# American Institute of Mathematical Sciences

January  2006, 6(1): 111-128. doi: 10.3934/dcdsb.2006.6.111

## On a well-posed turbulence model

 1 University of Pittsburgh, Department of Mathematics, Pittsburgh, PA 15260, United States 2 Université Rennes 1, IRMAR, UMR CNRS 6625, F-35000 Rennes, France

Received  August 2004 Revised  August 2005 Published  October 2005

This report considers mathematical properties, important for practical computations, of a model for the simulation of the motion of large eddies in a turbulent flow. In this model, closure is accomplished in the very simple way:

$\overline{u u}$˜ $\overline{\bar {u} \bar {u}}$, yielding the model
$\nabla \cdot w= 0, \quad w_{t} + \nabla \cdot (\overline{w w}) - \nu \Delta w + \nabla q = \bar {f}$.

In particular, we prove existence and uniqueness of strong solutions, develop the regularity of solutions of the model and give a rigorous bound on the modelling error, $||\bar {u} - w||$. Finally, we consider the question of non-physical vortices (false eddies), proving that the model correctly predicts that only a small amount of vorticity results when the total turning forces on the flow are small.

Citation: W. Layton, R. Lewandowski. On a well-posed turbulence model. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 111-128. doi: 10.3934/dcdsb.2006.6.111
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