Spatial and temporal discretization schemes

Tim Broeckhoven, Jan Ramboer, Sergey Smirnov, and Chris Lacor

Introduction to discretization schemes

In contrast to standard computational fluid dynamics (CFD) applications, for which second-order accuracy in space is sufficient for engineering purposes, the requirements on the schemes are much more stringent in the computation of aeroacoustical applications. There are several reasons for this.

First, the amplitudes of acoustic waves are several orders of magnitude smaller than the average aerodynamic field amplitudes. In addition, the length scales of acoustic waves, typically the principal acoustic wavelengths, are some orders of magnitude larger than the dimensions of the sound-generating perturbations (vortices and turbulent eddies). Thirdly, sound generated by turbulence is broadband noise with often three orders of magnitude difference between the largest and the smallest acoustic wavelengths. Finally, acoustic waves propagate at the speed of sound (which is not necessarily comparable to the mean flow velocity) over large distances in all spatial directions, whereas aerodynamic perturbations are only convected by the mean flow. Moreover, one is usually interested in the noise level at the far field, implying that the waves have to be traced accurately over long distances.

This requires numerical methods with higher accuracy than routinely applied in CFD codes. In particular, the discretization of the convective operator (i.e., the first-order derivative) requires special attention. In this respect, the actual order of the scheme, which can be determined based on a Taylor expansion analysis, is not the primary concern.