CIRCULATION DEPOSITION ON SHOCK-ACCELERATED PLANAR AND CURVED DENSITY-STRATIFIED INTERFACES - MODELS AND SCALING LAWS

Vorticity is deposited baroclinically by shock waves on density inhomogeneities. In two dimensions, the circulation deposited on a planar interface may be derived analytically using shock polar analysis provided the shock refraction is regular . We present analytical expressions for GAMMA', the circulation deposited per unit length of the unshocked planar interface, within and beyond the regular refraction regime. To lowest order, GAMMA' scales as GAMMA' is-proportional-to (1 - eta-1/2)(sin alpha)(1 + M-1 - 2M-2)(M - 1)(gamma1/2/gamma + 1), where M is the Mach number of the incident shock, eta is the density ratio of the gases across the interface, alpha is the angle between the shock and the interface and gamma is the ratio of specific heats for both gases. For alpha less-than-or-equal-to 30-degrees, the error in this approximation is less than 10% for 1.0 < M less-than-or-equal-to 1.32 for all eta > 1, and 5.8 less-than-or-equal-to eta less-than-or-equal-to 32.6 for all M. We validate our results by quantification of direct numerical simulations of the compressible Euler equations with a second-order Godunov code. We generalize the results for total circulation on non-planar (sinusoidal and circular) interfaces. For the circular bubble case, we introduce a 'near-normality' ansatz and obtain a model for total circulation on the bubble surface that agrees well with results of direct numerical simulations. A comparison with other models in the literature is presented.