Interaction of oblique shock waves with regions of variable pressure, entropy, and energy

Equations are derived for computing the form of an oblique shock wave as it passes through supersonic regions in which static pressure, stagnation pressure, stagnation temperature, or combinations of these are continuously variable. Rigorous portions of the analysis are limited to shock strengths for which the flow downstream of the shock remains supersonic. When no downstream waves other than those generated by the interaction process are present, the rate of change of shock angle with upstream Mach number is found to be a function only of the local shock angle and upstream Mach number; hence, the propagation through a nonuniform region depends only on the initial shock strength and Mach number. A procedure is described for computing the supersonic portion of the flow field downstream of the shock wave. For the special cases of supersonic shear flow and Prandtl-Meyer flow, charts of the shock angle as a function of upstream Mach number are presented so that the passage of a shock wave through these types of nonuniform regions can be easily traced. For a prescribed initial shock strength and initial Mach number, a minimum upstream Mach number is found below which no physically realistic solution can be obtained with the equations for simple propagation. This result serves as a sufficient condition for the avoidance of separated flow, reversed flow, or other upstream effects. An example is computed of the propagation of a shock wave through a wake-type supersonic shear profile and the flow field downstream of the shock is constructed.