- Source: Sod shock tube
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.
(
ρ
L
P
L
u
L
)
=
(
1.0
1.0
0.0
)
{\displaystyle \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)}
,
(
ρ
R
P
R
u
R
)
=
(
0.125
0.1
0.0
)
{\displaystyle \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
ρ
{\displaystyle \rho }
is the density
P
{\displaystyle P}
is the pressure
u
{\displaystyle u}
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 the
various regions of the system. Namely the rarefaction wave, the contact discontinuity and
the 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 discontinuities
and 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 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
c
s
1
=
γ
P
L
ρ
L
{\displaystyle cs_{1}={\sqrt {\gamma {\frac {P_{L}}{\rho _{L}}}}}}
c
s
5
=
γ
P
R
ρ
R
{\displaystyle cs_{5}={\sqrt {\gamma {\frac {P_{R}}{\rho _{R}}}}}}
where
γ
{\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:
Γ
=
γ
−
1
γ
+
1
{\displaystyle \Gamma ={\frac {\gamma -1}{\gamma +1}}}
,
β
=
γ
−
1
2
γ
{\displaystyle \beta ={\frac {\gamma -1}{2\gamma }}}
The states after the shock are connected by the Rankine Hugoniot
shock jump conditions.
ρ
4
=
ρ
5
P
4
+
Γ
P
5
P
5
+
Γ
P
4
{\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
P
4
=
P
3
{\displaystyle P_{4}=P_{3}}
Unfortunately the pressure in region 3 can only be calculated iteratively, the right
solution is found when
u
3
{\displaystyle u_{3}}
equals
u
4
{\displaystyle u_{4}}
u
4
=
(
P
3
′
−
P
5
)
1
−
Γ
ρ
R
(
P
3
′
+
Γ
P
5
)
{\displaystyle u_{4}=\left(P_{3}'-P_{5}\right){\sqrt {\frac {1-\Gamma }{\rho _{R}(P_{3}'+\Gamma P_{5})}}}}
u
3
=
(
P
1
β
−
P
3
′
β
)
(
1
−
Γ
2
)
P
1
1
/
γ
Γ
2
ρ
L
{\displaystyle u_{3}=\left(P_{1}^{\beta }-P_{3}'^{\beta }\right){\sqrt {\frac {(1-\Gamma ^{2})P_{1}^{1/\gamma }}{\Gamma ^{2}\rho _{L}}}}}
u
3
−
u
4
=
0
{\displaystyle u_{3}-u_{4}=0}
This function can be evaluated to an arbitrary precision thus giving the pressure in the
region 3
P
3
=
calculate
(
P
3
,
s
,
s
,
,
)
{\displaystyle P_{3}=\operatorname {calculate} (P_{3},s,s,,)}
finally we can calculate
u
3
=
u
5
+
(
P
3
−
P
5
)
ρ
5
2
(
(
γ
+
1
)
P
3
+
(
γ
−
1
)
P
5
)
{\displaystyle u_{3}=u_{5}+{\frac {(P_{3}-P_{5})}{\sqrt {{\frac {\rho _{5}}{2}}((\gamma +1)P_{3}+(\gamma -1)P_{5})}}}}
u
4
=
u
3
{\displaystyle u_{4}=u_{3}}
and
ρ
3
{\displaystyle \rho _{3}}
follows from the adiabatic gas law
ρ
3
=
ρ
1
(
P
3
P
1
)
1
/
γ
{\displaystyle \rho _{3}=\rho _{1}\left({\frac {P_{3}}{P_{1}}}\right)^{1/\gamma }}
References
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.
See also
Shock tube
Computational fluid dynamics
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