Implementation of the Variational Riemann Problem Solution for Calculating Propagation of Sound Waves in Nonuniform Flow Fields

The variational Riemann problem (VRP) is defined as the first variation of the solution to Riemannʹs initial-value problem, also known as the problem of breakup of an arbitrary discontinuity in a gas, when the initial data undergo small variations. We show that the solution to the VRP can be analytically obtained, provided that the solution to the baseline Riemann problem is known. This solution describes the interaction of two abutting parcels of small disturbances against the background of a given base flow and therefore can be efficiently implemented in numerical methods for aeroacoustics. When the spatial distribution of disturbances and base flow parameters are given at a time moment at mesh points of a computational grid, one can exactly determine the disturbance evolution for a short lapse of time by solving the VRP at mesh interfaces. This can then be applied to update disturbance values to a new time moment by using the standard finite-volume scheme. In other words, the VRP can be used in computational aeroacoustics in the similar way to the Riemann problem used in Godunov-type methods for computational fluid dynamics. The present paper elaborates on this idea and adopts the solution to the VRP as a building block for a finite-volume Godunov-type method for aeroacoustics.