Qualitative analysis of some evolutionary problems of gas dynamics

We consider a class of evolutionary problems for hyperbolic equations of gas dynamics, describing planar stationary supersonic flows and one-dimensional non-stationary flows. Properties of the solution of evolutionary problem with specific initial and boundary conditions are analyzed. Properties of the solution of evolutionary problem are determined based on qualitative analysis using the isoentropic hypothesis and Prandtl-Meyer and Riemann, without constructing analytical or numerical solution. For isoentropic flows general variables and general functions are introduced, allowing to simultaneously analyze both planar and one-dimensional non-stationary flows. The equivalence rule is formulated, by which the qualitative properties of evolutionary problems for planar supersonic flow are formulated for corresponding evolutionary problem for one-dimensional non-stationary flow and vice versa. Analytical solutions that describe physical properties of considered problems are constructed. An asymptotic solution for high values of evolutionary variable is obtained, which demonstrate the evolution of initial alterations. It is shown analytically that for the considered class of evolutionary problems continuous isoentropic flow given in the initial state can lead to the formation of the shock waves and for high values of evolutionary variable be transformed into non-isoentropic flow with constant pressure and one of the velocity components in the direction of another independent variable being equal to zero. Demonstrated numerical examples confirm the main flow properties obtained in the qualitative analysis.