Mathematical Modeling, Numerical Methods and Software Complexes

Construction of a self-similar solution to the system of gas dynamics equations describing the outflow of polytropic gas into vacuum from an inclined wall in the inconsistent case

E. I. Pon'kin

Snezhinsk Physic Institute of the National Research Nuclear University MEPhI, Snezhinsk, 456776, Russian Federation

Abstract: 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 $t<0$, 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.

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

UDC: 517.958:531.332

MSC: 76N15, 35C06, 35Q35

Received: February 9, 2023
Revised: May 18, 2023
Accepted: May 25, 2023
First online: June 26, 2023

DOI: 10.14498/vsgtu1999

Bibliographic databases:

• 
• 
• 
• 
•