Thermocapillary Bubble Migration at High Reynolds and Marangoni Numbers: 3D Numerical Study

Yousuf Alhendal1*, Ali Turan2, Abdulrahim Kalendar1, Hosny Abou-Ziyan1, Rafik El-shiaty1

1 College of Technological Studies (CTS), Public Authority for Applied Education and Training (PAAET). P. O. Box 5351 Hawally, zip code 32084, Kuwait. Email: ya.alhendal@paaet.edu.kw, Office: (+965) 22314202.2 School of Mechanical, Aerospace and Civil Engineering, the University of Manchester, George Begg Building, Sackville Street, Manchester, M13 9PL, UK.

Abstract

Thermocapillary motion of initially spherical bubbles due to the constant temperature gradient in a liquid bounded medium is simulated numerically for low, intermediate, high Reynolds and Marangoni numbers using a three dimensional model. The volume of fluid (VOF) method was used to track the liquid/gas interface utilizing a geometric reconstruction scheme based on the piece-wise linear interface calculation (PLIC) method of Ansys (2013) to capture the bubble interface. The simulation results are in good agreement with the earlier experimental observations, and the migration velocity of the bubble is greatly influenced by the temperature gradient which thrusts the bubble from cold to hot regime. The results indicate that the scaled velocity of bubbles decreases with an increase of the Marangoni number, which agrees with the results of previous space experiments. Thermal Marangoni number (MaT) of single bubble migrating in the zero gravity condition ranged from 106 to 904620, exceeding that in the previous reported experiments and numerical data that was limited to 10,000. In addition, an expression for predicting the scaled velocity of the bubble has been developed based on the obtained data in the present numerical study.Keywords: bubble; Two-phase; Zero-gravity; Thermocapillary; Marangoni; Surface Tension gradient; VOF-Ansys

1. Introduction

Thermocapillary migration involves the motion of fluid from the cold region to the hot region in a non-uniform temperature field. This motion arises due to a net imbalance of forces acting upon the fluid particles as the surface tension varies according to the local temperature conditions, with a greater surface tension existing in colder regions than in hotter regions. In 1959, Young et al. (1959) conducted a pioneer work with co-authors Block and Goldstein, holding a small bubble stationary and moving it downwards against the buoyant rise of gas bubbles by applying a temperature gradient between the lower and upper sides. This

experiment commenced research and experimentation on thermocapillary bubble and droplet motion. Their linear model is termed YBG model and the linear bubble velocity is termed VYBG. Both model and bubble velocity were related to the names of the three authors Young, Block, and Goldstein. The topic of thermocapillary migration including experimental, analytical, and numerical studies up to 2000 was reviewed by Subraminan and Balasubramaniam (2001) and Subraminan et al. (2001). More recently, Zhao et al. (2010) extended the review of the topic up to 2008.

Microgravity experiments on thermocapillary migration of bubbles and droplets have been performed in zero gravity conditions, such as onboard of the microgravity sounding rocket, Spacelab, and China’s ShenZhou-4 (Kang et al., 2008). These studies have noted that there is much need for numerical results and investigations into the complex behaviour of thermocapillary bubble migration to aid in the evaluation and further investigation of their data (Kang et al. 2008). Experiments under microgravity conditions are limited to low Reynolds and Marangoni numbers because of the difficulties in obtaining experimental results in microgravity, and the requirements of continuous observation of the moving fluid during a test. These difficulties make the design of an experiment, to accommodate the objectives, challenging in space based experiments. It seems that it is also difficult to do such an experiment, considering various limitations to the experiments. Xie et al. (2005) also confirmed that relatively long experimental time in microgravity conditions is necessary for a bubble/droplet to approach its steady thermocapillary velocity. The authors further elaborated the difficulty to obtain complete information about the behaviour of bubbles/droplets in space. Radulescu and Robinson (2008) stated that bubble and/or drop dynamics have become a hot point of research.

However, due to the practical difficulties in experimental studies, numerical investigations have been undertaken by many researchers in order to compare and analyze their experimental results (Treuner et al., 1996). Larkin (1970) was possibly the first to numerically investigate

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Marangoni convection caused by the presence of a bubble situated on a heated wall (O’Shaughnessy and Robinson 2008). Shankar and Subramanian (1988), and Oliver and DeWitt (1994), used the finite difference method to investigate the thermocapillary motion of gas bubbles at Marangoni number up to 200. Szymczyk and Siekmann (1988), Balasubramaniam and Lavery (1989), Chen and Lee (1992), Nas and Tryggvason (1993), Ma (1998), and Welch (1998) extended the range of the Marangoni number up to 1,000. Welch (1998), used a finite volume method with interface tracking to study the effect of bubble deformation on the transient thermocapillary migration under microgravity conditions. In addition, Hadland et al. (1999) provided numerical solutions for the thermocapillary migration of bubble to extended Marangoni numbers of 5812. Recently, Alhendal and coworkers (2010, 2013, 2015 and 2016) conducted a comprehensive program of bubble and droplet migration, using CFD analysis; however, their work was limited to small and medium MaT and was not extended to large values. The advancement in numerical simulation has allowed a much better understanding of thermocapillary flow and the field has undergone a considerable change. Numerical simulations have consequently become an important tool in studies of two-phase flows in a microgravity environment and can help clarify the basic fluid physics, as well as assist in the design of future experiments or systems for zero-gravity environments.

It is clear from the above discussion that the stated limitations for space experiments have made thermocapillary migration at large Marangoni number complex. No experiment has been performed in which sufficient velocity development of bubble migration at large Marangoni numbers could be observed. Therefore, further appropriate numerical simulation studies aimed at identifying the behaviour of bubbles in microgravity are really needed. This will help to observe the developing trend of bubble migration velocity with an increase in the values of the Marangoni number as stated by Wölk et al. (2000) and Colin et al. (2008). Clear and profound understanding of the flow patterns, including the shape and the area of the varying complex interfaces is of vital importance to the understanding

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and prediction of the physics behind these flow systems. As the flow patterns of some regimes remain largely undiscovered, consequently, accurate predictions of flow patterns are highly desirable (Alhendal et al. 2010; Alhendal et al. 2015).

2. Mathematical Analysis

The first investigated thermocapillary migration of bubbles and droplets, using a linear model, were conducted by Young et al. (1959). Their model and migration velocity are commonly called the YGB model, which is suitable for small Reynolds and Marangoni numbers:

1

where and ', and λ and λ' are the dynamic viscosity and thermal conductivity of continuous phase fluid and bubble, respectively, and rb is

the radius of the bubble. The constant or is the rate of change of interfacial surface tension and is the temperature gradient imposed

in the continuous phase fluid. On the other hand, the velocity derived from the tangential stress balance at the free surface is given by:

2

Thermal Reynolds (ReT) and Marangoni (MaT) numbers for thermocapillary bubble migration are defined by equations 3 and 4, respectively.

3

4

Since Prandtl number (Pr) is the ratio of kinematic viscosity to thermal diffusivity as defined by equation 5, equation 4 can be written in the form of equation 6:

4

5

6

In the above equations ν is the kinematic viscosity in m²/s and 𝜌 is the density of the fluid in kg/m3.

7

3. Numerical Simulation and Results Validation

Unsteady 3D geometry was formulated using the commercial software package Ansys (2013) for modelling the rise of an isolated bubble in a column filled with ethanol liquid in zero gravity (Marangoni flow). The cold wall of the test cell is held at a constant temperature of Tc = 300 K, hot wall has a temperature of Th = 325 K, and the side walls are held at a steady linear temperature profile between the cold and hot walls resulting in a temperature gradient. All simulations were run at time t=0 with an initial stationary liquid and gas, with an applicable surface tension between those of ethanol and nitrogen (σ) = 27.5 (dyn/cm), and surface tension gradient (σt) = -0.09 (dyn/cm ºC). The domain was defined as a cavity with bounded walls. For all simulations, the initial state was set with no velocity at the inlet or the outlet and the pressure was taken to be equal the atmospheric pressure. The physical properties of the liquids used in the simulation of Ethanol and Nitrogen are given in Table 1 (Thompson et al., 1980). Thermal Reynolds and Marangoni numbers varied, depending on the diameter of the bubble and the temperature gradient. Both heat convective and heat conduction inside the bubble were ignored, note that in all cases in which the physical properties of the Matrix liquid are assumed constant.

Numerical prescription for the surface tension variation against temperature is provided via user defined functions (UDFs). These UDFs are dynamically linked with the FLUENT solver. The initial rise velocity for

5

the bubble is set to zero. The upper surface (top wall) of the model is hotter than the bottom surface (bottom wall); both top and bottom walls are set to no-slip solid walls, as seen in Fig.1, and the size of the computational wall bounded domain was chosen as 120 × 60 mm with zero permeability "no inflow or outflow” from the sides.

The thermocapillary motion of an isolated nitrogen bubble with a diameter of 6 mm placed 10 mm in a confined apparatus was the first multiphase test case, which will serve as a stringent test for our treatment of surface tension. Grid sensitivity analysis was conducted by increasing the number of region adaption cells from 40 to 304 per bubble by decreasing the grid size in X, Y, and Z directions and checking the convergence of certain parameters of interest such as bubble migration time toward the hotter side and migration velocity. Three sizes of meshes were tested, specifically 25×100×25 (192,400 cells), 30×120×30 (324,000 cells) and 40×160×40 (768,000 cells). Table 2 lists the bubble migration velocity, towards the hotter side, against time for the three tested meshes. It is clear that no difference is noted in the bubble migration velocity between the two fine meshes (324 and 768kCell). Therefore a mesh size of 324k Cell was considered sufficient and was used throughout the course of the present work.

At the beginning of this research study and for the verification and validation process, investigations based on the computational fluid dynamics (CFD) concept have been carried out and results were compared with the experimental measurements of Thompson et al. (1980). Fig. 2 shows a comparison between the present numerical results and the measurements of Thompson et al (1980), using the 340kCell mesh, for the same Pr number (small Pr = 16.3). The difference between the results is mainly attributed to the difference in initial conditions between the experiment and the model as a spherical bubble is injected in the experiment whereas the model assumed a stagnant sphere. It is really difficult to duplicate the initial condition of bubble injection process in the numerical simulation. Therefore, the numerical results over predict the

6

bubble velocity in the early stage of the process compared to the measured values by Thompson et al. (1980). The numerical model gets closer with the experiment later in the process. It is to be stated that a small difference (less than ±4mm) is observed in the bubble migration distance between the numerical and experimental results.

Based on the results reported in Figs. 2, it was found that CFD predictions, using the VOF model and the UDF, are in good agreement with the experimental data. The results proved that the surface tension coefficient was well modeled, suggesting that the UDF is an appropriate choice to solve thermocapillary. The fully three-dimensional model ascertains the accuracy of the simulation that is mostly dependent on the mesh size, convergence criteria and discretisation schemes.

4. Results and Discussion

Decreasing or increasing the temperature gradient of the liquid medium could decrease or increase the migration velocity of the bubble, causing it to move towards the warmer region. The upper and lower walls were kept constant (Tc = 300 K, and Th = 325 K) for all cases to avoid the effect of fluid property variation on the results. For such constant upper and lower wall temperatures, the bubble diameter and temperature gradient varied while the aspect ratio between the domain size and bubble diameter remained constant.

Thus, the CFD data for each case reach different average velocity levels. Sample of these results are listed in Table 3 for a range of thermal Reynolds (6.51 < ReT < 6943) and thermal Marangoni numbers (106 < MaT < 113,000) and the full range of results is presented in Fig. 3. It is to be noted that the Prandtl number (Pr), reference velocity (VT) and (VYGB) were taken as average and constant values for all cases and the normalized velocity is defined as Vbubble = VCFD/VYGB. However, Table 3 lists the CFD results of the bubble migration velocity for different temperature gradients between the upper and lower walls. The temperature gradient applied, for the first case in Table 3 for MaT=106, equals 6.67 K/mm, and the bubble CFD migration velocity VCFD was found to be equal to 15.73

7

mm/sec, which is higher than the rest of the simulation’ velocities. Effect of the temperature gradient is substantial for MaT <1000. The CFD data for this group reach different scaled velocity¿) between 0.5 – 0.3. It is apparent that the calculated bubble velocity, VCFD, changes due to the varying temperature gradient of the host liquid and the size of the bubble; a direct relation between the temperature gradient and bubble velocity is, thus, observed. Therefore, it can be concluded that different temperature gradients lead to different bubble migration velocities for constant bubble size.

Figure 3 displays more clearly a continuous decrease in the scaled bubble velocity, Vbubble, with increasing MaT. The bubble velocity becomes almost constant at high Marangoni numbers. In order to prove the robustness of the numerical model, bubble migration velocity was obtained for bubble diameter of 9mm and various temperature gradients (0.1-0.28K/mm) and also for different bubble diameters (9 and 12mm). The results of those cases are plotted as solid symbols in Fig. 3. It is clear that the solid symbols are well aligned with the other data at the same MaT proofing that the results are independent on the temperature gradient or bubble diameter. In addition, the numerical results of scaled migration velocity, V CFD

V YGB=V

bubble,was correlated with Marangoni number and the correlation of a

single bubble was also plotted in Fig. 3. The obtained correlation for a differentially heated fluid cell is given in equation 8.

V bubble=1.85 MaT−0.25 8

where Vbubble is the bubble’s scaled migration velocity and VYGB was used for scaling the calculated bubble velocity in Eq.8. The above equation has a correlation coefficient of 0.9184 with maximum residual of 0.0716 and minimum residual of -0.0867.

Figures 4a-4f show the liquid-phase streamlines (on the left) and temperature contours (on the right) for a thermocapillary isolated Nitrogen bubble migrating in a stagnant Ethanol at the end of bubble

8

migration, under microgravity effect, for Pr=16.2 and different Marangoni numbers. The bubble moves from the colder side to the warmer one in microgravitational environment under the thermocapillary migration. This is caused by the decrease in surface tension as the temperature increases and the non-uniform surface tension at the fluid interface which leads to shear stresses that act on the outer fluid by viscous forces. The flow due to surface tension gradient, at the liquid-gas interface (meniscus), is referred to as Marangoni flow. In the case of temperature dependence, this phenomenon may be called thermocapillary convection. Figs. 4a-4f show this phenomenon as motion of the Nitrogen bubble triggers motion of the ethanol liquid around it according to the temperature gradients that control Ma numbers. Also, the variations in both streamlines and temperature contours as Marangoni number increases are clearly observed. As the temperature contours varies considerably as Ma increases, particularly at Ma equal or greater than 3392. Also, the streamlines are different for each case of the presented Ma values (see Fig. 4). It was noticed that the bubble remained spherical in shape and no deformation was noticed up to Ma = 424 (Fig. 4c), afterward, the bubble deformed greatly (Figs. 4d-4e).

Figures 5 and 6 shows the positions of the bubble migrating under thermocapillary effect for MaT= 424 and 904621, respectively. Fig. 5a (top) shows the bubble position versus the time that indicates the bubble needs only 1.25s to reach the top wall at low MaT. On the other hand, Fig. 6 shows that the bubble needs over 2600s to reach the hotter wall. In Fig. 5, the thermal Marangoni number (MaT) is small, and the bubble is subjected to a large temperature gradient at its surface, resulting in strong thermocapillary flow. Interestingly, the migration of the bubble in the continuous phase is very promptly in a straight line and exactly followed the direction of the local temperature gradient during the traverse. Also, it shows that no shape deformation was found and the bubble had a spherical shape for this thermal Marangoni number. In the case of bubble migration for high thermal Reynolds and Marangoni numbers, the numerical results show that the scaled bubble velocities,

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Vbubble, is very low compared to that given by the linear prediction of the YGB model. The reasons for this are due to decrease in the movement source - in this case, temperature difference, besides the bubble’s migration direction slightly deviated from a vertical straight line and displaying an oblate ellipsoidal shape as seen in Fig. 6.Figure 7 shows a comprehensive comparison between the obtained numerical results of the present work, for a Nitrogen bubble and ethanol as a host liquid, and the experimental (Balasubramaniam et al. 1996 and Hadland et al. 1999) and numerical results (Hadland et al. 1999). The present results are compared with the most compelling results in literature that were attained from the experiments performed in the space flights. Namely, the International Microgravity Laboratory Mission (IML-2) in 1994 (Balasubramaniam et al. 1996) and the Life and Microgravity Science (LMS) Mission in 1996 (Hadland et al. 1999) on the Columbia Space Shuttle. In the LMS mission, Re was extended from 0.839 to 87.2 and Ma from 51.7 to 5780 of bubble thermocapillary migration. Fig. 7 shows bubble’s scaled velocity (V bubble¿ versus thermal Marangoni number (MaT) limited to 10,000 which is the maximum MaT, reported in literature. Typical examples of the various behaviours of bubble migration are depicted in Fig. 7 in dimensionless presentation. At MaT larger than 1000, the present numerical results (showed in Fig. 7 as solid symbols) are aligned with the upper range of the experimental data of Hadland et al. (1999), at T=1K/mm, and lower than the numerical data of Hadland et al. (1999). For MaT lower than 1000, the present results are in agreement with the experimental data of Balasubramaniam et al. (1996) and Hadland et al. (1999), at T=0.33K/mm. Also, the present data lie between the numerical results of Hadland et al. (1999). In conclusion, the results of the present work are in good agreement with previously reported data in literature. It is to be noted that the results are verified up to the maximum limit of MaT reported in literature. However, the continuous trend of results shown in Fig. 3 supports the results for MaT larger than 10,000.

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4. Conclusions

This paper utilizes a computational approach to calculate the transient thermocapillary migration of an isolated bubble in a zero-gravity environment in a large range of the Marangoni number that was not investigated before. This is because most experiments in microgravity have constraints such as time limitations. On the other hand, computer simulation using CFD tools is not restricted in such a way, and can simulate any arbitrary geometry. Thus, numerical simulations prove to be a valuable tool to study such complex problems under the conditions of zero and reduced gravity. The work has shown conclusively that the current VOF technique is a robust numerical method for the simulation of gas–liquid flow, and the ability to simulate surface tension as a function of temperature (thermocapillary flow), using a UDF for routine design and development engineering.

The current results are validated up to the limit of MaT examined in the literature and the fundamental trend of the data supports the results for larger MaT. The results show conclusive existence of Marangoni bubble flow phenomena in a zero-gravity environment of such high MaT. The present CFD results show that different temperature gradients lead to different bubble migration velocities, and it was proven that bubble migration velocity varies linearly with the temperature gradient for the given condition, generally speaking, as MaT increases, the scaled velocity of a single bubble decreases and steadily approaches its asymptotic value. The bubble experiences deformable shape and the bubble’s migration direction slightly deviates from a vertical straight line and displaying an oblate ellipsoidal shape at large MaT. In addition, the obtained data of scaled bubble migration velocity was correlated for the tested range of MaT.

References:Alhendal, Y., Turan, A.: Thermocapillary bubble dynamics in a 2d axis swirl domain. Heat and Mass Transfer, 51, 529-542 (2015).

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Alhendal, Y., Turan, A.: Microgravity Sci. Technol. 28, 639 (2016). https://doi.org/10.1007/s12217-016-9521-xAlhendal, Y., A. Turan, M. Al-mazidi.: Thermocapillary bubble flow and coalescence in a rotating cylinder: A 3D study. Acta Astronautica 117, 484-496 (2015).Alhendal, Y., Turan, A., Aly, W.I.A.: Vof simulation of Marangoni flow of gas bubbles in 2d-axisymmetric column. Procedia Computer Science, 1, 673-680 (2010).Alhendal, Y., Turan, A., Hollingsworth, P.: Thermocapillary simulation of single bubble dynamics in zero gravity. Acta Astronautica, 88, 108-115 (2013).Ansys-Fluent 2013. Balasubramaniam, R., Lavery, J.E.: Numerical simulation of thermocapillary bubble migration under microgravity for large Reynolds and Marangoni numbers. Numer. Heat Transf. A 16(2), 175–187 (1989).Balasubramaniam, R., Lacy, C.E, Wozniak, G.: Thermocapillary migration of bubbles and drops at moderate values of the Marangoni number in reduced gravity. Phys Fluids, 8(4) 872-880 (1996).Chen, J.C., Lee, Y.T.: Effect of surface deformation on thermocapillary bubble migration. AIAA J. 30(4), 993–998 (1992).Colin, C., Riou, X., Fabre, J : Bubble coalescence in gas–liquid flow at microgravity conditions. Microgravity Sci. Technol. 20(3), 243–246 (2008).Hirt, C. W. & Nichols, B. D.: Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics, 39, 201-225 (1981).Hadland, P.H., Balasubramaniam, R., Wozniak, G., Subramanian, RS.: Thermocapillary migration of bubbles and drops at moderate to large Marangoni number and moderate Reynolds number in reduced gravity, Experiments in Fluids 26(3): 240–248 (1999).Kang, Q., Cui, H.L., Duan, L.: On-board experimental study of bubble thermocapillary migration in a recoverable satellite. Microgravity Sci. Technol. 20(2), 67–71 (2008).Larkin, B.K.: Thermocapillary flow around hemispherical bubble. AICHEJ. 16, 101–107 (1970). Ma, X.J.: Numerical simulation and experiments on liquid drops in a vertical temperature gradient in a liquid of nearly the same density. PhD thesis, Clarkson University, Potsdam, New York, USA (1998).Nas, S., Tryggvason, G.: Computational investigation of the thermal migration of bubbles and drops. In: Proceedings of the ASME Winter Annual Meeting (AMD-174/FED-175), 71–83 (1993)Oliver, D.L.R., De Witt, K.J.: Transient motion of a gas bubble in a thermal gradient in low gravity. J. Colloid Interface Sci. 164, 263–268 (1994).O’Shaughnessy, S.M., Robinson, A.J.: Numerical investigation of bubble induced marangoni convection: some aspects of bubble geometry. Microgravity Sci. Technol. 20(3), 319–325 (2008).Radulescu, C., Robinson, A.J.: The influence of gravity and confinement on marangoni flow and heat transfer around a bubble in a cavity: a numerical study. Microgravity Sci. Technol. 20(3), 253–259 (2008).

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Shankar, N., Subramanian, R.S.: The stokes motion of a gas bubble due to interfacial tension gradients at low to moderate Marangoni numbers. J. Colloid Interface Sci. 123(2), 512–522 (1988).Subramanian, R.S., Balasubramaniam, R.: The motion of bubbles and drops in reduced gravity. Cambridge University Press, London (2001).Subramanian, R.S., Balasubramaniam, R.,Wozniak, G.: Fluid mechanics of bubbles and drops. In: Physics of fluids in microgravity. Gordon & Breach, Amsterdam, 149–177 (2001). Szymczyk, J.A., Siekmann, J.: Numerical calculation of the thermocapillary motion of a bubble under microgravity. Chem. Eng. Commun. 69(1), 129–147 (1988).Thompson, R.L., Dewitt, K.J., Labus, T. L.: Marangoni bubble motion phenomenon in zero gravity. Chemical Engineering Communications, 5, 299-314 (1980).Treuner, M., Galindo, V., Gerbeth, G., Langbein, D., Rath, H.J.: Thermocapillary bubble Migration at high Reynolds and Marangoni numbers under low gravity. Journal of Colloid and Interface Science, 179, 114-127 (1996).Welch, S.W.J.: Transient thermocapillary migration of deformable bubbles. J. Colloid Interface Sci. 208, 500–508 (1998).Wölk, G., Dreyer, M., Rath, H.J.: Flow patterns in small diameter vertical non-circular channels. International Journal of Multiphase Flow, 26, 1037-1061 (2000).Xie, J.-C., Lin, H., Zhang, P., Liu, F., Hu, W.-R.: Experimental investigation on thermocapillary drop migration at large Marangoni number in reduced gravity. Journal of Colloid and Interface Science, 285, 737-743 (2005).Young, N.O., Goldstein, J.S., Block, M. J.: The motion of bubbles in a vertical temperature gradient. Journal of Fluid Mechanics, 6, 350-356 (1959).Youngs, D.L.: Time-dependent multi-material flow with large fluid distortion. Numerical Methods For Fluid Dynamics, Academic Press, 273–285 (1982).Zhao, J-F., Li, Z-D., Li, H-X., Li, J.: Thermocapillary migration of deformable bubbles at moderate to large Marangoni number in microgravity. Microgravity Science and Technology, 22, 295-303 (2010).

Table 1.Physical properties of the liquids employed in the simulation at 300K for (Pr=16.28)

Properties Unit Ethanol

Nitrogen (N2)

Density (ρ) kg/m3 790 1.138Specific heat (Cp) J/(kg.K) 2470 1040.7

Thermal conductivity (k) W/(m.K) 0.182 0.0242

Viscosity (μ) kg/(m.s)

1.2×10-3 1.66×10-5

Surface tension (σ0) N/m 0.0275 ------

Surface tension N/ 9. ------

13

coefficient (σT) (m.K) 0×10-5

Temperature gradient () K/mm 0.208 ------

Prandtl number (Pr) ------- 16.28 0.79

Table 2. Grid sensitivity check for a bubble diameter=10 mm at Ma = 3770.0

Grid

(∆x×∆y×∆z)

Number of

grids

Grid cells per

bubbleMigration time (sec)

Bubble speed (mm/s)

(1) 25×100×25 96000 40 8.5 12

(2) 30×120×30 324000 128 8.2 12.5

(3) 40×160×40 768000 304 8.2 12.5

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Table 3 Calculated ReT and MaT and Vbubble corresponding to the fluid properties given in Table 1 and the results of CFD test for N2 (Pr=0.79) bubble migrating in Ethanol (Pr=16.3)∇ T

(K/mm) radius(m)

VCFD(mm/s) Vbubble ReT MaT

6.67 0.00014 15.73 0.49 6.51 106.013.33 0.00028 14.68 0.45 13.02 212.021.67 0.00056 12.96 0.40 26.04 424.04

0.833 0.00113 12.37 0.38 52.08 848.080.417 0.00225 11.65 0.36 104.15 1696.160.208 0.0045 10.09 0.31 208.30 3392.330.139 0.0068 9.28 0.29 312.45 5088.490.104 0.009 8.78 0.27 416.60 6784.650.083 0.01125 8.27 0.26 520.75 8480.820.069 0.0135 7.48 0.23 624.90 10176.9

80.059 0.01575 7.06 0.22 729.05 11873.1

40.052 0.018 6.69 0.20 833.20 13569.3

10.046 0.02025 6.51 0.20 937.35 15265.4

70.042 0.0225 6.26 0.19 1041.5

016961.6

40.028 0.03375 4.98 0.15 1562.2

625442.4

50.0208 0.045 4.43 0.14 2083.0

133923.2

70.0140 0.0675 3.64 0.11 3124.5

150884.9

10.0104 0.09 3.17 0.10 4166.0

267846.5

40.0083 0.1125 2.87 0.089 5207.5

284808.1

80.0063 0.15 2.71 0.080 6943.3

6113077.

57Note that: VT = 0.07 and VYGB = 32.2 for all simulations

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Side

wal

l

Side

wal

l

Bubble

Hot wall

Cold wall

Axis

sym

met

ry

D=20xd

20xd

d

10 20 30 40 500

1

2

3

4

5

6

7

8

Thompson et al. (1980)Present CFD Results

Bubble displacement (mm)

Tim

e (s

)

16

Figure 2. Validation of present CFD model with experimental data for 6 mm diameter Nitrogen bubble in Ethanol (Pr= 16.3).

Figure 1. Schematic diagram of solution domain for bubble migration in a uniform temperature

gradient.

100 1000 10000 100000 10000000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Present Results

CFD Results for various bubble diameters and temperature gradients

Vbubble = 1.85 Ma -̂0.25

MaT

Vbub

ble

Figure 3. Normalized bubble migration velocity (Vbubble = VCFD/VYGB) versus Marangoni numbers obtained for Nitrogen bubble in

Ethanol (Pr= 16.3).

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a. Ma=13.25 b. Ma= 212 c. Ma= 424

d. Ma= 3392 e. Ma= 42404 f. Ma= 169616

Figure 4. Streamlines (Kg/s) and temperature contours (K) for a Nitrogen bubble rising in a stagnant Ethanol at the end of bubble migration under thermocapillary effect, for different Marangoni

numbers.

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Figure 5. Sequence of bubble positions during the simulation for ReT = 26 and MaT =424.

19

t=600s t=1600s t=2600s

Figure 6. Sequence of bubble positions during the simulation for ReT=55547, MaT

=904621.

10 100 1,000 10,0000.0

0.2

0.4

0.6

0.8

1.0

1.2IML-2 (Balasubramaniam et al. 1996)Present CFD Results (different domain sizes)LMS T=1 K/mm (Hadland et al. 1999)∇Numerical solution (Hadland et al. 1999)Asymptote solution (Hadland et al. 1999)LMS T=0.33 K/mm (Hadland et al. 1999)∇Kang et al. (2008)Present CFD for different dimeter & T (120 x 60 mm)∇

MaT

VCFD

/VYG

B

Figure 7. Comparison between present results and literature for MaT up to 10000

20

21