Behavior of the vorticity vector in a supersonic nonuniform swirl flow of a combustible gas behind a moving curvilinear shock or detonation wave

Abstract

Expressions for components of the vorticity vector behind a curvilinear shock or detonation wave propagating in a supersonic nonuniform flow of a combustible gas are derived. Plane and axisymmetric gas flows are considered. The free stream in the general case is a vortex flow with a specified distribution of parameters. Formulas for the vorticity components in the plane of the flow for axisymmetric flows are found to have the same form as formulas for steady axisymmetric flows. As in the case of steady flows, the normal-to-wave component of vorticity is demonstrated to remain continuous across the discontinuity surface; in the case of axisymmetric flows, the ratio of the tangential component of vorticity aligned in the plane of the flow to density also remains continuous, though the quantities themselves become discontinuous.In the case of steady flows, the expression for vorticity behind a curved shock wave for a flow with constant parameters was first derived by Truesdell [1] and later by other authors. Lighthill [2] obtained formulas for components of vorticity behind an arbitrary curved wave in the general case, under the assumption of an infinite shock-wave intensity. A generalized formula was given by Hayes [3]. Maikapar [4] and Rusanov [5] obtained formulas for vorticity components behind a shock wave of an arbitrary intensity with constant free-stream parameters. Levin and Skopina [6] studied the behavior of the vorticity vector in supersonic axisymmetric swirl flows behind a steady curvilinear shock or detonation wave. In the general three-dimensional case, expressions for the components of the vorticity vector behind a discontinuity surface generated by a steady supersonic nonuniform flow of a combustible gas around a solid were derived in [7]. For unsteady flows, Levin and Skopina [8] obtained formulas for vorticity on a cylin-1 Institute of Automation and Control Processes, Far-East Division, Russian Academy of Sciences, Vladivostok 690041; levin@imec.msu.ru.drical discontinuity surface propagating in an axisymmetric swirl flow of an ideal gas away from the axis of symmetry. In the present activities, we determine the vorticity directly behind a curvilinear shock or detonation wave propagating over a nonuniform swirl flow of a combustible gas. We study plane-parallel and axisymmetric unsteady motions of the gas, which depend only on two coordinates: x and y. The plane of the flow is the plane (x, y). The axis of symmetry coincides with the straight line x = 0. The main examined quantities (velocity vector V , pressure p, density ρ, and vorticity 2ω = rotV ) are considered as functions of the Cartesian coordinates (x, y) and time t.The velocity vector V in the coordinate system (x 1 , x 2 , x 3 ) has the components u, υ, and w. For planeparallel motion, we have x 1 = x, x 2 = y, and x 3 = z; for axisymmetric motion, we have x 1 = x, x 2 = y, and x 3 = ϕ. For a plane-parallel flow, we have w = 0.