Curved shock theory

Abstract

Curved shock theory (CST) is introduced, developed and applied to relate pressure gradients, streamline curvatures, vorticity and shock curvatures in flows with planar or axial symmetry. Explicit expressions are given, in an influence coefficient format, that relate post-shock pressure gradient, streamline curvature and vorticity to preshock gradients and shock curvature in steady flow. The effect of pre-shock flow divergence/convergence, on vorticity generation, is related to the transverse shock curvature. A novel derivation for the post-shock vorticity is presented that includes the effects of pre-shock flow non-uniformities. CST applicability to unsteady flows is discussed. History and introductionThere is a long, albeit thin, history of research, stretching over 75 years, on curved shocks, in steady flow with bounding curved streamlines and varying pressure, from Crocco [2] and Thomas [22] to the modern treatment of Emanuel [5]. These efforts, focusing on shock curvature and the resulting flow property gradients, have been largely analytical. Crocco [2] showed that, on a curved, planarly symmetric (planar) shock wave, there is a shock angle where the streamline behind the shock is straight, irrespective of shock curvature. This location on the shock surface is called the Crocco point. Thomas Communicated by B. Skews. B S. Mölder[24] derived the curved shock equations for steady flow of an ideal gas with planar shocks in uniform flow. He found an expression for the curvature of the streamlines behind a curved shock. Any influence of upstream vorticity was not considered. Lin and Rubinoff [16] re-derived the equations of Crocco and Thomas to show that a normal shock can sit on a continuously curving surface only if the Mach number exceeds a certain supersonic value. Thomas [23] extended the curvature notion to higher derivatives of shock and streamline shape, giving extensive graphs of the first-derivative relations. Algebraic complexities prevented Thomas from examining higher derivatives. Today's computerized algebra manipulators such as Matlab and Maple could be used to advance Thomas' early efforts. Thomas [24,25] also considered the motion of a shock attached to the leading edge of a planar, curved surface and developed total differential equations for the first, second and third approximations for the surface pressure. Truesdell [28] derived the formula for the vorticity jump across a curved shock wave, but erroneously concluded that "when a uniform flow of any fluid breaks across a shock the pressure gradient cannot vanish on the rear side of the shock at any point where the shock is curved and oblique". A simple physical argument shows otherwise and so does the correct theory. The shock angle and place on the shock wave where the pressure gradient vanishes is called the Thomas point. An application of curved shock theory (CST) to the propagation and decay of spherical blast waves is found in Thomas [27]. Gerber and Bartos [6] presented coefficients for the curved shock equations for determining the ...

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