A formally exact wall-boundary condition in large eddy simulations using volume filtering

Turbulent wall-bounded flows, although present in many practical applications, are particularly challenging to simulate because of their large velocity gradients near the walls. To avoid the necessity of an extremely fine mesh resolution in the near-wall regions of wall-bounded turbulent flows, large eddy simulation (LES) with specific modelling near the wall can be applied. Since filtering close to the boundaries of the flow domain is not uniquely defined, existing wall-modelled LES typically rely on extensive assumptions to derive suitable boundary conditions at the walls, such as assuming that the instantaneous filtered velocity behaves similarly to the unfiltered mean velocity. Volume filtering constitutes a consistent extension of filtering close to the boundaries of the flow domain. In the present paper, we derive a formally exact expression for the wall-boundary conditions in LESs using the concept of volume filtering applied to wall-bounded turbulent flows that does not make any a priori assumptions on the flow field. The proposed expression is an infinite series expansion in powers of the filter width. It is shown in an a priori study of a turbulent channel flow and an a posteriori study of the turbulent flow over periodic hills that the proposed expression can accurately predict the volume-filtered velocity at the wall by truncating the infinite series expansion after a few terms.