On Theoretical Broadband Shock-Associated Noise Near-Field Cross-Spectra

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1 On Theoretical Broadband Shock-Associated Noise Near-Field Cross-Spectra Steven A. E. Miller The National Aeronautics and Space Administration NASA Langley Research Center Aeroacoustics Branch AIAA Aeroacoustics Conference 5 June nd 6 th 5

2 Acknowledgements NASA Advanced Air Vehicles Program Commercial Supersonic Technology Project Alessandro Savarese of The Airbus Group James Bridges of NASA Glenn Research Center Krishnamurthy Viswanathan of The Boeing Company Many previous curious researchers June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov

3 Introduction June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov 3

Aerospace Flight Vehicles Need for ﬁrst principles

observers and on the ﬂight vehicle airframe F- 8

Sources in Model Scale Tests, Journal of Spacecraft

4 Aerospace Flight Vehicles Need for ﬁrst principles mathema;cal theory to predict cross- spectra at observers and on the ﬂight vehicle airframe F- 8 Carrier Takeoﬀ Image Credit: NASA/Boeing Website: NASA.gov June 5 Panda, J. and Mosher, R., `Microphone Phased Array to Identify Liftoff Noise Sources in Model Scale Tests, Journal of Spacecraft and Rockets, Vol. 5, No. 5., 3. DOI:.54/.A3433 Steven A. E. Miller, Ph.D., s.miller@nasa.gov 4

5 Modeling Goals Predict auto-spectra and cross-spectra of BBSAN Capture acoustic near-field and far-field effects Include the flight vehicle Mach number Desire simple closed-form equation Evaluate quickly Validate for Multiple jet Mach numbers and temperature ratios Far- and near-field auto-spectra and coherence Gain insight into the physical process June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov 5

6 Mathematical Approach June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov 6

7 Mathematical Overview Governing equations are Navier-Stokes Derive Lighthill analogy for pressure with ambient meanflow, and solve in most general fashion Form two-point correlation, subsequent crosspower spectral density, simplify, obtain Cross- Power Spectral Density Acoustic Analogy Model T ij T lm for BBSAN source, simplify, and obtain model dependent on turbulent statistics Model turbulent statistics Reduce model equation to simple double summation using reasonable assumptions June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov 7

8 Forming a Two-Point Pressure Correlation Form an acoustic analogy from the Navier-Stokes equations Solving and expanding the source terms Z Z p (x,t)= 4 " r i M,j T ij r c r + 3( Z p (x,t) p (x,t )= 6 A = r ir j r l r m r r ( r i r j r M ) T ij r 3 Z " Tij c r + 3(... " Tij c r + 3( # Z ij M ) T ij c r + 3( M ) T ij r 3 # " T ij c r + ( Finding the two-point correlation of the pressure where r i r j rr Other terms are similar M )T ij r 3 (A + A + A 3 + A 4 ) d d M ) T ij c r + 3( M ) T ij r 3 r l M,m r r i r j r " Tij c r + 3( lm " Tij c r + 3( #" T lm c r + 3( M ) T ij c r + 3( M ) T ij r 3 + r ir j M,l M,m r r 3 M ) T ij c r + 3( M ) T ij r 3 " Tij c r + 3( June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov 8 # +M,i M,j T ij r 3 d M ) T lm c r + 3( M ) T r 3 #" T lm c r + 3( M ) T r 3 #" T lm c r + ( M )T r 3 M ) T ij c r + 3( M ) T ij r 3 # lm lm lm # # # T lm

9 G (x, x,!)= 6 Z... where the far-field term is, and the mid-field term is, Cross-Spectral Acoustic Analogy Z 8 >< >: apple exp Far-Field Term z } { F t Tij T lm + i! F t = r ir j rl r m r r Mid-Field Term z } { M t T ij T lm + + x y c x y Near-Field Term z } { N t T ij T lm June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov 9 apple c 4 rr c M t = r ir j rl apple r m 9( M ) 3( M) 3( M) r r c r r c rr 3 c r 3 r r i r j rl apple 6( M ) M,m M,m r i r apple l r m 6( M ) r r r r rr 3 rr c + r ir l rr r i r j r r l r m r apple 6M,j M,m c r r c 9 >= >; d d d M,j M,j r r r 3 r apple 3( M ) lm c r r + ( M ) lm c rr 3 + M,lM,m c rr 3 apple 3( M ) ij c r r + ( M ) ij c r 3 r + M,iM,j c r 3 r + r apple apple i 4M,j lm r c r r + r l 4M,m ij ij lm r c r r + c r r Miller, S. A. E., of Near-Field Jet Cross Spectra, AIAA Journal, 5. doi:.54/.j5364.

10 Source Modeling Two point correlation is modeled as T ij T 4 Rv ijlm(y,, ) For broadband shock-associated noise a great choice is Rijlm v = k A ijlm p s p s, / s 4! 4 c R ijlm That is compatible with previous modeling efforts Z apple R ijlm =exp u 4 apple Aijlm p s p s, 4 / s 4! 4 c exp u s apple ( u ) exp l s apple exp June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov l sy apple exp Equivalent source model and cross-spectral acoustic analogy requires multiple integrations with retarded time. One is = A ijlm 4 s p s p s, k c apple l s u exp apple ( u ) exp ls u s apple exp lsy apple exp lsy apple exp lsz apple exp exp l sz l sz exp [ "! 4iu + l s! 4u i! ] d Double spatial integrals performed analytically in cylindrical coordinates Note sources are located at relatively discrete locations #.

11 Cross-Spectrum of BBSAN Model Equation The resulting model equation for the cross-spectrum of BBSAN is G (x, x,!) = S xsr XN s XN s apple A ijlm p s,m p s,n kl s z m z n l 4c 4 c u 4 m= n= s r exp s! r 4u apple apple apple 4iu(y y )! y y (z exp 4u exp exp m + zn) u s lsr apple x y exp i! x y rm r n I c c lsr Cross-Power Spectral Density Double summation over shock wave shear layer positions, N s Grouped equivalent source term Exponential equivalent source terms Descriptions of terms Modified Bessel function of the first kind Note if x = x then evaluation guarantees power-spectral density June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov

12 Results June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov

13 Far-Field Auto-Spectra s of BBSAN auto-spectra Far-field at R/D = Observers on arc M d =. M j =.4 TTR =. D =.58 m s of, Bridges, J. and Brown, C. A., `Validation of the Small Hot Jet Acoustic Rig for Aeroacoustic Research, th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 5-846, 5. doi:.54/ SPL per unit St June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov 3 3 db 3 db =. db Ψ = 3 o =. db Ψ = o =.3 db Ψ = 9 o =.5 db Ψ = 7 o =. db Ψ = 5 o St - -

14 Far-Field Auto-Spectra M d =., M j =.57, TTR = 3. M d =.5, M j =.94, TTR =. = 4. db Ψ = 3 o = 9.4 db Ψ = 3 o 3 db SPL per unit St = 6. db Ψ = o = 7.9 db Ψ = 9 o 3 db SPL per unit St =. db Ψ = o = 4.3 db Ψ = 9 o = 5.9 db 3 db Ψ = 7 o 3 db = 5. db Ψ = 7 o = 7.4 db Ψ = 5 o = 5.5 db Ψ = 5 o - - St - - St June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov 4

15 4 Near-Field Auto-Spectral Contours Contours of SPL at St =.86 (6 khz) from M d =.5, M j =.67, and TTR =. jet r/d - / x/d r/d - / 5 5 June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov db 8 db 36 db 4 db 44 db x/d Experiment s of, Yu, J. C., `Investigation of the Noise Fields of Supersonic Axisymmetric Jet Flows, Ph.D. Thesis, Syracuse University, db db 4 db

16 Near-Field Auto-Spectral Contours Contours of SPL at St = 4.67 (4 khz) from M d =.5, M j =.67, and TTR =. jet db Experiment r/d - / x/d r/d - / 5 5 s of, Yu, J. C., `Investigation of the Noise Fields of Supersonic Axisymmetric Jet Flows, Ph.D. Thesis, Syracuse University, 97. June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov db 34 db 4 db x/d 38 db

17 33 3 Near-Field Auto-Spectral Contours s Only Contours of SPL from M d =.5, M j =.67, and TTR =. jet Near-field detail showing coalescence of BBSAN radiation St =.86 (6 khz) St = 4.67 (4 khz) r/d - / r/d - / x/d x/d June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov 7

18 Far-Field s of BBSAN coherence Far-field at R/D = Observers at sideline Observer pairs vary azimuthally M d =. M j =.4 TTR = φ = o - - St φ = 3 o φ = o - - St φ = 4 o - - St - - St s of, Viswanathan, K., Alkislar, M. B., and Czech, M. J., `Characteristics of the Shock Noise Component of Jet Noise, AIAA Journal, Vol. 48, No.,, pp doi:.54/.385. June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov 8

19 Far-Field s of BBSAN coherence Far-field at R/D = Observers at sideline Observer pairs vary azimuthally M d =. M j =.57 TTR = φ = o - - St φ = 3 o φ = o - - St.5 φ = 4 o - - St - - St s of, Viswanathan, K., Alkislar, M. B., and Czech, M. J., `Characteristics of the Shock Noise Component of Jet Noise, AIAA Journal, Vol. 48, No.,, pp doi:.54/.385. June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov 9

20 Near-Field s of BBSAN coherence Near-field at R/D = 4 Reference observer at sideline Observer pairs on linear array and vary axially M d =. M j =.6 TTR = June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov St St x = -.5D x =.65D St St x = -.65D x =.5D - s of, Savarese, A., Jordan, P., Royer, A., Fourment, C., Collin, E., Gervais, Y., and Porta, M., `Experimental Study of Shock- Cell Noise in Underexpanded Supersonic Jets, 9th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 3-8, 3. doi:.54/6.3-8.

21 Near-Field s of BBSAN coherence Near-field at R/D = 4 Reference observer at sideline Observer pairs on linear array and vary axially M d =. M j =.8 TTR = June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov St St x = -.5D x =.65D St St x = -.65D x =.5D - s of, Savarese, A., Jordan, P., Royer, A., Fourment, C., Collin, E., Gervais, Y., and Porta, M., `Experimental Study of Shock- Cell Noise in Underexpanded Supersonic Jets, 9th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 3-8, 3. doi:.54/6.3-8.

22 Summary and Conclusion Formed equation for near-field BBSAN cross-spectra using cross-spectral acoustic analogy Equivalent source term captures physics and consistent with other models Validated near- and far-field, auto-spectra and coherence, with measurements Near-field auto-spectra show formation of increasing numbers of directional lobes Large variations observed in source intensity at each shock wave shear layer interaction Large broad lobes of coherence due to sound radiation from shock wave shear layer interactions, at lower and higher frequencies are due to other effects June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov

23 Questions June 5 Steven A. E. Miller, Ph.D., s.miller@nasa.gov 3

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

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