Sod shock tube

NOTE: The equations provided below are only correct when rarefaction takes place on left side of domain and shock happens on right side of domain. The different states of the solution are separated by the time evolution of the three characteristics of the system, which is due to the finite speed of information propagation. Two of them are equal to the speed of sound of the left and right states

${\displaystyle cs_{1}={\sqrt {\gamma {\frac {P_{L}}{\rho _{L}}}}}}$

${\displaystyle cs_{5}={\sqrt {\gamma {\frac {P_{R}}{\rho _{R}}}}}}$

where ${\displaystyle \gamma }$ is the adiabatic gamma. The first one is the position of the beginning of the rarefaction wave while the other is the velocity of the propagation of the shock.

Defining:

${\displaystyle \Gamma ={\frac {\gamma -1}{\gamma +1}}}$, ${\displaystyle \beta ={\frac {\gamma -1}{2\gamma }}}$

The states after the shock are connected by the Rankine Hugoniot shock jump conditions.

${\displaystyle \rho _{4}=\rho _{5}{\frac {P_{4}+\Gamma P_{5}}{P_{5}+\Gamma P_{4}}}}$

But to calculate the density in Region 4 we need to know the pressure in that region. This is related by the contact discontinuity with the pressure in region 3 by

${\displaystyle P_{4}=P_{3}}$

Unfortunately the pressure in region 3 can only be calculated iteratively, the right solution is found when ${\displaystyle u_{3}}$ equals ${\displaystyle u_{4}}$

${\displaystyle u_{4}=\left(P_{3}'-P_{5}\right){\sqrt {\frac {1-\Gamma }{\rho _{R}(P_{3}'+\Gamma P_{5})}}}}$

${\displaystyle u_{3}=\left(P_{1}^{\beta }-P_{3}'^{\beta }\right){\sqrt {\frac {(1-\Gamma ^{2})P_{1}^{1/\gamma }}{\Gamma ^{2}\rho _{L}}}}}$

${\displaystyle u_{3}-u_{4}=0}$

This function can be evaluated to an arbitrary precision thus giving the pressure in the region 3

${\displaystyle P_{3}=\operatorname {calculate} (P_{3},s,s,,)}$

finally we can calculate

${\displaystyle u_{3}=u_{5}+{\frac {(P_{3}-P_{5})}{\sqrt {{\frac {\rho _{5}}{2}}((\gamma +1)P_{3}+(\gamma -1)P_{5})}}}}$

${\displaystyle u_{4}=u_{3}}$

and ${\displaystyle \rho _{3}}$ follows from the adiabatic gas law

${\displaystyle \rho _{3}=\rho _{1}\left({\frac {P_{3}}{P_{1}}}\right)^{1/\gamma }}$

• Sod, G. A. (1978). "A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws" (PDF). J. Comput. Phys. 27 (1): 1–31. Bibcode:1978JCoPh..27....1S. doi:10.1016/0021-9991(78)90023-2. OSTI 6812922.
• Toro, Eleuterio F. (1999). Riemann Solvers and Numerical Methods for Fluid Dynamics. Berlin: Springer Verlag. ISBN 3-540-65966-8.

• Shock tube
• Computational fluid dynamics