Theoretical study of the self-similar traits, hysteresis phenomenon, and stability of reflected shock waves associated with inlet flow patterns

Of interest to the analytical study of Mach stem heights is the shock patterns that form in a simple inlet flow. A simple inlet flow is defined as a flow in which the inlet height is of such a size such that the incident wave does not interact with the expansion fan, and that the inlet wedge face is small enough that a reflected shock does not impinge upon it. The condition for which the incident wave intersects the leading edge of the expansion fan, and for the case where a reflected wave intersects the trailing edge of the inlet wedge were found in the form of a ratio between the inlet height and wedge face length. The various wave angle limits for both cases were also studied and presented.; Once the limits for a simple inlet flow were defined, the problem of solving for Mach stem heights was studied. Quantitatively, past analytical studies were able to achieve moderate success at predicting stem heights for a very limited range of Mach numbers. Inherent to a quantitative study regarding Mach stems is the lack of a physical length scale. Therefore, the present study approached the problem of Mach Reflections in a qualitative way. The self-similar aspects of the Mach stem were revealed and a solution for Mach stems was found. The explicit dependence of the Mach stem on the freestream Mach number, wedge angle, and ratio of specific heats were noted. Additionally, the theoretical lower and upper limits of the Regular and Mach Reflection flowfields were studied.; In addition to solving qualitatively for Mach stem heights, a study was done on the stability of the shock waves pertaining to the reflection patterns that form. In the region betaN ≤ beta ≤ beta D it is possible for either a Regular Reflection or a Mach Reflection to form for a given freestream Mach number. Of practical interest, besides the stability of the shock pattern, is the transition from a Regular Reflection to a Mach Reflection and vice versa. The transition between the two reflection patterns undergoes a hysteresis which depends on whether the wave angle increases or decreases between the limits betaN and betaD. The results from the analysis of both the stability and the hysteresis are presented.