Estimates for the energy cascade in three-dimensional turbulent flows

The phenomenological theory of turbulence in three dimensions postulates th at 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 wa venumber given in that range decays principally by breaking down into small er ones, with each of those smaller ones eventually breaking down into stil l smaller eddies, and so on, a process conventionally called a cascade in t he wavenumber space. Such a cascade proceeds until the size of the descenda nt eddies is sufficiently small to enter the so-called dissipation range an d disappear by the direct action of molecular viscosity. In this note, whic h is a continuation of [5], we prove the existence of the inertial range pr ovided the Taylor wavenumber is sufficiently large. More precisely, we prov e that the energy flux to higher modes is nearly equal to the energy dissip ation rate throughout a certain range of wavenumbers much smaller than the Taylor wavenumber. These rigorous results show that the Taylor wavenumber i s such that below it the conditions prevailing in the inertial range for th e energy cascade are strictly satisfied. Moreover, we obtain several estima tes concerning characteristic numbers and nondimensional numbers related to turbulent flows. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.