Keywords

1 Introduction

Turbulent dispersed multiphase flows with bubbles occur in many places such as ocean waves, chemical reactors, and ship hydrodynamics, where bubble breakage and coalescence play a major role. This breakup of bubbles by turbulence is important as its usually decides the size and distribution of the bubble phase in these different flows, with the bubble size in turn being an important factor in deciding mass, momentum, and energy transport between the primary and the dispersed phase. The phenomena of breakup and coalescence of bubbles has attracted considerable attention over the past few decades, with numerous theoretical, numerical, and experimental studies [5,6,7, 13,14,15,16,17, 20, 23]. A turbulent flow is populated with vortical structures [1], and these structures are highly coupled in nature. The presence of bubbles results in a highly complex multiscale coupled interaction between vortical structures and bubbles. An idealized study of such a complex interaction can be thought to be the interaction of a vortex ring with a bubble. In the present study, we take a single air bubble and study its breakup as it interacts with a vortex ring (see Fig. 1a) generated in the water medium.

Fig. 1
figure 1

a A schematic of an air bubble of diameter Db and a vortex ring of ring diameter 2R and core diameter of 2a, traveling vertically upward in z direction with velocity UC. b Schematic of the experimental setup for a single bubble interacting with a vortex ring. A piston cylinder arrangement was used to generate the vortex ring, and the tube was connected to an air supply to generate the air bubble

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There have been very few investigations on the interaction of a bubble with vortex structures as stated by Magnaudet and Eames [18]. The numerical work by Ferrante and Elghobashi [9] on the interaction of bubbles with a Taylor–Green vortex showed that the presence of a bubble can reduce vorticity and enstrophy of the vortex. Oweis et al. [22] adopted one-way coupling method to predict the capture of a bubble in a line vortex. Rastello et al. [24], Van Nierop et al. [27], Bluemink et al. (2009), and Sridhar and Katz [26] experimentally determined correlations for drag and lift coefficients of bubbles in vortex structures. Mazzitelli et al. [19] numerically explored the two-way coupled interaction between bubbles and homogeneous isotropic turbulence where they stated lift force to be responsible for the clustering of bubbles on the vortex on the side having downward velocity. Deng et al. [8] experimentally investigated bubble’s behavior in a Taylor vortex. Finn et al. [10] explored the equilibrium position of few microbubbles in a traveling vortex-tube. Higuera [11] and Revuelta [25] analyzed the interaction of a bubble and a vortex ring where they observed distinctive modes of bubble breakup depending on the Weber number and the initial vortex-to-bubble size ratio. Recently Jha and Govardhan [12] studied the interaction of a large bubble on a thin vortex ring at volume ratio of about 0.1 and observed significant permanent fragmentation of the vortex core at low Weber number. They have also presented different stages of bubble–vortex interaction along with some interesting aspects of bubble dynamics which includes bubble’s capture time in the vortex core, final number of broken bubbles, and their mean size at different Weber numbers.

As an extension of Jha and Govardhan [12], some more intriguing bubble dynamics aspects related to bubble–vortex interaction, which are bubble’s equilibrium position inside a vortex core, the critical aspect ratio of an elongated bubble and simultaneous breakup and coalescence of broken bubbles inside a vortex ring are explored in the present study. There are several parameters that govern the dynamics of interaction between a vortex and a bubble as given in Jha and Govardhan [12] and Biswas and Govardhan [3, 4], which include parameters on both the vortex ring side and the bubble side. Parameters from ring’s side includes the vortex ring radius (R), vortex core radius (a), non-dimensional core radius (a/R), circulation strength of the ring (Γ), and ring Reynolds number (ReΓ = Γ/ν, ν is kinematic viscosity of water). On the bubble side, these are the bubble diameter (Db) and surface tension (σ) at air–water interface. Two more parameters associated with the coupled interaction of the bubble and the ring are the ratio of the bubble’s volume to the ring’s core volume (VR = (π/6)D3/(2π2Ra2)), and Weber number (We) defined here as the ratio of the pressure difference (ΔP = 0.87ρ(Γ/2πa)2) of ring’s core and outside far-field, to the Laplace pressure (σ/Db).

The construction of the paper is as follows. In Sect. 2, we discuss about the methods and experimental approaches used in the present work. In Sect. 3, we present bubble dynamics during the interaction with the ring obtained for a wide regime of Weber numbers (We ~ 12–763) keeping VR fixed at about 0.1. Finally, in Sect. 4, we present a brief overview of the main results.

2 Data/Methodology/Experimental Set Up

A vortex ring was generated using a piston cylinder arrangement in a water tank with transparent side walls providing optical access from all sides. The radius of the vortex ring (R), its core radius (a), and the circulation of the core (Γ) were measured from particle image velocimetry (PIV) measurements after the complete formation of the vortex ring. A large range of vortex ring circulations (Γ) was obtained by changing the impulse of the piston. In the present study, results are shown for a wide range of ring Reynolds numbers (6003–67,376); this covers laminar, transitional, and turbulent vortex rings. A large bubble of diameter (Db) about 5.7 mm was generated next to the vortex ring generator by a small tube connected to an external air supply.

The uncertainties in the measurement of the vortex ring radius (R), core radius (a), bubble diameter, (Db) and vortex ring strength (Γ) were estimated to be ±0.2 mm, ±0.15 mm, ±0.1 mm, and ±5%, respectively. The vortex ring used for this study was relatively thin with the ratio of the core radius of the vortex ring (a) to the radius of the ring (R) being around 0.2 (a/R ~ 0.2) [21]. Micron size bubbles were generated at the tip of the vortex ring nozzle using electrolysis and were used to trace the ring. We also traced the vortex ring using dye, with both these measurements being within experimental uncertainties. Due to separate requirement of illumination, both front and top view were captured separately using a Photron SA5 FASTCAM high-speed camera at a frame rate of 3000. The origin of the coordinate was taken at the center of the face of the vortex ring generator nozzle (see Fig. 1b), which is the exit plane of the vortex ring, x and y represent the axis in the horizontal plane, and z represents the vertical direction.

3 Results

In this section, we present bubble dynamics of a 5.7 mm bubble interacting with a thin vortex ring for a range of Weber numbers (see Table 1) keeping the volume ratio fixed at about 0.1. The circulation strength (Γ) of the vortex ring is varied by changing the impulse of the piston, and we achieved different ring Reynolds numbers and their corresponding Weber numbers (We) (see Table 1). In all cases, the vertical position of the vortex ring (z) was traced with time (t) from the side view visualization to obtain information about the convection speed of the ring. The top view visualization was also performed, and this was useful to see interesting aspects of bubble capture and breakup process within the ring. The dimensional position (z) of the ring was normalized with the ring radius (R), and dimensional time (t) was normalized with the ring radius (R) and the convection speed (UC) of the ring, with both R and UC being taken just before capture of the bubble. We present here both front view (Fig. 2a) and top view (Fig. 2b) of the time sequence images of visualization at We = 160. As discussed by Jha and Govardhan [12], there are broadly four stages of interaction: (i) Before capture of the bubble, (ii) capture due to the low pressure inside the vortex core followed by elongation and breakup of the bubble by the azimuthal pressure gradient within the ring, (iii) bubble breakup is complete but the vortex core can fragment, and (iv) the last stage where the bubble(s) leave the ring as the ring becomes weaker. Further detailed information of these stages can be found in Jha and Govardhan [12] and Biswas and Govardhan [4].

Table 1 Parameters in our study
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Fig. 2
figure 2

a Front view of bubble–vortex interaction: time sequence of front view visualization images of the interaction of single bubble with a vortex ring at ReΓ = 29,395 and We = 160. (i) Before capture, (ii) capture, (iii) azimuthal elongation, (iv) and (v) and (vi) breakup. Time instances (tUC/R) are: (i) 3.32, (ii) 3.87, (iii) 4.19, (iv) 4.62, (v) 5.68, (vi) 6.73. b Top view of bubble–vortex interaction: time sequence of top view visualization images of the interaction of single bubble with a vortex ring at ReΓ = 29,395 and We = 160. (i) Before capture, (ii) and (iii) elongation, (iv) and (v) and (vi) breakup. Time instance (tUC/R): (i) 3.17, (ii) 3.77, (iii) 3.98, (iv) 6.13, (v) 6.48, (vi) 8.72

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Now, we are going to discuss some interesting bubble dynamics aspects gleaned from the top view visualizations. We start with the bubble’s radial equilibrium location inside the core of the vortex, which is the radial position of the bubble after which only azimuthal elongation occurs. In the experiment, this is measured as the radial location of the bubble within the core after bubble capture and just before the bubble starts elongating azimuthally. Once bubble’s entrainment into the ring’s core is completed, it tends to settle around a mean radial location within the core, where there is radial force equilibrium between the radially inward pressure-gradient-induced force, and the radially outward lift force (Sridhar and Katz 1999) [12]. As the bubble entrains and gets settled, its presence results in an observable distortion of the vortex core, which is a function of the bubble’s settling location. In the present study, the mean radial location (r/a) of the bubble from the vortex core’s center (r/a = 0) is seen to be dependent on Weber number, as shown in two top view visualizations shown in Fig. 3. At We of 12 (and ReΓ = 6003), the center of the bubble is offset by a distance of about r/a ~ 0.9 from the vortex core’s center, and as We is increased to 763, the bubble sits much closer to the core’s center. Also, the figure shown is a plot of this radial location as a function of the We taken from many such visualizations, showing that this offset gradually reduces as the We is increased, and tends toward zero (center of the core) at large We.

Fig. 3
figure 3

Bubble’s radial equilibrium position (r/a) inside the vortex core after capture at different Weber numbers. a, b Represent the top view of the vortex ring with the bubble being seated inside the ring at We = 12 and 763, respectively. At We = 12, bubble’s radial location is offset from the ring’s axis (or center of the core) by a distance of about r/a ~ 0.9. For the other case at We = 763, bubble sits very close to the vortex ring’s axis (or center of the core). Here, r/a = 0 indicates the axis (or core’s center) of the vortex ring

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After the bubble reaches a mean settling location inside the vortex, the bubble is subjected to azimuthal pressure gradient due to the presence of the bubble inside the vortex. This azimuthal pressure gradient stretches the bubble azimuthally which can be seen in Fig. 2b. At the same time, the bubble interface is subjected to interfacial instabilities/perturbations seen as roughness of the bubble’s surface (Fig. 2b (ii) and (iii)). These perturbations grow over time and cause the elongated bubble to break at several azimuthal locations.

As discussed by Jha and Govardhan [12], the breakup of the azimuthally elongated bubble inside the vortex core could be due to Kelvin waves and Rayleigh–Plateau instability. In the present study, the aspect ratio (Lb/W) of the elongated bubble which is calculated as the ratio of the elongated length (Lb) of the bubble to the azimuthally averaged width (W) of the stretched bubble has a critical value beyond which the extended bubble starts breaking. We call this the critical aspect ratio ((Lb/W)critical), and this seems to increase with Weber number till We of about 160 and then starts decreasing, and eventually it appears to become independent of We as seen in Fig. 4. As We increases, the bubble elongates more due to higher azimuthal pressure gradient forcing larger extension of the bubble. The critical Lb/W at which breakup begins increases till certain We (~160) as seen in the figure. Increasing the We further, we observe more number unstable waves on the bubble’s surface which occurs due to the turbulent nature (large ReΓ) of the vortex ring. At very large We (or large ReΓ) these surface waves are seen to grow faster and penetrating the elongated bubble causing early breakup of the bubble by means of multiple fragmentation happening at several azimuthal locations of the extended bubble. Due to this early breakup of the elongated bubble at larger We (or ReΓ), the critical Lb/W is seen to reduce beyond certain We (~160) and then gradually becoming independent of We (or ReΓ).

Fig. 4
figure 4

Critical aspect ratio (Lb/W)critical of azimuthally elongated bubble at different Weber numbers. Top view images of maximum azimuthal elongation (till breakup) bubbles for three We cases (12, 160, and 763) are shown here

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Once reaching the critical aspect ratio, the elongated bubble starts breaking and produces several broken bubbles which can further undergo breakup depending on Weber number and their size. Now we focus on the transient changes in the number of broken bubbles inside the vortex presented in Fig. 5. In the present study, from top view visualizations, we observe that the mother/main bubble which elongates azimuthally can undergo binary/tertiary breakup till We of about 321, and beyond this, rupture/fragmentation of the mother bubble is seen to happen producing relatively larger number of broken bubbles. Revuelta [25] in their numerical study of bubble breakup outside a vortex ring due to induced velocity of the vortex ring observed binary bubble breakup at relatively low We and fragmentation at large We, which is in line with our observations. In the present work, we observe breakup and coalescence of broken bubbles and depending on the We, we can classify this broadly into three categories. The first one comes under low We regime which is of about We ~ 12 in the present study, where the main bubble can undergo binary or tertiary breakup and those broken bubbles can further undergo breakup but no chances of coalescence as they are situated azimuthally sufficiently far from each other. In the second category for which We is broadly of the range of ~46–321, after binary/tertiary breakup of the main bubble, broken bubbles stay inside the ring’s core relatively closer to each other and can undergo collisions which can lead to coalescence and reduction of Nb as seen in Fig. 5. Finally in the third category which has much higher We (>321), although there are relatively larger number of broken bubbles situated more closely to each other inside the vortex ring and have higher chances of collisions, these broken bubbles are less likely to coalesce as they offer more resistance to coalescence due to relatively higher surface tension force.

Fig. 5
figure 5

Number of broken bubbles (Nb) with time (tUC/R)

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4 Conclusions

Motivated by the complex mechanism of bubble breakup in a turbulent environment, we have experimentally investigated an idealization of this, which is the study of bubble dynamics in a bubble–vortex interaction. The focus of the work has been to explore some aspects of bubble dynamics in this interaction as a function of the many parameters of the interaction. We have presented some important bubble dynamics aspects which are the variation of the mean equilibrium location of the bubble inside vortex, bubble’s critical aspect ratio where breakup initiates, and several distinctive bubble breakup patterns starting from binary and tertiary breakup at low and intermediate We and fragmentation at large We, which are comparable with several observations on bubble breakup in bubbly turbulent flows. This idealized study can help us advance our understanding of the breakup mechanism of bubbles interacting with vortical structures in turbulent environments.