NUMERICAL METHOD FOR CALCULATING THE INTERNAL STRUCTURE OF SHOCK WAVES

The internal flow of shock waves is governed by a set of hydrodynamic equations with asymptotic boundary conditions. It is usually converted into an initial value problem and solved iteratively by the shooting method. Yet the flow variables always diverge from their correct values in the test, showing the failure of the shooting method. In this paper, the dynamics of the shock system are analyzed qualitatively by means of the topology of the phase portrait, which suggests the importance of the integral direction. It is found that the downstream point is a saddle point, and any tiny error will be significantly multiplied when the phase point approaches it, leading to divergence. The problem can be solved by a backward marching method, in which an initial value point near the downstream point is first determined using L’Hôpital’s rule and the Euler scheme, and then integrate backward from the downstream to the upstream. This method is unconditionally convergent because the integral curve is always directed to the upstream point. To verify it, calculations were performed for shocks over a wide range of Mach numbers (1.01 ~ 100), and the results show that this procedure can solve the shock structure problem correctly and efficiently.