Estimates for the energy cascade in three-dimensional turbulent flows

Abstract

The phenomenological theory of turbulence in three dimensions postulates that at large Reynolds numbers there exists an interval of wavenumbers within which the direct effects of the molecular viscosity are negligible. Within that interval, the so-called inertial range, an eddy characterized by a wavenumber given in that range decays principally by breaking down into smaller ones, with each of those smaller ones eventually breaking down into still smaller eddies, and so on, a process conventionally called a cascade in the wavenumber space. Such a cascade proceeds until the size of the descendant eddies is sufficiently small to enter the so-called dissipation range and disappear by the direct action of molecular viscosity. In this note, which is a continuation of [5], we prove the existence of the inertial range provided the Taylor wavenumber is sufficiently large. More precisely, we prove that the energy flux to higher modes is nearly equal to the energy dissipation rate throughout a certain range of wavenumbers much smaller than the Taylor wavenumber. These rigorous results show that the Taylor wavenumber is such that below it the conditions prevailing in the inertial range for the energy cascade are strictly satisfied. Moreover, we obtain several estimates concerning characteristic numbers and nondimensional numbers related to turbulent flows.