Semi-Empirical Prediction of Noise from Non-Zero Pressure Gradient Turbulent Boundary Layers

Transcription

1 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 Engineering Theoretical Fluid Dynamics and Turbulence Group June 2017 UF MAE, Steve Miller, Ph.D., 1

2 Outline Introduction and Background Mathematical Theory Results Flow Aeroacoustics Summary and Conclusion June 2017 UF MAE, Steve Miller, Ph.D., 2

3 Introduction and Background June 2017 UF MAE, Steve Miller, Ph.D., 3

4 Turbulent Boundary Layers 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, doi: /1.j Objective: Develop acoustic analogy for prediction of TBL noise with NZPG Lee, J. H., Kwon, Y. S., Monty, J. P., and Hutchins, N., Tow-Tank Investigation of the Developing Zero-Pressure-Gradient Turbulent Boundary Layer, 18th Australasian Fluid Mechanics Conference, June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 4

5 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 K = ( u 3 1) 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

6 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 to volumetric integral of second time derivative of Lighthill stress tensor multiplied by 1 1 c1 5 Powell showed that pressure on wall is an aerodynamic imprint of turbulence and Naka et al. (2015) supported this viewpoint Howe (1991) related the wall wavenumber pressure spectrum to the acoustic spectrum Glegg et al. (2007) created a model that depends on the wavenumber spectrum of the surface pressure fluctuations Hu et al. (2003, 2006), performed DNS combined with an acoustic analogy 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 high speed Mach numbers. Predictions showed excellent agreement but had considerable computational cost Miller (2017) used an acoustic analogy with mathematical models of the source terms to predict noise from turbulent boundary layers using an acoustic analogy and the results agreed excellently with Gloerfelt and Berland (2013) and Gloerfelt and Margnat (Gloerfelt2014) June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 6

7 Previous Scaling Analysis Previous paper of Miller (AIAAJ 2017) we have showed scaling of TBL noise goes as S / c 2 f 2 1 w 2 l sxl syl sz s u 4 u 4 1 c 4 1r 2 l 4 sx + u2 u 4 1 c 2 1r 4 l 2 sx + u4 1 r 6 V and within the far-field as S far-field / 1 r 2 c 2 f c 4 1 s 1 w 2 u 8 1V June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 7

8 Mathematical Theory June 2017 UF MAE, Steve Miller, Ph.D., 8

9 Theoretical Approach Lighthill s acoustic analogy, We solve, expand, convert to pressure, simplify, and obtain, Z 1 Z 1 Z " 1 r i r j Tij r 2 c 2 1r + 3(1 p (x,t)= c 2 1 i 1 2r i M 1,j T j M 1) 2 T ij c 1 r 2 + 3(1 M 1) 2 2 T ij r 3 " ij 2T ij c 1 r 2 + 3(1 " T ij c 1 r 2 + (1 M 1) 2 T ij r 3 M 2 1)T ij r 3 # # # where, +2M 1,i M 1,j T ij r 3 d r = x y + M 1 (x y) and, y = c 1 M 1 t + M 1 x y June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 9

10 Theoretical Approach Forced to take a statistical approach We perform the two-point cross-correlation of p at x and x and t and t A 1 = r i r j r 0 l r0 m r 2 r 02 p (x,t) p (x 0,t 0 )= where for example, " Tij c 2 1r + 3(1 2r i r j rr Z M 1) 2 T ij c 1 r 2 + 3(1 M 1) 2 2 T ij r 3 r 0 l M 1,m r 0 r i r j r 2 " Tij c 2 1r + 3(1 lm " Tij c 2 1r + 3(1 Z 1 1 (A 1 + A 2 + A 3 + A 4 ) d 0 d, #" T 0 lm c 2 1r 0 + 3(1 M 1) 2 T ij c 1 r 2 + 3(1 M 1) 2 2 T ij r 3 + 2r ir j M 1,l M 1,m r 2 r 03 M 1) 2 T ij c 1 r 2 + 3(1 M 1) 2 2 T ij r 3 " Tij c 2 1r + 3(1 M 1) 2 T 0 lm c 1 r (1 M 1) 2 2 T 0 r 03 #" 2 T 0 lm c 1 r (1 M 1) 2 T 0 r 03 #" T lm 0 c 1 r 02 + (1 M 1)T 2 0 r 03 M 1) 2 T ij c 1 r 2 + 3(1 M 1) 2 2 T ij r 3 # lm lm lm # # # T 0 lm and unfortunately A 2, A 3, and A 4 are just as complicated! June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 10

11 Theoretical Approach We now group terms according to their contribution to the far-field, mid-field, and near-field, p (x,t) p (x 0,t 0 )= Z 1 where the far-field term is, and the mid-field term is, 1... Z 1 1 F t Tij T 0 lm + M t T ij T 0 lm + N t T ij T 0 lm d 0 d, F t = r ir j r 0 l r0 m r 2 r 02 June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 11 apple 1 c 4 1rr 0 M t = r ir j rl 0 apple r0 m 9(1 M 2 1 ) 2 3(1 M1) 2 2 3(1 M1) 2 2 r 2 r 02 c 2 1r 2 r 02 c 2 1rr 03 c 2 1r 3 r 0 r i r j rl 0 apple 6(1 M 2 1 ) 2M1,m M 1,m r i r 0 apple l r0 m 6(1 M 2 1 ) 2M1,j M 1,j r 2 r 0 c 2 1 r 2 r 02 rr 03 rr 02 c 2 1 r 2 r 02 r 3 r 0 apple r i r j 3(1 M 2 1 ) lm r 2 c 2 1r 2 r 02 + (1 M 1) 2 lm c 2 1rr M 1,lM 1,m c 2 1rr 03 rl 0 apple r0 m 3(1 M 2 1 ) ij r 02 c 2 1r 2 r 02 + (1 M 1) 2 ij c 2 1r 3 r 0 + 2M 1,iM 1,j c 2 1r 3 r 0 + r irl 0 apple 16M1,j M 1,m rr 0 c 2 1r 2 r 02 + r apple apple i 4M1,j lm r c 2 1r 2 r 02 + r0 l 4M1,m ij ij lm r 0 c 2 1r 2 r 02 + c 2 1r 2 r 02 The near-field term is even larger.

12 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 apple ( u ) 2 R =exp l 2 sx apple (1 tanh[ ]) u exp l sx apple exp l sx apple exp l sy apple exp l sz We estimate the length scale within R by adopting the model of Efimtsov (1982) l s = a a1 f 4 + u c 2 f u a a 2 a 3 where a 1 = 0.1, a 2 = 72.8, a 3 = 1.54, and a 4 = 6. The spanwise length scale uses an alternative set of coefficients, where a 1 = 0.1, a 2 = 548, a 3 = 13.5, and other values of a i remain the same June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 12

13 Final Model Equation The spectral density of acoustic pressure is Z 1 Z 1 S (x,!) = {A ijlm l sy l sz F t I}d d 1 1 The coefficient matrix A ijlm is A ijlm P f 0 u i u j u 0 l u0 m, and I = 8 >< >: 12u 4 l 4 sx 12u 4 l 4 sx h 1/2 l sx u 2u exp 2 2iu(l sx +2 )! l 2 sx!2 +u tanh[ ]( 2(u+il sx!)+u tanh[ ]) h i h 4u 2 i ilsx! tanh[ ] u ilsx! u tanh[ ] exp and 1/2 l sx 2u h u erfc u erfc 2u +exp il sx h! u erfc u+ilsx! h i u exp 2 2iu(l sx +2 )! l 2 sx!2 +u tanh[ ](2(u il sx!)+u tanh[ ]) i h 4u 2 i h ilsx!+u tanh[ ] 2u +exp ilsx!(1+tanh[ ]) u erfc u+ilsx!+u tanh[ ] 2u i i u tanh[ ] 2u i for 0 for < 0. June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 13

14 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 p/@x = ( u 3 1) 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

15 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 flowfield 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., 15

16 Results June 2017 UF MAE, Steve Miller, Ph.D., 16

17 Flow-Conditions Theory based flow conditions M 1 Re x x l [m] w [Pa] [m] u [ms 1 ] y + Distance [m] Steady RANS inlet boundary conditions M 1 p t p1 1 T t T June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 17

18 Numerically Derived Flow Properties M 1 Re x x l [m] w [Pa] [m] u [ms 1 ] y + Distance [Pa m June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 18

19 Computational Domain Contours of M for M = 0.50 and p / x = 0 June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 19

20 Example Residual History Variation of residual of field-variables for M = 0.90 and p / x = 0. June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 20

21 Variation of u + in Inner Coordinates as Function of M and p / x M = 0.30 M = 0.50 June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 21

22 Variation of u + in Inner Coordinates as Function of M and p / x M = 0.70 M = 0.90 June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 22

23 Variation of ρ and T in Inner Coordinates for Various M M = 0.30 M = 0.50 June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 23

24 Variation of ρ and T in Inner Coordinates for Various M M = 0.70 M = 0.90 June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 24

25 Normalized Variation of u rms in Inner Coordinates M = 0.30 M = 0.50 June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 25

26 Normalized Variation of u rms in Inner Coordinates M = 0.70 M = 0.90 June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 26

27 Normalized Variation of the Root Mean of uv in Inner Coordinates M = 0.30 M = 0.50 June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 27

28 Normalized Variation of the Root Mean of uv in Inner Coordinates M = 0.70 M = 0.90 June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 28

29 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., 29

30 Predictions of SPL per unit f with Various M and p / x M = 0.30 M = 0.50 June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 30

31 Predictions of SPL per unit f with Various M and p / x M = 0.70 M = 0.90 June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 31

32 Summary and Conclusion June 2017 UF MAE, Steve Miller, Ph.D., 32

33 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

34 Thank You Questions? June 2017 UF MAE, Steve Miller, Ph.D., 34

35 Modeling the Equivalent Source We need to create models for, T ij T 0 lm and, T ij Tlm 0 and, T ij T 0 lm We define, T ij T 0 lm = R ijlm(y 1,, ) and argue based on the principles of Millionshchikov, M. D., 1 T ij T 2 2 T 0 4 T 4 R ijlm(y 2 T 0 2 T 2 R ijlm(y 1,, ) Thus, we only need to model what is R ijlm )?, T ij T 0 lm as they are inter-related, (eg June 2017 UF MAE, Steve Miller, Ph.D., saem@ufl.edu 35