A general objective shock wave detection from a geometric singular perturbation approach

For the general unsteady multi-dimensional flow, the non-linear non-equilibrium nature of shock waves is investigated from the geometric singular perturbation theory. With the introduction of a pressure non-equilibrium term, the modified Euler equation can be reduced to systems of ordinary differential equations(ODEs) along carefully constructed curves. Along each curve, a slow-fast system is derived from the governing ODEs, and the geometric singular perturbation theory is then applied. The motion of the slow-fast system is decomposed to two parts, the quasi-equilibrium slow motion where the non-equilibrium effect is negligible and the fast motion where the non-equilibrium effect plays a dominating role. It is then shown that a shock wave can be recognized as the fast motion of a slow-fast system in an objective manner, and this shock detection method can serve as a rational foundation for practical shock detection problem.