Construction of a self-similar solution to the system of gas dynamics equations

E. I. Pon'kin

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 27 (2023) no. 2, pp. 336-356

Résumé

The present paper is devoted to an initial-boundary value problem for the system of gas dynamics equations in the formulation of the characteristic Cauchy problem of standard form, which describes, at $t>0$, the expansion of a polytropic gas into vacuum on an inclined wall in the space of physical self-similar variables $\xi=x/t$, $\eta=y/t$, and at $t0$, strong compression of gas in the prismatic volume. The solution of the initial-boundary value problem is constructed in the form of series of functions $c( \xi, \vartheta )$, $u( \xi,\vartheta )$ and $v( \xi,\vartheta )$ with powers $\vartheta$, where $\vartheta$ is the known function of independent variables. Finding the unknown coefficients $c_1( \xi )$, $u_1( \xi )$ and $v_1( \xi )$ of the series of functions $c( \xi, \vartheta )$, $u( \xi,\vartheta )$ and $v( \xi,\vartheta )$ is reduced to solving the transport equation for the coefficient $c_1( \xi )$. The study deals with construction of an analytical solution of the transport equation for the coefficient $c_1( \xi )$ of the solution of the system of gas dynamics equations, which describes the isentropic outflow of a polytropic gas from an inclined wall, in the general inconsistent case, when $\tg^2 \alpha \ne (\gamma+1)/(3-\gamma)$. When $\gamma=5/3$, which is the case of hydrogen, an analytical solution of the transport equation is constructed for the coefficient $c_1 ( \xi )$ in explicit form for the first time. The obtained solution has been applied to the description of the compression of a special prismatic volume, which is a regular triangle in cross section. The specific feature of the obtained solution $c_1( \xi )$ indicated in the article is that the value $ c_1 \to \infty $ as $ \xi \to \xi_* $, where the value $\xi_*$ is given by the equation $c_0 (\xi_* )=3.9564$. It is concluded that at the sound characteristic, which is the interface between the flows of centered and double wave types, a gradient catastrophe occurs at the point with coordinates $\xi=\xi_*$ and $\vartheta =0$, which results in development of strong discontinuity in the shock-free flow and formation of a shock wave.

Détail

Keywords: characteristic Cauchy problem of standard form, analog of the Kovalevskaya’s theorem, inclined wall
Mots-clés : transport equation.