Globally conservative high-order filters for large-eddy simulation and computational aero-acoustics

In computational aero-acoustics, large-eddy simulations (LES) or direct numerical simulations (DNS) are often employed for flow computations in the source region. As part of the numerical implementation or required modeling, explicit spatial filters are frequently employed. For instance, in LES spatial filters are employed in the formulation of various subgrid-scale (SGS) models such as the dynamic model or the variational multi-scale (VMS) Smagorinsky model; both in LES or DNS, spatial high-pass filters are often used to remove undesired grid-to-grid oscillations. Though these type of spatial filters adhere to local accuracy requirements, in practice, they often destroy global conservation properties in the presence of non-periodic boundaries conditions. This leads to the incorrect prediction of the flow properties near hard boundaries, such as walls. In the current work, we present globally conservative high-order accurate filters, which combine traditional filters at the internal points with one-sided conservative filters near the wall boundary. We test these filters to remove grid-to-grid oscillations both in a channel-flow case and in 2D cavity flow. We find that the use of a non-conservative filter leads to erroneous predictions of the skin friction in channel flows up to 30%. In the cavity-flow simulations, the use of non-conservative filters to remove grid-to-grid oscillations leads to important shifts in the Strouhal number of the dominant mode, and a change of the flow pattern inside the cavity. In all cases, the use of conservative high-order filter formulations to remove grid-to-grid oscillations lead to very satisfactory results. Finally, in our channel-flow test case, we also illustrate the importance of using conservative filters for the formulation of the VMS Smagorinsky model.