Triple points and sign of circulation

Abstract

Interaction of multiple shock waves generally produces a contact discontinuity whose circulation has previously been analyzed using “thermodynamic” arguments based on the Hugoniot relations across the shocks. We focus on “kinematic” techniques that avoid assumptions about the equation of state, using only jump relations for the conservation of mass and momentum but not energy. We give a new short proof for the nonexistence of pure (no contact) triple shocks, recovering a result of Serre. For Mach reflection with a zero-circulation but nonzero-density-jump contact, we show that the incident shock must be normal. Nonexistence without contacts generalizes to two or more incident shocks if we assume that all shocks are compressive. The sign of circulation across the contact has previously been controlled with entropy arguments, showing that the post-Mach-stem velocity is generally smaller. We give a kinematic proof assuming compressive shocks and another condition, such as backward incident shocks, or a weak form of the Lax condition. We also show that for 2 + 2 and higher interactions (multiple “upper” shocks with clockwise flow meeting multiple “lower” shocks with counterclockwise flow in a single point), the circulation sign can generally not be controlled. For γ-law pressure, we show that 2 + 2 interactions without contacts must be either symmetric or antisymmetric, with symmetry favored at low Mach numbers and low shock strengths. For full potential flow instead of the Euler equations, we surprisingly find, contrary to folklore and prior results for other models, that pure triple shocks without contacts are possible, even for γ-law pressure with 1 < γ < 3.