Skip to main content Accessibility help

A volume integral implementation of the Goldstein generalised acoustic analogy for unsteady flow simulations

  • Vasily A. Semiletov (a1) (a2) and Sergey A. Karabasov (a1) (a2)


A new volume integral method based on the Goldstein generalised acoustic analogy is developed and directly applied with large-eddy simulation (LES). In comparison with the existing Goldstein generalised acoustic analogy implementations, the current method does not require the computation and recording of the expensive fluctuating stress autocovariance function in the seven-dimensional space–time. Until now, the multidimensional complexity of the generalised acoustic analogy source term has been the main barrier to using it in routine engineering calculations. The new method only requires local pointwise stresses as an input that can be routinely computed during the flow simulation. On the other hand, the new method is mathematically equivalent to the original Goldstein acoustic analogy formulation, and, thus, allows for a direct correspondence between different effective noise sources in the jet and the far-field noise spectra. The implementation is performed for conditions of a high-speed subsonic isothermal jet corresponding to the Rolls-Royce SILOET experiment and uses the LES solution based on the CABARET solver. The flow and noise solutions are validated by comparison with experiment. The accuracy and robustness of the integral volume implementation of the generalised acoustic analogy are compared with those based on the standard Ffowcs Williams–Hawkings surface integral method and the conventional Lighthill acoustic analogy. As a demonstration of its capabilities to investigate jet noise mechanisms, the new integral volume method is applied to analyse the relative importance of various noise generation and propagation components within the Goldstein generalised acoustic analogy model.


Corresponding author

Email address for correspondence:


Hide All
Afsar, M. Z., Goldstein, M. E. & Fagan, A. 2011 Enthalpy-flux/momentum-flux coupling in the acoustic spectrum of heated jets. AIAA J. 49 (11), 25222531.
Afsar, M. Z., Sescu, A. & Leib, S. J.2016 Predictive capability of low frequency jet noise using an asymptotic theory for the adjoint vector Green’s function in non-parallel flow. AIAA Paper 2016–2804.
Afsar, M. Z., Sescu, A., Sassanis, V. & Lele, S. K. 2017 Supersonic jet noise predictions using a unified asymptotic approximation for the adjoint vector Green’s function and LES data. In 23rd AIAA/CEAS Aeroacoustics Conference, AIAA AVIATION Forum. AIAA Paper 2017–3030.
Bogey, C. & Bailly, C. 2010 Influence of nozzle-exit boundary-layer conditions on the flow and acoustic fields of initially laminar jets. J. Fluid Mech. 663, 507540.
Bogey, C., Bailly, C. & Juve, D.2001 Noise computation using Lighthill’s equation with inclusion of mean flow–acoustic interactions. AIAA Paper 2001-2255.
Bogey, C., Marsden, O. & Bailly, C. 2012 Influence of initial turbulence level on the flow and sound fields of a subsonic jet at a diameter-based Reynolds number of 105 . J. Fluid Mech. 701, 352385.
Brentner, K. S. & Farassat, F. 1998 Analytical comparison of the acoustic analogy and Kirchhoff formulation for moving surfaces. AIAA J. 36 (8), 13791386.
Bres, G. A., Jaunet, V., Le Rallic, M., Jordan, P., Colonius, T. & Lele, S. K.2015 LES for jet noise: the importance of getting the boundary layer right. AIAA Paper 2015-2535.
Bres, G. A., Nichols, J. A., Lele, S. K., Ham, F. E., Schlinker, R. H., Reba, R. A. & Simonich, J. C. 2012 Unstructured large eddy simulation of a hot supersonic over-expanded jet with chevrons. In 18th AIAA/CEAS Aeroacoustics Conference, 33rd AIAA Aeroacoustics Conference. AIAA Paper 2012-2213.
Bridges, J.2010 Establishing consensus turbulence statistics for hot subsonic jets. AIAA Paper 2010-3751.
Bridges, J. & Wernet, M.2003 Measurements of aeroacoustic sound sources in turbulent jets. AIAA Paper 2003-3130.
Cavalieri, A. V. G., Rodriguez, D., Jordan, P., Colonius, T. & Gervais, Y. 2013 Wavepackets in the velocity field of turbulent jets. J. Fluid Mech. 730, 559592.
Chintagunta, A., Naghibi, S. E. & Karabasov, S. A. 2018 Flux-corrected dispersion-improved CABARET schemes for linear and nonlinear wave propagation problems. Comput. Fluids 169, 111128.
Curle, N. 1955 The influence of solid boundaries upon aerodynamic sound. Proc. R. Soc. Lond. A 231 (1187), 505514.
Depuru Mohan, N. K., Dowling, A. P., Karabasov, S. A., Xia, H., Graham, O., Hynes, T. P. & Tucker, P. G. 2015 Acoustic sources and far-field noise of chevron and round jets. AIAA J. 53 (9), 24212436.
Faranosov, G. A., Goloviznin, V. M., Karabasov, S. A., Kondakov, V. G., Kopiev, V. F. & Zaitsev, M. A. 2013 CABARET method on unstructured hexahedral grids for jet noise computation. Comput. Fluids 88, 165179.
Ffowcs Williams, J. E. 1963 The noise from turbulence convected at high speed. Phil. Trans. R. Soc. Lond. 255, 469503.
Ffowcs Williams, J. E. & Hawkings, D. L. 1969 Sound generation by turbulence and surfaces in arbitrary motion. Phil. Trans. R. Soc. Lond. A 264, 32142.
di Francescantonio, P. 1997 A new boundary integral formulation for the prediction of sound radiation. J. Sound Vib. 202 (4), 491509.
Freund, J. B. 2003 Noise-source turbulence statistics and the noise from a Mach 0.9 jet. Phys. Fluids 15 (6), 17881799.
Fureby, C. & Grinstein, F. F. 2002 Large eddy simulation of high-Reynolds-number free and wall-bounded flows. J. Comput. Phys. 181, 6897.
Goldstein, M. E. 1975 The low frequency sound from multipole sources in axisymmetric shear flows, with application to jet noise. J. Fluid Mech. 70 (3), 595604.
Goldstein, M. E. 2002 A unified approach to some recent developments in jet noise theory. Intl J. Aeroacoust. 1 (1), 116.
Goldstein, M. E. 2003 A generalized acoustic analogy. J. Fluid Mech. 488, 315333.
Goldstein, M. E. 2010 Relation between the generalized acoustic analogy and Lilley’s contributions to aeroacoustics. Intl J. Aeroacoust. 9 (4–5), 401418.
Goldstein, M. E. 2011 Recent developments in the application of the generalized acoustic analogy to jet noise prediction. Intl J. Aeroacoust. 10 (2–3), 89116.
Goldstein, M. E. & Leib, S. J. 2008 The aero-acoustics of slowly diverging supersonic jets. J. Fluid Mech. 600, 291337.
Goldstein, M. E. & Leib, S. J.2016 Azimuthal source non-compactness and mode coupling in sound radiation from high-speed axisymmetric jets. AIAA Paper 2016-2803.
Goldstein, M. E., Sescu, A. & Afsar, M. Z. 2012 Effect of non-parallel mean flow on the Green’s function for predicting the low-frequency sound from turbulent air jets. J. Fluid Mech. 695, 199234.
Goloviznin, V. M. & Samarskii, A. A. 1998 Finite difference approximation of convective transport equation with space splitting time derivative. J. Matem. Mod. 10 (1), 86100.
Hussein, H. J., Capp, S. P. & George, W. K. 1994 Velocity measurements in a high-Reynolds-number, momentum conserving, axisymmetric, turbulent jet. J. Fluid Mech. 258, 3175.
Ingraham, D. & Bridges, J. E.2017 Validating a monotonically-integrated large eddy simulation code for subsonic jet acoustics. AIAA Paper 2017-0456.
Karabasov, S. A. 2010 Understanding jet noise. Phil. Trans. R. Soc. Lond. A 368, 35933608.
Karabasov, S. A., Afsar, M. Z., Hynes, T. P., Dowling, A. P., McMullan, W. A., Prokora, C. D., Page, G. J. & McGuirk, J. J. 2010 Jet noise: acoustic analogy informed by large eddy simulation. AIAA J. 48 (7), 13121325.
Karabasov, S. A., Bogey, C. & Hynes, T. P. 2013 An investigation of the mechanisms of sound generation in initially laminar, subsonic jets using the Goldstein acoustic analogy. J. Fluid Mech. 714, 2457.
Karabasov, S. A. & Goloviznin, V. M. 2009 Compact accurately boundary adjusting high-resolution technique for fluid dynamics. J. Comput. Phys. 228, 74267451.
Karabasov, S. A. & Hynes, T. P.2006 Adjoint linearized Euler solver in the frequency domain for jet noise modelling. AIAA Paper 2006-2673.
Karabasov, S. A. & Sandberg, R. D. 2015 Influence of free stream effects on jet noise generation and propagation within the Goldstein acoustic analogy approach for fully turbulent jet inflow boundary conditions. Intl J. Aeroacoust. 14 (3–4), 413430.
Lau, J. C, Morris, P. J. & Fisher, M. J. 1979 Measurements in subsonic and supersonic free jets using a laser velocimeter. J. Fluid Mech. 93 (1), 127.
Leib, S. J. & Goldstein, M. E. 2011 Hybrid source model for predicting high-speed jet noise. AIAA J. 49 (7), 13241335.
Leib, S. J., Ingraham, D. & Bridges, J. E.2017 Evaluating source terms of the generalized acoustic analogy using the jet engine noise reduction (JENRE) code. AIAA Paper 2017-0459.
Lighthill, M. J. 1952 On sound generated aerodynamically. Part I. General theory. Proc. R. Soc. Lond. 211 (1107), 564587.
Lilley, G. M. 1958 On the noise from air jets. Aero. Res. Counc. R&M 20, 376.
Markesteijn, A. P. & Karabasov, S. A. 2018 CABARET solutions on graphics processing units for NASA jets: grid sensitivity and unsteady inflow condition effect. C. R. Méc. doi:10.1016/j.crme.2018.07.004.
Morris, P. J. & Zaman, K. B. M. Q. 2010 Velocity measurements in jets with application to noise source modeling. J. Sound Vib. 329, 394414.
Najafi-Yazdi, A., Bres, G. A. & Mongeau, L. 2011 An acoustic analogy formulation for moving sources in uniformly moving media. Proc. R. Soc. Lond. A 467 (2125), 144165.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
Samanta, A., Freund, J. B., Wei, M. & Lele, S. K. 2006 Robustness of acoustic analogies for predicting mixing-layer noise. AIAA J. 44, 27802786.
Semiletov, V. A. & Karabasov, S. A. 2013 CABARET scheme with conservation-flux asynchronous time-stepping for nonlinear aeroacoustics problems. J. Comput. Phys. 253 (15), 157165.
Semiletov, V. A. & Karabasov, S. A.2014a Adjoint linearised Euler solver for Goldstein acoustic analogy equations for 3D non-uniform flow sound scattering problems: verification and capability study. AIAA Paper 2014-2318.
Semiletov, V. A. & Karabasov, S. A. 2014b CABARET scheme for computational aero acoustics: extension to asynchronous time stepping and 3D flow modelling. Intl J. Aeroacoust. 13 (3–4), 321336.
Semiletov, V. A. & Karabasov, S. A. 2017 On the similarity scaling of jet noise sources for low-order jet noise modelling based on the Goldstein generalised acoustic analogy. Intl J. Aeroacoust. 16 (6), 476490.
Semiletov, V. A., Karabasov, S. A., Chintagunta, A. & Markesteijn, A. P.2015 Empiricism-free noise calculation from LES solution based on Goldstein generalized acoustic analogy: volume noise sources and meanflow effects. AIAA Paper 2015-2536.
Shur, M. L., Spalart, P. R. & Strelets, M. Kh. 2005 Noise prediction for increasingly complex jets. Part I: methods and tests. Part II: applications. Intl J. Aeroacoust. 4 (3–4), 21366.
Shur, M. L., Spalart, P. R., Strelets, M. Kh. & Travin, K. T. 2015 An enhanced version of DES with rapid transition from RANS to LES in separated flows. Turbulence Flow Combust. 95 (4), 709737.
SILOETProgramme. Rolls-Royce private data.
Tam, C. K. W. & Auriault, L. 1998 Mean flow refraction effects on sound from localized sources in a jet. J. Fluid Mech. 370, 149174.
Tam, C. K. W., Viswanathan, K., Ahuja, K. K. & Panda, J. 2008 The sources of jet noise: experimental evidence. J. Fluid Mech. 615, 253992.
Viswanathan, K. 2004 Aeroacoustics of hot jets. J. Fluid Mech. 516, 3982.
Viswanathan, K. 2009 Mechanisms of jet noise generation: classical theories and recent developments. Intl J. Aeroacoust. 615, 253992.
Welch, P. D. 1967 The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust. 15, 7073.
Witze, P. O. 1974 Centerline velocity decay of compressible free jets. AIAA J. 12 (4), 417418.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed