Analytical investigation of the interaction of a shock with a diffuse contact

Shock propagation through fluids of continuously changing properties, specifically temperature and density gradients, are observed in a variety of high speed flow scenarios including shock tube experiments, shocks through atmospheric layers, and detonation waves. As a shock propagates through the gradient, the strength of the wave changes due to shock refraction within the continually changing medium. Numerically, this diffusion is represented by a finite-thickness interface layer across which temperature and density change discontinuously. However, approximate numerical Riemann solvers often require high resolution discretization in areas of steep gradients and can introduce unrealistic (numerical) diffusion.

In this study, we compare analytical and numerical solutions of a shock propagating through a gas with a diffuse contact profile. Both low-to-high and high-to-low density configurations are investigated. The method of characteristics for an ideal gas is employed to analytically model the shock and resulting wave structures through the domain. The interface is represented by a finite number of contact waves that separate regions of varying temperature and density. As the shock interacts with each interface, an exact Riemann problem is solved to determine the resulting wave structure. Subsequent wave interactions are treated similarly, resulting in continually reflected and transmitted waves at each interface. The final transmitted and reflected wave strength and speed are determined analytically and compared to numerical results of similar discretization. The effect of discretization and numerical diffusion are explored. The method utilized offers an analytical solution to the resulting wave structures with a reduction in compute time, allowing for higher resolution in the diffuse region.