On a Leray–α model of turbulence

Abstract

In this paper we introduce and study a new model for three–dimensional turbulence, the Leray–α model. This model is inspired by the Lagrangian averaged Navier–Stokes–α model of turbulence (also known Navier–Stokes–α model or the viscous Camassa–Holm equations). As in the case of the Lagrangian averaged Navier–Stokes–α model, the Leray–α model compares successfully with empirical data from turbulent channel and pipe flows, for a wide range of Reynolds numbers. We establish here an upper bound for the dimension of the global attractor (the number of degrees of freedom) of the Leray–α model of the order of (L/l_d)^12/7, where L is the size of the domain and l_d is the dissipation length–scale. This upper bound is much smaller than what one would expect for three–dimensional models, i.e. (L/l_d)³. This remarkable result suggests that the Leray–α model has a great potential to become a good sub–grid–scale large–eddy simulation model of turbulence. We support this observation by studying, analytically and computationally, the energy spectrum and show that in addition to the usual k^−5/3 Kolmogorov power law the inertial range has a steeper power–law spectrum for wavenumbers larger than 1/α. Finally, we propose a Prandtl–like boundary–layer model, induced by the Leray–α model, and show a very good agreement of this model with empirical data for turbulent boundary layers.