Behavior of the vorticity vector in supersonic axisymmetric swirl flows behind a detonation wave

Abstract

The behavior of the vorticity vector on a discontinuity surface arising in a supersonic nonuniform combustible gas flow with the formation of a shock or detonation wave is studied. In the general case, it is a vortex flow with prescribed distributions of parameters. It is demonstrated that the ratio of the tangential component of vorticity to density remains continuous in passing through the discontinuity surface, while the quantities proper become discontinuous. Results calculated for flow vorticity behind a steady-state detonation wave in an axisymmetric supersonic flow of a combustible mixture of gases are presented.Formulas for components of vorticity behind a steady-state shock wave for flows with constant parameters in the general case were obtained in [1] under the assumption of an infinite shock-wave intensity. Formulas for components of vorticity behind a shock wave of an arbitrary intensity for a uniform incoming flow were derived in [2,3]. The vorticity behind a curvilinear steady-state detonation wave in a supersonic vortex flow is obtained in the present paper. Calculation of Vorticity behind a Steady-State Detonation Wave.Let there be a detonation wave in a steady-state supersonic axisymmetric vortex flow of a combustible gas. The detonation wave is considered as a strong discontinuity surface on which combustion of a unit mass of the gas releases an amount of heat Q, which may be a variable function. For Q = 0, there is a usual shock wave. In this case, the gas motion is described by the following system of equations:Here u(x, r), v(x, r), and w(x, r) are the components of velocity V in a cylindrical coordinate system (x, r, ϕ) and ρ and p are the density and pressure of the gas, respectively. The vorticity 2ω = rot V has the components 2ω r = − ∂w ∂x , 2ω x = 1 r ∂rw ∂r , 2ω ϕ = ∂v ∂x − ∂u ∂r .(2)

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

“…For D = 0, this formula coincides with the expression for vorticity behind a curvilinear steady shock or detonation wave with constant values of the initial gas parameters, which was obtained in [1][2][3][4][5][6][7].…”

“…The form of the formulas for the vorticity components ω 1 and ω 2 lying in the plane of the flow, which differ from zero in the case of axisymmetric flows, coincides with the form of the formulas for the vorticity components behind a steady axisymmetric curvilinear detonation wave in a supersonic vortex flow of a combustible gas, which were obtained in [6]. Let us decompose the vorticity vector into the normal ω n and tangential components ω τ and ω 3 : ahead of the wave, we have…”

“…Thus, the normal component of vorticity remains a continuous function across the discontinuity surface, as in the case of steady flows [6,7]. For unsteady axisymmetric flows, the law of conservation of the quantity ω τ /ρ across the discontinuity surface is valid for all distributions of free-stream gas parameters, as in the case of steady axisymmetric flows [6].…”

“…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 axisymmetric flows, the law of conservation of the quantity ω τ /ρ across the discontinuity surface is valid for all distributions of free-stream gas parameters, as in the case of steady axisymmetric flows [6].…”