Hydrodynamic interaction of bubbles rising side-by-side in viscous liquids
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Abstract
Detailed experiments are conducted to study hydrodynamic effects of two simultaneously released bubbles rising in viscous liquids. Different types of interactions are observed as a function of the liquid viscosities, leading to different bubble shapes, ranging from rigid spheres and spheroids to deformable spheroids. Bubble velocities are obtained by an automated smooth spline technique, which allows for an accurate calculation of the lift and drag forces. The results obtained for spherical bubbles are in agreement with predictions of Legendre et al. (J Fluid Mech 497:133–166, 2003). The observations of deformed bubbles show that a very small equilibrium distance can be established due to the induced torque arising from the deformation. In terms of the lateral interaction, different separation distances can be observed depending on the initial distance. For deformable bubbles, the results are limited to a qualitative analysis due to limitations of the processing technique to handle strong shape irregularities. Nevertheless, the observations reveal that the deformation plays an important role with respect to bubble interactions and path instability of which the latter can be triggered by the presence of other bubbles.
Graphic abstract
List of symbols
Dimensionless numbers
- \(\mathrm {Ca}\)
\(Ca=\mu U/\sigma \). Capillary number
- \(\mathrm {Eo}\)
\(\rho _l g D_{\mathrm{eq}}^2/ \sigma \). Eötvös number
- \(\mathrm {Oh}\)
\(\mu /( \rho _l \sigma D_{\mathrm{eq}}/2)^{1/2}\). Ohnesorge number
- \(\mathrm {Re}\)
\(\rho _l U D/ \mu \). Reynolds number
- \(\mathrm {We}\)
\(\rho _l U^2D/ \sigma \). Weber number
- \(\mathrm {Mo}\)
\(Mo=g\mu _l^4/( \rho \sigma ^3)\). Morton number
- \(\upchi \)
\(R_1/R_2\). Bubble aspect ration
Greek symbols
- \(\rho \)
Density
- \(\epsilon \)
Measurement error
- \(\theta \)
Pitch angle
- \({\varvec{\Omega }}\)
Angular velocity
- \({\varvec{\Gamma }}\)
Torque
- \(\sigma \)
Surface tension
Latin symbols
- a
Acceleration
- \(\mathbf{A}\)
Added mass tensor
- A
Surface area
- \(C_\mathrm{D}\)
Drag coefficient
- \(C_\mathrm{l}\)
Lift coefficient
- \(\mathbf{D}\)
Rotational added mass tensor
- \(D_{\mathrm{eq}}\)
Equivalent diameter
- E
Energy
- \(\mathbf{F}\)
Force
- \(f_{\mathrm{req}}\)
Recording frequency
- \(R_1\)
Semi-major axis
- \(R_2\)
Semi-minor axis
- s
Distance between two centres
- S
Dimensionless distance between two centres
- t
Time
- U
Bubble velocity
- V
Bubble volume
1 Introduction
Bubbles are often encountered in industrial, biochemical, and environmental processes. In these processes, bubbles interact with the liquid phase and other bubbles and significant experimental and computational efforts have been made to obtain closures for drag and lift forces. However, the behaviour of rising gas bubbles (isolated or in swarms) is very complex and even the simplest case such as the rise of a single deformable air bubble is quite complex. Some of the behaviour is still under investigation and not completely understood, such as the path instability of rising bubbles (Magnaudet and Eames 2000; Ern et al. 2012). In industrial applications, bubbles in dense swarms are encountered revealing complex interactions and additional effects of effective drag and lift and earlier onset of path instability. Thus, the behaviour of multiple bubbles has attracted more and more attention due to its practical significance. For instance, the disturbance from a neighbouring bubble plays a significant role in bubble rising behaviour such as coalescence and clustering. Coalescence is usually unfavourable in industrial applications, because it decreases the overall surface area and hence deteriorates mass and heat transfer rates. In addition, coalesced bubbles alter the flow due to stronger deformation and a larger moving interface. The enhanced interaction of multiple bubbles often leads to local clustering, which consequently alters the flow field and the mass and heat transfer characteristics. Despite extensive research efforts, a lot of open questions remain.
To understand the physics underlying bubble swarms, a lot of effort has been made as well. Analytical studies (Van Wijngaarden and Jeffrey 1976) and simulations based on irrotational flow (Sangani and Didwania 1993; Smereka 1993; Yurkovetsky and Brady 1996) have revealed that bubbles aggregate in the horizontal plane due to the interaction. However, experiments show that low volume fraction bubbly flows tend to be homogeneously dispersed. Discrepancy between simulation and experiment has been attributed to the improper assumption that bubbles bounce elastically. Alternatively, taking deformability of the bubbles into account, front tracking (FT) simulations (Bunner and Tryggvason 2003; Esmaeeli and Tryggvason 2005) have revealed that spherical bubbles tend to align horizontally, whereas deformable bubbles tend to uniformly distribute over the volume under consideration. Apart from the clustering, Cartellier and Rivière (2001) and Risso and Ellingsen (2002) have shown that wake interaction among multiple bubbles causes bubble-induced turbulence, which has a strong impact on the efficiency of heat and mass transfer. Studies of Roghair et al. (2011a, b, 2013a, b) have resulted in several correlations quantifying the swarm effect on the effective drag coefficient. Recently, Loisy et al. (2017) studied bubbles rising in configurations of ordered and freely rising bubble arrays. In addition, the velocity fluctuation of rising bubbles due to the swarm effect was studied experimentally in spite of the experimental limitation at high gas volume fraction (Martínez-Mercado et al. 2007; Riboux et al. 2010; Colombet et al. 2015). Moreover, bubbles rising in a thin gap were studied as a model system (Bouche et al. 2012, 2014; Roig et al. 2012), in which the turbulence is suppressed.
From the studies reported in literature, it is clear that the interaction between neighbouring bubbles alters the bubbly flow globally. However, bubble-pair interaction has drawn a lot of attention due to its evident relevance to understand more complex systems. The interaction of either inline bubble pairs or side-by-side pairs has been frequently studied. Harper (1970) calculated the interaction between a pair of spherical inline bubbles and found that an equilibrium distance between the bubbles exists based on potential theory. Yuan and Prosperetti (1994) numerically investigated a similar configuration and confirm the presence of the equilibrium distance indicating the importance of the viscous force. Experimental research by Katz and Meneveau (1996), however, was not in agreement with these predictions, showing that pairs of small bubbles tend to collide and coalesce instead. Experiments by Sanada et al. (2005) have revealed the presence of an equilibrium distance. However, in their study, the equilibrium distance was not stable and larger than predicted by the previous studies. Finally, recent 3D DNS simulations by Gumulya et al. (2017) again revealed the existence of an equilibrium distance, leaving the discussion open to debate.
Apart from studies focusing on bouncing and coalescence, the only experimental study reported in the literature is due to Kok (1993). Unfortunately, due to experimental limitations , trajectories were the only obtainable quantitative data. The opposing findings reported in the literature and the advancement of experimental techniques prompted us to undertake the present study. Moreover, to the best of our knowledge, the simulation study of Tripathi et al. (2017) and Zhang et al. (2019) is the only work addressing the interaction between deformed bubbles. In the present study, a spherical bubble pair and a deformed bubble pair are experimentally studied to investigate their hydrodynamic interaction. This paper is organised as follows: the experimental setup and processing procedures will be presented in Sect. 2. Results and discussion will be reported in Sect. 3. The conclusions will be presented in Sect. 4.
2 Setup and processing procedures
2.1 Experimental setup and liquid properties
The measurement system incorporates a high-speed camera (PCO, dimax HD+) mounted with a Nikon lens, an LED lamp with a diffuser plate as the illumination source and a glass column sized \(100\times 100\times 500\) mm (see Fig. 1). The bubble pair was generated at the nozzle with two orifices submerged at the bottom of the column. The nozzle was made of stainless steel and the surface was polished to minimise the effect of roughness. The diameter of the orifices was 1 mm. The distance between the orifices was varied from 4 mm, 6 mm to 7 mm. The orifices were connected to a two-channel syringe pump (KD Scientific LEGATO 100) mounted with two gas-tight syringes (Hamilton).
For the experiments, water–glycerol mixtures were used to vary the liquid viscosity (see Table 1) and as a consequence the behaviour of the rising bubble pair. To minimise the effect of preceding bubbles, a flow rate of 1 ml/min was chosen, based on earlier tests (Kong et al. 2019).
Physical properties of liquid used in the experiments
Liquid | \(\rho \) (\(\mathrm{kg}\; \mathrm{m}^{-3}\)) | \(\mu \) (\(\mathrm{kg}\; \mathrm{m}^{-1}\mathrm{s}^{-1}\)) | \(\sigma \) (\(\mathrm{N}\mathrm{m}^{-1}\)) |
---|---|---|---|
80 wt% glycerol | 1208.5 | \(60.1 \times 10^{-3}\) | \(6.5\times 10^{-2}\) |
60 wt% glycerol | 1153.8 | \(10.8 \times 10^{-3}\) | \(6.77\times 10^{-2}\) |
40 wt% glycerol | 1099.3 | \(3.72 \times 10^{-3}\) | \(6.95\times 10^{-2}\) |
20 wt% glycerol | 1046.9 | \(1.76 \times 10^{-3}\) | \(7.09\times 10^{-2}\) |
2.2 Data-processing procedures
Unfortunately, the fitting function is very sensitive to the choice of the smoothing parameter. Moreover, it is very challenging to choose the approximate value of the smoothing parameter (Aydın et al. 2013). To establish the aforementioned balance in a rational fashion, the method to determine the smoothing parameter was based on studies of De Boor et al. (1978), Hutchinson (1986) and Wahba (1983).
3 Results and discussion
Studies on bouncing and coalescence have revealed that wakes play an important role (Duineveld 1998; Sanada et al. 2005). According to the study of Magnaudet and Mougin (2007), rising bubbles can be categorised into three groups (Fig. 5). Glycerol solutions with pre-selected viscosities are employed to obtain the desired bubble rise regime. Bubbles of small size (Re) and (nearly) spherical shape (regimes a and b) have no vortex in the wake and rise in a rectilinear path; bubbles of larger size (Re) and moderate deformation (regime c) have an axisymmetric standing vortex wake and rise following a rectilinear path; bubbles of even larger size (Re) and pronounced deformation possess unstable wakes and rise following a complex three-dimensional path.
How a bubble pair behaves is a fundamental question for understanding the large-scale behaviour of bubble swarms. In the present study, the rising dynamics of pairs of spherical bubbles and spheroidal bubbles are experimentally studied to investigate the interaction. The dimensionless distance of two centres of mass is defined as \(S=2s/D_{\mathrm{eq}}\). It should, however, be noted that the approach or separation of the pair of bubbles does not always reveal the interaction of the bubbles, because for large bubbles, the unstable path can result in the relative movement of a pair of bubbles, which obviously is not due to the existence of the second bubble. It should be noted that the bubble sizes are carefully checked to make sure that the size difference between the two bubbles of the pair as well as the size of the corresponding single bubble is approximately \(\pm 0.03\) mm.
Characteristic values and dimensionless numbers of bubbles
\(D_{\mathrm{eq}}\) | Re | log(Mo) | Eo | We | Oh | \(\chi \) | Ca | Type | |
---|---|---|---|---|---|---|---|---|---|
80 wt% glyc. | 3.15 | 7.5 | \(-\) 3.4 | 1.81 | 0.8 | 0.17 | 1.08 | 0.1 | a |
60 wt% glyc. | 3.18 | 84.9 | \(-\) 6.4 | 1.69 | 3.4 | 0.03 | 1.5 | 0.04 | b |
40 wt% glyc. | 3.23 | 286.3 | \(-\) 8.3 | 1.62 | 5.0 | 0.01 | 2.1 | 0.016 | c |
20 wt% glyc. | 3.22 | 574.6 | \(-\) 9.6 | 1.50 | 4.3 | 0.005 | 2.5 | 0.007 | d |
3.1 Spherical bubbles (type a)
The interactions of spherical bubbles are represented in Fig. 6a. It is clear that irrespective of the initial separation distance, all the bubbles repel each other after an initial short period of attraction. The repulsion is more prominent for a smaller initial distance.
Velocities and deformation are calculated by implementing the image and data-processing procedure, as discussed in Sect. 2.2. Figure 7 displays the rise velocity, deformation, and separation distance as a function of the height for different initial separation distances. The mirrored horizontal velocity and overlapped deformation reveal the symmetry of the experiments. As shown in Fig. 6a, the distance between the bubbles comprising the pair all increase with the rising height. It can be seen that a smaller initial distance leads to a more pronounced separation and larger horizontal velocity. The curves for deformation of the bubbles are identical and equal to that of a single bubble, which reveals that the interaction has no impact on the shape for such relatively rigid bubbles. Furthermore, the vertical velocities of both bubbles are identical. However, surprisingly, the bubble pairs rise faster in comparison with a single rising bubble.
3.2 Deformed bubbles (type b)
Type b, deformed bubbles are no longer spherical, but have a relatively stable ellipsoidal shape. The rising behaviour of these bubbles is presented in Fig. 6b. For initial distances of 6 mm and 7 mm, the interaction is very weak and both bubbles rise almost identically to a single rising bubble, which is especially apparent from the vertical rise velocity and the deformation. However, the interaction is prominent for the case of an initial distance of 4 mm. The bubbles initially approach before separating and during the interaction, especially during separation these bubbles seem to rotate slightly. Strikingly, all bubbles seem to attain the same final properties.
3.3 Deformable bubbles (types c, d)
Compared with deformed bubbles and spherical bubbles, deformable bubbles are less rigid. The bubbles will experience shape oscillations (Kong et al. 2019) due to the wake behaviour. As shown in Fig. 6c, d, multiple bubble encounters occur for deformable bubble pairs and the deformation itself is strongly altered by the interaction.
Based on the discussion of Sects.3.1 and 3.2, we can conclude that the interaction is negligible if both the vertical velocity and deformation are identical to the corresponding single bubble, and arguably if only the deformation is identical. In Fig. 9, for all the cases of initial separation distance, the bubbles first approach each other before separating. The velocity and deformation are identical to that of a single rising bubble, up to the point, where the bubbles meet and separate again. After this encounter and during the subsequent separation, the shape oscillations and rise velocity start to deviate from the single bubble rise. This onset of bubble interaction occurs earlier for smaller initial distance. Both the deformation and the rise velocity are suppressed with the interaction of the two bubbles.
As a complementary case, a bubble pair rising in 20% glycerol (see Fig. 6d) shows the interaction of a bubble pair with an unstable wake. Instead of the development of a 3D trajectory for a single bubble rise, the bubble pair has an extended period of an in plane movement.
3.4 Drag and lift coefficients
The drag and lift coefficients are shown in Fig. 12 for spherical bubbles, deformed bubbles, and deformable bubbles. It is stressed here that the results obtained for deformable bubbles should be regarded as qualitative, as we have discussed in Sect. 3.3. In addition, the acceleration phase will be excluded from the analysis, as the obtained data are prone to error amplification, as discussed in Sect. 2.2.
In this case, the magnitude of the lift coefficient declines exponentially with a long tail. This reveals that the repulsive interaction decays until a certain separation distance has been reached. In addition, the study of Legendre et al. (2003) reported a drag coefficient ratio of 90% for interacting bubbles, which is in good agreement with our results.
Together with the altered deformation described by a ratio of aspect ratios (\(\chi /\chi _{\mathrm{single}}\)), the aligned tendency of the curves reveals that the deformation is related to the bubble interaction.
As shown in Fig. 13, most of the energy from the interaction feeds into a suppression of the surface energy, where additionally the lateral kinetic energy \(E_{\mathrm{kx}}\) and rotational energy \(E_{\mathrm{rot}}\) are orders of magnitude smaller. In particular, the rotational energy \(E_{\mathrm{rot}}\) is negligible. Based on the energy analysis, we find out that even though it does not seem very significant in Fig. 6, the deformation instead of the repulsion dominates in the case of strong interaction. Risso (2018) pointed out that the bubble agitation is surprisingly independent of bubble interaction, even at significant gas volume fraction. This has been found in several studies (Martínez-Mercado et al. 2007; Riboux et al. 2010; Colombet et al. 2015). In the present study, the dominance of the deformation could be a factor that leads to that phenomenon. Moreover, the time scale of relaxation of the altered deformation is similar to the decay of the surface oscillation in the order of \(10^{-1}\,\mathrm{s}\), which is correlated with the Ohnesorge number (Kong et al. 2019). This might indicate that the change of surface energy is due to viscous dissipation enforced by bubble interaction, after which it returns to the initial value when the stable separation distance is reached.
However, the lateral motion is significant as well, because it determines the separation distance which is a key parameter for the bubble swarm effect. For lift and drag force calculations, the deformation effect was neglected. Therefore, we provide an estimate of the deviation due to this neglect. From Fig. 8, it follows that the deformation is around 10% of its terminal shape during the interaction. A 5% deviation of the calculation of the lift and drag force is caused consequently by the neglect of the deformation. A similar estimate can be found in the study of Shew et al. (2006).
Comparing the lift coefficients of the 4 mm case and 6 mm case, the distinctions of bubble lift coefficients at the same separation distance reflect the complexity of interaction between deformed bubbles. The larger lift coefficient of 4 mm at the same separation distance as that of 6 mm case should be attributed to the torque effect. This also suggests that the equilibrium distance of side-by-side-deformed bubbles depend on the initial distance as well as the orientation.
In addition, as shown in Fig. 15, the magnitude of the lift coefficients surprisingly is of the same order of magnitude as for the spherical bubbles.
However, the lift coefficient of the deformed bubbles declines with a higher rate in comparison with the slow decay observed for the spherical bubbles. In this sense, deformed bubbles can establish a smaller separate distance between bubbles in comparison with spherical bubbles. Moreover, the ratio of the drag coefficient follows the opposite trend in comparison with spherical bubbles. The drag is higher for the rising bubble pair.
Finally, the results for the deformable bubbles (type c) will be discussed qualitatively. As indicated before, the results of deformable bubble pairs are less accurate. As shown in Fig. 12, the lift and drag coefficients are superimposed against the separation distance to some extent. The lift coefficients decline fast in a similar fashion as deformed bubbles (type b), even though with oscillations. Moreover, sign changes of the lift coefficient occur due to the path instability, which has been discussed in Sect. 3.3. On the other hand, the ratio of drag coefficients is generally above 1, similar to deformed bubbles (type b), which reveal that the drag increases due to the bubble interaction.
4 Conclusion
In the present study, hydrodynamic interactions of side-by-side rising bubble pairs are studied experimentally. Four different bubble types were studied by varying the liquid viscosity.
For spherical bubbles (type a), which have been studied prior to the present study, the behaviour of the bubble pair depends on the Re number. Bubbles at low Re number rise slightly faster than a single rising bubble and tend to repel each other. This behaviour is exactly opposite to rising pairs of spherical bubbles at high Re numbers, whose behaviour can be explained on the basis of potential flow theory. The repulsion of low Re number spherical bubble pairs exhibits exponential decay, leading to a larger equilibrium distance confirming earlier predictions by Legendre et al. (2003).
Interactions of deformed bubbles (type b) have been studied far less. The observations in this study show that deformed bubbles rise slower than a single rising bubble and overall bubble pairs repel each other. The lateral interaction of deformed bubbles is complex due to the torque acting on the deformed bubble, which depends on the initial distance and orientation. This lateral interaction declines faster in general, which reveals that a smaller equilibrium distance of the bubble pair is possible.
The results obtained for deformable bubbles are far more difficult to interpret. The results, however, do show the importance of shape deformation on the general pattern of bubble interaction. Interaction of the bubbles suppresses both the extent of deformation as well as the rise velocity. Our observations also reveal that the path instability can be triggered by the bubble interaction.
Notes
Acknowledgements
We would like to thank the financial support from NWO TOP grant OND1356157 First-principles-based multi-scale modeling or transport in reactive three-phase flows. We would also like to thank the Industrial Partnership Programme i36 Dense Bubbly Flows that is carried out under an agreement between Akzo Nobel Chemicals International B.V., DSM Innovation Center B.V., SABIC Global Technologies B.V., Shell Global Solutions International B.V., Tata Steel Nederland Technology B.V.and Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO).
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