A new approach to the problem of turbulence, based on the conditionally averaged Navier-Stokes equations

Abstract

A new approach to the problem of turbulent flows is presented. This approach is based on the conditional averaging of equations for local characteristics of motion, which have a mechanism of self-amplification. In three-dimensional (3-D) turbulence the local characteristic is the vorticity field. In 2-D turbulence the corresponding local characteristic is the vorticity gradient (vg). The exact closed equation (or relation) for the conditionally averaged 3-D vorticity field (with fixed vorticity at a certain point) is derived from the Navier-Stokes equations. For 2-D turbulence a corresponding closed equation for the conditionally averaged vg is obtained. Solutions of these equations are presented, although realizability of solutions is not yet proved. On the basis of these solutions, the high order two-point moments of vorticity and vg are calculated. The obtained conditionally averaged tensor of strain rates is in accord with the appearance of "vortex strings" in 3-D turbulence. The presented new quantitative information about the structure of turbulent flows can be verified by laboratory and field measurements, as well as in numerical experiments. The concept of conditional averaging opens a new perspective for the modeling of small-scale turbulence in large-eddy simulations.