NOTE ON A HYPERSONIC TALL-SHOCK PROBLEM
RAND CORP SANTA MONICA CA
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The formation of a shock by a steady gas stream converging toward an axis of symmetry is investigated. The flow is assumed to be a conical field with vertex at the point where the inner boundary of the stream first reaches the axis and where the shock starts. A detailed treatment is given for the case where the flow downstream of the shock is uniform, axial, and of large Mach number M, so that the semivertex angle Tau of the shock is small, but M Tau is not. It is also assumed that the shock is at least fairly strong. An additional small parameter then appears, and a uniform first-order approximation with respect to it is derived. The treamlines are found to be nearly straight and parallel, but the gas is found to suffer a precompression, upstream of the shock, such that the inner boundary of the inviscid gas stream is a vacuum line. Similarly, vacuum is approached asymptotically on all streamlines with distance upstream from the shock. It is shown that the result concerning the occurrence of vacuum must be expected to be independent of all but one assumption--that the gas impinges on the axis at a non-zero angle. Accordingly, tail-shock formation in two-dimensional and axially symmetrical flows must be expected to differ in significant, still-unknown ways.
- Fluid Mechanics