The flow gradients in the vicinity of a shock wave for a   thermodynamically imperfect gas
  Supersonic vortex plane and axisymmetric flows of non-viscous non-heatconductive gas with arbitrary thermodynamic properties in the vicinity of a steady shock wave are studied. The differential equations describing the gas flow exterior to the discontinuity surface and the dynamic compatibility conditions at this discontinuity are used. The gas flow nonuniformity in the shock vicinity is described by the space derivatives of the gasdynamic parameters at a point on the shock surface. The parameters are the gas pressure, density, velocity vector. The derivatives with respect to the directions of the streamline and normal to it, and of the shock surface and normal to it are considered. Space derivatives of all gasdynamic parameters are expressed through the flow nonisobaric factor along the streamline, the streamline curvature, and the flow vorticity and non-isoenthalpy factors. An algorithm for these factors of the gas flow downstream a shock wave determination is developed. Examples of these factors calculation for imperfect oxygen and thermodynamically perfect gas are presented. The influence coefficients of the upstream flow factors on the downstream flow factors are calculated. As an illustration for flows with upstream Mach number 5 it is shown that the flow vorticity factor is the most influenced by the thermodynamical gas properties. The gas flow in the vicinity of the shock is described by the isolines of gasdynamic parameters. Uniform plane and axisymmetric flows on different distances from the axis of symmetry are examined; the isobars, isopycnics, isotachs and isoclines are used to characterize the downstream flow behind a curved shock in an imperfect gas. 
Vladimir N. Uskov and Pavel S. Mostovykh
physics.flu-dyn