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| 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 self-preserved 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 behavior, the spectrum, and also the large scale dynamics. Integral turbulence length scales based on the exact solutions are discussed, and used to derive rigorously a new recursion equation, which would be helpful in understanding the dynamical process of turbulence, especially for the Markov property and turbulence cascade. The general energy spectra and their behavior in the lowest wave number range are investigated. According to the present theory, the Loitsiansky integral is not an invariant in general cases. |
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| Keywords:isotropic turbulence, Karman-Howarth equation, exact solution, dynamics of large scales |
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