The transitory response of a perfect gas in a long tube separated into two parts using a diaphragm is studied. The
problem is well-known as the Riemann problem. The numerical results based on the finite element method with the Lagrangian
and Eulerian formulations, are compared to the analytical solution.
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.
The following criteria is used to compare the results:
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
Iale =
2
Second order integration scheme, /ALE/MUSCL
Input Files
Before you begin, copy the file(s) used in this problem
to your working directory.
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.
At the beginning of the experiment, the separation is removed, and a compression
shock runs into the low-pressure region, while a rarefaction wave moves into the
high-pressure part of the tube. A contact discontinuity usually occurs between the
two gases when the diaphragm is removed.
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).
The physical properties of the detonation gas products in the right and left sections
are:
表 1. Initial Conditions
High Pressure Section
Low Pressure Section
Pressure P
10.0 Mbar
1 Mbar
Mass density
1.7
1.0
Velocity
0.0
0.0
The following system is used: cm, ms, g, daN, Mbar
Model Method
Each section is modeled by one component.
Both are meshed with 3D /BRICK elements. As the problem is
one-dimensional, the relevant mesh size is the width of the mesh in the direction of
interest (here, the direction of propagation of the compression and rarefaction
waves). The size of the mesh in the other directions is of no importance (図 3).
The model is meshed with 512 elements (which accounts for a mesh size in the
direction of interest of roughly 2 mm).
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:(1)
Where,
Pressure
Relative volume
,, ,
Parameters specific to the explosives
Gruneisen coefficient
Detonation energy per unit volume
The /MAT/JWL cards used for the detonation gas products are
defined using the following parameters:
表 2. Values of Input Parameters for the
/MAT/JWL Cards
High Pressure Section
Low Pressure Section
Density
Initial density
1.7
1.0
Reference density
1.84
1.84
JWL parameters
A
8.545 Mbar
B
0.205 Mbar
R1
4.6
R2
1.35
0.25
Chapman-Jouguet parameters
Detonation velocity DCJ
0.693
Detonation pressure PCJ
0.21 Mbar
Detonation Energy E0
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:(2)
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
PCJ and velocity DCJ. 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 E0 is
computed using the JWL equation of state expression to get an initial pressure for the high-pressure gas, and 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
Boundary conditions around the 1D shock tube are sliding walls, which is the default
when using (/MAT/MULTIFLUID).
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.
It can be modified using the keyword
/DT/ALE:
/DT/ALE
0.5 0.0
Results
Results are examined 12 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.
Profiles of pressure and mass density along the tube after 12 s are visible on 図 8 and 図 9, as well as similar profiles for velocity and specific energy on
図 10 and 図 11, with the curve
obtained from the analytical solution.
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 and the exact solution is defined as:(3)
The £2-norm error to the analytical solution on pressure, mass density,
velocity are:
表 3. £2-norm error to the analytical solution with
and without /ALE/MUSCL
/ALE/MUSCL
Pressure
Mass Density
Velocity
Specific Energy
£2-norm
relative error
enabled
2.0%
6.5%
3.9%
4.5%
disabled
2.5%
9.2%
4.7%
6.7%
As shown on 表 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