Oscillation dynamics of a bubble rising in viscous liquid
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Abstract
In the present work, we study the oscillation dynamics of 3 mm-diameter bubbles generated through an orifice submerged in viscous liquids. The viscosity of those liquids is varied to change the behavior of the rising bubble. The details of the rising motion and shape oscillation of the bubbles are measured using a combination of high speed, high-resolution imaging, and an accurate digital image processing technique. Direct Numerical Simulations that mimic the experimental conditions are also performed using a front-tracking technique, called the Local Front Reconstruction Method. The predictions of the bubble shape and rising velocity obtained by the numerical simulations show good agreement with the experimental results. Our experimental and numerical results show that the oscillation frequency and the damping rate at lower modes can be predicted using available theoretical models found in the literature. However, discrepancies arise between our results with the theoretical predictions at higher order oscillation modes. We conclude that the discrepancies are due to the influence of rising motion and the vortex wave, which is not considered in the theoretical models.
Graphic abstract
1 Introduction
Because of its relevance in many industrial and natural phenomena, the dynamics of a rising gas bubble has been studied for many centuries and continues to be a problem of significant interest nowadays. This includes for example boiling, flotation, and intensification of heat and mass transfer in bubble column reactors. However, the fundamental understanding of bubble dynamics is still incomplete due to the large number of parameters, the non-linearity, and the fully three-dimensional transient nature of the problem.
Interestingly, Wu and Gharib (2002), Tomiyama et al. (2002), and Laqua et al. (2016) have independently reported that the method of bubble release from the orifice, i.e., the initial condition before detachment, affects the terminal rise velocity and the bubble shape. This unexpected experimental finding is attributed by the authors to the initial shape of the bubble at the beginning of its rise, which is in turn influenced by the way that the bubble is formed. Upon release, a bubble created by a small capillary undergoes strong shape oscillations and is found to reach a higher terminal velocity than a bubble of equal volume released from a large capillary, whose detachment is a much gentler process. This behavior is in strong contrast with the more common assumption of a unique terminal rise velocity for a bubble, which is often used in literature (Clift et al. 1978).
The correlation between rising motion and shape oscillations has been investigated by several authors (Meiron 1989; Lunde and Perkins 1997; De Vries et al. 2002; Veldhuis et al. 2008; Lalanne et al. 2013, 2015). Meiron (1989) developed a numerical model to study the steady rise and stability of inviscid bubbles and showed that the interaction of hydrodynamic pressure and surface tension forces does not lead to linear instability of the bubble path. Lunde and Perkins (1997) performed an experimental study on bubbles rising in still tap water. They showed that the low modes of the shape oscillations account for the wobbly and rocking nature of the shape and motion of intermediately sized bubbles. De Vries et al. (2002) performed experiments with 2–4 mm-diameter bubbles interacting with a hot-film anemometer probe in ultra-clean water to study whether bubble shape oscillation affects the rising velocity. Their experiments showed that the oscillating bubbles do not have higher mean velocities than the non-oscillating bubbles, which is in contrast with the results of Wu and Gharib (2002) and Tomiyama et al. (2002). Veldhuis et al. (2008) investigated the surface oscillations on bubbles rising in water. They showed that shape oscillation is linked to the path instability by checking the frequency of the main mode and the vortex shedding. Recently, Lalanne et al. (2015) studied the influence of the rising movement on the shape oscillations of small bubbles using experiments and numerical simulations. They showed that the effect of the rising motion on the oscillations is negligible, provided that the mean shape of the oscillation remains close to a sphere. Their finding indicates that the shape oscillation and the rising motion is one-way coupled. Lalanne et al. (2013) studied the distinct mechanism of the influence of rising motion on the oscillation of drops and bubbles. The oscillation dynamics taken at the transient stage are studied by DNS simulations, in which drops and bubbles are with initially prescribed deformations.
Limited by the requirements of high spatial and temporal resolutions and sophisticated data treatments of such a highly dynamic physics process, there have been only a few detailed studies. This motivates us to investigate larger bubbles, where the interface dynamics becomes even more pronounced. In the present work, we study the rising motion and shape oscillation of 3 mm bubbles generated through an orifice submerged in viscous liquids. The details of rising motion and shape oscillation of the bubbles are measured using a combination of a high-speed, high-resolution camera and an accurate digital image processing technique. Numerical simulations that mimic the experimental results are also performed using a front-tracking model, the Local Front Reconstruction Method (LFRM). Experiments and simulations have different strengths and weaknesses, related to control and accuracy. Therefore, the correspondence of experimental and simulation results makes the presented results more robust. This paper is organized as follows. The description of the experimental setup and measurement techniques is given in Sect. 2. Section 3 provides a detailed description of the numerical model. In Sect. 4, an extensive analysis of oscillation dynamics of a rising bubble in a viscous liquid is given. Finally, a summary of the main conclusions of the present work is provided.
2 Experimental method
2.1 Experimental setup
Fluid physical properties for air bubble formation in glycerol–water mixture
Fluid | \(\rho _{l}\) (\(\mathrm {kg} \, \mathrm {m}^{-3}\)) | \(\mu _{l}\) (\(\mathrm {kg} \, \mathrm {m}^{-1}\,\mathrm {s}^{-1}\)) | \(\sigma\) (\(\mathrm {N}\mathrm {m}^{-1}\)) |
---|---|---|---|
20 wt% glycerol | \(1.047\times 10^{3}\) | \(1.76\times 10^{-3}\) | \(7.09\times 10^{-2}\) |
40 wt% glycerol | \(1.099\times 10^{3}\) | \(3.72\times 10^{-3}\) | \(6.95\times 10^{-2}\) |
60 wt% glycerol | \(1.153\times 10^{3}\) | \(1.08\times 10^{-2}\) | \(6.77\times 10^{-2}\) |
2.2 Measurement and image processing techniques
The experimental images are captured using a pco.dimax HD high-speed digital video camera with a frame rate of 2000 Hz and a resolution of \(1.5 \times 10^{-2}\) mm/pixel. A calibration plate with known size and distance was used to convert pixel distances to mm. On the plate, the center distance of two horizontal/vertical neighboring white dots is 5 mm and the size of the dots is 1.2 mm (Kong et al. 2018). The spatial resolution is high enough to capture the bubble interface without the help of any interpolative reconstruction. In addition, the selected temporal resolution ensures that the bubble moves 3–4 pixels on two consecutive images. The recordings are performed with the help of a back lighting, which is only switched on during the imaging to reduce the heating of the liquid by illumination of the channel. A calibration plate with specific markers is used to obtain the pixel size.
3 Numerical model
The numerical model used in the present work is based on the Local Front Reconstruction Method (LFRM), originally developed by Shin et al. (2011) and adjusted by Mirsandi et al. (2018, 2019). In the sections below, the main characteristics of the numerical model are described.
3.1 Governing equations and solution methodology
Due to the advection of the interface, the size of the marker elements changes, decreasing the quality of the interface mesh. To remedy this situation, the LFRM procedure is periodically performed to ensure that each part of the interface is represented with a sufficient resolution. The details of this procedure can be found in the work of Mirsandi et al. (2018).
3.2 Computational setup
4 Results and discussion
4.1 Bubble shape and rising velocity
Equivalent diameter of the detached bubble and the corresponding dimensionless numbers for different fluids
Fluid | \(D_\mathrm{{eq}}\) (m) | Eo (–) | log (Mo) (–) | Oh (–) | Re (–) | We (–) |
---|---|---|---|---|---|---|
20 wt% glycerol | \(3.22 \times 10^{-3}\) | 1.50 | \(-9.60\) | \(0.5 \times 10^{-2}\) | 574.6 | 4.28 |
40 wt% glycerol | \(3.16 \times 10^{-3}\) | 1.55 | \(-8.29\) | \(1.1 \times 10^{-2}\) | 252.1 | 3.64 |
60 wt% glycerol | \(3.11 \times 10^{-3}\) | 1.62 | \(-6.43\) | \(3.1 \times 10^{-2}\) | 73.1 | 2.57 |
Figure 7 shows typical photographs of bubble detachment and rise from the orifice for three different liquids (Table 2). It can be seen that once the bubble detaches, the bubble undergoes shape deformation while rising. The shape evolves from a pendant shape at the time of detachment to an oblate spheroid. This shape deformation becomes more pronounced with decreasing liquid viscosity.
The corresponding bubble shapes obtained using the numerical model, which are represented with red lines, are superimposed on top of the photographs. The spatial evolution of the numerically predicted bubble shapes is very close to the corresponding experimental results. The deviations in the detached bubble diameter in all cases are less than 1%, meaning that the total volume detaching from the orifice plate is roughly the same and unaffected by fluid viscosity.
Comparison of oscillation frequencies for three different fluids
Fluid | \(f_\mathrm{{exp}}\) (Hz) | \(f_\mathrm{{DNS}}\) (Hz) | \(f_\mathrm{{Lunde}}\) (Hz) | \(f_\mathrm{{Meiron}}\) (Hz) | \(f_\mathrm{{Lamb}}\) (Hz) |
---|---|---|---|---|---|
20 wt% glycerol | 55.6 | 55.1 | 55.2 | 56.0 | 70.2 |
40 wt% glycerol | 57.2 | 57.3 | 57.3 | 58.8 | 69.8 |
60 wt% glycerol | 60.6 | 60.9 | 58.6 | 62.6 | 68.9 |
4.2 Analysis and discussion
Figure 12 also reveals that mode 2 is the main mode. The \(a_2\) is basically the representative of main shape of bubble and a similar to the aspect ratio in that it describes the extent of shape deformation. The value starts from a positive value due to the prolateness of the initial shape, while it decreases gradually to negative when the shape evolves to an oblate shape. For all three set of comparisons, the results of experiments and simulation match very well on mode 2, which was also evident in the qualitative visualization in Fig. 7. The match deteriorates for higher modes and longer time. This might be attributed to the spatial resolution of the simulation and the inherent in-stable nature of the higher modes. Furthermore, clearly, all the curves of the modes show that the deformations consist of a main shape evolution and a dampened oscillation. The later one is of our interest.
Moreover, we observed that most of the surface oscillation energy (mainly mode 2) damps only for a short period of time. In the case of 60% glycerol solution, the oscillations are damped after \(t=0.08\) s. In contrast, for bubbles rising in 20% glycerol solution, path instability occurs after \(t=0.08\) s, producing an increase of the oscillation amplitude. On the other hand, in the case of 40% glycerol solution, for each mode, the oscillations maintain their amplitude. Here, our results differ from the theoretical results of Prosperetti (1980) and simulation studies of Lalanne et al. (2013). They reported that the oscillation damping of bubbles is close to the theoretical prediction at the initial stages (or transient stages). In other words, the oscillation behavior of the rising bubble at the initial stages is similar to the oscillation of bubble in zero gravity, i.e., all modes dampen gradually. However, in our study, the oscillations are not damped at lower Oh numbers for any of the modes after the damping at the initial stage of bubble rise, which has also been reported by Lunde and Perkins (1997). Similarly, in the study of Gordillo et al. (2012), the agreement between the analytical model and DNS simulations deteriorates for longer times. The authors attribute this to the viscous dissipation including the boundary layer and wake, which are not considered in the analytical model.
Because the initial surface energies are close for all the cases (see Fig. 13), any oscillation energy should eventually dampen if the system is isolated. In the case of low Re number (\({Oh}=3.1 \times 10^{-2}\)), there is no strong wake effect beneath the bubble and the surface energy is eventually damped. Whereas in bubbles with higher Re numbers (\({Oh}=1.1 \times 10^{-2}\), \(0.5 \times 10^{-2}\)), there are standing vortices and vortex shedding appears in the wakes. In addition, our results also suggest that this energy is mainly added to higher modes instead of mode 2 at the initial stage of rising. Later, mode 2 also gains energy, which makes the oscillation system at all modes a forced oscillation system. Therefore, the oscillation behavior of bubbles at lower Oh numbers are different from bubbles in zero gravity in our results.
The dimensionless frequency is shown in Fig. 14 (right). It can be seen that the frequency increases with Oh number. The trend of mode 2 is consistent with Fig. 11, which correlates the frequency with the terminal shape of the bubble. Because the aspect ratio is a result of fluid properties and hydrodynamics, it is meaningful to correlate the dimensionless frequency with Oh number.
5 Conclusions
In the present work, we study the oscillation dynamics of 3 mm-diameter bubbles generated through an orifice submerged in viscous liquids by means of experiments and detailed numerical simulations. First, our results confirm that the potential flow theory can still predict the oscillation frequency of large size bubbles, where the dynamics are pronounced. This finding is also in accordance with several experimental studies reported in the literature (Lunde and Perkins 1997; van Wijngaarden and Veldhuis 2008; Gordillo et al. 2012; Lalanne et al. 2015). Second, it is found that the damping factor at mode 2 can still be described by a theoretical model (Prosperetti 1980), which takes into account viscosity effects and neglects the rising motion effect. However, at higher modes, the damping factor cannot be explained by the theoretical model and so far no available theoretical model can be used to interpret the damping behavior. Wake effects seem to play a very important role as an external energy source driving the shape oscillations of rising bubbles, particularly for the higher modes.
Notes
Acknowledgements
We would like to thank the financial support from NWO TOP grant. This work is also part of 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 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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