Semi-Empirical Prediction of Noise from Non-Zero Pressure ... · Semi-Empirical Prediction of Noise...

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

[email protected]

https://faculty.eng.ufl.edu/fluids

June 2017 UF MAE, Steve Miller, Ph.D., [email protected] 1

Outline

• Introduction and Background• Mathematical Theory

• Results• Flow

• Aeroacoustics

• Summary and Conclusion


Introduction and Background


Turbulent Boundary Layers


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


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)


⇢�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


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


Theoretical Approach


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



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!



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


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


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


@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


Results


Flow-Conditions


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


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


Contours of M for M∞ = 0.50 and ∂p /∂x = 0

Example Residual History


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


M∞ = 0.30 M∞ = 0.50

Variation of u+ in Inner Coordinates as Function of M∞ and ∂p/∂x


M∞ = 0.70 M∞ = 0.90

Variation of ρ and T in Inner Coordinates for Various M∞


M∞ = 0.30 M∞ = 0.50

Variation of ρ and T in Inner Coordinates for Various M∞


M∞ = 0.70 M∞ = 0.90

Normalized Variation of urms in Inner Coordinates


M∞ = 0.30 M∞ = 0.50

Normalized Variation of urms in Inner Coordinates


M∞ = 0.70 M∞ = 0.90

Normalized Variation of the Root Mean of uv in Inner Coordinates


M∞ = 0.30 M∞ = 0.50

Normalized Variation of the Root Mean of uv in Inner Coordinates


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


Predictions of SPL per unit f with Various M∞ and ∂p/∂x


M∞ = 0.30 M∞ = 0.50

Predictions of SPL per unit f with Various M∞ and ∂p/∂x


M∞ = 0.70 M∞ = 0.90

Summary and Conclusion


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


Thank You

Questions?


Modeling the Equivalent Source


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

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