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On a Leray-\(\alpha\) model of turbulence. (English) Zbl 1145.76386
Summary: We introduce and study a new model for three-dimensional turbulence, the Leray-\(\alpha \) model. This model is inspired by the Lagrangian averaged Navier-Stokes-\(\alpha \) model of turbulence (also known Navier-Stokes-\(\alpha \) model or the viscous Camassa-Holm equations). As in the case of the Lagrangian averaged Navier-Stokes-\(\alpha \) model, the Leray-\(\alpha \) 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-\(\alpha \) 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-\(\alpha \) 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 \(1/\alpha \). Finally, we propose a Prandtl-like boundary-layer model, induced by the Leray-\(\alpha \) model, and show a very good agreement of this model with empirical data for turbulent boundary layers.

76F02 Fundamentals of turbulence
35Q35 PDEs in connection with fluid mechanics
37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76D05 Navier-Stokes equations for incompressible viscous fluids
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