Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

On a well-posed turbulence model

Pages: 111 - 128, Volume 6, Issue 1, January 2006      doi:10.3934/dcdsb.2006.6.111

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W. Layton - University of Pittsburgh, Department of Mathematics, Pittsburgh, PA 15260, United States (email)
R. Lewandowski - Université Rennes 1, IRMAR, UMR CNRS 6625, F-35000 Rennes, France (email)

Abstract: 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.

Keywords:  turbulence, large eddy simulation.
Mathematics Subject Classification:  Primary: 76F65; Secondary: 35Q30.

Received: August 2004;      Revised: August 2005;      Available Online: October 2005.