Numerical solution of a free-boundary problem in hypersonic flow theory: Nonequilibrium viscous shock layers

Abstract

Non equilibrium viscous shock layer flows produced near the stagnation point of a blunt body constitute a standard problem in hypersonic flow theory. This flow configuration is usually modeled with the viscous shock layer (VSL) equations derived by writing the full Navier-Stokes equations in boundary-layer coordinates and performing an order of magnitude analysis. The terms retained in this process assure that for moderate Reynolds numbers the resulting equations are uniformly valid between the body surface and the shock, which may be treated as a thin discontinuity. The VSL equations are generally solved by starting with the thin layer approximation (TVSL). Boundary conditions are specified at the body surface and the Rankine-Hugoniot relations are imposed at the shock. Because the position of the shock is not known, one has to solve a free boundary problem. This paper presents a novel solution procedure for this situation. A reduced coordinate is introduced and the free boundary problem is transformed into a nonlinear eigenvalue problem. The new problem for an augmented set of variables is then solved with Newton iterations and adaptive gridding. The method is illustrated in the paper with a solution of the thin shock layer equations at the stagnation streamline. The solution obtained on this line is then used as an initial condition for a two-dimensional marching procedure. The complete axisymmetric problem is solved by performing fully coupled Newton iterations; moreover, we consider a new enlarged unknown function including the shock location. The model includes detailed transport properties and complex kinetics for air dissociation and ionization. However, in order to focus on the numerical method and for the sake of simplicity, thermodynamic equilibrium is assumed.