Three-dimensional curved shock theory based on orthogonal frame of shock surface and its applications

By deriving the Euler equations and Rankine–Hugoniot equations in the orthogonal frame field of the shock surface, the three-dimensional curved shock theory based on orthogonal frame of shock surface (3D-CST-boos) is established. In steady flow, this theory can be applied to three-dimensional (3-D) shocks without constraints on the incoming flow conditions. The derived equations elucidate the relationship between the first-order gradients of the preshock and postshock flow parameters and the geometric properties (curvature) of the 3-D curved shock. The correctness of 3D-CST-boos is verified for two-dimensional plane shocks and axisymmetric shocks. The analysis is then extended to the flow patterns of 3-D elliptical convex/concave shocks. Variations in the flow field behind a 3-D elliptical convex shock are explained based on different incoming flow conditions. Simultaneously, the fundamental mechanics underlying the differences between the flow fields of elliptical concave shocks and axisymmetric concave shocks are revealed using 3D-CST-boos. Finally, a concise analysis of the first-order flow parameters is presented for more complex 3-D shocks, including saddle-shaped shocks and cubic surface shocks.