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Brookhaven Science AssociatesU.S. Department of Energy
ARIES Project Meeting on Liquid Wall Chamber DynamicsMay 5-6, 2003, Livermore, CA
Numerical Simulation of Hydrodynamic Processes in High
Power Liquid Mercury Targets
Roman SamulyakCenter for Data Intensive Computing
Brookhaven National LaboratoryU.S. Department of Energy
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Brookhaven Science AssociatesU.S. Department of Energy
Talk outline
Theoretical and numerical ideas implemented in the FronTier-MHD code. Numerical example: the 3D Rayleigh-Taylor instability problem.
Numerical simulation of hydro and MHD processes in the Muon ColliderTarget.
Cavitation modeling and numerical simulation of CERN neutrino factory target experiments.
Further development of cavitation models and the simulation of hydrodynamic processes in the SNS target.
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Brookhaven Science AssociatesU.S. Department of Energy
The system of equations of compressible magnetohydrodynamics:
an example of a coupled hyperbolic – parabolic/elliptic subsystems
( )
( )
( )
2
2
1
1
40
t
Pt c
U Pt
ct
ρ ρ
ρ ρ
ρσ
πσ
∂= −∇ ⋅
∂∂ + ⋅∇ = −∇ + + × ∂ ∂ + ⋅∇ = − ∇ ⋅ + ∂
∂= ∇× × −∇× ∇× ∂
∇ ⋅ =
u
u u X J B
u u J
B u B B
B
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Brookhaven Science AssociatesU.S. Department of Energy
Constant in time magnetic field approximation
The distribution of currents can be found by solving Poisson’s equation:
( )
1
1 ,
1with ( )
c
c
c
σ φ
φ
φΓ
= −∇ + ×
∆ = ∇⋅ ×
∂= × ⋅
∂
J u B
u B
u B nn
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Brookhaven Science AssociatesU.S. Department of Energy
Numerical methods for the hyperbolic subsystem. The FronTier Code
The FronTier code is based on front tracking. Conservative scheme.
Front tracking features include the absence of the numerical diffusion across interfaces. It is ideal for problems with strong discontinuities.
Away from interfaces, FronTier uses high resolution (shock capturing) methods
FronTier uses realistic EOS models:- SESAME- Phase transition (cavitation) support
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Brookhaven Science AssociatesU.S. Department of Energy
Resolving interface tangling by using the grid based method
3D: We reconstruct the interface using micro-topology within each rectangular grid cell. There are 256 possible configurations for the crossings of the cell edge by the interface. Through elementary operations of rotation, reflection and separation these can be reduced to the 16 cases shown on the left.
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Brookhaven Science AssociatesU.S. Department of Energy
Methods for the parabolic/elliptic subsystem
• Finite elements based on vector (Whitney) elements.
• Dynamic finite element grid conforming to the moving interface. Point shifting method with rectangular index structure.
Triangulated tracked surface and tetrahedralized hexahedra conforming to the surface. For clarity, only a limited number of hexahedra have been displayed.
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Brookhaven Science AssociatesU.S. Department of Energy
Whitney elements
Let be a barycentric function of the node i with the coordinates xi
iλ
Whitney elements of degree 0 or “nodal elements”:
inijw λ=
Whitney elements of degree 1 or “edge elements”:
ijjieijw λλλλ ∇−∇=
Whitney elements of degree 2 or “facet elements”:
( )jikikjkjifijkw λλλλλλλλλ ∇×∇+∇×∇+∇×∇= 2
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Brookhaven Science AssociatesU.S. Department of Energy
Elliptic/Parabolic Solvers
• 3D version of the Chavent -Jaffre mixed-hybrid finite element formulation.
• Instead of solving the Poission equation,
we solve for better accuracy.×−=⋅∇ BE )u(1
( )1 , c
φ φ∆ = − ∇ × = ∇u B E
φ∇=Ec
• The parallel solver is based on the domain decomposition. Linear systems in subdomains are solved using direct methods and the global wire basket problem is solved iteratively.
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Brookhaven Science AssociatesU.S. Department of Energy
FronTier simulation of a 3D Rayleigh-Taylor mixing layer
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Brookhaven Science AssociatesU.S. Department of Energy
Applications: Muon Collider Target
Numerical simulation of the interaction of a free mercury jet with high energy proton pulses in a 20 T magnetic field
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Brookhaven Science AssociatesU.S. Department of Energy
Simulation of the Muon Collider target. The evolution of the mercury jet due to the proton energy deposition is shown.
No magnetic field.
t = 0
t = 80
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Brookhaven Science AssociatesU.S. Department of Energy
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Brookhaven Science AssociatesU.S. Department of Energy
MHD simulations: stabilizing effect of the magnetic field.
a) B = 0b) B = 2Tc) B = 4Td) B = 6Te) B = 10T
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Brookhaven Science AssociatesU.S. Department of Energy
Velocity of jet surface instabilities in the magnetic field
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Brookhaven Science AssociatesU.S. Department of Energy
Evolution of a liquid metal jet in 20 T solenoid
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Brookhaven Science AssociatesU.S. Department of Energy
Equation of state: the problem of cavitation
Material properties strongly influence the wave dynamics. The wave dynamics is significantly different in the case of cavitating flows.
The one-phase stiffened polytropic EOS for liquid led to much shorter time scale dynamics and did not reproduce experimental results at low energies.
An important part of our research is EOS modeling for cavitating and bubbly flows.
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Brookhaven Science AssociatesU.S. Department of Energy
Thermodynamic properties of mercury (ANEOS data)
Thermodynamic properties of mercury were obtained using the ANEOS data. Isotherms of the specific internal energy, pressure and entropy as functions of density are shown in a large density – temperature – pressure domain which includes liquid, vapor and mixed phases.
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Brookhaven Science AssociatesU.S. Department of Energy
Analytical model: Isentropic EOS with phase transitions
• A homogeneous EOS model
• Gas (vapor) phase is described by the polytropic EOS reduced to an isentrope.
1 1, , ,1( 1)where exp
S const
P E TR
SR
γ γ γη ηηρ ρ ργ
γη
− −
= ⇒
= = =−
− =
0
00
( 1) ,
,
(log log ) .1
P EPTR
RS P
γ ρ
ρ
γ ργ
= −
=
= −−
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Brookhaven Science AssociatesU.S. Department of Energy
The mixed phase
• Mixed phase is described as follows:
( ) ( )( )( )( )
( )
( )
.
,
,
,
2222
22
2
222
2
lv
l
llvv
lvllvvvl
llvvvvl
lvlvvvl
satl
aaaaP
dPE
aaaaPPP
satv
ρρρραα
ρρρρρρ
ξρ
ρ
ρραρρρραρρρ
ρ
ρ
−−
=
−−
=
=
−−−+
+=
∫
:fraction void the isand
where
log
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Brookhaven Science AssociatesU.S. Department of Energy
The liquid phase
• The liquid phase is described by the stiffened polytropic EOS:
( ) .1
log)(log
,
,)()1(
00 −
−+=
+=
−+−=
∞
∞
∞∞
γργ
ρ
γργ
RPPS
RPP
T
PEEP
1 1
,
, ,1
( 1)where exp
S constP P
PE E TR
SR
γ
γ γ
ηρη ηρ ργ ρ
γη
∞
− −∞∞
= ⇒
= −
= + − =−
− =
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Brookhaven Science AssociatesU.S. Department of Energy
CERN neutrino factory target experiments
Schematic of the experimental setup
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Brookhaven Science AssociatesU.S. Department of Energy
Mercury splash (thimble): experimental data
CERN neutrino factory target experiments
0.88 1.25 7t ms t ms t ms= = =
Energy deposition:
5 J/g
30 J/g
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Brookhaven Science AssociatesU.S. Department of Energy
Mercury splash (thimble): numerical simulation
Energy deposition = 15 J/g
250 530 730 1t s t s t s t msµ µ µ= = = =
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Brookhaven Science AssociatesU.S. Department of Energy
Hydrodynamic processes in the SNS target
The mercury target for SNS will consist of a sealed stainless steel chamber filled with mercury interacting with 20 kJ proton pulses at frequency 60 Hz. The desired target lifetime implies that the target should withstand ~1.e+8 proton pulses. The pitting of walls was observed experimentally after 200-1000 pulses.Future targets will require much higher beam intensities.
An effective approach capable of reducing the strength of rarefaction and shock waves is to use bubbly layers near flanges and bubbles in the bulk mercury.
Direct (tracked bubbles) and continuum (new eos model) simulations.
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Brookhaven Science AssociatesU.S. Department of Energy
Further development of continuum homogeneous equation of state models for bubbly flows
The (modified) Rayleigh-Plesset equation gives a dynamic closure for the fluid dynamics equation:
( )23 3
3 1 2 1 1 11 0,2 2
where is the effective dumping coefficient is the Weber number
is the cavitation number is the pressure term
tt t D t p
D
p
RR R R CR We R R R
We
C
γ γ
σδ
δ
σ
+ + + − + − + =
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Brookhaven Science AssociatesU.S. Department of Energy
Direct simulation approach: a system of tracked bubbles
Mean particle radius = 2mmOne phase mercury EOS for the liquid, the ideal gas EOS for the bubble
gasUniform in x and gaussian in y initial energy deposition with the center at
the container top resulting in the maximum pressure 500 bar.
Initial density Initial pressure
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Brookhaven Science AssociatesU.S. Department of Energy
Direct simulation: the pressure evolution
80 bubbles 130 bubbles
106 , 4t s p barµ= = 106 , 2t s p barµ= =
~72.1130 bubbles~103.880 bubbles3223221 phase fluid
Total max pressure at the
bottom
Max pressure at the bottom, t = 37
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Brookhaven Science AssociatesU.S. Department of Energy
Future research
Futher work on the EOS modeling for cavitating and bubbly flows.
Futher studies of the muon collider target issues. Studies of the cavitation phenomena in a magnetic field.
Studies of hydrodynamic issues of the cavitation induced erosion in the SNS target.
Studies of the MHD processes in liquid lithium jets in magnetic fields related to the APEX experiments.