The non-linear refraction of shock waves by upstream disturbances in steady supersonic flow

The general problem studied is the propagation of an oblique shock wave through a two-dimensional, steady, non-uniform oncoming flow. A higher-order theory is developed to treat the refraction of the incident oblique shock wave by irrotational or rotational disturbances of arbitrary amplitude provided the flow is supersonic behind the shock. A unique feature of the analysis is the formulation of the flow equations on the downstream side of the shock wave. It is shown that the cumulative effect of the downstream wave interactions on the propagation of the shock wave can be accounted for exactly by a single parameter Φ, the local ratio of the pressure gradients along the Mach wave characteristic directions at the rear of the shock front. The general shock refraction problem is then reduced to a single non-linear differential equation for the local shock turning angle θ as a function of upstream conditions and an unknown wave interaction parameter Φ. To lowest order in the expansion variable θΦ, this equation is equivalent to Whitham's (1958) approximate characteristic rule for the propagation of shock waves in non-uniform flow. While some further insight into the accuracy of Whitham's rule does emerge, the theory is not a selfcontained rational approach, since some knowledge of the wave interaction parameter Φ must be assumed. Analytical and numerical solutions to the basic shock refraction relation are presented for a broad range of flows in which the principal interaction occurs with disturbances generated upstream of the shock. These solutions include the passage of a weak oblique shock wave through: a supersonic shear layer, a converging or diverging flow, a pure pressure disturbance, Prandtl–-Meyer expansions of the same and opposite family, an isentropic non-simple wave region, and a constant pressure rotational flow. The comparison between analytic and numerical results is very satisfactory.