Mach Configuration in Pseudo-stationary Compressible Flow

When a shock front in unsteady compressible flow is moving along an inclined ramp, the incident shock will be reflected by the ramp, as it moves forward. In the process of reflection, various reflected wave pattern may occur depending on the slope of the ramp and the strength of the incident shock [14, 23]. This problem is one of the most important multidimensional prototype problems in gas dynamics. Generally, if the angle of inclination is larger than a critical value, then the reflected shock starting from the intersection of the incident shock with the ramp forms a smooth bubble, which expands when time goes on; such a reflection is called regular reflection. On the other hand, if the angle of inclination is small, then the intersection of the incident shock and the reflected shock will not touch the ramp, and an additional shock front called Mach stem appears, which connects the ramp with the intersection of the incident shock and the reflected shock. In this case three shock fronts meet at one point and form a triple shock configuration. An important fact is that the structure solely containing three shock fronts separating the neighborhood of the triple intersection into three zones with different continuous states does not exist [14, 27]. Many physical experiments indicate that there is a slip line issuing from the triple intersection. Such a local wave pattern is called Mach configuration or Mach structure. Correspondingly, the whole process of reflection is called Mach reflection. Because of the complexity of the whole picture of Mach reflection, it is natural and necessary to start work on getting a clear understanding of the local Mach configuration. Mach reflection also occurs in various other cases for compressible flow. For instance, in the study on oblique shock reflection in stationary compressible flow von Neumann indicated that (see [14, 25]) if the incident shock hit a wall with an angle larger than a critical value, then the reflection must be Mach reflection. Many physical experiments, approximate analysis and numerical computations verified the appearance of Mach configuration, and a tremendous amount of literature is devoted to study such problems (see [3, 4, 11, 19, 20, 24] and the references therein). However, for the rigorous proof on the stability of Mach configuration we can only mention a related result on stationary flow in [13].