Theory of two-dimensional oblique dispersive shock waves in supersonic flow of a superfluid

Abstract

Dispersive shock waves (DSWs) are studied theoretically in the context of two-dimensional (2D) supersonic flow of a superfluid. Employing Whitham averaging theory for the repulsive Gross-Pitaevskii (GP) equation, suitable jump and entropy conditions are obtained for an oblique DSW, a fundamental building block for 2D flows with boundaries. In analogy to oblique viscous shock waves (VSWs), these conditions yield analytic relations between Mach number $(M)$, velocity deflection angle $(\ensuremath{\theta})$, and wave angle $(\ensuremath{\beta})$. Unlike VSWs, the $M\text{\ensuremath{-}}\ensuremath{\theta}\text{\ensuremath{-}}\ensuremath{\beta}$ phase diagram for DSWs displays four distinct regions associated with phase transitions in supersonic flow over a corner which are predicted and verified by numerical computations of the GP equation. Quasistationary DSWs, shock detachment due to transonic flow, spontaneous excitation of vortices, and the onset of turbulent dynamics associated with cavitation of the superfluid are observed.