The renormalized eddy-fragmentation equation and its exact solutions

摘要

The fragmentation equation describes the evolution in time of particles sys-tem, when particles break up. The turbulent eddy decay is an example of such fragmentation. A collection of tangled trajectories of fluid particle, asso-ciated with a turbulent concentrated structure, resembles a "wool ball" of a typical scale r. Once a fluid particle is subjected to an intense acceleration, a new "wool ball" is formed, containing a part of total energy of flow. The population of newly appeared "wool balls" is assumed to be governed by the fragmentation equation, requiring conservation of the total kinetic energy in-jected on large scales. The question raised is how this energy is distributed in statistical ensemble of such "wool balls". In this paper, the renormalized form of the fragmentation equation is obtained for arbitrary functions for the spec-trum and for frequency of fragmentation. If the frequency of fragmentation is a power function of size, a simple exact solution to this equation is obtained, providing for stationary flux of energy, from large scales towards zero scales. A simple stochastic generation of random field with presumed fractal properties is illustrated. Also, presuming the spectrum of breakup and its frequency in the form of power functions, the exact self-similar solution is obtained on the basis of specifically introduced scaling transformations. Here the specific case is considered, when the breakup frequency is decreasing with decreasing of r. This work contributes to the group-theoretical description of statistically homogeneous turbulence, developed recently by authors in [1, 2, 3].