Journal of Fluid Mechanics

Asymptotic properties of Burgers turbulence

Shigeo  Kida a1
a1 Research Institute for Mathematical Sciences, University of Kyoto, Kyoto 606, Japan

Article author query
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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 [rightward arrow] [infty infinity]. 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 [not equal] 0.

(Published Online April 19 2006)
(Received May 22 1978)