Transfer function of low-pass filters on unstructured grids

Low-pass filters designed on unstructured grids are investigated in terms of a transfer function in the wavenumber space. The transfer functions on unstructured grids are derived, and the properties of the low-pass filters for removing high-wavenumber components and inducing phase errors are investigated through the derived transfer functions. The transfer function reveals that the low-pass filters on unstructured grids can achieve the property that higher-wavenumber components are removed more by adjusting a filter coefficient to a small value, whereas large filter coefficients induce unfavorable amplifications of high-wavenumber components. The presence of phase errors induced by the low-pass filters on triangle unstructured cells is also found by the transfer function. Furthermore, the transfer function shows that the numerical methods for evaluating the gradients that appear in the filter formulation affect the characteristics of the low-pass filters and that the simplest central scheme can have an advantage in terms of retaining numerical robustness by removing high-wavenumber components compared to the edge-normal augmentation scheme. The numerical experiments of inviscid Taylor–Green vortex and shock-vortex interaction are also conducted with the low-pass filter coupled with the non-dissipative kinetic energy and entropy preserving (KEEP) scheme on unstructured grids, demonstrating the validity of the present transfer function of the low-pass filter.