A Cylindrical Arbitrary Lagrangian Eulerian MHD Code
T.Goffrey
Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
t.goffrey@warwick.ac.uk
A two dimensional r-z Arbitrary Lagrangian Eulerian MHD code is under
development in order to study problems with such a symmetry. To this end a two
dimensional hydrodynamics ALE code has been produced.
An ALE code is a combination of Lagrangian and Eulerian techniques for
modelling fluid flow
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In combining the two models the ALE method attempts to mitigate to problems
associated with each method.
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However the strength and accuracy of an ALE code us underpinned by how
robust and long running it's Lagrangian phase is.
There are a number of methods for improving this factor, which seek to minimise
non-physical modes of grid movement, retain physical symmetry independently
of grid symmetry and capture shocks accurately and automatically.
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These methods are easily applicable to a number of coordinate systems, the
results presented here are set in Cartesian space for simplicity.
Shock Viscosity
Shock viscosity is a numerical method for automatically capturing and
treating shocks. It is shown in the equations of MHD as q.
For simplicity, q is denoted a scalar here, but in general it can be more
complex.
Caramana et al. [1] outlined five key qualities the shock viscosity must have.
1. Dissipativity
2. Galilean invariance.
3. Self-similar motion invariance
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Ultimately, however, the final code will be an r-z MHD code, the governing
equations of which are:
4. Wavefront invariance
5. Viscous force continuity.
To this end they derived a viscous force defined along the edges of each cell
of the form.
Subzonal Pressure Forces
This viscous force has been implemented in place of the scalar q in the
code.
To test some of the abilities of the chosen viscosity a circularly convergent
shock wave was driven by applying an external unit velocity to a unit density
disk of perfect gas, that was initially cold.
Plotted below is the density (left) and final grid (right) just before the shock
wave reaches the origin.
Symmetry is preserved.
In two dimensions a quadrilateral element
has six degrees of freedom.
There are also two non-physical modes,
which do not illicit a response from the
equations of hydrodynamics.
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These are known as hourglass modes,
show to the right.
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These can cause grid distortion, leading
to inaccurate, or incomplete solutions.
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Further Considerations
Subzonal pressures occur naturally from
the modelling of the fluid as Lagrangian.
The above plots represent a success for the scheme described in preserving
cylindrical symmetry in Cartesian coordinates, an equivilent problem to the
preservation of spherical symmetry in cylindrical coordinates.
However, the resulting forces are often
neglected.
This success is not however is not repeated for an initial grid that lacks
unequal angular zoning, this requires a modification of the gradient operator.
These forces do respond to hourglass
motion, and act against it.
The viscous force presented here also shown to maintain cylindrical
symmetry. However Shashkov et al point out problems with this viscous
force. Specifically;
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Presented are the result of running the
Saltzman piston problem without(above)
and with(below) subzonal pressures. The
result should be one dimensional.
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1. The size of the force can vary significantly over a cell.
2.The viscous force has too strong a dependence on the grid itself.
Both of these stem from the fact that the viscous force is not derived from a
discrete version of a true continuous tensor. To this end they devised a true
tensor based viscosity.
Coupling this tensor based viscosity with the additional considerations of an
MHD tensor shock viscosity is being investigated.
The introduction of subzonal densities essentially introduces a discrete
version of a subzonal continuous variation in density within the cell. ALE
schemes need to carry out some form of approximation to density variation
within the cell, the existing ALE scheme will need to be modified to account
for this.
[1] T. D. Arber et al. J. Comp. Phys. 171 (2001) 151
[2] G. A. Sod. J. Comp. Phys. 27 (1978,1
[3] M. Brio and C.C.Wu. J. Comp. Phys. 75(1988) 400
[4] P. Woodward and P. Colella. J. Comp. Phys. 54(1984) 115
This work acknowledges the financial support of AWE