Three-dimensional shock-weak-discontinuity interactions: curvature deposition and post-shock flow response

Abstract

A weak acoustic, entropy or vortical sheet can cross a finite-strength three-dimensional shock without producing a kink, yet it can deposit a discontinuity of curvature along the intersection curve. We derive a local Euler junction law that separates the active normal-plane scattering problem from its three-dimensional geometric realization. The scattering selects one scalar curvature amplitude, while the orientation and pre-existing shape of the shock determine the full curvature tensor and the downstream flow response. A controlled pair of saddle-shock interactions has identical incident and outgoing modal amplitudes but produces post-shock pressure-gradient vectors separated by 31.6 degrees and vorticity vectors separated by 25.6 degrees. For a perfect gas, acoustic incidence also possesses curvature-neutral branches: outgoing acoustic, entropy and shear waves remain finite while the curvature channel cancels. Convected entropy and in-plane vortical sheets deposit curvature with opposite signs, and their coefficients obey an exact relation imposed by total-enthalpy conservation and acoustic orthogonality; line-tangent vorticity is curvature-transparent. An expanding spherical shock provides an unsteady example with non-zero shock speed and non-zero intersection-line tracking speed, separating the smooth blast-wave acceleration from the deposited acceleration jump. Near an umbilic, the same junction selects or rotates the principal-curvature frame. The resulting law supplies the interface data needed to connect smooth curved-shock reconstructions across a continuously differentiable, piecewise twice-differentiable front.