The Navier-Stokes-alpha model of fluid turbulence

We review the properties of the nonlinearly dispersive Navier-Stokes-alpha (NS-alpha) model of incompressible fluid turbulence - also called the visco us Camassa-Holm equations in the literature. We first re-derive the NS-alph a model by filtering the velocity of the fluid loop in Kelvin's circulation theorem for the 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, in stead 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 th e inertial range for the NS-alpha model and thereby makes it more computabl e. We also explain how the NS-alpha model is related to large eddy simulati on (LES) turbulence modeling and to the stress tenser for second-grade flui ds. We close by surveying recent results in the literature for the NS-alpha model and its inviscid limit (the Euler-alpha model). (C) 2001 Published b y Elsevier Science B.V.