Existence of weak solutions for a scale similarity model of the motion of large eddies in turbulent flow

Abstract

In turbulent flow, the normal procedure has been seeking means $\overline{u}$ of the fluid velocity $u$ rather than the velocity itself. In large eddy simulation, we use an averaging operator which allows for the separation of large- and small-length scales in the flow field. The filtered field $\overline{u}$ denotes the eddies of size $O\left(\delta \right)$ and larger. Applying local spatial averaging operator with averaging radius $\delta$ to the Navier-Stokes equations gives a new system of equations governing the large scales. However, it has the well-known problem of closure. One approach to the closure problem which arises from averaging the nonlinear term is the use of a scale similarity hypothesis. We consider one such scale similarity model. We prove the existence of weak solutions for the resulting system.