Flow behind a Concave Hyperbolic Shock

Abstract

In the design of supersonic airplane and air intake shapes, for specific performance, it is useful to begin with a known shock wave shape and flow-field and deduce the required wall shapes - design methods referred to as “wave rider” or “wave trapper”. Questions then arise as to the nature and existence of flow behind a given shock shape. The left lobe of a hyperbola of revolution shape is proposed as a particular example of a doubly curved, concave axisymmetric shock surface. It offers an analytically simple surface for the study of pressure gradient and flow curvature effects on shock detachment and reflection where the cumulative effects of both shock curvatures are present. Such shock shapes are physically plausible for internal, converging flow and Mach disk shapes. In this paper the concave, hyperbolically shaped shock in both planar and axial flow is investigated analytically with oblique shock theory as well as curved shock theory to discover any tendency towards the formation of a shock wave in the flow immediately behind the hyperbolic shock. If such a shock appears, and impinges on the back of the shock, then there would have to be a kink in the originally posed smooth shock and a Mach interaction would ensue. The onset of Mach interaction, at the sonic point is shown to depend on the free stream Mach number and the ratio of shock curvatures. Critical roles are attributed to both the subsonic patch of flow behind the strong portion of the shock and the orientation of the sonic surface at the shock. There is much experimental evidence of the existence of strong concave shock waves in the studies of Mach reflection where such shocks constitute the Mach stem. No experimental or CFD examples of continuously curved concave shocks that span both the weak and strong shock range have been found; probably because the enclosing ducts have to have very special shapes. Such surface shapes (both planar and axial) are suggested here.