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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 $\left(NS-\alpha \right)$ 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).

##### MSC:
 76F02 Fundamentals of turbulence 76D05 Navier-Stokes equations (fluid dynamics) 76F65 Direct numerical and large eddy simulation of turbulence