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Zbl 1037.76022
Foias, Ciprian; Holm, Darryl D.; Titi, Edriss S.
The Navier-Stokes-alpha model of fluid turbulence.
(English)
[J] Physica D 152-153, 505-519 (2001). ISSN 0167-2789

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 2000:
*76F02 Fundamentals
76D05 Navier-Stokes equations (fluid dynamics)
76F65 Direct numerical and large eddy simulation of turbulence

Keywords: incompressible turbulence; viscous Camassa-Holm equations; Kelvin's circulation theorem; wavenumber spectrum; translational kinetic energy; scaling law; large eddy simulation; stress tensor

Cited in: Zbl 1185.35179 Zbl 1064.76058

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