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

Meryem Kaya

doi:10.1155/s1110757x03301111

Abstract

In turbulent flow, the normal procedure has been seeking means 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 denotes the eddies of size O(δ) and larger. Applying local spatial averaging operator with averaging radius δ 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.

Highlights

The turbulent flow of an incompressible fluid is modelled by solution (u, p) of the incompressible Navier-Stokes equations: ut + ∇ · − Re−1∆u + ∇p = f, in Ω, for 0 < t ≤ T, ∇ · u = 0, in Ω, for 0 < t ≤ T, u(x, 0) = u0(x), in Ω, u = 0 on ∂Ω, for 0 < t ≤ T, and pdx = 0 where Ω ⊂ Rd (d = 2 or d = 3), u : Ω × [0, T ] → Rd is the fluid velocity, p : Ω → R is the fluid pressure f(x,t) is the body force, u0(x) the initial flow field and Re the Reynolds number
One of the most promising current approaches is large eddy simulation (LES) in which approximations to local spatial averages of u are calculated
In large eddy simulation (LES), the filtered quantities and fluctuations are defined as u(x, t) = gδ ∗ u = gδ(x − x )u(x, t)dx u = u − u

Summary

Introduction

Let consider the model of the subgrid tensor T =uu − u u ∼ w w + w(w − w) + (w − w)w − (csδ)2 ∇w + ∇wt (∇w + ∇wt) We consider the question of existence of weak solutions to the following systems. Before we prove of the existence of weak solutions of (2.1)- (2.3) we give the following Lemma.

Results

Conclusion