Colloque des sciences mathématiques du Québec

10 janvier 2010 de 16 h 00 à 18 h 00 (heure de Montréal/HNE) Sur place

Some features and challenges of the Navier-Stokes-alpha-beta equation

Colloque par Eliot Fried

The Navier-Stokes-alpha equation regularizes the Navier-Stokes equation by including additional dispersive and dissipative terms. The former term is proportional to the divergence of the corotational time-rate of the symmetric part of the gradient of the filtered velocity. The latter term is proportional to the bi-Laplacian of the filtered velocity. Both terms involve factors of the square of alpha where, roughly, alpha represents the characteristic size of the smallest resolvable eddy. Combining dispersion and dissipation yields a model with certain attractive features. In particular, the Navier-Stokes-alpha equation possesses circulation properties analogous to those of the Navier-Stokes equation and allows for simulations with less artificial damping than those arising from more conventional subgrid-scale and Reynolds stress models. One drawback concerns boundary conditions. Except for flows in periodic domains, the additional dissipative term entering the Navier-Stokes-alpha equation necessitates additional boundary conditions. Unfortunately, the conventional method used to derive the Navier-Stokes-alpha equation does not provide such conditions. The absence of physically meaningful boundary conditions limits the applicability of the model. Using a framework for fluid-dynamical theories with gradient dependencies, we have derived a flow equation — the Navier-Stokes-alphabeta equation — that includes the Navier-Stokes-alpha equation as a special case. Aside from alpha, this equation involves an additional length scale beta. For beta=alpha, our flow equation reduces to the Navier-Stokes-alpha equation. Our formulation also yields boundary conditions at walls and free surfaces. We will consider the effects of alpha and beta on the energy spectrum and the alignment between the filtered vorticity and the eigenvalues of the filtered stretching tensor in three-dimensional homogeneous and isotropic turbulent flows in a periodic cubic domain, including the limiting cases of the Navier-Stokes-alpha and Navier-Stokes equations. We will also discuss some ongoing work and open mathematical challenges associated with the model.


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