18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
s13-1
Characteristisation of a recirculating flow using ultrasonic Doppler
velocimetry
Xiaodong Wang, Yves Fautrelle & Jacqueline Etay
EPM/SIMAP/CNRS/INPG/ENSEEG laboratory,
B75, 38402 Saint Martin d?H?res cedex, France
Corresponding author: Xiaodong Wang, E-mail: xiaodong.wang@hmg.inpg.fr
Abstract:
We deal with a metallic GaInSn flow driven by a traveling magnetic field. The flow field in a
parallelipedic box was measured by ultrasonic Doppler velocimetry. The mean velocity profiles and
fluctuating behavior are obtained. The mean velocity is influenced by the pole pitch of the linear motor.
The magnetic field distribution is measured and the Lorentz force is analyzed. The turbulent character of
the flow is confirmed by discrete Fourier analysis of the registered signal, the flow exhibits continuous
frequency spectrum.
R?sume:
Nous traitons un ?coulement m?tallique de GaInSn dans une bo?te parallelipedic mise en mouvement par
un champ magn?tique glissant. Le champ de vitesse a ?t? mesur? par v?locim?trie Doppler ultrasonique.
Les profils de vitesse moyenne et le comportement de fluctuation sont obtenus. La vitesse moyenne est
influenc?e par le pas polaire du moteur lin?aire. La distribution de champ magn?tique est mesur?e et la
force de Lorentz r?sultante est calcul?e. Le caract?re turbulent de l'?coulement est confirm? par l'analyse
de Fourier discr?te du signal enregistr?. L??coulement se pr?sente un spectre fr?quence continu.
Key-words:
traveling magnetic field; liquid metal; Ultrasonic Doppler velocimetry
Mot-cl?s:
champ magn?tique glissant; m?tal liquide ; Doppler Ultrasonique Velocimetry
1 Introduction
The application of electromagnetically driven convection on liquid metal, the so-called
electromagnetic stirring, has been widely used in metallurgy. Electromagnetic stirring may be
generated by two kinds of devices, namely the single?phase or polyphase inductors. Concerning
polyphase inductors, they can generate either traveling or rotating magnetic fields.
Electromagnetic stirring is used for example in ladle metallurgy for liquid metal refining to
promote transfer reactions through the interface between a liquid metal and a covering slag or
molten salt (Fautrelle Y., Perrier D. & Etay J. 2003) . A widespread application also concerns
the continuous casting of steel where electromagnetic stirring, rotating or traveling magnetic
fields were used either near the mold or in the mid-part of the liquid sump. One of the general
goals of the application of stirring is to act on the solidification of the liquid metal.
We present hereafter measurements of the flow field in a rectangular liquid metal pool
submitted to a linear motor. We are interested in the turbulent flow driven by the traveling
magnetic field.
Flow measurements are carried out by using of UDV-Ultrasonic Doppler Velocimetry
(DOP2000), which is a very efficient tool to measure velocity in opaque fluid such liquid metals
as mercury (Takeda Y. 1987), gallium (Brito D. & Nataf H.C. 2001) or sodium (Eckert S. &
Gerbeth G. 2002). We will introduce the experiment configuration, the description of the
velocimetry method, and then the characteristics of flow are analyzed.
18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
s13-2
2 Experimental set-up and measurement procedure
2.1 Experimental set-up
The experimental set-up is presented on FIG.1. A liquid metal of 67wt.%Ga-20.5wt.%Sn-
12.5wt.%In is contained in a transparent rectangular cavity of W=1cm in width, L=10cm in
length, and H=6cm in height. The origin of the coordinates is located at the left-bottom corner
within the meridian plane. The physical properties of the alloy are given in table 1.
The Linear Induction Motor (LIM) used to generate a traveling magnetic field is a
three-phase linear motor (LMS13, HIWIN?), which is installed at 5mm under the bottom of the
cavity. The nominal electric current used in the stirrer is I =3, 4, 5A (r.m.s. value) at a frequency
of 50Hz. 22 yx BBB +=
r
in different z positions have been measured in the absence of
metallic alloy as illustrated in FIG.2. The magnetic field distribution closed to the bottom of the
liquid bulk is strongly affected by the magnetic poles positions, as z increases, the influence
becomes weak. The horizontal position of the liquid bulk also noted in FIG.2.
FIG.1 ? Schematic view of the experimental arrangement
(a) Front view; (b) transverse view
L=10cm
17cm
Vx
(a)
Liquid metal
Vortex
Container
Linear motor
Transducer
o
x
y
z
y
z
o
(b)
Transducer
Lm=1cm
W=1cm
H=6cm
0
20
40
60
80
100
-40 -20 0 20 40 60 80 100 120 140
x (mm)
B
(m
T)
z=0mm
z=10mm
z=20mm
z=30mm
18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
s13-3
FIG.2 ? Measured distribution of the magnetic field amplitude Br in the z-direction, the inductor with
six magnetic poles and the location of the liquid bulk are illustrated. I=4A (r.m.s. values).
Table1. - Physical properties of GaInSn alloy
Properties and symbols Values and Units
Melting point, Tm 10.5 ?C
Density, ? 6.4?103 kg/m3
Electrical conductivity, ? 3.4 ?106 (??m)-1
Kinematic viscosity, ? 3.4?10-7 m2/s
Sound speed, c 2860 m/s
Surface tension, ? 0.533 N?m-1
2.2 Description of the velocimetry method
The ultrasonic transducer is placed in one of the sidewalls in the horizontal direction,
which is a 4-MHz probe, 8mm in diameter. According to relationship between wavelength and
frequency cf =? , c being the sound speed in the liquid metal, we deduce that the wavelength
is 0.715mm. Then, the spatial physical resolution in the shooting direction reaches 2?0.715 =
1.43mm in the liquid metal, each profile consists of 64 points, thus the total length of
investigation is mm52.916443.1 =? , therefore a measuring blind zone exists near the wall in
the presence of the transducer. We adjust the parameters of Doppler apparatus, namely the pulse
repetition frequency (PRF), the cycles, the profiles, and the emissions per profiles to obtain
optimal velocity signal. PRF is an essential parameter, the time tPRF between two emissions not
only determines the length of the profile, but also controls the velocity resolution (Brito D. &
Nataf H.C. 2001). Only the velocity component parallel to the ultrasonic beam is accessible in
present study. The velocity component in the x-direction Vx is measured and analyzed. Note that
small heterogeneities (i.e. oxide particles) in the bulk liquid are necessary in order to obtain
significant echoes.
Table 2. lists the main control parameters. The Vx velocity component is measured along a
horizontal line (y = 0, z = 10mm). On the other hand, we have observed that the fringe effect
exists near the left vertical wall in which ultrasonic transducer is installed. It causes a
perturbation near the wall but does not affect significantly the measured velocity value in the
centre part.
Table 2. - Ultrasonic Doppler velocimetry parameters for liquid metal GaInSn
Physical properties values
Wavelength 0.715 mm
Pulse cycles 2
tPRF 728 ?s
points per profile 64
Total measurement time 64.7s
We can estimate the turbulence kinetic energy from the root-mean-square velocity Then if
we assume that the turbulence kinetic energy distribution k is isotropic in space (the latter
assumption may strongly overestimate k), its order of magnitude is given by
2~
2
3
xVk ? (1)
18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
s13-4
Fast Fourier Transform is used to analyze the frequency content of the signals. Fourier
spectrum indicates the fluid flow characters (turbulence or not). Since we obtain the velocities
values in discrete data, the velocity value are adopted among each interval time mst 7.64=? ,
the DFT (Discrete Fourier Transform) method was adapted by mean of software
MATLAB7.0?.
3 The flow analysis
3.1 Lorentz force generated by a single linear inductor
Firstly, let us estimate the electromagnetic force amplitude in the present study. To that
purpose, we consider the similar case of an ideal linear stirrer (Dubke M. & Tacke K.H. 1988).
In case of the inductor being considered as infinitely long and neglecting the back influence of
the motion on the current density, the x-component of the electromagnetic force density is such
that:
( )( )xteFF zx ??? 22cos120 ?+= ? With 200 2 BF ???= , ( )2????? +?= ie (2)
In our case, the liquid metal flow is limited in a narrow domain, which increase ? so much,
therefore xF decrease significantly than the case in infinite liquid bulk width; ? is the wavelength
of the magnetic field, due to mm322 =? , thus 11962/2 ?== m?pi? , ? magnetic
permeability, ? electric conductivity, efpi? 2= , fe magnetic frequency, B0 applied magnetic
field strength, i unit of complex number 12 =i . The ratio 017.02 =???? is much less than
unit, therefore the value of the skin depth ? is imposed mainly by the pole pitch rather than by
the frequency. The skin depth is
mm1.511 =?= ??? (3)
When the current I is equal to 4.0A for example, then we have B0 = 25 mT (cf. FIG.2), fe =
50Hz, and thus the order of the corresponding amplitude of the electromagnetical force is
about: 332 /)10~O(10~ mNFx .
3.2 Mean flow behavior
FIG.3(a) illustrated the time evolution of the instantaneous velocity for I = 4A at P(0.05, 0,
0.1), The instantaneous velocities vary in a large range from -18 to -35 mm/s, the mean velocity
xV is -26.4 mm/s. On FIG.3(b), xV profile one are plotted for three values of the inducting
electrical current I = 3, 4 and 5A. These three profiles are similar in shape, when intensity
magnitude increases as I increase.
From the shape of the profile we may say that the mean flow exhibits a large central vortex
associated to two smaller and weaker vortices located near the bottom corners. It is not clear if
the vortices are driven by viscosity or are generated by electromagnetic force gradient
associated to the geometry of the polar pitch of the inductor.
18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
s13-5
FIG.3 - Velocity behaviors: (a) instantaneous velocity at P(0.05, 0, 0.1) I = 4A; (b) longitudinal mean
velocity profiles for various magnetic field intensities.
3.3 Turbulence characteristics
We plotted dimensionless quantity xx VV /~ as illustrated in FIG.4(a), the collected profiles
explicit that the fluctuation intensity is proportional to the mean xV , and they have xx VV %10~ ? .
Duo to fringe effect, the fluctuation near the right sidewall seems stronger, but we have not
observed the ideal symmetrical situation on the left side, it may cause the registered signal of
UDV.
The turbulence kinetic energy k computed according to Equ.(1~3). DFT analysis indicates
that the energy spectrum of Vx is continuous as illustrated in FIG.4(b), which confirms that the
flow is turbulent. This point can also be proved by high value of Reynolds number in the
present case:
4590Re == ?HVxx (4)
Where the typical order of the x-component velocity xV is chosen as 25 mm/s.
Regarding to the theoretical model of Kolmogorov spectrum for sufficiently
developed turbulence flow, as in Ref.(Felten F. & Fautrelle Y. 2004), the curve of the
typical frequency energy spectrum E(f) yields a short plateau, and decays with a slope
(-5/3) as the function of the bulk liquid frequency in inertial zone. In a like-manner, here,
a mono-directional spectrum of the velocity fluctuation is calculated with 15 experiment
series (namely 15 events). As illustrated in figure6, E(f) curve of the present calculating
result. There exist a sharply decays with approximately exponent slope (-1.9). The value
of this slope is a little bigger than that (-5/3) in the Ref.[19], that may cause of
mono-spectrum, the turbulence did not fully developed in other direction. After that, it
reaches to the viscosity zone; we observed an energy peak near one hertz, which is
corresponding to the flow turnover time. The order of flow turnover time in the cavity
can be estimated by the ratio between the typical length of the box and the typical
velocity of the flow:
(a)
-40
-35
-30
-25
-20
-15
-10
-5
0
0 10 20 30 40 50 60 70
time (s)
ve
loc
ity
(m
m/
s)
smmVx /4.26?=
(b)
P
-40
-35
-30
-25
-20
-15
-10
-5
0
0 10 20 30 40 50 60 70 80 90 100
x (mm)
me
an
ve
loc
ity
(m
m/
s)
I=3A
I=4A
I=5A
18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
s13-6
??
?
?
???
?=
xV
LO? (5)
The energy peak shows this turnover time is about 1.2 s.
FIG.4-Turbulent flow behaviors: (a) dimensionless quantity xx VV /
~
comparison for various
electromagnetical intensities; (b) mono-directional spectrum of the velocity fluctuation.
4. Conclusion
The flow of the metallic liquid GaInSn alloy submitted to a traveling magnetic field was
studied. The x-component velocities measured by mean of UDV, and its mean values and
fluctuations are analyzed. The magnetic force is calculated base on the electromagnetic
parameters, and its order is estimated. Analysis of Discrete Fourier transform indicated the
energy spectrum of this type flow yields turbulent characteristics. Such flow is used to control
solidification for instance by [Zaidat K and Moreau R].
References:
Brito D., Nataf H.C. 2001 Ultrasonic Doppler velocimetry in liquid gallium. Experiments in
Fluids. 31, 653-663
Dubke M., Tacke K.H and Schwerdtfeger K. 1988 Flow field in electro, magnetic stirring of
rectangular strands with linear inductors: part 1. Theory and experiments with cold models.
Met. Trans. B 19B, 581- 593
Eckert S. Gerbeth G. 2002 Velocity measurements in liquid sodium by mean of ultrasonic
Doppler velocimetry. Experiment in Fluids. Vol.32 No.5, 542-546
Fautrelle Y., Perrier D. & Etay J. 2003 Free surface controlled by magnetic fields. ISIJ
international. Vol.43 No.6, 801-806
Felten F., Fautrelle Y. 2004 Numerical modelling of electromagnetically-driven turbulent flows
using LES methods. Applied mathematical modelling. 28, 15-27
Takeda Y. 1987 Development of velocity profile of mercury flow by ultrasonic Doppler shift
method. Nuclear Technique. 79, 120-284
Zaidat K. 2005 The influence of the travelling magnetic field on the directional solidification of
the binary alloy. Ph.D. thesis, EPM, France, 43-67
(a) (b)
0
0,2
0,4
0,6
0,8
1
1,2
0,01 0,1 1 10
liquid metal frequency fl(Hz)
tur
bu
len
ce
en
er
gy
sp
ec
tru
m
E(
f)
-0,21
-0,18
-0,15
-0,12
-0,09
-0,06
-0,03
0,00
0 10 20 30 40 50 60 70 80 90 100
x (mm)
m
I=3A
I=4A
I=5A
xx VV /
~
Turnover frequency