Brookhaven Science Associates U.S. Department of Energy 1 General AtomicsJuly 14, 2009 Multiphase MHD at Low Magnetic Reynolds Numbers Tianshi Lu Department of Mathematics Wichita State University In collaboration with Roman Samulyak, Stony Brook University / Brookhaven National Laboratory Paul Parks, General Atomics

Brookhaven Science Associates U.S. Department of Energy 2 Tokamak (ITER) Fueling Fuel pellet ablation Striation instabilities Killer pellet / gas ball for plasma disruption mitigation Laser ablated plasma plume expansion Expansion of a mercury jet in magnetic fields Motivation

Brookhaven Science Associates U.S. Department of Energy 3 Equations for MHD at low magnetic Reynolds numbers and models for pellet ablation in a tokamak Numerical algorithms for multiphase low Re M MHD Numerical simulations of pellet ablation Talk Outline

Brookhaven Science Associates U.S. Department of Energy 4 Equations for MHD at low magnetic Reynolds numbers Full system of MHD equationsLow Re M approximation Maxwell’s equations without wave propagation Ohm’s law Equation of state for plasma / liquid metal / partially ionized gas Elliptic Parabolic

Brookhaven Science Associates U.S. Department of Energy 5 Models for pellet ablation in tokamak Full MHD system Implicit or semi-implicit discretization EOS for fully ionized plasma No interface System size ~ m, grid size ~ cm Tokamak plasma in the presence of an ablating pellet Pellet ablation in ambient plasma Global ModelLocal Model MHD system at low Re M Explicit discretization EOS for partially ionized gas Free surface flow System size ~ cm, grid size ~ 0.1 mm Courtesy of Ravi Samtaney, PPPL

Brookhaven Science Associates U.S. Department of Energy 6 Schematic of pellet ablation in a magnetic field Schematic of processes in the ablation cloud CloudPlasma Sheath boundary  (z) Sheath Fluxes

Brookhaven Science Associates U.S. Department of Energy 7 Local model for pellet ablation in tokamak 1.Axisymmetric MHD with low Re M approximation 2.Transient radial current approximation 3.Interaction of hot electrons with ablated gas 4.Equation of state with atomic processes 5.Conductivity model including ionization by electron impact 6.Surface ablation model 7.Pellet penetration through plasma pedestal 8.Finite shielding length due to the curvature of B field

Brookhaven Science Associates U.S. Department of Energy 8 1. Axisymmetric MHD with low Re M approximation Centripetal force Nonlinear mixed Dirichlet-Neumann boundary condition

Brookhaven Science Associates U.S. Department of Energy 9 2. Transient radial current approximation  r,z) depends explicitly on the line-by-line cloud opacity u . Simplified equations for non-transient radial current has been implemented.

Brookhaven Science Associates U.S. Department of Energy Interaction of hot electrons with ablated gas In the cloud On the pellet surface

Brookhaven Science Associates U.S. Department of Energy 11 Saha equation for the dissociation and ionization 4. Equation of state with atomic processes (1) Deuterium E d =4.48eV, N d =1.55×10 24,  d =0.327 E i =13.6eV, N i =3.0×10 21,  i =1.5 Dissociation and ionization fractions

Brookhaven Science Associates U.S. Department of Energy 12 High resolution solvers (based on the Riemann problem) require the sound speed and integrals of Riemann invariant type expressions along isentropes. Therefore the complete EOS is needed. Conversions between thermodynamic variables are based on the solution of nonlinear Saha equations of ( ,T). To speedup solving Riemann problem, Riemann integrals pre- computed as functions of pressure along isentropes are stored in a 2D look-up table, and bi-linear interpolation is used. Coupling with Redlich-Kwong EOS can improve accuracy at low temperatures. 4. Equation of state with atomic processes (2)

Brookhaven Science Associates U.S. Department of Energy Conductivity model including ionization by impact Ionization by Impact

Brookhaven Science Associates U.S. Department of Energy 14 Influence of Atomic Processes on Temperature and Conductivity TemperatureConductivity

Brookhaven Science Associates U.S. Department of Energy Surface ablation model Some facts: The pellet is effectively shielded from incoming electrons by its ablation cloud Processes in the ablation cloud define the ablation rate, not details of the phase transition on the pellet surface No need to couple to acoustic waves in the solid/liquid pellet The pellet surface is in the super-critical state As a result, there is not even well defined phase boundary, vapor pressure etc. This justifies the use of a simplified model: Mass flux is given by the energy balance (incoming electron flux) at constant temperature Pressure on the surface is defined through the connection to interior states by the Riemann wave curve Density is found from the EOS.

Brookhaven Science Associates U.S. Department of Energy Pellet penetration through plasma pedestal

Brookhaven Science Associates U.S. Department of Energy Finite shielding length due to the curvature of B field  The grad-B drift curves the ablation channel away from the central pellet shadow. To mimic this 3D effect, we limit the extent of the ablation flow to a certain axial distance.  Without MHD effect, the cloud expansion is three-dimensional. The ablation rate reaches a finite value in the steady state.  With MHD effect, the cloud expansion is one-dimensional. The ablation rate would goes to zero by the ever increasing shielding if a finite shielding length were not in introduced.

Brookhaven Science Associates U.S. Department of Energy 18 Equations for MHD at low magnetic Reynolds numbers and models for pellet ablation in a tokamak  Numerical algorithms for multiphase low Re M MHD Numerical simulations of the pellet ablation in a tokamak Talk Outline

Brookhaven Science Associates U.S. Department of Energy 19 Multiphase MHD Solving MHD equations (a coupled hyperbolic – elliptic system) in geometrically complex, evolving domains subject to interface boundary conditions (which may include phase transition equations) Material interfaces: Discontinuity of density and physics properties (electrical conductivity) Governed by the Riemann problem for MHD equations or phase transition equations

Brookhaven Science Associates U.S. Department of Energy 20 Front Tracking: A hybrid of Eulerian and Lagrangian methods Two separate grids to describe the solution: 1.A volume filling rectangular mesh 2.An unstructured codimension-1 Lagrangian mesh to represent interface Major components: 1.Front propagation and redistribution 2.Wave (smooth region) solution Main ideas of front tracking Advantages of explicit interface tracking: No numerical interfacial diffusion Real physics models for interface propagation Different physics / numerical approximations in domains separated by interfaces

Brookhaven Science Associates U.S. Department of Energy 21 Level-set vs. front tracking method 5 th order level set (WENO) 4 th order front tracking (Runge-Kutta) Explicit tracking of interfaces preserves geometry and topology more accurately.

Brookhaven Science Associates U.S. Department of Energy 22 FronTier is a parallel 3D multi-physics code based on front tracking n Physics models include n Compressible fluid dynamics n MHD n Flow in porous media n Elasto-plastic deformations n Realistic EOS models n Exact and approximate Riemann solvers n Phase transition models n Adaptive mesh refinement Interface untangling by the grid based method The FronTier code

Brookhaven Science Associates U.S. Department of Energy 23 Targets for future accelerators Supernova explosion Tokamak refuelling through the ablation of frozen D 2 pellets Liquid jet break-up and atomization Main FronTier applications Richtmyer-Meshkov instability Rayleigh-Taylor instability

Brookhaven Science Associates U.S. Department of Energy 24 Hyperbolic step Elliptic step Propagate interface Untangle interface Update interface states Apply hyperbolic solvers Update interior hydro states Generate finite element grid Perform mixed finite element discretization or Perform finite volume discretization Solve linear system using fast Poisson solvers Calculate electromagnetic fields Update front and interior states Point Shift (top) or Embedded Boundary (bottom) FronTier – MHD numerical scheme

Brookhaven Science Associates U.S. Department of Energy 25 Hyperbolic step Complex interfaces with topological changes in 2D and 3D High resolution hyperbolic solvers Riemann problem with Lorentz force Ablation surface propagation EOS for partially ionized gas and conductivity model Hot electron heat deposition and Joule’s heating Lorentz force and saturation numerical scheme Centripetal force and evolution of rotational velocity Interior and interface states for front tracking

Brookhaven Science Associates U.S. Department of Energy 26 Based on the finite volume discretization Domain boundary is embedded in the rectangular Cartesian grid. The solution is always treated as a cell- centered quantity. Using finite difference for full cell and linear interpolation for cut cell flux calculation 2 nd order accuracy Elliptic step Embedded boundary elliptic solver For axisymmetric pellet ablation with transient radial current, the elliptic step can be skipped.

Brookhaven Science Associates U.S. Department of Energy 27 High Performance Computing Software developed for parallel distributed memory supercomputers and clusters Efficient parallelization Scalability to thousands of processors Code portability (used on Bluegene Supercomputers and various clusters) Bluegene/L Supercomputer (IBM) at Brookhaven National Laboratory

Brookhaven Science Associates U.S. Department of Energy 28 Equations for MHD at low magnetic Reynolds numbers and models for pellet ablation in a tokamak Numerical algorithms for multiphase low Re M MHD  Numerical simulations of the pellet ablation in a tokamak Talk Outline

Brookhaven Science Associates U.S. Department of Energy 29 Previous studies Transonic Flow (TF) (or Neutral Gas Shielding) model, P. Parks & R. Turnbull, 1978 Scaling of the ablation rate with the pellet radius and the plasma temperature and density 1D steady state spherical hydrodynamics model Neglected effects: Maxwellian hot electron distribution, geometric effects, atomic effects (dissociation, ionization), MHD, cloud charging and rotation Claimed to be in good agreement with experiments Theoretical model by B. Kuteev et al., 1985 Maxwellian electron distribution An attempt to account for the magnetic field induced heating asymmetry Theoretical studies of MHD effects, P. Parks et al. P2D code, A. K. MacAulay, 1994; CAP code R. Ishizaki, P. Parks, 2004 Maxwellian hot electron distribution, axisymmetric ablation flow, atomic processes MHD effects not considered

Brookhaven Science Associates U.S. Department of Energy 30 1.Spherical model Excellent agreement with TF model and Ishizaki 2.Axisymmetric pure hydro model Double transonic structure Geometric effect found to be minor 3.Plasma shielding Subsonic ablation flow everywhere in the channel Extended plasma shield reduces the ablation rate 4.Plasma shielding with cloud charging and rotation Supersonic rotation widens ablation channel and increases ablation rate Our simulation results Spherical modelAxis. hydro modelPlasma shielding

Brookhaven Science Associates U.S. Department of Energy Spherically symmetric hydrodynamic simulation Normalized ablation gas profiles at 10 microseconds Polytropic EOSPlasma EOS Poly EOSPlasma EOS Sonic radius0.66 cm0.45 cm Temperature5.51 eV1.07 eV Pressure20.0 bar26.9 bar Ablation rate112 g/s106 g/s Excellent agreement with TF model and Ishizaki. Verified scaling laws of the TF model

Brookhaven Science Associates U.S. Department of Energy Axially symmetric hydrodynamic simulation Temperature, eVPressure, barMach number Steady-state ablation flow

Brookhaven Science Associates U.S. Department of Energy 33 Velocity distribution Channeling along magnetic field lines occurs at ~1.5 μs 3. Axially symmetric MHD simulation (1) Plasma electron temperatureTeTe 2 keV Plasma electron densitynene cm -3 (standard) 1.6x10 13 cm -3 (el. shielding) Warm-up timetwtw 5 – 20 microseconds Magnetic fieldB2 – 6 Tesla Main simulation parameters:

Brookhaven Science Associates U.S. Department of Energy Axially symmetric MHD simulation (2) Mach number distribution Double transonic flow evolves to subsonic flow

Brookhaven Science Associates U.S. Department of Energy 35 Dependence on pedestal properties -.-.-t w = 5  s, n e = 1.6  cm -3 ___t w = 10  s, n e = cm t w = 10  s, n e = 1.6  cm -3 Critical observation Formation of the ablation channel and ablation rate strongly depends on plasma pedestal properties and pellet velocity. Simulations suggest that novel pellet acceleration technique (laser or gyrotron driven) are necessary for ITER.

Brookhaven Science Associates U.S. Department of Energy 36 Supersonic rotation of the ablation channel 4. MHD simulation with cloud charging and rotation (1) Isosurfaces of the rotational Mach number in the pellet ablation flow Density redistribution in the ablation channel Steady-state pressure distribution in the widened ablation channel

Brookhaven Science Associates U.S. Department of Energy 37 G, g/s Pellet ablation rate for ITER-type parameters 4. MHD simulation with cloud charging and rotation (2)

Brookhaven Science Associates U.S. Department of Energy 38 Channel radius Ablation rate |ΔB/B|   Non-rotating2.3 cm195 g/s0.079 Rotating2.8 cm262 g/s0.088 Channel radius and ablation rate 4. MHD simulation with cloud charging and rotation (3) Normalized potential along field lines  G rot is closer to the prediction of the quasisteady ablation model G qs = 327 g/s  Magnetic β<<1 justifies the static B-field assumption Potential in the negative layer

Brookhaven Science Associates U.S. Department of Energy 39 Current work focuses on the study of striation instabilities Striation instabilities, observed in all experiments, are not well understood We believe that the key process causing striation instabilities is the supersonic channel rotation, observed in our simulations Striation instabilities: Experimental observation (Courtesy MIT Fusion Group) Striation instabilities

Brookhaven Science Associates U.S. Department of Energy 40 Plasma disruption mitigation Pressure distribution without rotation Gas ball R = 9 mm Killer pellet R = 9 mm

Brookhaven Science Associates U.S. Department of Energy 41 Plasma disruption mitigation Mach number distributions in the gas shell

Brookhaven Science Associates U.S. Department of Energy 42 Conclusions and future work Developed MHD code for free surface low magnetic Re number flows Front tracking method for multiphase flows Elliptic problems in geometrically complex domains Phase transition and surface ablation models Axisymmetric simulations of pellet ablation Effects of geometry, atomic processes, and conductivity model Warm-up process and finite shielding length Charging and rotation, transient radial current Ablation rate, channel radius, and flow properties Tracking of a shrinking pellet Future work 3D simulations of pellet ablation and striation instabilities Asymptotic ablation properties in long warm up time Natural cutoff shielding length Magnetic induction Systematic simulation of plasma disruption mitigation using killer pellet / gas ball Coupling with global MHD models