The Navier-Stokes-alpha model of fluid turbulence. (English) Zbl 1037.76022

Summary: We review the properties of nonlinearly dispersive Navier-Stokes-alpha \((NS-\alpha)\) model of incompressible fluid turbulence – also called viscous Camassa-Holm equations in the literature. We first re-derive the \(NS-\alpha\) 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 \(NS-\alpha\) model to roll off as \(k^{-3}\) for \(k\alpha>1\) in three dimensions, instead of continuing along the slower Kolmogorov scaling law, \(k^{-5/3}\), that it follows for \(k\alpha<1\). This roll off at higher wavenumbers shortens the inertial range for the \(NS-\alpha\) model and thereby makes it more computable. We also explain how the \(NS-\alpha\) 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 \(NS-\alpha\) model and its inviscid limit (the Euler-\(\alpha\) model).


76F02 Fundamentals of turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids
76F65 Direct numerical and large eddy simulation of turbulence
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