High frequency and numerical Eulerian methods for aeroacoustic problems

摘要

The aim of this work is the simulation of the acoustic propagation in a moving flow using the high-frequency approach. We linearize the Euler equations around a stationary state for which the resulting system of PDE cannot be in general reduced to a wave equation. We are however able to perform a high-frequency analysis of the acoustic perturbation, using the W.K.B. method, introducing a phase φ and an amplitude A. The phase φ is solution of a Hamilton–Jacobi equation that we solve by a numerical Eulerian method using a monotone scheme [S.J. Osher, C.W. Shu, High-order essentially nonoscillatory schemes for Hamilton–Jacobi equations, SIAM J. Numer. Anal, 28(4) (1991) 907–922] following Benamou et al. [A geometric optics method for high frequency electromagnetic fields computations near fold caustics Part I, J. Comput. Appl. Math. 156 (2003) 93–125]. Adopting the techniques of Lax and Rauch [Lectures on Geometric Optics, 〈http://www.lsa.umich.edu/rauch〉] for hyperbolic systems, we compute the leading order term of the amplitude A. Our results are still valid in the neighborhood of a fold caustic.