3.1. Description of the interactions explored

To evaluate the three-dimensional experimental approach, we focus on three specific bubble-vortex interaction cases: Case I, in which a weak interaction between the bubble and the vortex ring takes place; Case II where the bubble is captured and dragged by the vortex; and Case III with the bubble breaking up. The three cases studied are included in figure 3, which reproduces the transition map from Foronda-Trillo et al. (Reference Foronda-Trillo, Rodríguez-Rodríguez, Gutiérrez-Montes and Martínez-Bazán2021). In that study, numerical simulations were performed on bubble–vortex interactions at a fixed $\textit{Re} = 15\,000$ and a core ratio of $a_0/R_0 = 0.1$ (thin vortex rings), while experiments were performed at $\textit{Re} \in (9000, 20\,000)$ . The figure illustrates the breakup efficiency of experiments (hollow symbols) and numerical simulations (solid symbols), and transition curves are obtained depending on breakup probability. In this work, we intend to solve the three-dimensional flow field in three different characteristic interaction cases while keeping $R_0/R_b$ approximately constant. Thus, we first considered a low-Weber-number case, in which the bubble was neither trapped nor broken by the vortex. In the second case, we increased the Weber number while attempting to keep $a_0/R_0$ equal to that of Case I, to achieve vortex trapping of the bubble without breakup. To highlight the effect of the vortex core ratio, we aimed to induce bubble breakup at $\textit{We}\sim 1$ , by reducing $a_0/R_0$ . Thus, we observed bubble trapping and breakup in Case III. It should be noted that the three cases considered here are located within the transition region identified by Foronda-Trillo et al. (Reference Foronda-Trillo, Rodríguez-Rodríguez, Gutiérrez-Montes and Martínez-Bazán2021). They showed that, within this region, the interaction outcome is not uniquely determined by the nominal values of $\textit{We}$ , $R_0/R_b$ and $a_0/R_0$ , but small differences in the initial relative position between the vortex core and the bubble may lead to different interaction outcomes under otherwise similar operating conditions. Consequently, the experiments presented here should not be interpreted as repeated realisations of identical conditions. The goal of the present work is to provide a detailed three-dimensional characterisation of three representative interaction scenarios, rather than to determine the probability of occurrence of each regime.

Figure 4 depicts raw images captured at various time instants for each case. Both the bubble and the vortex ring can be observed in the figure: the bubble rises and the vortex ring descends until they interact. The type of interaction results from the values of the governing parameters, listed for each case in table 1. Additionally, the initial characteristics of the vortex rings are detailed in table 2. Note that the uncertainties in these parameters, and in the rest of the results throughout the manuscript, were estimated using standard propagation methods based on the literature (Ho Reference Ho1983; Stanislas et al. Reference Stanislas, Okamoto, Kähler, Westerweel and Scarano2008; Kähler et al. Reference Kähler, Astarita, Vlachos, Sakakibara, Hain, Discetti, La Foy and Cierpka2016; Schanz et al. Reference Schanz, Gesemann and Schröder2016; Sciacchitano & Wieneke Reference Sciacchitano and Wieneke2016). The error propagation analysis ensures that the cumulative relative uncertainty in velocity ranges between 2 % and 10 % in the region near or far from the vortex, respectively.

Table 1.

Dimensionless parameters describing the different types of interactions studied. Based on the sub-pixel accuracy of the measurement technique (described in Appendix B), the errors in the flow-related parameters and the bubble size are below $3\,\%$ and $1\,\%$ , respectively.

Table 1. Long description

A table with three rows and nine columns comparing dimensionless parameters for different types of interactions. The columns are labeled as Case, Description, R0/Rb, a0/R0, a0/Rb, Re, We, Bo, Ga, and Fr. The rows are labeled as Case I, Case II, and Case III. Row 1: Case I, Weak interaction, 4.3, 0.37, 1.59, 8900, 0.41, 1.57, 618.5, 1.35. Row 2: Case II, Capture, drag and release, 4.5, 0.38, 1.71, 14600, 1.00, 1.81, 690.5, 1.25. Row 3: Case III, Bubble break up, 4.2, 0.28, 1.18, 11800, 0.77, 1.75, 671.5, 1.28.

Figure 4.

Time sequence of the front view images (camera 1) showing representative cases of three different types of interaction covered in this work. (a) Case I, weak interaction; (b) Case II, capture, drag and release of the bubble by the vortex ring; (c) Case III, binary bubble breakup. Supplementary movies 2–4 are related to this figure, and are available at https://doi.org/10.1017/jfm.2026.12078.

Table 2.

Characteristic values of the vortex ring variables at $t=t_0$ for each case.

Table 2. Long description

A table comparing vortex ring variables for different cases. The table has 3 rows and 5 columns. Column headers are R 0 (mm), a 0 (mm), Gamma 0 (cm 2 s negative 1), Omega dot V r 0 (s negative 2), and E k V 0 (J m negative 3). Row 1: Case I, 15.9, 5.89, 94.1, 222.8, 2.68. Row 2: Case II, 16.6, 6.32, 145.6, 326.7, 5.60. Row 3: Case III, 15.2, 4.34, 118.6, 533.4, 5.87.

Figure 4(a) shows a weak bubble–vortex interaction (Case I). As both the bubble and vortex ring come into proximity, the bubble trajectory is slightly diverted by the flow induced by the vortex. Since $\textit{We}$ is well below unity, surface tension forces dominate over the inertial forces induced by the vortex. In configurations where the vortex ring is relatively larger than the bubble ( $R_0/R_b=4.3$ ), interaction events at such low Weber numbers are typically characterised as ‘free crossing’ events, where the bubble is not successfully trapped or broken by the vortex core (Foronda-Trillo et al. Reference Foronda-Trillo, Rodríguez-Rodríguez, Gutiérrez-Montes and Martínez-Bazán2021). Thus, the flow cannot produce a sustained stretching strong enough to deform or break the bubble. Under these conditions, the bubble undergoes only moderate, reversible deformation. Moreover, the relatively thick vortex core ( $a_0/R_0 = 0.37$ ) presents a smoother velocity distribution than thinner vortex rings. This further reduces the ability of the vortex to hold the bubble within its core. Thus, the bubble is accelerated towards the vortex core and reduces its rise velocity, but is not captured. After this, the bubble continues its rise upwards, restoring its path and terminal velocity after a short transient. In Case II, the interaction is more intense, as illustrated in figure 4(b), because both $\textit{We}$ and $\textit{Re}$ numbers are larger than in the previous case, while the values of the rest of the parameters remain practically unchanged. On one hand, a higher $\textit{Re}$ indicates that the ring is energetic and coherent, able to advect the bubble once it is captured. On the other hand, a higher $\textit{We}$ promotes bubble deformation. In particular, in this case, $\textit{We}=1$ , meaning that the inertial stresses of the vortex ring are comparable to surface tension. This leads to bubble deformation, but not necessarily to its fragmentation. Therefore, the bubble can be seen to be drawn towards and captured inside the core region, it remains moving downwards within the ring for a certain time interval until it is finally released and continues its rise.

Case III, illustrated in figure 4(c), corresponds to the most intense interaction: the bubble is trapped in the vortex core and remains inside for a relatively long time. In this case, inertial stresses are sufficient to overcome the restoring action of surface tension ( $\textit{We}\simeq 0.8$ ), so the bubble elongates azimuthally along the ring until it eventually collapses into two fragments that stretch along the core, as will be demonstrated later. Note that, although $\textit{We}$ is lower than in Case II (but still close to unity), it can lead to bubble fragmentation. The fact that the core is thinner ( $a_0/R_0=0.28$ and $a_0/R_b=1.18$ in Case III, while $a_0/R_b=1.71$ in Case II) is also a key feature promoting bubble breakup, since the ring’s straining region is more concentrated, producing sharper, more powerful localised stretching of the bubble due to more abrupt velocity gradients.

Figure 5.

Vertical velocity, $u_y$ , of the vortex ring ( $u_v$ , red) and the bubble ( $u_b$ , black) for the cases regarding (a) weak interaction, (b) capture, drag and release, and (c) bubble breakup. Temporal evolution of the bubble surface ratio, $S/S_b$ , for the cases regarding (d) weak interaction, (e) capture, drag and release, and ( f) bubble breakup. Dotted and dashed lines in panels (c) and ( f) show daughter bubbles’ velocities and surfaces, respectively. The interaction time is highlighted in grey.

As a first stage of the analysis of three cases, we present in figure 5 the evolution of the vertical velocities, $u_y(t)$ , of both the vortex ring, $u_v(t)$ , and the bubble, $u_b(t)$ (see figure 5 a–c), in dimensionless form, as well as the bubble surface ratio, $S(t)/S_b$ (see figure 5 d–f), with $S_b$ the initial bubble surface obtained from the bubble formation recordings of the additional camera in figure 2(a) and $S(t)$ the surface of the reconstructed bubble at time $t$ . In all cases, the vortex slows down as it moves downstream. The bubble accelerates quickly and suddenly, and then its velocity decreases sharply when it encounters the vortex ring. Note that the availability of time-resolved data for the trajectory of the bubble allows us to calculate the duration of the interaction: we define the beginning of the interaction, $t=0$ , from the moment the bubble deviates from its rising path due to the presence of the vortex ring, and its final, $t_f$ , when the bubble turns back to its stationary rise and leaves its influence, being the interaction time $t_i=t_f$ . We marked the starting and final points of the interaction time (shaded area in figure 5) by looking for peaks and inflection points in the bubble velocity, representing strong, fast deviations from the previous path and thus indicating the influence of the vortex ring on the bubble.

The bubble recovers its terminal velocity after being released by the vortex in Cases I and II (figure 5 a,b). In Case III (figure 5 c), the bubble is trapped by the vortex core, which dramatically slows its velocity and causes it to break into two smaller bubbles (dashed and dotted lines in figure 5 c, f). These daughter bubbles are then dragged by the vortex at its own velocity. Regarding the bubble surface, in Cases I and II (figure 5 d,e), it remains nearly constant because the bubble is barely deformed during its interaction with the vortex, as will be described later. Nevertheless, in Case III (figure 5 f), the mother bubble is strongly elongated within the vortex core before breaking up ( $S/S_b$ increases up to $\approx 2$ ). The bubble splits into two unequal daughter bubbles: one with approximately the same area as the mother bubble (dotted line) and another with nearly half the area (dashed line). The surfaces of both bubbles decrease slightly over time, suggesting that their surfaces are being restored, becoming more spherical, due to surface tension effects, as these bubbles are smaller than the mother bubble (and thus their corresponding Weber numbers are also smaller).

In the following, we will analyse in detail the kinematics and dynamics of each type of interaction obtained from the experimental data. While the overall outcomes of these interactions are consistent with the transition map proposed by Foronda-Trillo et al. (Reference Foronda-Trillo, Rodríguez-Rodríguez, Gutiérrez-Montes and Martínez-Bazán2021), thus supporting the validity of the present experimental conditions, the time-resolved three-dimensional measurements provide access to additional features of the interaction that cannot be captured using planar approaches. In particular, beyond the global classification of the interaction regimes, the present results allow us to resolve the spatial and temporal distribution of key quantities such as bubble deformation and enstrophy. For Case I, the interaction is shown to be highly localised in the azimuthal direction, with transient variations confined to the bubble plane. For Case II, the measurements reveal the progressive transition from a localised perturbation to a global deformation of the vortex ring during the capture and release process. For Case III, the three-dimensional reconstruction enables the quantification of the azimuthal elongation prior to breakup and provides insight into the redistribution of enstrophy, including the increase of radial and axial vorticity components associated with the formation of smaller-scale structures.

3.2. Case I: weak interaction

Figure 6.

(a) Three-dimensional and (b) top views of the bubble trajectory (thick solid line), and time evolution of the shape and position of the vortex ring core centre (thin dotted lines) in the weak interaction case (Case I). Both are coloured by time (see colourbar), with the bubble travelling upwards and the vortex downwards (counterflow). (c) Temporal evolution of the azimuthally averaged radial position of the vortex core centre and inner/outer limits, together with the bubble centroid. The shaded area in panel (c) corresponds to the duration of the interaction.

Figure 6. Long description

Panel A: A three-dimensional scatter plot shows the trajectory of a bubble (thick solid line) and the time evolution of the shape and position of the vortex ring core center (thin dotted lines) in a weak interaction case. The axes are labeled y/R0, z/R0, and x/R0. The bubble travels upwards while the vortex moves downwards. The plot is colored by time, with a color bar indicating time values. Panel B: A top view of the same trajectory and time evolution as in Panel A, with the same axes and color coding. Panel C: A line graph shows the temporal evolution of the azimuthally averaged radial position of the vortex core center and inner/outer limits, along with the bubble centroid. The x-axis is labeled tΓ0/R0^2 and the y-axis is labeled r/R0. The shaded area corresponds to the duration of the interaction.

Figure 6 shows the temporal evolution of both the vortex ring core centre and the bubble position corresponding to a weak interaction. In particular, panels (a) and (b) illustrate the spatial evolution of both the vortex ring and the bubble, coloured by dimensionless time, $t\varGamma _0/R_0^2$ . Note that we are able to solve the three-dimensional bubble trajectory as well as the position and the shape of the vortex ring: while the three-dimensional evolution is depicted in panel (a), the top view is shown in (b). The vortex ring core (thin dotted lines) propagates downwards, while the bubble (thick solid line) travels upwards. The bubble, initially rising vertically along the $y$ -axis, is pulled laterally away from the axis towards the core of the approaching vortex ring. In particular, when both the bubble and the ring are close enough, the bubble quickly deviates to its core, but it escapes very soon. The interaction time is short according to figure 6(a,b), where it is observed that the interval of time when the bubble and vortex coincide in both colour and position is brief. This interaction time is highlighted with a shaded area in figure 6(c), where the time evolution of the radial position of the bubble centroid (blue), the outer and inner limits of the vortex core (black), and its centre (red) are plotted. The bubble interacts with the vortex at $t\varGamma _0/R_0^2=0$ and quickly moves towards the core centre, i.e. the radial position increases. Subsequently, after a very short period, its radial position deeply decreases and the bubble escapes the core at the interaction time, $t_i\varGamma _0/R_0^2\simeq 6$ . Regarding the vortex ring, it maintains an almost circular form at each moment (figure 6), with a radius $R(t)$ that slightly grows over time after the perturbation (red line in figure 6 c). Moreover, the ring centre deviates towards the interaction position (see figure 6 b). This minimal distortion is expected in a weak interaction scenario, which fails to induce significant and lasting azimuthal instability or fragmentation in the vortex core.

Figure 7.

Temporal evolution of the azimuthal distribution of the (a) circulation, $\varGamma /(U_{0}R_{0})$ ; (b) planar, specific kinetic energy, $E_{k,A}^{s}R^{2}_{0}/(\rho \varGamma ^{2}_{0})$ ; and (c) planar, specific azimuthal enstrophy, $\varOmega _{\theta ,A}^{s}R^{4}_{0}/\varGamma ^{2}_{0}$ , of the weak interaction case (Case I). The values correspond to planar azimuthal sections covering two cores with $\theta \in [0^\circ ,180^\circ ]$ (full plane), as it would be obtained in 2D-PIV measurements. The areas enclosed by black dashed lines mark the instants and azimuthal coordinates corresponding to the position of the bubble, $\theta _{\textit{bubble}}$ , while the solid ones correspond to its position while it remains inside the vortex core during the interaction. The dotted vertical lines indicate the instants corresponding to different snapshots throughout the experiment (panels d–g, matched by colour) for a better understanding of the effect of the interaction. The bubble surface is coloured in magenta, while blue represents the $\lambda _2$ -criterion iso-surfaces (opaque blue: $\lambda _2 R^{4}_{0}/\varGamma ^{2}_{0}=-1.08$ ; faded blue: $\lambda _2 R^{4}_{0}/\varGamma ^{2}_{0}=-0.0072$ ). Supplementary movie 5 is related to this figure.

Figure 7 presents dimensionless circulation, $\varGamma /(U_{0}R_{0})$ in panel (a), planar, specific kinetic energy, $E_{k,A}^{s}R^{2}_{0}/(\rho \varGamma ^{2}_{0})$ in panel (b) and planar, specific azimuthal enstrophy, $\varOmega _{\theta ,A}^{s}R^{4}_{0}/(\rho \varGamma ^{2}_{0})$ , in panel (c) as a $\theta$ -time map. We also include three-dimensional snapshots corresponding to specific time instants (panels d–g). The angle $\Delta \theta =\theta -\theta (t=0)$ describes the position relative to the azimuthal coordinate of the centroid of the bubble at the beginning of the interaction. This means that $\Delta \theta$ defines the azimuthal extension of the bubble. Note that planar kinetic energy and azimuthal enstrophy in a particular $\theta$ and time correspond to (2.5) averaged in $(y,r)$ planes covering $180^\circ$ (full planes). The orientation of the planes where these magnitudes are calculated is varied from $-90^\circ \lt \Delta \theta \lt 90^\circ$ to span the entire ring and locate the interaction at the centre of the domain. This planar analysis, covering the two cores of the vortex, is comparable to a classical 2D study.

It is important to observe that for any azimuthal location, after a rapid decrease following the interaction, all three magnitudes show a continuous, gradual decay over time due to the weakening of the vortex as it descends. This implies that the interaction is not highly disruptive to the overall strength and long-term coherence of the vortex ring, in agreement with previous studies (Biswas & Govardhan Reference Biswas and Govardhan2022). Moreover, it can be noted that, close to the bubble region, $\theta _{\textit{bubble}}$ , all magnitudes, but mostly enstrophy, are slightly perturbed along $\theta$ , thus revealing the three-dimensional nature of the interaction. The sharp localisation indicates that the most intense modification occurs exactly where the bubble momentarily resides (black line), failing to spread significantly along the circumference of the vortex core due to the weakness of the interaction.

The $\lambda _2$ -criterion iso-surfaces (Jeong & Hussain Reference Jeong and Hussain1995) plotted in figure 7(d–g) correspond to four different instants: slightly before (panel d), during (panel e), right after (panel f), and long after (panel g) the interaction. These instants are highlighted with vertical dashed lines in figure 7(a–c). The vortex core structure can be visualised in these snapshots, showing that the ring remains highly coherent throughout the interaction, although it loses intensity. However, a remarkable loss of vorticity at the core centre can be observed by the reduction of opaque blue structures, although the vortex ring structure remains almost intact. Even though the interaction with the bubble is brief and localised, it triggers a strong decay of the vorticity in the core. These snapshots also confirm that the bubble (magenta surface) is attracted towards the vortex core but appears intact and unbroken upon the closest approach. This absence of rupture is consistent with the interaction being subcritical, as bubble breakup generally requires $\textit{We} \gtrsim 1$ for $R_0/R_b \gt 1$ (Foronda-Trillo et al. Reference Foronda-Trillo, Rodríguez-Rodríguez, Gutiérrez-Montes and Martínez-Bazán2021). In fact, notice that the bubble covers a nearly constant azimuthal extension throughout the interaction (black solid lines in the maps of figure 7 a–c), indicating no remarkable shape deformations of the bubble surface. Note that a change of the extension of the azimuthal planes covered by the bubble could indicate not only a real shape alteration, but also a radial motion of the bubble: for a given shape, the bubble covers a larger extension in the azimuthal coordinate when it moves to a smaller radial position, and vice versa (see figure 1 b). In fact, the decrease that takes place prior to the interaction corresponds to the bubble motion towards the core (higher radial position), while the increase after the interaction corresponds to the migration away from the core towards the central vertical axis, in agreement with figure 6(c).

Figure 8.

Temporal evolution of dimensionless (a) circulation; (b) specific, kinetic energy and (c) specific enstrophy in Case I (weak interaction). Results in planes with or without bubble are depicted in red and blue, respectively. Panels (b) and (c) show the azimuthal average of the corresponding 2D variable averaged in half-planes (red and blue) together with their 3D volumetric counterpart (black). The shaded areas correspond to the duration of the interaction. The black dotted lines mark the instants corresponding to different events throughout the experiment. (d–g) Velocity field and vorticity contours of the four instants marked in the top-row figures, on a vertical plane at $\theta =56^\circ$ . The silhouette of the bubble is depicted in green. The bubble crosses this plane at all the depicted instants, but to a different extent depending on its azimuthal position and deformation at such an instant (see the different sizes and shapes of the silhouettes in panels d–f).

The previous maps indicate that the bubble effect in this case is localised around the collision area. To further contrast the effect on the flow variables in this area ( $\theta _{\textit{bubble}}$ ) to the rest of the planes, where no bubble is present, we averaged them azimuthally, making a distinction between bubble and no-bubble azimuthal planes (red and blue lines, respectively, in figure 8(a–c), calculated with (2.5)), and separating the information in half-planes ( $\theta \in [0^\circ ,360^\circ ]$ ). Note that, in these figures, we only include the components of the velocity in the measuring plane to compute the kinetic energy and vorticity component perpendicular to the plane in the enstrophy, like in two-dimensional PIV measurements. However, since we perform three-dimensional measurements, the three components of the velocity and vorticity could have been obtained. Although the interaction is brief, notice that the bubble alters the vortex characteristics while it occurs (red line). Circulation is observed to be slightly enhanced during the interaction in the planes where the bubble is present, while significantly larger values of enstrophy and, mainly, kinetic energy are observed at the beginning of the interaction (red lines). As soon as the bubble starts moving away from the vortex core due to buoyancy, there is a quick drop of the enstrophy peak, reaching the value of the no-bubble planes (blue lines). At the same time, a decrease in energy happens, which may be understood as the bubble slowing down the vortex due to a wake effect. Thus, we can appreciate in the three graphs an evident change of slope to a steeper decay of the flow variables when the bubble is present, indicating that, despite the encounter being brief, the vortex loses strength due to its collision with the bubble. This effect, where the vortex is weakened by the interaction, aligns with findings from other studies, such as the experimental work of Jha & Govardhan (Reference Jha and Govardhan2015). Here, they reported a large reduction in enstrophy and a significant drop in the vortex convection speed in low-Weber-number cases where the bubble was briefly trapped, contrasting with negligible effects seen at higher $\textit{We}$ . This highlights the importance of performing measurements precisely in the azimuthal planes where the interaction occurs. This is feasible in three-dimensional experiments, where the interaction plane can be identified and selected afterwards. In this regard, the volume-averaged 3D value in the kinetic energy and enstrophy, (2.6), are included in figures 8(b) and 8(c), respectively (black line). Note that the volumetric metrics integrate over the 3D fluid volume, accounting for all three components of velocity and, thus, vorticity. While planar magnitudes clearly show the impact of interaction with the bubble, three-dimensional magnitudes depict smoother evolution, although they display the changes in the kinetic energy and the total enstrophy when the bubble faces the vortex ring. Nevertheless, the decay of both magnitudes over time reflects the overall evolution of the vortex ring correctly.

The velocity field and vorticity of the four instants selected in figure 8(a–c), along with the vorticity contours and the bubble surface, are shown in figure 8(d–g). Note that we selected a plane containing the bubble at all times to characterise its effect. However, due to the azimuthal motion of the bubble over time, its presence may only be partial at certain instants, depending on whether the plane crosses it far from the centroid. As expected, high velocities are observed around the vortex core, while the liquid is almost stagnant far from the vortex ring. It can be observed that, in the pre-interaction and capture stages (figures 8 d and 8 e, respectively), the vortex structure remains almost unchanged in terms of its shape and amount of vorticity. However, because of the bubble interaction, the vorticity of the core is heavily distorted locally once the bubble escapes (figure 8 f). Finally, the vortex continues to translate downwards, but, although still coherent, with a weakened core (figure 8 g).

3.3. Case II: capture, drag and release

Figure 9.

(a) Three-dimensional and (b) top views of the bubble trajectory (thick solid line), and time evolution of the shape and position of the vortex ring core centre (thin dotted lines) in the capture and drag case (Case II). Both are coloured by time (see colourbar), with the bubble travelling upwards and the vortex downwards (counterflow). (c) Temporal evolution of the azimuthally averaged radial position of the vortex core centre and inner/outer limits, together with the bubble centroid. The shaded area in panel (c) corresponds to the duration of the interaction.

We will now describe a stronger interaction case, with higher $\textit{Re}$ and $\textit{We}$ than Case I, but similar vortex core-to-radius ratio, $a_0/R_0$ . Figure 9(a,b) describes the position of the vortex–bubble pair over time. As can be observed in the bubble path (thick time-coloured line in figure 9 a,b and blue line in panel c), the bubble travels upwards first and then is dragged laterally to larger radial positions when it is close enough to the vortex ( $t=0$ ). During a certain time, the bubble is trapped inside the core (see figure 9 b,c) and travels downwards together with the vortex (see the period where the bubble path and the vortex contour are the same colour in figures 9 a and 9 b). Meanwhile, the presence of the bubble weakens the vortex core, introducing shape distortions. When the vortex is no longer powerful enough to hold the bubble in its core, the bubble is released and rises again (figure 9 a). This interaction eventually leads to a destabilisation of the vortex ring structure. This is most evident in figures 9(a) and 9(b), where at later times (reddish colours), the vortex no longer shows a circular shape. Additionally, in figure 9(c), at the moment the bubble leaves the ring ( $t_i\varGamma _0/R_0^2\approx 15$ ), the position of the vortex core (shown by the red and black lines) exhibits some distortion due to the ring structure no longer being discernible. Note that the interaction time is significantly longer in this case than in Case I, due to the higher vortex circulation and lower pressure in the core.

The evolution of circulation, planar, specific energy and planar, specific enstrophy are depicted in figure 10, together with the bubble position (black line enclosed area). As in Case I, circulation decreases over time, with no notable effects of the bubble during the initial stage of the interaction. However, after some contact between the ring and the bubble ( $t\varGamma _0/R_0^2\gtrsim 10$ ), the circulation starts to decrease more rapidly. This effect is even more evident in kinetic energy and enstrophy (figures 10 b and 10 c, respectively), with a steep decrease in their magnitude, which started much earlier ( $t\varGamma _0/R_0^2\approx 3$ ). Unlike in the previous case, it is also notable that the effects on the vortex ring are not limited to the vicinity of the bubble, but extend more globally around $\theta$ . The value of the Weber number ( $\textit{We}=1$ ) is not high enough to cause the bubble to break up. This may be due to the thickness of the vortex ( $a_0/R_0=0.38$ ), which makes it weaker than a thinner vortex with the same $\textit{Re}$ . Note that the result of the interaction and some differences between the cases may also be caused by factors not included in the scope of this work, such as the relative position of the bubble and the vortex at the beginning of the interaction.

Figure 10.

Temporal evolution of the azimuthal distribution of the (a) circulation, $\varGamma /(U_{0}R_{0})$ ; (b) planar, specific kinetic energy, $E_{k,A}^{s}R^{2}_{0}/(\rho \varGamma ^{2}_{0})$ ; and (c) planar, specific azimuthal enstrophy, $\varOmega _{\theta ,A}^{s}R^{4}_{0}/\varGamma ^{2}_{0}$ , of the capture+drag+release case (Case II). The values correspond to planar azimuthal sections covering two cores with $\theta \in [0^\circ ,180^\circ ]$ (full planes), as it would be obtained in 2D-PIV measurements. The areas enclosed by black dashed lines mark the instants and azimuthal coordinates corresponding to the position of the bubble, $\theta _{\textit{bubble}}$ , while the solid ones correspond to its position while it remains inside the vortex core during the interaction. The dotted vertical lines indicate the instants corresponding to different snapshots throughout the experiment (panels d–g, matched by colour) for a better understanding of the effect of the interaction. The bubble surface is coloured in magenta, while blue represents the $\lambda _2$ -criterion iso-surfaces (opaque blue, $\lambda _2 R^{4}_{0}/\varGamma ^{2}_{0}=-0.72$ ; faded blue, $\lambda _2 R^{4}_{0}/\varGamma ^{2}_{0}=-0.0036$ ). Supplementary movie 6 is related to this figure.

Figure 10. Long description

Panel A: A heatmap with overlaid line graphs shows the temporal evolution of the azimuthal distribution of circulation. The x-axis is labeled tΓ₀/R₀² and ranges from -15 to 38. The y-axis is labeled θ - θ(t = 0) and ranges from -90 to 90. The color scale ranges from 1 to 4. Black dashed and solid lines mark the bubble’s position. Panel B: A heatmap with overlaid line graphs shows the temporal evolution of the planar, specific kinetic energy. The x-axis is labeled tΓ₀/R₀² and ranges from -15 to 38. The y-axis is labeled θ - θ(t = 0) and ranges from -90 to 90. The color scale ranges from 4 to 12 times 10^-3. Black dashed and solid lines mark the bubble’s position. Panel C: A heatmap with overlaid line graphs shows the temporal evolution of the planar, specific azimuthal enstrophy. The x-axis is labeled tΓ₀/R₀² and ranges from -15 to 38. The y-axis is labeled θ - θ(t = 0) and ranges from -90 to 90. The color scale ranges from 4 to 8 times 10^-2. Black dashed and solid lines mark the bubble’s position. Panel D: A 3D visualization shows the bubble and vortex ring at tΓ₀/R₀² = -5.5. Panel E: A 3D visualization shows the bubble and vortex ring at tΓ₀/R₀² = 4.1. Panel F: A 3D visualization shows the bubble and vortex ring at tΓ₀/R₀² = 9.9. Panel G: A 3D visualization shows the bubble and vortex ring at tΓ₀/R₀² = 20.6. The bubble surface is colored in magenta, and blue represents the Q-criterion iso-surfaces.

Regarding the bubble position, before the interaction (dashed lines in figure 10 a–c), we can see the bubble drifting back and forth from its initial position (covering different azimuthal extension), indicating it has an unstable path (see also figure 9 a). Right after that (see the change from the dashed to the solid line), the bubble suffers an attraction towards the vortex core at a higher radial distance (see also figure 9 c), covering thus a lower azimuthal extension. Once inside the core, the bubble drifts azimuthally and then travels back closer to $\Delta \theta \approx 0$ , where it stays until its release. This may be a consequence of the vortex ring losing intensity and coherence, as can be seen in figure 10(d–f) through the $\lambda _2$ criterion at four different instants. During capture and the initial bubble drag phase (figure 10 d–e), the vortex structure retains its overall shape and level of vorticity. This period marks the beginning of intense interactions that strongly affect instantaneous liquid vortex properties. During the sustained drag (figure 10 f), the shape and symmetry of the ring are distorted due to the presence of the trapped bubble, in agreement with figure 9(a,b). This distortion, while not yet causing major global circulation loss, reflects the strong perturbation introduced by the bubble. Figure 10(g) corresponds to the post-release stage, during which the vortex core loses overall strength and coherence due to structural weakening. Regarding the bubble shape (magenta), note that although it is initially deformed at the beginning of the interaction due to the stresses induced by the vortex, they are not sufficiently high to induce its breakup.

Figure 11.

Temporal evolution of dimensionless (a) circulation; (b) specific, kinetic energy and (c) specific, azimuthal enstrophy in Case II (capture+drag+release). Results in planes with or without bubble are depicted in red and blue, respectively. Panels (b) and (c) show the azimuthal average of the corresponding 2D variable averaged in half-planes (red and blue) together with their 3D volumetric counterpart (black). The shaded areas correspond to the duration of the interaction. The black dotted lines mark the instants corresponding to different events throughout the experiment. (d–g) Velocity field and vorticity contours of the four instants marked in panels (a)–(c), on a vertical plane at $\theta =61^\circ$ . The silhouette of the bubble is depicted in green. The bubble crosses this plane at all the depicted instants, but to a different extent depending on its azimuthal position and deformation at such an instant (see the different sizes and shapes of the silhouettes in panels d–g).

To further analyse how the position of the bubble impacts the flow over time, we may look at figure 11. Here, we cannot identify any significant differences in the decay of circulation over time between planes with and without a bubble (figure 11 a), which is consistent with the findings of Jha & Govardhan (Reference Jha and Govardhan2015) for high enough Weber numbers and thick core vortices. However, a slight increase in circulation can be observed at $t\varGamma _0/R_0^2 \approx 6$ , probably due to the vorticity shed by the bubble when it starts to interact with the vortex ring. In contrast, we see a strong effect of the bubble in kinetic energy and azimuthal enstrophy. In the former (figure 11 b), both the 3D evolution and the mean value of the 2D half-planes without a bubble have a similar trend: a monotonic decrease over time, with a slightly steeper slope during the interaction period. However, the planes where the bubble interacts with the vortex show a significant loss of kinetic energy from the start of the interaction. This is a direct consequence of the perturbations introduced by the presence of the bubble, slowing down the vortex. In fact, figures 11(e) and 11( f) show how the vortex tilts with the core on the left side lower than that on the right, with the latter containing the bubble. Shortly before the bubble escapes from the vortex, the two-dimensional specific kinetic energy in the planes containing the bubble recovers to a value similar to that of the planes without a bubble, exhibiting a bump (figure 11 b). Note that the rate of decay of the three-dimensional, specific kinetic energy is also higher during interaction, which is consistent with direct numerical simulations by Foronda-Trillo et al. (Reference Foronda-Trillo, Rodríguez-Rodríguez, Gutiérrez-Montes and Martínez-Bazán2021). Regarding the time evolution of the azimuthal enstrophy, shown in figure 11(c), different trends are observed. In the planes containing the bubble, it initially decreases when the bubble starts to interact with the vortex ring. However, during this interaction, $\varOmega ^s_{\theta ,A}$ remains almost constant for a period of time before increasing as the bubble escapes from the vortex. This increase coincides with the rise of $E^s_{k,A}$ shown in figure 11(b). Differently, in planes which do not contain the bubble, a monotonic decrease of $\varOmega ^s_{\theta ,A}$ is observed with time (blue line in figure 11 c), with a slightly steeper slope during the interaction period. However, the volumetric enstrophy (black line in figure 11 c) shows an increase at the beginning of the interaction, in agreement with the 3D numerical simulations by Foronda-Trillo et al. (Reference Foronda-Trillo, Rodríguez-Rodríguez, Gutiérrez-Montes and Martínez-Bazán2021).

Finally, the evolution of the vortex and how it loses energy can be further observed in the velocity and vorticity fields in a plane containing the bubble, represented in figure 11(d–g). Similarly to the previous case, here the bubble quickly approaches the ring and gets trapped (figure 11 d,e). Unlike in the weak interaction case, instead of rapidly escaping, it is dragged downwards. In this interplay, the vortex level of vorticity is almost unaltered (figure 11 f), but the shape and symmetry of the ring are heavily distorted. It is not until the bubble escapes (figure 11 g) that the level of vorticity is importantly diminished. In summary, this case depicts a combination of local effects in the bubble surroundings, which then evolve to a strong global distortion of the vortex due to the presence of the bubble.

3.4. Case III: bubble breakup

Figure 12.

(a) Three-dimensional and (b) top views of the bubble trajectory (thick solid line) and time evolution of the shape and position of the vortex ring core centre (thin dotted lines) in the breakup case (Case III). Both are coloured by time (see colourbar), with the bubble travelling upwards and the vortex downwards (counterflow). (c) Temporal evolution of the azimuthally averaged radial position of the vortex core centre and inner/outer limits, together with the bubble centroid. The shaded area in panel (c) corresponds to the duration of the interaction.

The last case in this work covers the strongest interaction: here, the vortex ring breaks the bubble into two daughter bubbles. The kinematics of this interaction are depicted in figure 12. The three-dimensional and top views show that initially, the bubble rises near the vertical axis of the vortex ring while the ring travels downwards (dark to light blue in figure 12 a,b). When they are close enough, the bubble is quickly deviated towards the ring core at $t=0$ (see also figure 12 c) and starts to travel downwards within the vortex core. In this case, the vortex is thinner than in Cases I and II, and although $\textit{We}$ is smaller than in Case II, it is large enough to overcome the confining surface tension forces acting on the bubble. Thus, after a very short time, the mother bubble splits into two daughter bubbles, which move in the azimuthal direction without leaving the vortex core for the duration of the experiment. Regarding the vortex ring, its circular shape remains largely unaltered, as shown in figure 12(a,b).

Figure 13.

Temporal evolution of the azimuthal distribution of the (a) circulation, $\varGamma /(U_{0}R_{0})$ ; (b) planar, specific kinetic energy, $E_{k,A}^{s}R^{2}_{0}/(\rho \varGamma ^{2}_{0})$ ; and (c) planar, specific azimuthal enstrophy, $\varOmega _{\theta ,A}^{s}R^{4}_{0}/\varGamma ^{2}_{0}$ , of the breakup case (Case III). The values correspond to planar azimuthal sections covering two cores with $\theta \in [0^\circ ,180^\circ ]$ (full planes), as it would be obtained in 2D-PIV measurements. The areas enclosed by black dashed lines mark the instants and azimuthal coordinates corresponding to the position of the mother bubble, and the solid ones correspond to its position while it remains inside the vortex core during the interaction. After the mother bubble breaks, each of the daughter bubbles is marked with a different colour of the solid line (large bubble in purple and small bubble in magenta). The dotted vertical lines indicate the instants corresponding to different snapshots throughout the experiment (panels d–g, matched by colour) for a better understanding of the effect of the interaction. The bubble surface is coloured in magenta, while blue represents the $\lambda _2$ -criterion iso-surfaces (opaque blue, $\lambda _2 R^{4}_{0}/\varGamma ^{2}_{0}=-3.80$ ; faded blue, $\lambda _2 R^{4}_{0}/\varGamma ^{2}_{0}=-0.0038$ ). Supplementary movie 7 is related to this figure.

Figure 13 shows how the flow variables evolve with the azimuthal position of the bubble. Similarly to the previous cases, the evolution of the flow variables shown in figure 13(a–c) is characterised by a monotonic decay over time, which accelerates when the interaction begins. No particular effects on the temporal evolution of the circulation are spotted in figure 13(a), only slight changes appear along the azimuthal direction due to the perturbations induced by the bubbles. The kinetic energy (figure 13 b) shows higher values in the area around the bubble at the moment of capture. After this peak, the value quickly stabilises globally around the azimuth. Finally, the azimuthal enstrophy (figure 13 c) is much more strongly affected by the presence of the bubble, particularly during its breakup. Before the interaction, it displays a nearly uniform distribution around $\theta$ . However, when the interaction begins, it concentrates at the area of the bubble, while decreasing in the surroundings. It even increases and peaks a few instants before breakup, after which it quickly drops. This increase in enstrophy is associated with the bubble elongation within the vortex core. Moreover, the enstrophy remains higher around the daughter bubbles than elsewhere in the ring throughout the experiment, particularly in the area of the largest daughter bubble (purple solid line in figure 13 c). This indicates that even though the vortex structure in this case remains coherent, as can be noticed in figure 13(d–g), and thus less affected by the bubble, it gets perturbed, shedding vorticity, especially due to large bubbles trapped in the core (see figure 13 g).

Figure 14.

Temporal evolution of dimensionless (a) circulation; (b) specific, kinetic energy; and (c) specific, azimuthal enstrophy in the Case III (bubble breakup). Results in planes with or without a bubble are depicted in red and blue, respectively. Panels (b) and (c) show the azimuthal average of the corresponding 2D variable averaged in half-planes (red and blue) together with their 3D volumetric counterpart (black). The shaded areas correspond to the duration of the interaction. The black dotted lines mark the instants corresponding to different events throughout the experiment and the green dotted line marks the breakup time. (d–g) Velocity field and vorticity contours of the four instants marked in panels (a)–(c), on a vertical plane at $\theta =25^\circ$ . The silhouette of the bubble is depicted in green. The bubble crosses this plane at all the depicted instants, but to a different extent depending on its azimuthal position and deformation at such an instant (see the different sizes and shapes of the silhouettes in panels d–g).

If we focus now on the bubble dynamics, it shows a motion around the central axis before the interaction (black dashed line in figure 13 a–c): initially, the bubble covers all the $\Delta \theta$ values because it is at the centre of the axial axis (see also figure 13 d). A few instants later (figure 13 e), the bubble rises to a vertical position a few bubble diameters below the vortex ring, where it starts to be influenced by its velocity. At $t=0$ , it is quickly dragged laterally into the vortex core (see also figure 12 c). In fact, note that the azimuthal area covered by the bubble (solid black line in figure 13 a–c) rapidly decreases to a minimum of $\theta _{\textit{bubble}}\sim 90^\circ$ . It is after this minimum in the spread of the bubble in $\theta$ (see figure 13 e) that it grows again due to an azimuthal elongation of the bubble along the ring core (see figure 13 f). The sequence of capture, elongation and subsequent breakup observed here aligns closely with the description given by Jha & Govardhan (Reference Jha and Govardhan2015), who detailed a four-stage interaction process in which a bubble is attracted to the low-pressure core, elongates due to azimuthal pressure gradients and eventually breaks. The elongation is a consequence of the bubble perturbing the pressure distribution inside the core: the pressure rises at the location of the bubble, creating a pressure difference along the azimuthal direction that causes the bubble to expand along $\theta$ (Martínez-Bazán Reference Martínez-Bazán2015; Biswas & Govardhan Reference Biswas and Govardhan2022, Reference Biswas and Govardhan2023; Zhang et al. Reference Zhang, Deng and Shao2023), covering approximately up to one third of the circumference of the ring (see figure 13 f). The elongation stage is very short, because the bubble quickly breaks into two parts (see purple and magenta solid lines in figure 13 a–c at $t\varGamma _0/R_0^2\simeq 6$ ). These daughter bubbles have different sizes and their spread in $\theta$ tends to reduce after breakup, given that the surface tension is strong enough to restore the spherical-to-ellipsoidal shape of the bubbles (see figure 13 g). They also exhibit a drift along $\theta$ , like in the interaction of a vortex ring with a solid particle (Biswas & Govardhan Reference Biswas and Govardhan2025).

Now, we summarise the effect of the bubble on the flow characteristics by examining how the averaged values of kinetic energy and enstrophy differ in half-planes where the bubble interacts with vortex motion, compared with half-planes free of interaction, alongside their three-dimensional counterparts. Figure 14(a) shows that, as seen before, circulation is hardly affected by the bubble, with a moderate decay in time. The planar, specific kinetic energy (see figure 14 b) also decays in both planes with and without the bubble. However, it undergoes a significant change in slope a few moments after the bubble breaks up (marked by a green dotted line in figure 14 a–c). In the planes with a bubble (red line), the energy is redistributed near the bubble, which results in a slight increase in energy, which quickly drops before breakup and then recovers to the same value as the rest of the ring, in agreement with figure 13(b). Note also the rapid decrease of $E^s_{k,A}$ around $t\varGamma _0/R^2_0 \approx 2.7$ in the planes containing the bubble. By looking at the snapshots in figure 14(d–f), we may observe that the liquid velocity (and thus the kinetic energy) does not show a strong difference when the bubble starts to enter the vortex (figure 14 d–e). However, the values of both velocity and energy are much smaller when the bubble is in its equilibrium radial position inside the core for a sufficiently long time (figure 14 f). Similarly to the previous cases, the volumetric kinetic energy is reduced and its temporal decay is similar but smoother than that of its planar counterpart, although it also shows the change of slope after the bubble breakup.

Finally, regarding enstrophy (figure 14 c), there is a significant increase from the moment the bubble is attracted by the vortex at $t=0$ (see also figure 14 e). This increase in enstrophy is associated with the kinetic energy decrease observed at around $t\varGamma _0/R^2_0 \approx 2.7$ . The growth in enstrophy is fairly maintained until the bubble reaches its maximum elongation (see figure 14 f). Afterwards, the enstrophy decreases. Following the breakup, the enstrophy in the bubble half-planes (red line) begins to decay more quickly than before the bubble breakup. This decay is even more pronounced in the no-bubble planes (blue line). This indicates not only the growth of vorticity near the bubbles, but also its redistribution from the rest of the ring towards the bubble areas, as shown on the map in figure 13(c). In figure 14(g), the level of vorticity has diminished notably after breakup. Regarding the evolution of three-dimensional enstrophy, its behaviour is a combination of planar enstrophy in planes with and without a bubble. It also shows an increase in enstrophy when the vortex ring traps the bubble. It is worth mentioning that the evolution of kinetic energy and azimuthal enstrophy in Cases II and III are very much alike, which may be expected due to their similar $\textit{Re}$ and $R_0/R_b$ values, and relatively high $\textit{We}$ . The main difference between them is $a_0/R_b$ , with Case III presenting a thinner core, which indicates that core thickness is a key feature to enable bubble breakup.

The results reported in this section show that two-dimensional measurements are often insufficient to capture the complex, transient and nonlinear three-dimensional dynamics governing these phenomena. Although classical two-dimensional particle image velocimetry (2D-PIV) can be used to investigate the interaction between a rising bubble and a vortex ring, its ability to fully characterise the flow is inherently limited, particularly when the interaction induces out-of-plane motion and a loss of axial symmetry. Moreover, the use of a fixed measurement plane restricts the observation of spatially evolving features of the interaction. The present results show that even weak interactions can produce measurable changes in the global structure of the vortex, which may not be fully captured by planar approaches, highlighting an important limitation of such techniques. Thus, the fixed measurement plane in 2D-PIV must coincide with the interaction region, a condition that is difficult to ensure in practice due to the continuous displacement of the bubble during its rise. As a result, the capture of such localised phenomena becomes highly sensitive to the positioning of the measurement plane. In contrast, time-resolved 3D-LPT, coupled with shadowgraphy, enables a volumetric characterisation of the flow within the measurement domain. This approach allows for the simultaneous tracking of three-dimensional bubble deformation and the reconstruction of the three components of the velocity field, from which the vorticity field can be estimated. Furthermore, it enables a posteriori identification of the interaction region and provides a more comprehensive description of the temporal evolution of global quantities such as total kinetic energy and enstrophy. Consequently, this methodology alleviates key limitations of planar measurements and enables a more comprehensive assessment of whether vorticity is dissipated or redistributed following the loss of axial symmetry, as well as capturing the inherently three-dimensional nature of vortex instabilities and fragmentation processes.

3.5. Comparative analysis

Figure 15.

(a,d,g) Temporal evolution of the azimuthal distribution of the dimensionless circulation, (b,e,h) planar specific kinetic energy (central column) and (c, f,i) planar specific azimuthal enstrophy for (a–c) Case I, (d–f) Case II and (g–i) Case III.

Figure 15 directly compares the latter magnitudes in the $\Delta \theta {-}t$ planes for three interaction cases. If we focus on circulation (panels a,d,g), Case II (figure 15 d) shows larger values compared with Cases I and III (figures 15 a and 15 g, respectively), which can be attributed to the initial strength of the vortex generated for that experiment. In fact, Case II was intentionally set up with the highest circulation strength, resulting in the largest Reynolds number among the three scenarios. Although all magnitudes decay over time, circulation is the magnitude least affected by the bubble interaction. This means that the ranking established by the initial vortex strength (Case II $\gt$ Case III $\gt$ Case I) is maintained throughout the measurement time.

Nevertheless, the planar, specific kinetic energy level (figure 15 b,e,h) is much higher in Case III (figure 15 h) compared with Cases I and II (figures 15 b and 15 e, respectively) primarily because Case III involves an interaction with a vortex ring that has initially a higher kinetic energy (see table 2) and a thinner core, which concentrates the flow variables more intensely. Moreover, kinetic energy in Case III (figure 15 h) shows higher values in the area around the bubble at the moment of capture. This greater kinetic energy is related to the vorticity emitted by the bubble during the capture process, which consists first in stopping the bubble rise and then being directed towards the bubble core (see figure 13 e). In this case, the bubble changes its velocity more abruptly, which could explain the raise of the kinetic energy. After this initial peak, the kinetic energy magnitude quickly stabilises globally around the azimuth, maintaining a high level compared with the rapidly decaying energy found in the strongly distorted vortex core of Case II (figure 15 e).

Levels of specific azimuthal enstrophy (figure 15 c, f,i) are observed to be the highest also in Case III (figure 15 i), compared with Cases I and II (figures 15 c and 15 f, respectively). This fact is mainly due to the specific vortex properties of Case III, and the mechanism of intense vorticity concentration and stretching leading up to the bubble rupture. A thinner core structure means that the vortex principal vorticity is concentrated in a smaller cross-sectional area (Biswas & Govardhan Reference Biswas and Govardhan2023). This concentration leads to higher vorticity (and enstrophy) when measured in a cross-sectional plane. In contrast, only a brief, localised perturbation occurs in Case I (figure 15 c). In Case II (figure 15 f), the azimuthal enstrophy is lower because the effect of the thin core is absent, which spreads the flow variables over a larger area. The expected influence of the vortex core can be accounted for by redefining the Weber number to include the effect of its size on the characteristic velocity scale. Thus, since for a vortex ring, the velocity far from the bubble is given by $U_0= \varGamma _0/(4\pi R_0)ln[8R_0/a_0- 1/4]$ (Lamb Reference Lamb1924), a characteristic velocity of the form $(\varGamma _0/4 \pi R_0) \, ln(8 R_0/a_0)$ can be used to define the Weber number, as shown by Higuera (Reference Higuera2004), $\textit{We}^* = \textit{We}/4 \, [ln(8 R_0/a_0)]^2$ . According to the above-mentioned definition of the Weber number, the values for Cases I, II and III are 0.97, 2.32 and 2.16, respectively. This indicates that the effective Weber numbers in Cases II and III are very similar, and that the final outcome (trapping or breakup) is associated with the distance between the bubble and the vortex core at the onset of their interaction.

The three interaction cases show distinct responses that may be defined in terms of variations in the vortex volumetric ring magnitudes. In this regard, figure 16 compares the evolution of (a) circulation ( $\varGamma$ ), (b) specific volumetric kinetic energy ( $E_{k,V}^s$ ) and (c) specific total volumetric enstrophy ( $\varOmega _V^s)$ for the base vortex ring (without a bubble) and the three distinct interaction cases. The duration of the kinematic interaction is also highlighted in figure 16 through the shaded areas. As explained before, the volumetric magnitudes reflect the overall average evolution and long-term coherence of the vortex ring. Note that circulation (figure 16 a) is the integral property least affected by the presence and interaction of the bubble, with the decay over time similar for all cases. However, in Case I, in which the vortex does not have enough energy to attract and disturb the bubble, the circulation decreases abruptly after the interaction due to the disturbance introduced by the bubble.

Figure 16.

Evolution of (a) circulation; (b) specific, volumetric, kinetic energy; and (c) specific, volumetric, total enstrophy over time for a base case of vortex ring without bubble ( $\textit{Re}=28\,000$ , $a_0/R_0=0.29$ ) and the different interaction cases studied: Case I (weak interaction), Case II (capture, drag, release) and Case III (bubble breakup). Coloured areas indicate interaction intervals. The vertical green dotted line denotes breakup time in Case III. Note that to compare the evolution of the different regimes, the flow variables are normalised to their initial value and the time scales start at $t-t_0$ , with $t_0$ the initial time of the experiment, when the vortex has completely entered our measurement volume.

Regarding kinetic energy (figure 16 b), all interaction cases exhibit an enhanced decay rate when compared with the base case, signifying that the bubble interaction leads to a general loss of strength in the vortex. In Case I, the interaction is characterised by a brief encounter with the bubble. However, the vortex still loses strength, resulting in a steeper decay slope than in the base case. This aligns with the observation that cases with a low Weber number result in overall weakening (Jha & Govardhan Reference Jha and Govardhan2015). The decay rate for Cases II and III is larger during the interaction, and it slows down after release or breakup, respectively, suggesting a significant loss of kinetic energy from the beginning of the capture process.

Volumetric total enstrophy ( $\varOmega _V^s$ ) is the integral property most strongly affected by the bubble interaction. Notice that, in the three cases, $\varOmega ^s_V$ increases when the vortex starts to interact with the bubble, showing a bump in the time evolution. This is more evident in Cases II and III, since the beginning of the vortex–bubble interaction in Case I is close to the start of the recording time. After this rapid increase, the total volumetric enstrophy decreases at a rate larger than the base case, i.e. the single vortex ring (see Foronda-Trillo et al. Reference Foronda-Trillo, Rodríguez-Rodríguez, Gutiérrez-Montes and Martínez-Bazán2021). The overall reduction in Case I highlights a loss of strength consistent with large enstrophy reductions seen in moderate-Reynolds and low-Weber-number cases, even in brief interactions (Martínez-Bazán Reference Martínez-Bazán2015).

Finally, since we are performing 3D experiments, we are able to characterise not only the contribution of the azimuthal component of the vorticity to the enstrophy, $\varOmega ^s_{\theta , V}$ , but also the radial and axial contributions, $\varOmega ^s_{r, V}$ and $\varOmega ^s_{y, V}$ , respectively. Indeed, as shown in figure 17, the azimuthal contribution is the major one to the total enstrophy, making the radial and axial ones less relevant. As commented before, the volumetric enstrophy is the vortex ring property most strongly affected by the interaction, providing a global measure of the dissipation rate of kinetic energy and reflecting the overall coherence of the vortex ring. In fact, the kinetic energy of the flow is lost mainly through the creation of small-scale vorticity, that is, through an increase of enstrophy. The analysis of the contribution of the three vorticity components is crucial for understanding the three-dimensional effects of the vortex–bubble interaction.

Figure 17.

Specific volumetric enstrophy contributions of each vorticity component, together with the specific, total, volumetric enstrophy for (a) Case I, (b) Case II and (c) Case III. Note that a different scale is used for the global and azimuthal enstrophy (left axes), and the radial and axial ones (right axes).

In Case I, an increase in both $\varOmega ^s_V$ and $\varOmega ^s_{\theta ,V}$ can be observed during the vortex–bubble interaction time, which then drops sharply afterwards. This drop $\varOmega ^s_{\theta ,V}$ is accompanied by a slight increase in $\varOmega ^s_{r,V}$ and $\varOmega ^s_{y,V}$ , indicating that there is a redistribution of low-scale vorticity due to the three-dimensionality introduced by the presence of the bubble. Nevertheless, the contributions of the radial and axial vorticity components to the total enstrophy are much lower than that of the azimuthal component, and they are unable to compensate for the loss of $\varOmega ^s_{V}$ . This is probably due to the limitations in the spatial resolution of the three-dimensional measurements, unable to resolve the smallest vorticity scales. In Cases II and III, there is also an increase of enstrophy during the interaction time, more evident in $\varOmega ^s_V$ than in $\varOmega ^s_{\theta ,V}$ due to the contributions of $\varOmega ^s_{r,V}$ and $\varOmega ^s_{y,V}$ . In these cases, the deformation and/or breakup of the bubble contributes to the destabilisation of the vortex ring by generating small-scale radial and axial vorticity, which increases $\varOmega ^s_{r,V}$ and $\varOmega ^s_{y,V}$ , and consequently the total enstrophy, $\varOmega ^s_{V}$ . This fact, which is especially remarkable in Case III, aligns with the results from three-dimensional numerical simulations (Foronda-Trillo et al. Reference Foronda-Trillo, Rodríguez-Rodríguez, Gutiérrez-Montes and Martínez-Bazán2021).