INSTABILITY OF ROTATING INSTABILITY OF ROTATING MAGNETIC FIELD DRIVEN FLOW MAGNETIC FIELD DRIVEN FLOW

IN A COUNTER-ROTATING IN A COUNTER-ROTATING CYLINDERCYLINDER

Alexander Pedchenko and Ilmars Grants

Institute of Physics, University of Latvia, Salaspils, Latvia

THE 15th RIGA AND 6th PAMIR CONFERENCE ON FUNDAMENTAL AND APPLIED MHD

Applications of Rotating Magnetic Field (RMF)

Continuous Casting of Steel, Aluminum etc

Semiconductor Crystal Growth

create variety of flows with different properties,

combining mechanical and RMF induced rotation

stabilize unstable convective flow

PROBLEM FORMULATIONPROBLEM FORMULATION

melt mixing and homogenization

BRMF

RMF & counter-rotation driven flow

RMF

side

-wal

l bo

unda

ry la

yer

R

z

r

PROBLEM FORMULATIONPROBLEM FORMULATION

instability may occur at high RMF

0)()( 222

r

rV

r

w

Rr

w-RMF

more stable flow

0)( 2

r

Data Translation DT98214-channel 24-bit A/D USB module

EXPERIMETAL SETUPEXPERIMETAL SETUP

RMF Inductor

Container with liquid metal (Hg)

Permanent magnets

Rotating table with adjustable rot. speed

Registering equipment

340 mm

40 mmNb-Fe-B magnets

B

PC & A/ D card

EXPERIMETAL SETUPEXPERIMETAL SETUP

Hg electrodes

magnetsRMF coils

PC & A/D c a rd

SYSTEM PARAMETERS

Container:

H/R = 40/20 (mm) = 2

melt: Hg

RMF:

B = 0…3.8 mT (0≤Tam≤107)

f = 45; 136 Hz

Static magnetic field:[at z = H/2, r = 0]

BSMF = 40 mT (Ha=20)

Mechanical rotation:

= 15 rpm (Re=5500);

= 45 rpm; (Re=16500);

RMF coils

Registeringequipment

Rotating table

Permanentmagnets

Containerwith Hg

EXPERIMETAL RESULTS:EXPERIMETAL RESULTS:c

4 5 6 7

,r

pm

100

10

1

0.1 10 10 10 10

Ta

Determination of the fluid rotation rate driven by RMF only:

* c=1.98(/Ro2)Ta5/9 P.A.Davidson, JFM 245, 1992

P.A.Davidson formula *for turbulent flow

numericalsimulationw/ DC field

numericalsimulationw/o DC field

experimentRMF freq 45Hz

experimentRMF freq 136Hz

0 60 120 180 240 300 360 420

-15

-10

-5

0

5

10

15

,

V

time, s

calibration of electrodes byapplying abrupt pulse of mechanical rotation= f (c) - ?

spin-up of fluid

spin-downof fluid

stable rotationof fluid

Container at restContainer at rest

EXPERIMETAL RESULTS:EXPERIMETAL RESULTS: Container at restContainer at rest

0 .2 0 .0-0 .2

0 .2 0 .0-0 .2

0 .2 0 .0-0 .2

0 .2 0 .0-0 .2

tim e , s

, V

, V

, V

, V

Fluctuating component of the registered electric potential fordifferent strengths of RMF

105

106

107

0.01

0.1

1

Ta m

c = 0

.23

×1

06

(0

.75

mT

)

(2.4

mT

) T

a mc =

2.3

×1

06

Ta m

c = 0

.45

×1

06 (

1.1

mT

)

V

Tam

= 0 = - 15 rpm

= - 45 rpm

Ta = oBo2Ro4/22

- electrical conductivityo - RMF frequencyBo - RMF inductionRo - container radius - density of the fluid - kinematical viscosity

EXPERIMETAL RESULTS:EXPERIMETAL RESULTS:

Intensity of fluctuations () vs. magnetic forcing (Tam)

NUMERICAL STUDY:NUMERICAL STUDY:

eυυυυ

rzrfTapt

),()( 2

(r,f,z,t) - flow velocity; = 0

Boundary conditions:

where:

Steady axisymmetric solution o (r, z) is linearly unstable to infinitesimal perturbations ’ (r, , z) when an eigenvalue problem

with ’ = 0 and has at least one eigenvalue r > 0

erS Re|

2

42

2 ooo RB

Ta

2

Re oR

0|' Sυ

')'(')(''2 υυυυυυ o op

f (r,z) - e.m. forceo - RMF frequencyBo - RMF induction - rotation rate of cavityRo - radius of cavity - viscosity - density

Ta c

R e

NUMERICAL RESULTS:NUMERICAL RESULTS:

Calculation with SMF

Experiment

Flow reversal Ta values

Tac (Re) calc.

Re3/2

NUMERICAL RESULTS:NUMERICAL RESULTS:

r

Ta/105

Re

Re

Re

~ exp(r+iI)t

t

NUMERICAL RESULTS:NUMERICAL RESULTS:

radial coordinateax

ial c

oord

inat

e

-1

0

1

Azimuthal flow

Meridionalflow

c< 0 (wall direction)

c= 0

c> 0 (RMF direction)

Ta = 1.5×104

RMF

side

-wal

l bo

unda

ry la

yer

instability may occur at high RMF

0)()( 222

r

rV

r

w

R

r

w-RMF

more stable flow

0)( 2

r

Conclusions

Still stable flow

0)( 2

r

unstable flow

0)( 2

r

Conclusions● Concurrent action of RMF and mechanical counter-rotation on the

instability onset in cylindrical container with aspect ratio H/R=2 observed experimentally and numerically

● Strong counter-rotation of the container stabilizes the flow driven by RMF and changes the direction of the meridional circulation

● Weak counter-rotation of the container (when the RMF driven rotation is comparable to the rotation of the container) destabilize the flow. Concentration of the differential swirl occurs near the axis and Rayleigh stability criterion violated in this area.

● Regime with rapid instability can be used in applications when additional stirring of the melt is required e.g casting of metals etc.