Shock interactions with heavy gaseous elliptic cylinders: Two leeward-side shock competition modes and a heuristic model for interfacial circulation deposition at early times

Abstract

We identify two different modes, types I and II, of the interaction for planar shocks accelerating heavy prolate gaseous ellipses. These modes arise from different interactions of the incident shock (IS) and transmitted shock (TS) on the leeward side of the ellipse. A time ratio tT/tI(M,η,λ,γ0,γb), which characterizes the mode of interaction, is derived heuristically. Here, the principal parameters governing the interaction are the Mach number of the shock (M), the ratio of the density of the ellipse to the ambient gas density, (η>1), γ0, γb (the ratios of specific heats of the two gases), λ (the aspect ratio). Salient events in shock–ellipse interactions are identified and correlated with their signatures in circulation budgets and on-axis space–time pressure diagrams. The two modes yield different mechanisms of the baroclinic vorticity generation. We present a heuristic model for the net baroclinic circulation generated on the interface at the end of the early-time phase by both the IS and TS and validate the model via numerical simulations of the Euler equations. In the range 1.2⩽M⩽3.5, 1.54⩽η⩽5.04, and λ=1.5 and 3.0, our model predicts the baroclinic circulation on the interface within a band of ±10% in comparison to converged numerical simulations.