Abstract
An invariant submodel of the two-dimensional equations of gas dynamics, constructed on an operator which is a combination of the time-shift, rotation and projective operators, is investigated using the PODMODELI program [1]. A canonical form of the submodel is constructed and a preliminary analysis of it is carried out (the group property, the hyperbolicity region and the first integrals). The self-similar solution of the submodel is investigated in detail. It determines the solutions of the submodel in question with closed invariant streamlines. Using a hierarchy of submodels, first integrals are obtained in the “second-level” submodel. A qualitative description of the nature of the motion is given (the contact characteristics and the particle trajectories). It is shown that the solution possesses discrete symmetry - invariance under rotation around the origin of coordinates by an angle that is a multiple of 2 π/ N, with a certain natural N. It is pointed out that for certain values of the parameters, solutions of this type describe the gas motion with vacuum regions. The features of the flows obtained are illustrated by examples — the exact solutions of the gas — dynamic equations, which describe the expansion of a gas to a vacuum in an infinite time.
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