Pergamon

www.elsevier.com/locate/asr

Available online at www.sciencedirect.com

SCI8NCE DIR8CT.

doi: lO.l016/SO273-1177(03)00389-2

VARIOUS FLOW PATTERNS IN THE~OCAP&LARY CONVEC- IN HALF-ZONE LIQUID BRIDGE

OF HIGH PRANDTL NUMBER FLm

I. Ueno, S. Tanaka, and H. Kawamura

Tokyo University of Science, 2641 fanuzaki, No& Chiba 278-8510, JAPAN

ABSTRACT Various flow pattems induced by a th~ocapillarydriven convection in a halho& liquid bri%e of a high F’randtl number fluid (Pr = O(10)) far beyond the critical condition were investigated experimentally. After the onset of oscillatory convection, one can observe several types of flow patterns with increasing a temperature difference between the both end surfaces of the bridge. The flow patterns were categorized through flow visualization, measurement of surface temperature variation and reconstruction of the pseudo-phase space. Q 2003 COSPAR. Publislied by Elsevier Ltd. All rights reserved.

INTRODUCTION

Thermocapillarydriven convection arises if a non-uniform temperature distribution exists over a tiee surface. It takes place inevitably even in the space environment, where a buoyant force can be almost negligible. The space is expected to be an ideal environment to process a high-quality materials or crystals. In order to realize such new material p-sing in the space, however, one must understand the flow fields induced by the thermocapillary forces. Although it is found that the oscillation of the thermocapillary-driven flow in high-Prandtl number fluids exhibits a different behavior from that of a low-Prandtl number ones (Smith and Davis, 1983), both are induced by the coupling of the surface temperature variation and the internal velocity field. In the present study, the authors focused upon the flow fields of a high Pram&l number of O( 10).

In a half zone (HZ) model, a liquid bridge is sustained between cylindrical hot and cold rods, that is, the half of a floating zone (FZ) method is mimicked. The thermocapillary. flow develops from the hotter to colder end rods with a temperature difference AT. The intensity of the induced flow can be described by using the Marangoni number defined as Ma = (crrAT.H)/(pvtc) = (~~ATI-I)/(~~)(v/K) = Re,,Pr, where or is the tempera&m coefficient of the surface tension (= I&rBTI), H the height of the liquid bridge, p the density, v the kinematic viscosity, K the thermal dif&sivity. The Marangoni number can be expressed as the product of the thermocapillary Reynolds number Re, and Prandtl number Pr. Experimental and numerical studies have been widely conducted on this system, indicating that there exists a critical temperature difference between the rods. When the temperature difference exceeds the critical value AT,, or Ma > Ma, the flow field exhibits a transition from a two-dimensional steady flow to a three-dimensional time-dependent ‘oscillatory’ one. With further increasing AT a&r the onset of the oscillation, the flow presents standing- and traveling-type convections.

The transition and the structures of the oscillatory flow field have been investigated extensively in the last three decades. These works were mainly concerned with the critical point of the onset of the oscillation and with the flow structures a little beyond the criticality. Little information has been obtained regarding the flow fields far beyond the critical point except by Velten et al. (1991) and Frank and Schwabe (1997). They conducted a series of experiments with several kinds of test fluids of moderate to high Prandtl number and reported the chaotic behavior of the surface temperamre variation by use of two dimensional Poincare map. In the present study the flow fields induced by an imposed large tempemture di&rence were focused. Several kinds of new flow regimes after the well-known standing- and traveling-type oscillations were found and categorized through flow visualization and

Adv. Space Res. Vol. 32, No. 2, pp. 143-148.2003 0 2003 COSPAR. Published by Elsevier Ltd. All rights resCrved Printed in Great Britain 0273-l 177lS30.00 + 0.00

144 I. Ueno et al.

surface temperature measurement. Reconstruction of the pseudo-phase space from the time series of the surface temperature led a clear distinction of the flow fields at high Marangoni numbers.

EXPERIMENT

The test section is described in Figure 1. A liquid bridge of 24% silicone oil was formed between coaxial rods. The Prandtl number of the test fluid was 28.1 at 25 “C. The physical properties of the fluid is listed in Table 1. The whole apparatus was installed in a refrigerator to maintain the ambient temperature at about -20 ‘C in order to attain a large temperature difference while suppressing evaporation of fluid and its condensation over the cold rod. The end-surface diameter of the rods was 5.0 mm. The aspect ratio F (= H / R) was varied from 0.60 to 0.70, where H and R are the height and the radius of the bridge, respectively. The volume ratio of the bridge was kept constant and as close to unity as possible. In order to observe the flow field in the bridge through the top, the transparent top rod made of sapphii was employed. The bottom rod was made of ahnninum, with a chemically coated sidewall to prevent wetting by the liquid. A temperature difference AT between the end surfaces

CCD camera fortopvkw

Fig. 1. experimental apparatus.

Table 1. Physical properties of 2cSt silicone oil at 25 “C. Wlb6 2 * oxld laoQ7Jxld /%X103 [m’/s] $$& E’i] m/m] [N/m.K] [W

2.0 8.70 7.12 1.83 7.15 1.24

was imposed by heating the top rod. With this configuration there exists both thermocapillary and buoyancy forces. The relative importance oftwo different forces can be evaluated by considering the dynamic Bond number Bad = Gr/Re,, where Gr is the Grashof number defined as gf3ATH3/?, g the gravity and 8 the thermal expansion coefficient. In a typical case of I’ = 0.64 and AT = 50 K, the system has a value of Boa = o(lOV’). Thus buoyancy-driven flow is negligibly small in this configuration.

The flow was visualized by suspending Polystyrene particles of 20 pm in diameter in the fluid as tracers. The flow field was observed through two CCD cameras simultaneously at a frame rate of 30 fps; one i+om the top and the other from the side. A continuous light source illuminated the entire bridge; thus the captumd views presented the integrated fields.

A time series of the temperature slightly inside of the bridge was measured at mid-height with a Cr-Ar thermocouple (T/C) of 0.025 mm in diameter at a sampling rate of 50 Hz. Preliminary tests were carried out in order to determine most suitable T/C-junction position at which the fluid tempemture could be exactly measured while maintaining the effect upon the particle motion minimal. Hemafter, the measured temperamre at this location is called the surface tempemture. The temperature oscillation was measured after an elapsed time of 5 minutes to make sure that the flow field was fully developed for each designated AT condition.

It is noted that the temperature dependence of the fluid viscosity cannot be neglected in experiments with a large AT. The fluid viscosity is evahutted at the mean temperature T = (T+T$2 as the reference v with the use of an empirical correlation (Shin-Etau Chem. Co., Ltd., 1991).

RESULTS AND DISCUSSIONS

Through the observed particle motion and the Fourier spectrum of the surface temperature variation, the induced flow structures can be categorized into eight regimes (Rg. 1 to 8). A regime Rg. with a larger number basically corresponds to a flow structure emerging at a larger temperature difference at a fixed I. The flow regime was primarily determined by the &npe&me di%rence AT and the aspect ratio I. It further depended upon the increasing rate of the tempemture difference and its history. Some of these regimes did not appear depending upon the experimental procedures and conditions.

An overview of typical flow strucmre in each regime is presented in Figure 2. Flow fields with an azimuthal wave number m = 3 was observed at this aspect ratio (Ueno et al., 2000). Note that a modal strucmm with diierent

Themocapillary Convection in a Liquid Bridge 145

wave number emerges when the aspect ratio is varied (Ueno et al., 2003). Figure 2 contains three columns of illustrations; (1) the top view of the flow field, (2) the power spectrum ‘density (PSD) of the surface temperature variation, and (3) the reconstructed pseudo-phase space (PPS) from the surface temperanne. The mconsuuction of PPS was conducted f?om a single time series of the fully developed surface tempera&m by applying the time-delayed coordinates. The {dimensional coordinate is defined as vg(t) = (T(t), T(t+t), T(t+2r), .+., T(t+({-l)r)), where T(t) is the surface temperature at time t, and z the delay time. Note that the PPS illustrated in Figures 2-(3) was reconstructed ii-am the tempera&m fluctuation around an averaged value. The t@ectories shown in Figure 2 are reconstructed with a constant delay time of z = 0.1 s for Rg. 2 to Rg. 6 and with r = 0.06 s for Rg. 7 and Rg. 8. They consist of 2 000 points. Several delay times around 0.1 s were examined to con&m that the trajectories had the same shape irrespective of the delay times in a primary analysis.

Definition of Flow Regime Rg. 1: Steadv Flow

With a small AT, the flow field shows 2-D axisymmetric ‘steady’ flow (see Figures 2-(a)).,The tracer particles show only radial motion if observed Corn the top. They circle in the r-z plane. The surface temperature at a fixed point maintains at a constant value at a fixed AT. Thus, the trajectory in the PPS becomes a fixed point (not shown).

Ra. 2: Fulsatine Flow I After the onset of transition, the particles exhibit azimuthal motion in addition to their movement in the r-z

plane. The Marangoni number at the critical point is usually called as the critical Marangoni number Iv& With increasing Ma above M&, the convective field indicates modal structures in Rgs. 2 and 3. The boundary between Rgs. 1 and 2 has already been recognized well as the critical point of the flow transition from 2-D steady to 3-D time-dependent flows. The critical condition is discussed in Ueno et al. (2000).

In these regimes of Rgs. 2 and 3, a polygonal particlefke region is observed through the top end surf=; that is, tiny particles disperse in the region between the free surface and the particle-Ike polygon, and hardly break into the inside of the polygonal region once the modal structure is established.

At the early stage of the oscillation, the modal structure first exhibits a pukating or standing-wave state (Rg. 2). Figure 2+>(l) shows a snapshot of Rg. 2 flow. This picture indicates the transient shape during the turning over of the triangle through the pulsation. The particles are trapped within a unit azimuthal sector (l/6 = 1/(2m) azimuthally-divided region in this case) and seldom penetrate into the neighboring one. The surface temperature variation of this regime Rg. 2 consists of the fundamental and its harmonics (Figure 2-(b)-(2)). The fundamental tkquency is about 1.9 Hz, which agrees well with the frequency of the pulsating motion of the triangle. The latter was obtained from the successive video Games. In this case the time series of the surface temperature was measured at the antinode of the pattern oscillation.

Re. 3: Rotating Flow I With increasing AT, the polygonal particle&e zone in the bridge starts to rotate azimuthally (Rg. 3). Figure

2-(c)-(l) shows a snapshot of the flow in Rg. 3. Boundary lines of the sectors dkppear and the particles migrate randomly in the whole field. In Rg. 3 the direction of rotation cannot be predicted. The periodic fluctuation of the temperature corresponding to the modal structure rotation is still observed. The surface-temperature variation of Rg. 3 is quite similar to that of Rg. 2. It should be noted that the surface temperatum oscillation in Rg. 3 consists of a lower fundamental frequency than Rg2 despite the higher Ma. This tendency was also predicted by a numerical analysis concerning a medium-Pr-fluid liquid bridge (Leypoldt et al., 2000). The PPSs of the first pair of pulsating and rotating flows exhibit periodic orbits as shown in Figures 2-(b)-(3) and 2-(c)-(3). Existing works have mainly focused upon the regimes up to Rg. 3.

Rg. 4: Transition Further increasing AT, a distinct modal structure disappears and the particles again disperse into the whole

field (Rg. 4) as shown in Figure 2-(d)-(l). The visualized flow field seems to be consisted of randomly mixed structure of romting (Rg. 3) and pulsating (Rg. 5) convections. A high-fkquency fluctuation appears in the surface tempemture variation, while the main frequency still persists as shown in Figure 2-(d)-(2). Although no distinct flow structures were observed, the trajectory in the PPS clearly indicates a periodic behavior. This regime Rg4 is thus considered as a transition region of the flow structures between the fmt (Rgs. 2 and 3) and the second (Rgs. 5 and 6) pairs of pulsating and rotating oscillatory flows.

146 I. Ueno ef al.

It should be noted that oscillatory flow field with a certain azimuthal wave number m emerges at a limited range of the aspect ratio; at the periphery of the range, the standing-wave type oscillation favorably appears. In increasing AT, the flow seldom exhibits a steady rotating flow. The flow changes from Rg. 2 to IXg. 4 and then Rg. 5. At the center of the range, on the other hand, prefd flow pattern is traveling-wave type oscillation. In

100 FWWW WI 10'

100 FWWW W 10’

2

a 8;

100 Frequency PI 10'

(1) top view (2) PSD of surface temp. (3) reconstructed PPS

Fig. 2. Flow regimes of (a) Rg. 1: steady flow (the data shown here indicate tha case of AT = 28.5 [K], Ma = 2.6x10’, Pr = 49.0) (b) Rg. 2: Pulsating flow I(34.2, 3.2x10’, 45.9) (c) Rg. 3: Rotating flow I(36.6, 3.4~1~,44.6), (d) Rg. 4: Transition (69.7, 6.4~104, 31.2) (e) Rg. 5: Putsating ffow II (67.5, 6.2~104, 31.9), (9 Rg. 6: Rotating flow II (63.7, 7.7x10’, 27.1) (g) Rg. 7: Chaotii flow I (90.7, 6.4~104, 25.3), and (h) Rg. 6: Chaotic fiow II (Turbulence) (103.9, 9.6x10’, 22.4); Column (1) indii top view of the flow field, (2) power spectrum density (PSD) of the surface temperature variation, and (3) reconstruct&d pseudo-phase space (PPS). Aspect ratio of (a) - (h) is 0.64 excapt (d): 0.62 and (e): 0.66.

Thermocapillary Convection in a Liquid Bridge

increasing AT, the flow shows a steady rotating flow I atter the pulsathg flow I, and then changes into the next traveling-wave type oscillation Rg. 6, described below, witbout experiencing Rg. 4.

Rg. 5: F%dsath Flow II Beyond the transition regime Rg. 4, ordered

sbructures again emerge as Rgs. 5 and 6. In Rg. 5, another standing-wave oscillation emerges as shown in Figure 2-(e)-(l). ‘Ihe particles are trapped witbin azimuthallydivided l/6 cells again and seldom move to the neighboring ones. The pattern observed through the top rod is quite different from that in the first pulsating flow Rg. 2. This flow never exhibits any polygonal structure, but forms a particle-Gee region with 6 (= 2m) branches. Two bran&es lie in every other cell and oscillate witbin it The temperature fluctuation in this regime is also diEerent from that in Rg. 2. The harmonics of higher order are seen in the spectrm (Figure 2-(e)-(2)).

RQ. 6: Rotat& Flow II With increasing AT, another rotating oscillatory flow

or traveling-wave oscillation emerges as Rg. 6. As aforemention~ the rotating oscillatory flow tends to emerge at the center of the existing range of the flow structure with wave number m in terms of the aspect ratio. In this regime the particles tend to accumulate along a

Fig. 3. Surface temperature wiation (top) and ita short-time Fourier spectwm (bottom) for the transitionfromRg.6toRg.7att-13s.

single closed path (see Figure 2-(f)-( 1)). Particle accumulation structure (PAS) in the HZ bridge was first observed as a side view by use of a vertical light sheet by Schwabe et al. (19%), who called this &ucture as ‘cloud.’ Its three dimensional structure was reconstructed by Tanaka et al. (2000) for the first time by simuhaneous observation of the top and side views, and was named as the twisted-loop particle accumulation structure (IL-PAS). The 8mdamental tkquency of Rg. 6 is lower than that of Rg. 5. Such features are also detected in the first pair of the regimes (Rgs. 2 and 3). As shown in Figures 2-(e)-(3) and 2-(f)-(3), their at&actors become more twisted than those of the first pair. In addition, the orbit is more twisted in Rg. 6 than in Rg. 5. This tendency is qualitatively the same in the relation between Rgs. 2 and 3. It was also found that the flow exhibits the TL-PAS at various modal number m with varying I as seen in Rgs. 2 and 3. Their features and dneedimensional &ucmresweredescribedinUenoet al. (2000) in more detail.

Rrr. 7: Chaotic Flow I And Rn. 8: Chaotic Flow II With further increase AT, the flow field exhibits a transition to chaotic state. The distinct particle accumulation

structure disappeam and the particles disperse in the whole field again (Figures 2-(g)-(l) and 2-(h)-(l)). The surface temperature variation mostly loses its periodic nature, and its power spectrum broadens. The peaks are ahnost buried in the continuous frequency spectrum (Figutes 2-(g)-(2) and 2+X2)). Although almost no distinct di&rence is apparently observed in the visualized flow fields nor in the surface kmpemm~ variation, an evident difference is indicated in the reconstructed PPS between Rgs. 7 and.8 as seen in Figures 2-(f)-(3) and 2=(g)-(3). In I@. 7 the trajectory scatters and does not form a sharp and definite route, but nevertheless retains a certain structure in the PPS. In Rg. 8, on the other hand, the trajectory in the PPS exhibits no ordered &ucture. JYhis indicates that no periodic structure exists, that is, that the flow exhibits a transition to the turbulent state. Such tendencies can be e also by applying the short-time Fourier transform. Figure 3 indicates the surface temperature variation (top) and its short-time Fourier spectrogram (bottom) for the transition region between Rgs. 6 and 7. In the spectrogram, one can observe the time series of the variation of the tkquency distribution. In the regime Rg. 6, a constant fundamental Cequenq of fo u 2.4 Hz and its harm onics are clearly observed until t N 13 s. The power of the fundamental fkquency has the largest value and those of harmonics decrease gradually as the fkquency increases. Such Mbrmation fkom the spectrogram corresponds to the one obtained from the normal Fourier transformation. The flow field exhibits a transition from Rg. 6 to Pg. 7 at t w 13 s. Sharp peaks of the flmdamental and harmonic frequencies gradually

148 I. Ueno et al.

disappear, but bmademd peaks still remain with time dependent fluctuations. This causes scatter of the at&actor in the PPS as seen in Figure 2-(g)-(3), while the fluctuating main frequency retains the quasi-periodic @ucture of the at&actor. In the more disordered flow regime of Rg. 8, on the other hand, broadband noise dominates the fi-equency space. One cannot distinguish any sharp main frequency during this observation period. Such W distribution of the kqucncy reflects the disordered behavior of trajectory in the PPS. Considering further the continuous spectrum of Rg. 8, the flow regime Rg. 8 may be regarded as a more developed turbulent state, while the regime Rg. 7 as a transient chaotic state+ between the ordered strudum Rg. 6 and the turbulent state of Rg. 8. It is revealed that theregimesofRgs.7and8retainbothofthefractalstructuteandthesensitivitytoinitial~~oftheattractor in the pseudo-phase space, by the tkctal values of the correlation dimension, and the positive maximum Lyapunov exponents, repxtively (Ueno et al., 2003). One can thus conclude that these flow regimes maintain chaotic nature.

The thermocapillary-driven flow in a half-zone liquid bridge was investigated experimentally. The regimes of convection fields were categorixed Erom the steady to the turbulent flows through the observation of the suspended particle motion, the surface teqm&re variation, its Fourier spectrum and trajectory in the reconsuucted pseudo-phasespace.

After the onset of the oscilktion, pukting and then rotating flows emerged. After once exhibiting a transition regime, another type of the pukting or rotating flow appeared again. The flow structures of these second pulsating and rotating flows were quite Wt from ‘the former ones. These flow regimes still retained quasi-periodic structures. In the second rotating flow, one can observe unique particle motion in the liquid bridge, which is known as the dynamic particle accumulation structure (dynamic PAS). Further inexeasing the tempemture diffkence from the second rotating flow, the chaotic and turbulent states were realized. The non-linear behaviors of the flow fields can be analyzed quantitatively by the reconsuuction of the pseudo-phase space and the evahtation of the correlation diiension and the maximum Lyapunov exponent.

ACKNOWLEDGEMENTS

Dedicated contributions in carrying out the experiments and data analysis by Mr. Dai NAGANO and Mr. Kohei NISHIHATA, the former and current graduate students of Tokyo University of Science are gratefully acknowledged. A part of this work was financially supported by Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology.

REFERENCES

Frank, S., and D. Schwabe, Temporal and Spatial Elements of Thermocapillary Convection in Floating Zones, Experiments in Fluids, 23,23425 1, 1997.

Leypoldt, J., H. C!. Kuhlmann, and H. J. Rath, Three-dimensional numerical simulation of thermocapillary flows in cylindrical liquid bridges, J Fluid Mech., 414,285-3 14,200O.

Shin-Etsu Chemical Co., Ltd., Technical Note -Silicone Oil KF96 -, 1991 (in Japanese). Schwabc, D., P. Hit&, and S. Frank, New Features of Thcrmocapillary Convection in Floating Zones Revealed by

Tracer Particle Accumulation Structures (PAS), Mcrogruvity Sci. Technol., 9, 163- 168, 19%. Smith, M, K., and S. H. Davis, Instabilities of Dynamic Thermocapillary Liquid Layers. Part 1. Convective

Instabilities,J. Fluidbfech., 132, 119-144, 1983. Tanaka, S., I. Ueno, and H. Kawamura, Dynamic Particle Accumulation Structum in Half-Zone Liquid Bridge --

Part I: Experimental Study --, J. Jpn. Sot. Micropwity Appl., 17 (Supplement), 11 l-l 12,200O. Ueno, I., Y. ‘Ono, D. Nagano, et al., Modal Oscillatory Structure and Dynamic Particle Accumulation in

Liquid-Bridge Marangoni Convection, Proc. of & kW,USJSME Thermal Eng. Cona 3,265-270,200O. Ueno, I., S. Tanaka, and H. Kawamura, Oscillatory and chaotic thermocapillary convection in a ha&zone liquid

bridge, P&w. Fluial, l&408-416,2003. Veltcn, R, D. Schwabe, and A. f!hhmmq The periodic instability of thermocapillary convection in cylindrical

liquid bridges, P&s. Fluid, A 3,267-279, 1991.

E-mail address of I. Ucno ichciirs.noda.tus.ac.iv Manuscript rcveived 19 October 2002; revised 13 February 2003; accepted 7 March 2003.