Moving Solid Materials with Magnetic Fields
Robert E. Peterkin Jr., James J. Havranek, and Dennis E. Lileikis
Air Force Research Laboratory: Phillips Site
Introduction
We illustrate a technique for computing the time-evolution of solid materials in response to forces that arise from material and magnetic pressures on a generally non-orthogonal computational grid of a time-dependent, arbitrary Lagrangian-Eulerian (ALE) magnetohydrodynamics (MHD) simulation code. The ability to perform accurate simulation of the time evolution of a conducting solid is of great utility in a wide variety laboratory situations. Of particular interest to the authors is the motion of initially solid shells in response to large pressures that arise from explosions and/or megagauss-level magnetic fields.
In this paper, we briefly describe the implementation of the well-known Simple Line Interface Calculation (SLIC) technique for the geometric approximation to constructing material interfaces in our 2-dimensional and 3-dimensional MHD codes mach2 and mach3. The SLIC algorithm constructs one-dimensional interfaces from the various materials in each generally mixed zone of the calculation. As applications of these codes, we choose a pair of interesting pulsed power driven concepts. The first is the implosion of an initially quasi-spherical solid shell by a pulsed magnetic field. The second is the explosion of an initially solid cylindrical shell filled with high explosive and imbedded in a magnetic field that is created by a helical current-carrying coil surrounding the explosive armature. The initial states of these concepts are illustrated in figures 1 and 2 respectively.
Fig. 1 Initial state of a 2-dimensional MHD calculation with mach2 of the magnetic-field driven implosion of an initially quasi-spherical solid aluminum shell onto an initially solid cylindrical copper target. This concept has been investigated experimentally by Degnan et al. In these experiments, the volume between the aluminum and copper shells is filled with an initially 50-100 torr, 1 eV, hydrogen "working fluid." The aluminum shell is driven to implode by a current that rises sinusoidally from 0 to 12 MA in 10
ms. As the outer shell implodes, the pressure of the working fluid increases to the Mbar range and implodes the copper target.
Fig. 2 Initial state of a 3-dimensional MHD calculation with mach3 of the explosively driven explosion of an initially cylindrical solid aluminum shell onto a magnetic field that is created by the helical field coils. As the armature explodes, the magnetic field is compressed. This concept is called a magnetic flux compression generator.
The mach magnetohydrodynamic codes
Simulation of plasma dynamics is often performed on a discrete three-dimensional mesh that is built from primitive polyhedra. A particularly simple polyhedron that is used by many code developers is the cube, but it is not possible to stack cubes to conform to the geometry of figures 1 and 2 without overlap. To simulate plasma dynamics within volumes of complex shape, we have developed the mach (Multiblock Arbitrary Coordinate Hydromagnetics) codes for simulating the unsteady behavior of collisional conducting material in any phase (solid, liquid, gas, plasma). Our codes use a multi-block mesh composed of generally non-orthogonal hexahedral cells. The mach codes are of the Arbitrary Lagrangian/Eulerian (ALE) variety which allows for flexibility in the physics options at the expense of some numerical complexity. In an ALE code, Faraday's law is advanced in two steps: a Lagrangian advance followed by a remap of the magnetic field from the Lagrangian grid to the computational grid. The computational grid can be at rest in a laboratory frame (Eulerian), at rest in the fluid frame (Lagrangian), or in some other arbitrary state.
For a conducting material of mass density
r, magnetic induction, B, specific electron and ion internal energies ee and ei, radiation energy density, er, moving with velocity, u, the mach codes solve the following dynamical equations for:fluid momentum vector
Dtu = (Ñ P + J ´ B + Ñ · s d) / r + Ñ · n Ñ u 1
mass density
Dtr + r Ñ · u = 0 2
magnetic induction vector
¶ tB = - Ñ ´ E 3
specific ion internal energy
r Dtei = - Pi Ñ · u + Ñ · (k iÑ Ti) + F ei 4
specific electron internal energy
r Dtee = - Pe Ñ · u + J· E + Ñ · (k eÑ Te) - F ei + F er 5
radiation energy density
Dter = - (4/3) erÑ · u + Ñ · (k rÑ er) + F er 6
and elastic stress deviator tensor
¶ ts d = 2 m dd - u· Ñ dd . 7
The electric field E in equations 3 is obtained from a generalized Ohm’s law
E = hJ - u ´ B + J ´ B/ene - Ñ P/ene 8
for a material resistivity h and an electron number density ne. The viscosity of the material is n and the s d are the 5 independent components of the deviatoric stress tensor which are related to the stretching deviator tensor dd . The various k are the thermal conductivities and the F are inter-species coupling terms.
The SLIC algorithm in the mach codes
The key advancement that we have made to the SLIC algorithm as applied to the mach ALE codes is the correct partitioning of the various material properties in mixed zones. Generally, the fraction of a conserved quantity U for material m in a mixed zone is
fum = Um / U 9
such that S Um = U. The U may represent, for example, the total mass of a zone, the total energy in a zone, etc. A concern for an MHD code is the correct partitioning of the Ohmic heating of a mixed resistive zone. This is done by recalling that the power dissipated by a voltage V across a resistor of resistance R is V2/R. Hence the if dE is the total Ohmic heat dissipated in a mixed zone, this energy is partitioned as
d
Em = dE / (hm(S1/hm)) . 10Magnetically-driven solid shell implosions
The state of the magnetically-driven solid shell evolves from the initial state illustrated in figure 1. The state at 14 ms as computed by mach2 is shown in figures 3 and 4 where the only difference is that the mass density of the solid aluminum shell was initially constant for one simulation and was initially given a 1% random density perturbation around the same constant density in the other. Not how the material interfaces are distinct for both calculations, even though the solid material has moved through many times its initial thickness on a fixed Eulerian mesh. The noticeable perturbations on the outside of the aluminum shell in the initially perturbed simulation are a consequence of the magnetic Rayleigh-Taylor instability. The central copper target also has begun to compress in response to the rising pressure of the working fluid between the aluminum shell and the cylindrical copper target.
Figure 3. The material interfaces 14 ms into the magnetically-driven implosion of an initially uniform density aluminum quasi-spherical shell.
Figure 4. The material interfaces 14 ms into the magnetically-driven implosion of an initially solid aluminum quasi-spherical shell with a random 1% density perturbation.
Explosively-driven magnetic flux compression generator
The state of the explosively-driven solid shell evolves from the initial state illustrated in figure 2. The state at 42 ms as computed with the 3-dimensional mach3 is shown in figure 5. Notice again how well the SLIC algorithm maintains a sharp edge on the various materials. In this calculation, the high-explosive is detonated at a point on top of the HE on the axis of the cylindrical armature. The explosion propagates outward and downward so that by this time, most of the HE has exploded. At this time, the aluminum armature is beginning to interact with the solid aluminum helical field coil. The magnetic field lines that were initially vertical in figure 2 are now pushed out by the exploding armature. The high conductivity of the armature and the helical field coils traps most of the magnetic flux in the compressing volume between the armature and the coils.
The HE is modeled with an analytic Grüneisen equation of state. The equation of state for the metal is from the SESAME database from Los Alamos National Laboratory.
Figure 5. The material interfaces 42 ms into the explosively-driven explosion of an initially solid aluminum cylindrical shell that is packed with the high-explosive PBX9501. The compressing magnetic flux lines induce a rapidly-rising voltage across a load.