Sod shock tube explained

The Sod shock tube problem, named after Gary A. Sod, is a common test for the accuracy of computational fluid codes, like Riemann solvers, and was heavily investigated by Sod in 1978.The test consists of a one-dimensional Riemann problem with the following parameters, for left and right states of an ideal gas.

\left( \begin{array}{c}\rho _{L\\P} _{L\\u} _{L\end{array}\right) = \left(} \begin{array}{c} 1 . 0 \\ 1 . 0 \\ 0 . 0 \end{array} \right) ,\left( \begin{array}{c}\rho _{R\\P} _{R\\u} _{R\end{array}\right) = \left(} \begin{array}{c} 0 . 125 \\ 0 . 1 \\ 0 . 0 \end{array}\right)

where

\rho is the densityP is the pressureu is the velocity

The time evolution of this problem can be described by solving the Euler equations,which leads to three characteristics, describing the propagation speed of thevarious regions of the system. Namely the rarefaction wave, the contact discontinuity andthe shock discontinuity.If this is solved numerically, one can test against the analytical solution,and get information how well a code captures and resolves shocks and contact discontinuitiesand reproduce the correct density profile of the rarefaction wave.

Analytic derivation

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 thethree characteristics of the system, which is due to the finite speedof information propagation. Two of them are equal to the speedof sound of the left and right states

cs _{1} = \sqrt{\gamma (P _{L})/(\rho _{L}) }cs _{5} = \sqrt{\gamma (P _{R})/(\rho _{R}) }where \gamma is the adiabatic gamma.The first one is the position of the beginning of the rarefaction wave whilethe other is the velocity of the propagation of the shock.

Defining:

\Gamma = (\gamma -1)/(\gamma +1) , \beta = (\gamma -1)/(2 \gamma) The states after the shock are connected by the Rankine Hugoniotshock jump conditions.\rho _{4} = \rho _{5} (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 byP _{4} = P _{3} Unfortunately the pressure in region 3 can only be calculated iteratively, the rightsolution is found when u _{3} equals u _{4} u _{4} = \left(P _{3'} - P _{5 \right)\sqrt{ (1- \Gamma)/(\rho _{R(P} _{3'+\Gamma} P _{5)})} }u _{3} = \left(P ^{\beta\right)} _{3 '} \sqrt{ (( 1- \Gamma ^{1 /\gamma} _{1})/(\Gamma ^{2} \rho _{L}) }u _{3} - u _{4} = 0 This function can be evaluated to an arbitrary precision thus giving the pressure in theregion 3P _{3} = \operatorname{calculate}(P _{3,s,s,,)} finally we can calculateu _{3} = u _{5} + ((P _{3} - P _{5)})/(\sqrt{ (\rho _{5})/(((\gamma +1 )P _{3} + (\gamma -1 )P _{5)} 2)) }u _{4} = u _{3} and \rho _{3} follows from the adiabatic gas law\rho _{3} = \rho _{1} \left( (P _{3})/(P _{1}) \right) ^{1/\gamma}

References

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

See also

• Shock tube
• Computational fluid dynamics