Triple points and sign of circulation

Abstract

Mach reflection 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 conservation of mass and momentum, but not energy. We give a new short proof for non-existence of pure (no contact) triple shocks, recovering a result of Serre. For MR 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 all shocks are compressive. The sign of circulation across the contact has previously been controlled with entropy arguments, showing the post-Mach-stem velocity is generally smaller. We give a kinematic proof assuming compressive shocks and another condition, for example backward incident shocks, or a weak form of the Lax condition. We also show that for 2+2 interactions (two "upper" shocks with clockwise flow meeting two "lower" shocks with counterclockwise flow in a single point) the circulation sign can generally not be controlled. For $γ$-law pressure we show 2+2 interactions without contact must be either symmetric or antisymmetric, with symmetry favored at low Mach number and low shock strength. 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 $1<γ<3$.