Construction of Commutative Filters for LESon Unstructured Meshes

Application of large eddy simulation (LES) to flows with increasingly complex geometry necessitates the extension of LES to unstructured meshes. A desirable feature for LES on unstructured meshes is that the filtering operation used to remove small scale motions from the flow commutes with the differentiation operator. If this commutation requirement is satisfied, the LES equations have the same structure as the unfiltered Navier Stokes equations. Commutation is generally satisfied if the filter has a constant width. However, in inhomogeneous turbulent flows, the minimum size of eddies that need to be resolved varies throughout the flow. Thus, the filter width should also vary accordingly. Given these challenges, the objective of this work is to develop a general theory for constructing discrete variable width commutative filters for LES on unstructured meshes. Variable width filters and their commuting properties have been the focus of several recent works. Van der Ven (1995) constructed a family of continuous filters which commute with differentiation up to arbitrary order in the filter width. However, this set of filters applies only to an infinite domain without addressing the practical issue of boundary conditions in a finite domain. More recently, a class of discrete commutative filters was developed by Vasilyev et al. (1998) for use on nonuniform structured meshes. Their formulation uses a mapping function to perform the filtering in the computational domain. Although this type of mapping is impossible for the unstructured case, the theory developed in Vasilyev et al. (1998) was used as a starting point for the present work. In this paper we present a theory for constructing discrete commutative filters for unstructured meshes in two and three dimensions. In addition to commutation, other issues such as control of filter width and shape in wavenumber space are also considered. In particular, we wish to specify a desired filter width at each point in space and obtain a discrete filter which satisfies this requirement regardless of the choice of the computational mesh.