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Flow Measurement and Instrumentation 19 (2008) 131137

Flow Measurementand Instrumentation

www.elsevier.com/locate/flowmeasinst

Ultrasonic visualization of thermal convective motion in a liquidgallium layer

Yuji Tasakaa,, Yasushi Takeda a, Takatoshi Yanagisawab

a Graduate School of Engineering, Hokkaido University, Kita-13, Nishi-8, Sapporo 060-8628, JapanbJapan Agency for Marine-Earth Science and Technology, Yokosuka 237-0061, Japan

Received 17 November 2006; received in revised form 12 January 2007; accepted 21 June 2007

Abstract

For low Prandtl number fluids including liquid metals, optical methods cannot be utilized to visualize convective motion, and in this paper the

velocity profile in a liquid gallium layer was measured using ultrasonic velocity profiling, UVP. The applicability of the measurement system was

confirmed with the rotating flow of liquid gallium and in the natural convection appearing in a glycerol solution layer. The vessel for the liquid

gallium layer was optimized for the acoustic properties of liquid gallium. The measured velocity profile shows a cell like convective motion. The

spatio-temporal behavior of large-scale convective motion in turbulent convection was observed as a temporal variation of the velocity profile with

two kinds of periodic fluctuations of the convection cell.c 2007 Elsevier Ltd. All rights reserved.

Keywords: Ultrasonic wave; Thermal convection; Liquid metal; Low Prandtl number; Visualization

1. Introduction

Thermal convection induced by a vertical temperature

gradient in a shallow fluid layer, RayleighBenard convection,

is a basic problem in fluid dynamics, thermal engineering, and

geophysics, but it has not been studied for low Prandtl number

(Pr) fluids. A phase diagram constructed by Krishnamurti [1]

suggests that the convection of low Pr fluids easily changes

from a two-dimensional steady state to a turbulent state

via a three-dimensional state and a time dependent state.

From theoretical considerations, Busse [2] suggested that

RayleighBenard convection is time-dependent at low Rayleigh

numbers, and temperature measurements of the convection ofmercury support this. Rossby [3] reported that the convection of

mercury shows periodic variations in the Nusselt number even

in the transition state of the phase diagram. Yamanaka et al. [4]

showed that the variation is induced by periodic fluctuations

Corresponding address: Division of Energy and Environmental Systems,Graduate School of Engineering, Hokkaido University, Kita-13, Nishi-8Sapporo 060-8628, Japan. Tel.: +81 11 706 6373; fax: +81 11 706 7889.

E-mail address: [email protected](Y. Tasaka).

in temperature in the fluid layer in an experimental study

using liquid gallium. This kind of periodic phenomenon

could be related to large-scale convective motion in the fluid

layer. However, these studies do not clarify the type of

convective motion because the studies only made temperature

measurements at a single point. Almost all studies of low Pr

convection relied on such temperature measurements because

optical visualization cannot be used to observe the convective

motion of opaque fluids such as liquid metals. This study

attempted to visualize the convective motion of low Pr fluid by

measuring the velocity profile in the fluid layer using ultrasonic

velocity profiling (UVP).

Initially, UVP was developed for medical purposes, and ithas become a powerful tool in experimental fluid dynamics [5,

6] and fluid engineering because it can measure instantaneous

velocity profiles, and can be applied to fluid flow of opaque

liquids including liquid metal [710]. Results with UVP are

superior to optical methods of velocity measurement such as

PIV (Particle Image Velocimetry). Thanks to the transmission

properties of ultrasonic waves UVP has been applied in

industry, for instance in flow metering in large pipes [11] where

the container wall is opaque, and also in quality control in food

processing [12,13] where fluids are generally opaque. Further,

0955-5986/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.flowmeasinst.2007.06.003
http://www.elsevier.com/locate/flowmeasinstmailto:[email protected]://dx.doi.org/10.1016/j.flowmeasinst.2007.06.003http://dx.doi.org/10.1016/j.flowmeasinst.2007.06.003mailto:[email protected]://www.elsevier.com/locate/flowmeasinst

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132 Y. Tasaka et al. / Flow Measurement and Instrumentation 19 (2008) 131137

advances with this technique using ultrasonic waves have been

developed in recent years to extend the range of measurable

values [14,15].

The applicability of the measurement system using UVP to

the problem investigated here was established with the rotating

flow of liquid gallium and thermal convection of a glycerol

solution. We measured instantaneous velocity profiles in arectangular vessel filled with liquid gallium (Pr of around 0.03)

heated from below and cooled from above. Comparisons of

temporally averaged velocity profiles measured at higher or

lower positions in the vessel showed that the convective motion

is similar to two-dimensional cells at extremely low Rayleigh

numbers but that the angular velocity of rotation is not uniform

like in two-dimensional cells. A spatio-temporal velocity map

is used to represent the two different temporal behaviors of

the cell motion; one is a meandering motion maintaining the

size of the cell and the other is a repetition of expansions and

contraction of the cell.

2. Experimental

2.1. Measurement technique

Ultrasonic velocity profiling (UVP) utilizes the Doppler

shift frequency and ultrasonic (US) echography to determine

an instantaneous velocity profile. Fig. 1 shows (a) the basic

configuration of the experimental setup, (b) the appearance of

the US signals, and (c) a measured velocity profile in the UVP

measurement. The US waves emitted from a US transducer

propagates in the fluid and a part of the waves is reflected by the

particles of the fluid. If there is a sufficient number of particles

reflecting waves in the fluid, the US echo returns from a wide

range of directions on the line of propagation of the US waves.

Assuming that particles move with the fluid, an echo contains

Doppler shift frequency, fD, velocity information of the fluid

flow. Therefore, the instantaneous velocity at a position in the

direction of propagation is determined by

u(, t) = c fD/2 f0, (1)

where c and f0 are the speed of sound in the fluid and the basic

frequency of the emitted US waves. The suffix in the equation,

, shows that the measured velocity is a velocity component of

the direction. The position on the axis is determined by the

time of flight of the US waves as

= c/2. (2)

Repeated wave bursts and reception of the US echos are

required to determine the instantaneous Doppler shift frequency

fD(t) accurately. The frequency of repetition of waves, fprf,

determines the maximum length where it is possible to measure

a velocity profile, , as

= c/(2 fprf). (3)

Further details of the UVP principles are described in Ref. [16].

Fig. 1. Schematicoutline of theUVP measurements; (a) thebasicconfiguration

of the experimental setup, (b) the appearance of the US signals, and (c) a

measured velocity profile in the UVP measurement.

Table 1

Physical properties of liquid gallium [17]

Symbol Unit Value

Density kg/m3 6.095

Bulk modulus K1 1.26 104

Thermal diffusivity m2/s 1.18 105

Kinematic viscosity m2/s 3.22 107

Prandtl number Pr 0.025

2.2. Liquid gallium

Liquid gallium was used as the low Pr fluid in this study.

An advantage of using liquid gallium as the working fluid is its

safety. It also has a higher vapor pressure than mercury and doesnot react with water like sodium. Table 1 shows the physical

properties of liquid gallium given by Brito et al. [17].

The UVP (ultrasonic velocity profiling) measurements

require the suspension of US (ultrasonic) wave reflecting

particles in the fluid. Here, a fine powder of ZrB2 was

used; particles are 50 m in diameter and have a density of

6.17 kg/m3. This kind of powder has also been used in other

work of UVP measurements in liquid gallium and has provided

good results [18]. Because of the large surface tension of liquid

gallium, mixing the ZrB2 particles into the liquid gallium is

more difficult than mixing them into water. To reduce the

surface tension coefficient, to enable mixing of the particles,

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Y. Tasaka et al. / Flow Measurement and Instrumentation 19 (2008) 131137 133

Fig. 2. Temperature dependence of the speed of sound in liquid gallium

determined by the time of flight method.

during the mixing, the liquid gallium was kept at a high

temperature, around 500 C, for 30 min in a furnace.As expressed in Eqs. (1) and (2), the speed of sound in

a medium is directly related to the measurement accuracy of

both velocity and position with the UVP. Further, the mean

temperature of the system depends on the Rayleigh number

in the system in this study. Only the speed of sound in liquid

gallium at 30 C, close to the melting point of the gallium

(29.8 C), has been reported [17,19]. The speed of sound

at several temperatures from 30 to 50 C was measured by

the time of flight method and the temperature dependence

of the speed of sound was determined. Fig. 2 shows that

there is a gradual decrease in the speed of sound c with

respect to temperature. By a least square estimate, the following

relationship is obtained

c(T) = 0.616T+ 2891.9. (4)

Testing liquid gallium was well deoxidized by 10% hydrochlo-

ric acidethanol solution before the measurement, and hence

there is little influence of gallium oxide on the relational

expression.We performed trial measurements of liquid gallium flow

in a simple configuration to confirm the usefulness of the

UVP system for measurements with liquid gallium. Fig. 3 is a

photograph of the experimental apparatus with a supplemental

illustration; the liquid gallium is contained in a 87 mm

inner diameter glass beaker, covered with a 10% hydrochloric

acidethanol solution to prevent oxidation. A magnetic stirrer, a

0.2 T magnet, is beneath the beaker and the liquid gallium layer

is driven by the Lorentz force induced by the rotating magnetic

field of the magnetic stirrer. It was expected that the flow inside

the beaker becomes like a Rankine vortex, which consists of

a rigid vortex and a free vortex. This system was originally

developed for deoxidization of liquid gallium.Ultrasonic (US) waves, with a 4 MHz basic frequency and

a 5 mm effective diameter, is emitted from the US transducer

outside of the beaker. The velocity profile of the liquid gallium

layer along the line of propagation of the US waves, the axis,

is obtained by the UVP on a UVP monitor model Duo (Met-

Flow S. A. [20]). The signal filter in this system is optimized

by Met-Flow for measurements of slow flows. Fig. 4 shows

an instantaneous velocity profile, where the horizontal and

Fig. 3. Photograph of the experimental apparatus for the trial measurements

and supplemental illustration; a rotating magnet is beneath the bottom of the

vessel to induce moves in the liquid gallium in the vessel.

Fig. 4. Instantaneous velocity profile in the liquid gallium layer measured by

UVP, where represents the distance from the ultrasonic transducer and u is

the velocity component to the axis.

the vertical axes are the distance from the transducer and the

velocity. The velocity is largest at the center and decreases

toward the wall of the vessel. The measurement axis is located

at a small distance from the center line of the beaker, and the

profile is consistent with expectations. The estimated rotating

speed near the center is 1 rps at the surface of the liquid gallium

layer. Assuming that the flow near the center confirms to rigid

body rotation, the maximum velocity on the measured line is 90

to 100 mm/

s, and would suggest that the measured velocity isquantitatively accurate.

2.3. Experimental setup

To determine the convective motion in a liquid gallium layer,

a rectangular vessel was constructed. The vessel for the liquid

gallium consists of three parts, the glass side walls, top and

bottom copper plates, 25 mm thick, with grooves for flowing

water to control the temperature. Fig. 5 shows a schematic

outline of the vessel, where the top and the bottom figures

show horizontal and vertical cross sections of the vessel. It

is 50 mm high (L), 200 mm wide (=4L) and 50 mm deep

(=L) with this shape, fluid layer flows would be restricted to a

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(a) Horizontal cross section. (b) Side view and vertical cross section.

Fig. 5. Schematic outline of the experimental apparatus holding the liquid gallium, units in mm.

Table 2

Acoustic properties of liquid gallium and Pyrex glass, where the reflection coefficient of liquid gallium is defined as Rg = (ZgZ)/(Zg+Z), with Zg the acoustic

impedance of liquid gallium

Symbol Unit Liquid gallium Pyrex glass Plexiglas

Acoustic impedance Z 106 kg/(m2 s) 17.4 13.1 3.2

Sound speed c m/s 2860 5640 2730(longitudinal wave)

Reflection coefficient Rg 0.14 0.69

(for liquid gallium)

convective flow pattern. It was expected that a cell like pattern

would appear with an axis of rotation parallel to the y axis.

The side walls of the vessel are made of Pyrex glass which

can be wetted with liquid gallium. The acoustic impedance of

Pyrex glass is very close to that of liquid gallium as shown

in Table 2, and hence the ultrasonic (US) waves pass through

the side wall, rather than the other parts of the vessel (the

reflection coefficient for US waves passing through the Pyrex

grass to the liquid gallium, Rg, is one fifth lower than that for

Plexiglas, as shown in the table). The top and the bottom plates

are made of copper and are held in place with 15 mm thick

acrylic plates. There is a circular 12 mm diameter channel at

the top of the copper plates. The temperature of the flowing

water in the channels was controlled by thermostatic baths. The

water flows kept the surface temperatures of the top and the

bottom plates constant. The cooling temperature T1 (top plate)

was kept a higher temperature than the melting temperature of

gallium (29.8 C), at 32 C, and the Rayleigh numbers were

varied in the R = 200800Rc range by changing the heating

temperature T2. The Rayleigh number R is defined as

R =g(T2 T1)L

3

, (5)

where g, , , and are the gravity acceleration, bulk modulus,

thermal diffusivity, and kinematic viscosity of liquid gallium

respectively; Rc is the critical Rayleigh number of Rayleigh

Benard convection in a shallow fluid layer, Rc = 1707.7 [21].

Liquid gallium oxidizes easily, and the air in the vessel was

removed using a vacuum pump through two 10 mm diameter

holes drilled in the top plate with a buffer tank for liquid gallium

which was poured into the vessel with pressurized argon gas

through the 10 mm diameter hole (Fig. 6). Gallium pouring in

to the vessel was well deoxidized. The US wave transducer was

Fig. 6. Schematic outline of a filling process of liquid gallium to the vessel: (1)

air in the vessel was removed using a vacuum pump through two holes drilled

in the top plate and a buffer tank (2) liquid gallium (Ga) was poured into the

vessel with pressurized by argon gas (Ar).

placed at one end of the vessel and US wave bursts emitted

by the transducer propagated in the gallium layer parallel to

the x direction. Silicon oil was used as a coupler between thetransducer and the Pyrex glass plate to prevent existing air layer

in the very thin gap. The US wave bursts were at a 4 MHz basic

frequency and a 5 mm effective diameter, resulting in a spatial

resolution of the measured velocity profile in the liquid gallium

layer, , of around 1.4 mm.

3. Results and discussions

3.1. Flow patterns

Liquid gallium is opaque and it is difficult to make direct

comparisons between a measured velocity profile and the actual

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Fig. 7. (a) Temporally averaged velocity profile of the convective motion in

glycerol solution, and (b) Sketch of observed convective motion in glycerol

solution. ux

represents the x axis velocity component of the convective motion.

convective pattern. To establish whether the profile and the

pattern are similar, UVP measurements and observations of the

convective motion were performed in a transparent glycerol

solution.

Fig. 7 shows the results of the measurements for a 28 wt%

glycerol solution (Pr 18). The fluid layer is 200 mm wide,

30 mm deep, and 20 mm high. The side walls are Plexiglas,

and the top and the bottom plates are aluminum and copper

respectively. Flowing, temperature controlled water maintained

a constant temperature at the top and bottom boundary of

the fluid layer. The Rayleigh number, determined from the

temperature difference between the top and bottom boundaries,

is around 800Rc. 80 m nylon powder was used as tracer

particle and its density is quite similar to the solution. Fig. 7(a)

shows a temporally averaged velocity profile obtained from

1024 instantaneous profiles, where ux represents the velocity

component on the x axis. Fig. 7(b) shows the location of the

transducer and illustrates the observed convection pattern. As

shown in the figure, the formed convection pattern is a quasi-

two-dimensional cell, with axis of rotation perpendicular to the

measurement direction. The convective motion is unsteady and

the size of the cells change temporally; however, the width of

cells do not become larger than the height of the fluid layer. The

measured velocities vary in the measured direction and attainalternately positive and negative values. In comparison with

the observed convection pattern, the variations between positive

and negative values correspond to the motion in individual cells.

At a set Rayleigh number, the convective motion is dominated

by separated thermal boundary layers at the top and the bottom

boundaries, and hence the rotation of a cell may be seen to be

similar to a rigid vortex. A velocity profile would show a flat

distribution without boundaries between cells when the rigid

vortex is measured by the UVP along a line. The obtained

velocity, however, includes both flat and sinusoidal profiles;

suggesting that some movement in some of the cells differs

from that of a rigid vortex.

Fig. 8. Temporally averaged velocity profile of the convective motion in liquid

gallium layer determined at (a) a low and (b) a high position. ux represents the

x axis velocity component of the convective motion.

Fig. 8 shows temporally averaged velocity profiles measured

at a high and a low position in the fluid layer of liquid gallium,

the horizontal axis x represents the distance from the ultrasonic

(US) transducer. The sampling period of the profile is 80 ms and

1024 profiles were used in the averaging, the spatial resolution

on the x axis is 1.44 mm. The Rayleigh number is 770Rc where

convective motion is turbulent. The profiles are not smooth

but somewhat broken. As mentioned in the next section, we

consider that it may be due to rather imperfection of seeding

than turbulent motion.Low in the fluid layer (Fig. 8(a)), the velocity profile has two

pairs of variations in the velocity between positive and negative

values in the fluid layer. The measured velocity is from 10

to 10 mm/s. At the higher position (Fig. 8(b)), the velocity

oscillates and the measured velocity profile is approximately

symmetrical to that measured at the lower position. The range

of velocities at the higher position is similar to that determined

at the lower position. A comparison with the results in the

glycerol solution (Fig. 7) suggests that there are two pairs of

cells with axis of rotation parallel to the y axis. The motion of

the cells is quite different for a rigid vortex because the velocitydoes not show a flat profile but has mainly a sinusoidal profile.

At the set Rayleigh number, 770Rc, the convection is

turbulent according to the flow regime diagram [1] and hence

the convection displays multi-scale motion. The obtained

velocity profile shows the large-scale motion of the cells, and

it is not possible to accurately determine small-scale motion

using the current UVP system because of the low speeds of the

convective motion.

3.2. Spatio-temporal behavior

The UVP (ultrasonic velocity profiling) can measure an

instantaneous velocity profile, and this makes it possible

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(a) R = 428Rc.

(b) R = 770Rc.

Fig. 9. Spatio-temporal velocity profile. Color represents velocity, and counter clockwise rotation of a cell is represented by yellow and clockwise rotation by green.

to investigate the spatio-temporal variation in large-scale

convective motion. Fig. 9 shows the spatio-temporal velocitydistributions measured at R = 428Rc and 770Rc, where the

horizontal and vertical axes are the time and position measuredlow in the fluid layer, the colors represent velocity values. Black

points in the figures represent data voids caused by a lack ofparticles to reflect the ultrasonic waves and the shape of the

particles. The ZrB2 powder is shaped like powder grains and itsirregular surface scattered the ultrasonic waves. In comparison

with the temporally average velocity profile (Fig. 8), i t ispossible to distinguish two cells in these figures.

At R = 428Rc, there are four convection cells in the

fluid layer (Fig. 9(a)); counter clockwise rotation of a cell isrepresented by yellow and clockwise rotation by green. The

cells sway as expressed by the movement of the boundarybetween yellow and green. Small-scale velocity fluctuations are

superimposed on the large-scale fluctuation, but it is difficultto evaluate the small-scale phenomena because the small-

scale fluctuations are only a few times the velocity resolutionof the measurements, O(1 mm/s), and are indistinguishable

from the noise in the signals. The band enclosed by thebroken red line shows repeated expansion and contraction of

the cell maintaining the position on the x axis as shown inFig. 10(a). This movement is very slow with a period of

approximately 60 s (corresponding to 0.017 Hz of frequency).At the higher Rayleigh number, R = 770Rc, the motion

of the convection cell is different (Fig. 9(b)). There arestill four cells of the same size as with the lower Rayleigh

number, but the cell enclosed by the broken red line movesperiodically on the x axis without changes in size as shown in

Fig. 10(b) different from the convective motion in Fig. 9(a).

Fig. 10. Schematic illustration of motion in a convection cell; (a) repeated

expansion and contraction of the cell maintaining the position on the x axis,

(b) meandering motion with maintaining the size of the cell.

Further, the motion of the neighbouring cell corresponds

to that observed at the lower Rayleigh number. Convective

motion is faster than at the lower Rayleigh number and

the frequency determined by Fourier analysis is 0.059 Hz.

Simultaneous measurements of temperature fluctuations by a

thermistor shows the corresponding frequency. Selection of the

fluctuation pattern shown in Fig. 9 is alternative; it cannot be

determined which fluctuation pattern is chosen at each Rayleigh

number because the selections of the fluctuation pattern and

the number of the cells are strongly dependent on initial

condition and boundary condition, e.g. initial temperature

distribution in liquid gallium, temperature difference between

the boundaries and the gallium, etc. This is typical in many

instability phenomena. It is confirmed that there are three states

of convection with different number of cells; two, three or four.

Generally, it is difficult to control the number of cells.

4. Concluding remarks

Ultrasonic velocity profiling, UVP, was used to investigate

large-scale convective motion of a low Prandtl number fluid.

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The applicability of the measurement technique was confirmed

in rotating liquid gallium and in the thermal convection of

a glycerol solution. The vessel for the liquid gallium was

designed to solve problems of wetting and permeability of

ultrasonic waves, and the velocity profile measured at lower

and higher positions in the fluid layer showed convection

cells in the liquid gallium layer. The spatio-temporal velocityfield measured by the UVP expressed different motions of

convection cells, including repeated expansion and contraction

and periodical meandering motion. The large-scale motion has

been discussed with temperature variations at a point in the

fluid layer [3]; however, the spatio-temporal behavior has not

previously been discussed for flow patterns and is shown in this

study for the first time. The measurement system here cannot

detect small-scale motion in the natural convection because of

a lack of data due to an insufficient number of tracer particles,

and the ZrB2 powder used in this study is shaped like powder

grains and not optimum for UVP measurements. It would be

possible to compensate for this by optimizing the shape of the

tracer particles.

Acknowledgement

This work is supported by Grant-Aided Research for Science

of the Japanese Ministry of Education & Science: No.18760116

and No.18204038. The authors express thanks for this support.

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