WIT Press

Exact Statistical Theory Of Isotropic Turbulence


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Z. Ran


The starting point for this paper lies in the results obtained by Sedov (1944) for isotropic turbulence with the self-preserving hypothesis. A careful consideration of the mathematical structure of the Karman-Howarth equation leads to an exact analysis of all possible cases and to all admissible solutions of the problem. This kind of appropriate manipulation escaped the attention of a number of scientists who developed the theory of turbulence and processed the experimental data for a long time. This paper revisits this interesting problem from a new point of view. Firstly, a new complete set of solutions are obtained, and Sedov’s solution is one special case of this set of solutions. Based on these exact solutions, some physically significant consequences of recent advances in the theory of selfpreserved homogenous statistical solution of the Navier-Stokes equations are presented. New results could be obtained for the analysis on turbulence features, such as the scaling behaviour, the energy spectra, and also the large scale dynamics. Keywords: isotropic turbulence, Karman-Howarth equation, exact solution. 1 Introduction Homogeneous isotropic turbulence is a kind of idealization for real turbulent motion, under the assumption that the motion is governed by a statistical law invariant for arbitrary translation (homogeneity), rotation or reflection (isotropy) of the coordinate system. This idealization was first introduced by Taylor [29] and used to reduce the formidable complexity of statistical expression of turbulence and thus made the subject feasible for theoretical treatment. Up to now, a large amount of theoretical work has been devoted to this rather restricted kind of turbulence. However, turbulence observed either in nature or in laboratory has much more complicated structure. Although remarkable progress


isotropic turbulence, Karman-Howarth equation, exact solution.