Post on 18-Aug-2020
Semi-Empirical Prediction of Noise from Non-Zero Pressure Gradient
Turbulent Boundary Layers
S. A. E. Miller
University of Florida
Department of Mechanical and Aerospace EngineeringTheoretical Fluid Dynamics and Turbulence Group
saem@ufl.edu
https://faculty.eng.ufl.edu/fluids
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 1
Outline
• Introduction and Background• Mathematical Theory
• Results• Flow
• Aeroacoustics
• Summary and Conclusion
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 2
Introduction and Background
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 3
Turbulent Boundary Layers
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 4
Lee, J. H., Kwon, Y. S., Monty, J. P., and Hutchins, N., “Tow-TankInvestigation of the Developing Zero-Pressure-Gradient Turbulent BoundaryLayer,” 18th Australasian Fluid Mechanics Conference, 2012.
Present within almost all flow-fields of aerospace flight vehicles
This paper based on recent publication - Miller, S. A. E., “Prediction of Turbulent Boundary-Layer Noise,” AIAA Journal, 2017. doi:10.2514/1.J055087.
Objective: Develop acoustic analogy for prediction of TBL noise with NZPG
Previous Investigations (Select)Boundary Layers with Pressure Gradients
• Investigations focusing on turbulence with NZPG are rare• Scaling statistics of turbulence in NZPG TBL is an open
problem• Kovasznay (1970) characterizes pressure gradient using
a non-dimensional approach
• Kline (1967) examined mean velocity profiles of various boundary layers characterized by K
• Castillo (1997), collapsed the meanflow for pipes and channel flows using power laws and showed collapse possible• Opens possibility of correctly predicting boundary layer
meanflow with a pressure gradient• No insight on effect of pressure gradient on turbulent statistics
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 5
K = ⌫(⇢u31)�1
@p/@x
Previous Investigations (Select)Boundary Layers with Pressure Gradients -
Aeroacoustics• Powell (1960) used Lighthill's acoustic analogy in conjunction with
mirror source and showed that acoustic power is proportional tovolumetric integral of second time derivative of Lighthill stress tensormultiplied by
• Powell showed that pressure on wall is an aerodynamic imprint ofturbulence and Naka et al. (2015) supported this viewpoint
• Howe (1991) related the wall wavenumber pressure spectrum to theacoustic spectrum
• Glegg et al. (2007) created a model that depends on the wavenumberspectrum of the surface pressure fluctuations
• Hu et al. (2003, 2006), performed DNS combined with an acousticanalogy and a half-space Green's function
• Gloerfelt and Berland (2013) and Gloerfelt and Margnat (2014)performed LES of compressible turbulent boundary layer at three highspeed Mach numbers. Predictions showed excellent agreement but hadconsiderable computational cost
• Miller (2017) used an acoustic analogy with mathematical models of thesource terms to predict noise from turbulent boundary layers using anacoustic analogy and the results agreed excellently with Gloerfelt andBerland (2013) and Gloerfelt and Margnat (Gloerfelt2014)
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 6
⇢�11 c�5
1
Previous Scaling Analysis
Previous paper of Miller (AIAAJ 2017) we have showed scaling of TBL noise goes as
and within the far-field as
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 7
Sfar-field / 1
r2c2fc41
⌧s�
✓⇢⇢1⇢w
◆2
u81V
S / c2f
⇢2✓⇢1⇢w
◆2
lsx
lsy
lsz
⌧s
⇢u4u4
1c41r2l4
sx
+u2u4
1c21r4l2
sx
+u41r6
�V
Mathematical Theory
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 8
Theoretical Approach
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 9
Wesolve,expand,converttopressure,simplify,andobtain,
p (x, t) =1
4⇡
Z 1
�1
Z 1
�1
Z 1
�1
rirjr2
"Tij
c21r+
3(1�M21)Tij
c1r2+
3(1�M21)2Tij
r3
#
�2riM1,j
r
"2Tij
c1r2+
3(1�M1)2Tij
r3
#
��ij
"Tij
c1r2+
(1�M21)Tij
r3
#
+2M1,iM1,jTij
r3d⌘
where,
r = |x� y|+M1 · (x� y) y = ⌘ � c1M1t+M1|x� y|
Lighthill’s acousticanalogy,
and,
@
2⇢
@t
2� c
21
@
2⇢
@xi@xi=
@
2Tij
@xi@xj
Theoretical Approach
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 10
p (x, t) p (x0, t0) =1
16⇡2
Z 1
�1...
Z 1
�1(A1 +A2 +A3 +A4) d⌘
0d⌘,
Forced to take a statistical approach…We perform the two-point cross-correlation of p at x and x’ and t and t’
A1 =
rirjr0lr0m
r2r02
"Tij
c21r+
3(1�M21)Tij
c1r2+
3(1�M21)2Tij
r3
#"T 0
lm
c21r0+
3(1�M21)T 0
lm
c1r02+
3(1�M21)2T 0
lm
r03
#
�2rirjrr
r0lM1,m
r0
"Tij
c21r+
3(1�M21)Tij
c1r2+
3(1�M21)2Tij
r3
#"2T 0
lm
c1r02+
3(1�M1)2T 0lm
r03
#
�rirjr2
�lm
"Tij
c21r+
3(1�M21)Tij
c1r2+
3(1�M21)2Tij
r3
#"T 0lm
c1r02+
(1�M21)T 0
lm
r03
#
+2rirjr2
M1,lM1,m
r03
"Tij
c21r+
3(1�M21)Tij
c1r2+
3(1�M21)2Tij
r3
#T 0lm
where for example,
and unfortunately A2, A3, and A4 are just as complicated!
Theoretical Approach
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 11
We now group terms according to their contribution to the far-field, mid-field, and near-field,
p (x, t) p (x0, t0) =1
16⇡2
Z 1
�1...
Z 1
�1FtTij T 0
lm +MtTij T 0lm +NtTijT 0
lmd⌘0d⌘,
Ft =rirjr0lr
0m
r2r02
1
c41rr0
�where the far-field term is,
Mt =rirjr0lr
0m
r2r02
9(1�M2
1)2
c21r2r02� 3(1�M2
1)2
c21rr03� 3(1�M2
1)2
c21r3r0
�
�rirjr0lr2r0
6(1�M2
1)
c21
✓2M1,m
r2r02� M1,m
rr03
◆�� rir0lr
0m
rr02
6(1�M2
1)
c21
✓2M1,j
r2r02� M1,j
r3r0
◆�
�rirjr2
3(1�M2
1)�lmc21r2r02
+(1�M2
1)�lmc21rr03
+2M1,lM1,m
c21rr03
�
�r0lr0m
r02
3(1�M2
1)�ijc21r2r02
+(1�M2
1)�ijc21r3r0
+2M1,iM1,j
c21r3r0
�
+rir0lrr0
16M1,jM1,m
c21r2r02
�+
rir
4M1,j�lmc21r2r02
�+
r0lr0
4M1,m�ijc21r2r02
�+
�ij�lmc21r2r02
and the mid-field term is,
The near-field term is even larger.
Two-Point Cross-Correlation
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 12
The model integrations are based upon the mixed Gaussian-exponentially decaying model of the two-point cross-correlation of the equivalent source
ls = a4�
2
64✓2⇡a1f
uc
◆2
+a22⇣
2⇡f�u⌧
⌘2+
⇣a2a3
⌘2
3
75
� 12
We estimate the length scale within R by adopting the model of Efimtsov(1982)
R = exp
� (⇠ � u⌧)2
l2sx
�exp
� (1� tanh[↵|⇠|])|⇠ � u⌧ |
lsx
�exp
� |⇠|lsx
�exp
� |⌘|lsy
�exp
� |⇣|lsz
�
where a1 = 0.1, a2 = 72.8, a3 = 1.54, and a4 = 6. The spanwise length scale uses an alternative set of coefficients, where a1 = 0.1, a2 = 548, a3 = 13.5, and other values of ai remain the same
Final Model Equation
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 13
The coefficient matrix Aijlm is
Aijlm ⇡ Pf⇢ ⇢0uiuj u0lu
0m,
and
I =
8>>>>>>>><
>>>>>>>>:
12u4
l4sx
⇡1/2lsx
2u exp
hu2�2iu(l
sx
+2⇡)!�l2sx
!2+u tanh[↵⇠](�2(u+ilsx
!)+u tanh[↵⇠])4u2
i
⇥⇣exp
hil
sx
! tanh[↵⇠]u
ierfc
hu�il
sx
!�u tanh[↵⇠]2u
i+ exp
⇥il
sx
!u
⇤erfc
hu+il
sx
!�u tanh[↵⇠]2u
i⌘for ⇠ � 0
and
12u4
l4sx
⇡1/2lsx
2u exp
hu2�2iu(l
sx
+2⇡)!�l2sx
!2+u tanh[↵⇠](2(u�ilsx
!)+u tanh[↵⇠])4u2
i
⇥⇣erfc
hu�il
sx
!+u tanh[↵⇠]2u
i+ exp
hil
sx
!(1+tanh[↵⇠])u
ierfc
hu+il
sx
!+u tanh[↵⇠]2u
i⌘for ⇠ < 0.
S (x,!) = 4⇡�2
1Z
�1
...
1Z
�1
{AijlmlsylszFtI} d⇠d⌘
The spectral density of acoustic pressure is
FUN3D Steady RANS Simulations
• NASA Langley FUN3D Solver• 40k iterations per flow-field• Closed by Wilcox Reynolds stress model• Mach number 0.3, 0.5, 0.7, and 0.9• Pressure gradient imposed within the flow
through
• Imposed via term placement on RHS of momentum equation
• Resultant pressure gradient found numerically from solution
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 14
@p/@x = ⌫(⇢u31)�1
@p/@x
Evaluation of Spectral Density
• The final equation does not directly account for the wall
• We adopt the approach used by Powell (1960), who used the concept of the `mirror' source
• Sources reside within the turbulent flow-field
• There is an acoustic propagation delay from the mirror source
• Numerical integration is performed using the CFD solution
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 15
Results
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 16
Flow-Conditions
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 17
M1 Re
x
x
l
[m] ⌧
w
[Pa] � [m] u
⌧
[ms�1] y
+ Distance [m]0.30 6.818⇥106 1 20.43 3.073⇥10�2 4.117 3.667⇥10�6
0.50 1.136⇥107 1 53.24 2.976⇥10�2 6.646 2.271⇥10�6
0.70 1.591⇥107 1 100.2 2.916⇥10�2 9.116 1.656⇥10�6
0.90 2.045⇥107 1 160.7 2.872⇥10�2 11.55 1.307⇥10�6
M1 ptp�11 TtT�1
10.30 1.06443 1.018000.50 1.18621 1.050000.70 1.38710 1.098000.90 1.69130 1.16200
Steady RANS inlet boundary conditions
Theory based flow conditions
Numerically Derived Flow Properties
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 18
M1 Re
x
x
l
[m] ⌧
w
[Pa] � [m] u
⌧
[ms�1] y
+ Distance [m] @p/@x @p/@x [Pa m�1]0.30 6.818⇥106 1 12.512 7.792⇥10�3 3.223 4.683⇥10�6 0 -120.30 6.818⇥106 1 21.812 2.709⇥10�3 4.282 3.525⇥10�6 0.01 3360.30 6.818⇥106 1 30.531 5.957⇥10�3 5.089 2.966⇥10�6 0.02 3460.30 6.818⇥106 1 2.7360 1.170⇥10�2 1.493 1.010⇥10�5 -0.01 -7700.30 6.818⇥106 1 2.4245 9.896⇥10�5 1.401 1.077⇥10�5 -0.02 -28550.50 1.136⇥107 1 36.468 4.388⇥10�3 5.528 2.730⇥10�6 0 -210.50 1.136⇥107 1 45.299 4.314⇥10�3 6.204 2.433⇥10�6 0.01 2720.50 1.136⇥107 1 53.394 4.250⇥10�3 6.771 2.229⇥10�6 0.02 4500.50 1.136⇥107 1 30.283 4.499⇥10�3 5.039 2.995⇥10�6 -0.01 -4830.50 1.136⇥107 1 22.445 4.762⇥10�3 4.322 3.492⇥10�6 -0.02 -8280.70 1.591⇥107 1 71.474 2.624⇥10�3 7.853 1.923⇥10�6 0 -290.70 1.591⇥107 1 79.688 2.677⇥10�3 8.357 1.807⇥10�6 0.01 1490.70 1.591⇥107 1 87.488 2.721⇥10�3 8.816 1.713⇥10�6 0.02 2510.70 1.591⇥107 1 66.807 2.583⇥10�3 7.622 1.981⇥10�6 -0.01 -7560.70 1.591⇥107 1 61.686 2.558⇥10�3 7.341 2.057⇥10�6 -0.02 -12500.90 2.045⇥107 1 117.434 1.976⇥10�3 10.398 1.452⇥10�6 0 -260.90 2.045⇥107 1 124.998 1.990⇥10�3 10.823 1.395⇥10�6 0.01 130.90 2.045⇥107 1 133.133 7.066⇥10�4 11.394 1.326⇥10�6 0.02 9390.90 2.045⇥107 1 112.932 1.968⇥10�3 10.247 1.473⇥10�6 -0.01 -12880.90 2.045⇥107 1 108.354 1.963⇥10�3 10.084 1.497⇥10�6 -0.02 -2352
Computational Domain
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 19
Contours of M for M∞ = 0.50 and ∂p /∂x = 0
Example Residual History
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 20
Variation of residual of field-variables for M∞ = 0.90 and ∂p/∂x = 0.
Variation of u+ in Inner Coordinates as Function of M∞ and ∂p/∂x
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 21
M∞ = 0.30 M∞ = 0.50
Variation of u+ in Inner Coordinates as Function of M∞ and ∂p/∂x
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 22
M∞ = 0.70 M∞ = 0.90
Variation of ρ and T in Inner Coordinates for Various M∞
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 23
M∞ = 0.30 M∞ = 0.50
Variation of ρ and T in Inner Coordinates for Various M∞
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 24
M∞ = 0.70 M∞ = 0.90
Normalized Variation of urms in Inner Coordinates
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 25
M∞ = 0.30 M∞ = 0.50
Normalized Variation of urms in Inner Coordinates
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 26
M∞ = 0.70 M∞ = 0.90
Normalized Variation of the Root Mean of uv in Inner Coordinates
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 27
M∞ = 0.30 M∞ = 0.50
Normalized Variation of the Root Mean of uv in Inner Coordinates
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 28
M∞ = 0.70 M∞ = 0.90
Comparison of the Newly Developed Prediction Approach with the Predictions of Miller and the
LES Predictions of Gloerfelt and Margnat
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 29
Predictions of SPL per unit f with Various M∞ and ∂p/∂x
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 30
M∞ = 0.30 M∞ = 0.50
Predictions of SPL per unit f with Various M∞ and ∂p/∂x
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 31
M∞ = 0.70 M∞ = 0.90
Summary and Conclusion
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 32
Summary and Conclusion• Acoustic analogy connected to RANS algebraic Reynolds
stress model• Predictions agree with previous analytical model and well
validated LES
• Important findings• At low Mach numbers the statistics of turbulence, meanflow,
and acoustic radiation are highly affected by pressure gradient relative to high Mach number subsonic flows
• Small negative incremental steps in non-dimensional pressure gradient produce lower energy acoustic power spectra
• Spectra shift to lower frequencies and sound pressure levels with favorable pressure gradients
• Development of composite meanflow profiles and similarity of turbulent statistics with pressure gradient would allow a fully statistical model to be developed that does not rely on CFD
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 33
Thank You
Questions?
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 34
Modeling the Equivalent Source
June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 35
Tij T 0lmTij T 0
lmTijT 0
lm
We define,
TijT 0lm = Rijlm(y1,⌘, ⌧)
and argue based on the principles of Millionshchikov, M. D.,
@2
@⌧21Tij
@2
@⌧22T 0lm =
@4
@⌧4Tijlm(⌘, ⇠, ⌧) =
@4
@⌧4Rijlm(y1,⌘, ⌧)
and,
@
@⌧1Tij
@
@⌧2T 0lm =
@2
@⌧2Tijlm(⌘, ⇠, ⌧) =
@2
@⌧2Rijlm(y1,⌘, ⌧)
We need to create models for,
Thus, we only need to model as they are inter-related, (egwhat is Rijlm)?,
and, and,
TijT 0lm