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The Navier-Stokes-alpha model of fluid turbulence. (English) Zbl 1037.76022
Summary: We review the properties of nonlinearly dispersive Navier-Stokes-alpha (NS-α) model of incompressible fluid turbulence – also called viscous Camassa-Holm equations in the literature. We first re-derive the NS-α model by filtering the velocity of the fluid loop in Kelvin’s circulation theorem for Navier-Stokes equations. Then we show that this filtering causes the wavenumber spectrum of the translational kinetic energy for the NS-α model to roll off as k -3 for kα>1 in three dimensions, instead of continuing along the slower Kolmogorov scaling law, k -5/3 , that it follows for kα<1. This roll off at higher wavenumbers shortens the inertial range for the NS-α model and thereby makes it more computable. We also explain how the NS-α model is related to large eddy simulation turbulence modeling and to the stress tensor for second-grade fluids. We close by surveying recent results in the literature for the NS-α model and its inviscid limit (the Euler-α model).

MSC:
76F02Fundamentals of turbulence
76D05Navier-Stokes equations (fluid dynamics)
76F65Direct numerical and large eddy simulation of turbulence