Shock wave focusing using geometrical shock dynamics

Abstract

A finite-difference numerical method for geometrical shock dynamics has been developed based on the analogy between the nonlinear ray equations and the supersonic potential equation. The method has proven to be an efficient and inexpensive tool for approximately analyzing the focusing of weak shock waves, where complex nonlinear wave interactions occur over a large range of physical scales. The numerical results exhibit the qualitative behavior of strong, moderate, and weak shock focusing observed experimentally. The physical mechanisms that are influenced by aperture angle and shock strength are properly represented by geometrical shock dynamics. Comparison with experimental measurements of the location at which maximum shock pressure occurs shows good agreement, but the maximum pressure at focus is overestimated by about 60%. This error, though large, is acceptable when the speed and low cost of the method is taken into consideration. The error is primarily due to the under prediction of disturbance speed on weak shock fronts. Adequate resolution of the focal region proves to be particularly important to properly judge the validity of shock dynamics theory, under-resolution leading to overly optimistic conclusions.