A triple level finite element method for large eddy simulations

Since large eddy simulation (LES) was introduced by Smagorinsky in 1963, it has been improved with various thoughts from many researchers. Unfortunately, despite that, the filtered approach that is widely used at present still suffers because quite empirical factors are used to determine non-closured subgrid Reynolds stresses. Based on a new definition of LES and multiscale finite element concepts, this work presents an attempt to remove such factors. Using direct sum decomposition of the solution space, we devised a hierarchical multi-level formulation of the Navier-Stokes equations for turbulence. The base-level, bearing the information of large eddies, is calculated by the conventional finite element method. The finer levels are for small scale eddies of turbulence. We address the solution methods for the small scale movements. In particular, a spectral element method is introduced for the finer level solutions. Thus large eddies and small eddies to some extent may be accurately obtained. The introduced approach offers not only access to calculate turbulence in complex geometries because of the nature of finite element method but also an effective tool for multiscale physical problems with turbulence, such as reaction flows. It is worth noting that the approach introduced here is similar to the implicit LES in finite volume and finite difference methods.