Simulation of magneto-hydrodynamic (MHD) flows: electric potential

KIT – die Kooperation vonForschungszentrum Karlsruhe GmbHund Universität Karlsruhe (TH)

FUSIONFZK - EURATOM ASSOCIATION

Chiara MistrangeloJune 23, 2010

Karlsruhe Institute of Technology

Simulation of magneto-hydrodynamic (MHD) flows: electric potential

formulation

Chiara Mistrangelo, Ola Widlund

5th OpenFOAM workshop Göteborg, June 22-24, 2010




Karlsruhe Institute of TechnologyOutline

Motivations for studying MHD flows

Why a formulation with electric potential?available mhdFoam new solver mhdEpotFoam

Magneto-hydrodynamic (MHD) equations

Issues for MHD flow simulations Mesh constraints Numerical algorithm

MHD flow in electrically conducting ducts

new solver conjugatemhdFoam

MHD flows in ducts: solver validation

Summary & Outlook





Fusion blanket

Radiation shielding

Breeding of tritium6Li + n He + T + energy

Heat removal: conversion of nuclear kinetic energy into electric energy

Requirements can be accomplished with Li-containing liquids as breeder and coolant

Liquid metal flows in fusion blankets

International Thermonuclear Experimental Reactor

Magnetic confinement of plasma

Magnetic coils

Fusion plasma (D + T He + n + energy)

Test blanket module (TBM)





Fusion blanket

Radiation shielding





Magnetic coils

Fusion plasma (D + T He + n + energy)

Test blanket module (TBM)

Liquid metal magneto-hydrodynamics (MHD)

Moving electrically conducting fluid magnetic field






Other MHD applications:

Development of measuring techniques in liquid metal flows, electromagnetic flow meters and pumps.

Metallurgical technology – continuous casting (surface treatment, MHD liquid metal stirring), industrial processes…

Focus on fusion applications: High magnetic fields (B = 4

11T) very thin MHD boundary layers

Coupled phenomena Complex geometries Strict numerical issues

and requirements Simulation is a

challenging task!

Fusion blanket

Radiation shielding




Liquid metal magneto-hydrodynamics (MHD)

Moving electrically conducting fluid magnetic field





Need of electric potential formulation

Induction equation(transport eq. for B) vBBBvB 1 2

t mRe

/10LuRem Magnetic Reynolds number Magnetic diffusivity

mhdFoam available solver

mhdFoam available solver mhdEpotFoam new solver







t mRe


mhdFoam available solver

Boundary conditionsFully developed MHD duct flow (induced magnetic field constant along duct axis)

Local Dirichlet or Neumann BCs can be set

Induced magnetic field serves as streamfunction for current in duct cross-section

3D MHD flow

No local BCs can be defined induced field in external space has to be considered(jext = 0 Ampère‘s law

0 constB at 0/ nB

)0,0 2 outside at define BB

mhdFoam available solver mhdEpotFoam new solver







t

MHD channel flow

externally applied B

Initial boundary value problem: B = f (v)

is determined depending on the flow field

for Rem << 1 (liquid metals, industrial applications)

B

f (v, t)

B

f (t)

t B = 0

E =

Inductionless Approximation (the magnetic field is not affected by the flow. The induced magnetic field can be neglected.)

x E =

mRe


mhdEpotFoam new solver mhdFoam available solver





Bjvvvv

2

2

11Ha

ptN

Bvj

Conservation of

Momentum

Mass & Charge

Ohm‘s law

0,0 jv

Lorentz force

Induced electric field

el.magn. forceviscous force

el.magn. forceinertia force0

2

uLBN

22

2 BLHa

Dimensionless groups

Interaction parameter

Hartmann number

inertia forceviscous forceNHaRe /2Reynolds number

Bv 2

Poisson eq. for

N ≈

105

Ha ≈

104

Fusion reactors

Magnetohydrodynamic equations (Rem <<1)





MHD flow features and numerical issuesMHD fully developed flow

u

yz

B

Hartmann layerHa ~ Ha -1

Core

s ~ Ha -1/2Side layers

Velocity distribution

s ~ Ha -1/2

Ha ~ Ha -1 L

B

Hartmann wall (

B)S

ide

wal

l (II

B)

y

zx

v





MHD flow features and numerical issuesMHD fully developed flow

u

yz

B

Hartmann layerHa ~ Ha -1

Core

s ~ Ha -1/2Side layers

MHD simulation issues: MESH (discretization error)

Suitable resolution of MHD boundary layers- Refinement in boundary layers- Smooth grid transition between various regions (core - layers)- Good cell aspect ratio has to be maintained

By increasing Ha (i.e. B) the total number of nodes becomes larger

Walls of finite electric conductivity: strong current turns in the thin wall The corner region has to be properly resolved

Velocity distribution

s ~ Ha -1/2

Ha ~ Ha -1 L

B

Hartmann wall (

B)S

ide

wal

l (II

B)

y

zx

v





MHD flow features and numerical issues

Hartmann layer

Side layers

MHD simulation issues: Numerics Physics (modeling error)

Error in current density j is amplified by N in mom. Eq. when used to calculate the Lorentz force

High accuracy required to compute the current density

Charge conservation has to be ensured: in FVM balance of fluxes through cell faces

Source term in Eq. comes from a part of the current density and has to be given at cell faces Lorentz force defined at cell center proper interpolation of j from cell face to center

Governing equations

Bjvvvv NRept21

Bvj

0,0 jv Bv 2

MHD fully developed flow

s ~ Ha -1/2

Ha ~ Ha -1 L

B

Hartmann wall (

B)S

ide

wal

l (II

B)

y

zx

v





MHD duct flow: electrically insulating wallsCurrent streamlines

Current closes its path in boundary layers (BLs)

Resolution of BLs critical for the accuracy of the solution (velocity and pressure gradient)Ha = 500

u

zy

Velocity profileVelocity distribution

z = 1

B

y = 1

y

zx

u





MHD duct flow: walls of finite electric conductivityconjugateHeatFoam solver (OF 1.5 - dev)

conjugatemhdFoam new solver

Bvj

Bv

Fluid

www j

0 ww

Solid / Wall

w

w

njnj

, w electric conductivity of fluid and wall

y

zx

InterF{

type regionCouple;...}

InterW{

type regionCouple;...}

w solid

fluid

Definition of the electric conductivity field Coupled boundary conditions for

and

Momentum Eq. solved only on fluid mesh and Eq. on both meshes (combined matrix for fluid-solid coupledFvScalarMatrix)





core

MHD duct flow: walls of finite electric conductivity

0

3.0

3000

S

wwHa

cLtc

Ha

Transverse velocity profile

B-aligned velocity profile

Ha ~ Ha -1

core

conjugateHeatFoam solver (OF 1.5 - dev)


tw ,w

0

L

B

Conducting Hartmann walls In

sula

ting

side

wal

ls

y

zx

v

s

Ha

Hartmann layers

Side layers core

0

3.0

3000

S

wwHa

cLtc

Ha

u

u

y

z





MHD duct flow: walls of finite electric conductivity

B-aligned velocity profile

Ha ~ Ha -1

core

conjugateHeatFoam solver (OF 1.5 - dev)


tw ,w

0

L

B

Conducting Hartmann walls In

sula

ting

side

wal

ls

y

zx

v

s

Ha

Hartmann layers

Side layers core

0

3.0

3000

S

wwHa

cLtc

Ha

u

B

y z

core

Current streamlines

tw ,w

0

Side wall w = 0

B

y

zx

u

y





Summary

Explanation of need of electric potential formulation to simulate 3D MHD flows difficult BCs for 3D MHD flows in case of induction equation approach

Description of magneto-hydrodynamic (MHD) equations Lorentz force in mom. Eq., Ohm‘s law, Poisson Eq.

Issues for MHD flow simulations:

MESH: proper resolution of thin boundary layers, BL ~ Ha –n

ALGORITHM-MODELING: accuracy of j prediction, interpolation of j from cell face to center, charge conservation

new solver mhdEpotFoam

Code validation: perfect agreement with analytical solutions up to Ha = 5000

Successful application to 3D MHD problems

Channels can have walls of arbitrary electric conductivity new solver conjugatemhdFoam from conjugateHeatFoam (OF 1.5-dev)





Outlook

CooperationOla Widlund, ABB AB, Corporate Research Center, Västerås, SwedenVincent Dousset, Coventry University, UKElisabet Mas de Les Valls, Technical University of Catalonia, Barcelona, Spain

Wall element with

Optimization of present solver version (speed, numerical scheme, grid sensitivity studies, mesh skewness…)

Development of wall functions (boundary layer models)

Implementation of thin wall condition

Simulation of MHD flow in ducts with walls of finite electric conductivity

1st approach based on conjugateHeatFoam solver (OF 1.5-dev) conjugatemhdFoam solver

2nd approach as in chtMultiRegionFoam solver (OF 1.6, 1.6.x) mhdMultiRegionFoam solver

tw

jt

jt

jt

jt

jn

wtcn

nj

Ltc ww

Documents

Simulation of magneto-hydrodynamic (MHD) flows: electric potential