Analytic solution for the zero-order postshock profiles when an incident planar shock hits a planar contact surface

An explicit analytical solution to calculate the profiles after the shock collision with a planar contact surface is presented. The case when a shock is reflected after the incident shock refraction is considered. The goal of this work is to present explicit formulas to obtain the quantities behind the transmitted and reflected shocks valid for arbitrary initial preshock parameters.

Figure 1

A space-time diagram of the five regimes of the problem. Each regime is characterized by its pressure, density, isentropic exponent, and velocity. The solid lines are shock fronts and the dotted line is the contact surface between fluids “$a$” and “$b$.” The velocity of each front is indicated next to its line.

Figure 2

Dependence of the numerical solution and the four independent roots ${\sigma }_1$, ${\sigma }_2$, ${\sigma }_3$, and ${\sigma }_4$ of Eq. (49) as a function of $M_i$; (a) for ${\gamma }_a=5/3$, ${\gamma }_b=7/5$, and $R_0=1.5$; (b) for ${\gamma }_a={\gamma }_b=5/3$ and $R_0=1.5$. In both cases, the physical solution for $M_r$ starts from the point $(M_i=1,M_r=1)$ and it is given piecewise by ${\sigma }_3$ (in the interval $1\le M_i\le M_{\text{sing}}$), and ${\sigma }_4$ (in the interval $M_i\ge M_{\text{sing}}$). The branches ${\sigma }_1$ and ${\sigma }_2$ do not correspond to physically realizable solutions of $M_r$.

Figure 3

Dependence of the numerical solution and the four independent roots ${\sigma }_1$, ${\sigma }_2$, ${\sigma }_3$, and ${\sigma }_4$ of Eq. (49) as a function of $M_i$ for ${\gamma }_a=5/3$, ${\gamma }_b=7/5$, and $R_0=6$. In this particular case, the interval defined by Eq. (74) is $0.9<R_0<5.4$, and, therefore, the physical solution for every $M_i$ is given by ${\sigma }_3$. The branches ${\sigma }_1$, ${\sigma }_2$ and ${\sigma }_4$ do not correspond to physically realizable solutions of $M_r$.