A Finite Element, Filtered Eddy-viscosity Method for the Navier-stokes Equations with Large Reynolds Number

Fluid turbulence is commonly modeled by the Navier-Stokes equations with a large Reynolds number. The simulation of turbulence model is known to be very difficult. We study artificial spectral viscosity models that render the simulation of turbulence tractable. The models introduce several parameters. We show that the models have solutions that converge, in certain parameter limits, to solutions of the Navier-Stokes equations. We also show, using the mathematical analysis, how effective choice for the parameter can be made. The direct computational simulation of turbulence flow is a formidable task due to the disparate scales that have to be resolved. Turbulence modeling attempts to mitigate this situation by accounting for the effects of small-scale behavior on that at large-scales without explicitly esolving the small scales. One such approach is to add viscosity to the problem; the Smagorinsky and Ladyzhenskaya models and other eddy-viscosity models are examples of this approach. Unfortunately, this approach usually results in over-dampening at the large scales, i.e., largescale structures are unphysically smeared out. To mitigate this fault of eddy-viscosity modeling, filtered eddy-viscosity methods that add the artificial viscosity only to the high-frequency modes were developed in the context of spectral methods. We apply the filtered eddy-viscosity idea to finite element methods with hierarchical basis functions. We prove the existence and uniqueness of the finite element approximation and its convergence to a weak solution of the Navier-Stokes system.