Bifurcation Analysis of the Oblique Shock Wave in the Supersonic Flow Over a Wedge from a Calculus of Variation Approach

Abstract

The stability of the oblique shock in the supersonic flow over a wedge is investigated from a calculus of variation approach. By taking the nonequilibrium effects inside the shock wave into consideration, a higher order term of bulk viscosity is introduced into the Euler equation, and the resulting system has two smooth solutions corresponding to the two possible attached shock waves, respectively. Additionally, the new system admits a variational formulation, and the stability of the two possible shock waves can then be determined by the second variation of each shock solution. Due to the parameter dependence of the functional, there is a stability transition at the intersection point of the two solution branches, and this gives a new explanation of the fact that only one of the two possible shock waves is physically observable. It is then shown that the critical solution with maximum deflection angle corresponds to a fold bifurcation point.