Repeated cascade theory of homogeneous turbulence

The problem of turbulent spectrum engenders two coupled hierarchies: one originates from the development of stress, leading to a transfer function, and the other from the development of an eddy viscosity. In order to incorporate physical roles among scales, the turbulent velocity fluctuation is decomposed into a series of ranks in the increasing order of randomness, contributing successively to energy or stress, eddy viscosity, relaxation frequency, and higher-rank frequencies in the memory chain. As a result, the first hierarchy mentioned above becomes closed at the quadrupole correlation. The second hierarchy governs the eddy viscosities of different ranks, related to relaxation frequencies of such ranks, in the form of a memory chain. It is cut off by an implicit viscous mechanism. For zero wind gradient, the spectrum in the inertial subrange recovers the Kolmogoroff k−5/3 law with a numerical constant 1.58, in good agreement with experiments. For a strong wind gradient, the spectrum in the production subrange has a k−1 law. In the viscous subrange, a law of approach confirming the Heisenberg k−7 power spectrum, and a viscous cut-off in the form of an exponential tail are obtained, insuring the convergence of high-order spectral moments. The critical wavenumbers characteristic of the production, inertia, viscous subranges, and the cutoff are determined, together with their numerical coefficients.