One-way approximation for the simulation of weak shock wave propagation in atmospheric flows

A numerical scheme is developed to simulate the propagation of weak acoustic shock waves in the atmosphere with no absorption. It generalizes the method previously developed for a heterogeneous medium [Dagrau, Rénier, Marchiano, and Coulouvrat, J. Acoust. Soc. Am. 130, 20-32 (2011)] to the case of a moving medium. It is based on an approximate scalar wave equation for potential, rewritten in a moving time frame, and separated into three parts: (i) the linear wave equation in a homogeneous and quiescent medium, (ii) the effects of atmospheric winds and of density and speed of sound heterogeneities, and (iii) nonlinearities. Each effect is then solved separately by an adapted method: angular spectrum for the wave equation, finite differences for the flow and heterogeneity corrections, and analytical method in time domain for nonlinearities. To keep a one-way formulation, only forward propagating waves are kept in the angular spectrum part, while a wide-angle parabolic approximation is performed on the correction terms. The numerical process is validated in the case of guided modal propagation with a shear flow. It is then applied to the case of blast wave propagation within a boundary layer flow over a flat and rigid ground.