ilmenauer mhd-woche - vom 20. bis 24. september 2004 7th mhd-days (20. - 21. september 2004)...

Ilmenauer MHD-Woche - vom 20. bis 24. September 2004Ilmenauer MHD-Woche - vom 20. bis 24. September 20047th MHD-Days 7th MHD-Days (20. - 21. September 2004) (20. - 21. September 2004)

Numerical modelling

of an MHD flow in the presence

of transverse non-uniform magnetic field

F. DUBOIS*, J. ETAY*, O. WIDLUND** and Y. DELANNOY**EPM-Madylam ENSHMG BP95 38402 S t Martin d'Hères Cedex (FRANCE)

** CEA DER/SSTH/LDAS (Bat. 10.05) Rue des Martyrs 38000 Grenoble (FRANCE)

Tool :Tool :Fluent + MHD module + wall functions

Goals:Goals:Compare numerical results with Ilmenau experimental results

At each iteration of the flow calculation

with boundary conditions

ϕ = ϕo

∂ϕ

∂ n

= U × B( ) ⋅ n

on chosen

on -o

11- solve the problem

2 2 - then calculate j and F

MHD modulusMHD modulus

MHD modulusMHD modulus - tricky points1 - calculation of gradients is done using a self-developed subroutine <-> sensitivity to the mesh

2 - calculation of near the wall <-> term is reconstructed

U × B( )

MHD modulusMHD modulus - validationanalytical solution of a Hartman flow : Ha = 10, 30 and 100

discrepancy 2%, 2.2% and 1.2%

axial velocity current density

parallellayer

Wall functions for laminar MHD flowWall functions for laminar MHD flow

wall

B

U n

Hyp : - elec. insulating wall - solid wall -

p ⊥ n

Xn

= X ⋅ n( ) n = Xn

n

Xp

= X − X ⋅ n( ) n

Ref : O.Widlund/Eur. J. Mech B/Fluids 22 (2003)

U pi n( ) =τi δH sinhnδH( )

ρν

+ C i +Ei( )

δH2

ρνcosh

nδH

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ −1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

+dJ DBi δH

3

ρνsinh

n

δH

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ −

n

δH

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

δH

=

1

Bn

ρν

σ

τ

i

= ρν

∂ Up

i

∂ n

0( )

C

i

= ∇ P − n ∇ P ⋅ n( )( )

i

E

i

= σ Bn

∇ ϕ × n( )

i

DB

i

= B × n( )

i

dJ

= σ ∇p

2

ϕ

GeometryGeometry

Ilmenau loop

Honey comb20 mm

100 mm

30 mm

x

y

z

Galinstan flow

GridGrid

Refined gridRefined grid (no wall function)

•Hartman layer : first cell = 8.82m 10 cells

•// layer : first cells = 16.07m 10 cells

•total = 79 650

Coarse gridCoarse grid (+ wall function on Ha)

•Hartman layer : first mesh = 1 mm 1 cells

•// layer : first mesh = 16.07m 10 cells

•total = 31 860

100 mm 30 mm 200 mm100

mm

x

y

magnetTop view

Input Input parametersparameters

magnetic field

B ≈ Bz

x , y( )

x y

ResultsResultshyp : laminarhyp : laminar

particles path

e.m. forcese.m. forces

xelec. currentselec. currentson symmetry face

x

y

x

ResultsResults

Are identical for both grids, thanks to the use of wall functions

Are in conformity with phenomenology

Magnetic field acts as a “semi-permeable body”

2 wall jets developed at the vertical walls

Elec. Currents paths are different upstream and

downstream of the magnet

Comparison with Comparison with experimental results 1/4experimental results 1/4

voltagevoltage on x=0, on symmetry plane

shapes and level are similar

difference near the // walls

built with ϕ Δϕy num.


voltagevoltage on symmetry plane


on ∆ϕy - in the centre : identical level - near the side wall : num M / exp no M

proposed explanation in experiments

∂ϕ

∂ y

= −

jy

σ

+ ux

Bz

⎛

⎝

⎜

⎞

⎠

⎟

in the center ux weak jy high

∂ϕ

∂ y

≈ −

jy

σ

in the M ux high jy weak

∂ϕ

∂ y

≈ − ux

Bz

when B->0 then ∆ϕy->0


velocity

- in the centre : same level- in the M-jet : different level and

width- lower velocity at the border of

the jetproposed explanation- calibration ? - size of the probe ?- in exp. no flow conservation ux(z)?

calculations

experiments

z

ux (m/s) - symmetry plane - x=19,5 mm

probe

Comparison of numerical results with Ilmenau experimental results are satisfactory.

Wall functions effective in reducing the cost of gridding in boundary layers

Discrepancies can be explained

Turbulence is weak => the turbulence model can not be checked in optimal conditions.

ConclusionsConclusions

ResultsResults

x

elec. potential

ilmenauer mhd-woche - vom 20. bis 24. september 2004 7th mhd-days (20. - 21. september 2004)...

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