The Basic Equations for the Large Eddy Simulation of Turbulent Flows in Complex Geometry

The equations for large eddy simulation (LES) of turbulent flows are derived by applying a "filtering" operation to the Navier-Stokes equations. LES of inhomogeneous turbulent flows requires the use of filters with variable filter width. The use of such filters invalidates the standard derivation of the basic equations for the filtered fields since the filtering operation in general does not commute with the operation of differentiation. In this paper an alternate definition of the filtering operation based on the mapping function of the nonuniform grid is introduced. It is shown that with this modified definition the filtering and differentiation operations commute up to an error which is second order in the filter width. It is also shown that the commutation error can be expressed in terms of the filtered field and its derivatives as an asymptotic series in the square of the filter width. These results are then applied to the Navier-Stokes equations to derive the basic equations satisfied by the filtered fields.