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A numerical method for solving the problem of three-dimensional viscous cascade flow is developed and its accuracy is estimated.

The results are presented on a numerical study of the propagation of strongly underexpanded jets of an ideal gas in a flooded space with Mach numbers at the nozzle cut slightly exceeding unity. In this case the angle of turn of the stream in the vicinity of the rim of the conical nozzle can exceed 90‡ and the boundary of the jet turns back beyond the cross section of the nozzle cut. It is shown that the square root of the expansion ratio of the discharge is a self-similarity parameter for the geometrical dimensions of the calculated jets. The dependence of the longitudinal dimensions of such jets on the Mach number at the nozzle cut is close to linear. The effect of the adiabatic index on the shape of the jet boundary and the suspended compression shock is illustrated. The integration of the steady system of gasdynamic equations describing the two-dimensional supersonic flow of a nonviscous and thermally nonconducting gas, performed by a monotonie finite-difference system of through calculation of the first order of accuracy, was proposed in [1, 2] and generalized to the case of arbitrary orientation of the velocity vector in the plane of flow in [3].

The results are presented of the numerical investigation of the interaction of a supersonic axisymmetrical jet of a nonviscous and nonthermally conductive gas, flowing from a conical nozzle into a space with reduced pressure, with a plane obstacle. The presence of a triple point of intersection of the shock wave issuing from the obstacle with the trailing and reflected oblique compression shock is characteristic for the conditions considered in the paper. The solution of the problem is obtained by numerical integration of the gasdynamic equations by means of monotonic difference schemes of a straight-through calculation with first-order accuracy. The interaction of supersonic gas jets with surfaces is a vast problem and is one of the trends being developed intensively in the theory of jet streams. Of the whole multiplicity of problems of practical interest, the two-dimensional case of the “normal” collision between a supersonic axisymmetrical jet and a plane obstacle has been studied in most detail. As a result of the investigations carried out, many characteristic mechanisms of these flows have been revealed. Together with the numerous experimental papers, several reports have been published (for example, [1–4]) in which various numerical methods are employed to solve this problem. In addition to the method of integral relations used in [1], an implicit difference scheme [2] and explicit schemes of straight-through calculation [3, 4] have been used to calculate the subsonic zone of increased pressure in front of the obstacle. However, an extensive investigation of the special features of the action of a supersonic underexpanded jet on a plane obstacle, at a very small distance from the nozzle exit, still has not been carried out up to the present. In this paper, a solution of this problem is undertaken by the numerical method described in [4] using difference schemes [5, 8].

The use of the time-establishment process in solving problems relating to the steady flow of an ideal (nonviscous and heat-insulating) gas around a cascade of profiles is considered. Chief attention is paid to the role of the Chaplygin-Zhukovskii condition, which is needed [1, 2] when solving these as steady-state problems. Results obtained during the time-establishment process with and without the application of the condition in question are presented. In both cases certain time-independent distributions of the parameters satisfying the equations and conditions of the steady-state problem (or more precisely their finite-difference analog) are produced during the establishment process. Comparison of the resultant distributions shows that the difference between the two cases lies within the limits of computing error, the results of the calculations agreeing quite well with experimental data.

An explicit divergence difference scheme of third order of approximation with respect to the spatial variables is used to calculate two-dimensional steady flows of an inviscid gas that does not conduct heat in contracting-expanding nozzles in the presence of nonequilibrium physicochemical processes. The flow structure is demonstrated for a three-component vibrationally nonequilibrium gas mixture in planar Laval nozzles with different radii of curvature of the nozzle profile in the neighborhood of the critical section.

A large part of the known results of Laval nozzle theory relates to the inverse problem, in which the velocity distribution on some line (usually the axis of symmetry) is given rather than the nozzle contour. Many important properties of transonic flows have been disclosed as a result of numerous studies, whose basic results were presented together with an extensive bibliography in Ryzhov's monograph [1]. The solution of the inverse problem has recently been used not only to analyze the qualitative characteristics but also to construct nozzles with rather marked variation of the slope of the generator, which are of practical interest. In this connection we note the work of Pirumov [2] and also the studies of Hopkins and Hill [3, 4]. The latter authors, in addition to the classical Laval nozzle, studied several nozzle schemes with a centerbody. Pirumov used a specially developed numerical method for the solution of the inverse problem (we note that in the subsonic part of the nozzle the corresponding Cauchy problem is incorrect), while Hopkins and Hill used a series expansion which was preceded by a change of variables.

A large number of studies have been published which define bodies of minimal drag and nozzles of maximal thrust. A study has been made of the region of existence of the various solutions in planar flow, and it has been shown that the optimal profile in the presence of a length limitation may have a blunt end, which is a segment of a boundary extremum. The results of these studies and also the required bibiliography are contained in [1–3].