On a Leray model of turbulence
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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/ld)12/7, where L is the size of the domain and ld is the dissipation length-scale. This upper bound is much smaller than what one would expect for three-dimensional models, i.e. (L/ld)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. © 2004 The Royal Society.
author list (cited authors)
Cheskidov, A., Holm, D. D., Olson, E., & Titi, E. S.