3.1. Bubble dynamics

We now begin by presenting results from the bubble dynamics perspective for the initial stages of interaction comprising stages I and II. We shall present here visualizations and analysis of the processes of bubble entrainment into the ring, and its subsequent deformation leading to break-up, as a function of the vortex ring strength or equivalently the Weber number ( $\mathit{We}$ ).

3.1.1. Bubble capture by the vortex ring (stage I)

Bubble capture is the beginning of the vortex ring–bubble interaction. The low pressure within the core of the vortex ring induces motion of the bubble into the vortex core in a spiralling path, as seen in figures 1 and 2. During this capture process, the bubble undergoes extreme elongation and wraps itself around the core, before reaching an equilibrium radial position within the vortex core. This type of bubble capture process has been observed for example in the numerical study of Chahine (Reference Chahine and Green1995), and in the experiments of Sridhar & Katz (Reference Sridhar and Katz1995) and Oweis et al. (Reference Oweis, Van der Hout, Iyer, Tryggvason and Ceccio2005).

An important quantity related to the capture process is the capture time $(t_{c})$ of the bubble. We define $t_{c}$ here along the lines of Oweis et al. (Reference Oweis, Van der Hout, Iyer, Tryggvason and Ceccio2005), as the time taken for the bubble to move to a location within the core ( $r=a/4$ ) from an initial radial location $r=3a$ from the centre of the core ( $a=\text{core radius}$ ). We have experimentally measured this capture time from time sequences of the bubble visualization (as in figure 2) for the different $\mathit{We}$ cases, and this is shown in figure 6 by the solid symbols. The data in the figure are normalized by the vortex ring time scale ( $R/U_{c}$ ) and also include error bars obtained from three repetitions of the experiment. We find as one might expect that $t_{c}$ decreases as the ring circulation or $\mathit{We}$ is increased.

Figure 6.

The variation of normalized capture time with Weber number. Filled circles show experimental data and open circles data from a simple model based on a radial force balance. Ring Reynolds number for the data points from left are 3430, 11 900, 21 200, 29 900, 45 200 and 61 500.

The capture time can be modelled by doing a radial balance of the dominant forces acting on the bubble, as done for example by Oweis et al. (Reference Oweis, Van der Hout, Iyer, Tryggvason and Ceccio2005). Along their lines, we neglect the bubble acceleration and shear lift force in this balance, as these are small compared to the pressure gradient and drag-induced forces, due to the low air density and the negligible vorticity outside the vortex core, respectively. Hence, the radial force balance reduces to a balance between the pressure-gradient-induced force $(F_{P})$ and the drag force $(F_{D})$ on the bubble outside the vortex core. Further, assuming that the largest component of bubble velocity is in the radial direction $(u_{b,r})$ , the radial force balance can be written as

(3.1) $$\begin{eqnarray}\frac{{\it\rho}u_{{\it\theta}}^{2}V_{b}}{r}=\frac{1}{2}{\it\rho}C_{D}\left(\frac{{\rm\pi}}{4}\right)D_{b}^{2}u_{b,r}^{2},\end{eqnarray}$$

where $u_{{\it\theta}}$ is the azimuthal bubble velocity and $V_{b}$ is the bubble volume. In our case, where we consider a relatively thin vortex ring, the azimuthal velocity $u_{{\it\theta}}$ may be approximated by the induced velocity of a line Gaussian vortex. The radial velocity, $u_{b,r}$ , at any given radial location $(r)$ , can then be obtained from the above radial equilibrium equation using an appropriate expression for the drag coefficient ( $C_{D}$ ) valid over a large range of bubble Reynolds numbers ( $\mathit{Re}_{b}=u_{b,r}D_{b}/{\it\nu}$ , where ${\it\nu}$ is the kinematic viscosity), $C_{D}=(24/\mathit{Re}_{b})(1+0.197\mathit{Re}_{b}^{0.63}+0.00026\mathit{Re}_{b}^{1.38})$ (Haberman & Morton Reference Haberman and Morton1953). The capture time $(t_{c})$ can then be found by integration of the thus obtained radial velocity $(u_{b,r})$ from an initial location, $r=3a$ , to $r=a/4$ .

The above simple model is similar to that of Oweis et al. (Reference Oweis, Van der Hout, Iyer, Tryggvason and Ceccio2005), except for the fact that we use a more complete expression for $C_{D}$ due to the much larger bubble Reynolds numbers $(\mathit{Re}_{b})$ in our study. The capture time $(t_{c})$ can be evaluated from this model, given the bubble size $(D_{b})$ , the vortex strength $({\it\Gamma})$ , and its core size (a). Shown in figure 6 in open symbols is $t_{c}$ evaluated using this model. As may be seen from the figure, the qualitative trend of the values obtained from this simple model are in reasonable agreement with the experimental values. The quantitative values are however not close, with the model always under-predicting the experimental values, this being similar to the case in Oweis et al. (Reference Oweis, Van der Hout, Iyer, Tryggvason and Ceccio2005). We find that at low $\mathit{We}$ , the model gives values about 25 % lower than the measured values, while at higher $\mathit{We}$ , the difference increases to about 50 %, consistent with the fact that bubble deformations, that have not been modelled, increase with $\mathit{We}$ . Besides bubble deformations, the observed differences are also likely to be linked to other features observed in the experiments that have not been modelled. This includes, for example, the effects of added mass and lift forces that become more important as the bubble approaches the vortex core (Oweis et al. Reference Oweis, Van der Hout, Iyer, Tryggvason and Ceccio2005), apart from the fact that the bubble modifies the vorticity distribution.

As the ring circulation, or equivalently $\mathit{We}$ , is varied, the appearance (or shape) of the captured bubble and the radial equilibrium location of the captured bubble also change. This is illustrated in figure 7, where we present top view visualization of the bubble after it has been captured by the vortex ring. In each case, the visualization corresponds to an instant in time when the bubble has reached a radial equilibrium position, with further deformation of the bubble with time being purely in the azimuthal direction. We shall first discuss the shape and surface waves on the bubble at this instant, and then discuss the radial equilibrium position. One can see from the images that the bubble in figure 7(a) has a relatively smooth surface, while the bubble in figure 7(b) has a lot of wrinkling or small-scale features, with the complete bubble surface having a wrinkled appearance in figure 7(c). It appears that this change in the nature of the bubble surface is linked to the vortex ring becoming unstable, as the change in appearance corresponds well to the critical or transition $\mathit{Re}$ for the vortex ring. Based on the experimental results of Glezer (Reference Glezer1988), and also from our own visualizations, the critical $\mathit{Re}$ for the rings occurs at $\mathit{Re}={\it\Gamma}/{\it\nu}\sim 11\,000$ , which corresponds to the $\mathit{We}\sim 33$ case in our studies. Hence, the ring in figure 7(a) at a $\mathit{Re}=3430$ is laminar, while the cases in figures 7(b) and 7(c), corresponding to $\mathit{Re}$ well above the critical value, are turbulent. It is thus clear that the nature of the bubble surface at capture depends on whether the ring is laminar or turbulent, with the latter showing a wrinkled surface with much smaller-scale features.

Figure 7.

Bubble visualization immediately after bubble capture for different vortex ring strengths $({\it\Gamma})$ , and correspondingly different ring Reynolds numbers $({\it\Gamma}/{\it\nu})$ : (a) $\mathit{Re}=3430$ , $\mathit{We}=3$ , $t^{\ast }=4.63$ , (b) $\mathit{Re}=29\,900$ , $\mathit{We}=131$ , $t^{\ast }=4.44$ , (c) $\mathit{Re}=61\,500$ , $\mathit{We}=406$ , $t^{\ast }=3.88$ , the times being with reference to figure 5. In each case shown, the bubble has reached its radial equilibrium position. In (a), one can see that the large bubble is displaced from the microbubbles that mark the centre of the vortex core, while in (b) and (c) the large bubble is at the centre of the microbubbles.

We shall now proceed to discuss the radial location of the bubble in figure 7(a–c). The picture in figure 7(a) is striking as it shows that the captured large bubble is distinctly displaced from the small bubbles that presumably mark the centre of the vortex core. It should be stressed here that the bubble shown in figure 7(a) has reached a radial equilibrium position with further motion/deformation of the bubble occurring only in the azimuthal direction, as may be seen from the corresponding time sequence shown in figure 8(c,d). The fact that larger bubbles have a radial equilibrium position away from the core has been seen earlier in the work of Sridhar & Katz (Reference Sridhar and Katz1999), who discuss this point based on the radial force balance. As discussed by them, the radial equilibrium location within the vortex core can be found from the balance between the radially inward pressure-gradient-induced force $(F_{P}={\it\rho}u_{{\it\theta}}^{2}V_{b}/r)$ and the radially outward lift force $(F_{L}=(1/2){\it\rho}C_{L}({\rm\pi}/4)D_{b}^{2}u_{{\it\theta}}^{2})$ , where $u_{{\it\theta}}={\it\Gamma}r/2{\rm\pi}a^{2}$ is the azimuthal velocity of the bubble, and $V_{b}$ is the bubble volume. Substituting for the azimuthal velocity into the expressions, it is clear that $F_{P}$ is proportional to $({\it\Gamma}^{2}rD_{b}^{3})$ while $F_{L}$ is proportional to $({\it\Gamma}^{2}r^{2}D_{b}^{2})$ . Hence, as the bubble gets larger, the pressure-gradient-induced force gets much stronger, and the balance requires that the equilibrium radial location $(r)$ of the bubble be larger. This of course does not take into account the modification of the vorticity field, but gives a broad indication that larger bubbles would have an equilibrium radial position away from the vortex centre.

As the strength of the ring increases, as in figure 7(b,c), the experimental visualization shows that the equilibrium radial location of the bubble shifts to the centre of the core, as marked by the smaller microbubbles. It is not possible to explain this using the radial force balance in the form discussed above, as both $F_{P}$ and $F_{L}$ have the same ${\it\Gamma}^{2}$ dependence. This is hence likely to be related to the modification of vorticity, as suggested above, and as will be shown later in this section. In summary, we find that at lower vortex strengths (lower $\mathit{We}$ ), the bubble is displaced from the vortex centre, while at higher vortex strengths, the bubble pierces through the vortex core and sits in the centre of the vortex.

Figure 8.

Time sequence of bubble visualization images showing the capture and deformation of a bubble for the low $\mathit{We}=3$ case. The non-dimensional time ( $t^{\ast }$ ) corresponding to each image shown is (a) 3.19, (b) 4.41, (c) 4.61, (d) 4.74, (e) 6.47, and (f) 7.27, the times being with reference to figure 5.

3.1.2. Bubble dynamics within the vortex ring (stage II)

We shall now proceed to present results of the bubble deformation following capture, in what we refer to as stage II (as defined in figure 5). Top view bubble visualizations beginning from bubble capture are shown in figure 8 for the $\mathit{We}=3$ case. Figure 8(a–c) show capture, with 8(c) being the same as figure 7(a); this figure shows that the captured large bubble is displaced from the micron-sized bubbles that mark the vortex core. Following capture in figure 8(c), one can see pure azimuthal growth of the bubble in figure 8(d) showing that figure 8(c) does indeed show the radial equilibrium position. After this, in figures 8(e) and 8(f), one can see further elongation of the bubble in the azimuthal direction. This elongation occurs as a result of the azimuthal pressure difference resulting from the modified vorticity distribution in the presence of the bubble, as discussed in more detail later in this section. This azimuthal pressure difference stretches the bubble, which in turn leads to gradual breaking of a small bubble from the large bubble through necking as seen in figure 8(e,f).

As the ring strength is increased, $\mathit{We}$ increases, and the same broad features of bubble capture and azimuthal expansion are seen. We have already discussed the $\mathit{We}=131$ case that was shown in figure 1, where we can see capture of the bubble and bubble elongation as seen in the $\mathit{We}=3$ case in figure 8. In the $\mathit{We}=131$ case (figure 1), one can however see surface perturbations on the bubble that are absent at lower $\mathit{We}$ , which continue to grow as the bubble expands following capture. Following expansion, the bubble also contracts azimuthally with the surface perturbations continuing to grow. This gradually leads to the formation of a number of small bubbles as seen in figure 1(h).

From visualizations such as those in figures 1 and 8, we can extract bubble size information at different times. In particular, we can extract the azimuthal length of the bubble $(L_{b})$ following capture, and this is shown in figure 9, where $L_{b}$ is normalized by the vortex ring circumference $(2{\rm\pi}R)$ . It may be noted that the azimuthal bubble expansion length shown for each of the vortex ring cases in figure 9 were obtained as an average of three independent runs with variation between runs indicated by error bars. In the figure, the time ( $t^{\prime }$ ) starts from the instant of bubble capture and is normalized by a pressure-gradient-based time scale ( $t_{p}$ ) discussed below. The pressure difference between the core of a single Gaussian line vortex with core size a and the far field is ${\rm\Delta}P=\{0.87{\it\rho}({\it\Gamma}/2{\rm\pi}a)^{2}\}$ (Oweis et al. Reference Oweis, Van der Hout, Iyer, Tryggvason and Ceccio2005), while it is expected to be closer to the far-field pressure when the bubble sits in the centre of the vortex core, as discussed in more detail in the next section (see figure 11). Broadly speaking, the bubble is thus exposed to a pressure difference in the azimuthal direction, which is of the order of ${\rm\Delta}P=\{0.87{\it\rho}({\it\Gamma}/2{\rm\pi}a)^{2}\}$ , that is responsible for elongation of the bubble after capture. A time scale $(t_{p})$ may be obtained from the above pressure difference, $t_{p}=(0.87{\it\rho}{\it\Gamma})/({\rm\Delta}P{\rm\pi})={\rm\pi}(2a)^{2}/{\it\Gamma}$ .

Figure 9.

The variation of bubble expansion length $(L_{b})$ along the azimuthal direction within the vortex ring with time starting from the instant of capture. The normalized bubble expansion length $(L_{b}/2{\rm\pi}R)$ is plotted with time non-dimensionalized with an azimuthal pressure-difference-based time scale $(t_{p})$ . The initial bubble expansion data for the different $\mathit{We}$ cases collapse onto a single line.

In figure 9, we plot the bubble elongation data against time ( $t^{\prime }$ ) non-dimensionalized by this time scale $t_{p}$ , and we can see that the initial azimuthal elongation ( $t^{\prime }/t_{p}<4$ ) in all the six $\mathit{We}$ cases from 3 to 406 lie nearly along a single line. The collapse of the data for $t^{\prime }/t_{p}<4$ with this non-dimensionalization does however validate the fact that the azimuthal elongation of the bubble is indeed determined by the azimuthal pressure difference given above. One can further see from figure 9 that the slope of the contraction in all cases is also similar with this non-dimensionalization. One can however see from the figure that there are differences in the data for $t^{\prime }/t_{p}>4$ , which corresponds to the time after the bubble reaches its maximum elongation. For $\mathit{We}$ of 3 and 33.3, the trend after first maximum bubble expansion is different compared to higher $\mathit{We}$ ( $\mathit{We}>50$ ). For $\mathit{We}>50$ , the trend of bubble expansion, contraction and re-expansion is found to be nearly similar, as shown by the dashed line in the plot.

At later times, in all $\mathit{We}$ cases, the bubble breaks up into smaller ones, whose combined length is shown in the plot, and is seen to reach an approximately constant value. It may also be seen from the figure that both the first maximum bubble elongation $(L_{b,max})$ at $t^{\prime }/t_{p}$ between 4 and 7 and the final steady bubble length at large time, increase with $\mathit{We}$ at low $\mathit{We}$ , and then saturate for $\mathit{We}>200$ . $L_{b,max}$ saturates at $L_{b}/(2{\rm\pi}R)$ of about 0.7, while the long time bubble length saturates at about 0.8, both corresponding to $\mathit{We}>200$ .

3.2. Vorticity dynamics

We shall now present results from the vortex dynamics perspective for the initial stages of interaction between the ring and the bubble comprising stages I and II. With this in mind, we present, as in figure 3, measurements of the azimuthal vorticity obtained from time-resolved PIV in the $x{-}z$ plane cutting through the centre of the vortex ring. The time sequence of the measured azimuthal vorticity for the $\mathit{We}=3$ case is shown in figure 10, with the bubble in each plot being marked by a hatched area (magenta online). In figure 10(a), the bubble is outside the vortex ring, and the vorticity distribution of the undisturbed vortex ring can be seen. In addition, vorticity can also be seen around the approaching bubble, as seen in rising single bubble studies (Brücker Reference Brücker1999; Mougin & Magnaudet Reference Mougin and Magnaudet2001). At these low $\mathit{We}$ , the vorticity shed from the bubble is of the same order as the strength of the ring and is hence clearly visible in the vorticity plots. As the bubble gets pulled into the ring, it elongates and pierces into the ring, as shown in figure 8(b), and in the process distorts the vorticity distribution as shown in figure 10(b). In this low- $\mathit{We}$ case, as discussed earlier, the radial equilibrium position of the large bubble is not at the centre of the vortex, but is displaced radially outward from the centre of the vortex core. This may be seen in figure 10(c), where the higher vorticity contours of the right-hand-side core are bunched to the left of the bubble, indicating that the larger bubble is not sitting at the centre of the modified vortex core. After the bubble is captured, it elongates in the azimuthal direction, reaching maximum expansion and then contracts back as may be seen in figure 9. The vorticity plot corresponding to maximum elongation is shown in figure 10(d), and figure 10(e,f) correspond to later times, where the bubble length $(L_{b})$ reaches a constant value as may be seen from figure 9. There is no noticeable change in bubble size in figure 10(e,f), as there is not much change in the azimuthal length of the bubble after contraction. One can also clearly see from the figures that the left-hand-side core is relatively unaffected throughout the interaction process, as the bubble in the right-hand core is not sufficiently stretched to reach the other side.

Figure 10.

Time sequence of principal azimuthal vorticity showing the different stages of vortex ring–bubble interaction for the low $\mathit{We}=3$ case. In each image, the bubble is marked as a hatched area (magenta online), while the solid and dashed line contours represent positive and negative vorticity, respectively. The non-dimensional time ( $t^{\ast }$ ) corresponding to each image is (a) 3.18, (b) 4.29, (c) 4.6, (d) 6.4, (e) 7.25 and (f) 7.8, the times being with reference to figure 5. Non-dimensional vorticity contour ( ${\it\omega}R/U_{c}$ ) levels shown are $\pm 1.6,2.2,2.9,\dots$  .

Figure 11.

Contour plots of the normalized pressure distribution $P^{\ast }$ around the vortex ring calculated from the measured PIV velocity field. The vortex cores are marked by a dashed line while the bubble is indicated by the hatched region. The normalized pressure $P^{\ast }$ is defined as $(P-P_{\infty })/{\rm\Delta}P$ , where ${\rm\Delta}P=0.87{\it\rho}({\it\Gamma}/2{\rm\pi}a)^{2}$ is the pressure reduction at the centre of a two-dimensional Gaussian vortex. Non-dimensional time ( $t^{\ast }$ ) corresponding to (a), (b) and (c) is 1.77, 3.85 and 5.25 respectively ( $\mathit{We}=406$ ). As seen in (a), the pressure $P^{\ast }$ at the centre of the vortex core before bubble capture is close to $-1$ indicating that it is similar to an equivalent two-dimensional Gaussian vortex. In (b) and (c), one can see the large pressure difference between the two vortex cores, with the pressure in the left-hand core affected by the bubble being around $-0.2$ , and that of the unaffected right-hand core being around $-0.9$ .

As $\mathit{We}$ is increased, the time sequence of vorticity snapshots shows broadly similar features, although there are some important changes as well. The vorticity time sequence for the high $\mathit{We}$ of 131 was shown earlier in figure 3. A common feature in both the high- $\mathit{We}$ case (figure 3) and the low- $\mathit{We}$ case (figure 10) is the fact that the bubble pierces through the vortex core in plot (b), distorting the vorticity distribution as it enters the ring. Once the bubble is thus captured by the ring, the contours rejoin and the vorticity gets distributed in an annular form around the captured bubble. On the other hand, the differences between the high and low $\mathit{We}$ cases are seen for example in the radial location of the bubble after capture. In the high- $\mathit{We}$ case, the bubble is located roughly in the vortex centre, while in the lower- $\mathit{We}$ case, the bubble is clearly off the vortex centre, as seen earlier from the bubble dynamics in figure 7. The other difference is seen in plots (e) and (f) following maximum expansion of the bubble. In the higher- $\mathit{We}$ case (figure 3) the expansion of the bubble is larger, and hence the stretched bubble, or some fragments of it, reaches the other core as well, resulting in some noticeable distortion of the other core, unlike the lower- $\mathit{We}$ case (figure 10).

The pressure field around the vortex core can be calculated from knowledge of the velocity field using the Navier–Stokes equation. This can be done by substitution of the velocity field into the Navier–Stokes equation to determine the pressure gradient term, which can in turn be integrated with respect to a reference pressure, say the far-field value, to get the pressure field. As pointed out by Dabiri et al. (Reference Dabiri, Bose, Gemmell, Colin and Costello2013), there are issues in doing the above calculations with experimental PIV data that are inherently noisy, and they developed an appropriate algorithm to minimize these issues and implemented it into a code. Utilizing this code, we have determined the pressure field from our measured PIV velocity field, and this is shown in figure 11, the pressure being shown in non-dimensional form, $P^{\ast }=(P-P_{\infty })/{\rm\Delta}P$ , where ${\rm\Delta}P=0.87{\it\rho}({\it\Gamma}/2{\rm\pi}a)^{2}$ is the pressure reduction at the centre of a two-dimensional Gaussian vortex as discussed earlier and $P_{\infty }$ is the far-field pressure. The pressure fields shown correspond to the $\mathit{We}=406$ case, with the core (lowest vorticity contour) being marked in each case by a dashed line, and the bubble by the hatched area. In figure 11(a), the bubble is outside the ring, and the non-dimensional pressure ( $P^{\ast }$ ) in both cores is close to $-0.9$ , indicating that the core pressures are low and close to the values for an equivalent two-dimensional Gaussian vortex. On the other hand, the pressure field in figure 11(b) at bubble capture is significantly modified, with the pressure in the left-hand core being much closer to the far-field pressure at $-0.3$ , while that in the right-hand core remains nearly unaffected with a pressure of about $-0.8$ . There is thus a large non-dimensional pressure difference of about 0.5 in the azimuthal direction, which is responsible for the azimuthal elongation of the bubble, discussed earlier. Figure 11(c) corresponds to maximum elongation of the bubble in the azimuthal direction and hence a relatively smaller bubble radius. In this case, the pressure in the left-hand core with the bubble is a little lower with a value of about  $-0.4$ .

The pressure fields shown highlight the differences in pressure caused by the distorted vorticity distribution within the core, which is broadly in an annular form within the water phase around the bubble. It may be noted that these pressure fields were calculated with the fluid velocity within the bubble set to zero, thus giving no variation in pressure within the bubble area, as one would expect in the real case due to the much smaller density of air compared to that of water. Further, it should also be noted here that in these cases, the change in pressure across the interface due to surface tension is very small compared to ${\rm\Delta}P$ .

We have so far discussed the bubble and vorticity dynamics in stages I and II of the ring and bubble interaction. By the end of the second stage, the bubble has been completely broken up in to smaller bubbles, as shown for example in figure 1(h) for the high- $\mathit{We}$ case and figure 8(f) for the low- $\mathit{We}$ case. At the same instant in time, the vorticity fields in the high- and low- $\mathit{We}$ cases are shown in figures 3(f) and 10(f), respectively. In both cases, the vorticity is distributed in an annular form within the water phase, caused by the presence of the bubble in the centre. A simple and robust measure of the integral effect of the vorticity distribution is the convection speed of the vortex ring, which is the slope of the ring position data in figure 5. At this instant (end of stage II), the convection speed data show a distinct difference between the low- and high- $\mathit{We}$ cases. In the high- $\mathit{We}$ cases, the data for the ring with the bubble are nearly the same as the baseline case, while in the lower- $\mathit{We}$ case ( $\mathit{We}\sim 3$ ), there is a clear reduction in the convection speed of the ring. This is an indication of the fact that the ring has been considerably affected by the bubble at this time.

This naturally raises the question of what could be possible mechanisms that may be responsible for the reduction in convection speed of the ring. One possible mechanism is related to the distorted annular vorticity field caused by bubble capture, which occurs at the line demarcating stages I and II in figure 5. Around this time, bubble capture occurs at a rapid rate, but there is no significant change in convection speed at this time, as seen in figure 5. This suggests that the modified annular vorticity distribution caused by bubble capture does not by itself directly affect the convection speed. The reduction in convection speed may then be caused by an instability of the modified vortex core at bubble capture (such as figure 10 c), which then gradually develops. There are a few features of this modified vortex core at bubble capture that may be responsible for instability and reduction in the convection speed of the ring. These include the annular form of vorticity distribution within the water phase, and the presence of sharp density and viscosity changes at the interface. For example, Dritschel (Reference Dritschel1986) has shown that a single annular two-dimensional distribution of constant vorticity in an incompressible (single) phase can be unstable to two-dimensional perturbations, which can lead to the formation of multiple vortices, or what they refer to as ‘break-up’ of the vortex. We shall see in the next section that this is what happens in our case, although the precise reason for it in our case is not very clear.

4.1. Bubble dynamics

The break-up of the bubble into smaller bubbles is complete by the end of stage II, and hence there is almost no further break-up of the bubble within stage III. This stage may hence be referred to as a quasi-steady stage from the bubble dynamics perspective. This is shown by an example time sequence of bubble visualization within this stage in figure 12, that shows that the number of bubbles is almost the same. One of the smaller bubbles is marked in the first image, with the same bubble being marked in the other two images. This shows the gradual anti-clockwise azimuthal motion of the bubbles within the core during this stage, as indicated by the arrow in the first image.

Figure 12.

A time sequence of bubble visualization within stage III. The non-dimensional time ( $t^{\ast }$ ) corresponding to (a), (b) and (c) is 12.69, 25.3 and 47.59 respectively ( $\mathit{We}=131$ ). As may be seen, the number of bubble fragments remains nearly the same in all three cases. One of the smaller bubbles is marked in the first image, with the same bubble being marked in the other two images, showing the gradual azimuthal motion of the bubbles within the core.

In each of the Weber number cases studied, there is a fixed number of smaller bubbles within stage III, as no further break-up occurs within this stage. Shown in figure 13(a) is the number of smaller bubbles in this stage as a function of $\mathit{We}$ . As may be seen from the plot, the number of smaller bubbles formed as a result of break-up increases with $\mathit{We}$ , starting from 3 at low $\mathit{We}$ , and reaching about 16 at the higher $\mathit{We}$ of 406. It should be noted that all the values shown here are a result of the average of three experiments done for each $\mathit{We}$ case, with the variation between the three runs indicated by the error bars in the plot. Associated with this increase in the number of smaller bubbles is a reduction in the average diameter of the bubbles formed $(D_{m})$ , which is shown in figure 13(b) normalized by the nominal bubble size $D_{b}$ before entrainment. We can fit a power law to the data, and we find that the average diameter of the formed bubbles varies as $\mathit{We}^{-0.124}$ , as shown in the figure. This may be compared to the more-studied case of droplet/bubble break-up due to isotropic turbulence, where Shinnar (Reference Shinnar1961) showed that the average droplet/bubble size formed after complete break-up by isotropic turbulent flow varies as $\mathit{We}^{-0.6}$ based on simple arguments, $\mathit{We}$ here being defined as the ratio of kinetic energy of oscillating droplet/bubble to the surface energy.

Figure 13.

Variation of (a) the number of bubbles, and (b) their diameter $(D_{m})$ , normalized by the nominal bubble size $D_{b}$ before entrainment, with $\mathit{We}$ after complete break-up of the bubble (stage III). The number of Kelvin waves seen from the visualizations is also shown in (a), and is found to be different from the number of smaller bubbles. The number of Rayleigh–Plateau instability waves that fit on the cylindrical bubble is also shown in (a). The number of instability waves is close to the number of bubbles at low $\mathit{We}$ , but deviates at larger $\mathit{We}$ . ●, Experiment; ▫, Kelvin wave; ○, Rayleigh–Plateau instability.

In these cases of bubble break-up due to isotropic turbulence, as discussed by Martínez-Bazán et al. (Reference Martínez-Bazán, Montañés and Lasheras1999), break-up occurs as a result of the combined effects of inertia, surface tension and turbulent energy dissipation per unit mass. Risso & Fabre (Reference Risso and Fabre1998) further suggest that the time response of the bubble controls the maximum amount of energy which can be extracted from each turbulent eddy and the viscous damping limits the energy that the bubble can accumulate during its fluctuating deformation. This problem has received a lot of attention due to its importance in bubbly turbulent flow where the average diameter of the completely broken-up bubble determines the interfacial area, which in turn determines the rate of mass, momentum and energy transfer between the two phases. Our experiments suggest that the exponent is different in the present idealized case of a single vortex structure (ring) compared to Shinnar (Reference Shinnar1961), which is for isotropic turbulence. It should be noted here that the change in exponent is not unexpected, as in the isotropic turbulence case, the whole process of break-up is considerably more complex due to the multi-scale nature of turbulent structures. The differences between the two can be firstly attributed to the fact that bubble break-up in the turbulent case not only occurs due to an intense single eddy, but also through the combined action of a succession of different eddies (Risso & Fabre Reference Risso and Fabre1998). This is in addition to the obvious fact that in the turbulent case, the broken bubbles are always exposed to eddies with smaller scales that they can interact with, unlike in the present case.

A close-up of the bubble break-up in our present case from side visualizations is shown in figure 14; this occurring in stage II. The visualizations show both the large bubble being deformed within the ring, as well as the micron-sized bubbles tracking the vortex core. The smaller bubbles tracking the core clearly show the Kelvin waves in the core of the vortex ring, as discussed by Widnall et al. (Reference Widnall, Bliss and Tsai1974). It is natural to then ask whether the number of smaller bubbles is related to the number of Kelvin waves in the ring, which can be easily counted from the visualizations, besides getting an estimate from the theory based on the relative core size (Widnall et al. Reference Widnall, Bliss and Tsai1974). We present in figure 13(a), the number of Kelvin waves as counted from the visualizations at each $\mathit{We}$ , and it is clear that the number of Kelvin waves does not match the average number of bubbles formed after complete break-up. The Kelvin waves do however play some part in the break-up as seen in the images in figure 14, where the bubble appears to break up due to these azimuthal waves, as may be seen by following the time sequence of images. The process of necking can be seen in figure 14(b), which develops leading to the formation of a gas bridge in figure 14(e) and break-up in figure 14(f). However, the number of bubbles formed after complete break-up is clearly not determined by the Kelvin waves. Apart from the mismatch in numbers discussed earlier, there is also a time-scale issue, as these Kelvin waves are typically short lived as they rapidly grow and result in the ring becoming turbulent. The complete break-up of the bubble on the other hand takes much longer.

Figure 14.

Successive images of bubble break-up while interacting with the vortex ring illustrating ‘pinch-off’ of smaller bubbles ( $\mathit{We}=131$ ). Necking can be seen in (d). The non-dimensional time ( $t^{\ast }$ ) corresponding to each image is (a) 4.72, (b) 4.75, (c) 4.82, (d) 4.85, (e) 4.88 and (f) 4.91, the times being with reference to figure 5.

Apart from the Kelvin waves, another possible mechanism that could determine the average size or number of bubbles following break-up would be the classical Rayleigh–Plateau instability of the stretched cylindrical bubble. The wavelength corresponding to maximum growth rate for this instability is about 4.5 times the diameter of the cylindrical bubble. In our case, if we use the minimum diameter of the stretched bubble $(d_{br,min})$ , which corresponds to the first maximum azimuthal length $(L_{b,max})$ , we can get a wavelength for each one of the different $\mathit{We}$ cases, from the data set shown in figure 9. From this wavelength, we can get both the number of waves that can fit on a cylindrical bubble of length $L_{b,max}$ and the diameter of the corresponding broken-off bubble, as shown in figures 13(a) and 13(b), respectively (open circles). As may be seen from the values in the plot, the number of these waves that fit into the stretched bubble and their variation with $\mathit{We}$ is in reasonable agreement with experimental data at low $\mathit{We}$ ( $\mathit{We}<150$ ), although it is not very close. At $\mathit{We}>150$ , the visualized number of broken-off bubbles continues to increase, while the number of instability waves is fixed as the bubble length and diameter saturate, as discussed earlier with reference to figure 9. The diameter of the broken-off bubble estimated from these waves is also shown in figure 13(b), showing a similar trend to those seen in the experiments, although the actual values are again not very close. It should be noted here that the number of broken bubbles seen in the experiments is typically larger than from the instability wavelength, and the difference increases for $\mathit{We}>150$ , where the $\mathit{Re}$ of the ring is also larger. Although the reason for this is not very clear, the larger number of bubbles seen in the experiments may be caused by the increased level of turbulence as the ring $\mathit{Re}$ increases. In addition, differences may be caused by the presence of rotational flow around the bubble.

Figure 15.

Comparison of principal azimuthal vorticity distribution for a vortex ring (a) without and (b) with interactions with a bubble for $\mathit{We}=3$ at a later time ( $t^{\ast }$ with reference to figure 5 is 14.67). In (b), the vorticity distribution for the ring interacting with a bubble is markedly different from the identical ring in the absence of a bubble shown at the same time in (a). The vortex core in (b), particularly the one on the right with the bubble, appears fragmented with significant shedding from it. The bubble is marked by hatched area (magenta online), while the solid and dashed line contours represent positive and negative vorticity, respectively. Non-dimensional contour levels shown are $\pm 0.16,0.32,0.48,\dots ,1.29$ .

4.2. Vorticity dynamics

We present here the azimuthal vorticity measurements during the later stages of interaction between the bubble and the vortex ring. In figure 15, we present the measured vorticity field for the $\mathit{We}=3$ case, at non-dimensional time $t^{\ast }=14.67$ , where the convection speed of the vortex ring is substantially lower than the base case of the vortex ring without a bubble, as may be seen from figure 5. The figure shows both the base case, in figure 15(a), and the case with bubble interaction in figure 15(b) with only lower level contours shown in both cases to highlight the edges of the vortex and shedding from it. As may be seen from the figures, there is a striking difference in the measured vorticity fields in the two cases. The vorticity field in figure 15(a) shows a clear circular vortex core with no shedding of vorticity from the ring, consistent with the fact that the ring is laminar. On the other hand, the vorticity contours for the case with bubble interactions in figure 15(b) shows a fragmented right-hand core where the bubble is present, with significant shedding of vorticity from this core. The left-hand core is also affected, but not as significantly as the right, but the overall convection speed of the ring has been significantly reduced (about 30 %) as seen in figure 5. Shortly after this, the bubble escapes, leaving behind a ring with a fragmented core that travels significantly slower. We would like to stress here that Sridhar & Katz (Reference Sridhar and Katz1999) also noted vortex distortion in their study, but the vortex core regained its initial undistorted state in that case once the bubble escaped. On the other hand, the effect on the vortex core is a lasting one in our case and the core is left in a fragmented state even after bubble escape.

Figure 16.

Comparison of principal azimuthal vorticity distribution for a vortex ring (a) without and (b) with interactions with a bubble for $\mathit{We}=406$ at later time ( $t^{\ast }$ with reference to figure 5 is 14.02). In (b), the vorticity distribution for the ring interacting with a bubble is again significantly different from the identical ring in the absence of a bubble shown at the same time in (a). The main difference here is the increased shedding from the ring. However, at this higher $\mathit{We}$ , the core appears reasonably intact with no significant deformation, unlike in the lower- $\mathit{We}$ case in figure 15. The bubble is shown as the hatched area (magenta online), while the solid and dashed line contours represent positive and negative vorticity, respectively. Non-dimensional contour levels shown are $\pm 0.1,0.21,0.32,\dots ,1.06$ .

In contrast to the low- $\mathit{We}$ case, the vorticity distribution in the higher- $\mathit{We}$ cases show less influence of the bubble on the core, as shown in figure 16 for $\mathit{We}=406$ . As in the low- $\mathit{We}$ case, both the base ring in the absence of the bubble (in figure 16 a), and the ring after interactions with the bubble in figure 16(b) are shown. Both these fields are shown at non-dimensional time $t^{\ast }=14.02$ , where the convection speed is nearly the same as the base ring case, as may be seen from figure 5. In this case, the base ring vorticity field (in figure 16 a) shows inherent shedding from the ring, consistent with the fact that the ring in this case is turbulent (Gan, Dawson & Nickels Reference Gan, Dawson and Nickels2012), as discussed earlier. However, in the case of the ring with the bubble, as shown in figure 16(b), one can see a significant increase in the shedding from the ring compared to 16(a). Although the reason for this increased shedding from the vortex core with the bubble is not completely clear, it is likely to be caused by the shedding from the bubble present within the core, as has been discussed for example in Sridhar & Katz (Reference Sridhar and Katz1999). Despite this increased shedding, the core appears relatively intact in this high- $\mathit{We}$ case, unlike in the lower- $\mathit{We}$ case in figure 15. The convection speed in this high- $\mathit{We}$ case is also relatively unaffected, consistent with the almost intact vortex core.

Figure 17.

Principal azimuthal vorticity $({\it\omega})$ profile along a radial line through the vortex centre for (a) $\mathit{We}=3$ , and (b) $\mathit{We}=406$ , at later non-dimensional time ( $t^{\ast }$ ) with respect to figure 5 of 14.67 for (a), and 14.02 for (b). In each case, the vorticity profile in the absence of the bubble is shown by open symbols, while interacting case is shown by filled symbols. The data shown correspond to the vorticity fields shown in figures 15 and 16, respectively. All vorticity values in these plots are normalized by the peak vorticity of the base vortex ring $({\it\omega}_{peak})$ . The bubble is marked by the hatched area (magenta online).

The vorticity distribution within the (single) vortex core for the low-and high- $\mathit{We}$ cases is shown in figure 17 for both the ring interacting with the bubble, and the base case, to further highlight the differences. In figure 17(a), the vorticity distribution along a 45° line through the core with the bubble is shown for the $\mathit{We}=3$ case, as also marked on the image in figure 15(b). The vorticity distribution is clearly quite complex with the presence of multiple peaks: two positive vorticity peaks and one negative vorticity peak. As expected, the base vortex core in the absence of the bubble, shown by the dashed line, is a simple Gaussian profile. In the higher- $\mathit{We}$ case shown in figure 17(b), the vorticity is distributed in an annular way around the bubble, the distribution being in the form of two Gaussian profiles with reduced peak vorticity. This type of reduced peak vorticity for cases where the bubble resides within the core is a common feature in these flows, which as we shall see below has consequences.

Figure 18.

The variation of (a) enstrophy $(Z)$ and (b) circulation $({\it\Gamma})$ with time for a vortex ring interacting with a bubble and for the base ring case in the absence of the bubble. $Z$ and ${\it\Gamma}$ are non-dimensionalized by their respective values at $t=0$ , and time is non-dimensionalized by the ring time scale $(R/U_{c})$ .

The convection speed of the ring is an integral effect of the vorticity distribution within it. Hence, one way to understand the underlying physics of the reduction of convection speed is to look at the enstrophy or volume integral of the square of the vorticity of the vortex ring. In the current study, as two-dimensional time-resolved PIV measurements have been done, the enstrophy $(Z)$ is calculated as the area average of the square of the azimuthal vorticity, this being representative of the principal vorticity in the flow. For the purpose of enstrophy calculation, an area that is about 100 times the core area was used to ensure that all the vorticity corresponding to the vortex ring is included. There is some variation between runs, but this is within the experimental uncertainties, as may be seen by the variation in circulation between time instants and the variation between repetitions of the same experiment shown by the error bars in figure 18. Figure 18(a) shows the variation of enstrophy $(Z)$ with time for a low- $\mathit{We}$ and a high- $\mathit{We}$ case, along with their respective base ring cases; $Z$ is non-dimensionalized in each case by its respective initial value $(Z_{i})$ . In both the base cases, in the absence of a bubble, a gradual reduction in enstrophy is seen with time due to viscous diffusion of the core, reaching a value of about 0.8 at $tU_{c}/R\approx 12$ . In contrast, for the low- $\mathit{We}=3$ case, the enstrophy reduction with time is significantly enhanced in the case with a bubble. At small times, the enstrophy reduction is similar to the base case, until $tU_{c}/R\approx 4$ , where there is a sharp drop in enstrophy, due to the sudden capture of the bubble by the ring, which results in a sharp drop in peak vorticity. Subsequently, as the bubble expands azimuthally within the ring, the bubble diameter reduces, as may be seen in figure 9, and this leads to an increase in enstrophy, which almost reaches the enstrophy of the base vortex ring at $tU_{c}/R\approx 5.4$ . Following this, $Z/Z_{i}$ reduces reasonably rapidly reaching what appears to be an asymptotic value of about 0.4, which corresponds to a large reduction of about 50 % compared to the base case at the same time. As stated earlier, this state continues even after the bubble escapes, leaving behind a less coherent vortex ring, whose core appears fragmented as shown in figure 15. In contrast to this large reduction in enstrophy for the low- $\mathit{We}$ case, the reduction in enstrophy is relatively small for the higher- $\mathit{We}$ case of 263 also shown in figure 18(a). The enstrophy values for this case are in fact close to the base ring case, consistent with the negligible reduction in convection speed for this case, as may be seen from figure 5.

It should be noted here that reduction in enstrophy of a vortex due to interaction with bubbles has been observed earlier. For example, Ferrante & Elghobashi (Reference Ferrante and Elghobashi2007) in their numerical study of the interaction of a Taylor–Green vortex with a large number of entrained small bubbles report a reduction in enstrophy of around 4 % due to the modified vorticity distribution. For a turbulent channel flow with bubbles, Lu et al. (Reference Lu, Fernández and Tryggvason2005) report a large reduction of about 60 % in enstrophy in their DNS study, with suppression of structures. They attribute the observed drag reduction to the reduction in enstrophy and suppression of structures, with more drag reduction seen when their Weber number is closer to 1. Gils et al. (Reference Gils, Guzman, Sun and Lohse2013), for the case of bubbles in Taylor–Couette flow, show that the $\mathit{We}$ value close to the inner wall is very important in determining drag reduction, with more drag reduction observed when the bubbles deform. Our results show similar reductions in enstrophy in the more basic interaction of a single bubble with a vortex ring when $\mathit{We}\sim 3$ .

The circulation of the vortex core is another integral quantity of interest. In our case, as is typical of many experiments, we calculate the circulation as the line integral of velocity around a sufficiently large closed curve around a core. The resulting values of circulation for the low- and high- $\mathit{We}$ cases with corresponding base cases is shown in figure 18(b). As may be seen from the plot, there is some variation in circulation, but this is not significant compared to the variations in time, indicating that the observed variations are within the experimental error. Hence, at a broad level, we find that there is no significant change in the circulation around the core of the vortex ring, although some small variations may be present. It may be noted that Sridhar & Katz (Reference Sridhar and Katz1999) also reported no noticeable change in circulation in their experimental study of a ring interacting with a few smaller bubbles. On the other hand, Cihonski et al. (Reference Cihonski, Finn and Apte2013) in their numerical study of the same case showed an increase in circulation of about 10 %.

As discussed in the end of § 3, the convection speed of the ring was seen to have reduced significantly in the low- $\mathit{We}$ case by the end of stage II. We suggested that this reduction could be due to a possible instability of the modified vortex core that developed at bubble capture, with the vorticity being distributed in an annular form around the bubble. As stated there, Dritschel (Reference Dritschel1986), for example, has shown that a single annular two-dimensional distribution of constant vorticity in an incompressible (single) phase can be unstable to two-dimensional perturbations, leading to the formation of multiple vortices, which is what we observe in the present case at low $\mathit{We}$ (figure 15 b). We shall see in the following section that this deformation of the core leading to multiple vortices is even more dramatic in the thick ring case.