A model for characterization of a vortex pair formed by shock passage over a light-gas inhomogeneity

This work investigates the two-dimensional flow of a shock wave over a circular light-gas inhomogeneity in a channel with finite width. The pressure gradient from the shock wave interacts with the density gradient at the edge of the inhomogeneity to deposit vorticity around the perimeter, and the structure rolls up into a pair of counter-rotating vortices. The aim of this study is to develop an understanding of the scaling laws for the flow field produced by this interaction at times long after the passage of the shock across the inhomogeneity. Numerical simulations are performed for various initial conditions and the results are used to guide the development of relatively simple algebraic models that characterize the dynamics of the vortex pair, and that allow extrapolation of the numerical results to conditions more nearly of interest in practical situations. The models are not derived directly from the equations of motion but depend on these equations and on intuition guided by the numerical results. Agreement between simulations and models is generally good except for a vortex-spacing model which is less satisfactory. A practical application of this shock-induced vortical flow is rapid and efficient mixing of fuel and oxidizer in a SCRAMJET combustion chamber. One possible injector design uses the interaction of an oblique shock wave with a jet of light fuel to generate vorticity which stirs and mixes the two fluids and lifts the burning jet away from the combustor wall. Marble proposed an analogy between this three-dimensional steady flow and the two-dimensional unsteady problem of the present investigation. Comparison is made between closely corresponding three-dimensional steady and two-dimensional unsteady flows, and a mathematical description of Marble's analogy is proposed