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Magnetohydrodynamics of Free Surface Flows
R. Samulyak, J. Du, T. Lu, and J. Glimm
The purpose of this project is to develop novel mathematical models,
numerical algorithms, and computational software optimized for modern
supercomputers for the numerical simulation of free surface multiphase
magnetohydrodynamic (MHD) flows at low magnetic Reynolds numbers, and to
perform simulations in support of magnetically confined fusion and advanced
accelerator applications. Our primary goal is to gain through mathematical
modeling and numerical simulations a better understanding of physics, and
improve the design of experiments and devices critical for the DOE mission.
The research is being performed in close collaboration with General Atomics,
Neutrino Factory/Muon Collider Collaboration, and the SciDAC Center (ITAPS).
Specialized numerical algorithms, optimized for modern computer
architectures such as BlueGene, and new models for complex MHD processes in
multiphase systems are necessary in order to advance numerical simulations
of nuclear fusion devices such as the International Toroidal Experimental
Reactor (ITER) and future accelerators. Mathematical models, algorithms, and
software being developed under this project enable the simulation of ITER
fueling through the injection of small frozen deuterium-tritium pellets,
striation instabilities of the pellet ablation channel, liquid hydrogen or
lithium jets proposed for the tokamak plasma disruption mitigation, and the
simulation of a mercury jet interacting with intense proton pulses in the
target system for the Neutrino Factory/Muon Collider.
The system of free surface MHD equations in the low magnetic Reynolds number
approximation is a coupled hyperbolic - elliptic system in a geometrically
complex moving domain. We have developed a numerical method for solving such
equations along with the corresponding parallel software. The numerical
method treats the MHD system in an operator split manner. We use the front
tracking hydro code FronTier with free interface support for solving the
hyperbolic subsystem. The Poisson equation for the electric potential can be
solved using techniques for irregular domains described below. FronTier
represents interfaces as lower dimensional meshes moving through a volume
filling grid. The traditional volume filling finite difference grid supports
smooth solutions located in the region between interfaces. The location of
the discontinuity and the jump in the solution variables are defined on the
lower dimensional grid or interface. The dynamics of the interface comes
from the mathematical theory of Riemann solutions, which are idealized
solutions of single jump discontinuities for a conservation law. FronTier is
capable of tracking 3D interfaces and resolving their topological changes.
Some features of the FronTier hyperbolic solvers include the use of
high-resolution methods, exact and approximate Riemann solvers, and equation
of state models for real materials. The existence of a tracked surface,
across which physical parameters and the solution change discontinuously,
has important implications for the solution of an elliptic or parabolic
system. We have developed an algorithm and software for the dynamic
generation of finite element meshes conforming to the interface based on the
point shift technique. We recently enhanced capabilities of FronTier-MHD to
work with complex interfaces in 3D by developing an elliptic solver based on
the embedded boundary method [1]. The method uses a finite volume
discretization with control volumes being rectangular grid cells away from
interfaces, and partial cells near the interface (Figure 1). To achieve
second order accuracy of the solution, fluxes through irregular cell
boundaries are calculated using an interpolation of fluxes in neighboring
cells. The corresponding parallel iterative solvers for linear systems of
equations are based on PETSc libraries and the algebraic multigrid method
implemented in the Hypre software package.
An important class of modeling problems related to our targeted applications
are the thermodynamic and mechanic properties of materials interacting with
intense sources of external energies. These include nonlinear wave
phenomena, cavitation and vaporization of fluids, condensation of gases, the
ablation of solids by intense laser or electron beams, laser – plasma
interaction, and atomic processes (ionization, dissociation and
recombination) in gases. Such models have been developed and implemented in
the FronTier code. Namely, homo- and heterogeneous (direct numerical
simulation) methods for phase transitions such as cavitation, a technique
that would allow numerical simulation of the cavitation in liquid hydrogen
jet proposed for the mitigation of plasma disruptions, surface ablation
models, and models for the interaction of gas/plasma with hot electrons
through a simplified solution of kinetic equations have been implemented.
Details of the application of the FronTier code to the tokamak fueling and
accelerator target problems can be found in [2] and [3].
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a)
b) |
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Figure 1. Control volumes near the interface and
interpolation of fluxes through irregular cell boundaries in 2D
(a) and 3D (b) for the elliptic problem discretization. |
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| Figure 2. Formation of the pellet ablation channel in tokamak
magnetic fields ranging from 0 to 5 Tesla. |
Reference
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[1] Samulyak, R., Du, J., Glimm, J., and Xu, Z. A numerical algorithm
for MHD of free surface flows at low magnetic Reynolds numbers. J. Comp.
Phys., 2006 (submitted).
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[2] Samulyak, R., Lu, T., and Parks, P. A magnetohydrodynamic simulation
of pellet ablation in the electrostatic approximation. Nucl. Fusion.
Accepted, 2006.
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[3] Samulyak, R. and Prykarpatskyy, Y. Richtmyer-Meshkov instability in
liquid metal flows: influence of cavitation and magnetic fields.
Mathematics and Computers in Simulations 65: 431-446 (2004).
Last Modified: January 31, 2008
Please forward all questions about this site to:
Claire Lamberti
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