Hostname: page-component-5cf477f64f-pw477 Total loading time: 0 Render date: 2025-04-07T20:08:46.234Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic properties of Burgers turbulence

Published online by Cambridge University Press:  19 April 2006

Shigeo Kida
Shigeo Kida
Affiliation:
Research Institute for Mathematical Sciences, University of Kyoto, Kyoto 606, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

The asymptotic properties of Burgers turbulence at extremely large Reynolds numbers and times are investigated by analysing the exact solution of the Burgers equation, which takes the form of a series of triangular shocks in this situation. The initial probability distribution for the velocity u is assumed to decrease exponentially as u → ∞. The probability distribution functions for the strength and the advance velocity of shocks and the distance between two shocks are obtained and the velocity correlation and the energy spectrum function are derived from these distribution functions. It is proved that the asymptotic properties of turbulence change qualitatively according as the value of the integral scale of the velocity correlation function J, which is invariant in time, is zero, finite or infinite. The turbulent energy per unit length is shown to decay in time t as t−1 (with possible logarithmic corrections) or $t^{-\frac{2}{3}}$ according as J = 0 or J ≠ 0.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 93 , Issue 2 , 26 July 1979 , pp. 337 - 377
Copyright
© 1979 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Burgers, J. M. 1950 Correlation problem in a one-dimensional model of turbulence. Proc. Amsterdam 53, 393–406.Google Scholar
Burgers, J. M. 1974 The Nonlinear Diffusion Equation. D. Reidel Publishing Co.
Cole, J. D. 1951 On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math. 9, 225–236.Google Scholar
Crow, S. C. & Canavan, G. H. 1970 Relation between a Wiener-Hermite expansion and an energy cascade. J. Fluid Mech. 41, 387–403.Google Scholar
Frisch, U. 1978 Private communication.
Hopf, E. 1950 The partial differential equation ut + uux = uxx. Commun. Pure Appl. Mech. 3, 201–230.Google Scholar
Hosokawa, I. & Yamamoto, K. 1970 Numerical study of Burgers’ model of turbulence based on the characteristic functional formalism. Phys. Fluids 13, 1683–1692.Google Scholar
Jeng, D. T., Foerster, R., Haaland, S. & Meecham, W. C. 1966 Statistical initial-value problem for Burgers’ model equation of turbulence. Phys. Fluids 9, 2114–2120.Google Scholar
Kawahara, T. 1968 A successive approximation for turbulence in the Burgers model fluid. J. Phys. Soc. Japan 25, 892–900.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. Acad. Sci. U.R.S.S. 30, 301–305.Google Scholar
Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85.Google Scholar
Kraichnan, R. H. 1968 Lagrangian-history statistical theory for Burgers’ equation. Phys. Fluids 11, 265–277.Google Scholar
Lighthill, M. J. 1956 Viscosity effects in sound waves of finite amplitude. In Surveys in Mechanics (ed. G. K. Batchelor and R. M. Davies), pp. 250–351. Cambridge University Press.
Malfliet, W. P. M. 1969 A theory of homogeneous turbulence deduced from the Burgers equation. Physica 45, 257–271.Google Scholar
Meecham, W. C. & Siegel, A. 1964 Wiener-Hermite expansion in model turbulence at large Reynolds numbers. Phys. Fluids 7, 1178–1190.Google Scholar
Reid, W. H. 1957 On the transfer of energy in Burgers’ model of turbulence. Appl. Sci. Res. A 6 85–91.
Tatsumi, T. 1969 Nonlinear wave expansion for turbulence in the Burgers’ model of fluid. Phys. Fluids Suppl. II 12, 258–264.
Tatsumi, T. & Kida, S. 1972 Statistical mechanics of the Burgers model of turbulence. J. Fluid Mech. 55, 659–675.Google Scholar
Tatsumi, T. & Mizushima, J. 1979 Submitted to J. Fluid Mech.
Tatsumi, T. & Tokunaga, H. 1974 One-dimensional shock turbulence in a compressible fluid. J. Fluid Mech. 65, 581–601.Google Scholar
Tokunaga, H. 1979 To be published.
Yamamoto, K. & Hosokawa, I. 1976 Energy decay of Burgers’ model of turbulence. Phys. Fluids 19, 1423–1424.Google Scholar