mach number, M , for a Md = 1.50, Mj = 1.30,
T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet.
Contours of the turbulent kinetic energy, K m2 /s2 , for a Md =
1.50, Mj = 1.30, T T R = 1.00, D = 0.0127 m converging-diverging
axisymmetric jet. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contours of the dissipation, m2 /s3 , for a Md = 1.50, Mj = 1.30,
T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet.
Contours of the shock pressure, ps Pa, for a Md = 1.50, Mj = 1.30,
T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet.
Extracted values along y/D = 0.50 of a converging conical nozzle
operating at Mj = 1.50 and T T R = 1.00 or T T R = 2.20 using the
Menter SST or Chien K- turbulence models. . . . . . . . . . . . . .
Extracted values along the cross stream directions of a converging
conical nozzle operating at Mj = 1.50 and T T R = 1.00 or T T R =
2.20 using the Menter SST or Chien K- turbulence models. a)
Transverse at x/D = 5.00 b) Transverse at x/D = 10.00. . . . . . .
Comparison between the experimental (dots) and numerical (lines)
po1 /po of the Md = 1.00, Mj = 1.50, converging nozzle case. Each
set of data is separated by x/D = 0.20 starting at x/D = 0.0 at
the left and stopping at x/D = 2.0 on the right. . . . . . . . . . . .
Comparison between the experimental (dots) and numerical (lines)
p/(1/2?u2j ) of the Md = 1.00, Mj = 1.50, converging nozzle case.
Each set of data is separated by x/D = 0.20 starting at x/D = 0.0
at the left and stopping at x/D = 2.0 on the right. . . . . . . . . .
Comparison between the experimental (dots) and numerical (lines)
M of the Md = 1.00, Mj = 1.50, converging nozzle case. Each set
of data is separated by x/D = 0.20 starting at x/D = 0.0 at the
left and stopping at x/D = 2.0 on the right. . . . . . . . . . . . . .
vii
50
51
54
54
54
55
55
57
58
62
63
63
2.16 Comparison between the experimental (dots) and numerical (lines)
po1 /po of the Md = 1.50, Mj = 1.30, converging-diverging nozzle
case. Each set of data is separated by x/D = 0.20 starting at
x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. . .
2.17 Comparison between the experimental (dots) and numerical (lines)
p/(1/2?u2j ) of the Md = 1.50, Mj = 1.30, converging-diverging
nozzle case. Each set of data is separated by x/D = 0.20 starting
at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right. .
2.18 Comparison between the experimental (dots) and numerical (lines)
M of the Md = 1.50, Mj = 1.30, converging-diverging nozzle case.
Each set of data is separated by x/D = 0.20 starting at x/D = 0.0
at the left and stopping at x/D = 2.0 on the right. . . . . . . . .
2.19 Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50, Mj = 1.30, De = 0.01776
m jet along the major axis plane. . . . . . . . . . . . . . . . . . .
2.20 Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50, Mj = 1.30, De = 0.01776
m jet along the minor axis plane. . . . . . . . . . . . . . . . . . .
2.21 Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50, Mj = 1.70, De = 0.01776
m jet along the major axis plane. . . . . . . . . . . . . . . . . . .
2.22 Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50, Mj = 1.70, De = 0.01776
m jet along the minor axis plane. . . . . . . . . . . . . . . . . . .
2.23 Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.30, De = 0.01776 m
jet along the major axis plane. . . . . . . . . . . . . . . . . . . . .
2.24 Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.30, De = 0.01776 m
jet along the minor axis plane. . . . . . . . . . . . . . . . . . . . .
2.25 Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.70, De = 0.01776 m
jet along the major axis plane. . . . . . . . . . . . . . . . . . . . .
2.26 Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.70, De = 0.01776 m
jet along the minor axis plane. . . . . . . . . . . . . . . . . . . . .
viii
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64
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64
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65
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68
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69
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70
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71
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72
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73
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74
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75
2.27 Comparison between the experimental and numerical schlieren of a)
the converging nozzle case Md = 1.00, Mj = 1.50, b) the convergingdiverging nozzle case Md = 1.50, Mj = 1.30. The nozzle exit is at
x/D = 0.0 and the flow moves from x/D = 0.0 to the right. The
nozzle centerline is at y/D = 0.0 and the nozzle lips at y/D = 0.5.
2.28 Comparison between the experimental and numerical schlieren of
the converging nozzle case Md = 1.00, Mj = 1.50 a) T T R = 2.20
air, schlieren on top and numerical schlieren on bottom, b) top:
schlieren of T T R = 2.20 air. bottom: numerical schlieren of helium
/ air mixture. The nozzle exit is at x/D = 0.0 and the flow moves
from x/D = 0.0 to the right. The nozzle centerline is at y/D = 0.0
and the nozzle lips are at y/D = 0.5. . . . . . . . . . . . . . . . .
2.29 Comparison between the experimental and numerical schlieren of
the rectangular nozzle case Md = 1.50, Mj = 1.30, and T T R =
1.00. a) The minor axis plane top: schlieren, bottom: numerical
schlieren. b) The major axis plane, top: schlieren, bottom: numerical schlieren. The nozzle exit is at x/De = 0.0 and the flow moves
from x/De = 0.0 to the right. . . . . . . . . . . . . . . . . . . . .
2.30 Comparison between the experimental and numerical schlieren of
the rectangular nozzle case Md = 1.50, Mj = 1.70, and T T R =
1.00. a) The minor axis plane top: schlieren, bottom: numerical
schlieren. b) The major axis plane, top: schlieren, bottom: numerical schlieren. The nozzle exit is at x/De = 0.0 and the flow moves
from x/De = 0.0 to the right. . . . . . . . . . . . . . . . . . . . .
2.31 Comparison between the experimental and numerical schlieren of
the rectangular nozzle case Md = 1.50 and T T R = 2.20. The experimental results use a helium / air mixture and the simulations use
heated air. a) Mj = 1.30 Major axis plane top: schlieren, bottom:
numerical schlieren. b) Mj = 1.70 major axis plane. top: schlieren.
bottom: numerical schlieren. The nozzle exit is at x/De = 0.0 and
the flow moves from x/De = 0.0 to the right. . . . . . . . . . . . .
2.32 Variation of the logarithm of shock strength, K?p2 (dots), compared with the off-design parameter ? 4 (line) offset slightly. The
logarithm of ? 4 has a nearly identical slope with K?p2 . . . . . . .
2.33 Comparison of Mach number contours for the Md = 1.50, Mj =
1.30, T T R = 2.20, and D = 0.0127 m jet. Top: Fully turbulent
flow. Bottom: Laminar flow inside the nozzle and turbulent flow
outside the nozzle. . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
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78
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80
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82
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82
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83
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85
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87
2.34 Centerline values for a Md = 1.50, Mj = 1.30, T T R = 2.20, D =
0.0127 m jet. Top: Variation of Mach number. Bottom: Variation
of pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.35 Lipline values (y/D = 0.50) for a Md = 1.50, Mj = 1.30, T T R =
2.20, D = 0.0127 m jet. Top: Variation of turbulent kinetic energy.
Bottom: Variation of streamwise velocity. . . . . . . . . . . . . . .
2.36 Comparison of Mach number contours for Md = 1.00, Mj = 1.50,
T T R = 2.20 air or T T R = 1.00 helium/air, D = 0.0127 m jet. Top:
Contours of M of hot air. Bottom: Contours of M of the helium /
air mixture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.37 Centerline values for a Md = 1.50, Mj = 1.30, T T R = 2.20 or
T T R = 1.00 helium / air mixture, D = 0.0127 m jet. Top: Variation of density, ?. Bottom: Variation of centerline velocity, u. . . .
3.1
3.2
3.3
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88
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89
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90
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92
Flowchart of the BBSAN prediction code. . . . . . . . . . . . . . . 105
Flowchart of the BBSAN integration subroutine. . . . . . . . . . . . 107
Interpolation of u m/s onto the integration region from the CFD
solution of a Md = 1.00, Mj = 1.50, T T R = 1.00, D = 0.0127 m
jet. The red line encloses of the integration region. . . . . . . . . . . 110
3.4 The magnitude of the Fourier transform of the shock cell pressure p?s . 112
3.5 a) Tcf verses T T R for a Md = 1.00 and Mj = 1.50 jet. b) The
associated change in dB when heating a Md = 1.00 and Mj = 1.50
jet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.6 Comparisons of BBSAN predictions with experiments for Md =
1.00, Mj = 1.50, T T R = 1.00, R/D = 100. . . . . . . . . . . . . . . 125
3.7 The total BBSAN prediction and the accompanying contributions
from selective integrations over contributing wavenumbers of p?s representing different waveguide modes of the shock cell structure.
Md = 1.00, Mj = 1.50, T T R = 1.00, R/D = 100, ? = 120.0. . . . . . 126
3.8 Various plots of the flow-field region for Md = 1.00, Mj = 1.50,
T T R = 1.00, R/D = 100, ? = 90.0. a) contours of ps . b) contours
of K. c) The spatially distributed source of the BBSAN at the peak
frequency of fp = 20241 Hz or St = 0.653. . . . . . . . . . . . . . . 127
3.9 Comparisons of BBSAN predictions with experiments for Md =
1.00, Mj = 1.50, T T R = 2.20, and R/D = 100. . . . . . . . . . . . . 128
3.10 Comparisons of BBSAN predictions with experiments for Md =
1.00, Mj = 1.22, T T R = 1.00, and R/D = 100. . . . . . . . . . . . . 129
3.11 Comparisons of BBSAN predictions with experiments for Md =
1.00, Mj = 1.22, T T R = 2.20, and R/D = 100. . . . . . . . . . . . . 130
x
3.12 Comparisons of BBSAN predictions with experiments for Md =
1.00, Mj = 1.22, T T R = 3.20, and R/D = 100. . . . . . . . . . . . .
3.13 Comparisons of BBSAN predictions with experiments for Md =
1.50, Mj = 1.30, T T R = 1.00, and R/D = 100. . . . . . . . . . . . .
3.14 Comparisons of BBSAN predictions with experiments for Md =
1.50, Mj = 1.30, T T R = 2.20, and R/D = 100. . . . . . . . . . . . .
3.15 Comparisons of BBSAN predictions with experiments for Md =
1.50, Mj = 1.40, T T R = 1.00, and R/D = 100. . . . . . . . . . . . .
3.16 Comparisons of BBSAN predictions with experiments for Md =
1.50, Mj = 1.60, T T R = 1.00, and R/D = 100. . . . . . . . . . . . .
3.17 Comparisons of BBSAN predictions with experiments for Md =
1.00, Mj = 1.47, T T R = 3.20, and R/D = 100. . . . . . . . . . . . .
3.18 Comparisons of BBSAN predictions with experiments for Md =
1.00, Mj = 1.56, T T R = 3.20, and R/D = 100. . . . . . . . . . . . .
3.19 Comparisons of BBSAN predictions with experiments for Md =
1.50, Mj = 1.70, T T R = 1.00, and R/D = 100. . . . . . . . . . . . .
3.20 Comparisons of BBSAN predictions with experiments for Md =
1.50, Mj = 1.70, T T R = 2.20, and R/D = 100. . . . . . . . . . . . .
3.21 Integration regions for the BBSAN calculation of the dualstream
jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 1.04, T T Rp = 2.70. .
3.22 Comparisons of BBSAN predictions with experiments for the dualstream jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 1.04,
T T Rp = 2.70, R/D = 100. . . . . . . . . . . . . . . . . . . . . . . .
3.23 Comparisons of BBSAN predictions with experiments for the dualstream jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 0.96,
T T Rp = 2.70, R/D = 100. . . . . . . . . . . . . . . . . . . . . . . .
3.24 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 1.00, R/De = 100 in
the major axis direction. . . . . . . . . . . . . . . . . . . . . . . . .
3.25 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 1.00, R/De = 100 in
the minor axis direction. . . . . . . . . . . . . . . . . . . . . . . . .
3.26 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 1.00, R/De = 100 in
the major axis direction. . . . . . . . . . . . . . . . . . . . . . . . .
3.27 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 1.00, R/De = 100 in
the minor axis direction. . . . . . . . . . . . . . . . . . . . . . . . .
xi
131
132
133
134
135
136
137
138
139
142
143
144
148
149
150
151
3.28 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 2.20, R/De = 100 in
the major axis direction. . . . . . . . . . . . . . . . . . . . . . . .
3.29 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.30, T T R = 2.20, R/De = 100 in
the minor axis direction. . . . . . . . . . . . . . . . . . . . . . . .
3.30 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 2.20, R/De = 100 in
the major axis direction. . . . . . . . . . . . . . . . . . . . . . . .
3.31 Comparisons of BBSAN predictions with experiments for the rectangular jet Md = 1.50, Mj = 1.70, T T R = 2.20, R/De = 100 in
the minor axis direction. . . . . . . . . . . . . . . . . . . . . . . .
3.32 Comparisons of BBSAN predictions for the laminar / RANS and
fully RANS simulation with experiments for a Md = 1.50, Mj =
1.30, T T R = 2.20, jet at R/D = 100. . . . . . . . . . . . . . . . .
3.33 Comparisons of BBSAN predictions for heated air and helium /
air simulated jets with experiments using heated air and helium
/ air mixtures. The jet conditions are Md = 1.00, Mj = 1.50,
T T R = 2.20, R/D = 100. . . . . . . . . . . . . . . . . . . . . . .
3.34 Comparisons of BBSAN predictions using turbulence coefficients
that are optimized for a non-screeching jet with experiments for
Md = 1.50, Mj = 1.70, T T R = 2.20, R/D = 100. . . . . . . . . . .
xii
. 152
. 153
. 154
. 155
. 157
. 159
. 161
List of Tables
3.1
3.2
3.3
3.4
3.5
3.6
Summary of the parametric study of the integration and wavenumber regions for the circular converging jet operating at Mj = 1.50,
T T R = 1.00, D = 0.0127 m. . . . . . . . . . . . . . . . . . . . . .
Recommended integration and index ranges based on the integration study of Mj = 1.50, T T R = 1.00, D = 0.0127 m. . . . . . . .
Recommended non-dimensional integration values based on D and
index ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jet operating conditions of the RANS CFD and BBSAN predictions
for axisymmetric single stream jets. . . . . . . . . . . . . . . . . .
Jet operating conditions for the RANS CFD and BBSAN predictions of the dual stream jets. . . . . . . . . . . . . . . . . . . . . .
Jet operating conditions for the RANS CFD and BBSAN predictions of the 3D calculations. . . . . . . . . . . . . . . . . . . . . .
xiii
. 116
. 117
. 117
. 119
. 142
. 145
List of Symbols
a Local speed of sound
cl
Coefficient of the characteristic streamwise coherent large length scale of
the convecting turbulence
cp
Specific heat at constant pressure
c?
Coefficient of the characteristic coherent large time scale of the convecting
turbulence
c?
Coefficient of the characteristic cross-stream coherent large length scale
of the convecting turbulence
c?
Speed of sound in the far-field
D
Exit diameter of a nozzle
De
Equivalent exit diameter of a non-circular nozzle
Dj
Fully expanded diameter of a nozzle
Dje
Equivalent fully expanded diameter of a non-circular nozzle
D?
The diameter at the throat of the nozzle
e Specific internal energy
f
Frequency
fc
Characteristic frequency
fp
Peak Broadband Shock-Associated Noise frequency
fs
Screech frequency
xiv
fia
Unsteady force per unit volume related to the interaction of fluctuations
in the sound speed (or temperature), caused by the turbulence and the
shock cells, and the associated pressure gradients
fiv
Unsteady force per unit volume associated with interactions between the
turbulent velocity fluctuations and the velocity perturbations associated
with the shock cells
fxv
Component of fiv in the direction of the observer
K
Turbulent kinetic energy
k
Wavenumber
l
Characteristic streamwise coherent large length scale of the convecting
turbulence
l?
Characteristic cross-stream coherent large length scale of the convecting
turbulence
M
Mach number
Mc
Convective Mach number
Md
Design Mach number of the nozzle
Mj
Fully expanded Mach number of the jet
N P R Nozzle pressure ratio
N P Rp
Nozzle pressure ratio of the primary jet
N P Rs
Nozzle pressure ratio of the secondary jet
p Thermodynamic pressure
pref
po
po1
Reference acoustic pressure
Stagnation pressure
Stagnation pressure or total pressure at Pitot probe
ps
Shock pressure
p?s
Fourier transform in the streamwise direction of the jet of the shock pressure
xv
p?
Ambient pressure
R Gas constant or distance from nozzle exit to observer
Rv
v
Rnm
v
Rxx
r
Two-point cross correlation function of the turbulent velocity fluctuations
in the observer direction
Two-point cross correlation function of fnv
Two-point cross correlation function of the components of fnv in the direction of the observer
Radial direction
Re Reynolds number
S
Sv
Spectral density
Cross spectral density of the two-point cross correlation function of the
turbulent velocity fluctuations in the observer direction
St Strouhal number based on the characteristic frequency
SP L Sound Pressure Level
T
Tcf
To
T?
Thermodynamic static temperature
Tam temperature correction factor
Stagnation temperature
Thermodynamic static temperature in the far-field
T T R Total temperature ratio
t Time
uc
Convective velocity
uj
Fully expanded jet velocity
vg
Component of the vector Green?s function
x Vector from the center of the nozzle exit to the observer
y
Vector from the center of the nozzle exit to a source point in the jet
near-field
xvi
?f
Frequency bandwidth
?
Off-design parameter
??
?
Heat expansion coefficient
Ratio of specific heats
?() Dirac delta function
?ij
Kronecker delta function
Turbulent dissipation
?
Vector between two points in the source region
?
Observer angle from the downstream axis, [degrees], or alternatively the
dilatation rate
? Thermal conductivity
?
Kinematic viscosity
?t
Eddy viscosity
? The local density
?j
Fully expanded density of a jet
?
Logarithmic expression of pressure
?g
Component of the vector Green?s function
?
Retarded time
?s
Characteristic coherent large time scale of the convecting turbulence
?ij
Shear stress tensor
? Specific dissipation
?
Radial frequency (? = 2?f )
xvii
Acknowledgments
This research is sponsored by the NASA Cooperative Agreement NNX07AC88A
entitled, ?A Comprehensive Model for the Prediction of Supersonic Jet Noise,? a
multiyear project involving professors, students, industry, and government. The
Technical Monitor of the Cooperative Agreement is Dr. Milo Dahl of NASA.
Dr. Nick Georgiadis of NASA provided helpful discussions regarding the Wind-US
CFD solver.
Experimental data is provided mainly by the author?s colleague and experimental counterpart Dr. Jeremy Veltin of The Pennsylvania State University. Additional acoustic data is courtesy of Mr. Ching-Wen Kuo of The Pennsylvania
State University, Dr. K. Viswanathan of the Boeing Company, and also various
NASA sources. All experimental data presented in this dissertation is published
with permission.
The author?s parents, Mr. Arthur F. Miller and Mrs. Beth A. Miller, and
his sister, Dr. Tiffany A. Fidler, are acknowledged for encouraging studies in
Aerospace Engineering. Their continuous love and support is invaluable.
The author?s former advisor at Michigan State University, Professor of Aerospace
Engineering Mei Zhuang of Ohio State University, is acknowledged for encouraging
studies in Aeroacoustics.
The Ph.D. committee, Professor Dennis K. McLaughlin, Professor Kenneth S.
Brentner, Professor Daniel C. Haworth, and former member Professor Vigor Yang,
are acknowledged for committing their time to providing insight into the project.
Most importantly, Boeing/A.D. Welliver Professor of Aerospace Engineering
Philip J. Morris, is acknowledged for acting as the author?s adviser and committee
chair. His many years of guidance and support for this and many other research
projects is invaluable. He provided the best graduate and research opportunities
possible. Finally, his work ethic and dedication to scientific investigations will be
used as a guide for the author?s research career.
xviii
Dedication
Dedicated to Pierre-Simon, Marquis de Laplace.
?We ought then to regard the present state of the universe as the effect
of its anterior state and as the cause of the one which is to follow. Given
for one instant an intelligence which could comprehend all the forces
by which nature is animated and the respective situation of the beings
who compose it - an intelligence sufficiently vast to submit these data
to analysis - it would embrace in the same formula the movements of
the greatest bodies of the universe and those of the lightest atom; for
it, nothing would be uncertain and the future, as the past, would be
present to its eyes.? - Laplace 1820.1
?The curve described by a simple molecule of air or vapor is regulated
in a manner just as certain as the planetary orbits; the only difference
between them is that which comes from our ignorance.? - Laplace 1820.
?Were she then to watch me live through it, she might smile condescendingly, as one who watches a marionette dance to the tugs of strings
that it knows nothing about.?2
1
Laplace, P., 1820, Essai Philosophique sur les Probabilits, Paris: V Courcier; repr. F.W.
Truscott and F.L. Emory (trans.), A Philosophical Essay on Probabilities, New York: Dover,
1951.
2
Causal Determinism, The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.).
xix
Chapter
1
Introduction
This dissertation is a component of a larger research project for the development
of a comprehensive prediction method for noise generated by supersonic jets. The
project consists of a tightly combined effort between universities, industry, and government using analytical, experimental, and numerical methods to develop noise
models. The project uses some existing prediction methods, however, the focus
of this dissertation is the development of a new broadband shock-associated noise
(BBSAN) prediction mathematical model, which is based primarily on first principles and is free of the restrictions of past empirical and semi-empirical models.
However, much physical insight and mathematical approaches were adapted from
past efforts. The BBSAN model developed only uses the operating conditions and
geometry of the nozzle to form predictions from a single closed formula consisting
of integrations over the region containing noise sources in the flow-field of the jet
plume. This approach removes a large number of restrictions and forms a unique
and new model for BBSAN. BBSAN is one of three components of supersonic jet
noise and is always the dominant component at certain frequencies and angles to
the jet axis.
Past prediction methods of BBSAN are based in part on empirical correlations
of radiated noise and jet flow measurements. These methods are restricted to
circular single and dual stream jet nozzle geometries. These BBSAN models were
mainly developed based on observations with cold jets. However, BBSAN remains
very important in hot jets also, considering that engine exhaust is always heated.
In heated jets the mixing noise dominates in the downstream quadrant with respect
2
to the jet exit. However, the BBSAN is always higher in intensity than the mixing
noise at larger angles to the jet downstream axis. Since typical engine exhaust is
hot and has nozzles that may be non-circular in practice, it is difficult to apply
current BBSAN models in these situations.
Gathering the acoustic spectra of shock noise from experiments in jet facilities
is an alternative to prediction. However, performing experiments using model
nozzles in small or medium scale facilities using heated air is relatively expensive
compared to calculations. Furthermore, numerical calculations may be faster to
obtain than experiments, which increases the number of points in the design space
when creating new aircraft configurations. Various governments and industries
pursuing new designs of supersonic business and military planes using supersonic
jets are also in need of improved prediction capability. Furthermore, community
noise issues related to military fighter aircraft are becoming a serious concern for
residents living near military facilities. Unfortunately, no present capability exists
to make reliable predictions of noise produced by either existing jet engines or new
nozzle designs for future aircraft. The model presented herein provides industry
and government aircraft and engine developers with a low cost method of predicting
BBSAN.
Many of the current approaches to jet noise reduction are based on modifying
the jet flow field, which is mainly achieved through changes to the nozzle exit
geometry. However, there is no method that can relate the changes in the flow field
to the radiated noise. Therefore, there is a need to develop a model that predicts
BBSAN over a wide range of operating conditions, both hot and cold, and takes
into account the nozzle geometry, which essentially controls the development of the
flow field. This will yield more understanding between the jet flow field predictions,
experimental measurements, and BBSAN prediction and measurement.
Burns [2] states that since the beginning of civil and military air transportation there has been great interest in reducing the noise emitted from the jets. This
is mainly due to community noise around military facilities and civilian airports.
When aircraft take off the noise from the jet is dominant relative to other sources
and also problematic during cruise. During aircraft landing jet noise is less of a
concern relative to take-off and cruise. However, the intensities of jet noise are
still a major component of the total noise spectrum. In 1987 a renewed interest
3
was generated in supersonic jet noise research instituted by the United States Government as it recognized its need to remain competitive in aerospace research [3],
which occurred long after the termination of supersonic transport programs [4].
To guide noise reduction efforts, the Federal Aviation Administration [5] creates
long term noise suppression goals for noise reduction from jets. The BBSAN model
developed is designed to be compatible with next generation aircraft noise prediction tools as described by Golub and Posey [6] and will be implemented in future
versions of the Aircraft Noise Prediction code of NASA.
Early BBSAN prediction methods, such as those available in the NASA Aircraft
Noise Prediction Program (ANOPP) [7] and the SAE ARP876 [8], are based on
the work of Harper-Bourne and Fisher [1]. These noise prediction methods involve
master spectra and correlation functions based on experimental observations of
supersonic jets from a single stream convergent nozzle. While such schemes are
useful, they are only reliable within the range of jet conditions and geometries
included in the existing database.
Prediction methodologies for BBSAN based on Large Eddy Simulations (LES)
or similar large scale computations such as Direct Numerical Simulation (DNS) are
able to predict BBSAN. Recent research at Penn State by Paliath and Morris [9],
as well as related computations by Shur et al. [10], demonstrated the capabilities
of this prediction methodology. Complex geometries, including a beveled nozzle
and simulated chevrons were studied. The results are very encouraging and indicate the quality of noise predictions that can be achieved through a judicious
use of discretization methods, grid distribution, and turbulence modeling. Unfortunately, the computations are expensive and time-consuming. These methods
are so computationally expensive that they will remain a research based tool for
the foreseeable future. The BBSAN model developed in this thesis uses Reynolds
Averaged Navier-Stokes (RANS) solutions as input. These are generated knowing
only the geometry of the nozzle and operating conditions, and so are computationally more efficient than unsteady simulation-based methods such as LES, DNS, or
another computationally expensive scheme.
The BBSAN model development makes extensive use of experimental measurements of two-point statistical properties of turbulent fluctuations in supersonic jets.
As will be shown they are a key component of the BBSAN prediction. The far-field
4
measurements of fluctuation pressure are also key, and their breakdown into different components of noise is also helpful in development of a model. The coupling of
experimental methods with model development is a most effective approach of research, more so than either methods of research individually as there are strengths
and weaknesses associated with each.
There are two original contributions to science presented in this dissertation.
First is the validation of RANS solutions for off-design supersonic jets operating
over a wide range operating conditions using various nozzle geometries. These
comparisons show both the strengths and deficiencies of current state-of-the-art
modeling strategies for supersonic jets operating off-design. The second major
contribution is the development of a BBSAN model that is based on first principles and has no restrictions regarding nozzle geometry and operating conditions.
Only a precise RANS solution of the jet flow-field is needed for a BBSAN prediction. Unlike other models, the developed BBSAN prediction methodology is a
true prediction scheme and not calibrated for a finite range of operating conditions.
The BBSAN model developed in this dissertation represents the only prediction
method in existence that has no restrictions.
This dissertation is presented in the same order as the BBSAN model was
developed and is divided into two main parts. The first is the computational
fluid dynamics (CFD) development of the RANS solutions, which is described in
Chapter 2. The CFD results presented are compared with experimental data.
Since these RANS results are the input to the BBSAN model their accuracy is extremely important. Furthermore, RANS solutions that are compared with various
experiments give validity to the numerical output of the CFD code and associated
equations that model fluid. Chapter 2 is divided into two parts. The first part
introduces the CFD and associated equations and turbulence models. The second
part presents various CFD results and compares them with experimental data.
Chapter 3 shows the BBSAN development mathematically and then predictions
based on the model are compared to far-field spectra for various jet conditions.
Multiple studies of the effect of the variation of parameters for the BBSAN predictions are made to help establish the sensitivity of the model. Results for axisymmetric single and dual stream, and rectangular jets are compared with experiment.
Finally, conclusions and suggestions for future research are provided in Chapter 4.
5
Before the methods and results of the RANS solutions are presented, the remaining
part of this introductory chapter examines the parameters that describe off-design
supersonic jets and reviews the past BBSAN modeling efforts.
1.1
Supersonic Jets
Before the aerodynamic and acoustic properties of the flow-field of supersonic
jets are discussed, a few important parameters and conditions that are related to
BBSAN are presented. First, the nozzle pressure ratio is the ratio of the stagnation
pressure inside the jet plenum to the ambient pressure, N P R = po /p? . The
total temperature ratio, T T R, is related to the stagnation temperature, To in the
plenum, divided by the ambient temperature, T? , T T R = To /T? . The stagnation
or total conditions are found through simple isentropic gas dynamic theory. These
two parameters completely specify the operating conditions of a nozzle. The design
Mach number, Md , of a nozzle is only dependent on the geometry of the nozzle.
Specifically, the ratio of the exit diameter, D, to the choked throat, D? , forms a
relation for Md ,
(?+1)/(2?2?)
1 + ??1
D2
2
=
(?+1)/(2?2?)
??1
D?2
Md 1 + 2 Md2
(1.1)
where ? is the ratio of specific heats for an ideal gas. This equation is implicit
for Md and must be found by a simple iterative method once the ratio of the
diameters are given. Note that if the ratio of D/D? = 1.00 then Md = 1.00. The
fully expanded jet diameter, Dj , is defined as the diameter of the jet to which an
off-design jet adjusts downstream of the nozzle. It is given by,
"
1+
Dj
=
D
1+
(??1)
Mj2
2
(??1)
Md2
2
#(?+1)/(4??4) Md
Mj
1/2
(1.2)
where Mj is the fully expanded Mach number. The fully expanded Mach number,
Mj is related only to the N P R by,
1/2
??1
2 Mj =
NP R ? ? 1
??1
(1.3)
6
If the N P R is needed to produce a Mach number of Md then Md can be substituted
into the above equation for Mj . If Md 6= Mj then the jet is said to be operating
off-design.
The Reynolds number, introduced by Osborne Reynolds [11] [12], is the ratio
of the dynamic forces to the viscous forces. In general the Reynolds number, Re,
is defined by,
Re = uj D/?
(1.4)
Figure 1.1 shows all possible operating conditions of a jet operating both subsonically and supersonically on- and off-design. The left upper diagram shows the
cross-section of a converging-diverging nozzle where Md > 1.00. The upper right
diagram shows the cross-section of a converging nozzle where Md = 1.00. The
high pressure (and possibly high temperature) fluid moves from inside the nozzle
through the throat to the exit where it forms a plume in the ambient environment. Corresponding graphs below each nozzle show possible static pressure, p,
distributions normalized by the stagnation pressure, po , along the centerline of
the nozzle. Depending on the ratio of po to p? , which forms the N P R, various
flow-fields inside and outside the nozzle exit can be obtained. If the N P R is small
then a subsonic exit flow results as seen in the static pressure distributions of lines
A or G. If the stagnation pressure is raised, thus increasing the N P R, eventually
the flow will reach M = 1.0 at the throat of the nozzle but return to a subsonic
condition as seen in line B or H. Notice that line H also corresponds to the exit
of the converging nozzle where M = Md = 1.00 and is said to be operating ondesign. In the on-design case the pressure at the exit of the nozzle exactly matches
the ambient pressure, p = p? . For the flow to reach M = 1.0 which is deemed
choked flow, the ratio of the static pressure in the throat of the nozzle relative
to the stagnation pressure in the plenum is 0.528 for ? = 1.4. If the N P R is increased further in the converging-diverging case then a normal shock will form in
the diverging portion of the nozzle. As the ratio of po /p? grows, the normal shock
travels downstream and eventually will exit the interior of the nozzle?s diverging
section. Once this occurs there are only three possible operating conditions for
the nozzle. The first is on-design, where the nozzle is perfectly expanded and the
fluid exits the converging-diverging nozzle at a supersonic speed, as seen in line
7
E. In this case p? exactly matches p at the exit of the nozzle. Since there is no
pressure mismatch the nozzle is considered to be operating on-design. This is also
the case in line H of the converging nozzle except that the exit M = Md = 1.00.
If po /p? does not allow for the static pressure to match the ambient pressure and
Mj 6= Md then the nozzle operates off-design. This situation is shown by lines D
or F of the converging-diverging nozzle and line I of the converging nozzle. The
cases of interest of study in this dissertation focus on when the values of N P R
are such that the operating conditions of the converging-diverging or converging
nozzles operate off-design and supersonically as shown in lines D, F, or I. These
lines of static pressure in particular are drawn very jagged outside the nozzle as the
pressure mismatch causes a very unique flow-field to develop. For a more thorough
introduction to the subject, Anderson [13] provides an overview with a historical
perspective.
Figure 1.1. Possible operating conditions of converging and converging-diverging nozzles.
8
1.1.1
Structure of the Flow Field
The previous section described parameters necessary to specify the boundary conditions of the fluid dynamics problem and also some of the parameters needed to
quantify the flow. Now, the structure of the flow-field or jet plume is discussed.
This is accomplished by showing plots of numerical simulations of supersonic onand off-design jets. A single circular nozzle with Md = 1.50 (converging-diverging
nozzle) and T T R = 1.00 is used for this task. The details of the simulations
and their validation are given in Chapter 2. Figure 1.2 shows the three types
of supersonic jets that are of interest to this work, over-expanded a), on-design
b), and under-expanded c). The three cases are illustrative of those shown in
the converging-diverging nozzle of Figure 1.1 cases D, E, and F respectively. The
converging nozzle operating off-design is not shown because it is has the same characteristics as the under-expanded converging-diverging nozzle. The over-expanded
jet is operating with a N P R of 2.77 and Mj of 1.30. The on-design jet is operating with a N P R of 3.67 and a Mj of 1.50. Finally, the under-expanded jet is
operating with a N P R of 4.94 and a Mj of 1.70. In each of the contour plots the
scale has been normalized by the exit diameter of the jet, D. The x-axis is in the
downstream direction and the y-axis is in the cross-stream direction. x/D = 0.0
is at the nozzle exit plane and y/D = 0.0 is the center axis of the jet. The nozzle
lips are at x/D = 0.0 and y/D = ▒0.5. The large white region in the vicinity
of x/D < 0.0 is the interior of the nozzle. High pressure (and in aircraft engines heated, T T R > 1.00) fluid exits the nozzle supersonically. In this figure the
flow-field of the jet is axisymmetric and three dimensional, but only a slice of the
flow-field is shown through the center line of the nozzle. The contours of Figure 1.2
represent the slope of the density in the y direction, which simulates a numerical
Z-type schlieren. This is discussed in detail in Chapter 2. Region A represents the
inside of the nozzle and is the converging section of the nozzle. In this area the
Mach number of the fluid is all less than one. In Region B the fluid accelerates
past M = 1.00 and becomes supersonic exiting the nozzle, in this particular case
at approximately M = 1.50. Note that distribution of M in the y direction is
quadratic, as found by Meyer [14]. Waves in region B emit from the wall because
the nozzle is not perfectly expanding the flow by eliminating the characteristic
waves when they impinge on the opposite wall.
9
In the over-expanded case, a), conical oblique shock waves may be seen at
point C. This oblique shock wave always originates from the nozzle lip in overexpanded jets, and it happens that this particular nozzle is over-expanded enough
that the oblique shock waves terminate in a barrel shock or a Mach disc, which
is characterized by the vertical line near x/D = 0.25. Mach discs do not form
in slightly off-design jets. Oblique shock waves can also be found in the underexpanded jet, but they do not originate from the nozzle lip, and can be see also
at point C in case c). Region D in case a) and c) are two locations of PrandtlMeyer [14] expansion waves. In these regions the flow accelerates and the pressure
gradient is less than one in the streamwise direction. In case b) there are no strong
oblique shock waves, only relatively weak ones in comparison. This is because the
static pressure in the jet closely matches the ambient pressure at location E. In the
off-design jet cases either a Prandtl-Meyer expansion fan, region D of case c), or
an oblique shock wave must occur to match the ambient pressure. The repeating
pattern of oblique shock and Prantdl-Meyer waves form shock cells. This is the
so called shock cell structure1 and is present in some form in all supersonic offdesign jets. In 1950, Pack [15] gave an excellent analytical method for predicting
the pressure fluctuations in the shock cell structure, but it is not accurate for
jets operating highly off-design. The pattern of shock waves and Prandtl-Meyer
expansion waves repeat and become weaker as they interact with the turbulent
shear layer (initially at x/D = 0.0 and y/D ▒ 0.5) and finally dissipate in the
turbulent mixing region far-downstream (x/D ? 10).
The turbulent shear layer develops in all jets, subsonic and supersonic, overexpanded, under-expanded, or on-design, due to Kelvin-Helmholtz [16] instabilities
that arise in shear layers of jets. These instabilities form a toroid (or another threedimensional shape due to a non-circular nozzle) vortex. These toroid vortices are
also unstable and Widnall [17] instabilities form that eventually cause the shear
layer to become fully turbulent. It is the growth of the turbulent shear layer that
acts like a waveguide which contains the shock cell structure. The shear layer
eventually dissipates the shock cell structure due to turbulent dissipation. The
multiple oblique shock wave interactions with convecting turbulence in the shear
layer is the cause of BBSAN. Details of how the shock cell structure and turbulent
1
Some researchers refer to this structure in off-design supersonic jets as shock diamonds.
10
shear layer are predicted are given in Chapter 2. These results are illustrative of
the physics of the problems and it will be shown that almost identical plots can
be produced from experimental methods.
Figure 1.2. The three possible types of operating conditions in a supersonic on- and
off-design supersonic jet. Numerical contours of ??/?y are gradients of density in the
cross-stream direction representative of a schlieren image. a) An over-expanded jet. b)
An on-design jet. c) An under-expanded jet.
For supersonic jets operating off-design, one of the most important parameters
used to link the operating conditions of the jet with the BBSAN intensity is the
off-design parameter. The off-design parameter was first used by Harper-Bourne
and Fisher [1],
?=
q
Mj2 ? 1
where ? is the off-design parameter. This definition of ? works very well for
11
converging nozzles where Md = 1.00. However, it needs to take into account
the situations where converging-diverging nozzles are present. For Md > 1, the
traditional definition of Harper-Bourne and Fisher is modified as,
?=
q
|Mj2 ? Md2 |
(1.5)
This definition of ? is used throughout the dissertation and is equivalent to the
original definition for converging nozzles. If ? = 0 then the jet is operating ondesign and as ? increases the jet operates further off-design. Harper-Bourne and
Fisher made many observations of the effect of ? as it varies for converging nozzles,
Md = 1.00, and found that the intensity of sound from an off-design supersonic jet
increases as ? 4 until a Mach disc or barrel shock forms in the core of the jet near
? ? 1.00. Once the barrel shock forms, then the noise intensity no longer increases
in the same fashion. This observation is confirmed with numerical predictions in
Chapter 2. Another important observation regarding the off-design parameter is
that the shock-cell spacing is linearly proportional to ? for slightly or moderately
off-design jets.
1.1.2
Three Components of Supersonic Jet Noise
The flow-fields of supersonic off-design jets have been described in the previous
section. This section describes the acoustic properties associated with these flowfields. Excellent overviews of the subject are given by Ffowcs Williams [18][19],
Goldstein [20], and for supersonic jet noise, Tam [21]. Supersonic jet noise from
off-design jets is often shown as acoustic spectra. One such plot may be seen in
Figure 1.3. The x-axis of the figure is the St (Strouhal) number and is based on
the characteristic frequency, fc of the jet. The characteristic frequency of a jet is
simply the fully expanded velocity uj , divided by the fully expanded diameter Dj ,
fc = uj /Dj . The St number is the non-dimensional frequency, in this case the
St = f /fc , where f is the frequency of the sound in Hz. The y-axis represents the
Sound Pressure Level (SP L) per unit St where the bandwidth of the spectra is
one unit St. All the results in the present dissertation are presented as SP L per
unit St. SP L found experimentally is calculated from the spectral density of the
time history of the acoustic signal and is discussed in detail in Appendix A. The
12
jet conditions in Figure 1.3 are Md = 1.00, Mj = 1.50, N P R = 3.67, T T R = 1.00,
D = 0.0127 m, R = 100D, and various observer locations labeled as ?. R is the
distance from the nozzle exit to the observer and ? is the angle to the observer
measured from the downstream axis. The spectrum at each observer location is
labeled in the plot at various observer angles.
The three kinds of supersonic jet noise are mixing, screech, and BBSAN and are
labeled in Figure 1.3 at ? = 70.0 degrees. These different components of jet noise
are also apparent at the other observer angles. Additional strong directional waves
are also created by some supersonic jets. This is called Mach wave radiation and is
classified as a type of mixing noise and will be discussed later. The mixing noise,
in both subsonic and supersonic jets, is broadband in nature and contributes to all
frequencies of the spectrum. It is always the dominant component of noise in the
downstream arc of the jet, especially so in hot jets. Notice that as ? is decreased
from 70.0 to 60.0 degrees that the peak of the mixing noise grows. The screech,
which is the discrete component of shock associated noise, is the most obvious
noise component as it is located at discrete St and is often more intense than the
other noise components for cold jets. As ? varies, the screech frequency does not
change, and in this case remains at St = 0.30. The screech tones located at values
of St = 0.60 are harmonics, and sometimes as many as five or six harmonics can
be seen in experimental data. Screech is more intense in the upstream direction
to the jet. Finally, screech is suppressed when the jet is heated.
Finally, the BBSAN noise consists of multiple broadband peaks and dominates
the mixing noise for high values of ?. The peaks of the BBSAN are closer together
for high ? and spread out greatly for low ?. The BBSAN makes very little contribution to frequencies lower than its first broadband peak, only the screech and
mixing noise contribute to these frequencies. More details of research performed
for each of the types of noise observed in the acoustic spectra are given in the
following sections, along with some of the methods of prediction, giving insight
into the physical noise generation mechanisms.
13
Figure 1.3. SP L per unit St of a Md = 1.00, Mj = 1.50, N P R = 3.67, T T R = 1.00,
D = 0.0127 m, jet at R/D = 100 and various observer locations, ?. The experimental
data is courtesy of NASA.
14
1.1.2.1
Mixing Noise
Turbulent mixing noise is present in all types of jets, at all jet velocities uj , and is
associated with the turbulence in the jet shear layer and the downstream mixing
region. Extensive experimental measurements of the acoustic spectra from these
jets operating supersonically and on-design, as off-design jet spectra are contaminated by shock noise, were performed by Yu and Dosanjh [22] and cold and hot
jets by Seiner et al. [23]. Mixing noise is divided into two components, at least
by the majority of the aeroacoustics community, as fine scale and large scale. The
fine scale mixing noise is found at larger observer angles from the downstream
axis while the large scale mixing noise is more dominant at relatively low angles
to the downstream axis. The large and small scale mixing noise has nothing to
do with its frequency content and is broadband in nature. The large scale mixing
noise originates from the large turbulent eddies convecting in the jet while the
small scale mixing noise originates from the small scale turbulence. The fine scale
mixing noise is believed to create a very broad range of contributions to the spectrum, while the dominant mixing noise intensity is due to the large scale mixing
noise. All mixing noise depends primarily on the T T R and Mj . As Mj increases,
the relative intensities between the small and large scale mixing noise increase. If
one holds Mj constant while increasing the T T R of the jet then the mixing noise
spectrum broadens and the overall sound pressure increases.
Fine scale turbulent mixing noise occurs due to the turbulent mixing of the
fluid in the shear layer, transition region, and fully turbulent regions of the jet.
It may be predicted by various methods. A revolutionary method for jet noise
prediction was proposed by Lighthill [24][25] where the Navier-Stokes equations
are reorganized into a inhomogeneous wave equation. The inhomogeneous wave
equation can be solved by analytical methods such as with Green?s functions.
However, the inhomogeneous right hand sides are not easily assigned values because
the spatial and temporal variations of the flow-field of the jet are not known exactly.
The method, and many based on it, are known as acoustic analogies because the
sources of jet noise on the right hand side of the modified equations are analogous
acoustic sources. Generally, the right hand side of the equation is split up into
analogous acoustic sources that are like a monopole, dipole, and quadrapole. The
quadrapole term is analogous to the turbulent source term of fine scale mixing
15
noise. However, the right hand terms of acoustic analogies are only representative
of the source of flow induced noise and give no insight into the actual physical
mechanism of noise generation. A more recent method of predicting mixing noise
can be performed computationally with DNS or LES but these simulations at
realistic Re are extremely expensive. DNS and LES simulations are essentially a
numerical experiment and give no additional insight into the actual physics of jets.
Finally, predictions of mixing noise can be found based on RANS CFD solutions
such as those by Raizada and Morris [26] for high speed subsonic jets. This method
could be extended to the supersonic regime.
Traditional RANS turbulence models and acoustic analogies do not give any
indication of the large scales associated with large scale mixing noise. Instead,
models for the large scale structures were pioneered by Plaschko [27][28] and greatly
improved upon by Morris et al. [29] and then by Liou[30]. Spreading rates of
supersonic jets are much smaller than subsonic jets because of the large associated
Re and the turbulent statistics vary slowly in the streamwise direction. This
implies that the large scale turbulent structures in the jet may be modeled by a
linear superposition of the modes of instability waves. The modes of the instability
waves reference a Fourier decomposition with respect to the azimuthal angle. This
yields a general equation for flow-field variables in the jet. The pressure associated
with the instability wave can be written as, for example,
p (x, r, ?, t) =
? Z
X
n=??
?w
an (?) (eigenfunction) exp [in? ? ?w t] d?w
(1.6)
??w
where an is the amplitude of the instability wave and is considered to be a stochastic
function, n is the azimuthal mode number, and ? is the angular frequency of the
instability wave.
Statistically, the large turbulent eddies and the instability wave model of the
jet are identical. This model assumes that the turbulent structures are instigated
by ?scientifically random? fluctuations at the nozzle lip. Tam and Burton [31] used
matched asymptotic expansions to construct a prediction scheme to connect the
far-field spectra to the near-field model of the instability waves. A wavy wall
analogy suggests that the instability waves act as a wavy-wall, which has the
16
same characteristics of the instability wave such as wavelength and speed. If the
instability waves move faster than the ambient speed of sound, then Mach wave
radiation results and is characteristic of the large scale mixing noise from supersonic
jets in which the large scale turbulent structures move faster than the ambient
speed of sound. The directivity of the Mach wave radiation is arccos (c? /C), where
C is the wave speed of the mode of the instability wave. McLaughlin et al. [32]
confirmed this theory with their helium / air jet experiments by comparing the
largest instability wave modes predictions with peak sound intensity frequencies.
Since various instability waves have different speeds, C, then they radiate Mach
wave radiation in slightly different directions and have the affect of broadening the
peak of the large scale mixing noise. However, when the convection velocities
are subsonic, relative to the ambient speed of sound, the analogy of a wavy wall
is less applicable. Recently, Morris [33] connected the far field spectral densities
of pressure fluctuations from subsonic and supersonic jets to wavenumber / frequency spectrum of pressure fluctuations in the jet shear layer. It is shown that
for subsonic instability waves, C < 1, the noise in peak directions is from the
large turbulent structures in the streamwise direction and not due to small scale
mixing as previously believed. Tam [34] demonstrates that the growth and decay
of the large turbulent structures needs to be taken into consideration for subsonically traveling waves. Tam showed that the subsonically traveling large turbulent
structures may generate Mach wave radiation via broadening of the wavenumber
spectrum of the traveling waves.
1.1.2.2
Screech
The discrete component of shock-associated noise, the screech tone, was first observed by Powell [35][36] in his studies of choked jets. Powell observed that a
powerful screech tone has an associated wavelength that is related to the shock
cell spacing in the jet. Powell then proposed an empirical formula for its frequency
based the jet diameter and the nozzle pressure ratio. Indeed, he was the first to
attribute screech tones to a feedback loop of acoustic waves traveling upstream in
the subsonic portion of the shear layer and impinging on the jet lipline, causing
instability waves to propagate downstream in the jet.
Many more experimental studies follow the work of Powell. Davies et al. [37]
17
proposed the mechanism of the screech tones in rectangular jets. Sherman et
al. [38] found the screech frequency is inversely proportional to the length of the
first shock cell by varying the N P R of an axisymmetric vertical jet of air. Norum [39] studied screech from underexpanded off-design jets, found the maximum
amplitude of screech occurs near ? = 160 degrees, and used various methods of
suppression by changing the jet exit geometry. Rosfjord and Toms [40] studied
the temperature dependence of screech from an axisymmetric sonic jet with Powell?s empirical formula. Fisher and Morfey [41] proposed an improvement to the
model of Powell based on a phased array of acoustic sources. Tam et al. [42] made
further improvements to screech tone prediction with a semi-empirical relation for
the peak screech frequency,
"
#?1
?1/2
?1/2
0.67uj
To 1/2
??1 2
2
fs =
Mj
Mj ? 1
1 + 0.7Mj 1 +
Dj
2
T?
(1.7)
where fs is the screech frequency, Mj is the fully expanded Mach number of the
jet, To is the stagnation temperature in the plenum, T? is the ambient static
temperature, and uj is the fully expanded velocity of the jet.
Panda [43] found with his experimental observations that another length scale
may be present that is more representative of the average length scales than those
proposed by Powell [35], Fisher and Morfey [41], or Tam [42]. Panda used the
standing wavelength Lsw to predict a more precise screech frequency,
fs =
uc
Lsw (1 + Mc )
(1.8)
where Lsw is the standing wavelength length scale. The standing wavelength length
scale may be larger or smaller than the average shock cell spacing. Unfortunately,
Lsw can not be found analytically based on the jet operating conditions. It can be
shown that by using Equation 1.8, and assuming an average length scale for the
shocks along with Tam?s empirical approximations, Equation 1.7 may be obtained.
Unfortunately, no empirical or analytical model exists that predicts the amplitude of screech. The screech phenomenon itself makes the jet?s flow-field structure
oscillate. The different modes of oscillation present makes the problem of predict-
18
ing amplitude even more difficult. Various factors affect the intensity of screech.
Larger lip thicknesses of the nozzle greatly increase screech. Increasing D or Mj
also increases the amplitude of screech. Surfaces near the jet may also increase the
intensity as they work like an additional acoustic reflector. However, as T T R is
increased, then the intensity of screech is greatly reduced to a point where other
sources of noise are dominant or the mechanism that produces screech is suppressed. Since jet engines operate with a high T T R then screech is often not a
concern. Furthermore, screech in cold jets may be eliminated by disrupting the
feedback loop via a single tab protruding into the flow from the nozzle lip.
1.1.2.3
BBSAN
BBSAN is observable in acoustic spectra in the far-field as shown in Figure 1.3.
The first acoustic measurements that were conducted to specifically study BBSAN
were performed by Harper-Bourne and Fisher [1] using converging round nozzles
with unheated air. Tanna [44] made more extensive measurements of shock related
noise using convergent round nozzles. Seiner and Norum [45] [46] examined plumes
of various jets performing cross correlations of the turbulence and shock positions.
Norum and Seiner [47] [48] studied a wide range of conical and contoured nozzles
operating over a wide range of pressure ratios to study the mean static pressure
and far-field acoustic spectra. This greatly extended the available experimental
database of shock related noise. Tanna [49] performed additional experiments
on a Md = 1.67 nozzle and formed a basic peak frequency dependence formula.
Seiner [50], Seiner and Yu [51], and Yamamoto et al. [52] carried out additional
extensive studies of BBSAN. Viswanathan [53] has shown that turbulent mixing
noise can be of the same level as BBSAN at high frequencies and care must be
exercised in trying to extract the contributions from each noise source for the
total spectrum. Generally BBSAN is omnidirectional except for very low observer
angles. Recently, Bridges [54] showed that as the T T R ratio of the jet increases
then the BBSAN will increase slightly. However, as T T R continues to increase
then the BBSAN remains constant. This is most likely because increasing the
T T R in supersonic jets stabilizes the magnitude of the instability waves or large
scale structures in the shear layer.
19
1.2
Current BBSAN Prediction Methodology
The first experimental investigation and semi-empirical BBSAN model was developed by Harper-Bourne and Fisher [1]. Because of the importance of this model,
it is discussed further in the following section. Their work explained some of the
fundamental physics that causes BBSAN. Following Harper-Bourne and Fisher,
an extremely simple empirical model developed by Mani et al. [55] that includes
some of the basic concepts of Harper-Bourne and Fisher. The model by Mani et
al. [55] yields only the peak sound pressure level at the peak frequency and the peak
frequency as a function of observer position, and consist of two simple algebraic
formulae and uses heavy empiricism. A simple model such as this is important
for rough calculations but isn?t useful in predictions outside the model?s range of
operating conditions. These types of purely empirical models give no insight into
the physics of the problem and only serve to replace interpolations of tables of
experimental data.
More recently, Stone [56] developed an empirical BBSAN model based on correlations of data from conical nozzle geometries. This model further includes some of
the changes in the shock cell structure when Mach disks form when the off-design
parameter becomes large. A direct formula is given by Stone for the Overall SPL
(OASPL) and also the sound pressure level in 1/3 octave bands. Unfortunately, this
model relies heavily on a sizable table of values of measured OASPL and SPL and
therefore contains the same limitations as the model by Mani. Finally, Tam [57]
developed a stochastic BBSAN model for circular nozzles operating slightly off
design and unheated, which is discussed further in the following sections because
of its importance for BBSAN prediction. Tam [58] extended this model for moderately off-design jets operating with slight heating. This is perhaps the most
widely used BBSAN model. This list of empirical and semi-empirical models is
incomplete, however, it is sufficient to describe the range of models thus developed.
Comparisons of these notable models for various nozzle and operating conditions
are given by Kim et al. [59].
20
1.2.1
Harper-Bourne and Fisher?s Model
Harper-Bourne and Fisher were the first to propose a model for BBSAN that
included both physical modeling as well as empirical data. They performed a
combination of experimental and analytical studies to make the first accurate BBSAN predictions. Their research methods should be replicated by other groups for
other studies in fluid dynamics. This section is based on their research and will
only briefly reproduce the important development and conclusions of their model.
Many details are intentionally left out, but may be found in their paper [1]. They
first observed that when converging nozzles choke Mj > 1.00, significant increases
in noise occur above ? > 90. They postulated, based on experimental observations, that this additional noise source is relatively unrelated to the jet stagnation
temperature To , and the angle of the observer to the jet.
Harper-Bourne and Fisher extended Powell?s model for discrete components
of shock noise. The model employs an array of sources on the nozzle lip line
that are equally spaced by a constant shock cell spacing, L. It was subsequently
assumed that each subsequent shock cell after the first one decreased by 6% in
relative strength. Another assumption is that as the turbulent eddies convect on
the lip line of the nozzle they interact with the tip of each shock cell and emit
acoustic waves. Experiments show that the turbulent eddies between successive
shock cells have correlation coefficients of 0.60, and two shock cell regions apart
have correlation coefficients of 0.20. This reduces greatly as correlations are made
further apart between different shock cell structures. Also, the time it takes for a
turbulent eddy to travel between successive shocks in the shear layer is the distance
between successive shock cell structures divided by the convection velocity of the
turbulent eddies, u»c . Another important observation that was taken into account
while developing the model, is that as the N P R increases, the shock cell spacing
increases in a linear fashion and that as the shear layer growth increases, that the
turbulent structures decay faster [60].
Harper-Bourne and Fisher first stated that each source contributes to the farfield acoustic pressure from a random source fluctuation evaluated at a retarded
time. The sound intensity, p2 (x, t) can be formed by summing the contributions
from each source, taking the square, and then the time average,
21
N X
N
X
1
r
r
m
n
p2 (x, t) = 2
Fm t ?
Fn t ?
x m=1 n=1
a?
a?
where Fm is a random source fluctuation, N is the number of sources, and rm is
the distance from each source location to the observer. If an assumption is made
that the fluctuations of the acoustic pressure are statistically stationary and that
the turbulent convection speed controls the phase between the sources then with
some algebraic detail omitted Harper-Bourne and Fisher state,
?(xn ? xm )
rm
rn
(1 ? Mc cos ?)
Fm t ?
Fn t ?
= Gmn (?)cos
a?
a?
uc
where Gmn is the source cross spectral density between the sources Fm of shocks
at positions xm and xn . Inserting the cross correlation between sources into the
equation for the sound intensity immediately yields an equation for the spectral
density of the pressure time history at the observer position,
N
N
1 XX
?(r1n ? r1m )
S (x, ?) = 2
(1 ? Mc cos ?)
Gmn (?)cos
x m=1 n=1
uc
(1.9)
S (x, ?) is the spectral density per Hz, and r1n is the distance downstream from
the nozzle to the nth shock cell. If the shock cells decrease a small amount after
the previous shock cell then the nth shock cell may be approximated by Ln =
L1 ? (n ? 1)/?L. If this relation is inserted into Equation 1.9 then,
N
N
n?1
X
1 XX
?L1
S (x, ?) = 2
Gmn
(1 ? Mc cos ?) |n ? m| ? ?L/L1
K
x m=1 n=1
uc
m
!
(1.10)
where L1 is the length of the first shock cell, and K is an integer from m to n ? 1.
Equation 1.10 is the Harper-Bourne and Fisher model for BBSAN [1]. All that
is left to do is to model Gmn . Equation 1.9 may be expanded in terms of group
source spectral densities, Sn ,
22
?L
S (x, ?) = So (x, ?) + S1 (x, ?) cos
(1 ? Mc cos(?)
u?c
2?L
+S2 (x, ?) cos
(1 ? Mc cos(?)
u?c
2?L
(1 ? Mc cos(?) + ...
+S3 (x, ?) cos
u?c
(1.11)
where So , represents the summation of the source spectral densities from a single
shock, S1 is the summation of the source spectral densities from all the adjacent
shocks, S2 represents the source spectral densities of shocks separated by 2L, and so
on. Si , for i > 0 is representative of the source spectral densities that are either destructive (negative summations) or constructive (positive summations) to the final
far-field spectral density and are the source of the peaks of BBSAN above the peak
frequency, fp . Harper-Bourne and Fisher show that values of Si may be related to
values of So recursively through Si (x, ?) = 2(N ? i)/ci (?)So (x, ?) /N . ci is the
group cross-correlation coefficient for shocks separated by iL and is Smn / (Smm Snn ).
It is relatively independent of frequency, but at very high frequencies it drops off.
This is not surprising as small scale fluctuations decorrelate before they convect
to downstream shocks. Finally So ,
Z?
So (x, ?) d? ?
D2 ? 4
x2
(1.12)
0
where ? is the off-design parameter. Essentially, Harper-Bourne and Fisher found
So through an empirical fit by a method of least squares and dimensional arguments
for various values of ?, D, and x, and experimental data. Example results from
Equation 1.10 appear in Figure 1.4. They have been generated for a Md = 1.00
converging nozzle with a N P R = 3.25, T T R = 1.00, Mj = 1.42, D = 0.025m, and
uj = 410.0m/s, fc = 16, 404 measured at R = 118D with off-design parameter
? = 1.00. The number of shock cells taken into account is N = 8. There is a 6%
decrease in shock cell length per shock cell as observed from experiment, and a
convective Mach number of uc = 0.7uj is assumed.
Using Equation 1.9 and assuming that the shock cell spacing in an off-design
supersonic jet is constant, then the maximum spectral density, S (x, ?), will occur
when the cosine terms have an argument of zero or an integer multiple of 2?. When
23
Figure 1.4. The Harper-Bourne and Fisher model prediction compared with experimental results of a Md = 1.00, Mj = 1.42, D = 0.025m jet at 118 D, and ? = 135
degrees. The experimental data is digitized from Harper-Bourne and Fisher [1].
the argument of the cosine is zero then the peak of the BBSAN may be found for
(n 6= m),
fp =
uc
L (1 ? Mc cos ?)
(1.13)
where fp is the peak frequency of the BBSAN. It is a combination of the convective
speed of the turbulence in the shear layer, the shock cell spacing, and the angle to
the observer. At fp the sound radiation from each BBSAN source is constructive
and at other frequencies there is at least some destructive interference or purely
destructive interference where lower spectral densities are expected. This behavior
24
yields the characteristic broadband peaks of BBSAN.
1.2.2
Tam?s Model
The stochastic models of Tam [57] [58] are summarized here. Because of the
complexity and length of the development of the model, only a summary of its
development is given. For full details about the model, including some of the
notation and definitions of the constants, the reader is directed to the original
source. In the first prediction method, Tam [57] models large turbulent structures
in the jet shear layer as instability wave modes of the mean flow set in a cylindrical
coordinate system. These instability waves interact with the quasi-periodic shock
cell structure and this interaction is considered the source of BBSAN. The form of
the stochastic instability waves used in the model are based on the work of Tam and
Chen [61]. The solution of the shock values and magnitudes of the instability waves
are matched with the solutions of Pack [15]. The model development starts with the
non-linear Euler equations consisting of budgets of mass, momentum, and energy.
Field variables are split into a contribution of the mean flow, a perturbation due
to the shock cell structure, a perturbation due to the turbulence, and a final term
that is a perturbation due to the interaction of the turbulence and shock cells. This
final term is arranged on the left hand side of the governing equations and the other
terms are arranged on the right hand side as source terms. It is important to note
that both the shock cell structure and the turbulence alone satisfy the linearized
Euler equations (LEE). The system of equations now forms the LEE with source
terms on the right hand side. Tam and Tanna [49] observed that only source
terms with a supersonic phase velocity radiate sound while sources with subsonic
phase velocity are inefficient acoustic sources. The LEE with corresponding source
terms are then expanded as a Fourier series and the method of matched asymptotic
expansions is used to solve for the field variable pressure using the method of Van
Dyke [62]. This process forms an equation for pressure due to the mth waveguide
mode and the turbulence in the shear layer in cylindrical coordinates,
25
? Z
X
pm =
n=??
?
?
Z
??
an (?)gnm Hn(1) exp [in? ? i?t + ix? + i(n + 1)?/2] d?d?
??
(1.14)
where an is a stochastic amplitude function, Hn1 is the nth order Hankel function
of the first kind, and
1
gnm (?, ?) =
2?
Z
?
Anm (x, ?) exp [i (?n ? ?m ) /] dx
(1.15)
??
where Anm is an amplitude function, and is a small valued constant that characterizes the rate of spreading of the jet. The autocorrelation of the pressure is
found and then the spectral density is formed after some simplification by performing the inverse Fourier transform. Anm (x, ?) exp [i (?n ? ?m ) /] remains to
be modeled and could be calculated numerically but this involves extensive computational cost. Instead, the function is approximated by an analytical model of
a Gaussian curve assuming that the shock cells are steady,
Anm (x, ?) exp [i (?n ? ?m ) /]
uj 1/2
?
? exp ? ln 2 (?x/uj ? Xm )2 /L2 + i (kt ? km ) x
a?
(1.16)
where L is the half-width of the Gaussian function, and Xm is representative of
the maximum location of the instability wave. Using this analytical approximation
and performing some simplification, a model for the far-field spectral density is
obtained,
S (R, ?, f ) =
CL2 Mj2 ? Md2
2
Aj ?2? a2? u2j
2
1 + ??1
Md2 R2 f
2
?
X
exp ? (fm /f ? 1)2 (1 + Mc cos ?)2 / (uc /uj )2 (L2 /2 ln 2)
О
2
?m
m=1
(1.17)
where C, L, fm , are constants or are given by simple formulas relating the physics
of the jet to the model, ? is the observer angle from the upstream axis of the jet,
Aj is the fully expanded jet area based on Dj , and ?m is the mth zero of the Bessel
26
function of the first kind. Full details of the constants and the model are given in
Tam [57], along with a formula for the spectral density in the near-field.
To take into account the effect of slightly heating the jet or having the jet operate moderately off-design some extensions were made to the model of Tam [58].
In this modified model a slight change is made to the instability wave spectrum
of Equation 1.16 to allow for high frequency predictions. A semi-empirical model
is also used to scale the shock cell strength depending on whether the jet is operating over- or under-expanded. Finally, to account for heating effects, a simple
multiplication factor to the spectra density is introduced, given by,
?j
??
??1 2
1+
Mj
2
?1
(1.18)
where ?j is the fully expanded density of the jet. The paper published by Tam [58]
shows no comparisons with heated jets, however, results of the model compare well
with experimental data for moderately heated and moderately off-design jets.
Example results of the model by Tam [57] are shown in Figure 1.5 evaluated
from Equation 1.17. They have been generated for a Md = 1.00 converging nozzle
with a N P R = 3.67, T T R = 1.00, Mj = 1.50, D = 0.0127 m, fc = 16, 404
measured at R/D = 100 and ? = 1.12 with the following additional conditions,
c? = 343.20 m/s, R = 287.00, ? = 1.40, To = 293.15 K, Tj = 202.17 K, T? =
293.15 K. The number of shock cells taken into account is N = 8. The experimental
results are courtesy of NASA and are represented by a thick black line. The
prediction itself is the orange line and components of the prediction are the red lines
underneath the total spectrum. In the figure, the total prediction is a summation
of individual BBSAN predictions corresponding to the mth waveguide mode of the
shock cells.
1.2.3
Current Model Limitations
Existing BBSAN models are based on empirical correlations of the measured aerodynamic characteristics of the jet plume and radiated noise; further, they are
restricted to circular geometries. This is due to the ease of creating semi-empirical
models and the abundance of past experimental investigations of cold axisymmetric
jets. Perhaps the biggest shortcoming is that they were developed predominantly
27
Figure 1.5. Tam?s model prediction compared with experimental results of a Md = 1.00,
Mj = 1.50, N P R = 3.67, T T R = 1.00, D = 0.0127 m jet at R/D = 100 and ? = 130.
The experimental data is courtesy of NASA. Note that the prediction has been translated
by -5 dB to better match experiment.
for cold jets, for which the shock-associated noise dominates the jet mixing noise.
The models for BBSAN of Harper-Bourne and Fisher [1] and Tam [58] use a number of assumptions in their development that restrict their application to certain
nozzle geometries and operating conditions. However, these two models are much
more advanced than their empirical counterparts.
Harper-Bourne and Fisher restricted their model development and experiments
in their study to converging nozzles operating with a T T R = 1.00. Although
no data shown supporting heated jets, their model does support slightly heated
conditions with some success. Furthermore, their model is developed only for
28
circular nozzle geometries. The model is mainly empirical in that it uses a master
spectrum that is formed with a least squares analysis based on measured data.
BBSAN is observed to be lower in intensity at small observer angles with respect
to the downstream axis of the jet. Unfortunately, the Harper-Bourne and Fisher
BBSAN model has a relatively constant amplitude of its spectral density with
respect to observer angle. This constant BBSAN amplitude is not observed in
experimental data. With these restrictions of the model considered, the model
development of Harper-Bourne and Fisher is an excellent example of a combined
experimental and theoretical approach that should be followed for other studies in
science.
The first stochastic BBSAN model of Tam [57] supports both converging and
converging-diverging nozzles operating at over- and under-expanded conditions.
Nozzle geometries are required to be circular and the temperature ratios are also
required to be T T R = 1.00 or slightly heated. The off-design parameter is required
to be small so the amount that the nozzle is operating off-design must be small.
These restrictions are partially overcome by Tam [58] by updating the approximation of the shock cell structure and using an empirical temperature correction
factor to multiply the final spectral density. This allows prediction of jets from
circular converging-diverging nozzles operating at over- and under-expanded conditions while moderately off-design and heated. However, no results were presented
for heated jets, and only comparisons with Norum and Seiner [47] are shown.
The BBSAN models developed thus far have either been entirely empirical
or semi-empirical with restrictions on having circular nozzles, mainly unheated
or slightly heated flow, and operating slightly or moderately off-design. In the
present dissertation a BBSAN model is developed that overcomes the limitations
of past modeling efforts by removing the restrictions on the jet operating conditions
and the shape of the nozzle. Any shape of nozzle may be considered, operating
at any N P R or T T R, and is applicable to a large range of ?. Only the nozzle
geometry and operating conditions need to be specified to make a prediction. This
is accomplished by drawing heavily on the modeling efforts of Harper-Bourne and
Fisher and Tam in conjunction with validation between the new predictions, models
of Tam, and experiments conducted at the Pennsylvania State University (PSU)
and the Boeing Company. The BBSAN model developed draws on the steady fluid
29
dynamic flow-field produced by a steady RANS simulation using any CFD solver.
The BBSAN model developed here consists of an integration over the physical
sources of BBSAN and its corresponding wavenumber spectrum of perturbation
pressure due to the shock waves in the jet. The turbulent statistics of the large scale
coherent eddies in the jet shear layer are modeled with an analytic function based
on traditional descriptions of turbulent scales from the turbulent kinetic energy,
K, and the turbulent dissipation, . A list has been constructed to summarize the
accomplished goals of this research project that will be presented in the proceeding
chapters. Each item in the list represents a goal and accomplishment that has been
achieved.
? Develop the first BBSAN model that...
- is only dependent on nozzle geometry and operating conditions.
- uses RANS solutions of the flowfield.
- can make predictions from any nozzle geometry.
- can use any off-design parameter.
- is able to plot the spatial distribution of the source strength for a given
frequency and observer angle.
? Perform converging and converging-diverging circular jet BBSAN predictions.
? Perform the first dual stream jet BBSAN predictions.
? Perform the first non-circular jet BBSAN predictions.
? Validate off-design...
- circular cold jet RANS solutions with Pitot measurements.
- circular cold and hot jet RANS solutions with schlieren.
- rectangular cold jet RANS solutions with Pitot measurements.
- rectangular cold and hot jet RANS solutions with schlieren.
? Validate the first helium / air mixture RANS solutions for off-design jets and
compare with experiment.
30
? Compare heated air and helium / air mixture BBSAN predictions with helium / air and heated experimental measurement.
? Obtain RANS solutions for a fully turbulent and laminar flow nozzle and
make corresponding BBSAN predictions.
The developed BBSAN prediction scheme enables supersonic jet noise to be
predicted based only on the nozzle geometry and operating conditions. In this
dissertation, the connection to the jet mean flow is made, enabling BBSAN to be
predicted for general geometries based on knowledge of the jet operating conditions
alone. This provides a significant advance in the state-of-the-art in supersonic jet
noise predictions.
Since the developed model relies heavily on accurate CFD input, the next
chapter of this dissertation explains in detail how the RANS solutions are obtained
and validated closely with experiments in the near-field. The third chapter gives
details of the BBSAN model development, and its validation with experimental
data and comparisons with other models for both circular and rectangular jets at
many different operating conditions. Finally, conclusions are drawn for the project
along with suggestions for future improvements.
Chapter
2
Computational Fluid Dynamics
This chapter contains two parts. The first section gives an overview of CFD and
how it is applied to the BBSAN prediction. This brief introduction is meant to give
the reader the necessary background into the subject. The second part, section 2.2,
shows various CFD calculations. These calculations are validated against experimental results in both a qualitative and quantitative sense with experimental data
from the PSU and Boeing. Validation of the CFD with experimental measurement
is important because these results are used as the input for the BBSAN model.
Fluids deform continuously under any force. Generally systems of partial differential equations model fluids. Unfortunately, the vast majority of mathematical
descriptions of fluids contain no analytical or general solution. The most popular method to obtain an approximate solution to a system of partial differential
equations is to use a discretization method which approximates the derivatives of
the equations and forms a set of algebraic equations that are discrete. These discrete algebraic equations can be solved by a digital computer on a computational
grid. Discretization schemes vary widely and are an area of intense mathematical
research. The quality of the discretization scheme chosen will affect the quality of
the solution. The process of solving systems of partial differential equations which
model the physics of various kinds of fluid dynamic situations is termed CFD. A
good entry level text that covers many of the ideas and algorithms for the vast
field of CFD is given by Ferziger and Peric [63].
32
2.1
Introduction to CFD
In general there are well defined methods for finding solutions to CFD problems.
First, equations need to be developed to model the phenomena of interest. These
are most often the Navier-Stokes equations or a simplified version of them. The
equations that describe the fluid dynamics of the off-design supersonic jets are
introduced in section 2.1.1. These equations are discretized by a CFD system onto
computational grids described in section 2.1.2. The CFD system used to generate
solutions in this dissertation is the Wind-US solver [64].
2.1.1
Governing Equations
The equations of motion that model fluid motion are the Navier-Stokes equations,
named after Claude-Louis Navier and George Gabriel Stokes. These equations are
conservative in the sense that they preserve the amount of mass, momentum, and
energy in a system. They can be used to model many different fluids and flow
types. One of these flow types is the off-design supersonic jet. The Navier-Stokes
equations usually refer to a statement of the conservation of momentum of a fluid
dynamic system. However, without additional supporting equations, the system
of equations is not closed. The equation for the conservation of mass is,
?? ??ui
=0
+
?t
?xi
(2.1)
where ? is the density, and ui is the velocity vector. Einstein summation is implied
over repeated indices. The compressible Navier-Stokes equations, stating that
momentum is conserved in orthogonal directions is,
??ui ??ui uj
?p
??ij
+
=?
+
?t
?xj
?xi
?xj
(2.2)
where p is the static pressure, and ?ij is the shear stress tensor. A final partial
differential equation is needed. It is the energy equation,
33
??uj cp T
??cp T
+
= ?T
?t
?xj
?p
?p
+ uj
?t
?xj
?ui
?
+ ?ij
+
?xj ?xj
?T
?
?xj
(2.3)
where cp is the specific heat at constant pressure, T is the temperature, ? is the heat
expansion coefficient, and ? is the thermal conductivity. The ideal gas equation
is usually used to close the system of equations. An excellent reference for the
development of these equations and their description is found in the famous work by
Schlichting [65]. Unfortunately, solutions of coupled partial differential equations
are difficult or impossible to find and analytic solutions to the compressible NavierStokes equations remain unknown. Not only is a general solution to the NavierStokes equations unknown but it is also unknown if such a solution exists or is
unique. However, with the advent of modern computers, numerical solutions of
the Navier-Stokes equations can be found for many realistic problems.
It is possible to simply solve the discretized Navier-Stokes equations with appropriate boundary conditions for an off-design supersonic jet and find the associated
BBSAN using a computer. Directly solving the Navier-Stokes equations in this
fashion with computers is called DNS. The amount of computing power needed
to perform a DNS simulation of a high Reynolds number supersonic jet operating
off-design is not available for the foreseeable future.
A very conservative estimate of the computer power needed to solve the NavierStokes equations can be made. As stated in the introduction, the Reynolds number
of a jet is Re = uj D/?. As an approximation, the integral length scale l of the
turbulence in the supersonic off-design jet is the exit diameter of the jet, therefore
D = l with associated Rel = ul/?. If the integral scale of the velocity in the jet
is approximated as the square root of the turbulent kinetic energy, ? K 1/2 , and
?3/4
the space between each grid point, ?x, behaves as ?x/l ? Rel
Kolomogorov [66], then u?t/l ?
obtain a solution goes as
Re3l .
?3/4
Rel .
according to
Therefore, the computer time required to
As an example of the number of grid points needed
for a DNS simulation, let u ? 0.2uj , l ? D, then Rej = 5 О 105 for a very simple
jet experiment in a laboratory environment, the number of grid points in l3 , Rel ?
9/4
5 О 105 , is Rel
? 2 О 1011 , for DNS. If a Large Eddy Simulation is conducted,
34
?1/2
then ?/l = Rel
is the number of grid points needed which equates roughly
to 32 million. Therefore, it is necessary to model the turbulence by modifying
the Navier-Stokes equations because DNS is currently impossible to perform for
realistic jet Re and LES is extremely expensive. There are various approaches
that have been used to model the turbulence. However, the approach used in
this dissertation makes use of Reynolds averaging of the Navier-Stokes equations
(RANS). Tannehill et al. [67] show the well known development of the Reynolds
Averaged Navier-Stokes equations. The RANS solutions in this work are found
by numerically by solving the density weighted RANS equations as presented by
Wilcox [68] as,
?
? ??
+
(??u?i ) = 0
?t ?xi
(2.4)
?
?
??P
?
(??u?i ) +
(??u?j u?i ) =
+
??ij ? ?u00j u00i
?t
?xj
?xi
?xj
(2.5)
????ij = ??u00i u00j
(2.6)
where,
and,
" #
?
u?i u?i
?u00i u00i
?
u?i u?i
00 00
?? e? +
+
+
??u?j h? +
+ u?j ?ui ui
?t
2
2
?xj
2
i
? h
=
?qL ? ?u00j h00 + ?ij u00i ? 1/2?u00j u00i u00i
?xj
? +
u?i ?ij ? ?u00i u00j
?xj
(2.7)
where the tilde represents a mass average, the bar represents an ensemble average,
the double prime is a fluctuation associated with the Favre averaging, and ? is
the Kronecker delta function. Once the mass weighted averaging is performed, a
turbulence model is needed (additional equations) to close the modified equations
because of additional terms such as the Reynolds stresses that appear. There are
various ways to close the modified system of equations. A thorough review of two
35
equation turbulence models, their development, and their implementation is given
by Wilcox. [68]
To find the appropriate turbulence statistics and mean flow quantities required
for the BBSAN model the Menter Shear-Stress Transport (Menter SST) model
is used. Menter used an empirical approach to combine the traditional strengths
of the K ? and K ? ? models. The Menter SST turbulence model uses the
traditional K ? ? of Wilcox in the inner region of the boundary layer. Additional
information may be found on the traditional K ? ? model in Wilcox [69]. The
Menter SST model automatically switches to a traditional K ? in the outer region
of the boundary layer and in free shear flows. This is done because the K ? ?
model is much better than the K ? model in the inner region of the boundary
layer. Another additional modification in the Menter SST turbulence model is
made to the definition of the eddy-viscosity which better accounts for the effects
of transport of turbulent shear stress. Along with the Reynolds Averaged NavierStokes equations the following two transport equations are used, which define the
Menter SST turbulence model,
D?k
?
?k
?ui
?
?
= ?ij
? ? ?? k +
(х + ?k хt )
Dt
?xj
?xj
?xj
(2.8)
?k ?ui
?
D?? ?
1 ?k ?? ?
= ?ij
? ??? ?2 +
[(х + ?? хt )] + 2 (1 ? F1 ) ??w2 ?
(2.9)
Dt
vt ?xj
?xj
? ?xj ?xj
The constants1 in the Menter SST turbulence model, ? ? , ?k , ?k , ?, ?? , F1 , ??2 are
defined by Menter [70]. The eddy viscosity is defined as,
vt =
0.31k
max (0.31?, |? О ~u|F )
(2.10)
where ? is the magnitude of the vorticity, and F is a function of K, ?, and
1
Note that the constants represented here for the Menter SST turbulence model share notation
found elsewhere in this dissertation. However, this is the notation chosen by Menter and is kept
here only for consistency.
36
the distance to the wall. The distance to the wall has been set to a very large
value. Both K and ? are very important parameters for modeling turbulence in
the BBSAN model and these spatially varying quantities can only be provided by
solving the RANS. The K ? turbulence model of Chien [71] is also available and
works in the same fashion as the Menter SST turbulence model. However, the
Menter SST turbulence model is used because it has been found to be more robust
in obtaining RANS solutions.
2.1.2
Grid Generation
The governing equations that model fluid flow behavior need to be discretized.
Since these equations can not be solved analytically they need to be solved with
a computer. The vast majority of modern computers are digital, meaning that
values must consist of finite discrete quantities. These quantities are found in both
space and time. A set of points are referred to as grid points and fill a solution
space of the domain where the solution is desired to be found. Typically these
points are connected in some fashion with connectivity data. There are generally
two systems that connect the grid points in the domain. These are referred to as
unstructured and structured grids. Structured grids represent an orderly implied
connection between grid points. Unstructured grids have grid points connected
in a non-implied orderly fashion, but have more variety in the ways they may be
connected.
The computational grids used in this work were created with the grid generation
software package, Gridgen version 15. Gridgen was first developed in 1984 and
started to be commercially available for purchase in 1994. Gridgen runs on a
number of platforms including Windows, Mac, Linux, and Unix. Gridgen supports
the creation of both structured, unstructured, and hybrid meshes in two and three
dimensions. It has the capability to import a large number of computer aided
design files or simple coordinates in a text file. More importantly, it has the ability
to directly export grids into binary file formats for many commercial, industrial,
and government CFD programs. Some of the more notable formats include Fluent,
Star-CD, and Wind-US.
The computational grids used in this research are all structured. This is ad-
37
vantageous for the CFD in that the finite difference methods can be used. The
computational grids used in this research are either two or three dimensional. Two
dimensional meshes consist of a list of x and y points while three dimensional
meshes are lists of nodal coordinates in the x, y, and z directions. Connectivity
information is assumed from the order in which the points are written in files for
structured grids. Unstructured computational grids could also be used, without
limitation in the BBSAN model or for finding RANS solutions, however the CFD
solver chosen has much more flexibility with structured grids. Many numerical
techniques and turbulence models are not implemented in the unstructured portion of the code.
The computational grids are created by importing the nozzle geometries into
Gridgen, which subsequently is used to create the computational domain inside the
nozzle and downstream in the jet plume. Gridgen also has the ability to export files
into formats for CFD codes to read in their desired format. It also has the ability
to prescribe elementary information to help guide the CFD solver. Wind-US is
the chosen CFD RANS solver for the present work and it has the capability to
set boundary condition information based on Gridgen output. This saves a large
amount of time when setting up the simulations because there is no other step
necessary before solving the RANS equations. For supersonic jet simulations a
number of boundary conditions can be prescribed. The first is an inflow boundary
condition which prescribes the total pressure and total temperature in the plenum
of the nozzle. These values correspond directly to the operating conditions of the
nozzle, N P R and T T R. An initial Mach number must also be set, however, this
can vary as only the total pressure and total temperature must remain constant.
Freestream boundary conditions are applied in the far-field of the domain. Far-field
boundary conditions require the specification of an ambient pressure p? , ambient
temperature T? , and freestream Mach number M? . These may fluctuate and
are not held constant, although they are not expected to change greatly as flow
gradients near these boundary conditions are expected to be small. The freestream
Mach number must never be zero for stability purposes and is set to a very small
value of 0.0001. A freestream boundary condition is never expected to be used
where a large mass flux of fluid will be moving across it. Regions with large masses
and gradients of fluid exiting the computational domain cannot be handled by the
38
freestream boundary condition and are handled by an outflow boundary condition.
A subsonic outflow boundary condition requires that a single quantity be specified,
the back pressure, which in the present simulations is set to the ambient pressure
p? . To simulate the walls of the nozzle there needs to be a boundary in the
computational domain between the fluid and the solid wall. There are typically
two kinds of boundary conditions for walls in traditional CFD. The viscous or
no-slip wall boundary condition represents an adiabatic wall where the fluid at the
wall must have a velocity of zero. The inviscid, Eulerian, or slip wall boundary
condition has the same parameters as the viscous wall boundary condition except
that it allows the velocity at the wall to have a component tangent to the wall.
The velocity normal to the wall must be zero. This implies that the flow may
?slip? while not allowing mass flux to move through the wall. The final boundary
condition used is the interface. The interface boundary condition allows the fluid
flow to be continuous across various sub domains. The computational domain,
since it is structured, is made of multiple zones or smaller sub domains. These
zones need to exchange information and an interface boundary condition performs
this task.
Figure 2.1 shows the location of the boundary conditions for a two dimensional
computational grid designed for axisymmetric jet RANS calculations. The sub domains of the computational grid are separated by black lines. The zone in the lower
left hand corner represents the interior of a convergent divergent nozzle. The zone
just above it represents the area outside the nozzle surface. Finally, the largest
zone represents the area into which the nozzle will exhaust and the jet plume
forms. Different boundary conditions are labeled on all the black lines which were
described in the previous paragraph. The lower inviscid wall represents the centerline of the jet. Even though the domain is in two dimensions, the equations being
solved are in an axisymmetric form. Gridlines from the computational domain
have been removed to better illustrate the sub zone interfaces and the locations of
the boundary conditions.
39
Figure 2.1. Example placement of boundary conditions for CFD of an axisymmetric
jet simulation.
40
2.1.3
Solver
The CFD solver finds solutions to either the Navier-Stokes equations or the RANS
equations closed by the Menter SST or Chien turbulence models discussed in the
previous sections of this Chapter. It can approximate the solution to the NavierStokes equations or the Menter SST equations in the entire computational domain
or a combination of the sub domains in different physical regions of the computational grid. The Navier-Stokes or RANS simulations are conducted using
the Wind-US 2.0 code [64]. Wind-US is a product of the National Project for
Application-oriented Research in CFD (NPARC) Alliance, a partnership between
the NASA Glenn Research Center and the USAF Arnold Engineering Development
Center. Wind-US is the merger of four different CFD systems, NASTD, NPARC,
NXAIR, and ICAT. The code has been extensively tested and numerous validation
cases for benchmark flow regimes have been established. It can find solutions to
the Euler and Navier-Stokes equations including turbulence models and chemically
reacting flows [72]. In relation to the present research, the code is used to predict
the development of single and dual stream supersonic jets operating off-design for
both axisymmetric and non-axisymmetric jets.
Wind-US can solve the field equations for perfect gases, real gases, or flows undergoing chemical reactions. The solutions are found by iterative schemes for time
integration, including Runge-Kutta [73], Jacobi iteration, Gauss-Seidel iteration,
MacCormack method, or Eulerian. Solutions in this work used a combination of
time dependent explicit Euler iterations with final implicit Euler iterations to find
the final steady solution. In the Spatial differencing for structured grids can be
either central or upwind. Wind-US is written almost entirely in Fortran 90. The
code is compiled with an Intel Fortran compiler and the GNU compilers. For the
simulations in the present work, Roe second order upwind differencing is used for
spacial discretization and a basic iterative solver is used to march the solutions in
time until a steady state solution is reached. The Total Variational Diminishing
(TVD) method is also used in some solutions for stability, both in the computational domain and at boundaries.
The Wind-US solver has the capability to run in parallel using the Message
Passing Interface (MPI) or the Parallel Virtual Machine (PVM). MPI is by far the
most popular method for parallel processing today on distributed computers. The
41
most complete reference for MPI is the recent standards publication [74]. However, due to technical limitations of the PSU development Linux cluster2 system,
Wind-US can not run in MPI mode, unlike many other programs developed at the
PSU. PVM can run on the development Linux cluster and alternatively a heterogeneous computer environment connected by a simple network. The PVM system
automatically distributes the load of the problem to different computer architectures if necessary to achieve the fastest possible solution time. Sunderam [75]
gives an overview of PVM and its framework. The capabilities of PVM to run on
a non-homogeneous computer network are not needed as the cluster consists of a
homogeneous computing environment.
PVM is used for the Wind-US calculations which are too large for a single
computational processor on the development Linux cluster or to lower the time
required to obtain a solution. In this work, PVM is only applied to the threedimensional cases that simulate rectangular jets. Wind-US unfortunately has no
automatic grid decomposition such as METIS for unstructured grids or even an
automatic algebraic grid decomposition for structured grids. For this reason, when
a PVM or MPI Wind-US simulation is performed, the grid must be manually
divided into separate sub domains, each assigned its own processor. Unfortunately,
Wind-US 2.0 has a maximum number of indices that are allowed in any direction
for a structured grid. This artificial limitation is set at 1000 grid points in an
indice direction. Fortunately, the NPARC Alliance has included a utility called
?Decompose? to help divide larger domains into ones that are smaller for either
PVM or for bypassing the artificial restriction on zone size. Still, with the program
Decompose the user has to manually specify which zone to split up, which index
to split, and into how many sub domains. This is often time consuming with
multiple large zones in large three-dimensional simulations. Besides some of these
extra limitations imposed by the code, PVM has proved to be very versatile as a
library for distributed computing.
The primary environment for the Wind-US solver used in this dissertation is
a Linux cluster. The server of the cluster consists of 4 Intel Xeon 1.5 Gigahertz
(Ghz) processors with 4 Gigabytes (Gb) of RAM. Each of the 64 nodes of the cluster
contains dual 2.4 Ghz Intel Xeon processors and 2 Gb of RAM. Gigabit and Fast
2
The name of the computer cluster is Cocoa3.
42
Ethernet are used for connectivity between nodes. Serial jobs are performed to
find the steady solutions of axisymmetric jets and the parallel jobs using PVM
are performed for the three-dimensional cases. When performing an axisymmetric
solution using approximately 500,000 grid points, solutions on a single processor
are obtained in approximately five days. The three dimensional simulations are
more expensive. A typical three dimensional solution is found using three to four
processors over the course of 20 days. This is taking into account that only one
fourth of the solution is calculated in three dimensions by taking advantage of the
planes of symmetry of a rectangular jet.
2.1.4
Post Processing
Post processing is a term to describe the analysis of the solution after Wind-US
completes or partially completes calculating a solution to the RANS or NavierStokes equations. When the Wind-US solver terminates it produces solution files
containing all the CFD information on the computational grid points. A serial or
parallel run will produce two files, one is the CFL file containing the solution and
the other is the CGD file containing the computational grid. These files are used
to view the CFD solution and also as input to the BBSAN prediction code. More
details about how the CFL and CGD file are integrated with the BBSAN code are
given in the next chapter. For the CFD solutions of this section they are used for
post-processing.
CFPOST is a post-processing program that reads common grid and common
flow files. It can produce listings, results, reports, files for other post-processing
programs, and many other miscellaneous functions [76]. CFPOST is a command
line driven program. Each time it is run the user must specify commands just
as at a command prompt to instruct CFPOST to perform various options. Since
CFPOST has been used so much in the process of producing results in this dissertation, various script files were developed to automatically generate results from
CFD simulations. By using scripts for producing different kinds of plots a large
amount of time is saved. CFPOST, even though it has a large amount of capability, has only been used in the scope of this dissertation to produce PLOT3D
files for interpretation by the Tecplot program, which is used to calculate other
43
variables and produce figures.
PLOT3D files, just as CFL and CGD files, are often split up as a solution and
grid file. A single grid file may be used for several solution files. PLOT3D files
can only contain structured computational grids and do not support unstructured
computational grids. The files themselves may be formatted, unformatted, or in
ASCII formats. PLOT3D has been in use for the storage of numerical simulation
results, and not just for CFD simulations, for a considerable time and is an industry
standard. More information about the development of PLOT3D, its format, and
application can be found in Walatka et al. [77]. PLOT3D files in this dissertation
are used for interpretation of the CFD results, as input to another post-processor,
and are not used to transfer data to the BBSAN prediction code. The CFD results
and later the BBSAN results, used for comparison with experimental measurement
in this dissertation have been plotted with Tecplot. Tecplot has the ability to read
a multitude of government and industrial CFD solutions.
2.2
CFD Results
Since the ultimate goal of the BBSAN model is to make predictions for both axisymmetric and three dimensional off-design supersonic jets, CFD cases have been
developed to validate these jets. In particular two axisymmetric jets operating
off-design at a single condition and one rectangular jet operating off-design at two
different conditions are chosen for validation. Two methods for validation are performed for these jets. The first is a quantitative approach comparing various flow
quantities in the jet with experimental data. The second is to further validate the
CFD, a study of how the shock wave strength scales with the off-design parameter changes is conducted. This is done to help validate the CFD with respect to
Harper-Bourne and Fisher?s observations [1]. Sometimes the flow transitions to
turbulence inside or outside the nozzle when the Reynolds number is low. A single
study has been produced where the flow in the nozzle is laminar and outside is
turbulent. Finally, since some of the experimental measurements for low density
jets have been conducted with helium / air mixtures to simulate hot jets, CFD
comparisons are made between a helium / air jet and a hot air jet.
Axisymmetric simulations are run until both convergence is achieved and a
44
visual inspection of the solution yields satisfactory results. Convergence is achieved
when the convergence residual of the system of RANS equations level reaches 10?15
universally or the residual of the various equations being solved becomes steady as
the number of iterations increase. Grid sequencing is used in both cases to speed up
convergence. Sequencing, in terms of Wind-US, means systematically skipping grid
points generating a coarsened grid for the initial iterations, until later iterations
where all the grid points are used. The transition from a sequenced domain to
an un-sequenced domain uses linear interpolation so initial iterations on the unsequenced domain require a temporary low CFL number. Sequencing is also used to
study the grid independence of the axisymmetric solutions. By sequencing the grid
in each index direction and comparing the solution with non-sequenced solutions,
grid independence is proven. This is performed for both axisymmetric validation
cases and it is shown that the solutions are independent of the computational grid.
Axisymmetric simulations use approximately 20,000 iterations until convergence if
there are no stability problems. Axisymmetric simulations are run in serial mode
because the benefit from parallelism would be small due to the relatively low
number of grid points. For the initial iterations a global CFL number is utilized
with a value of 0.75 in conjunction with a basic explicit Euler method. This drives
the solution towards a steady state. After the initial disturbance waves exit the
domain then the CFL number is not used and a constant time step is used at
each computational grid point. This constant time step is chosen so that the CFL
number at any grid point is less than one and is used in conjunction with the same
explicit Euler method. This process creates a very smooth steady solution which
demonstrates a constant residual with increasing time step.
To validate the axisymmetric CFD simulations of off-design supersonic jets,
two sets of CFD simulations were compared with experimental results. These
included one converging nozzle, Md = 1.00, and one converging-diverging nozzle,
Md = 1.50. The converging nozzle operated at N P R = 3.67 and the convergingdiverging nozzle operated at N P R = 2.77. Both of the nozzle operating conditions
used T T R = 1.00. This implies that the total temperature specified in the plenum
is the ambient temperature. The exit diameter of both nozzles is D = 0.0127m.
The nozzle walls are contoured to reduce the number of Prandtl-Meyer waves in
the supersonic region of the nozzles.
45
Validation of the three dimensional CFD results from Wind-US are conducted
with a rectangular jet. The design Mach number of the rectangular jet is Md =
1.50. The inside walls of the rectangular jet are contoured to minimize the number
of Prandtl-Meyer waves originating from the wall of the nozzle in the supersonic
region. The rectangular jet can not be modeled with a two dimensional grid as
it is not axisymmetric. However, there are two planes of symmetry in a rectangular jet, one through the major axis and one through the minor axis, parallel
to the streamwise direction of the jet. Therefore, the computational domain can
be represented by one fourth of the entire domain. When creating the computational grids, the two symmetric planes are used by specifying them as symmetric
boundary conditions or slip-boundary conditions.
Many CFD simulations have been conducted to assess the capabilities of the
Wind-US system, to produce more input for the BBSAN prediction code, and
for other research projects. However, this chapter focuses on the CFD results
used only for validation. The creation of the additional CFD results that are
not presented, but noted here and later in the BBSAN prediction chapter, follow
the same procedure. Results are also presented for axisymmetric dual stream
calculations, however, there is no experimental data of the flow-field available, but
an overview of the CFD of these jets is presented.
2.2.1
Computational Grids
For the simulations of the axisymmetric jets in the present research, a two dimensional grid is created. For the converging and converging-diverging nozzles with
design Mach numbers of Md = 1.00 and Md = 1.50, the y + distance is set to one
and the corresponding distance from the first grid point to the wall is then found.
The total number of grid points for the Md = 1.00 and Md = 1.50 axisymmetric
nozzles are 759, 609 and 485, 323 points, respectively. The number of grid points
used is quite large for axisymmetric supersonic jet simulations. This is in part
needed to make sure that there are enough grid points to fully resolve the shock
cell region of the jet. This is discussed more in the next chapter, where studies are
performed regarding the needed grid density for precise noise prediction. The computational domain downstream from the nozzle exit extends 75 nozzle diameters.
46
The range of the computational domain in the cross-stream direction is 50 nozzle
diameters from the centerline axis. Figure 2.2 shows the computational grids near
the nozzle exit for the two axisymmetric benchmark cases. The centerline is along
y/D = 0 and the nozzle exit in both cases is at x/D = 0. Part a) shows the
computational grid around the Md = 1.00 nozzle itself and the body of the nozzle
can be seen faintly. Part b) shows a close-up view of the computational grid in the
region of the nozzle Md = 1.00 exit. Part c) shows the converging-diverging nozzle
computational grid. The white region in the middle of part c) is the structure of
the nozzle. Finally, in part d) a closeup of the computational grid near the nozzle
exit can be seen for the Md = 1.50 nozzle. The lines in the computational grid
in the cross-stream direction do not vary in the streamwise direction. This makes
extracting cross-stream profiles of different field variables of the RANS solution
simple and avoids interpolation.
Figure 2.2. Nozzle profiles and closeup of the computational grid near the nozzle
exit plane. a) Contour of the converging nozzle. b) Closeup of the nozzle exit of the
converging nozzle. c) Contour of the converging-diverging nozzle. d) Closeup of the
nozzle exit for the converging-diverging nozzle.
47
An additional axisymmetric computational grid needs to be mentioned because
it has additional important features not present for single stream jet calculations.
A dual stream nozzle contains a primary nozzle, just like a single stream jet, but
with an additional secondary annular jet stream around the primary jet. Figure 2.3
illustrates this geometry by showing a computational grid that has been created
for RANS CFD calculations. The entire computational domain has been divided
into various sub domains that have been colored for identification. The red region
represents the inside of the primary nozzle. The orange region represents the
interior of the secondary nozzle. The green color is an exterior region outside
the nozzle that is upstream from the exit of the primary nozzle and the black
region is downstream from the exit of the primary nozzle. Once again, the xaxis represents the centerline of the nozzle and the entire domain is considered
to be axisymmetric about it. There are 15,390 grid points in the interior of the
primary nozzle and 12,150 grid points inside the secondary nozzle. There are
228,943 exterior grid points present and the total number of grid points in the
computational domain is 256,483. The y + distance at the exit of the nozzle is set
to one and the corresponding wall distance from the first grid point to the wall is
found. Both the primary and secondary nozzle are converging, so the design Mach
numbers are 1.00 for both jets. Note that the two nozzle exits are off-set. The
secondary jet exit is close to x/D = ?1.6 while the primary jet exit is at x/D = 0.
This is often the case with commercial aircraft engines. The geometry for the dual
stream nozzle and associated experimental data is courtesy of Dr. K. Viswanathan
of the Boeing Company.
The rectangular jet with design Mach number Md = 1.50 has an exit width of
the nozzle exit of w = 0.0208 m and height of h = 0.0119 m. For the validation
case a three dimensional grid is created. Noting the advantage of two planes of
symmetry, one through the major axis and one through the minor axis of the
rectangular exit, only one quadrant of the flow-field is modeled. Taking advantage
of the planes of symmetry would not be possible if the simulation were unsteady.
This is highly advantageous because it reduces the run time of the CFD solver
by approximately four or more and saves a large amount of memory and PVM
communication. The number of grid points in the computational domain of the
rectangular jet is 19, 768, 058. Since the nozzle is highly three-dimensional and the
48
Figure 2.3. Dual stream nozzle profile and closeup view of computational grid near the
jet exit planes.
number of grid points so large, the domain needs to be further broken down into
additional sub domains. Six different zones, or separate ordered pairs of i, j, k,
indices are created. The number of grid points in the jet plume region is 501 by
190 by 190. 501 points are used in the streamwise direction. This value is still
lower than the axisymmetric case, but increasing it adds unreasonable runtimes
when calculating solutions with the available Linux development cluster. It is
advantageous for noise calculations to have more grid points in the plume region,
compared with the interior of the nozzle and the freestream region of the flow.
Therefore, 91.41% of the grid points are located downstream of the nozzle. The
number of grid points allocated for the three-dimensional calculations is much
greater than needed.
Figure 2.4 shows the entire computational grid used for the rectangular jet calculations. The exterior of the computational domain consists of six planes but only
three are visible. The x direction is from the left to the right and represents the
streamwise direction of the jet. The other two Cartesian directions are cross-stream
directions. The z direction corresponds to the major axis and the y direction corresponds to the minor axis of the rectangular nozzle. The computational grid has
49
been colored to represent different regions of the flow for clarity. The red region
represents the grid points that are inside the nozzle. The black region represents
grid points that are downstream of the nozzle exit. The green points represent
the grid points that are upstream of the nozzle exit, but not inside the nozzle.
The planes have been labeled as symmetry and freestream. The symmetric planes
represent where cuts have been made in a much larger fictitious computational
domain. These symmetric planes are locations of symmetric boundary conditions.
The freestream label only applies to the green domain as these grid points of this
plane are freestream boundary conditions. The red grid points on the freestream
plane are representative of an inflow boundary condition. The planes on the other
side of the computational domain that are not seen are freestream boundary conditions and an outflow boundary condition in the y ? z plane. To obtain the entire
flow-field, the RANS solution is mirrored across the planes of symmetry to fill in
the other quadrants.
Another view of the rectangular grid is shown in Figure 2.5. The representative
coloring of the computational grid remains the same and additional thick black lines
have been placed to separate each of the sub domains of the computational domain.
Various grid clustering can be seen near these thick black boundaries. There
are many grid points near the walls of the nozzle and these extend downstream
from the nozzle exit into the far-field both in the cross-stream and streamwise
directions. This is very beneficial in the near-field of the jet to properly resolve the
three dimensional shock waves that form. At the inflow boundary there is no grid
clustering, as it is not needed because this region of the flow-field is solved with
no-slip walls: essentially solving the Euler equations (inviscid equations of motion).
This is done for the stability of the calculations. Note that the grid spacing near
the center line of the rectangular jet is also very small. This is to ensure that the
oblique shock waves in the jet core do not artificially become Mach disks at the
centerline.
50
Figure 2.4. The computational domain of the rectangular nozzle with Md = 1.50. The
domain extends 75 equivalent diameters downstream and 50 equivalent diameters in the
cross-stream directions from the nozzle exit center plane. The width of the nozzle exit
is w = 0.0208 m and the height is h = 0.0119 m.
51
Figure 2.5. An enlarged view of the rectangular nozzle computational grid.
52
2.2.2
General CFD Results
In chapter 3 it will be shown that the BBSAN noise model is based on an integration over space and wavenumber. Therefore, before detailed comparisons are made
between experiment and numerical prediction of the CFD results, contour plots of
variables that the BBSAN prediction model will use are presented. This should
give the reader a better physical understanding for the values of the variables in
the BBSAN model when it is developed later in this dissertation. This task is
performed by presenting contour plots of the converging-diverging axisymmetric
validation case. This nozzle operates at Md = 1.50, Mj = 1.30, T T R = 1.00, and
D = 0.0127 m. Distances have been normalized by the exit nozzle diameter. Since
the BBSAN model is dimensional, the variables have been left in their dimensional
form for these plots only, to better facilitate understanding of the quantities involved. The simulation is axisymmetric, so for a whole plane of the flow-field to be
shown, the results are mirrored across the x-axis by the operation ymirror = ?y.
The RANS solution variables in the BBSAN noise model are the average velocity of the fluid in the streamwise direction u, the local Mach number M , the
turbulent kinetic energy K, the viscous dissipation rate , and the shock pressure
ps . All these values are dependent on space alone y = (x, y, z), and not time t,
as the RANS solution is steady. Figure 2.6 shows the velocity in the streamwise
direction, u m/s. The centerline of the jet is located at y/D = 0 and the jet exit
plane is located at x/D = 0. The lip of the nozzle is located at x/D = 0 and
y/D = ▒0.50. The high pressure fluid (air) is expanded out of the nozzle in the
positive x direction, moving from the left of the page to the right. One can see
immediately that in the over-expanded case of Mj = 1.30 a conical oblique shock
originates at the lip of the nozzle and terminates in a barrel shock at x/D ? ?0.25.
Prandtl-Meyer expansion waves originate in the second shock cell and again form
another conical oblique shock. The slip lines downstream of the barrel shock propagate though the entire shock cell structure. As expected, u drops significantly
after shocks and accelerates through Prandtl-Meyer expansion waves. Clearly, the
values of u near the nozzle exit and above the lipline are close to zero, however
there is some entrainment of the fluid.
Figure 2.7 shows the same simulation and spatial ranges of the plot of u. In this
case the local contours of M are plotted. The local Mach number is found from
53
the ratio of the magnitude of the velocity vector and the local speed of sound. The
local speed of sound is a function of the local static temperature. For the BBSAN
model input, this is done beforehand by extracting the static temperature from the
common flow library solution file. The contours of M are consistent with respect
to the shock waves, both conical oblique and the barrel shock. Figures 2.8 and 2.9
show contour plots of the turbulent kinetic energy K, and viscous dissipation rate,
. The turbulent kinetic energy contour plot shows that it grows steadily from the
lips of the nozzle in the downstream direction. The maximum value of K is at
approximately x/D = 4.00. Further downstream the fully turbulent mixing region
is present. There the annular regions of K meet and subsequently disappear very
far downstream from the nozzle exit. There are slight waves in the contours of K
in the core of the jet which are due to the expansions and shock waves of the shock
cell structure. The viscous dissipation rate, , shown in Figure 2.9, is not a field
variable that may be extracted from the Menter SST solution. Instead, the specific
dissipation ? is converted to viscous dissipation rate as described by Wilcox [68]
or Menter [70]. The relationship between and ? used in this dissertation is
= 0.09K?. A large amount of dissipation is seen close to the nozzle lips. As
the nozzle lips in all the CFD simulations are finite, small recirculation regions
occur here. The same wavy contours of dissipation are also apparent like those
of K because of the shock cell structure. Finally, Figure 2.10 shows contours of
shock pressure, ps . The shock pressure perturbation is the thermodynamic static
pressure minus the ambient static pressure, p? . The shock pressure perturbation
is not close to zero at the exit of the nozzle because the jet is operating off-design.
The regions where the shocks and expansions occur are very apparent in this plot
because it removes the shear layer present in the other plots. Clearly, the first few
shock shear layer interactions are very strong relative to ones further downstream
in the jet. This is apparent from the large values of ps and because the spacing
between the contour lines is small.
54
Figure 2.6. Contours of the velocity component, u m/s, for a Md = 1.50, Mj = 1.30,
T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet.
Figure 2.7. Contours of the Mach number, M , for a Md = 1.50, Mj = 1.30, T T R =
1.00, D = 0.0127 m converging-diverging axisymmetric jet.
Figure 2.8. Contours of the turbulent kinetic energy, K m2 /s2 , for a Md = 1.50,
Mj = 1.30, T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet.
55
Figure 2.9. Contours of the dissipation, m2 /s3 , for a Md = 1.50, Mj = 1.30, T T R =
1.00, D = 0.0127 m converging-diverging axisymmetric jet.
Figure 2.10. Contours of the shock pressure, ps Pa, for a Md = 1.50, Mj = 1.30,
T T R = 1.00, D = 0.0127 m converging-diverging axisymmetric jet.
56
2.2.3
Menter SST and Chien K- Comparisons
The scaling of u and K with T T R for jets controls the scaling of BBSAN intensity. There is some skepticism regarding the ability of the Menter SST turbulence
model to correctly predict correct K and values of a jet when unheated and
especially when heated. Georgiadis et al. [78] compared various turbulence model
implementations in Wind-US, the CFD RANS solver of choice for this work, for
subsonic jets. In general, the Chien K- turbulence model displayed slightly better spreading rates and values of K closer to experimental data than the Menter
SST turbulence model. This is mainly because the implementation of the Chien
K ? turbulence model contains the ?PAB? temperature corrections introduced by
Abdol-Hamid et al. [79] in the PAB3D finite volume code. The PAB temperature
correction essentially implements a Sarkar compressibility correction and modifies
a closure coefficient based on gradients of total temperature in the jet. The PAB3D
code shows superior mean flow prediction capabilities relative to standard K ? RANS solutions for supersonic heated jets. Because of the improvements of predicted values of K from the Chien K ? turbulence model relative to experimental
jet data presented by Georgiadis, K comparisons are made of the Menter SST and
Chien K ? turbulence models for the same jet with two temperature ratios.
The converging benchmark nozzle of Md = 1.00 and D = 0.0127 m is utilized
to compare how K varies with temperature for both turbulence models. Four total
simulations are conducted with a Mj = 1.50 and T T R = 1.00 or T T R = 2.20 using
either the Menter SST or Chien K- turbulence models. Unfortunately, the Chien
K- with heating at T T R = 2.20 used a coarsened computational grid relative
to the other three simulations. This was necessary for the Chien K- model to
converge to a steady solution. Since the BBSAN sources in an axisymmetric jet
are located spatially along the lipline of the jet, lipline values of u and K are shown
in Figure 2.11. Note that there are multiple y axis in this figure representing two
variables separately. Clearly, the shock cell structure prediction from the Chien
model is stronger in the velocity plot than the Menter model. This is because the
shear layer growth in the Chien model is greater than the Menter model. Cross
stream plots of the values of u and K show more detail of both models capability.
Figure 2.12 shows cross stream values at x/D = 5.00 in a) and x/D = 10.00
in b). The values near x/D = 5.00 are located in a region with strong BBSAN
57
sources for this particular nozzle and operating conditions. The maximum values
of u at x/D = 5.00 are 477.0, 671.0, 461.0, and 646.0 m/s for the Menter SST
model at T T R = 1.00/2.20 and Chien model T T R = 1.00/2.20 respectively. The
maximum values of K at x/D = 5.00 are 5527, 11547, 4249, and 8567 m2 /s2 for
the Menter SST model at T T R = 1.00/2.20 and Chien model T T R = 1.00/2.20
respectively. Ratios of the maximum velocity from T T R = 1.00 to T T R = 2.20
are 1.406 and 1.401 for the Menter and Chien models. Ratios of the maximum
K from T T R = 1.00 to T T R = 2.20 are 2.09 and 2.02 for the Menter and Chien
models. Similar ratios are present at other x/D locations. Predicted values of u
are in close agreement between the two models operating at different temperature
ratios, although the spreading rate is slightly different. Even though the predicted
magnitude of K is different between the two models, the scaling of K at various
streamwise and cross-stream locations is exactly the same. This scaling is the same
even though the Chien model has superior prediction capabilities of K. Since the
scaling of u, K, or other field variables is the same for supersonic jets operating
off-design, and the Menter SST turbulence model implementation in Wind-US is
extremely robust relative to the Chien K-, the Menter SST RANS CFD solutions
are utilized for all other CFD simulations and BBSAN predictions.
Figure 2.11. Extracted values along y/D = 0.50 of a converging conical nozzle operating at Mj = 1.50 and T T R = 1.00 or T T R = 2.20 using the Menter SST or Chien K-
turbulence models.
58
Figure 2.12. Extracted values along the cross stream directions of a converging conical
nozzle operating at Mj = 1.50 and T T R = 1.00 or T T R = 2.20 using the Menter
SST or Chien K- turbulence models. a) Transverse at x/D = 5.00 b) Transverse at
x/D = 10.00.
59
2.2.4
Pitot Probe Comparisons
In this section, comparisons of the RANS CFD results produced from the Wind-US
calculations introduced in this chapter are compared directly with experimental
results. Specifically, these experimental results are produced with Pitot and static
pressure probes. The experimental data was collected in The Pennslvania State
University High Speed Jet Facility by Dr. Jeremy Veltin. For more details on
this facility and its capabilities see Doty and McLaughin [80]. The static pressure
probe and the Pitot pressure probe are separate. Also, some experimental data was
collected with a Pitot rake. The experimental data produced from these probes
provides values of total pressure, static pressure, and using oblique shock tables
and isentropic theory, Mach number. These values can be directly or indirectly
extracted at the same locations where the probe resides from the CFD simulations.
The experimental results are compared at various downstream and cross-stream
locations near the jet exit in the region where the BBSAN sources reside.
Predicted results are directly compared with experimental results at various
downstream locations. Comparisons are made for the axisymmetric jets of Md =
1.00, Mj = 1.50, T T R = 1.00, and Md = 1.50, Mj = 1.30, T T R = 1.00. Rectangular jet pitot probe comparisons are made at the following conditions: Md = 1.50,
Mj = 1.30, T T R = 1.00, and Md = 1.50, Mj = 1.70, T T R = 1.00. These CFD
results have been discussed in the previous sections of this chapter. Dual stream
comparisons are not made because experimental data are not available. In general,
the predictions and experimental comparisons are made in the first few diameters
of the jet. This is not to imply that comparisons far downstream are not important, but the source of BBSAN is in the shear layer close to the exit of the nozzle
relative to the fully turbulent region of the jet.
Before various results are presented, a short discussion of the analysis needed to
obtain them is necessary for their full understanding. There are some choices made
in the combination of equations used to evaluate the flow parameters (such as the
local Mach number). In these choices, the Pitot pressure is always used. For this
reason the simulation data are manipulated to produce profiles of local stagnation
pressure at a point that would be measured directly behind a hypothetical Pitot
probe. This is done because when a probe is physically put into a supersonic flow
it creates a normal shock in front of it, effectively lowering the total pressure. This
60
manipulation involves using the Mach number and local static pressure ahead of
the hypothetical shock that would be formed in the presence of the probe. For the
full details of how the various comparisons are conducted see Miller et al. [81]. The
total pressure measured by a Pitot probe, po1 , resides behind a normal shock wave
which is created by the probe interfering with the supersonic portion of the flow.
Alternatively, the Pitot probe measures the total pressure in a subsonic region of
the jet flow where no normal shock exists. This quantity, po1 is compared directly
with the experimental Pitot pressure as additional evidence of the quality of the
computations.
Figure 2.13 shows a comparison of the ratio of the Pitot pressure and the
plenum pressure, po1 /po , data for the axisymmetric converging nozzle, Md = 1.0.
The uncertainty estimate shown graphically on the figure is approximately 6% of
the mid range static pressure measurement, but does not include errors caused by
the distortion of the flow in a non-uniform flow field. Such an effect is evident in
specific regions of the flow including the region near where the oblique shock wave
originates at approximately x/D = 1.0. The data in this region near the centerline
of the jet shows larger differences between the experimental and numerical solutions
(larger than the pressure transducer uncertainty). It is likely that the discrepancies
observed can be attributed to the interaction of the flow within the oblique shock
created by the tip of the probe with other flow gradients (such as the mixing layer
or the jet shock waves) that reflects back to the probe body upstream of the static
pressure hole. This can produce an error difficult to predict due to the complex
shape of the mixing layer and the three dimensionality of the problem.
Figure 2.14 shows the next detailed profile comparison, namely a plot of the
ratio of static pressure and plenum pressure, p/po data for the converging nozzle
case. The static pressure at almost all locations shows very good agreement. However, the region near where the oblique shock wave originates at approximately
x/D = 1.10 shows some difference between the numerical and experimental solutions when the probe is near the centerline of the jet. This is again due to
the oblique shock wave interacting with the probe inside the flow. As before, as
the probe moves away from this region, the numerical results again agree with
experiment.
Figure 2.15 shows the local Mach number, M , calculated as described above
61
and presented at the same locations as the previous figures. Some discrepancies
appear near the jet centerline starting at x/D = 1.2 due to the presence of the
normal shock observed in the experiment but not in the numerical predictions.
Some discrepancies are also apparent in the outer part of the mixing layer, where
the experimental results exhibit a less smooth appearance. At these locations, the
computed velocities are very low. These local speeds correspond to very low pressure values measured by the Pitot probe pressure transducers. As discussed earlier,
the uncertainty is estimated at 4% and 6% of the mid range value respectively for
the static and the Pitot pressure. Therefore, these low speed measurements are
below the precision achievable with the experimental setup.
An additional problem appears near the nozzle exit where the static pressure
probe cannot traverse to the nozzle lip-line where it would encounter the nozzle.
Therefore, the M experimental plot at x/D = 0.0 shows a smaller radial range than
its numerical counterpart. Furthermore, the numerical results at this location show
that the velocity is zero on the nozzle lip; this accounts for the large gradient in
the region near y/D = 0.5 compared to the experimental data. The spreading rate
of the jet, shown by the lower gradients of pressure in the shear layer, is consistent
with the numerical results for the range of measured values.
The results of the probe traverses for the converging-diverging nozzle case are
shown in Figure 2.16. These predicted Pitot pressure profiles show overall good
agreement with the experiment. Figure 2.17 shows the corresponding static pressure measurements. Note that the static pressure has been non-dimensionalized
with respect to (1/2)?u2j for a clearer representation. These also have good agreement. However, as in the convergent nozzle case, there is one small disagreement in
the static pressure data. This is at approximately x/D = 0.2, which is the region
near the bottom of a barrel shock. The reasons for this discrepancy are probably
due to the same ones related to the converging nozzle case.
The final profile comparison, of the local Mach number in the convergingdiverging case, is shown in Figure 2.18. These Mach number estimates use the
data from the static pressure probe with the Rayleigh pitot formula to produce
quite good agreement between computations and experiment. However, the error
due to the static pressure probe interaction with the barrel shock at x/D = 0.2 is
apparent. The static pressure measurement has altered the solution for the Mach
62
number at this point. By inspection of Figure 2.18, the predicted growth rate of
the shear layer is lower than the experimental data. This is most likely a problem
with the turbulence model in the simulations.
Figure 2.13. Comparison between the experimental (dots) and numerical (lines) po1 /po
of the Md = 1.00, Mj = 1.50, converging nozzle case. Each set of data is separated by
x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right.
63
Figure 2.14. Comparison between the experimental (dots) and numerical (lines)
p/(1/2?u2j ) of the Md = 1.00, Mj = 1.50, converging nozzle case. Each set of data
is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0
on the right.
Figure 2.15. Comparison between the experimental (dots) and numerical (lines) M of
the Md = 1.00, Mj = 1.50, converging nozzle case. Each set of data is separated by
x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0 on the right.
64
Figure 2.16. Comparison between the experimental (dots) and numerical (lines) po1 /po
of the Md = 1.50, Mj = 1.30, converging-diverging nozzle case. Each set of data is
separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0
on the right.
Figure 2.17. Comparison between the experimental (dots) and numerical (lines)
p/(1/2?u2j ) of the Md = 1.50, Mj = 1.30, converging-diverging nozzle case. Each set
of data is separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at
x/D = 2.0 on the right.
65
Figure 2.18. Comparison between the experimental (dots) and numerical (lines) M
of the Md = 1.50, Mj = 1.30, converging-diverging nozzle case. Each set of data is
separated by x/D = 0.20 starting at x/D = 0.0 at the left and stopping at x/D = 2.0
on the right.
66
Pitot probe measurements have also been conducted for the rectangular jet. A
five probe Pitot static rake is used to gather total and static pressure at various
downstream and cross-stream locations in the rectangular jet just as in the circular
jet comparisons. The rake probe has five probes spaced evenly on a single device
allowing for multiple data to be collected at each downstream location. More
details about the rake Pitot probe can be found in Veltin and McLaughlin [82].
The exit geometry of the rectangular nozzle is described by the height h and width
w of its geometry. In the circular calculations the downstream and cross-stream
directions have been normalized by the diameter, D. A length scale is chosen for
the rectangular jet as the equivalent diameter, De , which is formed by matching
p
the exit area of the rectangular nozzle to a fictitious circular one, De = 4wh/?.
The fully expanded diameter of the rectangular nozzle can also be found as before
by replacing D by De . For the rectangular jet tested, De = 0.01776 m and the
aspect ratio of the nozzle is 1.75.
Comparisons between the simulations and experiment are made for two rectangular jets with Md = 1.50, T T R = 1.00, and Mj = 1.30 or Mj = 1.70. The
variation of po1 and M are determined at various downstream x/De locations in
the minor and major axes. The major and minor axes are in the z and y directions
respectively. Just as in the axisymmetric comparisons, the numerical results use a
simulated normal shock, if the local M is greater than one, to correspond to the
experimental results. The major and minor axis Pitot pressure traverse results for
the rectangular jets are shown in Figures 2.19 through 2.22 for both the underand over-expanded cases. Each of the figures has a corresponding downstream
location labeled as x/De and an integer n is used to separate the data on the axis.
The dots represent the experimental measurements while the lines represent the
numerical predictions. Generally, the agreement in the Mj = 1.30 case is excellent for the Pitot pressures except farther downstream where the spreading rate
of the experiment is higher. The same good agreement can also be seen in the
Mj = 1.70 case. However, on the minor axis there is more discrepancy in the Pitot
comparisons than in the other three figures. Comparisons of M at various x/De locations for the same rectangular jet operating conditions are found in Figures 2.23
through 2.26. Generally the agreement between the predictions and measurements
for the Mach number are good near the nozzle exit. Comparisons downstream
67
demonstrate less agreement. Recall that the experimental results use a combined
approach of analytical and experimental measurement to find the M data and this
needs to be considered when comparing results. The direct comparison of the Pitot
pressure is a much better metric for comparing experimental and numerical predictions because it does not include assumptions for the jet static pressure variation
across the jet. Overall the rectangular pitot probe comparisons are consistent with
the circular jet comparisons. This is encouraging because both simulations used
the same simulation parameters and it infers a degree of quantitative confidence
in the RANS results.
Figure 2.19. Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50,
Mj = 1.30, De = 0.01776 m jet along the major axis plane.
68
Figure 2.20. Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50,
Mj = 1.30, De = 0.01776 m jet along the minor axis plane.
69
Figure 2.21. Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50,
Mj = 1.70, De = 0.01776 m jet along the major axis plane.
70
Figure 2.22. Pitot probe comparisons between the experimental (dots) and numerical (lines) of po1 /po of the Md = 1.50,
Mj = 1.70, De = 0.01776 m jet along the minor axis plane.
71
Figure 2.23. Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.30,
De = 0.01776 m jet along the major axis plane.
72
Figure 2.24. Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.30,
De = 0.01776 m jet along the minor axis plane.
73
Figure 2.25. Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.70,
De = 0.01776 m jet along the major axis plane.
74
Figure 2.26. Pitot probe comparisons between the experimental (dots) and numerical (lines) of M of the Md = 1.50, Mj = 1.70,
De = 0.01776 m jet along the minor axis plane.
75
76
2.2.5
Schlieren Comparisons
Schlieren3 visualization techniques are widely used in the Aerospace industry for
flow visualization in all flow regimes and were invented by Robert Hooke. An excellent introductory overview of visualization techniques for fluid dynamics using
schlieren techniques, with a historical perspective, including other visualization
techniques and their strengths is presented by Settles. [83] Schlieren techniques
are characterized by producing parallel rays of visible light. This light then travels through a flow-field that has density gradients (compressibility). The density
gradients bend the beams of parallel lights due to refraction. A mirror focuses the
resultant altered beams and some of them are removed with a ?knife edge? because
their trajectory has been changed due to the density gradients. The light rays that
do not collide with the knife edge are captured with a visualization device such
as a camera. For the schlieren images shown below a Z-Type schlieren system is
used. The specific techniques that produce schlieren for these comparisons, and
a detailed overview and setup of the experiments themselves, are given by Veltin
and McLaughlin. [84] The schlieren shown in this dissertation were produced by
Dr. Jeremy Veltin.
For a qualitative comparison of the experimental schlieren images it is necessary to produce the same images from the Wind-US CFD solutions. The Z-type
schlieren essentially produces density gradients in the cross-stream direction of the
fluid flow. Therefore, it is a simple process to take the derivative of the density
in the cross-stream direction from the numerical solutions. Essentially, if y is the
cross-stream direction of the fluid flow then the data for an artificial or numerical schlieren are produced by performing the simple operation of d?/dy using the
Tecplot CFD analysis package. The numerical solutions, which are either axisymmetric or have planes of symmetry, are mirrored on the center axis or a plane of
symmetry so that the entire flow-field is visible for a comparison. This allows for
plotting of contours across the centerline or the plane of symmetry. The sign of the
gradient of density below the centerline or plane of symmetry is then changed. This
is done to simulate the inversion of the light gradients observed in the experiments.
The numerical contours of d?/dy are plotted over a very small range of negative
3
Schlieren from the German word schliere, often confused as the inventors name. It is the
plural of schliere meaning streak.
77
one to one with one hundred intervals. This is performed to emphasize the smallest
features of the density gradients and to magnify any possible imperfections in the
numerical simulation.
The converging axisymmetric jet case of Md = 1.00, Mj = 1.50, and T T R =
1.00, is shown in Figure 2.27 part a). The top image in the figure represents the
experimental schlieren result and the bottom image shows the numerical schlieren.
The axes of the figures are non-dimensionalized by the diameter of the nozzle D.
The flow moves from left to right in the positive x direction. The lips of the nozzle
at y/D = ▒0.5 are evident by the shear layer spreading from these locations. The
sharp demarcation between dark and light regions emanating from the nozzle lip
and ending on the jet centerline indicates the end of the Prandtl-Meyer expansion
fan and the initiation of compression waves caused by the interaction of the expansion waves with the free shear layer. Additionally, the shock waves and their
corresponding angles can be seen as strong gradients of d?/dy. The first conical
oblique shock originates at the same location in both experiment and simulations
at approximately x/D = 1.1 near the centerline. The second oblique shock wave
occurs at approximately x/D = 2.5. In both the experimental and numerical
schlieren, the origin of the shock waves are the same. By inspection, the angles
of both the experimental and numerical oblique shock waves agree. This shows
that the calculated and experimental shock wave strengths are equal. However,
a slight discrepancy occurs at approximately x/D = 1.1, at the jet center. The
experimental visualization shows a very short normal shock, prior to the oblique
shock waves. This normal shock creates a small subsonic region downstream, delimited by slip lines. This phenomenon can also be referred to as a barrel shock.
Any increase in the nozzle pressure ratio shows an increase of this subsonic region.
The numerical results only start showing this phenomenon when the simulated
nozzle pressure ratio is increased slightly to produce Mj = 1.53. This value was
found by incrementing the N P R until a normal shock was noticeable. It could
also be created artificially by increasing the grid spacing along the centerline. The
sensitivity to grid spacing near the centerline is found to be as important as the
lipline grid density for this reason.
Figure 2.27 part b), shows a comparison between the experimental and numerical schlierens for the converging-diverging axisymmetric jet of Md = 1.50,
78
Mj = 1.50, and T T R = 1.00. The arrangements of the shocks are very different
because, unlike the converging nozzle case, an oblique shock wave originates at
the nozzle lip and terminates as a barrel shock. The appearance of the more typical conical oblique shocks downstream ensues. A qualitative comparison of the
position and angles of the shock waves show good agreement. In particular, the
location of the normal shock just downstream of the exit of the nozzle are in the
same position. The slip stream downstream of the normal shock is seen clearly in
the simulations.
Figure 2.27. Comparison between the experimental and numerical schlieren of a) the
converging nozzle case Md = 1.00, Mj = 1.50, b) the converging-diverging nozzle case
Md = 1.50, Mj = 1.30. The nozzle exit is at x/D = 0.0 and the flow moves from
x/D = 0.0 to the right. The nozzle centerline is at y/D = 0.0 and the nozzle lips at
y/D = 0.5.
The PSU small scale jet anechoic facility does not have the ability to heat the air
in the plenum that subsequently emerges from the nozzle. Instead, helium is mixed
with compressed air in the plenum before it exits from the nozzle. This creates
a lower density mixture of helium and air, which simulates heated jets, where
T T R > 1.00. To help validate the acoustic results of heat-simulated jets from this
facility, and to ensure that they compare favorably to hot jet simulations, CFD
79
calculations have been carried out between a Md = 1.00, Mj = 1.47, N P R = 3.52,
T T R = 2.20 and a Md = 1.00, Mj = 1.47, T T R = 1.00 helium / air mixture.
This helium / air simulation for supersonic jets operating off-design is the first
ever conducted. The specific values of the helium / air mixture jet are p? = 98.00
kP a, T? = 294.3K, R = 703.5 m2 K/s2 , ? = 1.55, and the density of the mixture
is ? = 0.756 kg/m3 . Both of these simulations used the converging computational
grid discussed in section 2.1.2. Note that the helium / air mixture is used to
simulate a T T R = 2.20 but the T T R specified in the numerical simulation is 1.00.
In the heated air simulation only hot air exits from the converging nozzle, while
in the helium / air mixture simulation the mass fraction of helium to air in the
nozzle is 0.2326. Figure 2.28 shows schlieren comparisons of these two numerical
simulations against a Z-type schlieren of the helium / air experimental jet. The
top of a) and b) are the same image: the helium / air simulated jet. The bottom
image of a) is a numerical schlieren of the hot air jet while the bottom image
of b) is the numerical schlieren of the helium / air jet. Clearly, the numerical
result of the helium / air mixture jet compares extremely well with that of the
experimental schlieren. There is a slightly different shock cell structure in the hot
air simulations relative to the helium / air numerical result and the schlierens. The
shock cell structure in the helium / air mixture jet has been shortened compared
to the hot air results. However, in section 2.2.8 it will be shown that more subtle
differences occur according to contour plots of M and plots of various quantities.
Z-type schlieren comparisons are made for the rectangular jets with Md = 1.50
that correspond to the same conditions described for Pitot probe comparisons.
In addition, a comparison is made through the major axis for the same overand under-expanded cases with a T T R = 2.20. These schlierens are conducted
through the highly three-dimensional flow-fields of the rectangular jets while the
numerically generated schlierens are generated from the major and minor axis
plane data alone. This causes some discrepancy in the qualitative comparisons
because the numerical schlierens are not able to capture the effect of the density
gradients of the main major and minor axis. It is not possible to plot contributions
from multiple planes without transparency. The numerical schlieren in the major
axis plane is generated by taking density gradients in the z direction and numerical
schlieren in the minor axis plane is formed by taking density gradients in the y
80
Figure 2.28. Comparison between the experimental and numerical schlieren of the
converging nozzle case Md = 1.00, Mj = 1.50 a) T T R = 2.20 air, schlieren on top and
numerical schlieren on bottom, b) top: schlieren of T T R = 2.20 air. bottom: numerical
schlieren of helium / air mixture. The nozzle exit is at x/D = 0.0 and the flow moves
from x/D = 0.0 to the right. The nozzle centerline is at y/D = 0.0 and the nozzle lips
are at y/D = 0.5.
direction. These directions should conform to those of the experimental schlieren.
Numerical data is mirrored across the centerline axis to form an entire plane and
then the density gradient in the lower half of the corresponding plane is multiplied
by negative one for the gradients of density to conform to experimental ones.
The major and minor axis experimental and numerical schlieren of the rectangular Md = 1.50, Mj = 1.30, T T R = 1.00 jet is shown in Figure 2.29. Part a)
shows the minor axis plane with the numerical schlieren under the experimental
one and part b) shows the major axis plane with the same configuration. The
major axis plane shows excellent qualitative agreement of spreading rate, shock
position, and shock angles with respect to the experimental result. Less agreement can be seen in the minor axis plane. There is significant interference in the
picture of density gradients of the minor axis plane in the z direction. Figure 2.30
shows the rectangular Md = 1.50, Mj = 1.70, T T R = 1.00 schlieren comparisons
81
in the same layout as Figure 2.29. Once again the major axis plane in part b)
shows overall excellent agreement in terms of the spreading rate, shock position,
and shock angles. Comparisons for the minor axis plane in part a) show much
better agreement than the corresponding over-expanded cold rectangular jet. In
particular, the first expansion waves and corresponding downstream oblique shocks
through both the major and minor axis compare almost perfectly.
Finally, comparisons are shown for experimental and numerical schlieren of
the helium / air simulated T T R = 2.20 rectangular jets operating at Md = 1.50,
Mj = 1.30 and Mj = 1.70. Numerical schlierens are produced only for the major
axis because experimental data is only available for this plane. Figure 2.31 shows
both the over- and under-expanded qualitative comparisons of the helium / air
simulations. Part a) shows the Mj = 1.30 rectangular jet major axis and part b)
shows the Mj = 1.70 rectangular jet major axis. Note that the simulations are
performed with an actual T T R = 2.20, unlike the experiments, which are heat
simulated with the helium / air mixture. Unfortunately, the experimental plots
in the top half of the figure are heavily populated by the turbulent eddies as long
time averaging of the helium / air jets is not possible due to the relative expense
of helium / air experiments. Very little can be observed of the jet shock cell
structure by comparing the plots in part a). However, in part b) the first oblique
shock wave is easily discernable in the experiment and the numerical result and
compares favorably.
82
Figure 2.29. Comparison between the experimental and numerical schlieren of the
rectangular nozzle case Md = 1.50, Mj = 1.30, and T T R = 1.00. a) The minor axis plane
top: schlieren, bottom: numerical schlieren. b) The major axis plane, top: schlieren,
bottom: numerical schlieren. The nozzle exit is at x/De = 0.0 and the flow moves from
x/De = 0.0 to the right.
Figure 2.30. Comparison between the experimental and numerical schlieren of the
rectangular nozzle case Md = 1.50, Mj = 1.70, and T T R = 1.00. a) The minor axis plane
top: schlieren, bottom: numerical schlieren. b) The major axis plane, top: schlieren,
bottom: numerical schlieren. The nozzle exit is at x/De = 0.0 and the flow moves from
x/De = 0.0 to the right.
83
Figure 2.31. Comparison between the experimental and numerical schlieren of the
rectangular nozzle case Md = 1.50 and T T R = 2.20. The experimental results use a
helium / air mixture and the simulations use heated air. a) Mj = 1.30 Major axis
plane top: schlieren, bottom: numerical schlieren. b) Mj = 1.70 major axis plane. top:
schlieren. bottom: numerical schlieren. The nozzle exit is at x/De = 0.0 and the flow
moves from x/De = 0.0 to the right.
84
2.2.6
Off-Design Study
To further validate the CFD solutions for the BBSAN prediction model, a shock
strength study has been conducted. Using the same geometry as the convergingdiverging validation case, simulations are conducted at off-design conditions from
Mj = 1.60 to Mj = 2.50 in 0.1 increments, creating a total of 10 new cases for
study. Besides a different pressure specification at the inlet boundary condition
seen in Figure 2.1, which corresponds to the appropriate NPR, all other simulation
parameters are the same. By varying Mj , the off-design parameter ? changes
accordingly as Md = 1.50 remains constant. As in all off-design cases of supersonic
jets oblique shock waves originate from inside the core of the jet. When Mj is larger
than Md an expansion originates at the nozzle exit followed by an oblique shock.
The pressure difference across this first oblique shock can easily be found when
post-processing the CFD results. A simple model for shock associated noise, such
as the model of Harper-Bourne and Fisher [1], suggests that the noise should scale
depending on both the shock cell strength, where it intersects the shear layer, and
the typical velocity fluctuation in the shear layer. The resulting intensity would
depend on the square of this product. Using the pressure difference across the
first oblique shock and the turbulent kinetic energy in the shear layer, K, the
strength of the shock-associated noise source can be estimated as K?p2 . If the
logarithm of K?p2 at the first oblique shock/shear layer intersection is compared
with the logarithm of the variation of the off-design parameter then a nearly linear
relationship results. Figure 2.32 shows the logarithm of K?p2 as a function of
? for the validation converging-diverging nozzle. The slope of this relationship is
almost exactly 4.0 as Harper-Bourne and Fisher [1] and Tam and Tanna [49] have
shown for cold jets.
85
Figure 2.32. Variation of the logarithm of shock strength, K?p2 (dots), compared
with the off-design parameter ? 4 (line) offset slightly. The logarithm of ? 4 has a nearly
identical slope with K?p2 .
86
2.2.7
Laminar and Turbulent Flow Nozzle
Some of the conditions at which the nozzles operate have a low Reynolds number
because of their small size and inflow conditions. This implies that the flow inside
the nozzle or exiting the nozzle may not be turbulent but laminar. One such
nozzle and condition that has been tested is the convergent divergent nozzle with
Md = 1.00, Mj = 1.30, T T R = 2.20, and D = 0.0127 m. Under these conditions
the Reynolds number of the flow is Re = 337, 000. This low Reynolds number
can allow for laminar flow to exist fully inside the nozzle and to transition to
turbulence downstream of the nozzle exit due to the Kelvin-Helmholtz instability.
Since it is not known exactly where the flow transitions from laminar to turbulent,
it is assumed as a test case that transition occurs at the nozzle exit. If a flow is
laminar then it is modeled by the Navier-Stokes equations. If a flow transitions
from laminar to turbulent at the nozzle exit, the need arises to solve both the
RANS and Navier-Stokes equations in different regions of the flow. In this case, the
laminar equations of motion are solved inside the nozzle and the RANS equations
are solved everywhere outside the nozzle.
Figure 2.33 shows contours of Mach number for the jet with transition at the
nozzle exit. The top contour plot shows the solution given by RANS in the entire
domain and the bottom contour plot shows the laminar / RANS jet. The most
obvious difference between the two numerical solutions is that the oblique shock
wave in the RANS solution is outside the nozzle and formed at the nozzle lip. In
the laminar / RANS case characteristic waves are able to coalesce and form an
oblique shock inside the nozzle at approximately x/D = ?0.1. This difference in
origin of the first oblique shock changes the entire shock cell structure. The laminar
/ RANS case has a separation or recirculating region after the oblique shock forms
on the inside wall of the nozzle near the exit. This is in contrast to the RANS
simulation, which has an attached turbulent boundary layer along the entire nozzle
wall. The slipstream in the fully RANS case, that is created by the barrel shock,
is much stronger than in the laminar / RANS case. This observation is due to
the lower effective Dj in the transition case. By carefully comparing both cases, it
can be observed that the shock cell structure itself is very similar but shifted by a
small amount x/D in the negative streamwise direction for the RANS case relative
to the laminar / RANS case. Interestingly, nozzle exit velocities of the laminar /
87
RANS case are over 140 m/s lower than the RANS case on the centerline.
Figure 2.33. Comparison of Mach number contours for the Md = 1.50, Mj = 1.30,
T T R = 2.20, and D = 0.0127 m jet. Top: Fully turbulent flow. Bottom: Laminar flow
inside the nozzle and turbulent flow outside the nozzle.
Figure 2.34 shows two plots along the centerline of the jet. The top plot is M
and the bottom is p/po . In each case both the fully turbulent RANS and laminar
/ RANS solution are shown. Note, that the exit of the nozzle is at x/D = 0.0, so
part of each plot is inside the nozzle itself. The qualitative observation made in
Figure 2.33 that the shock cell structures appear very similar is even more apparent
in the plots of M and p/po . The very sharp vertical lines of M and p/po inside
the nozzle of the laminar / RANS jet and outside the nozzle of the RANS jet are
due to shock waves. As expected, in the first plot the Mach number is greater
than one before a shock and reduces to a value lower than one after the shock, and
a corresponding jump in static pressure also occurs at the same x/D location. If
either variable, or another flow-field variable of the laminar / RANS case, is shifted
by a value of x/D = 0.30 then the two solutions, at least in terms of the shock cell
structure, would be almost identical. This would exclude very small changes in
amplitudes between the two cases. Both sets of data show that the solutions inside
88
the nozzle are identical up until x/D = ?0.1. Not coincidentally, this is where
the waves coalesce in the laminar nozzle. The spacing of the shocks is generally
unchanged between the laminar and turbulent simulations on the centerline.
The values of turbulent kinetic energy and streamwise velocity on the lipline
are important for BBSAN intensity scaling. Figure 2.35 shows values of K and u
on the lipline of the nozzle, y/D = 0.50. Both K and u have identical trends just
after the nozzle exit but with the same shift as seen on the centerline. However,
the turbulent nozzle simulation shows higher K and u at the exit of the nozzle.
This is due to the recirculating region just inside the nozzle lip at the nozzle wall
of the laminar nozzle and the existence of the oblique shock at the nozzle lip of
the fully RANS jet.
Figure 2.34. Centerline values for a Md = 1.50, Mj = 1.30, T T R = 2.20, D = 0.0127
m jet. Top: Variation of Mach number. Bottom: Variation of pressure.
89
Figure 2.35. Lipline values (y/D = 0.50) for a Md = 1.50, Mj = 1.30, T T R = 2.20,
D = 0.0127 m jet. Top: Variation of turbulent kinetic energy. Bottom: Variation of
streamwise velocity.
90
2.2.8
Helium and Hot Jet Comparisons
Additional comparisons are made of the helium / air mixture jet and hot air jet
of T T R = 2.20. The parameters and details of this simulation were presented
in section 2.2.5. Here, additional quantitative comparisons are made between the
numerical simulations for the two jets. Figure 2.36 shows contours of Mach number
of the two jets. The top half plane above y/D = 0.0 represents the contours of
M of the hot air jet and the bottom plane below y/D = 0.0 shows the helium /
air mixture contours of M . The contours are almost identical near the nozzle exit,
especially at the centerline. Far downstream the two numerical simulations show
some subtle differences in spreading rate and positions of the shock waves on the
center line. This is most likely due to a slight variation in the rates of diffusion
between a helium / air mixture and a hot air jet.
Figure 2.36. Comparison of Mach number contours for Md = 1.00, Mj = 1.50, T T R =
2.20 air or T T R = 1.00 helium/air, D = 0.0127 m jet. Top: Contours of M of hot air.
Bottom: Contours of M of the helium / air mixture.
Centerline data has been extracted from the two numerical simulations of the
helium / air and heated air jets. Two important variables in particular have
been extracted from the centerline and are shown in Figure 2.37. The top plot
shows the density variation in the jet and the bottom plot shows the variation
of the streamwise velocity component, u. These values have not been normalized
and remain dimensional. The raw values in this case help to illustrate how very
similar these jets are. Data produced from these simulations are used to perform
a prediction of BBSAN to illustrate the ability to use helium / air jets to simulate
91
hot jets for aeroacoustic applications.
This chapter has presented various equations of motion that model steady fluid
flow of off-design supersonic jets. These equations have been solved by Wind-US
and subsequently validated against experimental results from various sources. The
validation involved comparisons of predicted values of pressures and Mach number
with experimental data found from Pitot measurements. Qualitative comparisons
of predicted and Z-type schlieren were also shown for various jets. The jets examined included heated and unheated, helium / air, and transitional. Also, both rectangular and circular nozzles were validated. In the next chapter the development,
implementation, and calibration of the BBSAN model is shown. Predictions based
on the validated RANS solutions of this chapter, along with additional RANS solutions for other jets, are used by the developed BBSAN model. These predictions
are compared with experimental data from various sources and the predictions of
Tam for a wide range of jet conditions and nozzle geometries.
92
Figure 2.37. Centerline values for a Md = 1.50, Mj = 1.30, T T R = 2.20 or T T R = 1.00
helium / air mixture, D = 0.0127 m jet. Top: Variation of density, ?. Bottom: Variation
of centerline velocity, u.
Chapter
3
Broadband Shock-Associated Noise
Thus far an overview of the current prediction models of BBSAN and validation
for the RANS CFD of supersonic jets operating off-design have been presented.
This chapter is presented in three main parts. The first part of the chapter is the
mathematical development of the BBSAN model where the prediction method is
developed into a single mathematical formula for the spectral density in the farfield. The second part discusses the implementation of the mathematical model
and subsequent parametric studies. Finally, in the third part, the prediction results
over a wide range of jet conditions are presented both for the CFD validation cases
developed in the previous chapter and also using additional RANS CFD solutions.
These BBSAN predictions are compared with the prediction methods of Tam and
with experimental data. Conditions are chosen to include a wide range of nozzle
shapes, pressure ratios, and temperature ratios to test the capability of the model.
3.1
Model Development
The BBSAN model builds on the analysis developed by Tam [57]. Tam?s analysis is
considerably simplified if the following form of the inviscid compressible equations
of motion are used.
D? ?vi
+
=0
Dt
?xi
(3.1)
94
Dvi
??
+ a2
=0
Dt
?xi
(3.2)
D
?
?
=
+ vi
Dt
?t
?xi
(3.3)
where,
where a is the local speed of sound, t is time, and vi are the velocity components
in the xi directions of a Cartesian coordinate system. ? is related to the logarithm
of the pressure,
?=
1
ln (p/p? )
?
(3.4)
where p is the pressure, p? is the ambient pressure, and ? is the ratio of specific
heats of an ideal gas. Following Tam [57], the instantaneous flow-field properties
are separated into four components. That is,
? ? ?
?
?
?? + ?s + ?t + ? 0
? ?=?
?
0
vi
v?i + vsi + vti + vi
(3.5)
where the overbar denotes the long time averaged value, the subscript s denotes the
perturbations associated with the shock cell structure, the subscript t denotes the
fluctuations associated with the turbulence, and the primes denote the fluctuations
generated by the interaction of the turbulence and the shock cell structure. In addition, it is assumed that the unsteady linearized version of these equations is also
satisfied by the turbulent velocity fluctuations. This is justified if the important
components of the turbulence, so far as the BBSAN is concerned, are coherent over
relatively large axial distances. These components are described well by a linear
instability wave model. It will be assumed that the shock cell structure satisfies
the steady linearized version of Equations 3.1 and 3.2. That is,
??s ?vsi
+
=0
?xj
?xi
(3.6)
?v?i
?vsi
??s
+ v?j
+ a?2
=0
?xj
?xj
?xi
(3.7)
v?j
vsj
95
The shock cell structure is assumed to be steady in the model even though the
shock cell structure oscillates slightly in the jet. This could be improved later if
necessary. It is also assumed that the mean static pressure is constant throughout
the flow-field. The mean pressure in the jet is the sum of the ambient pressure, p? ,
and the steady perturbations, ?s . Furthermore, it is assumed that to linear order
?? = 0. Finally, it is assumed that the turbulence responsible for the generation of
BBSAN consists only of the large-scale structures, as BBSAN requires significant
coherence lengths, and they also satisfy the linearized equations. This assumption
is consistent with the ?instability wave? model of the turbulence. Making these
assumptions yields,
??t
??t ?vti
+ v?j
+
=0
?t
?xj
?xi
(3.8)
?vti
?vti
?v?i
??t
+ v?j
+ vtj
+ a?2
=0
?t
?xj
?xj
?xi
(3.9)
Next the decomposition given by Equation 3.5 is substituted in Equations 3.1
and 3.2. Use is also made of Equations 3.6 through 3.9. This gives,
?v 0
?? 0
?? 0
??t
??s
+ v?j
+ i = ?vsj
? vtj
?t
?xj ?xi
?xj
?xj
?vi0
?v 0
?v?i
?? 0
?vti
?vsi
??t
??s
+ v?j i + vj0
+ a?2
= ?vsj
? vtj
? a2s
? a2t
?t
?xj
?xj
?xi
?xj
?xj
?xi
?xi
(3.10)
(3.11)
The additional terms contribute to the development of the turbulence and the
mean flow, which are found from the RANS CFD solutions. Clearly, this also
requires the inclusion of viscous terms in the mean flow calculation though they
are not crucial for the BBSAN generation. The operators on the left hand side of
Equation 3.10 and 3.11 are the linearized Euler equations. The following definitions
are introduced,
?vsj
??t
??s
? vtj
=?
?xj
?xj
(3.12)
96
?vsj
?vti
?vsi
? vtj
= fiv
?xj
?xj
(3.13)
?a2s
??s
??t
? a2t
= fia
?xi
?xi
(3.14)
Making these definitions, the inhomogeneous equations for the fluctuations associated with the interaction of the turbulence with the shock cells can be written,
?? 0
?? 0
?v 0
+ v?j
+ i =?
?t
?xj ?xi
(3.15)
?vi0
?v 0
?v?i
?? 0
+ v?j i + vj0
+ a?2
= fiv + fia
?t
?xj
?xj
?xi
(3.16)
where ? is a dilatation rate generated by the interaction between the pressure
gradients and the turbulent velocity perturbations and the shock cells. fiv is the
unsteady force per unit volume associated with interactions between the turbulent velocity fluctuations and the velocity perturbations associated with the shock
cells. Finally, fia is the unsteady force per unit volume related to the interaction
of fluctuations in the sound speed (or temperature), caused by the turbulence and
the shock cells, and the associated pressure gradients. In traditional approaches to
turbulence mixing noise models these equivalent sources have been treated separately and the same assumption will be made here. The solution to Equations 3.15
and 3.16, can be written in terms of the vector Green?s function that satisfies the
equations,
n
??gn
??gn ?vgi
+ v?j
+
= ? x ? y ? (t ? ? ) ?0n
?t
?xj
?xi
(3.17)
n
n
?vgi
?vgi
??gn
n ?v?i
2
+ v?j
+ vgj
+ a?
= ? x ? y ? (t ? ? ) ?in
(3.18)
?t
?xj
?xj
?xi
n
n
where ?gn = ?gn x, y, t ? ? and vgi
= vgi
x, y, t ? ? are the components of the
vector Green?s function, x denotes the observer position, y denotes the source
location, ?( ) is the Dirac delta function, and ? is the source emission time. ?ij is
the Kronecker delta function. In the far-field,
97
1
? = ln
?
p? + p0
p?
(3.19)
since ?s and ?t vanish for the jet. Also, if p0 << p? ,
p0
p0
=
?p?
?? a2?
?0 '
(3.20)
Then the solution for p0 (x, t) in the far-field is given by,
0
p (x, t) =
?? a2?
Z
?
Z
...
??
+
3
X
?
?gn
?g0 x, y, t ? ? ? y, ?
??
x, y, t ? ?
(3.21)
[fnv
+
fna ]
y, ? d? dy
n=1
Let the periodic Green?s function be defined by,
?gn
1
x, y, t ? ? =
2?
?gn
Z
Z
?
?gn x, y, ? exp [?i? (t ? ? )] d?
(3.22)
??
?
x, y, ? =
?gn x, y, t ? ? exp [i? (t ? ? )] dt
(3.23)
??
From this point, only the source term associated with the velocity perturbations
will be considered. It is expected that the scaling of the other source terms would be
similar. The exception would be the source term associated with the temperature
fluctuations. However, the importance of this term remains the subject of debate
in the prediction of turbulent mixing noise in heated jets. So, for the moment, this
term will not be considered further. Then, the far-field pressure is given by,
?
Z? X
3 X
3
2 4 Z
?
a
?
?
?gn x, y, ?1 ?gm (x, z, ?2 )
p0 (x, t) p0 (x, t + ? ? ) =
...
2
(2?)
??
?? n=1 m=1
v (z, ? ) exp [?i? (t ? ? ) ? i? (t ? ? ) ? i?? ? ]
Оfnv y, ?1 fm
2
1
1
2
2
Оd?1 d?2 d?1 d?2 dydz
The spectral density is related to the autocorrelation by,
(3.24)
98
Z
?
p0 (x, t) p0 (x, t + ? ? ) exp [i?? ? ] d? ?
S (x, ?) =
(3.25)
??
The spectral density is found from the autocorrelation. Making use of the
integral,
Z
?
exp [i (? ? ?2 ) ? ? ] d? ? = 2?? (? ? ?2 )
(3.26)
??
yields the spectral density,
?2 a4
S (x, ?) = ? ?
2?
Z
?
Z
?
...
??
3 X
3
X
?gn x, y, ?1 ?gm (x, z, ?2 )
?? n=1 m=1
Оfnv y, ?1 fnv (z, ?2 )? (? ? ?2 ) exp [?i (?1 + ?2 ) t]
(3.27)
О exp [i?1 ?1 + i?2 ?2 ] d?1 d?2 d?1 d?2 dydz
fnv is dependent on the strength of the shock cells and the turbulent fluctuations
and its product is significant in regions where the shocks and expansions intersect with the turbulent shear layer. That is, if there is no turbulence present or
pressure perturbation due to shock cells, then the term is small. Furthermore,
the amplitude of fnv is proportional to the shock cell pressure perturbations and
the turbulent velocity fluctuations. The two-point cross correlation function of the
BBSAN source term is given by,
v
v (z, ? )
Rnm
(y, ?, ? ) = fnv y, ?1 fm
2
(3.28)
where ? is a vector between two source positions, ? = z ? y, and ? = ?2 ? ?1 .
This is consistent with the statistics of the turbulence as a local function of the
separation distance and time delay between two source locations. This yields,
?2 a4
S (x, ?) = ? ?
2?
Z
?
Z
...
??
?
3 X
3
X
?gn x, y, ?1 ?gm (x, z, ?2 )
?? n=1 m=1
v
ОRnm
y, ?, ? ? (? ? ?2 ) exp [?i (?1 + ?2 ) t]
О exp [i (?1 + ?2 ) ?1 + i?2 ? ] d?1 d?2 d?1 d?2 dydz
(3.29)
99
Now, the integrations with respect to ?1 , ?1 , and, ?2 can be performed using,
S (x, ?) =
?2? a4?
Z
?
Z
?
...
??
3 X
3
X
?gn x, y, ?? ?gm x, y + ?, ?
?? n=1 m=1
(3.30)
v
ОRnm
y, ?, ? exp [i?? ] d?dy
BBSAN is known to radiate at larger angles to the jet downstream axis. In those
directions the propagation of the sound through the jet flow is minimally affected by
the mean velocity and temperature gradients. In addition, the regions of strongest
interaction between the turbulence and the shocks cells will occur at the center
of the jet shear layer. The Green?s functions could be calculated numerically
for a given mean flow. This could involve a locally parallel approximation or
the full diverging flow. Also, the problem could be formulated in terms of the
adjoint Green?s function for the linearized Euler equations as described by Tam
and Auriault [85]. However, BBSAN is radiated predominantly at large angles to
the jet downstream axis where the refractive effects of the mean flow would be small
or absent. In view of this, the Green?s function is approximated with the absence
of a mean flow as shown in Appendix C. The components of the vector Green?s
function are readily related to the Green?s function of the Helmholtz equation.
This gives,
?gn x, y, ? =
i? xn
x ? y /a?
exp
i?
4?a3? x x
(3.31)
where xn is the nth component of x, and x is the magnitude of x. Also, if y << |x|,
?gm
x, y + ?, ? =
?gm
?x
x, y, ? exp ?i
и?
a? x
(3.32)
With the use of these approximate forms for the far field Green?s functions, the
spectral density can be written as,
?2? ? 2
S (x, ?) =
16? 2 a2? x2
?
xn xm v
...
R
y,
?,
?
nm
x2
??
??
?x
и ? d? d?dy
О exp i?? ? i
co x
Z
?
Z
(3.33)
100
In Equation 3.33 the summation is implied over the indices n and m.
v
. From the continuity equations for the shock
All that remains to model is Rnm
cell structure Equation 3.6,
v?j
??s ?vsi
+
=0
?xj
?xi
(3.34)
Thus, it is assumed that the velocity perturbations associated with the shock
cell structure are proportional to the local mean velocity multiplied by the perturbation in ?. That is,
u?
?s vti
l
fiv ?
(3.35)
where l is the characteristic length scale, taken to be the same for the shock cell
structure and the turbulence. From the form of fiv given by Equation 3.13, and on
dimensional grounds, it is assumed to scale as,
fiv ?
ps v t
? ? a? l
(3.36)
where ps represents the shock cell strength, and vt is a characteristic turbulent
velocity fluctuation. These variables will be determined from the RANS CFD
solution. Also, for simplicity, use is made of the Proudman form for the cross
correlation. That is,
v
where Rxx
x n xm v
v
R
y,
?,
?
=
R
y,
?,
?
(3.37)
nm
xx
x2
y, ?, ? = fxv y, t fxv y + ?, t + ? , and fxv is the component of fiv
in the direction of the observer. This simplification amounts to assuming that the
source term is isotropic. Since the BBSAN radiation is dominant at large angles to
the jet axis, this is a reasonable assumption. But it could be relaxed in the future
with other models for the source statistics. Use of these relationships gives,
v
Rxx
y, ?, ? =
where,
1
ps
?2? a2? l2
y ps y + ? Rv y, ?, ?
(3.38)
101
Rv (y, ?, ? ) = vx y, t vx y + ?, t + ?
(3.39)
is the two-point cross correlation function of the turbulent velocity fluctuations in
the observer direction. Then the far field spectral density can be written as,
Z?
?2
S (x, ?) =
16? 2 a4? x2
Z?
1
p
y
+
?
p
y
s
s
l2
??
??
xи?
v
ОR (y, ?, ? ) exp i? ? ?
d? d?dy
xa?
иии
(3.40)
It is convenient to introduce the cross spectral density of the turbulent velocity
fluctuations. This enables the turbulent velocity statistics to be characterized
either in terms of the cross correlation or cross spectral density. Both have been
used in the modeling of turbulent mixing noise. The cross spectral density is given
by,
S
v
Z?
y, ?, ? =
Rv y, ?, ? exp (i?? ) d?
(3.41)
??
with,
R
v
1
y, ?, ? =
2?
Z?
?
S v y, ?, ? ? e?i? ? d? ?
(3.42)
??
Then,
?2
S (x, ?) =
32? 3 a4? x2
Z?
Z?
1
ps y ps y + ? S v (y, ?, ? ? )
2
l
??
??
?i?x
?
О exp [?i? ? ] exp [i?? ] exp
и ? d? ? d? d?dy
a? x
...
(3.43)
The integration with respect to ? and then ? ? can be performed as before since,
102
Z?
exp [?i (? ? ? ?) ? ]d? = 2?? (? ? ? ? )
(3.44)
??
So that,
?2
S (x, ?) =
16? 2 a4? x2
Z?
Z?
1
ps y ps y + ?
2
l
??
??
?i?x
?
y, ?, ? exp
и ? d?dy
a? x
ОS v
...
(3.45)
In order to emphasize the quasi-periodic nature of the shock cell structure and
to assist in the implementation of the model, the axial spatial Fourier transform
of the shock cell?s pressure perturbation is defined. It is given by,
1
ps y =
2?
Z?
p?s (k1 , y2 , y3 ) exp [ik1 y1 ] dk1
(3.46)
??
with
Z?
p?s (k1 , y2 , y3 ) =
p?s y exp [?ik1 y1 ] dy1
(3.47)
??
where k1 is the wavenumber in the axial direction, y1 . These relationships can be
substituted into Equation 3.40. It should be noted that the axial Fourier transform
of the shock cell pressure perturbation is only applied to one of the two terms in
the integrand. Performing only one transform of the shock pressure results in
the evaluation of the BBSAN to be more convenient in terms of computational
efficiency. After some simplification, the spectral density is found to be given by,
?2
S (x, ?) =
32? 2 a4? x2
Z?
Z?
иии
??
??
ОS v
1
ps y p?s (k1 , y2 , y3 ) exp [ik1 (y1 + ?)]
2
l
xи?
y, ?, ? exp i? ? ?
dk1 d?dy
xa?
with, ? = (?, ?, ?). A model is now proposed for Rv y, ?, ? in the form,
(3.48)
103
Rv y, ?, ? = K exp [? |? | /?s ] exp ? (? ? u?c ? )2 /l2
h
2 2 i
О exp ? ? 2 + ? 2 /l?
(3.49)
where ?s is the turbulent time scale, l? is the turbulent length scale in the crossstream direction, and K is the turbulent kinetic energy. The scales, ?s , l, l? , are
found directly from the CFD RANS solution. Then,
?2
S (x, ?) =
32? 3 a4? x2
Z?
Z?
K
p
y
p?s (x, y, k) exp [ikz + ik? + i?? ]
s
l2
??
??
h
2i
2
2
2
2 2
О exp ? |? | /?s ? (? ? u?c ? ) /l ? ? + ? /l?
?i?x
v
ОS y, ?, ? exp
и ? dkd?dy
a? x
...
(3.50)
If x is written x/x = (sin ? cos ?, sin ? sin ?, cos ?) then the following integral needs
to be evaluated,
I??
Z? Z? i?
2 2
=
sin ? cos ?
exp ?? /l? ?
a?
?? ??
i?
2 2
О exp ?? /l? ?
sin ? sin ?? d?d?
a?
(3.51)
The evaluation of I?? is shown in Appendix B. Making the substitution of the
evaluated integral I?? yields,
Z? Z? 2
Kl?
?2
S (x, ?) =
...
ps y p?s (x, y, k) exp [ikz]
2
4
2
2
32? a? x
l
??
??
"
#)
? |? | (? ? u?c ? )2
i? cos ??
О exp
?
+ i?? ?
+ ik?
dkd? d?dy
?s
l2
a?
(3.52)
The integral over ? and ? now needs to be evaluated. This integral is defined as
I?? and is shown in Appendix B. Once this integration is performed and substituted
into the above equation the final model for the spectral density is found,
104
Z?
Z? 2
1
Kl?
ps y p?s (k1 , y2 , y3 ) exp (ik1 y1 )
S (x, ?) = ? 4 2
иии
l?s
? ?a? x
??
??
)
2
? 2 ?s2 exp ?l2 (k1 ? ? cos ?/a? )2 /4 ? ? 2 l?
sin2 ?/4a2?
О
dk1 dy
1 + (1 ? Mc cos ? + u?c k1 /?)2 ? 2 ?s2
(3.53)
Equation 3.53 provides the prediction formula for the BBSAN. All of the parameters can be determined from a RANS CFD solution. The equations developed thus far can be applied to three-dimensional flow fields. In the case of an
axisymmetric jet, the integrations with respect to the cross stream direction can
be reduced to a single integration in the radial direction. The implementation of
this prediction model is described in the next section.
3.2
Implementation
The BBSAN integration model, shown in Equation 3.53, is implemented in a Fortran 90 program and linked with the common flow libraries. The Fortran 90 code
conforms to the Fortran 95 standard ISO 1539-1997 [86] outlined by the J3 Fortran
committee. The program is compiled with the Intel Fortran compiler for 32-bit applications, version 9.1 build 20060323z. The BBSAN code is a stand-alone program
that is designed to load Wind-US structured grid solutions in their native format
(or from another CFD program that provides output format in common flow libraries as structured grids). It outputs values of SP L per unit St (dB/[20хP a2 ])
at each prescribed observer location. Figure 3.1 shows an overview of the BBSAN
program structure.
The BBSAN program starts with a main function that calls the various subroutines depending on if the jet is axisymmetric or three-dimensional. Immediately
the main program prints a comment card to the standard output detailing the
version of the code and its associated library versions. This also lists the primary
contact for the BBSAN program. Next the BBSAN program calls a subroutine that
reads in the parameter input file. The parameter input file is unique to each jet
simulation and contains information the program needs to successfully find spec-
105
Figure 3.1. Flowchart of the BBSAN prediction code.
tral densities. It contains a flag that defines if the solution is three dimensional or
axisymmetric, if the CFD solution is a dual stream jet, the names of the CFD grid
and solution files, the number of the sub-zones to load from the CFD simulation,
the ranges of the integration regions, the number of indices to be used in each
direction of the integration regions, the jet or jet?s D, Md , Mj , and To , the number
of observers and observer angles from the nozzle exit from the downstream axis,
the number and ranges of frequencies to calculate, and an origin for the observer
positions. Each CFD solution for each calculation is stored in sub-directories. This
allows for a unique parameter input file to be stored in each sub-directory where
the corresponding common flow library solutions are stored for a specific jet. This
eliminates the need to enter the simulation parameters more than once and yields
sub-directories with self contained executables and case files.
The sub-zones of the CFD solution are specified as much of the solution is not
needed for a successful calculation. Only the sub-zones that contain the shock
waves and the shear layers are loaded. This saves a large amount of computer
memory when examining large flow-field solutions from three dimensional jets. In
fact, three dimensional integrations would not be possible if the ability to load subzones were not present in the serial implementation of the BBSAN model. Ranges
of the interpolated integration regions are specified so that the BBSAN code only
integrates over a small area of the loaded sub-zones. The sources of BBSAN are
compact relative to the entire flow-field. Integration studies and reasons that an
106
interpolated integration regions are used instead of directly integrating over the
CFD computational grid are discussed in the following sections. The diameter and
operating conditions of the off-design supersonic jet are specified only for plotting
purposes and are not used in the actual integration procedure. The BBSAN model
evaluation is completely independent of these parameters. The BBSAN program
only makes one assumption about the CFD solution: the positive axial streamwise
direction of the jet is in the positive x direction and that the cross-stream direction
is either r or y and z. This assumption could easily be removed.
After the subroutine that reads the parameter input file finishes, the main
function calls a subroutine that reads the common flow library grid and solution
files. This subroutine, as seen in the third step of Figure 3.1, reads in specific subzones of the computational grid and solution files using the common flow library
subroutines. It automatically detects if the variables written by Wind-US are single
or double precision and stores them in double precision Fortran arrays. Also, the
far-field quantities such as p? , T? , and ?? are read automatically. The field data
and computational grid are immediately converted to the SI system as the common
flow libraries store data in a non-dimensional format. This conversion is performed
because the BBSAN program is written in dimensional form. Since the majority
of the CFD simulations use the Menter SST K ? ? turbulence model and the
BBSAN model requires the viscous dissipation rate, ? needs to be converted to .
The relation between the two variables is = 0.09?K as specified by Wilcox [68]
or Menter [70]. If a K ? model is being read from the common flow libraries no
conversion is necessary. After the subroutine reads the common flow libraries it
echoes each sub-zone?s dimensional data to individual Tecplot files for inspection to
ensure that they are stored in memory correctly and that the solution is reasonable.
These files can be compared directly to those produced by the Wind-US post
processing program CFPOST. CFPOST can also perform the conversion of ? to
and convert the non-dimensional solution to dimensional SI units. If the echoed
files from the developed subroutine and CFPOST agree perfectly, this demonstrates
that the subroutines that read the solution files are implemented correctly. The
fourth step in Figure 3.1 involves calculating the metrics of the computational subdomains from the common flow libraries. These metrics were originally used to
calculate derivatives of flow-field quantities. However, no derivatives are needed in
107
the current model. Finally, the main function calls either the axisymmetric single
and dual stream BBSAN subroutine or the three dimensional BBSAN subroutine
which calculate the spectral densities. The general form of the axisymmetric or
three-dimensional BBSAN spectral density subroutine is shown in Figure 3.2.
Figure 3.2. Flowchart of the BBSAN integration subroutine.
The BBSAN subroutine immediately calculates the values of Tj , uj , and Dj
based on D, Md , Mj , and To . These values are used to plot the results. The
equations for this task are based on isentropic theory and are,
Tj =
1+
uj = Mj
To
??1
Mj2
2
p
?RTj
(3.54)
(3.55)
108
1 + (? ? 1)Mj2 /2
Dj = D
1 + (? ? 1)Md2 /2
(?+1)/(4??4) Md
Mj
1/2
(3.56)
The integration of the prediction equation is performed on structured grids that
are derived from the CFD solution with constant spacing between grid points. This
is done for two reasons. First, this method allows grid independence studies using
the highly-resolved CFD solution databases. Second, it simplifies the calculation
of the Fourier transform of ps since the grid spacing is constant with respect to
x, r or y, and z. Thus a standard discrete Fourier transform or fast Fourier
transform library can be used. Also, this ensures that the radial locations of p?s
and the other field variables of the integration region are the same. Arrays that
hold interpolated values in the integration regions are allocated based on the values
in the parameter input file. These arrays include values in the integration region
of K, , p, x, r or y, z, area, u, ps , and xk . These arrays hold the values in
the integration regions, which are created and interpolated to based on the CFD
solution. There are two interpolated regions that correspond to a jet. The first is
the integration region and consists of a structured computational grid in two- or
three-dimensions over the sources of BBSAN. The spacing between grid points in
each direction is constant and the grid lines are parallel to the x-, y-, r-, or z-axes.
The second interpolated region is the wavenumber region and it also has constant
grid point spacing with its grid lines parallel to the x-, y-, r-, or z-axes. The
spatial coordinates of the integration and wavenumber regions are found based on
the parameter input file integration ranges and the respective number of indices.
The range of the wavenumber region in the axial direction is based on the minimum
distance between points in the axial direction.
The interpolation scheme selected for calculating data on the integration and
wavenumber regions is the inverse weighted distance interpolation algorithm of
Shepard [87]. This algorithm is not a high order interpolation algorithm. It was
found that higher order interpolation algorithms such as those based on radial
basis functions of Li [88] are not needed and do not affect the accuracy of the BBSAN calculations. These high order interpolation algorithms create an additional
unneeded computational cost. The inverse weighted distance formula for u, for
example,
109
N
P
u(y) =
wn (y)un
n=0
N
P
(3.57)
wn (y)
n=0
where N is the maximum number of points close to the interpolated point, n is an
integer representing the nth closest point, and wn is a weighting function given by,
wn (y) =
(x ? xn
)2
1
+ (y ? yn )2 + (z ? zn )2
(3.58)
Clearly, the interpolation scheme is simple to implement and inexpensive. However, it requires finding the nearest N points, which can be an expensive operation.
For axisymmetric BBSAN calculations N = 3 and for three-dimensional BBSAN
calculations N = 4. An algorithm is implemented that calculates the distances
between the interpolation location and each of the computational grid points on
the CFD solution. The nearest N points can easily be identified in this fashion, although in three-dimensions it is very expensive because the search space
is extremely large, even if it is restricted to one of the sub-zones. Conveniently,
the structured computational domains used in the CFD are ordered. Therefore,
some amount of computational logic is used in the three-dimensional subroutine
to overcome the limitations of the brute force algorithm. This is accomplished by
restricting the search space to planes in the cross-stream direction that are always
parallel to the nozzle exit. The search algorithm for the nearest N points could
be generalized by sorting the points of the CFD solution. Entire computational
libraries exist for this purpose alone. However, the search algorithm is sufficient for
structured CFD data but would need improvement if unstructured CFD data were
being used as input to the BBSAN program. The variables that are interpolated
onto the integration region are K, , p, and u. Figure 3.3 shows the integration
region superimposed on the CFD solution with contours of u for the axisymmetric
jet with Md = 1.00, Mj = 1.50, T T R = 1.00, and D = 0.0127 m. The CFD
solution is calculated only in the top half plane. However, it has been mirrored in
the bottom half plane to illustrate the accuracy of the interpolation scheme. Other
variables show the same agreement.
The interpolation from the CFD solution onto the wavenumber region uses the
110
Figure 3.3. Interpolation of u m/s onto the integration region from the CFD solution
of a Md = 1.00, Mj = 1.50, T T R = 1.00, D = 0.0127 m jet. The red line encloses of the
integration region.
same interpolation scheme. However, the wavenumber region generally extends
two core lengths downstream relative to the integration region. Note the integration region, in both the axisymmetric calculations and the three-dimensional
calculations is chosen to enclose only the BBSAN sources. This extends from the
nozzle exit to the end of the core or beyond and in the cross-stream direction from
the centerline to roughly a single jet diameter.
The wavenumber region is chosen to extend downstream of the integration
region and shock cell structure by approximately one to two core lengths. The
grid points of the wavenumber region in the y or r and z directions correspond to
the same y or r and z grid points of the integration region. This is because in the
BBSAN model equation the wavenumber spectrum must be known at the same
cross-stream locations as the other field variables. The locations of the interpolated
points of the wavenumber region in the streamwise direction do not correspond
to streamwise points of the integration region. This is because the streamwise
direction of the wavenumber integration region is transformed from x (streamwise
direction) to wavenumber k. The index values of the wavenumber spectrum are now
k, y or r, and z. When interpolating using inverse weighted distance interpolation
only the interpolated values of p are evaluated on the wavenumber region. The
shock pressure in the wavenumber region is then found by subtracting the ambient
111
pressure, p? . The variation of ps is constructed as an even function by mirroring
the values across the nozzle exit plane. Now, values of shock pressure are even
across x/D = 0.0 and the Hanning window [89], wn , is applied at each crossstream location in the stream-wise direction to ps ,
1
2?n
wn =
1 ? cos
2
N ?1
(3.59)
where N is the maximum number of points in the streamwise direction of the
wavenumber region and n is an integer from zero to N . Performing this operation
helps to smooth the final wavenumber spectrum. After the Hanning window has
been applied, the Fast Fourier Transform (FFT) or the Discrete Fourier Transform (DFT) is applied in the streamwise direction at each cross-stream location.
Three different Fourier transform schemes have been implemented in the code
and all produce the same result. The first is a simple discrete Fourier transform
programmed directly from the definition of the Fourier transform. The second is
the FFT written by Ferziger [90]. Since the Ferziger implementation is an FFT,
it requires that the number of points in the streamwise direction be an integer
power of two. Finally, the scheme normally used in the BBSAN code is the Single
Precision Complex Fast Fourier Transform (SPCFFT) written by Ferguson [91].
The subroutine is further modified to use double precision values and also to use
dynamic arrays that are available to Fortran 90. SPCFFT is unique because it
reverts automatically to an FFT if the number of points in the streamwise direction is an integer power of two and works as a DFT otherwise. Therefore, there is
no need to ensure that the number of points chosen in the streamwise direction is
a power of two when specifying the indices in the parameter input file. SPCFFT
has been benchmarked extensively after its modification by transforming common
functions such as cos and exp in the forward and backward directions. After the
DFT of the shock pressure has been calculated, the correction for the Hanning
window is applied by multiplying the result by 8/3. The Fourier transform of the
shock pressure in the axial direction, p?s , has now been constructed. An example
result of p?s is shown in Figure 3.4 for Md = 1.00, Mj = 1.50, and T T R = 1.00.
However, the absolute value of the quantity is shown to make the contours more
easily identifiable.
112
Figure 3.4. The magnitude of the Fourier transform of the shock cell pressure p?s .
Each of the peaks of the wavenumber spectrum, p?s , contribute to a peak in
the predicted BBSAN spectrum. It is desirable to have a very large axial range
for the shock pressure relative to the core length of the jet so that the spacing
between wavenumber points in the spectrum is small. This helps resolve each
peak?s contribution to the BBSAN. Note the values of p?s are real, symmetric about
k = 0, and are a function of space only in the cross-stream directions. In the three
dimensional BBSAN implementation p?s is a function of k, y, and z, and is also
three-dimensional. Once p?s is calculated, files are created that are readable by
Tecplot that contain all the interpolated data for the integration and wavenumber
regions. The BBSAN program now has all the values ready to integrate as seen in
Figure 3.2 using the developed model.
In the evaluation of Equation 3.53 there are essentially three or four integrations
to be performed over x, r or y, z, and k. These integrals are performed in a
group of computational ?do? loops as seen in Figure 3.2. The outside do-loop
is over the observer angle ? and the inside do-loop is over wavenumber, k. In
the axisymmetric version of the BBSAN subroutine the fourth do-loop is skipped
because the data is in two-dimensions. In the outer do-loop, ?, the observer angle
is calculated from the parameter input file and a vector from the nozzle exit to
the observer is also calculated. Inside the integration number do-loop the radian
frequency ? is calculated. Inside the spatial do-loops of x, r or y, and z the
three turbulence coefficients ? , l, and l? are found based on the values in the
113
integration region. These loops and corresponding inner loops use the Fortran
cycle command if K < 0.10Kmax . Cycle essentially skips the evaluation of the
do-loop at a particular value under certain conditions. This saves a great amount
of time evaluating the spectral density because spatial regions that do not possess
sufficient K will not contribute to the spectral density. Finally, the innermost
do-loop is over the wavenumber k. By examining Figure 3.4 it is clear that certain
ranges of wavenumber have very small values of p?s that do not contribute to the
spectral density. Therefore, cycling is used in the innermost do-loop when p?s is not
in the range of wavenumbers of interest to BBSAN. Performing a full wavenumber
integration is more expensive and no difference in the final solutions are apparent.
The integrand of Equation 3.53 is in the center of the seven do-loops. After the
loops finish, the spectral densities at each observer location are multiplied by the
term in front of the integrals of Equation 3.53 and also by a prefactor that controls
the amplitude of the spectral densities, Pf . This is discussed in a later section.
After the integral is multiplied by the prefactor, an adjustment to the spectral
density is made if the jet is heated. This is necessary because of the deficiency in
scaling of K with T T R as discussed in Chapter 2 of the RANS CFD solutions. To
make the adjustment for heated jets the fully expanded jet density is calculated,
po
?j =
RTo
??1 2
Mj
1+
2
?1
??1
(3.60)
and the correction to account for a heated jet is,
Tcf
?j
=
??
??1 2
1+
Mj
2
?1
(3.61)
where Tcf is the temperature correction factor proposed by Tam [58]. Tcf = 1.00
if T T R = 1.00 and is less than one if T T R > 1.00. The variation of Tcf with T T R
is shown in Figure 3.5 part a) for a Md = 1.00 and Mj = 1.50 jet. Part b) of the
figure shows the associated correction in dB applied to the SP L for the same jet.
Because the Menter SST turbulence model is used to close the RANS equations the Tcf is applied to all the BBSAN calculations shown in this dissertation.
By using Tcf better agreement is found with heated jets in terms of magnitude,
although, the spectrum shapes are not affected by either Tcf or Pf .
114
Figure 3.5. a) Tcf verses T T R for a Md = 1.00 and Mj = 1.50 jet. b) The associated
change in dB when heating a Md = 1.00 and Mj = 1.50 jet.
Once the spectral density is found using Equation 3.53 and appropriate correction factors applied, it is written to files corresponding to each observer location
as an ordered list of St, SP L, and f . Since Equation 3.53 uses radian frequency,
the radian frequency is converted to St as,
St =
?Dj
2?Uj
(3.62)
and the SP L is written to the file as,
?
SP L = 20 log10
S
2 О 10?5
!
+ 10 log10 (Uj /Dj )
(3.63)
where S is the summation of the spectral density contributions from each integration region. The predictions can now be compared directly with experimental
data. See Appendix A for additional details regarding the evaluation and scaling
of the spectral density.
115
3.3
Parametric Studies
A parametric study has been performed for the various computed jet flow-fields to
determine the effects of changing the integration range and the grid point densities
of the interpolated regions. Table 3.1 summarizes this work. The first column is the
case number that is used for identification. The last column identifies if the case is
a base case, or if no change, a minor change, or a very large change to the BBSAN
is observed from the preceding base case. The bold values are changes in the
specification of the interpolation regions relative to the base case. An integration
region is defined by two points for both the three dimensional and axisymmetric
cases. For the axisymmetric cases, the spatial point is (xmin , ymin ) and the second
is (xmax , ymax ), forming a rectangle with sides parallel to the x- and r-axis. The
three dimensional cases have an associated zmin and zmax value. As described in
the previous section, the integration region is divided by a number of indices in
the x- and r-axis directions that correspond to iindex and jindex respectively. The
range of the wavenumber region always starts with the center of the nozzle exit
plane and forms a rectangle with the second point at kmax and ymax . The number
of indices in the x- and r-axis directions for the wavenumber region is jindex and
kindex respectively. The first base case in Table 3.1 matches the experimental SPL
well. Cases 2 through 8 vary the range and number of indices in the wavenumber
region, thereby finding a range and grid density that is independent of the BBSAN
solution. A new base case is constructed as number 8 and the ranges and indices
are varied in the streamwise direction for the integration region. Cases 14 through
24 are conducted to examine the affects of varying the grid density and ranges of
the integration region in the cross-stream direction. Finally, cases 25 through 29
are conducted partly to determine how close the integration region needs to be to
the nozzle lip.
Table 3.2 presents similar results to Table 3.1 but the ?case? column is replaced.
The first row, ?Fast,? shows recommended dimensional values to obtain a BBSAN
solution using the least number of recommended grid points in the integration
and wavenumber regions. The last row uses the least number of grid points to
get the most accurate solution from the model and achieve grid independence.
116
Case
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
xmin
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0100
0.0195
ymin
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0020
0.0040
0.0001
0.0001
0.0001
0.0001
0.0001
xmax
0.1137
0.1137
0.1137
0.1137
0.1137
0.1137
0.1137
0.1137
0.1421
0.1421
0.1421
0.1421
0.1421
0.1421
0.1421
0.1421
0.1137
0.1421
0.1421
0.1421
0.1421
0.1421
0.1421
0.1421
0.1421
0.0102
0.0722
0.1421
0.1421
ymax
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0100
0.0200
0.0200
0.0100
0.0100
0.0200
0.0127
0.0100
0.0100
0.0100
0.0127
0.0127
0.0127
0.0127
0.0127
iind.
400
400
400
400
400
400
400
400
100
300
200
100
150
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
jind.
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
50
100
200
50
100
50
75
75
75
75
75
75
75
75
kmax
0.4000
0.4000
0.6000
0.4000
0.3000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
0.4000
kind.
300
500
300
400
300
200
300
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
Change
Base
Minor,1
Minor,2
No,2
No,4
Yes,5
Minor,5
Base
No,8
Minor,9
Minor,9
Minor,9
Minor,9
Base
Yes,14
Yes,14
No,16
Base
Minor,18
Minor,18
Minor,18
Minor,18
Minor,22
Yes,22
Base
Minor,25
Minor,25
Yes,25
Yes,25
Table 3.1. Summary of the parametric study of the integration and wavenumber regions
for the circular converging jet operating at Mj = 1.50, T T R = 1.00, D = 0.0127 m.
Finally, the ?Fast/Accurate? row shows a trade off between the two extremes of
Fast and Precise. This middle row represents recommended values that allow fast
solutions without sacrificing any accuracy of the BBSAN solutions, while allowing
small deviations from the precise solution at very high and very low frequencies,
well away from the BBSAN peaks. Clearly, the number of indices needed for a
BBSAN solution is much less than those in the computational grids used in the
117
CFD solutions. Table 3.3 shows the ranges of the grid in a non-dimensional format
based on the exit diameter of the nozzle. The ranges in the downstream direction
roughly correspond to just more than the core length of the jet for the integration
region and roughly three times the core length for the wavenumber region. The
cross-stream direction range is recommended to be a single jet diameter and easily
encompasses the BBSAN sources. In general, the chosen ranges and number of
indices for the BBSAN solutions shown later in this chapter are based on the exit
or equivalent diameter of the nozzle geometries based on the ?Fast/Accurate? row
of Table 3.3.
Case
Fast
Fast/Accurate
Precise
xmin
0.0001
0.0001
0.0001
ymin
0.0001
0.0001
0.0001
xmax
0.1421
0.1421
0.1421
ymax
0.0127
0.0127
0.0127
iindex
150
300
500
jindex
75
75
100
kmax
0.4000
0.4000
0.4000
kindex
300
300
400
Table 3.2. Recommended integration and index ranges based on the integration study
of Mj = 1.50, T T R = 1.00, D = 0.0127 m.
Case
Fast
Fast/Accurate
Precise
xmin
0.0001
0.0001
0.0001
ymin
0.0001
0.0001
0.0001
xmax
11.189
11.189
11.189
ymax
1.00
1.00
1.00
iindex
150
300
500
jindex
75
75
100
kmax
31.5
31.5
31.5
kindex
300
300
400
Table 3.3. Recommended non-dimensional integration values based on D and index
ranges.
The scaling coefficients in the BBSAN model control the shape of the spectrum
and in part its magnitude. c? is set to 1.25 and controls the relative magnitude of
the spectrum and is the primary means to control the sharpness of each BBSAN
peak. cl is set to 3.25 and determines the turbulent length scale in the streamwise
direction. It controls to some degree the width of the peaks, the smoothness of
the spectra, and the relative magnitude of the BBSAN as a function of observer
angle. Increasing cl smooths the BBSAN peaks, increases the width of each peak
in the BBSAN spectrum, and lowers the relative magnitude between the peaks and
118
troughs. It should be noted that the value of cl of 3.25 is much larger than that
used by Tam and Auriault [85] in their model of fine-scale turbulent mixing noise.
This reflects the fact that BBSAN is controlled by the large scale structures in
the jet shear layer that are coherent over relatively large axial distances. c? is set
according to experimental observations, which show that the cross-stream length
scale is approximately 30% of the streamwise length scale and is therefore set to
0.30. c? controls the rate at which the high frequency predictions decay. These
values were chosen based on the spectrum of a jet at ? = 100 and operating at
Md = 1.00, Mj = 1.50, and T T R = 1.00. Finally, Pf is chosen by matching the
SPL at various frequencies of the same jet at ? = 100. The same values of c? , cl ,
c? , and Pf are used for every calculated spectral density.
3.4
Single Stream Axisymmetric Jets
BBSAN predictions have been made for the jet conditions shown in Table 3.4
based on corresponding Wind-US CFD solutions using the Menter SST turbulence
model. The cases selected include both unheated and heated jets operating at over(Mj < Md ) and under-expanded (Mj > Md ) conditions for two different nozzle
geometries with design Mach numbers, Md = 1.00 and Md = 1.50. The CFD
simulations for all the cases use a jet exit diameter, D = 0.0127 m. However, other
CFD simulations for the same operating conditions but with different diameters
have been performed and results are similar for the BBSAN prediction.
The BBSAN predictions are made at various angles ? from the downstream jet
axis at a radial polar distance of 100 D from the center of the jet exit. Predictions
are lossless and corresponding experimental data in each case has had humidity
and atmospheric absorption corrections applied so that the presented SPL are
also lossless. The experimental data provided by Boeing was measured at 97.5
D from the nozzle exit while the Pennsylvania State University (PSU) data was
measured at 150 D. Both sets of data have been corrected to 100 D. The PSU
experimental data has its origin centered at x/D = 5.0 or 0.0635 m, but this is
of less importance for BBSAN since the source region is relatively close to the jet
exit and the dominant radiation is at large angles to the jet downstream axis. The
119
Md
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.50
1.50
1.50
1.50
1.50
1.50
Mj
1.22
1.22
1.22
1.47
1.56
1.50
1.50
1.30
1.30
1.40
1.60
1.70
1.70
NP R
2.50
2.50
2.50
3.50
4.50
3.67
3.67
2.77
2.77
3.18
4.25
4.94
4.94
TTR
1.00
2.20
3.20
3.20
3.20
1.00
2.20
1.00
2.20
1.00
1.00
1.00
2.20
D (m)
0.0508
0.0508
0.0508
0.0508
0.0508
0.0127
0.0127
0.0127
0.0127
0.0127
0.0127
0.0127
0.0127
Dj (m)
0.05176
0.05176
0.05176
0.05460
0.05604
0.01377
0.01366
0.01209
0.01209
0.01236
0.01309
0.01354
0.01209
fc
07115
10552
12728
13791
14004
31040
45784
31897
47191
32936
34106
34294
56826
?
0.700
0.700
0.700
1.077
1.077
1.118
1.118
0.748
0.748
0.539
0.557
0.800
0.800
Tj (K)
225.6
496.4
722.1
655.8
631.5
202.2
450.3
219.1
479.6
210.6
193.9
185.8
406.6
uj (m/s)
368.3
546.2
658.8
753.0
784.8
427.5
625.3
385.7
570.6
407.2
446.6
464.5
687.2
Table 3.4. Jet operating conditions of the RANS CFD and BBSAN predictions for
axisymmetric single stream jets.
Boeing experiments were performed with heated air while the PSU experiments
were performed with helium / air mixtures to simulate heated jets. The diameter
of the convergent nozzle in the Boeing experiments is 0.0622 m while the diameter
of the PSU nozzle is 0.0127 m. Experimental data, measured in the Small Hot Jet
Aeroacoustic Research facility at NASA Glenn Research Center, have also been
used for evaluation of the predictions. The SHJAR jets have a 0.0508 m diameter
nozzle with microphones on a polar arc at R/D = 50 centered at the center of the
nozzle exit. This data has also been extended to R/D = 100.
The integration regions chosen for the converging jets with D = 0.0127 m
are 0.0001 ? x ? 0.1421 m and 0.0001 ? r ? 0.0127 m, with 300 and 75 grid
points in the x and r directions respectively. The spacing between these grid
points are constant in both directions. These parameters were chosen based on
the integration studies. The shock pressure used for the Fourier transform was
taken from 0.0001 ? x ? 0.4000 m using 512 grid points. Since the data is
mirrored about x/D = 0.0, the total number of points in the Fourier transform
is 1024. This yields a wavenumber spacing, ?k, of approximately 7.86 m?1 . The
integration regions for the convergent-divergent jets are 0.0001 ? x ? 0.1421 m
and 0.0001 ? r ? 0.0127 m with 400 and 100 grid points in the x and y directions
120
respectively. The wavenumber integration range remains the same. More grid
points are used because the source regions are more compact relative to the underexpanded cases.
The model developed in the present dissertation is only for the BBSAN, which
is only one of the components of off-design supersonic jet noise. Thus, it is useful to
separate the individual noise components from the total spectrum. Viswanathan [53]
has developed a framework that separates the individual components of the total
noise spectrum into mixing noise and BBSAN. The Boeing experimental data presented for the converging nozzle predictions shows this breakdown to help illustrate
the capability of the prediction scheme. Specifically, the proposed model for BBSAN is compared directly against the extracted shock component from the total
noise for the two convergent nozzle cases.
The BBSAN predictions are presented in terms of SP L per unit St. The case
that has been used to fix the coefficients for the turbulence scales is the underexpanded converging jet with Mj = 1.50 and T T R = 1.00 at ? = 100 degrees.
Results of the model prediction in this case, in addition to various experiments,
are shown in Figure 3.6. There are eight comparisons of the predictions with
experimental data for different observer angles ? with respect to the downstream
jet axis. Each comparison is labeled with its corresponding observer angle and the
maximum SPL level of the experiments of Boeing. The screech tones are not used
to find the maximum value. Experimental data from Boeing, NASA, and PSU
are shown for each observer angle. Though there are minor differences between
the different sets of experimental data, the overall agreement is good. Also shown
are predictions based on the BBSAN prediction formulas provided by Tam [57]
or Tam [58] where appropriate. The breakdown of the total Boeing spectra into
the BBSAN and mixing noise components is also shown. At the different observer
angles the predictions are in good agreement with respect to all the experiments.
The predictions capture the multiple peaks in the BBSAN spectra as well as the
broadening in the spectral shape with decreasing angle to the jet downstream axis.
For these operating conditions the amplitude of the primary peak in the BBSAN is
relatively insensitive to the observer angle. This is captured by the predictions. It
is important to emphasize that the predictions at every angle use the same scaling
coefficients. As the angle to the jet downstream axis decreases, the primary peak
121
in the BBSAN spectrum moves to higher Strouhal numbers. This is also predicted
in the BBSAN models of Harper-Bourne and Fisher [1] and Tam [57]. The increase
in the relative importance of the mixing noise at lower Strouhal numbers is evident
in the experiments.
Before showing predictions for other jet operating conditions it is interesting to
show how the predicted BBSAN spectrum is constructed from various wavenumber
components. The wavenumber integration in Equation 3.53 can be limited to contributions associated with individual components of the shock cell?s wavenumber
spectrum. This is equivalent to examining the contributions of the interactions
of the turbulence with the Fourier modes in a waveguide model of the shock cell
structure. Figure 3.7 shows these contributions and the corresponding peaks they
generate in the BBSAN spectrum. This selection of wavenumber ranges replaces
the summation over Fourier components of the waveguide model for the shock cell
structure in Tam?s model [57]. However, in the present model, the spectral width is
controlled by both the spectrum shape associated with the turbulent velocity fluctuations and the finite bandwidth of the dominant lines in the Fourier transform
representation of the shock cell structure.
The source regions of the BBSAN according to the model equation can be
specifically identified. It was postulated that the regions of BBSAN occur where
there are large values of |ps | and large values of turbulent kinetic energy, K. Figure 3.8 shows contour plots of the Md = 1.00, Mj = 1.50, T T R = 1.00 converging
nozzle case. The spatial ranges of these contour plots do not correspond to the entire integration region and the values in the contour plots are of interpolated values
from the RANS CFD solution. Figure 3.8 part a) shows a contour plot of the shock
pressure, the axisymmetric shock cell structure is readily visible. Part b) shows
contours of K that are strongest in the spatially growing turbulent shear layer of
the jet. By evaluating the integrand of the model equation at ? = 90.0 degrees,
St = 0.653, f = 20241 Hz, and R/D = 100 and plotting values spatially over x/D
and r/D, the source distribution of the BBSAN is shown in part c). This observer
angle and frequency are chosen because they correspond to the first dominant BBSAN peak and have the smallest effect due to the shear layer on the predicted
frequency. Part c) shows that the sources of BBSAN are strong peaks that are relatively compact compared to the flow-field of the jet and are located where shock
122
waves interact with the turbulent shear layer. The dominant BBSAN source for
this angle and frequency occurs at the first oblique shock wave at x/D = 1.50, the
second dominant source is at x/D = 3.00, the third at x/D = 4.50, etc. Clearly,
the largest contribution to the BBSAN occurs at x/D = 1.50 and the sources are
relatively equally spaced. Since the shock cell structure is confined by the shear
layer, the shocks themselves terminate at lower r/D locations as x/D increases.
BBSAN sources also follow this trend at a relatively constant rate. That is, as x/D
increases the positions of the BBSAN sources decrease at a relatively constant rate
towards r/D = 0.0. The strength of the BBSAN sources far downstream becomes
smaller as the shock pressure decreases. Finally, there is no BBSAN source contribution near the lip of the nozzle or a positive contribution in regions where the
Prandtl-Meyer expansion fans interact with the shear layer. Only weak negative
contributions exist in this region.
The remaining BBSAN prediction results are presented in order of increasing
Mj and then associated T T R as shown in Table 3.4. Using the same N P R as the
previous case, heating is added to the converging nozzle flow at a T T R = 2.20.
The BBSAN predictions are shown in Figure 3.9. Overall, the spectral shape of
the predictions is excellent and the peaks of each BBSAN component are captured.
However, the predicted BBSAN magnitude at all frequencies is too high by 6 dB
without applying Tcf . Tam [58] argued that as the jet temperature increases, for
supersonic conditions, the growth rate and amplitude of the large scale turbulent
structures should decrease. This is supported by instability theory. Thus it is expected that the turbulence levels should decrease as the jet temperature increases.
This is not found in either the K ? or K ? ? RANS simulations. In the absence of an improved RANS turbulence model for the effect of jet heating, Tam?s
correction factor has been adopted here as an empirical correction for the effects
of jet heating. Tcf is applied to every prediction in this dissertation. Turbulence
modeling may be one of the issues that contributes to poor scaling of BBSAN intensity with increasing T T R. It could also be attributed to a failure to capture this
phenomena with the acoustics model. Unfortunately, the reason why the BBSAN
intensity does not continue to increase with increasing T T R is not understood by
the aeroacoustics community and is an area of current research.
To explore the effect of heating the jets further, three over-expanded cases
123
are conducted with the converging nozzle and T T R = 1.00, T T R = 2.20, and
T T R = 3.20. The noise predictions in these three simulations are shown in Figures 3.10, 3.11, and 3.12. In each case the predictions and experiment agree very
well both in magnitude and peak frequencies. The higher frequency BBSAN peaks
are captured more accurately as the T T R is increased. However, if the factor Tcf is
not applied to the predicted spectral densities, the higher values of T T R will correspond to very high predicted BBSAN magnitudes relative to experiment. The Tcf
corrected dB for the three cases is 0.0, -3.45, and -5.07 dB, respectively. As T T R
increases even more, the overshoot of the BBSAN and the associated corrected dB
due to heating increases even further.
Two over-expanded cases of Md = 1.50, Mj = 1.30, and T T R = 1.00 and
T T R = 2.20 are shown in Figures 3.13 and 3.14 respectively. In the unheated case
the peak levels are underpredicted and the spectral peaks are somewhat narrower
than in the experiments and the BBSAN is evident at all angles to the jet axis.
However, in the heated case, the mixing noise is clearly dominant at angles less
than 80 degrees to the jet downstream axis. There is also close agreement between
the present predictions and Tam?s model in the heated case. In the unheated
case, Tam?s model overpredicts the levels, whereas the present model underpredicts
them.
Two slightly off-design cases have been predicted with over- and under-expanded
jets for Md = 1.50 and T T R = 1.00. This enables an important illustration of the
differences between Tam?s model and the current one to be shown. Predictions
for jets with Mj = 1.40 and Mj = 1.60 are presented in Figures 3.15 and 3.16
respectively with SHJAR data at Mj = 1.38 and Mj = 1.61 also shown. The
general shape of the dominant BBSAN peak agrees well between the two prediction schemes, with the current implementation having a slightly narrower peak.
When the observer angle is less than ? = 90, the present prediction scheme gives
levels that are lower than the peaks predicted by Tam?s model. However, the overall amplitude of Tam?s model at low observer angles is generally higher than the
experimental data while the present predictions are lower. These two cases illustrate that computationally the model is more robust as the off-design parameter
is increased.
Two final BBSAN predictions from converging nozzles are conducted with mod-
124
erately off-design parameters and jet heating of T T R = 3.20. These are shown in
Figures 3.17 and 3.18. The fully expanded Mach numbers are very close to one
another and the temperature correction factor applied to both cases is -5.07 dB.
The small change in Mj between the two cases increases both the experimental
data and predictions by approximately the same amount of 2 to 3 dB. At lower angles the peak frequencies are slightly higher than experiment and the magnitudes
increase too much as the observer angle increases. However, the higher frequency
peaks of the BBSAN are captured extremely well.
BBSAN predictions are now made for Md = 1.50, Mj = 1.70 with T T R = 1.00
and T T R = 2.20. The unheated and heated predictions with accompanying experimental data are shown in Figures 3.19 and 3.20 respectively. In the unheated
case both prediction schemes give similarly good agreement with the experimental
data. The multiple peaks in the BBSAN predictions are more evident in Tam?s
model and the spectral shape given by the present model is closer to the measured
spectrum. In the heated case, the peak BBSAN levels are slightly overpredicted in
the present model and again corrected by Tcf . One interesting feature of the effect
of jet temperature on the BBSAN can be seen in Figures 3.19 and 3.20. From the
measured spectra it is clear that the peak amplitude of the BBSAN is only slightly
higher in the heated case. However, the fully expanded jet velocity has increased
by 48%. Since the amplitude of the BBSAN depends on the turbulence levels as
well as shock cell strength, a larger increase in levels would be expected. A close
examination of the prediction formula 3.53 suggests a dependence on jet velocity
of u3j . In the present case, assuming that the shock cell strength is relatively independent of T T R, this would give an increase of 5-6 dB. This increase is consistent
with the predictions and also with the predicted effects of heating in the other
cases shown previously, but it is not evidenced in the experiments.
125
Figure 3.6. Comparisons of BBSAN predictions with experiments for Md = 1.00,
Mj = 1.50, T T R = 1.00, R/D = 100.
126
Figure 3.7. The total BBSAN prediction and the accompanying contributions from
selective integrations over contributing wavenumbers of p?s representing different waveguide modes of the shock cell structure. Md = 1.00, Mj = 1.50, T T R = 1.00, R/D = 100,
? = 120.0.
Figure 3.8. Various plots of the flow-field region for Md = 1.00, Mj = 1.50, T T R = 1.00, R/D = 100, ? = 90.0. a) contours of
ps . b) contours of K. c) The spatially distributed source of the BBSAN at the peak frequency of fp = 20241 Hz or St = 0.653.
127
128
Figure 3.9. Comparisons of BBSAN predictions with experiments for Md = 1.00,
Mj = 1.50, T T R = 2.20, and R/D = 100.
129
Figure 3.10. Comparisons of BBSAN predictions with experiments for Md = 1.00,
Mj = 1.22, T T R = 1.00, and R/D = 100.
130
Figure 3.11. Comparisons of BBSAN predictions with experiments for Md = 1.00,
Mj = 1.22, T T R = 2.20, and R/D = 100.
131
Figure 3.12. Comparisons of BBSAN predictions with experiments for Md = 1.00,
Mj = 1.22, T T R = 3.20, and R/D = 100.
132
Figure 3.13. Comparisons of BBSAN predictions with experiments for Md = 1.50,
Mj = 1.30, T T R = 1.00, and R/D = 100.
133
Figure 3.14. Comparisons of BBSAN predictions with experiments for Md = 1.50,
Mj = 1.30, T T R = 2.20, and R/D = 100.
134
Figure 3.15. Comparisons of BBSAN predictions with experiments for Md = 1.50,
Mj = 1.40, T T R = 1.00, and R/D = 100.
135
Figure 3.16. Comparisons of BBSAN predictions with experiments for Md = 1.50,
Mj = 1.60, T T R = 1.00, and R/D = 100.
136
Figure 3.17. Comparisons of BBSAN predictions with experiments for Md = 1.00,
Mj = 1.47, T T R = 3.20, and R/D = 100.
137
Figure 3.18. Comparisons of BBSAN predictions with experiments for Md = 1.00,
Mj = 1.56, T T R = 3.20, and R/D = 100.
138
Figure 3.19. Comparisons of BBSAN predictions with experiments for Md = 1.50,
Mj = 1.70, T T R = 1.00, and R/D = 100.
139
Figure 3.20. Comparisons of BBSAN predictions with experiments for Md = 1.50,
Mj = 1.70, T T R = 2.20, and R/D = 100.
140
3.5
Dual Stream Axisymmetric Jets
The axisymmetric implementation of the BBSAN model can also be applied to
axisymmetric dual stream jets. The only physical difference between the single
stream and dual stream axisymmetric jets is the addition of a secondary stream
surrounding the core flow. The dual stream jets of interest could have a supersonic
primary and subsonic secondary flow, subsonic primary or supersonic secondary
flow, or both supersonic flows operating off-design. The case where the core of
the jet is operating supersonically and the secondary flow operates subsonically is
essentially a problem similar to the single stream jet but with a high speed co-flow.
If the secondary flow is supersonic while the primary flow is subsonic then the
integration regions need only be constructed over the shock cell structures that
interact with the turbulent shear layers. Finally, the most difficult case occurs
when both streams are supersonic, as two separate integration and wavenumber
regions must be placed over the regions of corresponding shock wave shear layer
interactions.
Table 3.5 shows the dual stream experimental conditions available to the author. The characteristic frequency, fcp , is based on the primary jet only. The
method to obtain the RANS CFD solution for these cases was outlined in the
previous chapter. Clearly, both the primary and secondary nozzle are converging
which yields Md = 1.00. The secondary flow is cold and the primary flow is heated.
The characteristic frequency is based on the fully expanded diameter and velocity
of the primary jet. The off-design parameters for both flows are listed. The offdesign parameter does not apply to the subsonic flow of one of the cases and is
listed as not applicable.
For the dual stream case with two supersonic flows, multiple integration regions must be used to calculate the BBSAN contribution from each shock cell /
shear layer interaction in the primary and secondary stream. Figure 3.21 shows
the shock pressure contours and the integration regions. Note that the axes of the
figure have been normalized by the primary nozzle diameter, Dp . The integration
regions? interpolated data is overlaid on the flow-field of the RANS CFD solution
and the associated boundaries are represented by red lines. The first integration
region covers the primary stream while the second integration region covers the
141
secondary stream. There are two wavenumber regions that correspond to both of
the integration regions and to the respective radial locations of the interpolated
data of the integration regions. The spectral densities calculated from each individual integration and wavenumber region are summed to give a total BBSAN
prediction. In one case, the off-design parameter of the secondary stream is relatively weak and the BBSAN is dominated by the primary stream. Both integration
regions extend from just after the nozzle lips at x/Dp = 0.0 and x/Dp = ?1.36, as
there is no BBSAN source at the nozzle lips in under-expanded jets, to 9.64 Dp .
The wavenumber regions extend in the streamwise direction from their respective
nozzle lips to 19.28 Dp .
Figures 3.22 and 3.23 show BBSAN predictions for the dual stream jets with
conditions shown in Table 3.5. The addition of the supersonic secondary stream
increases the maximum SPL at all angles by 3 to 4 dB. This is reflected in the
predictions as the maximum BBSAN at all observer angles is also increased by
approximately 3 to 4 dB. Peak frequencies agree with experiment best at ? = 90.0
degrees for both cases. This implies that the high speed secondary flow, both
subsonic or supersonic, has the effect of lowering the peak BBSAN frequencies.
Mismatch of predicted frequency becomes quite extreme at high and low observer
angles. The peak frequencies of the predictions could be improved by using a
better approximation of the vector Green?s function to include secondary flow
effects. The BBSAN predicted amplitudes compare very well with experiment. At
high observer angles of Figure 3.22 there are regions where no prediction is present
near St = 0.50. This is in contrast with the results of Figure 3.23 where there
is a BBSAN contribution across the same St range. The higher speed secondary
stream of Figure 3.22 causes a stronger Doppler shift to the predicted BBSAN and
shifts the peaks to lower frequency. The prediction could be improved by lowering
c? which would have the effect of broadening the BBSAN peaks, thus eliminating
the region where no BBSAN is predicted.
142
Mdp
1.00
1.00
Mjp
1.19
1.19
Mds
1.00
1.00
Mjs
0.96
1.04
T T Rp
2.70
2.70
T T Rs
1.00
1.00
Dp (m)
0.0622
0.0622
fcp
9398
9398
?p
0.645
0.645
?s
N/A
0.578
Table 3.5. Jet operating conditions for the RANS CFD and BBSAN predictions of the
dual stream jets.
Figure 3.21. Integration regions for the BBSAN calculation of the dualstream jet
Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 1.04, T T Rp = 2.70.
143
Figure 3.22. Comparisons of BBSAN predictions with experiments for the dualstream
jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 1.04, T T Rp = 2.70, R/D = 100.
144
Figure 3.23. Comparisons of BBSAN predictions with experiments for the dualstream
jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 0.96, T T Rp = 2.70, R/D = 100.
145
3.6
Three Dimensional Jets
Thus far axisymmetric single and dual stream CFD RANS solutions have been used
to perform BBSAN predictions. These used the axisymmetric form of the BBSAN
model equation and its implementation. The flow-fields produced by rectangular
or non-axisymmetric jets are highly three-dimensional and are not applicable to
the axisymmetric BBSAN model. Here, the BBSAN model is evaluated with the
three dimensional implementation of the BBSAN code using the rectangular jet
RANS solutions validated for rectangular jets in Chapter 2. Both cold rectangular
jets with Md = 1.50 were validated with Pitot and schlieren comparisons. The two
off-design rectangular jet simulations with T T R = 2.20 were validated by comparing schlieren of a simulated hot jet using a helium / air mixture. Table 3.6 shows
the rectangular nozzle operating parameters that are used for the corresponding
BBSAN predictions. The rectangular nozzle chosen has Md = 1.50, and operates
either over- or under-expanded at Mj = 1.30 or Mj = 1.70 and T T R = 1.00 or
T T R = 2.20. The experimental results were provided by Veltin and the experimental details are documented in Goss et al. [92]. Microphone positions were placed at
1.905 meters from the jet at various ? in the minor and major axis planes and at
an azimuthal angle of 45 degrees. The experimental data was extended to 100 De
to be consistent with the predictions presented in the previous sections. Only predictions are presented in the major and minor axis planes because the aspect ratio
of the rectangular jet is relatively small, therefore the intensity of the BBSAN will
vary only a little with respect to azimuthal angle. The characteristic frequencies
of the rectangular jets are based on the fully expanded equivalent diameter found
from the De and the N P R.
Md
1.50
1.50
1.50
1.50
Mj
1.30
1.70
1.30
1.70
NP R
2.77
4.94
2.77
4.94
TTR
1.00
1.00
2.20
2.20
De (m)
0.01778
0.01778
0.01778
0.01778
Dej (m)
0.01693
0.01896
0.01693
0.01896
fc
22888
24499
33792
36334
?
0.748
0.800
0.748
0.800
Tj (K)
219.1
185.8
482.0
408.7
uj (m/s)
385.7
464.5
572.1
688.9
Table 3.6. Jet operating conditions for the RANS CFD and BBSAN predictions of the
3D calculations.
146
Unlike the axisymmetric version of the code, the three-dimensional BBSAN
implementation evaluates the integrals over the three dimensional integration region and wavenumber region. Since there is an additional integral that needs to be
evaluated in the z-axis direction, the run time of the computer code can increase
greatly. To minimize the amount of time needed to make a three-dimensional BBSAN prediction, careful choices are made of the range and number of indices of the
integration and wavenumber regions. Based on the conclusions for the integration
studies of the axisymmetric jet shown in Table 3.3, integration ranges for the four
rectangular cases are chosen as 0.001m < x < 0.1437m, 0.001m < y < 0.011m,
and 0.001m < z < 0.021m with corresponding index values of 300, 75, and 75
in the streamwise, minor-axis, and major-axis directions respectively. Since the
RANS CFD simulations uses planes of symmetry of the nozzle, the implemented
BBSAN code automatically integrates across the planes. There is no restriction
in the implemented version of the BBSAN model regarding the flow-field, even
though planes of symmetry were used to construct the RANS solution. The same
implementation of BBSAN may be applied to any three-dimensional jet flow-field.
The range of the wavenumber region is chosen as 0.001m < xk < 0.250 m with 512
indices. Summation of the integration region is selective by only including contributions from 10% of K using cycling. This cycling technique saves a large amount
of computational time just as it had in the axisymmetric calculations. Also, the
solution is found relatively quickly by specifying a limited range for the integrations over wavenumber. The wavenumber integration range is carefully chosen to
only encompass peak values of the wavenumber spectrum. For example, in the
first rectangular BBSAN condition of Table 3.6, the spacing between each value
of wavenumber in the wavenumber spectrum is ?k = 12.57 and only wavenumber
contributions are used in the range of ?25 < k < ?166 and 25 < k < 166.
BBSAN predictions of the over-expanded cold rectangular jet in the major and
minor axis planes are shown in Figures 3.24 and 3.25. By comparing the BBSAN
predictions with the experimental data, it is seen that only a small 2 dB difference
is apparent between the BBSAN peaks in the major and minor axis. Additional
strong screech tones are present in the minor axis direction. The predictions in both
figures are the same as the azimuthal angle is not included in the current prediction
formula. This azimuthal dependance could easily be included if necessary. The
147
peak BBSAN at all observer locations matches well with respect to frequency and
magnitude. As in the circular nozzle case operating at Md = 1.50 and Mj = 1.30,
the overall prediction is slightly low. The width of the BBSAN peaks is also too
narrow relative to experimental measurements.
Figures 3.26 and 3.27 show the under-expanded unheated rectangular jet operating at Mj = 1.70. In this case, the variation of the magnitude of the BBSAN
is invariant with azimuthal angle, unlike the previous over-expanded case. The
predictions at all observer angles agree extremely well with the experimental data.
The first BBSAN peak and subsequent minor peaks at higher frequencies generally
align with those of experiment and the amount of fall-off in the spectrum is not
significant. The small high frequency peaks of the BBSAN are more pronounced
in the experimental data in the major axis plane. The predictions could be slightly
improved for this jet if c? is lowered to flatten out the strong dominant BBSAN
peak.
BBSAN predictions are performed for an over-expanded rectangular jet operating at Mj = 1.30 and T T R = 2.20. These predictions are shown in Figures 3.28
and 3.29 in the major and minor axes planes respectively. Variation of maximum
dB in the experiment with azimuthal angle only varies by up to 2 dB. Like the
heated axisymmetric case, the predictions are slightly lower than the experimental
data. The magnitude of BBSAN matches the experiment at high observer angles,
and also matches the peak frequencies. At ? = 90.0 degrees and lower the mixing
noise dominates the BBSAN. The width of the peaks, as with the axisymmetric jet,
are too narrow. This is especially true for low St when the prediction has extreme
fall-off. Note that for this heated prediction, along with all the other rectangular
jet predictions, the temperature correction factor has been applied.
The final rectangular predictions are shown in Figures 3.30 and 3.31 for the
major and minor axis directions respectively, for the Mj = 1.70 and T T R = 2.20
jet. The peak frequencies match those of the experiment very well at most angles
except for large values. BBSAN magnitudes align well perpendicular to the jet,
however, are over-predicting at high observer angles. At low observer angles the
BBSAN is dominated by the mixing noise because the jet is heated. The predicted
values of BBSAN do not contribute to the total spectra just as in the corresponding
heated circular jet case. The variation of the peak BBSAN magnitude is partly
148
controlled by cl . If cl is smaller, then the variation of BBSAN magnitude with
angle would better match the experiment.
Figure 3.24. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.30, T T R = 1.00, R/De = 100 in the major axis direction.
149
Figure 3.25. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.30, T T R = 1.00, R/De = 100 in the minor axis direction.
150
Figure 3.26. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.70, T T R = 1.00, R/De = 100 in the major axis direction.
151
Figure 3.27. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.70, T T R = 1.00, R/De = 100 in the minor axis direction.
152
Figure 3.28. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.30, T T R = 2.20, R/De = 100 in the major axis direction.
153
Figure 3.29. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.30, T T R = 2.20, R/De = 100 in the minor axis direction.
154
Figure 3.30. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.70, T T R = 2.20, R/De = 100 in the major axis direction.
155
Figure 3.31. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.70, T T R = 2.20, R/De = 100 in the minor axis direction.
156
3.7
The Effect of Laminar Flow in the Nozzle
BBSAN predictions have been made for the laminar / RANS and fully RANS jet
flows and compared to experimental data. This required the two CFD simulations
that were conducted to examine the effect of laminar or turbulent flow inside the
nozzle for a Md = 1.50, Mj = 1.30, T T R = 2.20 jet. The first CFD simulation
solved the RANS equations in the entire domain, and the second CFD simulations
solved the Navier-Stokes equations in the interior of the nozzle and the RANS
equations in the exterior. The second simulation thereby forces the flow to be
laminar inside the nozzle and turbulent outside the nozzle. In the laminar / RANS
case, a conical oblique shock originates from inside the nozzle. This is more similar
to an under-expanded jet than an over-expanded jet. The boundary layer in the
laminar nozzle separates due to the oblique shock originating inside the nozzle.
This is in contrast to the boundary layer in the RANS simulation where it remains
fully attached to the nozzle wall. BBSAN predictions based on the model formula
are conducted using no modification to the BBSAN code, as turbulence values are
available in the exterior region in both cases. Results are found by performing the
same spatial and full wavenumber integration of the model as previously discussed.
The spectral densities of both solutions are multiplied by Tcf .
Figure 3.32 shows the measured SPL at various microphone positions from
the PSU anechoic jet chamber, the prediction with Tam?s [58] model, and the
BBSAN predictions based on the two RANS solutions. The laminar nozzle case
shows slightly lower peak frequencies of BBSAN than the fully turbulent case,
and the SPL is 2 to 3 dB higher at all angles. Also, the BBSAN fall-off in the
laminar nozzle case is much steeper at high frequencies. These differences are due
to a lower effective exit diameter of the laminar / RANS jet due to separation of
the boundary layer. This lower effective diameter lowers the initial speed of the
turbulent shear layer of the jet thus lowering the predicted frequencies. In addition,
there are initially higher values of K and the existence of an oblique shock wave
at the nozzle lip of the fully RANS solution. This is in contrast where K near the
nozzle lip of the laminar / RANS jet is almost zero and no shock wave exists. This
explains the magnitude difference between the two simulations as the shock wave
turbulence interaction is the extra source of BBSAN in the fully RANS simulation.
157
Figure 3.32. Comparisons of BBSAN predictions for the laminar / RANS and fully
RANS simulation with experiments for a Md = 1.50, Mj = 1.30, T T R = 2.20, jet at
R/D = 100.
158
3.8
Helium / Air Mixture and Hot Air
BBSAN predictions were performed for helium / air mixtures and hot air using
the two RANS solutions previously presented in Chapter 2 Section 2.2.8. The two
RANS solutions were constructed from a hot air jet operating at Md = 1.00, Mj =
1.50, and T T R = 2.20 and a helium / air mixture operating at T T R = 1.00. The
helium / air jet simulates a jet with a T T R = 2.20. The local densities and Mach
numbers of the two simulations are very similar. BBSAN predictions based on the
model formula are conducted here with no modification to the implementation.
The integration, and wavenumber ranges and indices are the same in both cases.
Experimental data from Boeing used heated air at T T R = 2.20 while PSU used a
helium / air mixture to simulate heating.
Plots of BBSAN predictions for the jets with operating conditions of the PSU
and Boeing experiments are shown in Figure 3.33. Tcf is applied to the heated air
and the helium / air mixture predictions. Even though the helium / air mixture
is not heated, Tcf must be applied to the simulations to be consistent because the
mixture in the plenum is low density and Tcf will not be unity. The overall amplitude of both predictions, including that of the second model of Tam [58], are too
high relative to the Boeing experimental data and less so relative to the PSU data.
However, the magnitudes of the peaks of all three predictions are relatively the
same, though higher than the experimental data. Peak frequencies of the heated
prediction agree better with experimental data at observer angles perpendicular to
the jet, however, the helium / air mixture prediction shows much better agreement
in peak frequencies at high observer angles. Experimental data from Boeing, which
is heated, shows a lower peak frequency amplitudes than the helium / air mixture
experimental data of PSU. This is also reflected in the corresponding predictions.
The helium / air prediction also shows slower fall-off compared to the heated case.
Differences between the two predictions are partially due to the use of the Menter
SST turbulence model which does not have the PAB temperature correction of
the K ? model of Chien. Also, the hot air simulation uses an ideal gas model
while the helium / air simulation uses a three species model of helium, nitrogen,
and oxygen that does not account for other species. Furthermore, even though the
two simulations have similar streamwise velocities and Mach numbers as seen in
159
Chapter 2, the maximum K in the hot air simulation is 12,847 m2 /s2 while the
maximum K in the helium / air simulation is 15,047 m2 /s2 . This is primarily the
reason that the BBSAN magnitudes are slightly higher for the predictions.
Figure 3.33. Comparisons of BBSAN predictions for heated air and helium / air
simulated jets with experiments using heated air and helium / air mixtures. The jet
conditions are Md = 1.00, Mj = 1.50, T T R = 2.20, R/D = 100.
160
3.9
Turbulent Scale Coefficients
The coefficients, c? , cl , and c? , have been calibrated for a Md = 1.00, Mj =
1.50, T T R = 1.00 jet at ? = 100.0 degrees. These coefficients, and associated
scaling factors Pf and Tcf , have been used to make all predictions thus far in
this dissertation. The same coefficients are applied to each observer angle, nozzle
geometry, and jet conditions creating a true prediction scheme. The calibration
jet spectral density possesses screech tones. A screech tone changes the velocity
cross-correlation. Cross-correlations of screeching jets are very different from crosscorrelations of non-screeching jets. Screech is not often found when jets operate
with T T R > 1.00 or on large scale engines. Since the axisymmetric Md = 1.50,
Mj = 1.70, T T R = 2.20 jet does not possess a screech tone, the coefficients chosen
are not applicable because of the large difference in the cross-correlation of the
velocities in the shear layer. The coefficients connecting the various length and
time scales to the RANS solutions for the Md = 1.50, Mj = 1.70, T T R = 2.20 jet
have been recalibrated to illustrate this point. The angle chosen for calibration is
? = 100 degrees as before. Coefficients for a jet without screech are recommended
to be c? = 0.85, cl = 3.00, c? = 0.30, Pf = 101.3 . These coefficients yield good
predictions for a wide range of jets without screech.
Figure 3.34 shows a Md = 1.50, Mj = 1.70, and T T R = 2.20 jet at R/D = 100.
Experimental data from the PSU, Tam?s model, and two BBSAN predictions using
the developed model are shown. The black line shows BBSAN predictions using the
coefficients, c? = 1.25, cl = 3.25, c? = 0.30, that were calibrated for the screeching
jet operating at Md = 1.00, Mj = 1.50, and T T R = 1.00. The predictions using
the new coefficients are shown in red. Since the jet is heated the temperature
correction factor has been applied to both BBSAN predictions. Using the new
coefficients to predict the BBSAN shows that more precise scaling of BBSAN is
captured with increasing ?, the peak BBSAN magnitudes are captured successfully,
and the width of the primary and sometimes secondary BBSAN peaks matches the
experimental data successfully.
161
Figure 3.34. Comparisons of BBSAN predictions using turbulence coefficients that
are optimized for a non-screeching jet with experiments for Md = 1.50, Mj = 1.70,
T T R = 2.20, R/D = 100.
162
In this chapter a model for the BBSAN from off-design jets has been developed.
This model overcomes many of the restrictions of the models developed by HarperBourne and Fisher [1] and Tam [57] [58]. The BBSAN model has been implemented
in a Fortran 90 code that directly reads the RANS solutions produced by Wind-US.
These RANS solutions were validated with experiment in Chapter 2. The BBSAN
program is used to calibrate the coefficients connecting the RANS solutions with
the turbulent scales in the model. Various parametric studies were performed to
find the optimal integration and wavenumber region ranges. These studies were
performed for circular jets and applied successfully to dual stream and rectangular
jets. Most importantly, the BBSAN model coefficients were only calibrated against
one jet condition and observer angle and applied to many other jets as a true
prediction methodology. BBSAN predictions were compared with the two models
of Tam for circular, dual stream, and rectangular nozzles. All the nozzles had
varying diameters. The operating conditions of the nozzles varied greatly, both in
terms of N P R and T T R. The model was also used to make predictions of helium
/ air jets and a laminar nozzle. These results agreed favorably with corresponding
heated and fully RANS simulations. Finally, modeling coefficients were suggested
for jets operating without a screech tone, although, some jets without a discrete
shock tone had BBSAN predictions that compared very well with experimental
data. The next chapter will review the methods and results presented in this
dissertation. Future developments of the BBSAN and code will be discussed.
Chapter
4
Conclusion
This chapter is divided into three sections. The first gives a summary of the
research presented in the previous chapters. The second section draws conclusions
from the findings based upon the RANS solutions and BBSAN model. The final
section discusses future investigations to be completed after the publication of this
dissertation.
4.1
Summary
Convergent or convergent divergent nozzles that operate supersonically and offdesign contain a shock cell structure that is confined by the jet shear layer. This
flow-field can be very complicated as it is often produced by nozzles with complex
geometries. Three different types of noise are created by the flow-field of an offdesign supersonic jet: Mixing noise, screech, and broadband shock-associated noise
(BBSAN). Mixing noise occurs due to the presence of the large and small scale
turbulent structures in the shear layer, transitional region, and mixing region of
the jet. Noise due to shocks consists of the discrete screech tone that is caused by
a feedback loop of acoustic waves propagating upstream in the subsonic region of
the shear layer and turbulence convecting downstream in the jet shear layer. This
feedback loop of screech is easily disturbed and is generally not supported in hot
jets. Finally, the BBSAN is due to the turbulence in the jet shear layer interacting
with the shock cell structure and causes constructive and destructive sound wave
interference in the far-field. It is characterized by multiple broad peaks in the
164
spectral density in the far-field. BBSAN is the dominant noise source in cold and
hot jets, operating off-design, and with a wide variety of nozzle geometries, mainly
in the upstream quadrant of the jet. The noise due to the presence of shocks is a
cause of concern for both military personnel and civilian residents in the vicinity
of an airport and for passengers in an aircraft at cruise.
Various models, both empirical and semi-empirical, have been developed for
BBSAN prediction. Harper-Bourne and Fisher [1] performed the first study dedicated to BBSAN and constructed the first prediction methodology in a combined
experimental and numerical investigation. Their work was built on by Tam [57],
who formed a stochastic model for BBSAN from circular nozzle jets operating moderately off-design and slightly heated. The models of Harper-Bourne and Fisher
and Tam have significant restrictions on their application. These restrictions include the need to have cold or moderate total temperature ratios, circular and more
recently dual stream axisymmetric nozzles, and operation at slightly or moderately
off-design conditions. Alternatively, a LES or DNS could be used to perform a numerical simulation of the jet. Unfortunately, these methods are impractical for the
foreseeable future due to their extremely high computational demands, especially
when modeling jets with realistic Reynolds numbers. The need for an approach using a three-dimensional steady RANS solution is apparent so that these limitations
can be overcome without expending large amounts of computer resources. Steady
RANS CFD solutions obtained from many commercial, government, or university
CFD solvers are fast to obtain relative to LES or DNS solutions, but still allow for
any geometry and operating conditions of the nozzle to be specified.
The Wind-US solver developed by the NPARC alliance is used to obtain steady
flow-field solutions using the RANS equations closed by the Menter SST turbulence
model. The RANS solution is the only input that the developed BBSAN model
requires, thus verification of the accuracy of the RANS solutions with experiment
is extremely important. Extensive Pitot probe measurements have provided experimental data for static and total pressure. Axisymmetric and rectangular jet
RANS simulations at various operating conditions have been compared with these
measurements. Furthermore, Mach numbers derived from the simulations were
directly compared with experiment. Overall the agreement between the predicted
and measured Mach number, static pressure, and total pressure measured is good.
165
The total pressure comparisons agree most favorably in all cases because no assumptions are made. This is unlike the calculation of the Mach number, where
an estimation of the static pressure is required. The static pressure simulations
often diverge from experiments. This is due to the shock waves generated by the
probe tip reflecting off the shear layer and interfering with the downstream static
pressure port. Furthermore, simulated helium / air mixtures and simulated hot
air schlieren are compared directly with experimental schlieren. Generally, the
spreading rate in the jet shear layer and the shock cell structure near the jet exit
agreed very well with the experiment. Farther downstream from the nozzle exit
the experimental and numerical data diverged to a greater degree. This is most
likely due to an error in the positioning of the experimental probes far from the
nozzle exit and also due to a lack of accurate turbulence models for cold turbulent
shock containing jets.
When jets are heated the prediction capability of the turbulence models in CFD
codes are less satisfactory for calculating turbulent kinetic energy or dissipation.
The Chien K ? and Menter SST K ? ? turbulence models have been used to
examine the effects of the scaling of K with increasing T T R both in an overand an under-expanded jet. Even though the Chien model contains temperature
corrections for jets and predicts levels of K more precisely than that of the Menter
SST model, the scaling of K and u with temperature remains the same in both
models. The coefficients of the RANS models available have been optimized for
wall-based flows and not three-dimensional shear layers. Scaling of the turbulent
kinetic energy with total temperature ratio for both the Menter SST and Chien
K ? turbulence models is not appropriate for BBSAN prediction. This suggests
development of a turbulence model that can improve the prediction of the flow-field
of heated supersonic jets.
When the Reynolds number of the jet is low, it is possible that the flow will
remain laminar in the nozzle and transition to turbulent flow in the region downstream from the nozzle exit due to the Kelvin-Helmholtz and Widnall instabilities.
Since this location is often unknown and difficult to predict it has been assumed
for practical purposes to occur at the nozzle exit. A single simulation has been performed where the Navier-Stokes equations have been solved inside the nozzle and
the RANS equations were solved outside the nozzle, thus simulating the transition.
166
Also, helium / air jets have been used to simulate heated jets. A RANS solution
has been constructed that simulates both the hot air and helium / air jet for a
total temperature ratio of 2.20. Theoretically, both of these CFD results should
yield the same BBSAN. Finally, Harper-Bourne and Fisher observed that BBSAN
intensity scales with the 4th power of the off-design parameter for cold jets. An
off-design study has been performed for a series of jets using a converging-diverging
nozzle by varying the off-design parameter, while keeping the total temperature
ratio constant at 1.00. It has been shown that the BBSAN sources scale with the
off-design parameter as observed by Harper-Bourne and Fisher.
A BBSAN prediction model has been developed that requires only the specification of the operating conditions and geometry of the nozzle. The RANS solution is
the only input that the BBSAN model implementation requires. A vector Green?s
function is used to find the acoustic pressure in the far-field from the compressible linearized Euler equations. The final prediction formula involves a spatial and
wavenumber integration over the BBSAN sources. It does not possess many of
the restrictions of the Harper-Bourne and Fisher and Tam models. Namely, it
can be applied to any kind of jet flow-field, nozzle shape, off-design parameter,
or temperature ratio. An exponential / Gaussian based model for the two-point
cross-correlation of the velocity perturbation of the turbulence has been used. The
model does require the knowledge of the turbulent length and time scales as a
function of spatial position. These values are linked to the RANS CFD simulation
by simple formulas and scaling coefficients.
The BBSAN model has been implemented in a Fortran 90 computer program.
The program has been used to conduct parametric studies of the integration and
wavenumber regions, along with grid density studies of these regions, to find grid
independent solutions. The model has been calibrated by adjusting the scaling
coefficients that represent the relevant turbulent length and times scales at one
operating condition only. Furthermore, many simulations have been performed to
find the coefficients relating the turbulence scales for the model from the Menter
SST turbulence model. This has yielded accurate results for a wide range of
experimental conditions. These coefficients are only calibrated against the single
operating condition of a cold under-expanded jet. The coefficients of the turbulence
scales have not been optimized for a wide range of jet conditions. This optimization
167
has not been performed in order to help better illustrate the model?s capability.
A wide range of calculations of single stream jets with various Md , Mj , and
T T R have been chosen to test the developed model and corresponding implementation. Selective integration over small ranges of the wavenumber region has
shown how the peaks of the wavenumber spectrum contribute to different peaks
in the predicted BBSAN spectral density. The dominant peak of the wavenumber spectrum corresponds in frequency to the dominant broadband peak of the
BBSAN spectrum adjusted by the Doppler factor. The predicted absolute levels
for unheated under-expanded jets are slightly low in the calculations. The effect
of jet heating is to increase the noise levels, for the same N P R, by 5-6 dB for a
T T R = 2.20. This can be associated with a jet velocity scaling given by u3j , as
seen in the noise prediction formula. However, the measured experimental changes
are of the order of 1 dB. A temperature correction factor suggested by Tam [58]
has been implemented for all predictions, heated and unheated. This adjustment
does not change the spectral density shape.
A limited number of dual stream calculations have been performed. This is
partly due to the limited amount of experimental data and only a single set of dual
stream nozzle geometry available to the author. However, the single stream axisymmetric implementation of the BBSAN model applied to the dual stream cases
with multiple integration and wavenumber regions is shown to give initially satisfactory results. However, the high speed co-flow can lower the predicted BBSAN
peak frequencies below those observed in the experiment. This could be corrected
by numerically evaluating the Green?s functions based on the RANS solution to
account for sound propagation through the high speed secondary stream.
The general form of the BBSAN model developed is fully three dimensional with
wavenumber integration in the streamwise direction only. This three-dimensional
version of the model, like the axisymmetric one, supports multiple integration
regions. The RANS solutions of the rectangular jets have been conducted by taking
advantage of nozzle symmetry about the major- and minor-axes. Therefore, the
integration region of the BBSAN model in the predictions presented here only uses
one fourth of the flow-field. The integration and wavenumber regions may have
ranges across the entire BBSAN source region or exist only in one fourth of the flowfield. The rectangular jet BBSAN predictions compare very favorably in both the
168
over- and under-expanded cold jet calculations. However, as for the axisymmetric
jet predictions, the over-expanded BBSAN peaks are slightly low relative to the
under-expanded counterparts. When the same rectangular jet conditions are used
but with a heating of T T R = 2.20, the same over-prediction of the magnitudes is
apparent as in the axisymmetric cases. However, with the temperature correction
factor, the magnitudes and spectra are in-line with experimental spectra for the
hot three-dimensional calculations.
4.2
Conclusion
The large number of comparisons of the RANS solutions with experimental data for
off-design supersonic jets showed good agreement overall. Agreement of the results
are better in the near field region of the jet as turbulent mixing plays a smaller
role relative to the convective terms of the equations of motion. Far downstream
from the nozzle exit the prediction capability is not as accurate, specifically with
respect to the spreading rates and field variables of the equations. Fortunately,
sources of BBSAN are located in the shear layer where strong shocks exist near
the nozzle exit. However, the predicted RANS solutions were sufficient to develop
and create a BBSAN model that is accurate.
As the T T R increases from unity to larger values the BBSAN intensity increases
slightly then ceases to increase. The reason that BBSAN ceases to increases with
increasing T T R is unknown to the aeroacoustics community. This may be due to
a possible limitation of the instability waves ability to scale with a linearly increasing T T R or the onset of the barrel shock. Nonetheless, additional experimental
and numerical investigations need to be conducted to determine why the BBSAN
intensity is limited with increasing T T R. The developed BBSAN model and the
model developed by Tam [58] have similar BBSAN intensity scaling with increasing
T T R. Unfortunately, both models display different BBSAN intensity scaling with
T T R than observed in experiment. This deficiency in both models was corrected
by an empirical temperature correction factor, Tcf , proposed by Tam.
The developed BBSAN model has better prediction capabilities than those of
Harper-Bourne and Fisher [1] and Tam [57]. Harper-Bourne and Fisher?s model
does not scale in intensity with observer angle, and only applies to convergent cir-
169
cular nozzles, and with unheated under-expanded jets. Tam?s model only makes
predictions for convergent or convergent divergent circular nozzles and moderately heated and moderately off-design jets. The developed BBSAN model has
been shown to make accurate predictions for convergent or convergent divergent
circular nozzles that operate highly off-design and with or without heating. In
addition, accurate predictions were made for dual stream circular nozzles operating with heating. Rectangular convergent divergent nozzles producing over- and
under-expanded jets with highly three dimensional flow fields produced accurate
BBSAN predictions. Tam?s model usually underpredicted the BBSAN contribution
between dominant BBSAN peaks. The developed model makes better predictions
relative to experimental data between these BBSAN peaks compared to Tam?s
model. The new model has no restriction on the type of nozzle geometry or associated operating conditions. Because of the lack of limitations of nozzle geometry
and operating conditions, relative to the existing BBSAN models, the newly developed BBSAN model represents a major improvement of prediction capabilities.
Unlike other models the developed BBSAN prediction methodology is a true prediction scheme and is not calibrated for a finite range of operating conditions. The
developed BBSAN model is almost non-empirical and has no restrictions regarding nozzle geometry and operating conditions. The BBSAN model developed in
this dissertation represents the only prediction method in existence that has no
restrictions.
4.3
Future Work
The RANS CFD solutions could be improved slightly with the use of a shock capturing scheme such as the Weighted Essentially Non-Oscillatory scheme for spatial
discretization. Currently, Wind-US does not support such a scheme and the current solutions are generated with a second order upwind Roe flux vector splitting
algorithm. This scheme does not preserve the discontinuities as well as a shock
capturing scheme would and tends to dissipate the strength of the shock waves
prematurely. Unfortunately, a turbulence model has not been developed by the
scientific community to properly simulate supersonic jets, in particular, heated supersonic jets operating off-design. Since the scaling of the BBSAN amplitude with
170
respect to increasing T T R is too large, and based on the investigation comparing
the PAB temperature corrections in the K ? model with no such corrections
in the Menter SST turbulence model, the scaling of K is wrong. This scaling
with temperature needs to be corrected in the RANS CFD solutions so that the
temperature correction factor Tcf is not needed. Furthermore, improved modeling
techniques will yield results that are in better agreement with the experimental
flow-field Pitot and schlieren data. The effects of heating jets on BBSAN is clearly
a topic for future research.
It is possible that the predictions could be improved by adding a frequency dependence to the characteristic turbulent length scale l. Experimental data shows
that the length scale decreases with increasing frequency while it remains constant
in the model. In addition, the assumed form for the two-point cross correlation
function of the turbulent velocity fluctuations is exponential / Gaussian in nature
and is based on similar forms used in the prediction of turbulent mixing noise.
However, the components of the turbulence that are most important for BBSAN
are the large scale coherent structures. The statistical properties of these structures would be expected to show significant positive and negative peaks in their
cross correlations. The effects of changes in the model for the turbulent velocity
cross correlation should be examined. Furthermore, the cross-stream length scale
has been assumed to be independent of azimuthal angle. The prediction formula
could be further improved by allowing for a dependence on azimuthal angle, thus
improving predictions for highly three-dimensional flow-fields with cross-stream
turbulent length scales that vary in the azimuthal direction. This could improve
the predicted magnitudes of BBSAN in rectangular jets or jets with other noncircular geometries.
The order of the computational loops that evaluate the model could also be
rearranged so as to speed up the process of prediction. However, this would only
lower the wall-clock time for a prediction and would not improve the accuracy.
A near field prediction formula should also be developed so that the model can
be implemented in noise prediction codes that take into account scattering by the
aircraft structures. Near field noise is also important for structural loading estimates as well as sound transmission studies for both civilian and military aircraft.
A vector Green?s functions could be evaluated numerically rather than assuming
171
that propagation effects through the mean flow are negligible. This is likely to
be most important for dual stream jets where the BBSAN from the primary jet
must propagate through the secondary stream. This would greatly increase the
prediction capability for dual stream jets.
Appendix
A
Correlation and Sound Pressure
Level
A correlation measures the similarity between two signals or functions that vary
with a dependent or independent variable. Finding correlation involves a spatial or
temporal analysis that may be useful for finding propagation times of quantities or
frequency content of fluid dynamic problems of interest. The correlation between
two signals x and y can be found by finding the average value of the product of
the signals and dividing by the number of products. This can be written as,
Z
T /2
hx (t) y (t + ? )i = lim
T ??
x (t) y (t + ? ) dt
(A.1)
?T /2
The correlation will be higher between two signals if they are more similar.
The auto-correlation is a measure between the same signals offset by a variable
amount ? ,
hx (t) x (t + ? )i
(A.2)
while the cross-correlation involves two different signals obtained at different physical locations or times. If ? = 0 then two identical signals are entirely correlated
and the autocorrelation evaluates to the mean square value of the signal. Furthermore, the autocorrelation is always symmetric about the origin. If the signal or
signals are scientifically random in nature then there is usually only a noticeable
173
correlation near ? = 0. Finally, if the signal or signals are periodic then the autocorrelation or cross-correlation will be periodic. In this work there is interest in
both auto and cross-correlations both for modeling turbulence and finding spectral
density. The mean square value of a periodic scientifically random signal may be
represented by the real part of a Fourier series as,
"
x (t) = Re
?
X
#
?
Cn exp (in?t) =
n=1
1X
(Cn exp (in?t) + Cn? exp (?in?t)) (A.3)
2 n=1
where Cn are the Fourier coefficients of the signal, n is an integer, ? is the radial
frequency, and t is an independent variable such as time. The mean square value
of x(t) is,
1
hx i = lim
4 T ??
2
ZT X
?
0
?
(Cn exp (in?t) +
Cn?
n=1
1X
Cn Cn?
exp (?in?t)) dt =
2 n=1
2
(A.4)
The contribution to the mean square in the frequency bandwidth, ?f , is defined
as,
1
G (fn ) = Cn Cn?
2
(A.5)
where G is the power spectrum of the signal. The power spectral density, S, is
defined as the power spectrum divided by a frequency bandwidth ?f . Substituting
the definitions of G into the relations for the mean square of the signal yields,
?
1X
hx i =
S (fn ) ?f
2 n=1
2
(A.6)
If the limit of the frequency bandwidth goes to zero then the mean square value
of the signal is,
2
Z
hx i =
?
S (f ) df
(A.7)
0
The advantage of using the spectral density instead of the power spectrum
174
is that the value of the spectral density is the same as the frequency bandwidth
varies. The spectral density can also be easily written as a summation of its Fourier
coefficients,
?
1X
S (?i ) =
Cn Cn? ? (? ? ?i )
2 n=1
(A.8)
It is useful in the development of the BBSAN model to express the spectral density
in terms of the autocorrelation of a signal. The inverse Fourier transform is,
?
Z
X (f ) exp (2?if (t + ? )) df
x (t + ? ) =
(A.9)
??
and the autocorrelation is,
1
R (? ) =< x (t) x (t + ? ) >= lim
T ?? T
Z
1
R (? ) = lim
T ?? T
?
Z
R (? ) =
??
x (t) x (t + ? ) dt
(A.10)
??
X (f ) exp (2?if (t + ? )) df dt
(A.11)
??
??
?
?
?
x (t)
Z
Z
1 ?
lim X (f ) X (f ) exp (2?if t) df
T ?? T
(A.12)
because,
1
hx i = lim
T ?? T
2
Z
T /2
Z
2
x (t) dt =
?T /2
?
1
X (f ) X ? (f ) df
T
??
T
??
lim
(A.13)
we obtain,
1
X (f ) X ? (f )
T ?? T
S (f ) = lim
(A.14)
This expression can be substituted directly into the auto correlation R (? ) to
obtain,
Z
?
R (t) =
S (f ) exp (2?if t) df
(A.15)
??
This Equation shows that the inverse Fourier transform of the Spectral density
is the auto correlation. The inverse of this operation shows that the spectral
175
density is the forward Fourier transform of the autocorrelation,
Z
?
R (t) exp (?2?if t) dt
S (f ) =
(A.16)
??
The equations which relate the Spectral density and the auto-correlation are
called the Wiener-Khintchine equations. These also hold for cross-correlations. If
the signal is the acoustic pressure in Pascals, then the dimensions of the autocorrelation are Pascal squared. If the forward Fourier transform is performed and
the variable of integration is time, then the power spectral density obtains units
of Pascals squared per second. So, the dimensions of the power spectral density is
mean square pressure per frequency bandwidth. The frequency bandwidth in this
dissertation is one Hertz. Therefore, the dimensions of power spectral density is
mean square pressure per Hertz.
In this work and those of Harper-Bourne and Fisher and Tam, the ultimate
goal of a BBSAN noise model is to produce an equation that is in terms of spectral
density per unit Hz. Often, aeroacoustic results are presented in Sound Pressure
level, as those that are shown in Chapter 3. To convert spectral density, S(x, ?)
[p2 /Hz] SPL per unit Hz,
SP Lf = 20 log10
!
p
S(x, ?)
pref
(A.17)
where SP Lf is the sound pressure level in decibels per Hz, and pref = 2.0 О 10?5 is
the reference pressure in pascals. Results produced in Chapter 3 are shown in SP L
per unit St, which requires an additional term applied to the previous equation to
preserve the energy in the spectral density,
SP L = 20 log10
!
p
S(x, ?)
+ 10log10 (fc )
pref
(A.18)
where fc is the characteristic frequency of the jet, uj /Dj . This also requires the
frequency to be changed to St,
St = f /fc
(A.19)
now, plots can be made of SP L per unit St by placing St on the x-axis and SP L on
176
the y-axis. To convert SP Lf to a sound pressure level other than unit frequency,
SP Lf b = SP Lf + 10 log10 (?f )
where ?f is the frequency bandwidth.
(A.20)
Appendix
B
Model Integrations
B.1
Integration of I??
The integral that needs to be evaluated is,
I??
Z? Z? i?
2 2
sin ? cos ?
=
exp ?? /l? ?
a?
?? ??
i?
2 2
О exp ?? /l? ?
sin ? sin ?? d?d?
a?
(B.1)
First, integrating over ? with the known integral and taking into account the
limits of ?? to ?,
Z?
?
2 2
exp ?p x ▒ qx dx =
2
?
q
exp
p
4p2
(B.2)
??
yields an integral only over ? that may be evaluated as,
Z?
exp ??
??
2
2
/l?
2
?
i?
?? 2 sin2 ? cos2 ?l?
?
sin ? cos ?? d? = l? ? exp
a?
4a2?
and the integral with corresponding limits of I?? yields,
(B.3)
178
I?? =
B.2
2
?l?
2
?? 2 sin2 ?l?
exp
4a2?
(B.4)
Integration of I??
The integral that needs to be evaluated is,
Z? Z?
I?? =
?? ??
"
#
? |? | (? ? u?c ? )2
i? cos ??
exp
?
+ ik? d?d?
+ i?? ?
?s
l2
a?
(B.5)
Expanding the integrand,
Z? Z?
I?? =
?? ??
i? cos ?? ? 2 2u?c ? ? u?2c ? 2
|? |
exp ik? ?
? 2 + 2 ? 2 + i?? ?
d? d? (B.6)
a?
l
l
l
?s
Performing the integration with respect to ? first with the known integral (integration tables),
Z?
?
2 2
exp ?p x ▒ qx dx =
2
?
q
exp
p
4p2
(B.7)
??
So,
Z?
i? cos ?? ? 2 2u?c ? ?
exp ik? ?
? 2 + 2
d?
a?
l
l
??
" 2 #
?
l2
? cos ?
2u?c ?
= ?l exp
i k?
+ 2
4
a?
l
"
#
2
u? ? 2
?
?l2
? cos ?
? cos ?
c
= ?l exp
k?
+ iu?c k ?
?+
4
a?
a?
l
(B.8)
Integration of ? is now complete and now ? must be integrated after substituting
the result from the integration of ?,
179
Z?
|? |
? cos ?
+? ? ?
d?
exp i u? k ?
a?
?s
??
Z0
=
??
Z?
+
?
? cos ?
exp
+ i u?c k ?
+ ? ? d?
?s
a?
??
? cos ?
exp
+ i u?c k ?
+ ? ? d?
?s
a?
0
=
+
=
(B.9)
1
1
?s
h
h
+ i u?c k ?
? cos ?
a?
? cos ?
a?
+?
i
+?
i
1
1
?s
? i u?c k ?
1
1
?s2
h
+ u?c k ?
? cos ?
a?
+?
i2 Thus,
?s2
?
?l exp
?l2
4
h 1 + u?c k ?
k?
? cos ?
a?
? cos ?
a?
+?
2 i2
(B.10)
?s2
Finally, simplifying the equation and reducing u?c /a? to M yields,
? 2
?l?s exp
I?? =
?l2
4
k?
? cos ?
a?
2 1 + [1 ? M cos ? + u?c k/?]2 ? 2 ?s2
(B.11)
Appendix
C
Green?s Function
Vector Green?s functions must be developed to allow the formation of the solution
for the time variation of acoustic pressure in the far-field relative to the jet plume.
It is assumed that the Green?s function is well approximated by the no flow solution.
This is a relatively good assumption because BBSAN emits primarily in the crossstream direction relative to the jet and the BBSAN sources are location roughly
in the middle of the shear layer, so only half of the shear layer has an effect on its
emission direction. Starting with,
?i??gn +
n
?vgi
= ? x ? y ?0n
?xi
(C.1)
??gn
= ? x ? y ?in
?xi
(C.2)
n
?vgi
= ?i?? x ? y ?0n
?xi
(C.3)
n
?i?vgi
+ a2?
is,
?? 2 ?gn ? i?
and
?i?
n
?vgi
? 2 ?gn
?
+ a2?
=
? x ? y ?in
2
?xi
?xi
?xi
If Equations C.3 and C.4 are combined then,
(C.4)
181
a2?
? 2 ?gn
?
2 n
+
?
?
=
i??
x
?
x
?
y
?
+
?
y
?in
0n
g
?x2i
?xi
(C.5)
or
? 2 ?gn ? 2 ?gn
1 ?
i?
+ 2 = 2 ? x ? y ?0n + 2
? x ? y ?in = R x, y, ?
2
?xi
a?
a?
a? ?xi
(C.6)
Now the solution to the problem,
?2
? 2 g (x, z, ?)
+
g (x, z, ?) = ? (x ? z)
?x2i
a2?
(C.7)
Is given by,
g (x, z, ?) = ?
exp [i? |x ? z| /a? ]
4? |x ? z|
(C.8)
?gn may now be formed and simplified as,
?gn
Z
?
Z
?
Z
?
g (x, z, ?)R x, y, ? dz
?? ?? ??
Z Z Z
i? ? ? ?
= 2
g (x, z, ?)? z ? y ?0n dz
a
?? ?? ??
Z ??Z ?
Z ?
1
? + 2
? z ? y ?in dz
g (x, z, ?)
a? ?? ?? ??
?zi
Z ?Z ?Z ?
i?
1
?
= 2 g (x, z, ?) ?0n 2
[g (x, z, ?)] ? z ? y ?in dz
a?
a? ?? ?? ?? ?zi
x, y, ? =
i?
1 ? i?gn x, y, ? = 2 g x, y, ? ?0n ? 2
g x, y, ? ?in
a?
a? ?yi
(C.9)
(C.10)
consider,
? ?
exp [i?R/a? ]
g x, y, ? = ?
?yi
?yi
4?R
p
R = x ? y = (xi ? yi ) (xi ? yi )
So,
(C.11)
182
?R
? (xi ? yi )
=
?yi
R
(C.12)
then,
?
g x, y, ? =
?yi
?1
i?
+
2
4?R
a? 4?R
(xi ? yi )
exp [i?R/a? ]
R
(C.13)
In the far-field, for this gives,
i? xi
?
g x, y, ? =
exp [i?R/a? ]
?yi
4?a? R R
(C.14)
Thus,
?i?
exp [i?R/a? ]
4?a2? R
(C.15)
i? xi
exp [i?R/a? ]
4?a3? R R
(C.16)
?g0 x, y, ? =
?gn x, y, ? =
Using these Green?s function results,
?gn x, y, ? =
i? xn
x ? y /a?
exp
i?
4?a3? x x
(C.17)
From definition,
?gn x, y, ?? = ?gn? x, y, ?
(C.18)
and,
p
p
x ? y = (xi ? yi ) (xi ? yi ) = xi xi ? 2xi yi + yi yi
r
xi y i y i y i
=x 1?2 2 + 2
x
x
xиy
'x?
x
Following a similar analysis,
xиy xи?
x ? y ? ? ' x ?
?
x
x
Yields,
(C.19)
(C.20)
183
?gm
x, y + ?, ? =
?gm
?x
и?
x, y, ? exp ?i
co x
(C.21)
?gn? x, y, ? ?gm x, y, ? may now be written as,
?gn? x, y, ? ?gm x, y, ?
i? xm
?i? xn
x ? y /a?
x ? y /a?
exp
?i?
exp
i?
=
4?a3? x x
4?a3? x x
xn xm
?2
=
2
6
2
16? a? x x2
and substituted in for the vector Green?s functions.
(C.22)
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Vita
Steven Arthur Eric Miller
Steven Arthur Eric Miller attended Michigan State University, studied Mechanical Engineering, and subsequently earned his Bachelor of Science in December
2003. While at Michigan State University he performed undergraduate research in
Aerodynamics and Aeroacoustics under Professor Mei Zhuang. He began graduate
studies in Aerospace Engineering at The Pennsylvania State University in January
2004 under the guidance of Boeing/A.D. Welliver Professor Philip J. Morris. He
was awarded his Master of Science degree in May 2006 with a thesis entitled, ?The
Aerodynamics of Wind Turbine Airfoils.? Immediately after earning his Master
of Science he began work on his Doctor of Philosophy in Aerospace Engineering
under the same adviser. In addition to research activities at The Pennsylvania
State University he acted as a teaching assistant or an instructor for a number
of Aerospace Engineering courses. After completion, he will accept a research
position at NASA Langley and continue research in the field of Aeroacoustics.
ecomes quite extreme at high and low observer
angles. The peak frequencies of the predictions could be improved by using a
better approximation of the vector Green?s function to include secondary flow
effects. The BBSAN predicted amplitudes compare very well with experiment. At
high observer angles of Figure 3.22 there are regions where no prediction is present
near St = 0.50. This is in contrast with the results of Figure 3.23 where there
is a BBSAN contribution across the same St range. The higher speed secondary
stream of Figure 3.22 causes a stronger Doppler shift to the predicted BBSAN and
shifts the peaks to lower frequency. The prediction could be improved by lowering
c? which would have the effect of broadening the BBSAN peaks, thus eliminating
the region where no BBSAN is predicted.
142
Mdp
1.00
1.00
Mjp
1.19
1.19
Mds
1.00
1.00
Mjs
0.96
1.04
T T Rp
2.70
2.70
T T Rs
1.00
1.00
Dp (m)
0.0622
0.0622
fcp
9398
9398
?p
0.645
0.645
?s
N/A
0.578
Table 3.5. Jet operating conditions for the RANS CFD and BBSAN predictions of the
dual stream jets.
Figure 3.21. Integration regions for the BBSAN calculation of the dualstream jet
Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 1.04, T T Rp = 2.70.
143
Figure 3.22. Comparisons of BBSAN predictions with experiments for the dualstream
jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 1.04, T T Rp = 2.70, R/D = 100.
144
Figure 3.23. Comparisons of BBSAN predictions with experiments for the dualstream
jet Mdp = 1.00, Mjp = 1.19, Mds = 1.00, Mjs = 0.96, T T Rp = 2.70, R/D = 100.
145
3.6
Three Dimensional Jets
Thus far axisymmetric single and dual stream CFD RANS solutions have been used
to perform BBSAN predictions. These used the axisymmetric form of the BBSAN
model equation and its implementation. The flow-fields produced by rectangular
or non-axisymmetric jets are highly three-dimensional and are not applicable to
the axisymmetric BBSAN model. Here, the BBSAN model is evaluated with the
three dimensional implementation of the BBSAN code using the rectangular jet
RANS solutions validated for rectangular jets in Chapter 2. Both cold rectangular
jets with Md = 1.50 were validated with Pitot and schlieren comparisons. The two
off-design rectangular jet simulations with T T R = 2.20 were validated by comparing schlieren of a simulated hot jet using a helium / air mixture. Table 3.6 shows
the rectangular nozzle operating parameters that are used for the corresponding
BBSAN predictions. The rectangular nozzle chosen has Md = 1.50, and operates
either over- or under-expanded at Mj = 1.30 or Mj = 1.70 and T T R = 1.00 or
T T R = 2.20. The experimental results were provided by Veltin and the experimental details are documented in Goss et al. [92]. Microphone positions were placed at
1.905 meters from the jet at various ? in the minor and major axis planes and at
an azimuthal angle of 45 degrees. The experimental data was extended to 100 De
to be consistent with the predictions presented in the previous sections. Only predictions are presented in the major and minor axis planes because the aspect ratio
of the rectangular jet is relatively small, therefore the intensity of the BBSAN will
vary only a little with respect to azimuthal angle. The characteristic frequencies
of the rectangular jets are based on the fully expanded equivalent diameter found
from the De and the N P R.
Md
1.50
1.50
1.50
1.50
Mj
1.30
1.70
1.30
1.70
NP R
2.77
4.94
2.77
4.94
TTR
1.00
1.00
2.20
2.20
De (m)
0.01778
0.01778
0.01778
0.01778
Dej (m)
0.01693
0.01896
0.01693
0.01896
fc
22888
24499
33792
36334
?
0.748
0.800
0.748
0.800
Tj (K)
219.1
185.8
482.0
408.7
uj (m/s)
385.7
464.5
572.1
688.9
Table 3.6. Jet operating conditions for the RANS CFD and BBSAN predictions of the
3D calculations.
146
Unlike the axisymmetric version of the code, the three-dimensional BBSAN
implementation evaluates the integrals over the three dimensional integration region and wavenumber region. Since there is an additional integral that needs to be
evaluated in the z-axis direction, the run time of the computer code can increase
greatly. To minimize the amount of time needed to make a three-dimensional BBSAN prediction, careful choices are made of the range and number of indices of the
integration and wavenumber regions. Based on the conclusions for the integration
studies of the axisymmetric jet shown in Table 3.3, integration ranges for the four
rectangular cases are chosen as 0.001m < x < 0.1437m, 0.001m < y < 0.011m,
and 0.001m < z < 0.021m with corresponding index values of 300, 75, and 75
in the streamwise, minor-axis, and major-axis directions respectively. Since the
RANS CFD simulations uses planes of symmetry of the nozzle, the implemented
BBSAN code automatically integrates across the planes. There is no restriction
in the implemented version of the BBSAN model regarding the flow-field, even
though planes of symmetry were used to construct the RANS solution. The same
implementation of BBSAN may be applied to any three-dimensional jet flow-field.
The range of the wavenumber region is chosen as 0.001m < xk < 0.250 m with 512
indices. Summation of the integration region is selective by only including contributions from 10% of K using cycling. This cycling technique saves a large amount
of computational time just as it had in the axisymmetric calculations. Also, the
solution is found relatively quickly by specifying a limited range for the integrations over wavenumber. The wavenumber integration range is carefully chosen to
only encompass peak values of the wavenumber spectrum. For example, in the
first rectangular BBSAN condition of Table 3.6, the spacing between each value
of wavenumber in the wavenumber spectrum is ?k = 12.57 and only wavenumber
contributions are used in the range of ?25 < k < ?166 and 25 < k < 166.
BBSAN predictions of the over-expanded cold rectangular jet in the major and
minor axis planes are shown in Figures 3.24 and 3.25. By comparing the BBSAN
predictions with the experimental data, it is seen that only a small 2 dB difference
is apparent between the BBSAN peaks in the major and minor axis. Additional
strong screech tones are present in the minor axis direction. The predictions in both
figures are the same as the azimuthal angle is not included in the current prediction
formula. This azimuthal dependance could easily be included if necessary. The
147
peak BBSAN at all observer locations matches well with respect to frequency and
magnitude. As in the circular nozzle case operating at Md = 1.50 and Mj = 1.30,
the overall prediction is slightly low. The width of the BBSAN peaks is also too
narrow relative to experimental measurements.
Figures 3.26 and 3.27 show the under-expanded unheated rectangular jet operating at Mj = 1.70. In this case, the variation of the magnitude of the BBSAN
is invariant with azimuthal angle, unlike the previous over-expanded case. The
predictions at all observer angles agree extremely well with the experimental data.
The first BBSAN peak and subsequent minor peaks at higher frequencies generally
align with those of experiment and the amount of fall-off in the spectrum is not
significant. The small high frequency peaks of the BBSAN are more pronounced
in the experimental data in the major axis plane. The predictions could be slightly
improved for this jet if c? is lowered to flatten out the strong dominant BBSAN
peak.
BBSAN predictions are performed for an over-expanded rectangular jet operating at Mj = 1.30 and T T R = 2.20. These predictions are shown in Figures 3.28
and 3.29 in the major and minor axes planes respectively. Variation of maximum
dB in the experiment with azimuthal angle only varies by up to 2 dB. Like the
heated axisymmetric case, the predictions are slightly lower than the experimental
data. The magnitude of BBSAN matches the experiment at high observer angles,
and also matches the peak frequencies. At ? = 90.0 degrees and lower the mixing
noise dominates the BBSAN. The width of the peaks, as with the axisymmetric jet,
are too narrow. This is especially true for low St when the prediction has extreme
fall-off. Note that for this heated prediction, along with all the other rectangular
jet predictions, the temperature correction factor has been applied.
The final rectangular predictions are shown in Figures 3.30 and 3.31 for the
major and minor axis directions respectively, for the Mj = 1.70 and T T R = 2.20
jet. The peak frequencies match those of the experiment very well at most angles
except for large values. BBSAN magnitudes align well perpendicular to the jet,
however, are over-predicting at high observer angles. At low observer angles the
BBSAN is dominated by the mixing noise because the jet is heated. The predicted
values of BBSAN do not contribute to the total spectra just as in the corresponding
heated circular jet case. The variation of the peak BBSAN magnitude is partly
148
controlled by cl . If cl is smaller, then the variation of BBSAN magnitude with
angle would better match the experiment.
Figure 3.24. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.30, T T R = 1.00, R/De = 100 in the major axis direction.
149
Figure 3.25. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.30, T T R = 1.00, R/De = 100 in the minor axis direction.
150
Figure 3.26. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.70, T T R = 1.00, R/De = 100 in the major axis direction.
151
Figure 3.27. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.70, T T R = 1.00, R/De = 100 in the minor axis direction.
152
Figure 3.28. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.30, T T R = 2.20, R/De = 100 in the major axis direction.
153
Figure 3.29. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.30, T T R = 2.20, R/De = 100 in the minor axis direction.
154
Figure 3.30. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.70, T T R = 2.20, R/De = 100 in the major axis direction.
155
Figure 3.31. Comparisons of BBSAN predictions with experiments for the rectangular
jet Md = 1.50, Mj = 1.70, T T R = 2.20, R/De = 100 in the minor axis direction.
156
3.7
The Effect of Laminar Flow in the Nozzle
BBSAN predictions have been made for the laminar / RANS and fully RANS jet
flows and compared to experimental data. This required the two CFD simulations
that were conducted to examine the effect of laminar or turbulent flow inside the
nozzle for a Md = 1.50, Mj = 1.30, T T R = 2.20 jet. The first CFD simulation
solved the RANS equations in the entire domain, and the second CFD simulations
solved the Navier-Stokes equations in the interior of the nozzle and the RANS
equations in the exterior. The second simulation thereby forces the flow to be
laminar inside the nozzle and turbulent outside the nozzle. In the laminar / RANS
case, a conical oblique shock originates from inside the nozzle. This is more similar
to an under-expanded jet than an over-expanded jet. The boundary layer in the
laminar nozzle separates due to the oblique shock originating inside the nozzle.
This is in contrast to the boundary layer in the RANS simulation where it remains
fully attached to the nozzle wall. BBSAN predictions based on the model formula
are conducted using no modification to the BBSAN code, as turbulence values are
available in the exterior region in both cases. Results are found by performing the
same spatial and full wavenumber integration of the model as previously discussed.
The spectral densities of both solutions are multiplied by Tcf .
Figure 3.32 shows the measured SPL at various microphone positions from
the PSU anechoic jet chamber, the prediction with Tam?s [58] model, and the
BBSAN predictions based on the two RANS solutions. The laminar nozzle case
shows slightly lower peak frequencies of BBSAN than the fully turbulent case,
and the SPL is 2 to 3 dB higher at all angles. Also, the BBSAN fall-off in the
laminar nozzle case is much steeper at high frequencies. These differences are due
to a lower effective exit diameter of the laminar / RANS jet due to separation of
the boundary layer. This lower effective diameter lowers the initial speed of the
turbulent shear layer of the jet thus lowering the predicted frequencies. In addition,
there are initially higher values of K and the existence of an oblique shock wave
at the nozzle lip of the fully RANS solution. This is in contrast where K near the
nozzle lip of the laminar / RANS jet is almost zero and no shock wave exists. This
explains the magnitude difference between the two simulations as the shock wave
turbulence interaction is the extra source of BBSAN in the fully RANS simulation.
157
Figure 3.32. Comparisons of BBSAN predictions for the laminar / RANS and fully
RANS simulation with experiments for a Md = 1.50, Mj = 1.30, T T R = 2.20, jet at
R/D = 100.
158
3.8
Helium / Air Mixture and Hot Air
BBSAN predictions were performed for helium / air mixtures and hot air using
the two RANS solutions previously presented in Chapter 2 Section 2.2.8. The two
RANS solutions were constructed from a hot air jet operating at Md = 1.00, Mj =
1.50, and T T R = 2.20 and a helium / air mixture operating at T T R = 1.00. The
helium / air jet simulates a jet with a T T R = 2.20. The local densities and Mach
numbers of the two simulations are very similar. BBSAN predictions based on the
model formula are conducted here with no modification to the implementation.
The integration, and wavenumber ranges and indices are the same in both cases.
Experimental data from Boeing used heated air at T T R = 2.20 while PSU used a
helium / air mixture to simulate heating.
Plots of BBSAN predictions for the jets with operating conditions of the PSU
and Boeing experiments are shown in Figure 3.33. Tcf is applied to the heated air
and the helium / air mixture predictions. Even though the helium / air mixture
is not heated, Tcf must be applied to the simulations to be consistent because the
mixture in the plenum is low density and Tcf will not be unity. The overall amplitude of both predictions, including that of the second model of Tam [58], are too
high relative to the Boeing experimental data and less so relative to the PSU data.
However, the magnitudes of the peaks of all three predictions are relatively the
same, though higher than the experimental data. Peak frequencies of the heated
prediction agree better with experimental data at observer angles perpendicular to
the jet, however, the helium / air mixture prediction shows much better agreement
in peak frequencies at high observer angles. Experimental data from Boeing, which
is heated, shows a lower peak frequency amplitudes than the helium / air mixture
experimental data of PSU. This is also reflected in the corresponding predictions.
The helium / air prediction also shows slower fall-off compared to the heated case.
Differences between the two predictions are partially due to the use of the Menter
SST turbulence model which does not have the PAB temperature correction of
the K ? model of Chien. Also, the hot air simulation uses an ideal gas model
while the helium / air simulation uses a three species model of helium, nitrogen,
and oxygen that does not account for other species. Furthermore, even though the
two simulations have similar streamwise velocities and Mach numbers as seen in
159
Chapter 2, the maximum K in the hot air simulation is 12,847 m2 /s2 while the
maximum K in the helium / air simulation is 15,047 m2 /s2 . This is primarily the
reason that the BBSAN magnitudes are slightly higher for the predictions.
Figure 3.33. Comparisons of BBSAN predictions for heated air and helium / air
simulated jets with experiments using heated air and helium / air mixtures. The jet
conditions are Md = 1.00, Mj = 1.50, T T R = 2.20, R/D = 100.
160
3.9
Turbulent Scale Coefficients
The coefficients, c? , cl , and c? , have been calibrated for a Md = 1.00, Mj =
1.50, T T R = 1.00 jet at ? = 100.0 degrees. These coefficients, and associated
scaling factors Pf and Tcf , have been used to make all predictions thus far in
this dissertation. The same coefficients are applied to each observer angle, nozzle
geometry, and jet conditions creating a true prediction scheme. The calibration
jet spectral density possesses screech tones. A screech tone changes the velocity
cross-correlation. Cross-correlations of screeching jets are very diffe
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