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HIMAG - a 3-D MHD solver for free surface flows
HIMAG : HHHHyPerComp IIIIncompressible MMMMHD solver for AAAArbitrary GGGGrids
Primary code objective:To obtain 3-D MHD solutions for incompressible flows with free surfaces
Principal code features:
• Parallel iterative solver, based on well developed base codes in CFD and CEM• Can use an arbitrary mesh structure, to resolve interfaces and complex geometry• Facility to use Volume of Fluid and Level Set methods for free surface capture• Implicit methods to ease stiffness and time step constraints
• Choice of 3-D MHD models: ϕ formulation and J-formulation
Team members: HyPerComp, Inc., Fusion Group at UCLA,Dr. Ali Hadid (Boeing)Sponsored by DoE as a Phase-II SBIR,contract # DE FG03-00ER83022
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3-DUnstructured and hybrid meshesParallel processingGraphical interfaces
Implicit incompressible flow solverColocated data storageLevel set technique for free surfaceMultiple MHD models
InductionlessCurrent / Induced field Hybrid
Adaptive meshing for higher accuracy
Existing technology, adapted to current application
Feature development
Development and validation of HIMAG - 1
Code attributes
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Main
Driver
Parallel data initialization •Upload / Download Input• Upload/Download Mesh• Upload/Download BCs• Upload/Download Patches• Prepare Grid• Memory allocations
Synchronize MPI
Solver Input/Output, restart files
Statistics, Post-processing
Mesh adaptation, repartitioning
Parallel task assignment
Pre-processing• Model Geometry Definition• Mesh Generation• Mesh partitioning for multiple processors (KMETIS)
Structure of the HIMAG Solver
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3-D Eddy current analysis : Summary of techniques
Legend:
A : Magnetic vector potential B = curl (A) A* : Modified A : A* = A + Ú grad φ dtφ : Electric scalar potential E = - ( ∂A/∂t + grad φ ) T : Current vector potential J = curl (T)W : Magnetic scalar potential H = - grad W µ : mag. perm., n = 1/ µ, σ = elec. cond.
A - f
A - f - W
A* - W
T - W
E - W
curl (n curl (A)) = -σ ( ∂A/∂t + grad φ )
div (-σ ( ∂A/∂t + grad φ )) = 0
curl (n curl (A)) = 0
curl (n curl (A*)) = -σ ( ∂A*/∂t)
curl(curl (T)/ σ) = - ∂/∂t (µ( T - grad W ))
div ( µ( T - grad W ) ) = 0
curl (n curl ( E )) = -σ ( ∂ E / ∂ t)
div(- µ grad W) = 0
A fA
A fW
A*W
T WW
EW
Formulation variable location in the current carrying region current free region
J ≠ 0
J = 0
Separate regions for storing variables :
(Stationary media)
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curl ( ν curl (A)) + σ ( grad φ + ∂A/ ∂ t - V × curl (A) ) = Jo
div (σ ( grad φ + ∂A/ ∂ t - V × curl (A) ) = 0
Using Coulomb’s gauge, ∇2 A - µ σ ( grad φ + ∂A/ ∂ t - V × curl (A) )= - Jo
Fem = σ ( grad φ + ∂A/ ∂ t ) × curl (A)
The A- φ formulation
div V = 0
ρ DV/D t = µv ∇2 V – grad p + ρ g + Fem + ρ KV
B = curl AE = - ( grad φ + ∂A/ ∂ t )
Boundary procedures:
B ⋅n and B ×n are continuous, J ⋅n = 0
Across interface At1 = At2
ν1 (curl A)t1 = ν2 (curl A)t2
Impedance BC n ×(curl A) = (1/Zs) n ×( n × ( i σ w A + σ grad φ))
Governing equations
A- φ equation set
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Validation procedures
Incompressible flow solverSpatial accuracy:
Driven cavity Rearward facing step
Temporal accuracy:2-D Karman vortex shedding
Non-orthogonality of mesh:stretched triangular meshes
MHD solver:Spatial accuracy:
2-D and 3-D rectangular channel flows using B and ϕ formulations, fully developed flow
Time accuracy:MHD Vortex shedding
Free surface:TBD - suggestions welcome!
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Time Loop t = t0 to tmax
Initialize: u, v, w, p, j, V, A at cell centers
Compute allowable dtInitialize mass flux at cell facesInitialize intermediate velocities (u*,v*,w*, etc.)
PN
nv
Sub-iterations between t and t + dt
Update state vector q = (u,v,w,p,j,V,A) at each cell center
For each interior cell face: Compute free surface and momentum flux-termsCompute Pressure Poisson flux-terms
For each boundary cell:Impose appropriate boundary flux
Update intermediate variables Use intermediate velocities to update potentials V,A
Repeat until t + dtquantities are converged(including BC values)
Schematic of Flow Solver
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• To convert HyPerComp’s existing adaptive hybrid unstructured mesh environment to problems in liquid metal MHD with free surfaces.• Modular code structure to the extent possible, for hierarchical MHD treatment• Interaction with research groups (LANL, LLNL, FZK, etc.)• Commercial applications to metallurgy (MAG/GATE: Concept Engineering Group, PA)
Goals of HyPerComp’s phase-II SBIR research
HIMAG : “HyPerComp Incompressible MHD( HyPEX) Adaptive Grid ” suite of codes
containing High Power Extraction code environment for nuclear fusion
CAD/Grid generation
Domain decompositionPC-cluster based parallel computing
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Code development status
The following systematic validation procedures have been carried out so far:1. Incompressible solver Spatial accuracy: Driven cavity problem, Rearward facing step Temporal accuracy: 2-D vortex shedding frequencies Grid-non-orthogonality: Compare results of above with stretched triangular cells2. MHD solver Assessment of J, B and ϕ formulations for channel flow problems 3-D square channel flows at a range of Hartmann numbers
Expected date for completion of this project: June 2003Expected date for completion of this project: June 2003Expected date for completion of this project: June 2003Expected date for completion of this project: June 2003
Flow in a square ductMagnetic field is ramped upfrom 0 to 1 at Ha = 1000,N = 1000
inflow
Region of B-gradientStreamline structure
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3-D meshes used in 2-D problems
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Driven cavity flow (Re = 1000)
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3-D Tetrahedral and Hexahedral meshes
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Adjustments for non-orthogonality
f
P
N
A
kD
d
DAk,AAd
dD −=
⋅= 2
Over-relaxed approach:
( ) fPN
fk
dD ϕϕϕϕ ∇⋅+−=∇⋅A
Treat non-orthogonal termsexplicitly in an iterative solver.
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Locating face-based quantities to face-centers
P Nf
c
r1
r2 r3
d
[ ] 321
21
2
1r
A
NPNP
c
facescP
⋅∇+∇+++=
=∇ ∑
ϕϕδδ
ϕδϕδϕ
ϕϕv
Iterate.(2-3 stepssufficient!)
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Highly stretched unstructured mesh
40 x 40 rectangular cells, each divided diagonally into 2 triangles
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Uniform channel flow
Inflow: u = 4*y*(1-y)
( )500Re,04.010
50u on basedRe,8.002.0
40,5
8
3
max
2
2
=−=∆→=
=−=∆→=−=∆=∆
∆−=∆⇒∂∂=
∂∂
− p
p
pxif
xp
y
u
x
p
µµ
µµ
µ
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Backward facing step flow at low Re
separation zonereattachment point
Inflow: parabolic u-profile
Re = Uavg*2*h*ρ/µ
Uavg = ("u dA)/Ah
h
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Flow past a cylinder
Farfield: u = 1, v = 0, p = 0
Farfield: u = 1, v = 0, p = 0
Outflow:u,v extrapolatedp = 0
Inflow:u=1,v=0, dp/dn=0
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Unstructured mesh with 20,000 cells
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Time averaged separation region and the vortex street
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Time evolution of lift and drag
Re = 100 Re = 200
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Change in lift and drag with Re
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Strouhal number computed on 3 grid levels
Grid-1
Grid-2
Grid-3
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( ) ( ) ( )We
k
x
u
xFr
f
x
puu
xt
u
j
i
j
i
iji
j
i
ρφφδφµ
ρρ∇−
∂∂
∂∂+−
∂∂−
∂∂−=
∂∂
Re
11
0=∂∂
i
i
x
u
0=∂∂
+∂∂
ii x
ut
φφ
2 Fluid0
Interface0
1 Fluid0
<=>
φφφ
CSF Model
The level set technique in free surface flows
• Initiated by Osher and Sethian (1988)
• Quickly becoming popular among various interfacial modeling areas
• Produces mass conservation errors at extreme deformations (corrections exist)
• Applied to unstructured meshes recently (Tornberg, 2000)
• Greatly simplified reinitialization procedure for unstructured meshes (vs. VOF)
• Combination of VOF-level set exist that combine their strengths
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the equation remains the interface unchanged.
Distance Function
Steady problem
0φφ
( )( )φφφ ε ∇−= 10signt
( ) ( )xxvv
00, φφ =
With the sign function, theoretically
has the same sign and zero level set as
1=∇φ
Away from the interfacewill converge to the actual distance.
HOWEVER
( )( )nnn signt φφ∆φφ ε ∇−+=+ 1001
1>∇φ
01 <⋅+ nn φφ
Discretized redistance equation
For the case of
It is very possible
Mass is not conserved
1,00 >∇> nφφ
10 −∇>∆
n
n
tφφ
0t∆ constant
φ
Reinitialization equations
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( ) ( )( )[ ] ( )( )( )( )∫
∫+++
=′ijk
ijk
aaasgn
asgnt
Ω
Ω
φφδ
φφδ∆
420
2
0
0111
( ) 11 00 −∇=′−= naattt φ∆∆∆
( )( )nnn tsign φφφφ ε ∇−∆+=+ 101
By preserving the volume of the bubble
Variable time step method
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3-D cylinder in a channel - test case under study
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Construction of current-free B distributions
• The applied field B0(x,y,z,t) must satisfy: div(B0) = 0 and curl (B0) = 0 such that there are no imposed currents.
• B0 can then be derived from a scalar potential such that B0 = -—j and —2j = 0
e.g.: for a flow entering a bore of a uniform magnet, if it is desired to obtain an accurate representation of B0 in the flow domain, the boundary conditions on j could be approximated as follows. (G1 , G2 , G3 represent flow, wall and external media respectively)
V B
j=0j=1
j=0
j=1
G1
G2
G3
Note: Given an analytical expression for B0, it is in general rather difficult to project out the div and curl free components simultaneously.
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Structured and unstructured discretization
Structured grid showing domain decomposition forparallel flow computation
Unstructured mesh in an identical geometry using tetrahedra
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Tokamak-like geometries
Solid model,domain definition
Sample unstructured mesh
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Single phase flow past an obstacle in a channel
Streamlines showing separationnear the inflow-bend and the formation
of the saddle-points near a cylinder
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2-D Channel flow Test case
Inflow: parabolic u-profile2 cm
20 cm
Bz0
0.35 Tesla
Channel flowWith parabolic inflow,Umax = 0.1 m/sρ = 500 kg/m3ν = 10^-6 m2/sσ= 10^6 mho/m
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3-D channel flow test case
- 4
4
x- 1
1
z
- 1 1y
By
0
1
Case based on Sterl (JFM, 1990)
By = 1/(1 + exp(-x/x0))Exit flow properties arecompared with fully developedflow results based on the B-formulation
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Sample calculation at Ha = 1000
Solution exhibits unsteady features, recirculation and an oscillating core flow
There is potential for numerical errors in the j formulation.
Perhaps a “Hybrid Formulation?”
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Convergence acceleration techniques
Solution to Poisson equation is iterative. (esp. Neumann BCs) A fast solver will make tremendous difference.e.g.: Conjugate Gradient technique,
Residual cutting technique,
Alternatives to wall computations: approximate or semi-analytical BCse.g., impedance BC, Hartmann layer velocity profile Green’s function type solvers for steady state cases
Also, a Newton-Krylov technique for overall flow solver will help.e.g., as in TELURIDE
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The induced magnetic field formulation
x
y
-4 -2 0 2 4-1
-0.5
0
0.5
1
0.14005 0.42015 0.70025 0.98035 1.26045 1.54055 1.82065 2.10075
x
y
-4 -2 0 2 4-1
-0.5
0
0.5
1
0.14005 0.42015 0.70025 0.98035 1.26045 1.54055 1.82065 2.10075
x
y
-4 -3 -2 -1 0 1 2 3 4
-1
0
1
0 0.22473 0.449459 0.674189 0.898919 1.12365 1.34838 1.57311 1.79784 2.02257
x
y
-4 -3 -2 -1 0 1 2 3 4
-1
0
1
0 0.22473 0.449459 0.674189 0.898919 1.12365 1.34838 1.57311 1.79784 2.02257
Validation cases involving induced magnetic field distributions are beingstudied. Effect of conducting walls and the computation of fields insidethe wall regions is included. Results are currently being validated.(above, Sterl’s problem for thin and thick conducting walls)