Compact Toroidal Hybrid Shock Tube Benchmark Simulations and Importing VMEC Equilibria Nick Roberds Thanks to Scott Kruger and Jake King at Tech-X for their valuable assistance. 8/13/2014 NIMROD Team Meeting, Logan, UT
Compact Toroidal Hybrid
Shock Tube Benchmark Simulations and Importing VMEC Equilibria
Nick Roberds
Thanks to Scott Kruger and Jake King at Tech-X for their valuable assistance.
8/13/2014 NIMROD Team Meeting, Logan, UT
Compact Toroidal Hybrid
Outline
1. Fluid and MHD Shock Tube Benchmarks
2. Importing Equilibria from VMEC
i. Objective
ii. Overview
3. Summary
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Compact Toroidal Hybrid
Shock Tube Benchmark Simulations
• Simulations of one-dimensional shock tube benchmark cases Sod (non-conducting fluid) shock tube1
Brio-Wu MHD shock tube2
• lamprof = “briowu” is implemented in my branch, in rect_init.f
1 G.A. Sod, Comp. Phys. (1978) 1.
2 M. Brio and C.C. Wu, Comp. Phys. (1988) 400.
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Compact Toroidal Hybrid
Sod Shock Tube Initial Conditions
Fluid dynamics problem commonly used in validation of CFD codes
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Sod Shock Tube Parameters
Units: 𝜇0 = 𝑘𝐵 = 𝑚𝑖 = 1
Physical and numerical parameters:
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t_dart_upw 0.01
nd_dart_upw 0.01
iso_visc 0.1
elecd 105
be0 10−15
𝛾 1.4
Δ𝑥 1
poly_degree 3
𝑐𝑓𝑙𝑚𝑎𝑥 0.9
lphi 1
Compact Toroidal Hybrid
Sod: Density and Pressure
Exact solution obtained from NPARC Alliance Validation Archive,
http://www.grc.nasa.gov/WWW/wind/valid/stube/stube.html
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Compact Toroidal Hybrid
Sod: 𝑣𝑥
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Brio-Wu MHD Shock Tube
Parameters for MHD shock tube that are different:
𝛾 2
nd_dart_upw 10
t_dart_upw 10
elecd 0
be0 1
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Compact Toroidal Hybrid
Brio-Wu Shock Tube Initial Conditions
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Compact Toroidal Hybrid
Brio-Wu: Density
Image taken directly from M. Brio
and C.C. Wu (1988)
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Leading edges of features (except ‘SM’) are consistent with characteristic speeds.
Compact Toroidal Hybrid
Brio-Wu: Pressure
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Image taken directly from M. Brio
and C.C. Wu (1988)
Compact Toroidal Hybrid
Brio-Wu: 𝑣𝑥
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Image taken directly from M. Brio
and C.C. Wu (1988)
Compact Toroidal Hybrid
Brio-Wu: 𝑣𝑦
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Image taken directly from M. Brio
and C.C. Wu (1988)
Compact Toroidal Hybrid
Brio-Wu: 𝐵𝑦
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Image taken directly from M. Brio
and C.C. Wu (1988)
Compact Toroidal Hybrid
Outline
1. Fluid and MHD Shock Tube Benchmarks
2. Importing Equilibria from VMEC
i. Objective
ii. Overview
3. Summary
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Compact Toroidal Hybrid
What is CTH?
• CTH is a stellarator-tokamak hybrid Has both a helical coil and an ohmic heating transformer
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Compact Toroidal Hybrid
Objective of Simulations
To gain insight into CTH low-q disruptions by whole device simulations in NIMROD.
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Will be loading the equilibrium informed by reconstructions sometime in the middle of the shot before a disruption would occur
Compact Toroidal Hybrid
Disruptions in CTH
• Three types of disruptions observed in CTH: Density Limit
VDE
Low-q
• Low-q disruptions don’t occur when relative strength of stellarator field is increased Can run with 𝑞𝑎 down to 1.25
Growing 𝑚, 𝑛 = (3,2) activity observed in external magnetics before
disruption
Why does stellarator field prevent disruption?
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Compact Toroidal Hybrid
Outline
1. Fluid and MHD Shock Tube Benchmarks
2. Importing Equilibria from VMEC
i. Objective
ii. Overview
3. Summary
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Compact Toroidal Hybrid
What is VMEC?
• VMEC is a 3D inverse equilibrium code Grad-Shafranov equilibrium codes cannot be used to study CTH
VMEC also used to study tokamaks and other devices that are nearly
axisymmetric but not exactly
• inverse code → nested flux surfaces A non-axisymmetric equilibrium with nested flux surfaces may have
singularities in the current density
Equilibrium ‘relaxation’ has been used in M3D simulations initialized with VMEC data1
1. L.E. Sugiyama et. al., 20th IAEA fusion energy conference 2004. Conference proceedings.
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More VMEC Details
• VMEC model: ideal MHD plasma surrounded by a vacuum region
• Does not output fields in
the vacuum region Vacuum region fields must be
computed to load into NIMROD
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Compact Toroidal Hybrid
nimset VMEC Import Module
• Capability to load VMEC equilibria into NIMROD dump file added
• User must provide VMEC ‘wout’ file
makegrid ‘mgrid’ file
• The evolving field arrays are loaded, and the equilibrium field arrays are set to zero Exception: the n=0 coefficient for number density 𝑛 is loaded into the
equilibrium field array to avoid divide by zero error
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Compact Toroidal Hybrid
Flux Surfaces
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Compact Toroidal Hybrid
Rotational Transform Profile
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Magnetic Axis
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Compact Toroidal Hybrid
Current Density has Artifacts and Spurious Features
𝑛 = 0 coefficient for 𝑗𝑧.
• Possible ‘quick fix’: Prescribing 𝑛 = 0 components
of 𝑗 and 𝐵 in equilibrium fields?
Will noise at edge ‘go away’
during relaxation process?
• Possibly a problem with my vacuum region field calculation Magnetic fields look smooth
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Compact Toroidal Hybrid
Fluxgrid Capability
• Can interface with fluxgrid to produce a flux aligned grid Useful for equilibria that are nearly axisymmetric
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Compact Toroidal Hybrid
Library Dependencies
• LIBSTELL: A VMEC library
• Functionality provided by LIBSTELL: Optimized rootfinding1 function for computing VMEC coordinates 𝑠, 𝑢, 𝑣
given 𝑅, 𝑍, 𝜙 .
Accurate extrapolation1 of 𝐵 near the magnetic axis.
Loads wout and mgrid files, accounting for file version.
1. S.E. Attenberger et. al., Comp. Phys. 72 (1987) 435.
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Compact Toroidal Hybrid
Outline
1. Fluid and MHD Shock Tube Benchmarks
2. Importing Equilibria from VMEC
i. Objective
ii. Overview
3. Summary
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Summary
• Shock tubes Demonstrate that NIMROD has some capacity to handle shocks and
rarefactions in 1D if upwind smoothing is used
Shock tubes in Fourier direction not explored
• Setting initial conditions in NIMROD for a VMEC equilibria Need to do more experimenting with trying to ‘relax’ equilibrium while
maintaining profiles
Some short simulations have been run and logged
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Compact Toroidal Hybrid
End
• Questions?
Compact Toroidal Hybrid
Reading
S.P. Hirshman and J.C. Whitson, “Steepest-Descent Moment Method for Three-Dimensional
Magnetohydrodynamic Equilibria”, Phys. Fluids 26 (1983) 3553.
S.P. Hirshman, W.I. van RIJ, P. Merkel, “Three-Dimensional Free Boundary Calculations Using a Spectral
Green’s Function Method”, Comp. Phys. Comm. 43 (1986) 143.
L.E. Sugiyama et. al., “Two-Fluid Limits on Stellarator Performance: Explanation of Three Stellarator
Puzzles and Comparison to Axisymmetric Plasmas”, 20th IAEA fusion energy conference 2004.
Conference proceedings.
H.R. Strauss et. al., “Simulation of Two Fluid and Energetic Particle Effects in Stellarators”, Nucl. Fusion 44
(2004) 1008.
S.E. Attenberger et. al., “Some Practical Considerations Involving Spectral Representations of 3D Plasma
Equilibria”, Comp. Phys. 72 (1987) 435.
G.A. Sod, “A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic
Conservation Laws”, Comp. Phys. (1978) 1.
M. Brio and C.C. Wu, “An Upwind Differencing Scheme for the Equations of Ideal
Magnetohydrodynamics”, Comp. Phys. (1988) 400.
“Shock Tube”, NPARC Alliance Validation Archive,
http://www.grc.nasa.gov/WWW/wind/valid/stube/stube.html, Accessed Aug. 8, 2014.
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