Numerical Simulation of Hydrodynamic Processes in High

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

[email protected]


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.


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


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


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


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.


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.


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


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.


FronTier simulation of a 3D Rayleigh-Taylor mixing layer


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


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


MHD simulations: stabilizing effect of the magnetic field.

a) B = 0b) B = 2Tc) B = 4Td) B = 6Te) B = 10T


Velocity of jet surface instabilities in the magnetic field


Evolution of a liquid metal jet in 20 T solenoid


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.


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.


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

= −

=

= −−


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


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

∞

− −∞∞

= ⇒

= −

= + − =−

− =


CERN neutrino factory target experiments

Schematic of the experimental setup


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


Mercury splash (thimble): numerical simulation

Energy deposition = 15 J/g

250 530 730 1t s t s t s t msµ µ µ= = = =


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.


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

+ + + − + − + =


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


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


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.

Numerical Simulation of Hydrodynamic Processes in High

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