Shock formation at the boundary of a sonic jet

Abstract

The similarity solutions of the equations of transonic gasdynamics make it possible to construct the flow in the vicinity of the point of the sonic contact discontinuity, at which the compression shock arises. The results of [1][2][3] show that if the sonic line is not a contact discontinuity, then the compression shock can arise on it only with reflection of a sufficiently strong singularity in the derivatives of the velocity vector. Such a flow may be realized only for bodies of a special form. In the case studied below the origin of the shock wave is not a consequence of the reflection of discontinuities in the derivatives, and flow of this type may arise with flow about arbitrary smooth bodies. In this sense it is an asymptotic flow in the considered vicinity of the flow sonic line.Let us consider a mixed sub-and supersonic flow of a two-dimensional sonic jet about a profile (Fig. 1). We shall assume that the sonic Line MN which is formed in the flow crosses all the streamlines and reaches the boundaries of the jet. The supersonic region between the profile and the edge of the jet must be closed by still another sonic line PQ. We can show that the ~eory of Nikol'skii and Taganov [4] on the nonexistence of continuous flow with rectification of any arbitrarily small segment of the profiIe is applicable to the flow in the resuking supersonic zone MNPQ. In fact, the result of [4] follows only from the theorem on the monotonic variation of the slope of the velocity vector with motion along the sonic line and nonexistence of the sLmple wave type of flow in the limited supersonic region. Both of these theorems are also applicable in the considered case. The result is instability of the continuous flow of a sonic jet past a profile. Therefore the construction of the flow in the vicinity of the point of origin of the compression shock in such a flow is of definite interest.Assume a compression shock occurs at the boundary of the sonic jet. We locate the coordinate origin at this point and direct the x axis along the streamline, and the y axis perpendicular thereto (Fig. 2). The flow is described by a system of transonic gasdynamic equations --uOu i Ox + O~ l Og = O,in which the coordinates x and y are considered dimensionless, and the dimensionless functions u and v are proportional to the components along the x and y axes of the vector of the disturbed particle velocity [5]. Fig. 1The discontinuous solutions of (1) describe flows with compression shocks, on the front of which the shock polar equation must be satisfied [6] 2(va--va) a = (u=--a,)=(a= + u,);(2) the relation [6] a2dx2 / dy -k v2 = usdx2 / dy q-u~,which follows from the condition of continuity of the tangential component of the velocity vector, must also be satisfied. In (2) and (3) the subscripts relate to quantities on the two sides of the shock front and x2 = x2 (y) is the equation specifying its position. We shall seek the problem solution in the class of similarity solutions [7] of the basic system (1) u = P("-"I(~), v = v~(--'>g(D, ~ = ~/~. Then the...