Compressible Homogeneous Isotropic Turbulence

  • Pierre Sagaut
  • Claude Cambon


This chapter is devoted to isotropic compressible turbulence dynamics and modelling. The four main regimes (quasi-isentropic regime, nearly linear thermal regimes, subsonic regime and supersonic regimes) are extensively discussed using the most recent results. Coherent structures dynamics, including shocklets dynamics, is addressed. The link with classical aeroacoustic theories, e.g. Lighthill and Howe theories for noise generation and noise scattering by turbulence, is discussed.


  1. Bailly, C., Lafon, P., Candel, S.: Subsonic and supersonic jet noise predictions from statistical source models. AIAA J. 35(11), 1688–1696 (1997)ADSCrossRefzbMATHGoogle Scholar
  2. Bataille, F.: Etude d’une turbulence faiblement compressible dans le cadre d’une modélisation Quasi-Normale avec Amortissement tourbillonnaire. Ph.D. thesis, Ecole Centrale de Lyon (in french) (1994)Google Scholar
  3. Bayly, B.J., Levermore, C.D., Passot, T.: Density variations in weakly compressible flows. Phys. Fluids 4(5), 945–954 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. Bertoglio, J.P., Bataille, F., Marion, J.D.: Two-point closures for weakly compressible turbulence. Phys. Fluids 13(1), 290–310 (2001)ADSCrossRefzbMATHGoogle Scholar
  5. Bos, W.J.T.: The temperature spectrum generated by frictional heating in isotropic turbulence. J. Fluid Mech. 746, 85–98 (2014)ADSMathSciNetCrossRefGoogle Scholar
  6. Bos, W.J.T., Chahine, R., Pushkarev, A.V.: On the scaling of temperature fluctuations induced by frictional heating. Phys. Fluids 27, 095105 (2015)ADSCrossRefGoogle Scholar
  7. Briard, A., Sagaut, P., Cambon, C.: Post-doctoral internal report. Available from the authors upon request (2017)Google Scholar
  8. Cai, X.D., O’Brien, E.E., Ladeinde, F.: Thermodynamic behavior in decaying, compressible turbulence with initially dominant temperature fluctuations. Phys. Fluids 9(6), 1754–1763 (1997)ADSCrossRefGoogle Scholar
  9. Cambon, C., Scott, J.F.: Linear and nonlinear models of anisotropic turbulence. Ann. Rev. Fluid Mech. 31, 1–53 (1999)ADSMathSciNetCrossRefGoogle Scholar
  10. Cambon, C., Coleman, G.N., Mansour, N.N.: Rapid distortion analysis and direct simulation of compressible homogeneous turbulence at finite Mach number. J. Fluid Mech. 257, 641–665 (1993)ADSCrossRefzbMATHGoogle Scholar
  11. Chu, B.T., Apfel, R.E.: Are acoustic intensity and potential energy density first- of second-order quantities? Am. J. Phys. 51(10), 916–918 (1983)ADSCrossRefGoogle Scholar
  12. Crow, S.C.: Visco-elastic character of fine-grained isotropic turbulence. Phys. Fluids 10(7), 1587 (1967)ADSCrossRefGoogle Scholar
  13. De Marinis, D., Chibbaro, S., Meldi, M., Sagaut, P.: Temperature dynamics in decaying isotropic turbulence with Joule heat production. J. Fluid Mech. 724, 425–449 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. Donzis, D.A., Jagannathan, S.: Fluctuations of thermodynamic variables in stationary compressible turbulence. J. Fluid Mech. 733, 221–244 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. Eckhoff, K.S., Storesletten, L.: A note on the stability of steady inviscid helical gas flows. J. Fluid Mech. 89, 401 (1978)ADSCrossRefzbMATHGoogle Scholar
  16. Erlebacher, G., Sarkar, S.: Statistical analysis of the rate of strain tensor in compressible homogeneous turbulence. Phys. Fluids 5(12), 3240–3254 (1993)ADSCrossRefzbMATHGoogle Scholar
  17. Erlebacher, G., Hussaini, M.Y., Kreiss, H.O., Sarkar, S.: The analysis and simulation of compressible turbulence. Theor. Comput. Fluid Dyn. 2, 73–95 (1990)zbMATHGoogle Scholar
  18. Fauchet, G.: Modélisation en deux points de la turbulence isotrope compressible et validation à l’aide de simulations numériques. Thèse de Doctorat, Ecole Centrale de Lyon (in french) (1998)Google Scholar
  19. Fauchet, G., Bertoglio, J.P.: A two-point closure for compressible turbulence. C.R. Acad. Sci. Paris, Série IIb 327, 665–671 (1999a). (in french)Google Scholar
  20. Fauchet, G., Bertoglio, J.P.: Pseudo-sound and acoustic régimes compressible turbulence. C.R. Acad. Sci. Paris, Série IIb 327, 665–671 (1999b). (in french)Google Scholar
  21. Favier, B., Godeferd, F.S., Cambon, C.: Modelling the far-field acoustic emission of rotating turbulence. J. Turbul. 9(30), 1–21 (2008)zbMATHGoogle Scholar
  22. Favier, B., Godeferd, F.S., Cambon, C.: On space and time correlations of isotropic and rotating turbulence. Phys. Fluids 22, 015101 (2010)ADSCrossRefzbMATHGoogle Scholar
  23. Ffwocs Williams, J.E.: The noise from turbulence convected at high speed. Phil. Trans. 255, 469–503 (1963)ADSCrossRefGoogle Scholar
  24. Ffwocs Williams, J.E., Hawkings, D.L.: Sound generation by turbulence and surfaces in arbitrary motion. Phil. Trans. A 264, 321–342 (1969)Google Scholar
  25. Ford, G.W., Meecham, W.C.: Scattering of sound by isotropic turbulence of large Reynolds number. J. Acous. Soc. America 32, 1668–1672 (1960)ADSMathSciNetCrossRefGoogle Scholar
  26. Goldstein, R.J., He, B.Y.: Energy separation and acoustic interaction in flow across a circular cylinder. J. Heat Transfer-Transactions of the ASME 123(4), 682–687 (2001)CrossRefGoogle Scholar
  27. Howe, M.S.: Multiple scattering of sound by turbulence and other inhomogeneities. J. Sound Vib. 27, 455–476 (1973)ADSCrossRefzbMATHGoogle Scholar
  28. Howe, M.S.: The generation of sound by aerodynamic sources in an inhomogeneous steady flow. J. Fluid Mech. 67(3), 597–610 (1975)ADSCrossRefzbMATHGoogle Scholar
  29. Jagannathan, S., Donzis, D.A.: Reynolds and Mach number scaling in solenoidally-forced compressible turbulence using high-resolution direct numerical simulations. J. Fluid Mech. 789, 669–707 (2016)ADSMathSciNetCrossRefGoogle Scholar
  30. Kaneda, Y.: Lagrangian renormalized approximation of turbulence. Fluid Dyn. Res. 39, 526–551 (2007).
  31. Kevlahan, N., Mahesh, K., Lee, S.: Evolution of the shock front and turbulence structures in the shock/turbulence interaction. In: Proceedings of the Summer Program, CTR, Stanford University (1992)Google Scholar
  32. Kida, S., Orszag, S.A.: Energy and spectral dynamics in forced compressible turbulence. J. Sci. Comput. 5(2), 85–125 (1990a)CrossRefzbMATHGoogle Scholar
  33. Kida, S., Orszag, S.A.: Enstrophy budget in decaying compressible turbulence. J. Sci. Comput. 5(1), 1–34 (1990b)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Kida, S., Orszag, S.A.: Energy and spectral dynamics in decaying compressible turbulence. J. Sci. Comput. 7(1), 1–34 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Kraichnan, R.H.: Comments on Space-time correlations in stationary isotropic turbulence. Phys. Fluids 2(3), 334–334 (1959)ADSCrossRefGoogle Scholar
  36. Lee, S., Lele, S., Moin, P.: Eddy shocklets in decaying compressible turbulence. Phys. Fluids 3(4), 657–664 (1991)ADSCrossRefGoogle Scholar
  37. Leslie, D.C.: Developments in the Theory of Turbulence. Clarendon Press, Oxford (1973)zbMATHGoogle Scholar
  38. Li, D., Zhang, X., He, G.: Temporal decorrelations in compressible isotropic turbulence. Phys. Rev. E 88(4), 021001(R) (2013)ADSCrossRefGoogle Scholar
  39. Lighthill, M.J.: On sound generated aerodynamically. I. General theory. Proc. Royal Soc. London/A 211, 564–587 (1952)Google Scholar
  40. Lighthill, M.J.: On the energy scattered from the interaction of turbulence with sound or shock waves. Proc. Cambridge Phil. Soc. 49, 531–551 (1953)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. Lighthill, M.J.: On sound generated aerodynamically. I. Turbulence as a source of sound. Proc. Royal Soc. London/A 222, 1–32 (1954)Google Scholar
  42. Lighthill, M.J.: The Bakerian Lecture, 1961. Sound generated aerodynamically. Proc. Royal Soc. London/A 267, 147–182 (1961)Google Scholar
  43. Lilley, G.M.: The radiated noise from isotropic turbulence. Theoret. Comput. Fluid Dyn. 6(5–6), 281–301 (1994)ADSCrossRefzbMATHGoogle Scholar
  44. Lilley, G.M.: The generation of sound in turbulent motion. Aeronaut. J. 112(1133), 381–394 (2008)CrossRefGoogle Scholar
  45. Marion, J.D., Bertoglio, J.P., Cambon, C.: Two-point closures for compressible turbulence. In: 11th International Symposium on Turbulence, Rolla (MO), Oct. 17–19. Preprints (A-89-33402 13-14) University of Missouri-Rolla, B22-1–B22-8 (1988a)Google Scholar
  46. Marion, J.D., Bertoglio, J.P., Cambon, C., Mathieu, J.: Spectral study of weakly compressible turbulence. Part II: EDQNM. C. R. Acad. Sci. Paris, série II 307, 1601–1606 (1988b)Google Scholar
  47. Miura, H., Kida, S.: Acoustic energy exchange in compressible turbulence. Phys. Fluids 7(7), 1732–1742 (1995)ADSCrossRefzbMATHGoogle Scholar
  48. Morfey, C.L.: Amplification of aerodynamic noise by convected flow inhomogeneities. J. Sound Vib. 31(4), 391–397 (1973)ADSCrossRefGoogle Scholar
  49. Orszag, S.A.: Lectures on the statistical theory of turbulence. In: Balian, R., Peube, J.L. (eds.) Fluid Dynamics, pp. 235–374. Gordon and Breach, London (1977)Google Scholar
  50. Pirrozoli, S., Grasso, F.: Direct numerical simulations of isotropic compressible turbulence: influence of compressibility on dynamics and structure. Phys. Fluids 16(12), 4386–4407 (2004)ADSCrossRefzbMATHGoogle Scholar
  51. Porter, D.H., Pouquet, A., Woodward, P.R.: Three-dimensional supersonic homogeneous turbulence: a numerical study. Phys. Rev. Lett. 68(21), 3156–3159 (1992a)ADSCrossRefzbMATHGoogle Scholar
  52. Porter, D.H., Pouquet, A., Woodward, P.R.: A numerical study of supersonic turbulence. Theor. Comput. Fluid Dyn. 4, 13–49 (1992b)CrossRefzbMATHGoogle Scholar
  53. Porter, D.H., Pouquet, A., Woodward, P.R.: Kolmogorov-like spectra in decaying three-dimensional supersonic flows. Phys. Fluids 6(6), 2133–2142 (1994)ADSCrossRefzbMATHGoogle Scholar
  54. Ribner, H.S.: On spectra and directivity of jet noise. J. Acous. Soc. America 35, 614–616 (1963)ADSCrossRefGoogle Scholar
  55. Samtaney, R., Pullin, D.I., Kosovic, B.: Direct numerical simulation of decaying compressible turbulence and shocklets statistics. Phys. Fluids 13(5), 1415–1430 (2001)ADSCrossRefzbMATHGoogle Scholar
  56. Sarkar, S., Erlebacher, G., Hussaini, M.Y., Kreiss, H.O.: The analysis and modelling of dilatational terms in compressible turbulence. J. Fluid Mech. 227, 473–493 (1991)ADSCrossRefzbMATHGoogle Scholar
  57. Simone, A., Coleman, G.N., Cambon, C.: The effect of compressibility on turbulent shear flow: a RDT and DNS study. J. Fluid Mech. 330, 307–338 (1997)ADSCrossRefzbMATHGoogle Scholar
  58. Tatarski, V.I.: Wave Propagation in a Turbulent Medium. McGraw-Hill Book Company (1967)Google Scholar
  59. Wang, J., Shi, Y., Wang, L.P., Xiao, Z., He, X., Chen, S.: Effect of shocklets on the velocity gradients in highly compressible isotropic turbulence. Phys. Fluids 23, 125103 (2011)ADSCrossRefGoogle Scholar
  60. Wang, J., Yang, Y., Shi, Y., Xiao, Z., He, X.T., Chen, S.: Statistics and structures of pressure and density in compressible isotropic turbulence. J. Turbul. 14, 21–37 (2013)ADSMathSciNetCrossRefGoogle Scholar
  61. Zank, G.P., Matthaeus, W.H.: Nearly incompressible hydrodynamics and heat conduction. Phys. Rev. Lett. 64(11), 1243–1245 (1990)ADSCrossRefGoogle Scholar
  62. Zank, G.P., Matthaeus, W.H.: The equations of nearly incompressible fluids. I. Hydrodynamics, turbulence and waves. Phys. Fluids 3(1), 69–82 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Laboratoire de Mécanique, Modélisation et Procédés Propres, UMR CNRS 7340, Ecole Centrale de MarseilleAix-Marseille UniversitéMarseilleFrance
  2. 2.Laboratoire de Mécanique des Fluides et d’Acoustique, UMR CNRS 5509Ecole Centrale de LyonÉcullyFrance

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