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In order to clarify the cause of oscillatory thermocapillary flow of high Prandtl number (Pr) fluid, various aspects of the oscillation phenomenon have been investigated, both experimentally and theoretically.
A numerical analysis is conducted to show the basic oscillation mechanism. By analyzing the variations in the temperature and velocity fields after a flow is disturbed, the oscillation process for high Pr fluid can be described as follows. The basic flow is mainly driven in a relatively small region near the hot wall, called the hot corner. The major viscous retardation also occurs in the hot corner. The hot corner changes its extent during oscillations. The retardation is large (small) when the hot corner is narrow (wide). One oscillation period is divided into an active period and a slow period. During the active period the hot corner is wide and since the retardation is small, the overall flow is strong. The strong flow eventually brings cold fluid from the cold wall region to the hot corner, which decreases the surface temperature in the hot corner. Consequently, the hot corner shrinks and the overall flow slows down. As the flow becomes slow, the flow into the hot corner becomes warmer and thus the hot corner becomes wider again, which starts the next cycle. According to the numerical analysis, this process eventually decays in the range of Ma where oscillations are found experimentally. Thus, an additional effect, such as dynamic free surface deformation, is needed to sustain the oscillation process.
We also analyzed oscillatory thermocapillary flow with concave free surfaces based on two-dimensional simulations. The oscillation mechanism is found to be different from the above mechanism for nearly flat free surfaces. With a concave free surface, the flow is retarded by the presence of a narrow flow passage, called the neck, near the mid-height of the liquid column. In the neck region, the flow cells interact across the liquid centerline. The flow cells compete strongly to pass through the neck beyond a certain Ma, which results in oscillatory flow. Macr is computed under various conditions.
It has been found experimentally that the cold wall temperature (TC) has an important effect on the onset of oscillations. Tests conducted under various cold wall temperatures show that the hot wall temperature at the critical condition is almost unchanged, which means that the critical Marangoni number (Macr) changes when TC is varied. Macr decreases with increasing TC. It is found that the buoyancy-driven airflow, which is induced by the heating-cooling arrangement of the experimental apparatus, is altered when TC is varied. Specifically, the change in the heat transfer rate from the liquid free surface to the air causes this change in Macr. In the past, the effect of the surrounding airflow has been mostly neglected because it is rather weak, but the present finding shows that it is very important to consider the effect, especially when we compare data taken in normal gravity and those taken in microgravity. This effect of TC is not observed when the free surface is concave. The airflow around the liquid bridge is simulated numerically in order to compute the heat transfer rate at the liquid free surface. The result shows that heat is generally lost from the free surface and the heat transfer rate is strongly affected by TC in the parametric range of the present experiment. The heat loss decreases when TC is increased. This decrease in the heat loss makes it easier for the liquid flow to become oscillatory, resulting in a reduction in Macr.
A series of experimental studies of oscillatory flow and transition behavior of high Prandtl number Marangoni convection has been performed. The study has started with the measurement of dynamic surface deformation (DSD) with a single-point-measuring device, i.e., a laser-focusing displacement meter. To measure DSD in more detail, a microscopic imaging displacement meter is developed and validated. It has a spatial resolution of 0.2 mm and a temporal resolution of 60 Hz. With this technique, it is revealed that the magnitude of DSD has a local maximum in the vicinity of the hot disk, called hot corner, particularly under the conditions of 10 cSt silicone oil, 62% in volume ratio, 0.6 in aspect ratio and about 48 oC in disk temperature difference. For larger volume ratios, the magnitude in the hot corner decreases with exhibiting a maximum at around the middle height of a liquid bridge. An axially monotonous variation of phase difference of travelling DSD indicates that the DSD propagates from the lower cold disk toward the upper hot disk for all experimental conditions examined. An interesting phenomenon, i.e., the phase inversion in the hot corner, is also observed in some experimental conditions.
To investigate physical relationship between DSD and surface temperature oscillation (STO), a long-wave-type IR thermometry is introduced into the DSD measurement system. The characteristics of the IR camera are exploited by the development of motion IR-image analyzing procedures that allow time resolved (30 Hz) STO measurements synchronized with DSD measurements. It is revealed that the phase difference between STO and DSD becomes approximately 180 degrees (i.e., out-of-phase) in the hot corner whereas the phase difference outside there is dependent on experimental conditions. This implies that the surface temperature lowers in the hot corner when the surface deforms itself outward and vice versa. The development of a surface velocity measurement technique based on photochromic flow visualization is initiated at this stage.
Careful measurements of STO with a fine thermocouple probe (13 mm in diameter) have shown that the above mentioned phase difference between STO and DSD holds true in the very vicinity of the edge of the hot disk. To understand better the oscillation phenomena in the hot corner, simultaneous observation of flow patterns and DSDs is made by using a sophisticated flow visualization technique. It is found that the temporal variations of flow patterns and DSDs in the hot corner are well correlated. It is conjectured that the local static pressure variations are responsible for the observed variations of flow patterns and DSDs. This encourages the measurement of surface velocity because its acceleration and deceleration should be related to the local pressure variations in the hot corner. An effort toward developing a reliable technique for measuring surface velocity is continued. As the phenomena observed in the hot corner is influenced by the heat transfer from the liquid surface to the surrounding gas, a precise heat loss measurement is performed. It is found that the heat loss measured is larger by a factor of about 3 than that estimated from a natural-convection heat-transfer correlation.
The fundamental equations and the boundary conditions were derived for the concerned configuration with considering the static 2-D and with the dynamic 3-D free surface displacements. Then, (1) code validation concerning the moderate and the high Prandtl number fluids and (2) a series of numerical simulation concerning the intermediate Prandtl number fluid were conducted.
The calculated values of the intermediate Prandtl number fluid were in good agreement with those obtained by linear stability analysis (LSA). Effect of the dynamic surface deformation upon the critical condition of transition was examined for the intermediate Prandtl number fluid; and its effect has been found rather small. The critical Marangoni number with dynamic surface deformation exhibited slightly higher value than the one obtained by the linear stability analysis. In addition, the calculation of the relative displacement induced by a weak thermocapillary convection of the high Prandtl number fluid was conducted in order to make a comparison with the experimental results. The numerical results exhibited a quite good agreement. As for the analysis concerning the high Prandtl number fluid, the effects of the numerical mesh size upon the flow field for the various Marangoni and Prandtl number fluids were investigated. As the results, a criterion was obtained for the required mesh size to resolve the large axial velocities and temperature gradients.
The critical Marangoni numbers with and without dynamic surface deformation were obtained for the intermediate Prandtl number fluid. In addition, the phase differences among the surface deformation, the axis and azimuthal velocity, and the surface temperature variation were analyzed. It was found that the rotating oscillatory flow retained a lower fundamental frequency than the pulsating one, which was also indicated by the terrestrial experiments.
Marangoni convection experiments have been conducted with acetone (Pr = 4.3) and methanol (Pr = 6.8) liquid bridges. In particular, the effects of liquid evaporation on the onset of oscillatory flow were investigated by performing the experiments under different evaporation rates. The effects of horizontal vibrations with small amplitudes were also investigated to understand the magnitude of free surface oscillation induced by such vibrations.
A test section with 7.0 mm diameter disks was constructed to conduct experiments under reduced evaporation conditions by providing a tight seal between the inside and outside of a quartz tube fitted around the liquid bridge. By removing the upper gasket, the evaporation rate could be significantly increased. The liquid bridge height was changed from 2.0 mm to 4.5 mm, covering aspect ratios from 0.55 up to 1.3. Additional experiments were also conducted using disk diameters of 5.0 and 10.0 mm, under strong evaporation rates.
Various advanced measurement techniques, such as photochromic dye activation method, particle image velocimetry, and infrared thermography were used to yield detailed experimental data useful for both understanding the physical mechanism responsible for transition as well as comparing with the predictions of a linear stability analysis and three-dimensional numerical simulation.
From the video images, the liquid bridge shape was digitized and a volume ratio was calculated from the shape profile. Because of evaporation, the shape and volume ratio changed continuously with time. The evaporation rate for the fully sealed system was found to be 1/4 - 1/5 of the rate for the partially sealed system.
For both the strong and reduced evaporation rates, the transition from steady to oscillatory Marangoni convection was characterized by the appearance of harmonic oscillations of small amplitude in the liquid temperature, azimuthal fluctuations in the surface velocity, vortex expansion and contraction, and surface temperature variations that suggested an instability due to hydrothermal waves.
For acetone under reduced evaporation rates, the transition from steady to oscillatory convection was found to occur at disk temperature differences of 0.5 ~ 1.1 K, much less than 1.6 ~ 2.3 K for the partially open system. This difference in the critical temperature difference and corresponding critical Marangoni numbers clearly showed the stabilizing effect of surface evaporation. The surface temperature profile for the partially open system also showed temperature inversion above the lower disk, where a cold region with a temperature below the cold disk temperature existed, as predicted by the linear stability analysis. On the other hand, the evaporation effect was found to be quite small for methanol, and the critical Marangoni numbers obtained agreed with the previous data for NaNO3 with a similar Prandtl number. For both acetone and methanol, the critical Marangoni numbers obtained for an aspect ratio of unity agreed with the predictions of a linear stability analysis, which also correctly predicted the stabilization effect of surface evaporation and an increase in the critical Marangoni number.
The effects of forced vibration on the stability of a liquid bridge surface was also investigated by subjecting the acetone liquid bridge of 7.0 mm diameter to horizontal vibrations under an isothermal condition. The acceleration level was kept at 5 mG but the vibration frequency and amplitude were varied in the experiments. The response of the liquid bridge surface was recorded by a video camera and the surface motion was analyzed for different volume and aspect ratios. The results indicated existence of resonance frequencies at which the surface oscillates with large amplitudes, and a strong effect of the volume ratio on the oscillation amplitude. The maximum surface oscillation amplitude as high as 250 mm was recorded at applied vibration frequencies of 4 and 12 Hz. The resonance frequencies at which the liquid bridge vibrated with large amplitudes decreased linearly with the height of the liquid bridge.
Selection of a fluid and solid material for sustaining a liquid bridge was carried out at preliminary works. The selected fluid was molten tin of which Pr number is identical with that of molten silicon (Pr=0.01). Since the melting point of tin is much lower than that of silicon, there is no need to use an infrared image furnace for melting a tin sample and an electric heater is applicable to melting it. Thus, the surface temperature of molten tin can be measured by a non-contact diagnostic with relatively low noise level. The high purity iron was selected for the solid material from the standpoint which moderate wettability and low reactivity against molten tin is required for the solid.
The difficult problem of oxidation at the tin surface has been successfully overcome. Consideration concerning the surface science of tin led us to design of a unique experiment apparatus where the clean surface of molten tin is able to obtain by an Ar+ ion etching and sustain under the high vacuum condition during an experiment. A non-contact measurement technique of the surface temperature has been also developed for detecting small amplitude of temperature fluctuation at around the Mac2. It was confirmed that the radiation thermometer equipped with the PbS photo detector had the sufficient temperature resolution to detect the Mac2 with high accuracy. The performances of the experiment apparatus equipped with those devices mentioned above have been already confirmed.
Surface temperature fluctuation at transition to oscillatory flow was successfully observed by the non-contact measurement. The flow transition was verified by comparison of the experimentally obtained Mac2 and frequency of the surface temperature fluctuation with the numerical results, and by a surface flow visualization experiment directly. The effect of the liquid bridge geometry on the Mac2 and oscillation frequency was further investigated and discussed.
For an internal flow measurement, a novel visualization technique using an ultrasonic transducer and a unique balloon-like tracer for the visualization has been studied experimentally. A critical condition on the Mac1 and a detail structure of oscillatory flow will be clarified by this measurement technique.
Numerical simulation of three-dimensional unsteady Marangoni convection in half-zone liquid bridges of medium and low Prandtl number fluids has been conducted. In the early stage of the project (1997 – 1998), modeling and numerical code were developed. In 1998 and 1999, numerical simulations had been conducted on the medium Prandtl number fluid cases: Pr=1.02. The mathematical model of the liquid bridge is rather primitive, i.e., a liquid bridge (length L, radius a) is suspended between two circular disks and surface of the liquid bridge is assumed to be cylindrical and adiabatic. Fundamental equations, such as: equation of continuity, Navier-Stokes equation and energy equation, were numerically solved by using the finite difference method with non-uniform grids and an explicit HSMAC scheme. Numerical results revealed that Marangoni flow develops quickly right after one of the discs is suddenly heated to and maintained at a constant temperature, while the other disc is kept at the initial temperature. At the beginning, flow field looks as if it is an axi-symmetric. However, soon three-dimensional oscillatory disturbances are automatically induced and increase their scale exponentially with time. At the early stage, most of them are pulsating. Later, they have been taken over by a rotating mode, where a three-dimensional flow and temperature fields are rotating in azimuthal direction with a constant r velocity. Numerical results also reveal that the wave number of the disturbances in azimuthal direction, m, is strongly dependent on the aspect ratio of the liquid bridge; As=L/a. Value of m varies between 1 in long bridges (As=1.5-1.8) and 3 for shorter bridges (As=0.66) in our simulations. We determined the critical condition for the incipience of flow transition from an axi-symmetric steady to an oscillatory three dimensional flow as the Marangoni number at which the growth rate constant became zero. Thus obtained critical Marangoni numbers showed very good agreements with those of linear stability analyses.
Since FY1999, we have been conducting numerical simulations of the Marangoni flow in low Prandtl number liquid bridges. The results for Pr=0.01 and 0.02 fluids confirmed that there occurs a first flow transition from an axi-symmetric steady to a three dimensional steady flow as was pointed out by linear stability analyses and also by non-linear numerical simulation of Levenstam and Amberg (1998). At higher Marangoni number, the second flow transition from three dimensional steady to an oscillatory flow occurs. We confirmed the first and second critical Marangoni numbers (or Reynolds number) for As=1.0 are very close to those of Levenstam et al. Further numerical works provided us with the first and second critical conditions for liquid bridges of different aspect ratios (As=0.8 –1.8).
In FY2000, we have developed a new numerical code in which temperature field is calculated by an implicit method. This modification provides reasonable calculation speed for small Prandtl number fluids. In FY2000, we also conducted series of numerical simulations for zero Prandtl number fluid. The results help understand general trend of the flow transition in liquid bridges of low Prandtl number fluids. Present work showed the first and the second critical Reynolds number for Pr=0 over a wide range of the aspect ratio, As=0.6 ~ 2.2. The wave number m of the steady is dependent on As; m=1 at As>1.5, m=2 for 1.5>As>0.8 and m=3 for As<0.8. The wave number of the oscillatory disturbances are slightly different from those, i.e., m=1 for As>2.0, m=1 for 2.0>As>0.8 and m=3 for As<0.8. Oscillation mode is classed into several types; rotating type, pendulum type, twisting type and more complex combined oscillations accompanied by slow alternation of wave numbers, such as between m=1 and m=2. At much larger Reynolds numbers, oscillatory flow becomes more complex and chaotic.
The main focus of the Marangoni-convection-modeling-research project is the exploration of the transition to oscillations of the thermocapillary flow in the half-zone model. To treat this problem a linear stability analysis of the basic two-dimensional flow has been employed. This method takes advantage of the normal-mode structure of the deviations from the two-dimensional flow at the critical threshold. Hence, the dependence of the velocity and temperature fluctuations on the azimuthal coordinate enters the problem via an integer azimuthal wave number and merely a two-dimensional problem remains to be solved by appropriate discretization techniques.
The first effort aimed at solving the stability problem for cylindrical liquid bridges by combinations of finite differences and Chebyshev collocation for high Prandtl numbers. Grid convergence of the linear stability analysis has been obtained for Prandtl numbers up to about 15. An a posteriori energy analysis showed that the instability is due to the hydrothermal-wave mechanism. For higher Prandtl numbers full grid convergence was not obtained by the present method which employs inverse iteration to determine the most unstable mode, even with 165 x 165 finite difference points (complex arithmetics) or 85 x 340 Chebyshev collocation and stretched finite difference points, respectively. The lack of suitable preconditioners prevented the efficient use of iterative methods by which much larger problems could have been treated. The results indicated, however, that for unit aspect ratio a change of modes from m = 2 to m = 1 occurs near Pr = 20, and that the critical Marangoni number should tend to an asymptotic value for Pr → ∞.
In addition, the effect of liquid evaporation has been calculated in the parameter range (Prandtl numbers) for which reliable results have been obtained. The most significant effect is not the mass loss, but rather the evaporative cooling at the free surface. The numerically obtained critical Marangoni numbers were found to be in good agreement with measurements for acetone which have been made in parallel by the group of M. Kawaji. It was found that the evaporative cooling stabilizes the basic flow. The favorable comparison with the experimental data allowed an interpretation of the instability mechanism that was operative in the experiments. The mechanism is primarily of hydrothermal-wave type, even though inertia effect become important due to the increase of the critical Reynolds number due to the basic-flow stabilization.
By using body-fitted coordinates the linear stability has also been calculated for moderately large-Prandtl-number flows in half zones with a free surface, deformed due to the volume of liquid and the hydrostatic pressure difference in a gravity field. The effect of various parameters like gravity, volume fraction, Prandtl number, and aspect ratio on the linear stability boundary has been investigated and the instability mechanism has been discussed. For 0 < Pr < 15, the mechanism is found to be of hydrothermal-wave type throughout. The energy analysis allowed a detailed understanding of how the above factors influence the stability boundary.
Finally, it was shown by an asymptotic analysis that dynamic surface deformations cannot play an active role in the instability process as long as the Capillary number is small, because the deformations decouple from the first-order linear stability problem. Combining the asymptotic analysis with the numerical linear-stability analysis the structure of the dynamic surface deformations has been predicted for the slightly supercritical oscillatory three-dimensional flow.
Following feasibility studies of on-board cell for Marangoni convection experiment were conducted.
Feasibility study of on-board experimental cell for low Prandtl number fluid that will be installed in Fluid Physics Experiment Facility (FPEF) was conducted.
Experimental requirements were to use Sn, Pb-Bi and Hg as low Prandtl number fluid and to purge Hydrogen gas or to get a high-vacuum atmosphere in experimental chamber. Based on these requirements, studies in the areas (system configuration and components, containment method of sample, liquid bridge formation method, compatibility with FPEF, safety, installation) of development were conducted.
From these studies, the focal points were extracted and development plan was constructed.
As the result of this study, the experiment cell satisfied the experimental requirements except for its envelope. With regard to envelope, it became clear that study of further miniaturization of components in experiment cell or negotiation with JEM system about the expansion of envelope for experiment cell is needed. For safety, since toxicity of low Prandtl number fluids and fire (or explosion) due to use of Hydrogen gas are highly critical, it became clear that detailed study of containment method of experimental chamber, for example triple seals or welding etc., is needed hereafter.
For measurement of a dynamic surface deformation for high Prandtle number liquid bridge, feasibility study of on-board experimental cell that will be installed in FPEF was conducted. Experimental cell for dynamic surface deformation measurement was studied based on on-going experimental cell development with surface flow velocity measurement.
Experimental requirements were to form a liquid bridge with diameters of 5mm, 10mm, 30mm, and 50mm, and to measure dynamic surface deformation and surface temperature at the same position of the liquid bridge. Based on these requirements, the studies (optical system configuration for dynamic surface deformation measurement, liquid bridge formation method for small diameters, compatibility with FPEF, safety, installation) were conducted.
From these studies, the focal points were extracted and development plan was constructed.
Although liquid bridge formation for small diameters will need to be established by tests, the study shows that the experiment cell for dynamic surface deformation measurement was confirmed to satisfy the experiment requirements. Therefore, this feasibility study demonstrated the possibility of conducting dynamic surface deformation measurement experiments on-board.
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