Triple-shock entropy theorem and its consequences

Abstract

For a convex equation of state, a general theorem on shock waves is proved: a sequence of two shocks has a lower entropy than a single shock to the same final pressure. We call this the triple-shock entropy theorem. This theorem has important consequences for shock interactions. In one dimension the interaction of two shock waves of the opposite family always results in two outgoing shock waves. In two dimensions the intersection of three shocks, such as a Mach configuration, must have a contact. Moreover, the state behind the Mach stem has a higher entropy than the state behind the reflected shock. For the transition between a regular and Mach reflection, this suggests that the von Neumann (mechanical equilibrium) criterion would be preferred based on thermodynamic stability, i.e. maximum entropy subject to the system constraint that the total specific enthalpy is fixed. However, to explain the observed hysteresis of the transition we propose an analogy with phase transitions in which locally stable wave patterns (regular or Mach reflection) play the role of meta-stable thermodynamic states. The hysteresis effect would occur only when the transition threshold exceeds the background fluctuations. The transition threshold is affected by flow gradients in the neighbourhood of the shock intersection point and the background fluctuations are due to acoustic noise. Consequently, the occurrence of hysteresis is sensitive to the experimental design, and only under special circumstances is hysteresis observed.