Exp Fluids (2012) 53:21–35 DOI 10.1007/s00348-011-1224-x
RESEARCH ARTICLE
Dynamic structures of bubble-driven liquid flows in a cylindrical tank Ju Sang Kim • Sang Moon Kim • Hyun Dong Kim Ho Seong Ji • Kyung Chun Kim
•
Received: 20 December 2010 / Revised: 16 October 2011 / Accepted: 1 November 2011 / Published online: 15 November 2011 Ó Springer-Verlag 2011
Abstract The spatial and temporal structures of turbulent water flows driven by air bubbles in a cylindrical tank were investigated. The time-resolved particle image velocimetry technique was adopted for quantitative visualization. Flow rates of compressed air were changed from 1 to 5 L/min at 0.5 MPa, and the corresponding range of bubble-based Reynolds number (Re) ranged from 8,300 to 21,100. The dynamics of flow structures was further investigated by the time-resolved proper orthogonal decomposition analysis technique. With increasing Re, mean velocity fields driven by the rising bubbles are almost same, but turbulence is dramatically enhanced. Both spatial and temporal modes were quite different with respect to the air flow rates. Three most dominant spatial structures are recirculating flow, bubble-induced motion, and sloshing of free surface, the bigger the latter the higher Re. We found the frequency of sloshing motion from flow visualization and the FFT analysis of temporal modes.
1 Introduction Bubble-driven flows comprise a large number of different flow situations, e.g., dispersed pipe flows, flows in multiphase agitated tanks, flows in multiphase fluidized-bed
J. S. Kim H. D. Kim H. S. Ji K. C. Kim (&) School of Mechanical Engineering, Pusan National University, Busan 609-735, Korea e-mail:
[email protected] J. S. Kim LG Electronics Co., Changwon, Korea S. M. Kim Doosan Heavy Industry and Construction, Changwon, Korea
reactors, and typical bubble-column flows. Especially in the chemical engineering application and industrial field, the mixing problems such as powder dispersion, solid blending, and gas dispersion into liquid have been important issues because product quality and productivity highly depend on the mixing process (Luewisutthichat et al. 1997; Tirto et al. 2001; Tu and Tra¨ga˚rdh 2002). Driven by high-volume moderate-velocity gas injection into liquid baths, such flows are often chaotic and turbulent and can feature a wide spectrum of interfacial length scales. The turbulence resulting from large bubble injection alters the characteristics of the heat and mass transfer in bubbling flows. There are flow situations where the bubble movement itself is the main source of momentum to the flow field and are often characterized by low superficial liquid velocities, relatively high superficial gas velocities, and no mechanical support of the flow. When gas-phase fluids issued into a liquid medium, bubbles are generated and vertically rise to the free surface due to buoyancy force. Recirculating flow is more often than not a predominant feature even at relatively low gas flow rates; gas fractions are generally high, leading to intense interaction between bubbles, often with rapid bubble coalescence and breakup and large variations in bubble size and shape. So even if the bubble-driven flow situations are probably of equal or even higher industrial importance than typical pipe flow problems, they seem to lag behind when it comes to fundamental understanding, modeling ability, and predictability. Mahalingam et al. (1976) performed velocity measurement in two-phase bubble-flow regime with laser Doppler anemometry (LDA) and observed that the errors in the measurements increase with increases in bubble size. Durst et al. (1984, 1986) conducted experimental and numerical studies on bubble-driven laminar flows by investigating
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liquid circulation and bubble’s motion with an LDA system. It was reported that the liquid-phase circulation pattern is not sensitive to the actual shape of the void fraction profiles. Johansen et al. (1988) studied the fluid dynamics in bubble-stirred ladles by employing an LDA to measure the axial and radial mean velocities and fluctuating velocities of the liquid phase. Air was supplied through a porous plug placed in the bottom wall of a cylindrical perspex model of a ladle. Hassan et al. (1992) experimentally studied a two-phase dispersed air bubble mixing flow within a rectangular vessel. The technique of particle image velocimetry (PIV) was utilized in order to obtain non-invasive velocity measurements of the resulting bubble flow field and its induced effects upon a surrounding liquid medium. Tokuhiro et al. (1998) investigated turbulent flow past a bubble and an ellipsoid in a square channel using shadow image and PIV techniques with 2 CCD cameras. Degaleesan et al. (2001) studied liquid recirculation and turbulence in bubble columns by using the computerautomated radioactive particle tracking (CARPT) technique. The quantitative analysis on the Lagrangian velocity data obtained using the CARPT technique provided insight into the fluids dynamics of the liquid phase in 3-D cylindrical bubble columns operated with different gas distributors and at different superficial gas velocities. Choi et al. (2002) investigated a single bubble motion in stagnant water by using flow visualization and image processing method. Two-dimensional water velocity fields and the motion of a rising bubble in the water were simultaneously measured by PIV and PTV, respectively. Lindken and Merzkirch (2002) introduced a new experimental procedure for performing simultaneous, phase-separated velocity measurements in two-phase flows. Their technique was a combination of the three most often used PIV techniques in multiphase flows: PIV with fluorescent tracer particles, shadowgraphy, and the digital phase separation with a masking technique. The advantage of this technique is that the signals of the two phases do not disturb each other. They demonstrated this technique for multiphase flows is able to measure velocities in two-phase flows with a precision that is high enough to compute turbulent derivations from the data. Fujiwara et al. (2004) conducted an experimental investigation into vertical pipe flow injected with dispersed bubbles using PIV, laser-induced fluorescence (LIF), and the projecting shadow image technique. They found that a high concentration of bubbles in the vicinity of the wall induces a reduction in the fluctuation velocity intensity of the liquid. However, the external force affected by the bubbles is expected to contribute to the production of turbulence energy. Recently, Montante et al. (2008) measured the turbulent gas–liquid flow and bubble size distribution in aerated stirred tanks using a two-phase PIV and a digital image
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processing method based on a threshold criterion. However, their experiments did not use a pure bubble-induced flow; rather, a Rushton turbine was included in the mixing tank. Kim et al. (2010a) experimentally studied the recirculation flow motion and turbulence characteristics of liquid flow driven by air bubble stream in a rectangular tank by using the time-resolved PIV technique. It was observed that the large-scale recirculation resulting from the interaction between rising bubble stream and side wall is the most dominant flow structure, and there are smallscale vortical structures moving along with the large-scale recirculation flow. Kim et al. (2010b) performed an experimental study to evaluate dynamic structures of flow and turbulence characteristics in bubble-driven liquid flows in a rectangular tank by using PIV and proper orthogonal decomposition (POD) method. With increasing Reynolds number (Re), bubble-induced turbulent motion becomes dominant rather than the recirculating flow near the side wall. Although they did extensive experimental studies to find dynamic structures in bubble-induced liquid flows, the rectangular tank they used has intrinsic corner flows, which cannot provide the flow is exactly axisymmetric nature. Moreover, they remained free surface oscillation interacting with the bubble-driven turbulence in question. Over the past decade, many techniques have been used for the identification and characterization of coherent structures. These include conditional sampling, wavelets, pattern recognition analysis, POD, stochastic estimation, and topological concept–based methods (Tabib and Joshi 2008). The most emphasized advantage of the POD technique is that it converges optimally fast in quadratic mean, compared with any other expansion. Due to the lack of detailed experimental data for bubbledriven mixing flow, accurate velocity field data with appropriate time and space resolution are needed. Since the mixing characteristics in bubble-driven turbulent liquid flow is strongly governed by large-scale motions in the flow field, analysis of the large-scale dynamics is necessary to understand the mixing mechanism. In addition, interaction between free surface sloshing and turbulent structures in bubble-driven liquid flow has not been studied in detail so far. In this study, we aim to measure the entire flow field and analyze the large-scale dynamic structures and turbulent characteristics in bubble-driven water flow in a cylindrical tank using time-resolved PIV and POD techniques for the bubbling mixer. The potential application of this study is to understand mixing characteristics in the vitrification process of radioactive wastes using an induction cold crucible melter (CCM) to find optimal operating conditions and design guidelines. In this application, the only possible way to mix molten glass with radioactive wastes is adopting the bubble mixing technique. It is known that increasing air flow rate may increase mixing.
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However, there must be an optimal condition for the most efficient mixing performance. To find the optimal mixing condition, it is needed to understand the mechanism of turbulence production in bubble-driven flows due to airbubbling.
2 Experiment Figure 1 shows a schematic diagram of the experimental setup and the coordinate system employed in this study. A cylindrical tank was made of transparent acrylic pipe with an inner diameter of 0.3 m. The cylindrical tank was placed in a rectangular tank to avoid optical refractions due to the curved surface of the inner tank. Tap water was used as the working fluid, and compressed air formed the bubble stream. The height of the free surface is 0.15 m from the bottom wall. An air injection nozzle with a nozzle diameter (DN) of 5 mm was placed at the center of the bottom wall. The exit of the nozzle was located 35 mm above the bottom wall. Compressed air was supplied to the nozzle through a regulator valve, and a flow meter was used to precisely control the flow rate of air. The flow rate of compressed air varied from 1 to 5 L/min at an air pressure of 0.5 MPa. To reduce the diffused reflections originating from the bubbles and the nozzle, fluorescent polymer particles (Dantec Dynamics, 10 lm) were used. The particles are based on melamine resin and fluorescent dye, and Rhodamine B is homogeneously distributed over the entire particle volume. The maximum excitation wavelength of the particle is 550 nm, and the maximum emission wavelength is 590 nm. A laser beam 5 mm in diameter originating from a 532-nm diode CW laser passed through a spherical lens and a cylindrical lens and turned into a 2-mm-thick sheet beam. The sheet beam irradiated the object plane through the transparent side walls. A 10-bit high-speed CMOS camera (1,280 9 1,024 pixels, pco. 1200 hs, PCO)
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was used with a 545-nm long-pass optical filter mounted its lens to eliminate diffused reflections. An image intensifier (UVI 507, Invisible Vision Ltd) was used to acquire enhanced clean particle images. Figure 1b shows the coordinate system defined in this study and the field-of-view of the measurement section. The size of the field-of-view is 130 mm 9 100 mm. It was chosen to acquire the whole field of bubble-driven water flow just below the oscillating free surface and beside the bubble stream to avoid optical disturbances. A typical twodimensional time-resolved PIV system was adopted. For each case, 1,000 images were acquired and evaluated. The time interval between each image was 10.0 ms [100 frames per second (FPS)], and the images were interrogated using a two-frame cross-correlation technique. Interrogation windows were taken to be 48 9 48 pixels, respectively; the FFT window size was set to 64 9 64 pixels, and a 50% window overlap was adopted. False vectors in the raw vector fields were eliminated by using magnitude difference technique with 80% threshold. The eliminated vectors were interpolated with 3 9 3 Gaussian convolution.
3 Results and discussion 3.1 Flow visualization Flow visualization studies were conducted to observe bubble formation, detachment, rising behaviors, shape evolution, free surface oscillation, and interaction between bubbles and free surface. Images were recorded in the rate of 100 FPS by a high-speed camera (10 bit, grayscale, CMOS, pco. 1200 hs) which has 1,280 9 1,024 pixels. The measurement volume was irradiated by a 230-W halogen lamp (Philips) which was located above the tank. To separate bubble images more clearly from the background, small amount of Rhodamine B solution was mixed into water during flow visualization. Since we have used a
Fig. 1 Schematic of experimental setup and measurement section. a Experimental setup, b coordinate system, and field-of-view
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Fig. 2 Flow visualization image of bubble stream and free surface fluctuation in the tank; a case I, QA = 1 L/min, b case II, QA = 3 L/min, c case III, QA = 5 L/min, and d ensemble-averaged gray level at the height of Y/DN = 19 (115 mm) from the bottom wall
545-nm low-pass optical filter, the gray level of bubble image is lower than that of the background as shown in Fig. 2. The mean diameters and the mean velocity of bubbles were evaluated through the recorded flow visualization images of each case. The sloshing frequency of the free surface was evaluated, and the frequency of bubble hitting on the free surface was also estimated. Dimensionless parameters, Re and Eo¨tvo¨s number (Eo), were calculated with the obtained flow parameters and confirmed the shape of bubble by the bubble shape chart from Clift et al. (1978). Figure 2 shows the flow visualization image of bubble stream and free surface fluctuation in the tank. The detached bubble size at the nozzle exit is about three times bigger than the nozzle diameter; however, bubbles are combined together, and then, the final size of the rising bubbles near the free surface becomes 4–6 times bigger than the nozzle diameter. The detached bubble size at the nozzle exit becomes bigger as the flow rate of compressed air is increased. It was observed that the bubbles moved with more vigorous agitation and the fluctuation of the free surface became stronger with increasing the air flow rate. In case III (Fig. 2c), the bubble keeps changing its rising direction toward a lowest surface level induced by the variation of the free surface.
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The flow field in the bubble stream is essentially classified into three regions with respect to the axial distance from the nozzle exit. The first region is located near the nozzle where the inertia force of injected gas plays an important role. The second region is located far from the nozzle in which the buoyancy force of bubbles governs the flow. The third region is near free surface where bubbles are destroyed and interacted with the free surface motions. Figure 2d shows the averaged diameter of bubbles. The detached bubble size can be measured and averaged by counting pixels from consecutive image; however, it is difficult to measure the size of bubbles beneath the free surface because of oscillation and deformation of bubbles. Therefore, after adopting the image processing techniques such as adaptive smoothing and binarization of the images, the gray levels are extracted at the line Y/DN = 19. From 1,000 visualization images, the gray levels are ensembleaveraged as shown in Fig. 2d. As the gas flow rate is increased, the width of profile at the same gray level is increased as expected. Through this process, the diameter of bubbles is estimated with the full width at half maximum (FWHM) and listed in Table 1. The diameters of bubbles beneath the free surface are increased about 50% of those detached at the nozzle exit due to agglomeration.
Exp Fluids (2012) 53:21–35 Table 1 Parameters and dimensionless numbers
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Case
DN (mm)
QA (L/min)
hDB i (mm) at the nozzle exit
hDB i (mm) beneath the free surface
h VB i (m/s)
I
5
1
12.2
18.1
0.460
8,326
3
16.7
24.2
0.596
14,423
79
III
5
21.0
29.6
0.747
22,111
118
inertial force qw VB DB ¼ viscous force lw
ð1Þ
Eo¨tvo¨s number: Eo ¼
Eo
II
The averaged rising velocity of bubbles is calculated by dividing the distance between the nozzle exit and the free surface by bubble rising time to reach the free surface after being detached from the nozzle. It should be noted that the rise velocities of the bubbles are not exactly proportional to the air flow rates due to the complicated relations in force balance. However, the bubble rising velocity near the free surface is almost same as the terminal velocity (Kim et al. 2010a, b). Dimensionless numbers, Re and Eo were calculated based on the averaged diameters of bubbles beneath the free surface and averaged rising velocity according to the Eqs. 1 and 2. Reynolds number:Re ¼
Re
lift force gdqD2B ¼ surface tension force r
ð2Þ
where qW is the density of working fluid, VB is the mean rise velocity of bubbles, DB is the mean diameter of bubbles, which is calculated from images, lW is the absolute viscosity of working fluid, g is the acceleration of gravity, Dq is the difference in density of the two phases, and r is a surface tension of the interface layer. The obtained dimensionless numbers are listed in Table 1. When Eo and Re numbers of three cases are plotted on the shape regime graph, all three cases are located in the regimes of ‘‘spherical-cap’’ shape (Clift et al. 1978), and Fig. 2 verifies that the shape of bubbles (cone shape) is similar to the ‘‘spherical-cap’’ shape literally. Because the air bubbles are disappeared at the free surface, there is no gas phase in the measurement volume. Gas bubble velocity in liquid is one of the parameters characterizing the bubble behavior, which decides the gasphase residence time and hence the contact time for the interfacial transport and subsequently contributes to the performance of the equipment. Gas bubbles experience a slip and the presence of viscous and inertial forces. The main forces, which act on a bubble during its motion in liquid, are gravity, buoyancy, drag force, history force (arising from the unequal distribution of vortices), and finally, lift force (due to either developed pressure gradients or unsteadiness of the flow). The
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added mass is defined as the acceleration of the fluid displaced by the bubble acceleration. This force is important for a bubble in unsteady motion when it accelerates or decelerates, e.g., near the nozzle or in a turbulent flow with swirl. History force is described as the integral of all past bubble accelerations, which represents the fading memory effect of the relative acceleration between the bubble and the surrounding liquid. The form of the history force arises from the generation of vorticity at the surface of the bubble to diffuse into the surrounding fluid to describe the initial motion of bubble. It becomes substantial when the bubble is accelerated at a high rate. The contribution of individual forces varies from case to case. As the superficial gas velocity increases, bubble rising velocity increases followed by increasing turbulence in the tank. The frequency of the free surface sloshing and the frequency of bubble hitting the free surface were evaluated by image processing techniques and FFT analysis. Figure 3a shows a raw image and the locations where the gray level was extracted. Figure 3b depicts the processed image by the adaptive smoothing technique and the binarization process. The adaptive smoothing scheme used in this study is the following equation: Pþ1 Pþ1 t t n¼1 1 Iiþm;jþn Ciþm;jþn tþ1 Ii;j ¼ ð3Þ Pþ1 Pþ1 t n¼1 1 Ciþm;jþn where 2 Ci;j rIi;j ¼ eðkrIi;j k=kÞ and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ffi rIi;j ¼ 1 Iiþ1;j Ii1;j þ Ii;jþ1 Ii;j1 : 2 The constant k value was chosen through some trials that showed a good result. Using 1,000 time series of binarized image level at the chosen locations [(a) and (b) denoted in Fig. 3b], the FFT analysis was performed. Figure 3c shows the result of FFT at the point (a) for case III, QA = 5 L/min to find the bubble hitting frequency. The primary dominant peak of the bubble hitting frequency is 6 Hz, and the secondary peak is 8 Hz. By the inspection of time-resolved raw image, the hitting frequency of the bubbles at the free surface in case III was about 7 Hz. Figure 3d depicts the result of FFT at the point (b) in
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Fig. 3 Data extraction points in the measurement section and FFT analysis results for the case III. a Raw image, b binary image, c bubble hitting frequency, d free surface sloshing frequency
Fig. 3b for case III to find the sloshing frequency of the free surface. We found a dominant peak at 3 Hz, which is the oscillating frequency of the free surface interacting with large air bubbles in case III. 3.2 Time-resolved PIV measurement Although the cylindrical tank is located inside the rectangular tank to minimize optical distortions, it should be checked before the PIV experiment. A grid target was placed in the image plane inside the cylindrical tank. The grid points were extracted by the centroid searching technique with a sub-pixel algorithm. The mean value of deviations is 0.512 pixel, and the variance is 0.284 pixel. To confirm the image distortion, the averaged distance between the neighborhood grid points along width and height were measured. The averaged distance along width is 23.651 pixels with a variance of 0.088 pixel, and the averaged distance along height is 23.681 pixels with a variance of 0.066 pixel. These values are comparable with the sub-pixel resolution (0.1 pixel), so that the image distortion can be ignored.
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Figure 4 shows the time-averaged velocity fields induced by bubbles for each case. A total of 1,000 instantaneous velocity fields were ensemble-averaged to obtain the mean velocity field. We note that there is upward flow driven by the bubble stream on the left half of the vector plots, and a clockwise large vortical structure is located at the upper right side. The rising bubbles are terminated at the free surface; therefore, the vertical moments are converted to the horizontal flows. The horizontal flows can be blocked by the tank wall, and then the flow moves down along the side wall. That is the reason why a clockwise vortex can be seen in the right-hand side of rising bubbles. We also observed inward flow at the leftbottom side of view, induced by rising air bubbles. When we compared the time-averaged velocity fields of each case, we observed the movement of the center of the vortical structure. As the flow rate of compressed air was increased, the center of the vortex moved downward. In case I, the core location of the structure is X/DN = 22 and Y/DN = 15, but in case III, the core location of the structure moved to X/DN = 22 and Y/DN = 13. We believe that the increased flow of compressed air transferred more
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Fig. 4 Mean velocity vector fields for each case and velocity profiles at Y/DN = 15 and 13. a Case I, QA = 1 L/min, b case II, QA = 3 L/min, c case III, QA = 5 L/min, d hV i component velocity profile
upward kinetic energy to the working fluid, the faster moving working fluid gained more inertial force, and this increased inertial force pushed the vertical structure downward. The mean vertical velocity profiles along the horizontal line crossing the vortex core for each case (Y/DN = 15 for case I and Y/DN = 13 for case II and III) are compared in Fig. 4d. With increasing air flow rate, the vertical velocity induced by the gas bubbles become stronger as shown in the left half of Fig. 4d; however, surprisingly, the mean velocity profiles across the vortex are the same in all cases. The power input from the compressed air is proportional to the air flow rate since the air pressure is constant. It means cases II and III provide 3 and 5 times higher energy input to water in the tank compared with case I. As shown in Fig. 5, the mean velocity field cannot reflect the increased energy input due to the increased air flow rate. Instead, the increased energy can be consumed by turbulence and sloshing motion of free surface. It should be noted that the
magnitude of vertical velocity is quite low (0.020– 0.030 m/s) compared with the bubble rise velocity (0.460–0.747 m/s). The contour plots of Fig. 5 show the time-averaged turbulent kinetic energy field of each case. The summation of turbulent kinetic energy in the field of view for cases I–III is 0.190, 0.393, and 1.082 m2/s2, respectively. As the gas flow rate is increased, the turbulent kinetic energy level is increased dramatically over the measurement section. In case I, the greatest turbulent kinetic energy is located at the right side of bubble stream as shown in Fig. 5a. Turbulence can be generated in the wake of the rising bubble. The effect of free surface seems weak in case I. In case II, the peak values of turbulent kinetic energy zone are found in the recirculating region near the free surface and in the vicinity of the bubble rising region as shown in Fig. 5b. In case III, the high values of turbulent kinetic energy are found near the free surface due to the strong sloshing motion as shown in Fig. 5c. We conclude that the
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Fig. 5 Contour plot of turbulent kinetic energy for each case and normalized values of integrated quantities with respect to air flow rates. a Case I, QA = 1 L/min, b case II, QA = 3 L/min, c case III, QA = 5 L/min, d variation of integrated quantities
dominant mechanism of turbulence generation strongly depends on the Re of the bubble-driven liquid flows. Figure 5d represents normalized values for averaged bubble rising velocity, overall kinetic energy, and turbulent kinetic energy according to the increment of the flow rate of compressed air. The overall kinetic energy in the measurement plane can be obtained by summating the addition of the mean kinetic energy and the turbulent kinetic energy over all grid points in the measurement section. The resulting overall kinetic energy for cases I, II, and III are 0.672, 1.033, and 1.675 m2/s2, respectively. The relative increases in the overall kinetic energy in cases II and III are 1.539 and 2.494 times higher compared with those in case I, respectively. These values are comparable with the square of the relative increments of averaged bubble rising velocity. Bubble rising velocities in cases II and III are 1.296 and 1.624 times higher than that in case I, respectively, so that the square values are 1.680 and 2.637, respectively.
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3.3 POD analysis Dynamic information of the flow field can be explained effectively by POD, from which the relative energy distribution is acquired. By using the POD technique as proposed by Lumley et al. (1993), the flow field can be decomposed into optimal orthogonal spatial modes and optimal orthogonal temporal modes. The POD for analytic study on the complex flow fields decompose a flow field U(x, t) into its POD mode, a(t) and u(x), which can reconstruct the original field by superimposing each POD mode. X Uðx; tÞ ¼ aðnÞ ðtÞuðnÞ ðxÞ ð4Þ n
where the u(n)(x)’s are eigenfunctions of the (Fredholm) integral equation. The optimality means that POD minimizes the mean square error of any partial sum of expansion and conversely the expansion in terms of these bases obtained by POD converges faster than the expansion
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in terms of any other basis such as Fourier decomposition. The spatial modes can be obtained by solving the following integral equation: Z 0 ðnÞ 0 Rðx; x0 Þu ðx Þdx ¼ kðnÞ uðxÞ ð5Þ x
where k(n) is the eigenvalue and is ensemble-averaged twopoint correlation function. The orthogonality of spatial basis and temporal basis is represented as followings as: D E 0 /ðnÞ ðxÞ; /ðmÞ ðx Þ ¼ dnm ; aðnÞ ðtÞ; aðmÞ ðtÞ ¼ kðnÞ dnm ð6Þ where hi denotes the ensemble average. Proper orthogonal decomposition minimizes the mean square error of any partial sum of expansion and conversely the expansion in terms of these bases obtained by POD converges faster than the expansion in terms of any other basis. The traditional direct method required a huge computation time, but the snapshot method proposed by Sirovich (1987) can decrease computation time. A high number of snapshots are required to evaluate the periodicity of the coherent structures in the flow because the systematic fluctuation occurs repeatedly. In this study, POD analysis is performed by snapshot method with fluctuating fields as the below equation. Uðx; tÞ ¼ hUðx; tÞi þ uðx; tÞ M X ¼ hUðx; tÞi þ aðmÞ ðtÞkðmÞ ðxÞ
ð7Þ
m¼1
In POD analysis of this study, the eigenvalue represents turbulent kinetic energy of each mode, and the total number of modes is 1,000 since all the PIV data sets are used for POD analysis. The time-mean flow field has 72.3% of total kinetic energy in case I. It means the dominant dynamic structure in the set of instantaneous velocity fields is the time-averaged recirculating motion. It is shown that most of turbulent kinetic energy is concentrated on the first few percent of the total number of modes. As an example, the first 20 modes have 88.6% of total kinetic energy in case I. With increasing Re, the energy-cascading process begins at higher modes, and the energy spectrum of turbulent transport becomes wider. In case III, 40.4% of total kinetic energy corresponds to the mean flow field, and the first 20 modes have 77.0% of total kinetic energy. Figures 6, 7 and 8 show the four spatial and temporal modes in order. They represent the dynamics of large-scale motions in turbulent mixing flow for three different flow rates, respectively. The temporal modes represent the time variation of the amplitude (or strength) of the corresponding spatial modes. Since the sampling time interval (s) of image acquisition was 0.01 s, the total time interval
for the POD analysis is 10 s, which is enough to resolve the free surface oscillation. Figure 6 demonstrates initial eigenmodes at the flow rate is 1 L/min. The first spatial mode resembles the mean recirculating flow, which contains the largest turbulent kinetic energy. Since the bubble rise is not so violent, instantaneous velocity vector fields are very similar with the first temporal mode. Several small-scale vortices are observed in the 2nd spatial mode as shown in Fig. 6c. The vortices are distributed along the bubble stream. In the 2nd spatial mode, a vertical upward flow is observed between the bubble stream and the recirculating flow. Figure 6e, g depicts the third and forth eigenmodes, which reveal the large-scale structure becomes smaller and smaller through the turbulent energy cascading process. Up and down motions of free surface are recognized where bubbles are reached. Higher modes appeared in Fig. 6e, g are not the dominant structures in the overall flow field compared with the 1st and 2nd modes; however, those motions certainly contribute to turbulent mixing behavior. The temporal mode represents the time variation of energy containing in the corresponding spatial mode. It can be conjectured that the temporal mode illustrates associated time scales with the corresponding spatial mode. The 1st temporal mode of case I shows very slow sinusoidal oscillation (Fig. 6b). The 2nd temporal mode has the same frequency of oscillation, but there is a phase lag between the 1st and 2nd temporal modes as shown in Fig. 6d. The 3rd and 4th temporal modes shown in Fig. 6f, h have discernible fluctuations, but the frequency of oscillation is twice than that of the 1st and 2nd mode modes. Figure 7 shows large-scale motions in case of the flow rate of 3 L/min. The 1st spatial mode represents the recirculating motion induced by the bubble stream. An interesting feature is that the high-momentum zone of the 1st spatial mode (Fig. 7a) coincides with the high-turbulent-kinetic-energy regions shown in Fig. 5b. The 2nd and 3rd spatial modes depict up and down vertical motion due to the free surface sloshing as shown in Fig. 7c, e. The 4th spatial mode (Fig. 7g) demonstrates a sink-like flow pattern associated with bubble wakes. As the flow rate of air increased, the velocity of bubbles increased, and then, the associated vibrating motion generates more turbulence. The increased kinetic energy of the bubbles creates free surface vibration followed by increased rising velocity near the free surface and randomly distributed smaller vortices in the higher eigenmodes as shown in Figs. 7e, g. There are several counter-rotating vortices inside the mean recirculating zone. Many saddle points are found in the 3rd and 4th spatial modes; these represent that 3-D motion of the large-scale eddies increased compared to case I. The 1st temporal mode in case II is similar to that in case I; however, high-frequency oscillation is added to the
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30 Fig. 6 The first four eigenmodes of case I; a 1st spatial mode, b 1st temporal mode, c 2nd spatial mode, d 2nd temporal mode, e 3rd spatial mode, f 3rd temporal mode, g 4th spatial mode, h 4th temporal mode
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Fig. 7 The first four eigenmodes of case II; a 1st spatial mode, b 1st temporal mode, c 2nd spatial mode, d 2nd temporal mode, e 3rd spatial mode, f 3rd temporal mode, g 4th spatial mode, h 4th temporal mode
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32 Fig. 8 The first four eigenmodes of case III; a 1st spatial mode, b 1st temporal mode, c 2nd spatial mode, d 2nd temporal mode, e 3rd spatial mode, f 3rd temporal mode, g 4th spatial mode, h 4th temporal mode
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Fig. 9 The FFT analysis result of temporal modes of the case III. a 1st temporal mode, b 2nd temporal mode, c 3rd temporal mode, d 4th temporal mode
low-frequency motion (Fig. 7b). The low-frequency oscillation might be due to the unsteady recirculating motion of water in the confined tank. The 2nd temporal mode shown in Fig. 7d is quite different than that of case I. Rather, the 3rd temporal mode (Fig. 7f) is similar with the 2nd temporal mode since those two modes are related with the free surface motion. The 4th temporal mode (Fig. 7h) represents the time scale of unsteady bubble rising motion. The fluctuating frequency is much higher than that of sloshing frequency. Figure 8 shows the four eigenmodes when the air flow rate is 5 L/min. Three different dynamic structures—recirculating flow, bubble-induced flow, and free surface oscillation—are also observed in case III; however, the order is different. The 1st spatial mode (Fig. 8a) represents high-momentum upward vertical flow due to the rising free surface, and the location is coincided with the highturbulent-kinetic-energy region shown in Fig. 5c. The 1st temporal mode (Fig. 8b) shows very sinusoidal variation, the dominant frequency of the motion is 3.0 Hz (see Fig. 9b), and this frequency corresponds with the
oscillation frequency of the free surface obtained from the visualization study. The 2nd spatial mode (Fig. 8c) represents the structure of the recirculating flow which is the 1st spatial mode in case II. The 2nd temporal mode (Fig. 8d) has a very low frequency of energy variation like the 1st temporal modes in cases I and II. The 3rd spatial mode (Fig. 9e) demonstrates the second mode of the free surface oscillation and the interaction with the recirculating motion. In the 3rd temporal mode (Fig. 8f), the oscillation of free surface is included. The 4th spatial mode (Fig. 8g) depicts the bubble-induced motion, which looks like a source flow due to the rising bubbles. It should be noted that there are higher-frequency modes in the 4th temporal mode (Fig. 8h) compared with the free surface sloshing. Higher-oscillation modes are thought to be the effect of bubble-induced motion. From the visualization of bubble images, we found that the bubble rising frequency is around 6–9 Hz according to the bubble formation pattern. To obtain more exact time scales associated with the free surface sloshing, the bubble-induced motion, and the recirculating motion, an FFT analysis of the temporal mode
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has performed. As shown in Fig. 9, the peak value of the amplitude spectrum illustrates the dominant frequency of the corresponding spatial POD modes. Figure 9a shows the FFT result of the 1st temporal mode in case III. The frequency of the sloshing motion is quantitatively found to be 3 Hz, which is exactly same as that obtained by the image processing method. The 2nd temporal mode of case III shows very slow sinusoidal oscillation with a frequency of 0.1 Hz (Fig. 9b). Due to the random nature of turbulent structures contributed to the recirculating motion, there is no distinct peak in the spectrum. This kind of lowfrequency unsteadiness was observed for a low-Re impinging jet in a confined wall (Kim et al. 2007). Since the bubble flow rising to the free surface is similar to the impinging jet to a normal wall, the low-frequency oscillation can be explained as a kind of sloshing motion of the free surface or the bubble jet. The FFT result of the 3rd temporal mode (Fig. 9c) represents the contribution of free surface oscillation mode with comparable magnitude of energy in the recirculating flow motion. The new peaks around 8 Hz are found in the spectrum of the 4th temporal mode. These peaks depict the presence of bubble-induced motion as shown in the 4th spatial mode (Fig. 8g). Since the final shape of bubbles just below the free surface has not a unique pattern due to irregular process of bubble agglomeration, the rising frequency has a wider spectrum compared with the free surface sloshing.
4 Conclusions Dynamic structures of bubble-driven turbulent water flow were investigated with three different air flow rates using the time-resolved PIV and POD techniques. Kinematic properties of bubble flows in the cylindrical tank were estimated by flow visualization experiment and image processing technique. Even though the time-averaged mean flow fields depict similar flow patterns for all the cases, the dynamic characteristics of large-scale turbulent motions are significantly different with respect to the bubble Re. With increasing Re, the turbulent kinetic energy level is increased dramatically over the measurement section. As the flow rates of compressed air were increased 3 and 5 times, the bubble rise velocities increased 1.296 and 1.624 times. The summation of turbulent kinetic energy over the field of view increased 2.069 and 5.695 times, and the magnitudes of the overall kinetic energy increased 1.539 and 2.494 times, respectively. Due to the presence of an oscillating free surface, energy transfer from the compressed air to the bubble-driven liquid flow had a distinct nonlinear manner.
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It was found that there are three dominant dynamic structures: recirculating flow, bubble-induced flow, and free surface oscillation. The contribution of each structure is different with respect to air flow rates. In case I, most of the turbulent kinetic energy and the large-scale motions are concentrated in the mean recirculating zone, while the sloshing motion of free surface turns out the dominant structure in case III. The frequency of the sloshing motion is estimated as 3 Hz based on the flow visualization study and the FFT analysis of temporal modes. The bubble rising frequency is different than the bubble detaching frequency at the nozzle due to the coalescence of bubbles during rise. The hitting frequency of bubbles to the free surface was estimated about 6–8 Hz based on visualization and temporal modes of POD, and the frequency seems similar in all cases. Acknowledgments This work was supported by the National Research Foundation (NRF) of Korea Grant funded by the Korean Government (MEST) (No. 2010-0000448, No. 2011-0030663, No. K20703001798-11E0100-00310).
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