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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
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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
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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
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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
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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
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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
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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
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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)
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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
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Ta c
R e
NUMERICAL RESULTS:NUMERICAL RESULTS:
Calculation with SMF
Experiment
Flow reversal Ta values
Tac (Re) calc.
Re3/2
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NUMERICAL RESULTS:NUMERICAL RESULTS:
r
Ta/105
Re
Re
Re
~ exp(r+iI)t
t
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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
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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
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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.