The Navier–Stokes-alpha model of fluid turbulence

We review the properties of the nonlinearly dispersive Navier–Stokes-alpha (NS-α) model of incompressible fluid turbulence — also called the viscous Camassa–Holm equations in the literature. We first re-derive the NS-α 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-α model to roll off as k−3 for kα>1 in three dimensions, instead of continuing along the slower Kolmogorov scaling law, k−5/3, that it follows for kα<1. This roll off at higher wavenumbers shortens the inertial range for the NS-α model and thereby makes it more computable. We also explain how the NS-α model is related to large eddy simulation (LES) turbulence modeling and to the stress tensor for second-grade fluids. We close by surveying recent results in the literature for the NS-α model and its inviscid limit (the Euler-α model).