## 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).

### MSC:

 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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