Application of Hill’s vortex in a problem of supersonic flow around bodies with a strong injection

A large interest to a problem of supersonic flow around bodies with an injection is determined by the numerous applications. The modern passive heat protection of space vehicles uses sublimation of special coatings. The projected systems of an active heat protection contain an injection of gas in spots of maximum heat transfer of a surface. The mathematical simulation of meteors and bolides in the atmosphere at velocities within the range 11 i V i 72 km/s widely uses supersonic flow with a strong injection. The similar configurations of flows are used in a modern astrophysics, for example, when the question is the interaction of an interstellar media flow with a solar wind. In the work, a new method of calculation of such current parameters is considered. For designing of the solution, it is offered to use the exact solution of hydrodynamical equations, known as Hill’s vortex [1]. Earlier this solution was successfully applied for the description of a shock layer on impermeable sphere and cylinder [2]. Here for the description of flow at presence of an injection, various versions of Hill’s vortex both in shock layer and in injection layer are used. The similar solution has been considered in the work [3], where the same flow function was used in the both layers. In [3] the flow parameters were determined only in the numerical solution, and the asymptotic formula had rather limited area of applicability on parameter of an injection. 1 Spherical Hill’s Vortex Below it will be considered only an axisymmetrical problem, therefore we shall give here a basic information on a spherical Hill’s vortex. Obviously, the generalization on plane case will not meet serious difficulties. The equation for an axisymmetrical flow function for incompressible fluid in spherical variables looks like: ∂ ∂r ( 1 sin θ ∂ψ ∂r ) + ∂ ∂θ ( 1 r2 sin θ ∂ψ ∂θ ) = r sin θf(ψ) (1) ρvr = − 1 r2 sin θ ∂ψ ∂θ , ρvθ = 1 r sin θ ∂|psi ∂r Here f(ψ) is proportional to a vortex intensity. The last two formulas give a connection of a field of speeds vr, vθ with derivatives of flow function. Received on November 15, 2000. † Institute of Mechanics, Moscow State University c ©Copyright: Japan Society of CFD/ CFD Journal 2001 At f(ψ) = A = const, the equation (1) has a particular class of the solutions ψ = (B r + cr + A 10 r ) sin θ. (2) The constants A,B,C are determined from boundary conditions in each particular problem. This class of the solutions is named by a spherical Hill’s vortex. 2 Statement and Solution of a Problem The prospective flow pattern is shown on Fig. 1. A homogeneous supersonic flow goes on a sphere of radius R. From the sphere surface, there is the injection of other gas with a given normal component of speed vr = Vw cos θ at r = R. (3) Initial and injected gases are divided by a tangential surface C, on which all variable currents except pressure have gaps. The velocity components normal to the surface are equal to zero, as the tangential surface is impermeable. On a head shock wave S, usual conditions of Hugoniot are put.