Application of DRP and compact schemes to computational aeroacoustics on curvilinear meshes

In this paper, an implementation of the dispersion- relation-preserving (DRP) scheme in curvilinear coordinate systems is presented and tested for aeroacoustic benchmark problems with known solutions. The theoretical spectral resolution of the 7-point DRP scheme lies between those of CD4 and CD6, these being the 4th- and 6th-order compact differencing schemes, respectively. Numerical experiments with the derivative of a sine function and the solution of the one- dimensional wave equation confirm this result. For these cases, the DRP scheme shows a higher accuracy than CD4 up to, and around, the resolution limit, but poorer performance when the grid is refined. In general, the DRP-based Runge Kutta (RK) time-marching scheme requires a much smaller time step size for stability compared to the standard RK method, using DRP for the spatial integration in both simulations. It has also been observed in the present work that failure to use the asymptotic far-field boundary conditions leads to the divergence of the DRP calculation for the case of reflection of an acoustic pulse off a plate. This boundary condition procedure was therefore implemented for all the test problems reported in this paper. For more difficult aeroacoustic problems, such as the scattering of periodic acoustic source, the correct solutions for sound directivity and the limit cycle pressure wave pattern, could beobtained only when damping terms are added to the equations, similar to the observation in Cartesian formulations of DRP. Furthermore, it was observed that one-sided, high-order (DRP) treatments of the damping terms lead to instability, suggesting the need for a centered stencil for the terms.