Numerical Methods for High-Speed Flows

We review numerical methods for direct numerical simulation (DNS) and large-eddy simulation (LES) of turbulent compressible flow in the presence of shock waves. Ideal numerical methods should be accurate and free from numerical dissipation in smooth parts of the flow, and at the same time they must robustly capture shock waves without significant Gibbs ringing, which may lead to nonlinear instability. Adapting to these conflicting goals leads to the design of strongly nonlinear numerical schemes that depend on the geometrical properties of the solution. For low-dissipation methods for smooth flows, numerical stability can be based on physical conservation principles for kinetic energy and/or entropy. Shock-capturing requires the addition of artificial dissipation, in more or less explicit form, as a surrogate for physical viscosity, to obtain nonoscillatory transitions. Methods suitable for both smooth and shocked flows are discussed, and the potential for hybridization is highlighted. Examples of the application of advanced algorithms to DNS/LES of turbulent, compressible flows are presented.