OR2-3
液柱マランゴニ対流内の粒子集合構造のトモグラフィック
再構築
Tomographic Reconstruction of Particle Accumulation
Structure in a Thermocapillary Liquid Bridge
矢野大志 1, 中西裕二 1, 西野耕一 2
Taishi YANO1, Yuji NAKANISHI1 and Koichi NISHINO212

神奈川大学，Kanagawa University
横浜国立大学，Yokohama National University

1. Introduction
Since the thermocapillarity—the force due to surface tension difference caused by the temperature gradientalong the liquid-gas interface—can be the primally source for the liquid motion under the reduced-gravity
environment, numerous hydrodynamic experiments related to this phenomenon have been conducted under
microgravity (μg, hereinafter) in the past few decades. The thermocapillarity-driven convection (referred tohereinafter as the thermocapillary convection) in a high-Prandtl-number liquid bridge is one of the most
famous research themes in the Japanese Experiment Module Kibo onboard the International Space Station(ISS), and several series of μg experiments—i.e., the so-called MEIS, Marangoni UVP, and Dynamic Surfprojects—had been performed in the period from 2008 to 2020 in the Kibo module 1,2). As shown in Figure 1,
there exists a pair of disks of diameter D with the gap H open, and a column of liquid (i.e., silicone oil in thiscase) is suspended between these disks. The upper disk is set warmer than the lower disk, and the temperature
difference ∆T (= TH−TC) given to these disks drives the toroidal thermocapillary convection as illustrated in
Figure 1, where TC and TH are the temperatures of cooler disk and warmer disk, respectively. Hereinafter, thedisk with lower/higher temperature is referred to as the cold/hot disk. One of the most interesting features ofthermocapillary convection in a high-Prandtl number liquid bridge is the transition of the flow from a steady

Figure 1. Liquid bridge formed in the μg experiment.
1 of 5state to an oscillatory state at a certain instability threshold 3,4). The thermocapillary convection after thetransition to the oscillatory state shows a variety of spatially complex structures. Furthermore, under somespecial conditions, oscillatory thermocapillary convections involve the structure of tracer particles
accumulating along a single or multiple narrow paths, which is called the particle accumulation structure (PAS,hereinafter)5–8). In this study, the PAS observed in the previous μg experiment is visualized by means of thetomographic reconstruction9) to understand its formation process.
2. Tomographic Reconstruction
In the experimental cells used in the μg projects mentioned above, there existed three CCD cameras on topof the liquid bridge2). Since the hot disk, along with the sensors and heaters embedded in it, were alltransparent, the CCD cameras could observe the flow inside the liquid bridge without being affected byrefraction and reflection at the free surface. There were a few dozen reference points marked on the liquidcontact surface of the cold disk. These reference points were used to obtain camera parameters that relate twodimensional (2-D, hereinafter) coordinate systems on images and a three-dimensional (3-D, hereinafter)
coordinate system in space pinned to the liquid bridge 10). In the μg experiments conducted aboard the Kibo
module, many fine tracer particles were seeded in a liquid bridge for flow visualization. Their 2-D positional
information taken by each CCD camera can be reconstructed into 3-D distributional ones by means of imagetomography9) using camera parameters.
In the μg experiments conducted aboard the Kibo, a rich variety of oscillatory thermocapillary convection
was observed under various experimental conditions; however, that involving the PAS was very limited. The
target μg experiment in this study is the third series of Dynamic Surf project conducted in 2015–2016 (i.e.,Dynamic Surf-3), in which PASs were observed in a few experimental runs. The test liquid was a 10-cSt
silicone oil with the kinematic viscosity of ν = 10.0×10−6 m2/s, the thermal diffusivity of κ = 8.94×10−8 m2/s, andthe resultant Prandtl number of Pr (= ν/κ) = 112 at 25 °C2). The PAS having an azimuthal mode number of m =1 was observed in the liquid bridge of D = 30 mm, Ar = 0.50, and Vr = 0.95, when the temperature conditions
were set at TC = 40.0 °C and TH = 48.2 °C, where Ar (= H/D) and Vr (= V/(πD2H/4)) are the aspect ratio and thevolume ratio (i.e., the ratio of the liquid volume V to the cylindrical gap volume πD2H/4), respectively. The
present study is targeted at reconstructing this PAS. To reach this goal, the 3-D space in which the liquid
bridge exists is discretized into many volume elements (i.e., voxels) as shown in Figure 2(a). We note that theFigure 2. Voxel configurations for the liquid bridge with Ar = 0.50 discretized in (a) the cylindrical coordinate
system and (b) the cartesian coordinate system.

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target domain is, in fact, discretized more finely, but the voxel configuration is illustrated roughly in this figure
for better visibility. The cylindrical domain is divided equally into (150, 360, 150) in the (r, θ, z) directions in acylindrical coordinate system, resulting in triangular prismatic and trapezoidal prismatic voxels in shape and8.1 million voxels in the total number. Actually, the triangular prismatic voxels exist only in the part adjacentto the z-axis, and the most part is occupied with the trapezoidal prismatic voxels. In the general tomographic
reconstruction9,11), the measurement domain is divided into rectangular prismatic voxels in a Cartesian
coordinate system as shown in Figure 2(b). As mentioned above, the present tomographic reconstruction is
performed with nonrectangular prismatic voxels, which is, to the best of authors’ knowledge, the first attempt
except for the earlier work of this study12). The use of a cylindrical coordinate system as well as voxels withaxisymmetric shape can reduce computational costs (or improve spatial resolution) by eliminating
unnecessary regions (i.e., the outer region of the cylinder in Figure 2(b)) because the target domain has
cylindrical shape. Additionally, since the thermocapillary convection involving the PAS has an axisymmetricstructure, it is favorable in postprocessing such as averaging in a rotating frame of reference.
The principal of tomographic reconstruction is basically independent on the shape of voxel; therefore, only
its brief introduction is provided here. Firstly, define the initial intensity of each voxel (i.e., E at n = 0), andupdate it by using the following multiplicative algebraic reconstruction technique (MART, hereinafter) 9):

Ncam Npx

𝐸𝑖 (𝑛) = 𝐸𝑖 (𝑛 − 1) ∏ ∏ [𝑘𝑗(𝑘)(𝑘)

𝐼𝑗𝜇𝑤𝑖𝑗](𝑘)vx∑N𝑖=1{𝑤𝑖𝑗 𝐸𝑖 (𝑛 − 1)},(1)where i, j, and k are, respectively, the voxel number, the pixel number, and the camera number; Nvx, Npx, andNcam are, respectively, the numbers of voxel, pixel, and camera; I is the image intensity; wij is the weighting
coefficient representing the contribution of the i-th voxel to the j-th pixel; and μ is the relaxation parameter.
As described above, Nvx = 8.1×106 and Ncam = 3. The size of image used in this study is 720×480 pixels, giving
Npx ≈ 3.5×105. The value of wij is evaluated by the Monte Carlo method13), but its details are not covered here.
We note that the contributions of camera parameters are contained in wij. The initial condition is arbitrary (e.g.,a uniform non-zero value). In this study, it is determined as
Ncam Npx(𝑘) (𝑘)𝐸𝑖 (0) = ∏ ∏{𝑤𝑗𝑖 𝐼𝑗 } ,𝑘(2)𝑗where wji is the weighting coefficient representing the contribution of the j-th pixel to the i-th voxel and is also
evaluated by the Monte Carlo method. Using equations (1) and (2), one can reconstruct the 3-D distribution oftracer particles in a liquid bridge from 2-D particle images.
3. Results and Discussion
Figure 3 shows an example of the PAS observed in Dynamic Surf-3, which are obtained by phase averaging
over 1000 original particle images after background subtraction. Since the flow field is rotating at a constantperiod (i.e., 32.8 s) in this case, the phase averaging can remove (or dilute) non-accumulated tracer particles
while leaving accumulated ones. We note that the actual tracer particles appear white in a dark background;
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Figure 3. Particle images used for the tomographic reconstruction. The black and white is reversed in each
image to improve visibility.

however, the monochrome is inverted in Figure 3 to improve the visibility. The 3-D distribution of tracer
particles reconstructed from the particles images in Figure 3 is shown in Figure 4(a). Although, the tracer
particles are elongated in the negative z-direction and the result seems noisy, it can be said that thetomographic reconstruction has been done adequately. Especially, the positional accuracy in the in-plane
direction (i.e., r and θ) is good. In order to make the structure of PAS much clearer, the tomographic
reconstruction is carried out for continuous data, and the time series of 3-D distributions of tracer particles areaveraged in a rotating frame of reference. The result is shown in Figure 4(b). This rotational averaging
enhances the contrast between accumulated and non-accumulated tracer particles, and one can recognize thatthe path of accumulating particles rises from the cold disk towards the hot disk while swirling in a clockwise
direction. This PAS seems to be opened in the present result because it is difficult to capture tracer particlesnear the free surface; however, it is expected to be closed according to the previous studies 6–8). In any case, it
can be said that the present tomographic reconstruction is effective for understanding the PASs.
4. Summary
The tomographic reconstruction technique dedicated to the thermocapillary convection in a cylindricalliquid bridge is developed, and the particle accumulation structure (PAS) is visualized. The measurement
volume is divided into millions of trapezoidal prismatic and triangular prismatic voxels in a cylindricalcoordinate system in the present tomographic reconstruction. The multiplicative algebraic reconstruction
technique (MART), which is customized for the present study, is used to obtain the distribution of tracer
particles in a thermocapillary liquid bridge. In Dynamic Surf-3—one of the series of μg experiments conducted

Figure 4. Distribution of tracer particles reconstructed by means of image tomography: (a) result for a certain
phase and (b) result averaged in a rotating frame of reference. Each of (1) and (2) show bird’s-eye view and rθ cross-sectional view, respectively. The green curves are eye guides for the particle-accumulated path.
4 of 5aboard the Kibo module onboard the International Space Station, the PAS was observed in a thermocapillaryliquid bridge of Ar = 0.50 and Vr = 0.95 under the temperature conditions of TC = 40.0 °C and TH = 48.2 °C, andthis PAS is reconstructed by the optical tomography developed in this study. The spiral path of accumulating
tracer particles rising from the cold disk toward the hot disk is well recognized, especially the result averaged
in a rotating frame of reference exhibits good observation. The developed tomographic reconstructiontechnique and the present results will hopefully help researchers to understand the details of PAS in athermocapillary liquid bridge.
References
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