# RD-V: 0500 Shyue Shock Tube (JWL EOS)

A 1D shock tube filled with detonation gas products is used as a test model for the JWL equation of state.

The shock tube problem is a standard problem in gas dynamics. It can be used as a verification test as an exact solution exists. The Jones-Wilkins-Lee equation of state is used to model the behavior of detonation product of high explosives. Here a 1D shock tube case with detonation gas products governed by the Jones-Wilkins-Lee equation of state is built, and the results compared to the analytical solution. The influence of the second order integration scheme in time and space (default in Radioss) is shown.

- Pressure
- Mass density
- Velocity
- Specific energy

## Options and Keywords Used

- Explosive material, /MAT/LAW5 (JWL)
- Multi-fluid law using the FVM solver, /MAT/LAW151 (MULTIFLUID)
- EULER property formulation with
`I`_{ale}= 2 - Second order integration scheme, /ALE/MUSCL

## Input Files

## Model Description

A shock tube consists of a long tube filled with gas, initially divided into two section separated by a diaphragm.

The gas in the two sections are in different physical states: there is usually a high-pressure and low-pressure section.

The interest of the shock tube problem is that it involves the three possible fundamental waves that can develop from uniform initial conditions: shock, rarefaction and contact waves.

Considered in one dimension, the shock tube is a Cartesian geometry Riemann problem,
for which an analytical solution exists. When applied to a gas governed by a JWL
equation of state, the shock tube case in 1D is called the test problem of Shyue. An
analytical solution of this problem is described with analytical solutions
available. ^{1}

A shock tube 100 cm long is filled with detonation gas products and is divided of two chambers of equal length. The gas pressure in the left section (the high-pressure section) is 10 times the pressure in the right section (low-pressure section).

High Pressure Section | Low Pressure Section | |
---|---|---|

Pressure P | 10.0 Mbar | 1 Mbar |

Mass density $\rho $ | 1.7 $\frac{g}{c{m}^{3}}$ | 1.0 $\frac{g}{c{m}^{3}}$ |

Velocity $u$ | 0.0 $\frac{cm}{\mu s}$ | 0.0 $\frac{cm}{\mu s}$ |

The following system is used: cm, ms, g, daN, Mbar

### Model Method

Each section is modeled by one component.

The detonation gas products in the two sections are modeled using /MAT/JWL which is based on the JWL equation of state. The JWL materials are then referenced by /MAT/MULTIFLUID which allows the use of the Radioss finite volume solver.

The two material laws (/MAT/JWL) that represent the high- and low-pressure gas are used in a multi-fluid law (/MAT/MULTILFUID) in order to use the Finite Volume solver.

### Detonation Gas Products Modeling

This shock tube case considers detonation gas products. In Radioss, explosive materials defined with
/MAT/JWL are considered solid until they detonate. To get
these materials in their gaseous states, a detonator is created with a
/DFS/DETPOINT card. The detonation time variable
`TDET` is set to -1E30s (meaning minus infinity), ensuring that
at the beginning at the simulation all the explosive material in the model consists
of detonation gas products.

Detonation gas product behavior is governed by the JWL equation of state. This EOS relates Pressure to the specific volume of the explosive.

The JWL EOS expression is:

- ${P}_{jwl}$
- Pressure
- $\upsilon $
- Relative volume $\upsilon =\frac{{\rho}_{0}}{\rho}$
- $A$ , $B$ , ${R}_{1}$ , ${R}_{2}$
- Parameters specific to the explosives
- $\omega $
- Gruneisen coefficient
- $E$
- Detonation energy per unit volume

High Pressure Section | Low Pressure Section | ||
---|---|---|---|

Density | Initial density $\rho $ | 1.7 $\frac{g}{c{m}^{3}}$ | 1.0 $\frac{g}{c{m}^{3}}$ |

Reference density ${\rho}_{0}$ | 1.84 $\frac{g}{c{m}^{3}}$ | 1.84 $\frac{g}{c{m}^{3}}$ | |

JWL parameters | A | 8.545 Mbar | |

B | 0.205 Mbar | ||

R1 | 4.6 | ||

R2 | 1.35 | ||

$\omega $ | 0.25 | ||

Chapman-Jouguet parameters | Detonation velocity D_{CJ} |
0.693 $\frac{cm}{\mu s}$ | |

Detonation pressure P_{CJ} |
0.21 Mbar | ||

Detonation Energy E_{0} |
42.88164777 Mbar | 7.233944805 Mbar |

The initial density corresponds to the density of the explosive in each compartment, while the reference density is the one used to compute the relative volume used in the JWL equation of state, as:

The JWL parameters comes from the reference article. As this model is a test case for computer code verification, they are not related to any existing explosive materials.

The Chapman-Jouguet parameters used for modeling the propagation of the detonation
are of no importance as this model only considers gaseous materials. The
Chapman-Jouguet parameters of the TNT have been used for detonation pressure
P_{CJ} and velocity D_{CJ}. The precise value of these two
parameters has no influence on the results, as they are only used to model the
detonation process of the solid explosive. The detonation energy E_{0} is
computed using the JWL equation of state expression to get an initial pressure
${P}_{left}=10\text{\hspace{0.17em}}Mbar$
for the high-pressure gas, and
${P}_{right}=1\text{\hspace{0.17em}}Mbar$
for the low-pressure gas.

Two instances of the multi-material law (/MAT/MULTILFUID) are created, each one corresponding to a section and integrating the gas defined using LAW5, with a volumetric fraction of 1.

/EULER/MAT should be defined for the /MAT/MULTIFLUID materials, to indicate that they are EULERIAN materials.

The /ALE/MUSCL option activates a full second order integration
scheme in time and space, which increases the accuracy of the results. The results
of the simulations with and without /ALE/MUSCL will be compared
to the analytical solution. ^{1}

### Boundary Conditions

### Engine Control

Since the Radioss 2019.0 release, the time step scale factor for all ALE and EULER elements is set by default to 0.5.

```
/DT/ALE
0.5 0.0
```

## Results

Results are examined 12 $\mu $ s after the removal of the separation between the two chambers.

Spatial profile of pressure, mass density, velocity and energy are compared to the analytical results given in the reference article. These profiles are obtained by drawing a node path along the tube. To draw node paths, the elemental results visible on the animation have to be averaged at the nodes, using for example the averaging method simple.

The simulation results for these four variables matches closely the analytical solution, and the fit is better for the results obtained with /MUSCL enabled (MUSCL can be disabled with /ALE/MUSCL/OFF Engine card).

The difference between numerical results and the analytical solution can be
quantified by computing the £_{2}-norm error between the curves.

The £_{2}-norm error (also named Eulerian norm error) between a numerical
result
${Y}_{num}$
and the exact solution
${Y}_{exact}$
is defined as:

_{2}-norm error to the analytical solution on pressure, mass density, velocity are:

/ALE/MUSCL | Pressure | Mass Density | Velocity | Specific Energy | |
---|---|---|---|---|---|

£_{2}-normrelative error |
enabled | 2.0% | 6.5% | 3.9% | 4.5% |

disabled | 2.5% | 9.2% | 4.7% | 6.7% |

As shown on Table 3, the use of /ALE/MUSCL allows for better precision in the numerical results.

### Conclusion

The numerical results show a good correlation with the analytical solution presented in the reference article. /MAT/JWL used within a /MAT/MULTIFLUID multi-material law allows for an accurate resolution use of the JWL equation of state.

There are clear benefits of using MUSCL scheme (default in Radioss) in terms of precision.

^{1}Kamm, J.R. An Exact, Compressible One-Dimensional Riemann Solver for General, Convex Equation of State. Los Alamos National Laboratory, 2015