1. MHD of Multiphase Flows: Numerical Algorithms and Applications
CSC Seminar March 9, 2006, BNL
MHD of Multiphase Flows: Numerical Algorithms and Applications
Roman Samulyak
Center for Data Intensive Computing, BNL
Collaborators:
Tianshi Lu, CSC/BNL, modeling, software development, fusion applications
Jian Du, Stony Brook University, software development, accelerator applications
Zhiliang Xu, Xiaolin Li, CSC/SBU, front tracking methods
Paul Parks, General Atomics, MHD theory, fusion applications
James Glimm, SBU/CSC, modeling, numerical algorithms

2. Talk outline Motivation
MHD system for free surface multiphase flow at low magnetic Reynolds numbers; approximations
Numerical algorithms for MHD equations
Applications
Numerical simulations of the pellet ablation in a tokamak

3. Motivation: Studies of Tokamak Fueling
Problems
Pellet ablation
Striation instabilities
Laser driven pellet acceleration
Gyrotron driven pellet acceleration
Liquid jet for plasma disruption mitigation
Laser driven pellet acceleration
ITER schematic
Fueling using a high speed gaseous jet

4. Motivation: Accelerator Design (Neutrino Factory/Muon Collider target)
The target has been proposed as a free mercury jet interacting with an intensive proton pulse in a 20Tesla magnetic field
Exparinemtal data (BNL), no B field
Future experiment at CERN (2007) will operate with
20 m/s mercury jet
15 Tesla pulsed solenoid
proton beam line

5. Full system of MHD equations Low magnetic Re approximation
MHD equations and approximations
Full system of MHD equations
Low magnetic Re approximation
MHD regimes, approximations, and discretization techniques
Astrophysics: full system, explicit time discretization, 7-wave Riemann problem
Magnetically Confined Fusion plasmas: full system, implicit or semi-implicit discretization
Liquid metal MHD: incomplessible fluid, low magnetic Re, implicit time discretization
Our approach: compressible fluid (shock waves), low magnetic Re, phase transitions, free surface flows, explicit time discretization

6. Numerical Approach
The low megnetic Re MHD is a copuled hyperbolic/elliptic system. Operator splitting.
The hyperbolic subsystem is solved on a finite difference grid in both domains separated by the free surface using flont tracking numerical techniques.
Implemented in FronTier code
Riemann problem for interface propagation
Complex interfaces with topological changes in 2D and 3D
High resolution hyperbolic solvers
Realistic EOS models
The elliptic subsystem is solved in geometrically complex domains
Dynamic grid generation, finite element diecretization of conforming grids
Embedded boundary finite volume discretization
Fast parallel linear solvers

7. FronTier-MHD numerical scheme
Elliptic step
Hyperbolic step
Point Shift (top) or Embedded Boundary (bottom)
Propagate interface
Untangle interface
Update interface states
Apply hyperbolic solvers
Update interior hydro states
Calculate electromagnetic fields
Update front and interior states
Generate finite element grid
Perform mixed finite element discretization or
Perform finite volume discretization
Solve linear system using fast Poisson solvers

8. Stencil and equations for the interface point propagate

9. Schematic of the interface point propagate algorithm

10. Comparison of techniques for elliptic equations
Point-shift grid generation
Compatible with mixed finite element formulation
The same order accuracy for the potential and gradients
Capable of generating grids for vector finite elements
Not robust (especially in 3D)
Colella’s embedded boundary method
Second order accurate for the potential
Robust
Simple implementation for parallel computing
Due to arbitrary shape elements, incompatible with mixed finite element formulation

11. Embedded Boundary Elliptic Solver
Main Ideas
Based on the finite volume discretization
Potential is treated as cell centered value, even if the center is outside the computational domain
Domain boundary is embedded in the rectangular Cartesian grid, and the solution is treated as a cell-centered quantity
Using finite difference for full cell and linear interpolation for cut cell flux calculation

12. Embedded Boundary Elliptic Solver
Area is corresponding to the computational cell on which to make
integration
f i is the flux across the non-boundary cell edges and f f is the boundary edge flux given by Neumann Conditions. All fluxes are calculated at the middle point of the edge and dl is the related edge length

13. Algorithm
Reconstruct the interface, record the crossings, and set the component type for grid point
Simplify the control volume configurations, remove small volume cells (better condition number)
Count the local number of blocks (both full and partial cells), set the matrix and vector dimension for the linear system solver,set global index for the counted blocks
For partial cells, the necessary information such as cell edge type, position of cell edge center and partial cell edge length are recorded.
A nine-point stencil is set for each partial cell. With the above information, related stencil value from flux integration is set and addeded into the corresponding matrix row
Set up the right hand side for the linear system, which is also calculated from flux integration by divergence theorem. Note that for partial cell, the boundary flux from both left and right side are cancelled out by Neumann boundary conditions.
Solve the linear system Ax = b with some fast parallel linear solver (PETSc or HYPRE)

14. Algorithm
Stencil Setting
Figure 3. Stencil for partial cells

15. Domain Boundary: Sin wave perturbed circle
Validation
Test Function:
Computational Domain
Domain Boundary: Sin wave perturbed circle
Convergence Rate:
Mesh Size
error (fx) R
32x32
2.496e-2
N/A
64x64
8.841e-3
1.497
128x128
3.093e-3
1.506
256x256
1.096e-3
1.503

16. Validation
Illustration of flux error (X direction)

17. 3D implementation
Parallel 3D implementation has been completed and fully tested
Same principle as 2D
Bilinear interpolation of flux

18. Muon Collider target: a brief summary of modeling and simulation
Simulation of the mercury jet target interacting with a proton pulse in a magnetic field
Studies of surface instabilities, jet breakup, and cavitation
MHD forces reduce both jet expansion, instabilities, and cavitation
Jet surface instabilities
Cavitation in the mercury jet and thimble

19. Mercury jet entering magnetic field. Schematic of the problem.
Magnetic field of the 15 T solenoid is given in the tabular format

20. Incompressible steady state formulation of the problem

21. Results: Aspect ratio of the jet cross-section
B = 15 T V0 = 25 m/s

22. Results: Aspect ratio of the jet cross-section
B = 15 T V0 = 25 m/s

23. Consequences of the jet distortion
The cross-section of the mercury jet interaction with the proton pulse is significantly reduced
This reduces the pion production rate
In order to avoid these undesirable consequences, the angle between the magnetic field and the solenoid axes was reduced to 0.3 rad. This implies some new constraints on the hardware design
Another possible solution was the use of an elliptic nozzle to compensate the MHD distortion. This option has not been explored due to time constraints on the final design.

24. Processes in the ablation cloud

25. Pellet ablation in tokamaks: previous works. Local models
Neutral gas shielding (NGS) model, P. Parks & R. Turnbull, 1978
Provides the scaling of the ablation rate with the pellet radius and the plasma temperature and density
1D steady state hydrodynamics model
Neglected effects: MHD, geometric effects, atomic effects (dissociation, ionization)
Theoretical studies of MHD effects, P. Parks
P2D code, A. K. MacAulay, 1994; CAP code R. Ishizaki, P. Parks, 2004
Non-spherical ablation flow (axial symmetry), proper treatment of scattering
Kinetic calculation of the electron heat deposition, atomic physics processes
MHD effects not considered

26. Pellet ablation in tokamaks: previous works. Global models
Simulations using MH3D code, H. Strauss & W. Park, 1998
Finite element version of the MH3D full MHD code
Details of the ablation are not considered
Pellet is given as a density perturbation of initial conditions
Smaller values of density and larger pellet radius (numerical constraints)
Simulations using MHD code based on CHOMBO AMR package, R. Samtaney, S. Jardin, P. Colella, D. Martin, 2004
Analytical model for the pellet ablation: moving density source
8-wave upwinding unsplit method for MHD
AMR package – significant improvement of numerical resolution

27. Improved model is needed:
Studies of the local pellet ablation physics are still missing
MHD
3D effects
Model for currents in the ablation cluod
Global plasma simulations in the presence of an ablating pellet need a better local model as input
Studies of striation instabilities, observed in all experiments, are not possible without a 3D detailed physics model
We are working on building and validations of such models

28. Physics Models for Pellet Studies : Electron Energy Deposition
In the cloud:
On the pellet surface:

29. Physics Models for Pellet Studies :
Surface Ablation
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.

30. Physics Models for Pellet Studies :
Equation of State with Atomic Processes. 1
Saha equation for the dissociation (ionization) fraction

31. EOS with Atomic Processes
Incomplete EOS
(known from literature):
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.
Second law of thermodynamics:
We found the complete EOS and showed that compatibility with the second law of thermodynamics requires:

32. Numerical Algorithms for EOS
For better numerical efficiency, FronTier operates with three pairs of independent thermodynamic variables:
For the first two pairs of variables, solve numerically nonlinear algebraic equation, and find T. Using , find the remaining state.
Such an approach is prohibitively slow for the calculation of Riemann integrals (involves nested nonlinear equations).
To speedup calculations, we precompute and store values on Riemann integrals as functions of the density and entropy. Two dimensional table lookup and bi-linear interpolation are used.

33. Conductivity of weakly ionized plasmas

34. EOS with Atomic Processes. Numerical Results

35. EOS with Atomic Processes. Numerical Results

36. Real gas EOS (future work)
Gas (deuterium vapor) near the pellet surface is cold and dense
Ideal EOS model is not accurate
A very good approximation is given by the Redlich-Kwong EOS:
We propose an extension of the Redlich-Kwong EOS to include atomic processes (dissociation and ionization)
The EOS will contain three terms; the partial pressure/energy of the molecular gas is written in the Redlich-Kwong form, and the partial pressure/energies of the dissociated and ionized components is written in the ideal EOS form.
Challenges to derive the complete EOS.

37. Simulation results B = 0 = 1 = 2.5 = 5 Tesla
Formation of pellet ablation channels in magnetic fields ranging from 0 to 5 Tesla

38. Simulation results B = 2.5 Tesla B = 5.0 Tesla P T P T
Developed ablation flow channels in magnetic fields of 2.5 and 5 Tesla.

39. Numerical results. Spherical model
G = 118 g/sec

40. Numerical results. Spherical model
G = 110 g/sec

41. Numerical results. Spherical model
G = 105 g/sec

42. Numerical results. 2D axisymmetric model
Pressure distribution
at t = 2 microseconds
Polytropic EOS Plasma EOS

43. Numerical results. 2D axisymmetric model

44. Striation instabilities
Striation instabilities have been observed in all pellet experiments; Important problem for the plasma confinement
Fluctuations of emitted light due to the cloud pellet separation; fingering of the ablated material
Several theories have been proposed, but they don’t satisfactory explain the phenomenon
Idea: Striation instabilities are caused by the Rayleigh-Taylor instability due to the cloud rotation. Linear model of the E x B rotation by Parks (1996).
Nonlinear studies (numerical simulations) are necessary

45. Let’s add missing physics
3D detailed physics model is critical for the study of cloud rotation, pellet – cloud separation, and striation instabilities. Work on the 3D model is in progress.

46. Conclusions and future work
Developed MHD code for free surface low magnetic Re number flows
Front tracking method multiphase/multimaterial flows
Elliptic problems in geometrically complex domains
Phase transition models
Numerical simulations of the Muon Collider target in magnetic fields
2D numerical simulations of the pellet ablation
Axially symmetric ablation flow
Kinetic calculation of the electron heat deposition, atomic physics processes
MHD effects; formations of ablation channels
Calculated pellet ablation rates, channel properties
Future work
3D simulations of the pellet ablation and striation instabilities (3D detailed physics model is critical for the study of cloud rotation, pellet – cloud separation, and striation instabilities)
Laser -- plasma interaction model with sharp absorption front
Laser acceleration of pellets