On statistical properties of shell models of turbulence

Abstract

We have conducted statistical analysis of the characteristics of the velocity field pulsations, spectral energy flux, and energy dissipation rate in the framework of two types of shell models of developed turbulence. It is shown that the probability distribution functions of the spectral energy flux of the considered models differ fundamentally: in one model the energy flux fluctuates but remains mainly positive, with a very small fraction of negative values, while in the other model the range of fluctuations is much wider and the probability of negative values is comparable to the probability of positive values. For the same initial conditions, only the first-order statistical moment coincides. Qualitative coincidence is demonstrated by the distribution function of the energy dissipation rate. It is important that, in spite of such different statistical properties of the cascade energy flux, both models reproduce equally well the statistics of the high-order moments for the velocity field pulsations. This fact is noteworthy because the behavior of the high-order structural functions is believed to be determined by the probability distribution function of the energy dissipation rate and/or the spectral energy flux density.