Efficient Runge-Kutta Discontinuous Galerkin Methods Applied to Aeroacoustics

摘要

The simulation of aeroacoustic problems sets demanding requirements on numerical methods, particularly in terms of accuracy. Runge-Kutta Discontinuous Galerkin (RKDG) schemes are increasingly popular for such applications, because they converge at an arbitrarily high order rate, they can deal with complex geometries, and they are amenable to parallel computing. However, they are still considered to be computationally costly. The work presented in this thesis aims at improving the computational efficiency of RKDG methods for linear aeroacoustic applications.The first part of the work is dedicated to the study of the stability and accuracy properties that affect the performance of RKDG methods applied to hyperbolic problems. An analysis technique inspired by the classical von Neumann method is used to determine the stability restrictions of the schemes, as well as their accuracy properties in terms of dissipation and dispersion. It is first used to investigate the influence of the element shape on CFL conditions with triangular grids, in order to improve the determination of the maximum allowable time step in practical simulations. Alternative methods to the CFL conditions are also devised for this purpose. Moreover, Runge-Kutta schemes specifically designed to maximize the computational efficiency of RKDG methods for wave propagation problems are derived.The second part of the work deals with the application of RKDG methods to linear aeroacoustics. RKDG formulations for the linearized Euler and Navier-Stokes equations are introduced, along with validation cases. Then, higher-order treatments of curved wall boundaries, needed to fully benefit from the efficiency of high-order RKDG methods in aeroacoustic propagation problems, are studied. Finally, the methods developed in this work are used in a hybrid approach to characterize the acoustic behaviour of orifices in plates under grazing flow. The results show a clear qualitative improvement over the existing analytical approaches.