The direct numerical simulation of the Navier-Stokes system in turbulent regimes is a formidable task due to the disparate scales that have to be resolved. Turbulence modeling attempts to mitigate this situation by somehow accounting for the effects of small-scale behavior on that at large-scales, without explicitly resolving 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., large-scale structures are unphysically smeared out. To overcome this fault of simple eddy-viscosity modeling, filtered eddy-viscosity methods that add 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 based on hierarchical basis functions. We prove the existence and uniqueness of the finite element approximation and its convergence to solutions of the Navier-Stokes system; we also derive error estimates for finite element approximations.
Bibliographical noteFunding Information:
This work is supported by the US Department of Energy Climate Modeling: Simulating Climate at Regional Scale program under grant number DESC00002624. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology 2010-0003654 and by WCU (World Class University) program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology R31-2008-000-10049-0.
All Science Journal Classification (ASJC) codes
- Applied Mathematics