Towards a stochastic multi-point description of turbulence

Abstract

It has been found in previous work that the multi-scale statistics of homogeneous isotropic turbulence can be described by a stochastic ‘cascade’ process of the velocity increment from scale to scale, which is governed by a Fokker–Planck equation. We show in this paper how this description can be extended to obtain the complete multi-point statistics of the velocity field. We extend the stochastic cascade description by conditioning on the velocity value itself and find that the corresponding process is also governed by a Fokker–Planck equation, which contains as a leading term a simple additional velocity-dependent coefficient in the drift function. Taking into account the velocity dependence of the Fokker–Planck equation, the multi-point statistics in the inertial range can be expressed by the two-scale statistics of velocity increments, which are equivalent to the three-point statistics of the velocity field. Thus, we propose a stochastic three-point closure for the velocity field of homogeneous isotropic turbulence.