Towards a stochastic multi-point description of turbulence

In previous work it was found 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 now show how this description for increments can be extended in order to obtain the complete multi-point statistics in real space of the turbulent velocity field (Stresing & Peinke, 2010). We extend the stochastic cascade description by conditioning on an absolute velocity value itself, and find that the corresponding conditioned process is also governed by a Fokker-Planck equation, which contains as a leading term a simple additional velocity-dependent coefficient, d10, in the drift function. Taking the velocity-dependence of the Fokker-Planck equation into account, the multi-point statistics in the inertial range can be expressed by two-scale statistics of velocity increments, which are equivalent to three-point statistics of the velocity field. Thus, we propose a stochastic three-point closure for the velocity field of homogeneous isotropic turbulence. Investigating the coefficient d10 for different flows, we find clear evidence that the multipoint structure of small scale turbulence is not universal but depends on the type of the flow.