Numerically nonreflecting boundary and interface conditions for compressible flow and aeroacoustic computations

Accurate nonreflecting or radiation boundary conditions are important for effective computation of aeroacoustic and compressible flow problems. The performance of such boundary conditions is often degraded upon discretization of the equations with finite difference and time marching methods. In particular, poorly resolved, spurious sawtooth waves are generated at boundaries due to the dispersive nature of the finite difference approximation. These disturbances can lead to spurious self-sustained oscillations in the flow (self-forcing), poor convergence to steady state, and long time instability of the numerics. Exact discretely nonreflecting boundary closures (boundary conditions for a downwind artificial boundary and an upwind physical boundary) are derived by considering a one-dimensional hyperbolic equation discretized with finite difference schemes and Runge-Kutta time advancements. The current methodology leads to stable local finite difference-like boundary closures, which are nonreflecting to an essentially arbitrarily high order of accuracy. These conditions can also be applied at interfaces where there is a discontinuity in the wave speed (a shock) or where there is an abrupt change in the grid spacing. Compared to other boundary treatments, the present boundary and interface conditions can reduce spurious reflected energy in the computational domain by many orders of magnitude.