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 viscous Camassa-Holm equations and the LANS 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 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, instead of continuing along the slower Kolmogorov scaling law, k^{-5/3}, that it follows for k \alpha < 1. This rolloff 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 (LES) 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).Comment: 22 pages, 1 figure. Dedicated to V. E. Zakharov on the occasion of his 60th birthday. To appear in Physica

A multiple mapping conditioning model for differential diffusion

This work introduces modeling of differential diffusion within the multiple mapping conditioning (MMC) turbulent mixing and combustion framework. The effect of differential diffusion on scalar variance decay is analyzed and, following a number of publications, is found to scale as Re. The ability to model the differential decay rates is the most important aim of practical differential diffusion models, and here this is achieved in MMC by introducing what is called the side-stepping method. The approach is practical and, as it does not involve an increase in the number of MMC reference variables, economical. In addition we also investigate the modeling of a more refined and difficult to reproduce differential diffusion effect - the loss of correlation between the different scalars. For this we develop an alternative MMC model with two reference variables but which also makes use of the side-stepping method. The new models are successfully validated against DNS results available in literature for homogenous, isotropic two scalar mixing

Tsallis Extended Thermodynamics Applied to 2-d Turbulence: Lévy Statistics and q-Fractional Generalized Kraichnanian Energy and Enstrophy Spectra

The extended thermodynamics of Tsallis is reviewed in detail and applied to turbulence. It is based on a generalization of the exponential and logarithmic functions with a parameter q. By applying this nonequilibrium thermodynamics, the Boltzmann-Gibbs thermodynamic approach of Kraichnan to 2-d turbulence is generalized. This physical modeling implies fractional calculus methods, obeying anomalous diffusion, described by Lévy statistics with q &lt; 5/3 (sub diffusion), q = 5/3 (normal or Brownian diffusion) and q &gt; 5/3 (super diffusion). The generalized energy spectrum of Kraichnan, occurring at small wave numbers k, now reveals the more general and precise result k−q. This corresponds well for q = 5/3 with the Kolmogorov-Oboukov energy spectrum and for q &gt; 5/3 to turbulence with intermittency. The enstrophy spectrum, occurring at large wave numbers k, leads to a k−3q power law, suggesting that large wave-number eddies are in thermodynamic equilibrium, which is characterized by q = 1, finally resulting in Kraichnan’s correct k−3 enstrophy spectrum. The theory reveals in a natural manner a generalized temperature of turbulence, which in the non-equilibrium energy transfer domain decreases with wave number and shows an energy equipartition law with a constant generalized temperature in the equilibrium enstrophy transfer domain. The article contains numerous new results; some are stated in form of eight new (proven) propositions