Information technology of mathematical modelling of a three-dimensional supersonic flow on the basis of particular solutions of hydrodynamic problems

Abstract

We propose the formalized methods of mathematical modelling of a three-dimensional supersonic flow on the basis of solutions of smaller dimensionality. We separate a base algorithm for constructing the solution and derive an equation for the streamlined space surface. We introduce concrete techniques, viz. the procedures of superposition, completion, and convergence. We construct test forms for debugging the multidimensional computational algorithms and study various parts of hypersonic aircraft.The methods of mathematical modelling and numerical experiment, which are used in research, find expanding application in engineering fields, where the mathematical models of processes and phenomena have not been adequately developed and tested. When passing to multidimensional complex problems we encounter the difficulties associated with the statement of the problems and the adequacy of the model to a real object. In particular, we mean the problems of high-speed aerodynamics. There are the most complicated aspects of the problem here, viz. on the one hand, the problem is multidimensional and on the other hand it is basically nonlinear for supersonic flows. At the same time the problem of a hypersonic three-dimensional flow has become most urgent because of the evolution of aerospace technology.When studying the hypersonic flow problems it is necessary to consider threedimensional forms. Motion with supersonic speeds in gaseous atmosphere inevitably gives rise to shock waves. When the supersonic speeds are high, these shocks are strong. Then the lifting properties of a body are mainly defined by the asymmetry of the shock configuration since the excessive pressure behind a bow shock is a substantial component when producing a lift force. Therefore three-dimensional problems are urgent and characteristic of a hypersonic velocity range. In 1969 Maikapar and Keldysh [24] introduced a special concept of gasdynamic design to construct a supersonic three-dimensional flow on the basis of particular solutions of smaller dimension. In this paper we try to formalize the technology of obtaining a three-dimensional solution, determine the main operations of the technology, and demonstrate a concrete application.In the procedure of superposition in which the base algorithm is combined with the same technique but this technique is applied to an extended undisturbed supersonic flow we obtain lifting systems whose lift force is produced by the pressure differential in the shock wave. That is why the gasdynamic design takes the features of the technology in hypersonic aerodynamics. The procedure of completion is introduced for constructing bodies with complicated, in particular, star-shaped cross-section.