Development of non-dissipative numerical schemes for computational aeroacoustics
The ability of popular computational fluid dynamic (CFD) methods to simulate the propagation of acoustic waves is assessed both theoretically and by numerical experiment. The schemes are used to model acoustic waves generated by a noise source situated at the aft end of a convergent-divergent nozzle. It is found that the acoustic signal that traverses up the nozzle has difficulty maintaining its correct amplitude as the frequency of the noise increases. When the sound is of sufficiently high frequency, the acoustic wave is dissipated so much that it does not reach the front end of the channel. Based on this experience, we argue for schemes that are inherently free of dissipation. In one dimension, we study an “upwind” form of the leapfrog scheme, and show how it can be used to compute a linearized perturbation of the nonlinear solution obtained from any Euler code. This method is non-dissipative and can accurately model the correct amplitude of the acoustic wave no matter what the frequency of its source. The scheme is extended to multiple space dimensions by employing bicharacteristic equations defined for a certain ’staggered’ storage of the unknowns. It remains free from dissipation, and the solution for the waves propagated into a half space by an oscillating piston is in excellent agreement with the analytical solution.
