Baroclinic circulation generation on shock accelerated slow/fast gas interfaces

Abstract

Vorticity is deposited baroclinically by shock waves on density inhomogeneities. In two dimensions, for a planar “slow–fast” interface, we present analytical results for σ, the circulation deposited per unit unshocked interface length, within the regular refraction regime. The parameters that describe the interaction are the Mach number (M), the density ratio of the two gases (η, η<1), the local angle between the shock and the interface (α), and the ratio of specific heats of the two gases (γ0,γb). For weak shocks σ scales as σ∝(1−η−1/2)ξ(M)sin α and for strong shocks σ→K(η,α,γ)/(1−ξ(M)). For scaling purposes, the gases are assumed to have the same γ. K(η,α,γ) is a function of the density ratio, the interface angle, and the ratio of specific heats γ [Eq. (4.6)] and ξ(M) is the normalized pressure gradient across the shock. The planar interface approach is used to find formulas to calculate the total circulation deposited on sinusoidal interfaces. To validate the formulas, numerical simulations of the compressible Euler equations were made using a second-order Godunov code. Simulations were done for 1.05⩽M⩽3.0 and η=0.14, 0.33 and 0.65.