3.1. Breakup morphology for different swirl strengths

The primary objective of our study is to explore the interaction of a swirling airflow with a freely falling droplet in an orthogonal configuration, where the droplet can encounter oppose-flow, cross-flow and co-flow situations in an unsteady manner, resulting in droplet morphologies that are distinct from those observed in the no-swirl configuration. As shown in figure 1, the air flows in the positive $x$ direction, while the droplet falls from the dispensing needle in the negative $y$ direction under the action of gravity. Figure 2 depicts the temporal evolution of morphology of an ethanol droplet in no-swirl ($Sw=0$), low-swirl ($Sw=0.47$) and high-swirl ($Sw=0.82$) conditions. As discussed in the previous section, the swirl number $(Sw)$ is varied by changing the vane angle of the swirler. To enable comparison of the present results with those of earlier studies, the droplet is ejected at ${\bar z}_d = 0$ (centre plane) in the no-swirl condition. But in swirl flow situations, the droplet is ejected at ${\bar z}_d = -0.13$ (negative), and thus the droplet interacts with the swirl flow in oppose-flow/cross-flow condition. The rest of the parameters are fixed at $We = 11.40$, ${\bar x}_d = 0.01$ and ${\bar y}_d = 0.89$. The results are presented at different dimensionless time, $\tau =t/T$, where $t$ and $T=d_0 \sqrt {(\rho _l/ \rho _a)}/U$ denote physical time and the time scale used in our study, respectively, and $\tau = 0$ represents the instant when the droplet dispenses from the needle.

Figure 2. Temporal evolution of the breakup dynamics for different values of swirl number, $Sw$: (a–f) $Sw= 0$ (no swirl), (g–l) $Sw= 0.47$ (low swirl strength) and (m–r) $Sw= 0.82$ (high swirl strength). In the no-swirl case, ${\bar z}_d=0$, but in the swirl cases, ${\bar z}_d = -0.13$. The rest of the parameters are fixed at ${\bar x}_d = 0.01$, ${\bar y}_d = 0.89$ and $We = 11.40$. The number at the top of each panel represents dimensionless time, $\tau =t/T$, where $t$ and $T=d_0 \sqrt {(\rho _l/ \rho _a)}/U$ denote the physical time and the time scale used in our study, respectively. The instant the droplet dispenses from the needle is represented by $\tau = 0$. An animation showing the vibrational (a–f), retracting bag breakup (g–l) and bag breakup (m–r) is provided as supplementary movie 1 available at https://doi.org/10.1017/jfm.2022.146.

For $Sw=0$ (shown in the first row of figure 2), the droplet exhibits a vibrational breakup mode. It can be seen that the droplet enters the aerodynamic field ($\tau = 2.38$) and deforms into a disk shape ($\tau =4.16$). In this case, the aerodynamic force is insufficient to overcome the viscous and surface tension forces, causing shape oscillations in the droplet (see $\tau =4.16$ to $5.45$). However, due to capillary instability, the droplet breaks into a few smaller droplets of comparable size at a later time. The dynamics observed for this set of parameters is qualitatively similar to that reported in earlier studies (e.g. Kulkarni & Sojka Reference Kulkarni and Sojka2014; Jain et al. Reference Jain, Prakash, Tomar and Ravikrishna2015).

A new type of breakup mode (termed as ‘retracting bag breakup’) is observed for an intermediate swirl strength ($Sw=0.47$) as shown in the second row of figure 2. In this case, when the droplet enters the swirl flow airstream, the lower part of the droplet first enters the swirling flow region, and the shape of the droplets is quickly converted into a slightly tilted disk shape from a near-spherical shape as the droplet encounters an oppose-flow condition (see $\tau =4.16$ in figure 2). The aerodynamic field has a high velocity in the shear zone, a low velocity in the wake of the vanes and a negative velocity in the centre region due to the recirculation created by the imposed swirl flow (refer to § 3.4). The disk shape that faces the strong swirl field bulges and creates a bag in the shear zone, while the bottom portion of the disk shape remains thicker as it encounters the low-velocity zone. The droplet exhibits the opposite condition when it first enters the vane's wake and migrates towards the shear layer. As a result, liquid accumulates in the wake region and forms a large node at the beginning of the bag breakup process ($\tau = 4.81$). Subsequently, under the influence of the aerodynamic force, the partial disk shape participates in the bag breakup process. As lower part of the droplet comes in wake of the vane, the velocity at the upper side of the disk becomes high, while the velocity at the lower side of the disk is low, causing the disk to rotate clockwise in the $x\text {--}y$ plane ($\tau =4.16\text {--}6.08$; see figure 17 in the Appendix). As the part of the droplet passes through the wake region, negative pressure drives the bag film in the direction opposite to the bag growth. Thus, while the bag retracts in the upper half of the drop, it remains intact in the lower half ($\tau =5.45$). The instability is induced on the drop surface in such a way that the nodes in the upper portion of the droplet move quicker while the nodes in the lower portion of the droplets move slower. Therefore, the ring is stretched and the bag is retracted by the airflow continuously. Further retraction of the bag ($\tau =6.08$) leads to the development of capillary instability, which causes the bag to burst in the opposite direction and the rim to disintegrate, which is distinct from the conventional bag breakup process. At later times, as the liquid enters the negative-velocity region generated by the recirculation zone, the lower part of the liquid disintegrates from the larger droplet. The retraction bag bursting process can be understood as the consequence of a change in the aerodynamic field, which is different from that in the no-swirl case. An animation showing the vibrational, retracting bag breakup and bag breakup is provided as supplementary movie 1.

As the swirl number increases from $Sw=0.47$ to 0.82, the mode changes from retracting bag breakup to bag breakup mode (third row in figure 2). The dynamics observed at the early times is similar to that of the low-swirl case ($Sw=0.47$). The drop deforms into a tilted disk shape ($\tau =3.51$) as it enters the swirl flow region in contrast to the straight disk confronting the airstream in the no-swirl case as shown in the first row of figure 2 (also see Guildenbecher et al. Reference Guildenbecher, López-Rivera and Sojka2009; Rimbert et al. Reference Rimbert, Escobar, Meignen, Hadj-Achour and Gradeck2020; Jackiw & Ashgriz Reference Jackiw and Ashgriz2021). In the swirl flow, the orientation of the disk depends on the strength of the swirl and its interaction with the swirl flow stream (refer to the discussion in § 3.4). It can be seen that the deformed disk is slanted more for $Sw=0.82$ than for $Sw=0.47$. Soni et al. (Reference Soni, Kirar, Kolhe and Sahu2020) also reported the deformed disk in various orientations when a drop interacts with a uniform airstream at oblique angles. In this case, the disk was unable to cross the shear-layer region. At $\tau =4.17$, the disk orients itself vertically enabling the entire disk to participate in the bag breakup process. Thus efficient energy transfer occurs and the bag expands in the direction of the swirling airflow resulting in a thinner rim as compared with the low-swirl case. The formation of multiple nodes is caused by the multi-directional stretching of the droplet, which is greater than that in the no-swirl and low-swirl cases. The formation of multiple nodes due to the swirling flow causes the faster stretching of the bag and rim ($\tau =4.84$). The bag and rim then disintegrate into tiny droplets ($\tau =5.35$). In this case, the droplet breakup is mostly caused by high shear velocity in the shear region when the swirl strength is strong.

The retracting bag breakup mode is apparent only when the droplet enters the vane's wake and the central recirculation zone, which can be explained as follows. Two important factors influencing the breakup process are the stretching rate and residence time of the drop in the high-airstream zone. Soni et al. (Reference Soni, Kirar, Kolhe and Sahu2020) reported that the obliquity of the airstream influences the stretching factor, which in turn influences the breakup phenomenon (also, refer to the discussion in § 3.6). As the angle of obliquity of the airstream increases, the droplet's rectilinear trajectory changes into a curvilinear trajectory. When a droplet is exposed to swirl air, it experiences a larger stretching factor than when it is exposed to no-swirl air. Moreover, the droplet encounters a radial centrifugal force in swirl flow, which prevents the droplet from penetrating the airstream. This force increases as the swirl strength increases. Thus, the droplet easily penetrates the shear region in the low-swirl case ($Sw=0.47$) and experiences a low velocity in the wake of the vanes and a forward velocity in the shear zone. Thus, the droplet develops the retracting bag breakup mode. On the other hand, in the case of strong swirl flow ($Sw=0.82$), the drop does not fully reach the wake region created by the vanes and the recirculation zone created by the swirl flow, thus resulting in a regular bag breakup mode.

The droplet breakup phenomenon for another set of parameters ($\bar x_d = 0.01$, $\bar y_d = 0.89$, $\bar z_d = -0.48$, $We = 7.72$ and $Sw = 0.82$) is presented in figure 3 (the corresponding animation is provided as supplementary movie 2). Here, both the front and top views captured using the two high-speed cameras are depicted. In this case, the disk easily enters and passes through the shear region as the aerodynamic force is low. The droplet exhibits the retracting bag breakup mode for this set of parameters as well, indicating that it is dependent on the swirl and Weber numbers and on the location of the dispensing needle (also see figure 17 in the Appendix). This point is further elaborated in § 3.2 where we present the regime map that demarcates different breakup modes for various parameters.

Figure 3. Temporal evolution of the retracting bag breakup morphology for $We = 7.72$ and $Sw = 0.82$ in two views. (a–f) Front $(x\text {--}y)$ and (g–l) top $(x\text {--}z)$ views. Here, $\bar x_d = 0.01$, $\bar y_d = 0.89$ and $\bar z_d = -0.48$. The dimensionless time ($\tau$) and the corresponding streamwise location of the drop are mentioned at the top and bottom, respectively. An animation showing the front and top views is provided as supplementary movie 2.

3.2. Regime map

Figure 4 depicts a regime map demarcating different modes, namely no breakup, vibrational breakup, retracting bag breakup and bag breakup modes, in $\bar z_d$–$We$ space. The rest of the parameters are fixed at $\bar x_d = 0.01$, $\bar y_d = 0.89$ and $Sw = 0.82$. Experiments are repeated five times for each set of parameters to ensure repeatability. The droplet encounters the airstream in oppose-flow, cross-flow and co-flow conditions as we move the needle from negative to positive values of $\bar z_d$ (see figure 9 and the associated discussion in § 3.4). For $\bar z_d=-0.63$ and 0.63, as the droplet falls under the action of gravity, it does not interact with the swirl flow and remains nearly spherical with minor shape oscillations but without breakup (circle symbols), indicating that these locations are outside the swirl airflow region. This implies that the aerodynamic potential and stretching rate are insignificant enough to disintegrate even the initial droplet. It can be observed that as we move the dispensing needle from $\bar z_d=-0.63$ to 0.63, the droplet only exhibits no breakup, vibrational breakup and retracting bag breakup modes for $We=7.72$ (low airstream velocity). This is owing to the resultant lower aerodynamic force and low stretching rate in these cases. Close inspection also reveals that there are three low-velocity zones created by the swirler's vanes near $\bar z_d \approx -0.33$, $0.08$ and $0.40$. For $We=11.8$ (increasing the velocity of the airstream), we observe bag breakup mode in addition to the other modes. The retracting bag breakup modes appear at the edges of low-velocity zones due to differential velocity interaction. The droplet shows bag breakup mode for negative $\bar z_d$ locations (oppose-flow and cross-flow conditions) and positive $\bar z_d$ locations slightly away from the centreline (co-flow condition). A further increase in the Weber number creates a more favourable condition for bag breakup, especially for negative $\bar z_d$. The following is an explanation for the behaviour. In oppose-flow conditions, efficient aerodynamic energy transfer occurs due to the long droplet–air interaction time; however, in co-flow conditions, the local migration of the droplet and the airflow are in the same direction, resulting in a short residence time for efficient energy transfer.

Figure 4. Regime map demarcating no breakup (circle), vibrational breakup (square), retracting bag breakup (triangle) and bag breakup (star) modes in $\bar z_d$–$We$ space. The rest of the parameters are fixed at $\bar x_d = 0.01$, $\bar y_d = 0.89$ and $Sw = 0.82$.

It is observed that the swirler vanes create three dead zones, and four high-airstream-flow zones between the dead zones and outer no breakup zones. To explore the influence of varying the coordinate of the dispensing needle on droplet breakup morphology, the high-airstream zones at $\bar z_d=-0.48$, $-0.13$, $0.23$ and $0.53$ are considered. Figures 5(a) and 5(b) show the various breakup modes observed by changing the location of the dispensing needle in the $\bar x_d\text {--}\bar z_d$ plane at $\bar y_d = 0.89$ and the $\bar y_d\text {--}\bar z_d$ plane at $\bar x_d = 0.01$, respectively. Here, $Sw = 0.82$ and $We=11.40$. It can be seen in figure 5(a) that there is a transition from the bag breakup mode to no breakup via the retracting bag breakup and vibrational breakup modes as $\bar x_d$ is increased (moving away from the nozzle in the streamwise direction) for any $\bar z_d$ location. The dilution of the flow field in the downstream direction from the tip of the air nozzle causes the various breakup modes (refer to our discussion in § 3.4). The droplet experiences an oppose-flow condition at negative $\bar z_d$ due to the helical motion of the swirl airstream, which is a key favourable condition for the bag fragmentation process. Due to the transition from favourable to unfavourable conditions from a bag breakup perspective, the axial extent downstream of the nozzle/swirler for bag breakup mode reduces when the droplet-dispensing needle traverses from $-\bar z_d$ to $\bar z_d$ direction. Near the centreline of the airstream, the droplet experiences a partial cross-flow with oppose-flow and co-flow conditions in the negative and positive sides of $\bar z_d=0$, respectively. As a suspended droplet in this zone is promptly ejected by the swirl flow field into the low-velocity region, the probability of the bag breakup is low in the downstream direction. Similarly, a suspended droplet in a peripheral location (see at $\bar z_d = 0.48$) encounters the airstream in a co-flow condition, and with the short interaction time, the deformed droplet goes out of the main airstream. Therefore, the bag breakup probability is less for the given operating parameters.

Figure 5. Regime map showing the no breakup (circle), vibrational breakup (square), retracting bag breakup (triangle) and bag breakup (star) modes in (a) $\bar x_d\text {--}\bar z_d$ plane at $\bar y_d = 0.89$ and (b) $\bar y_d\text {--}\bar z_d$ plane at $\bar x_d = 0.01$. The right-hand side panels show the directions with respect to the nozzle position in the $x\text {--}z$ plane (top view) and $y\text {--}z$ plane (side view). The rest of the parameters are $Sw = 0.82$ and $We=11.40$.

The height of the dispensing needle $\bar y_d$ can also play an important role in the breakup dynamics. Thus, we investigate the droplet breakup modes in the $\bar y_d\text {--}\bar z_d$ plane at $\bar x_d = 0.01$ (figure 5b). The inertial kinetic energy of the droplet migrating in the downward direction is related to the height of the dispensing needle, which increases as the height of the dispensing needle increases. A droplet with a high downward velocity (injected at a high $\bar y_d$ location) enters and travels quickly through the shear-layer zone without being influenced by the swirl airstream. Thus, at a fixed negative $\bar z_d$, i.e. in oppose-flow and cross-flow conditions (see, for instance, $\bar z_d=-0.48$ and $-0.13$), increasing the droplet-dispensing height leads to a change from the bag breakup to retracting bag breakup to vibrational breakup to no breakup mode. The transition from bag to retracting bag breakup mode occurs at a lower value of $\bar y_d$ when we move the dispensing needle from $\bar z_d=-0.48$ to $0.23$. This is because the swirler has a dome-like shape at the centre that results in an annular flow from the air nozzle. As the height of the dispensing needle is increased (see, for instance, at $\bar z_d=-0.13$ and $\bar y_d=2.09$), the droplet interacts with the upper part of the annular flow and pushes outside and downward. The droplet then enters the central recirculation zone produced by the dome and only exhibits the vibrational breakup mode even for higher values of $\bar y_d$. At $\bar z_d = 0.23$, the droplet experiences a cross-flow with co-flow condition. The upper annular flow pushes the droplet towards the annular flow on the right-hand side. In the co-flow configuration ($\bar z_d=0.53$), on the other hand, the bag breakup mode is not observed as it interacts with the airstream at the right-hand annular flow region. At lower heights, the potential energy of the droplet is low, and hence the droplet has been influenced by the flow and tries to be in the potential core so that the droplet exhibits bag breakup mode. At moderate $\bar y_d$ locations, we have observed the retracting bag breakup. In contrast, for greater heights, due to the associated higher kinetic energy, the droplet easily penetrates in the potential core, resulting in no breakup or vibrational breakup mode. From the above discussion, one can conclude that drop breakup morphology is strongly influenced by several key factors, such as aerodynamic field, initial momentum and residence time of the droplet.

3.3. Droplet trajectory

In order to gain a better understanding of the interaction of the droplet with the swirl airstream, in figure 6, we plot its trajectory for various conditions, such as different values of the swirl number and location of the dispensing needle. The Weber number is fixed at $We = 11.40$. The trajectory of the droplet injected at $\bar z_d = 0$ for the no-swirl case ($Sw = 0$) is shown (black circles) for reference displaying the vibrational breakup mode. In this case, the droplet enters the airstream but does not penetrate the potential core zone; instead, it deflects away from the shear-layer region. The conditions experienced by the droplet for this set of parameters favour the vibrational breakup mode. The high-swirl ($Sw = 0.82$) and low-swirl ($Sw = 0.47$) cases considered in figure 6 exhibit the bag breakup and the retracting bag breakup modes, respectively. As the swirler rotates in the clockwise direction, the droplet migrates from the negative $z$ to the positive $z$ direction (figure 6c,d). The droplet experiences an upward force due to swirling flow when $\bar z_d$ is negative ($\bar z_d = -0.13$), thus resulting in a higher curvilinear trajectory in the high-swirl case (figure 6b,d,f) as compared with the low- and no-swirl cases (figure 6a,c,e). Inspection of figure 6(e,f) also reveals that in contrast to the no-swirl case where the droplet follows mostly a rectilinear trajectory, in the swirl case ($Sw=0.47$), the droplet undergoes a curvilinear trajectory and approaches the central recirculation region through the dead zones of the vanes, which leads to retracting bag breakup mode for $\bar z_d = -0.13$. This curvilinearity in the trajectory is enhanced as we increase the swirl number. It can be seen that the droplet travels a compact path length as it undergoes a greater turning in the swirl cases compared to the no-swirl condition. By comparing the left- and right-hand panels of figure 6, we observe that the flow configuration (oppose-flow: $\bar z_d = -0.13$; and co-flow: $\bar z_d = 0.23$) has a significant effect on the droplet trajectory due to the change in the aerodynamic field for positive and negative values of $\bar z_d$. An oppose-flow condition limits the droplet penetration to the potential core region of the main airstream. Thus, the drop experiences a retarding velocity, resulting in a smaller drop penetration in the bag formation process than in the co-flow situation (see the trajectories marked by rectangles and triangles in figure 6).

Figure 6. Trajectories of the droplet for $We=11.40$ in different conditions. (a,c,e) $Sw=0.47$ and (b,d,f) $Sw=0.82$. (a,b) Three-dimensional views of the droplet's trajectories. Projections on (c,d) the $x\text {--}z$ and (e,f) the $x\text {--}y$ planes. The results represented by blue squares and red triangles in all panels are associated with $\bar z_d = -0.13$ and 0.23, respectively. The no-swirl case ($Sw=0$) with $\bar z_d=0$ is presented (black circles) for comparison purpose. Here, the locations $x, y$ and $z$ are in mm.

3.4. Analysis of the imposed swirl flow using stereo-PIV

As the imposed swirl flow field essentially influences the breakup phenomenon of the drop, we analyse the flow field for different swirl strengths (characterized by the swirl number), airflow rates in the nozzle (characterized by the Weber number) and the location of the droplet migration plane using a stereo-PIV set-up as shown in figure 15 in the Appendix. Figures 7 and 8 depict the contours of the resultant velocity $U_{net}$ ($= \sqrt {U_x^2+U_y^2+U_z^2}$) and $U_z$ overlapped with the velocity vectors for different Weber and swirl numbers, respectively. Here, $U_x$, $U_y$ and $U_z$ denote the components of the velocity vector in the $x$, $y$ and $z$ directions, respectively. It can be seen that the resulting flow is an annular flow. In figure 7, for a fixed value of the swirl number, we observe that increasing the Weber number increases the magnitude of velocity, leading to stronger annular flow with a recirculating zone in the centreline region. Thus, a droplet encounters a more aerodynamic force as the Weber number increases. The contours of $U_z$, which indicate the flow rotating clockwise, demonstrate the droplet movement from left to right when dispensing in the flow regime. The wake regions due to the vanes are also clearly evident in the flow field showing $U_z$ contours. Similarly, in figure 8, we present the contours of $U_{net}$ and $U_z$ for different swirl numbers for a fixed value of the Weber number ($We = 11.40$). It can be observed that while a symmetric potential core region is apparent in the no-swirl case (figure 8a,b), in the swirl cases (figure 8c–f), an annular flow due to the dome shape at the centre of swirler develops downstream, which markedly alters the breakup mode as discussed in § 3.1. The radial component of the velocity vector is negligible in the absence of swirl, but it becomes substantial in the presence of swirl. The outward radial component of the velocity is enhanced while the axial velocity component is decreased as the strength of swirl flow (i.e. the swirl number) is increased (figure 8c–f). The characteristics of the swirl flow (contours of $U_{net}$ and $U_z$) in the $x\text {--}y$ plane taken at different $\bar z$ locations (namely $\bar z_d=-0.48$, $-0.13$, $0.23$ and $0.53$) are presented in figure 9 for $We=11.40$ and $Sw=0.82$. Figure 9(a,b) (for $\bar z_d=-0.48$) corresponds to the $x\text {--}y$ plane at the extreme left-hand side of the swirler. In this case, the influence of the vane as an obstacle to the flow is clearly seen at $x \approx 16$ mm and $y \approx 2$ mm. The contours of $U_z$ for $\bar z_d=-0.48$ indicate that the droplet experiences an oppose-flow condition (see the upward-moving velocity vectors in figure 9a–d). As we move towards the central location (at $\bar z_d=-0.13$), the droplet still experiences an oppose-flow condition, albeit of lower strength as compared with that at $\bar z_d=-0.48$. On the other hand, when $\bar z_d$ is positive ($\bar z_d=0.23$ and 0.53), i.e. on the right-hand side of the swirler, the droplet experiences a co-flow configuration (as indicated by the downward velocity vectors in figure 9e–h). As noted in the introduction, a few researchers have previously used high-speed photography and PIV techniques to explore the features of swirl flow from a nozzle, albeit without droplets (Merkle et al. Reference Merkle, Haessler, Büchner and Zarzalis2003; Rajamanickam & Basu Reference Rajamanickam and Basu2017; Kumar & Sahu Reference Kumar and Sahu2019; Patil & Sahu Reference Patil and Sahu2021; Soni & Kolhe Reference Soni and Kolhe2021). They also reported a similar flow field.

Figure 7. The contours of the resultant velocity $U_{net}$ ($= \sqrt {U_x^2+U_y^2+U_z^2}$) (a,c,e,g) and $U_z$ (b,d,f,h) in the $x\text {--}y$ plane at ${\bar z}_d = 0$ for different Weber numbers: (a,b) $We = 7.72$, (c,d) $We = 11.40$, (e,f) $We = 15.77$ and (g,h) $We = 20.86$. In all cases, $Sw = 0.82$. The $x$ and $y$ labels are in mm, and $U_{net}$ and $U_z$ are in ${\rm m}\,{\rm s}^{-1}$.

Figure 8. The contours of the resultant velocity $U_{net}$ ($= \sqrt {U_x^2+U_y^2+U_z^2}$) (a,c,e) and $U_z$ (b,d,f) in the $x\text {--}y$ plane at ${\bar z}_d = 0$ for different swirl numbers: (a,b) $Sw = 0$, (c,d) $Sw = 0.47$ and (e,f) $Sw = 0.82$. In all cases, $We = 11.40$. The $x$ and $y$ labels are in mm, and $U_{net}$ and $U_z$ are in ${\rm m}\,{\rm s}^{-1}$.

Figure 9. The contours of the resultant velocity $U_{net}$ ($= \sqrt {U_x^2+U_y^2+U_z^2}$) (a,c,e,g) and $U_z$ (b,d,f,h) in the $x\text {--}y$ plane at (a,b) $\bar z_d = -0.48$, (c,d) $\bar z_d = -0.13$, (e,f) $\bar z_d = 0.23$ and (g,h) $\bar z_d = 0.53$. In all cases, $Sw = 0.82$ and $We = 11.40$. The $x$ and $y$ labels are in mm, and $U_{net}$ and $U_z$ are in ${\rm m}\,{\rm s}^{-1}$.

3.5. Theoretical modelling: RT instability

The RT instability occurs when a lighter fluid penetrates into a heavier fluid (Sharp Reference Sharp1983), which plays an important role in the secondary breakup process of a droplet interacting with an airstream (Guildenbecher et al. Reference Guildenbecher, López-Rivera and Sojka2009). The Atwood number, $At$ ($\equiv (\rho _{l}-\rho _{a})/(\rho _{l}+\rho _{a}))= 0.997$, characterizes the density contrast in our case. Thus, the interface of the droplet (heavier fluid) becomes unstable due to the penetration of the air phase into the liquid (ethanol) phase and perturbations (undulations) of a certain wavelength form at the droplet interface. These surface perturbations grow exponentially with time due to energy transfer from the airstream. Subsequently, the droplet undergoes fragmentation due to these instabilities. Several researchers have conducted linear stability analyses (Sahu et al. Reference Sahu, Ding, Valluri and Matar2009a) and direct numerical simulations (Sahu et al. Reference Sahu, Ding, Valluri and Matar2009b; Sahu & Vanka Reference Sahu and Vanka2011) to study the effects of viscosity contrast, density contrast and surface tension on the development of RT instabilities in shear flows. The RT instability is also commonly known as fingering instability as the small perturbations quickly develop into interpenetrating fingers of the heavier fluid (Sharp Reference Sharp1983). It has also been shown that when the wavelength of the disturbances created by the RT instability is smaller than the drop diameter, the fingers grow further and fragmentation of the drop occurs (Joseph, Belanger & Beavers Reference Joseph, Belanger and Beavers1999).

Figures 10(a) and 10(b) illustrate the development of fingers at the onset of a bag burst in two views, namely the front ($x\text {--}y$) and top ($x\text {--}z$) views, respectively. The parameters are $We=14.54$, $Sw=0.82$, $\bar x_d = 0.01$, $\bar y_d = 0.89$ and $\bar z_d = -0.13$. It can be observed that while just six fingers are visible in the front view, three additional fingers (total of nine fingers) are apparent in the top view. In the theoretical modelling (discussed below), as the total number of fingers are used to compute the maximum wavelength ($\lambda _m$) of the RT instability, we use both views to compute the total number of fingers. It should be noted that previous studies (e.g. Zhao et al. Reference Zhao, Liu, Li and Xu2010) solely employed the front view to estimate the number of fingers. The maximum wavelength ($\lambda _m$) is given by

(3.1)\begin{equation} \lambda_m = \frac{{\rm \pi} D_m}{N_f}, \end{equation}

where $D_m$ is the maximum disk diameter of the droplet and $N_f$ is the number of fingers counted experimentally at the onset of the breakup, as demonstrated in figure 10. In liquid jet breakup and droplet breakup under a straight airstream, $\lambda _m = \sqrt {3}\lambda _c$, where $\lambda _c = {2{\rm \pi} \sqrt {\sigma _l/\rho _l a}}$ is the critical wavelength of RT instability and $a$ is the acceleration of the drop (Varga, Lasheras & Hopfinger Reference Varga, Lasheras and Hopfinger2003; Marmottant & Villermaux Reference Marmottant and Villermaux2004; Zhao et al. Reference Zhao, Liu, Li and Xu2010; Gao et al. Reference Gao, Yang, Deng, Xu, Ji and He2015). In our study, the front ($x\text {--}y$) and top ($x\text {--}z$) views acquired using high-speed cameras and processed with the digital image processing software NIH IMAGE (ImageJ) are utilized to determine the acceleration of the drop. From our experiments, in the case of straight airflow ($Sw = 0$, $\bar z_d = 0$), we found that $\lambda _m = 4.47$ mm, which is close to that obtained using the classical analytical expression $\lambda _m = \sqrt {3}\lambda _c$. In the swirl cases, we observed that, for both the oppose-flow (negative $z_d$) and co-flow (positive $z_d$) configurations, increasing the swirl number decreases the maximum wavelength/increases the number of fingers. This indicates that the swirling airflow enhances the stretching factor in the droplet deformation process owing to the development of stronger RT instability as compared with the no-swirl condition. Thus, the correlation factor between the maximum unstable wavelength and the critical wavelength will be different in the swirl conditions and one cannot simply use $\lambda _m = \sqrt {3}\lambda _c$. In the case of liquid jet/sheet breakup, Vadivukkarasan & Panchagnula (Reference Vadivukkarasan and Panchagnula2017) found that the unstable wavelength/growth rate of disturbances is a complex Bessel function in swirl airstream. In order to calculate $\lambda _c$ one has to incorporate the swirl effect on the number of fingers/nodes. We will return to this point again in the next section.

Figure 10. Counting the number of fingers using the (a) front view ($x\text {--}y$) and (b) top view ($x\text {--}z$). The parameters are $We=14.54$, $Sw=0.82$, $\bar x_d = 0.01$, $\bar y_d = 0.89$ and $\bar z_d = -0.13$. While only six fingers (numbered in black circles) are visible in the front view, three more fingers (numbered in red circles) are apparent in the top view.

Further, we extend this analysis for different breakup modes observed in figure 4 for $We=11.40$ and 20.86. For a fixed $We$, as $\bar z_d$ location of the dispensing needle is shifted from $\bar z_d = -0.63$ to 0.63, the mode changes from no breakup to vibrational breakup to retracting bag breakup to bag breakup mode due to the change in the aerodynamic field and its interaction with the droplet, which in turn increases the number of fingers. Table 4 presents the total number of fingers observed in different breakup modes for $We=11.40$ and 20.86. The rest of the parameters are $Sw=0.82$, $\bar x_d = 0.01$ and $\bar y_d = 0.89$ (the same as for figure 4). It can be seen that increasing the Weber number (i.e. increasing aerodynamic force) for each mode increases the number of fingers in the majority of cases. The number of fingers also increases as we go from no breakup to bag breakup regions by changing the location of the dispensing needle. The above discussion is summarized in figure 11, which depicts how the number of fingers $(N_f)$ varies with the Weber number for different values of $Sw$ and the $\bar z_d$ location of the dispensing needle. The findings from Zhao et al. (Reference Zhao, Liu, Li and Xu2010) and Dai & Faeth (Reference Dai and Faeth2001) for the no-swirl condition ($Sw=0$, $\bar z_d=0$) are also presented in figure 11. Comparison of our results with those reported in previous studies for the no-swirl case reveals that while the trend in the variation of $N_f$ is similar, i.e. increasing $We$ increases the value of $N_f$, we observe a higher number of fingers since we have used both views in our analysis as demonstrated in figure 10. The following observations can be made from figure 11: (i) increasing the swirl number for a fixed Weber number increases the number of fingers and (ii) the oppose-flow (negative $\bar z_d$) and co-flow (positive $\bar z_d$) have a negligible effect for each swirl strength.

Table 4. The maximum number of fingers observed in different breakup modes for $We=11.40$ and 20.86. For each value of $We$, $\bar z_d$ is varied to obtain the maximum number of fingers associated with different breakup modes. The rest of the parameters are $Sw=0.82$, $\bar x_d = 0.01$ and $\bar y_d = 0.89$.

Figure 11. Comparison of number of fingers counted for different values of swirl number and position of dispensing needle. The fixed parameters are dispensing needle location at $\bar x_d = 0.01$ and $\bar y_d = 0.89$.

3.6. Evolution of drop topology

As discussed in the previous section, the mechanism and growth of the bag created in the swirl flow are different from those in straight airflow. While the bulging of the disk happens in the direction of the airstream in straight flow, it occurs in the direction of swirling action in swirl flow. In straight airflow, the droplet in the flow field deforms into a disk shape that is perpendicular to the cross-stream direction due to the aerodynamic force. As a result, the disk experiences an unequal pressure distribution across it, bulging outward towards the low-pressure side. The disk then morphs into a bag shape. Subsequently, the interface thins out and eventually bursts because it can no longer withstand the pressure force. The deformation and breakup phenomena in swirl airflow are more complex; the droplet also exhibits a curvilinear trajectory. Villermaux & Bossa (Reference Villermaux and Bossa2009) and Kulkarni & Sojka (Reference Kulkarni and Sojka2014) modelled the evolution of the drop topology in the case of straight airflow by considering cross-flow and oppose-flow configurations, respectively. Recently, Jackiw & Ashgriz (Reference Jackiw and Ashgriz2021) pointed out that the stretching factor $f$ in cross-flow and oppose-flow configurations will be different, which can have a significant impact on the fragmentation process. Depending on the swirl airstream characteristics, a droplet experiences all cross-flow, oppose-flow and co-flow configurations in swirl flow. In the following, the earlier approach used in straight airflow has been extended to droplet interaction with a swirling airstream.

A stagnation point flow can be considered for the velocity components of the airstream in the streamwise $x$ and radial $r$ ($\equiv \sqrt {(y^2+z^2)}$) directions, which are given by

(3.2a,b)\begin{equation} U_x ={-} f \frac{U x}{d_0}, \quad U_r = f \frac{U r}{2 d_0} , \end{equation}

where $f$ is the stretching factor, $U_r = \sqrt {U_y^2 + U_z^2}$ denotes the radial velocity and $U$ represents the mean velocity of the airflow field at the exit of the air nozzle. Thus, the resultant velocity of the airstream is $U_{net} = \sqrt {U_x^2+U_r^2}$. This is essentially the same as $U$ as verified by the flow field obtained from our stereo-PIV experiments.

Furthermore, using the Navier–Stokes and continuity equations, the pressure field $(p_a)$ around the droplet can be obtained by assuming the flow is inviscid, incompressible and quasi-steady (Kulkarni & Sojka Reference Kulkarni and Sojka2014):

(3.3)\begin{equation} p_a(r,x)=p_a(0) - \rho_a \frac{f^2 U^2}{8 d_0^2} r^2 + \rho_a \frac{f^2 U^2}{2 d_0^2} x^2, \end{equation}

where $r$ ($\equiv \sqrt {(y^2+z^2)}$) denotes the radial coordinate. Thus, at $x=0$:

(3.4)\begin{equation} p_a(r)=p_a(0) - \rho_a \frac{f^2 U^2}{8 d_0^2} r^2. \end{equation}

Here, $p_a(0) = {\rho _a U^2/2}$ is the stagnation pressure at $(r,x)=(0,0)$.

Assuming a fully developed airstream, the axisymmetric Navier–Stokes equations in the cylindrical coordinate system describe the dynamics of the droplet deforming from a spherical to a disk shape, which are given by

(3.5)\begin{gather} \rho_l\left ( \frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} \right ) ={-} \frac{\partial p_l}{\partial r} + \mu_l \left [ \frac{1}{r}\frac{\partial}{\partial r} \left ( r \frac{\partial u_r}{\partial r} \right ) - \frac{u_r}{r^2} \right ], \end{gather}

(3.6)\begin{gather} r \frac{\partial h}{\partial t} + u_r \frac{\partial (r u_r h)}{\partial r} = 0 , \end{gather}

where $h(t)$ is the time-dependent thickness of the disk (see figure 12). As the Reynolds number ($Re \equiv \rho _a U d_0/\mu _a$) in our experiments is in the range 1420–2130, assuming the interaction of the droplet with the airstream to be inviscid, the velocity field inside the droplet can be obtained from (3.6) as

(3.7)\begin{equation} u_r(r,t) = \frac{r}{R} \left ( \frac{{\rm d} R}{{\rm d}t} \right ), \end{equation}

where $R(t)$ is the radius of the disk-shaped droplet (figure 12). While the tangential stresses are negligible, the normal stress balance at the interface separating the air and liquid phase is given by

(3.8)\begin{equation} \sigma_l \kappa = T_{rr}(l) - T_{rr}(a), \end{equation}

where $\kappa$ is the curvature of the interface and $T_{rr}(l) = p_l(r)$ and $T_{rr}(a) = p_a(r)$ are the normal stress components at the interface ($r=R(t)$) associated with the liquid and air phases, respectively. Here, $\kappa = 2/h(t)$. Using (3.4), (3.8) becomes

(3.9)\begin{equation} p_l(R) = p_a(0) - \rho_a {\frac{f^2 U^2}{8 d_0^2} R^2} + \frac{2 \sigma_l}{h}. \end{equation}

Now we define $\alpha (t) = R(t)/(d_0/2)$. Non-dimensionalizing and integrating (3.5) from $r=0$ to $r=R(t)$ and using (3.9), we get

(3.10)\begin{equation} \frac{{\rm d}^2 \alpha}{{\rm d} \tau^2} - \left ( \frac{f^2}{4} - \frac{24}{We} \right ) \alpha = 0. \end{equation}

Solving (3.10) using the initial conditions $\alpha (0) = 1$ and $\alpha '(0) = 0$, which describe a sphere-shaped drop during its early interactions with the airstream, we get

(3.11)\begin{equation} \alpha = \exp\left({\tau \sqrt{\frac{f^2}{4} - \frac{24}{We}}}\right). \end{equation}

It can be observed from this equation that the value of the stretching factor $(f)$ in the cross-flow configuration $(We=12)$ is 2$\sqrt {2}$ (Kulkarni & Sojka Reference Kulkarni and Sojka2014), whereas $f=4$ in the oppose-flow configuration $(We=6)$ (Villermaux & Bossa Reference Villermaux and Bossa2009). Thus, we can use our theoretical model to determine the critical Weber number for a range of flow configurations by knowing the stretching factor.

Figure 12. (a–c) Various dimensions of the droplet obtained from the experiments to be used in the modelling. Here, $R(t)$, $l(t)$ and $h(t)$ represent the instantaneous radius, length in the streamwise direction and thickness of the disk of the deformed droplet, respectively.

We also found that $\alpha = \cosh (\tau \sqrt {{f^2}/{4} - {24}/{We}} )$ equally accurately predicts the experimental results. Figure 13 depicts the evolution of $\alpha (t)$ with dimensionless time $\tau$ for different values of the stretching factor $(f)$ and swirl flow conditions. The corresponding theoretical predictions (dashed lines) obtained using the above mentioned equation are also shown for different values of $f$. Our experimental results for the no-swirl case in the cross-flow configuration ($Sw=0, \bar z_d=0$) agree with the theoretically predicted exponential growth rate of the droplet evolution using $f=2\sqrt {2} \approx 2.82$ as suggested by Kulkarni & Sojka (Reference Kulkarni and Sojka2014) and Jackiw & Ashgriz (Reference Jackiw and Ashgriz2021). Although we present figure 13 on a linear scale to be consistent with previous studies, the corresponding figure on a logarithmic scale is shown as figure 18 (in the Appendix) to demonstrate exponential growth of $\alpha (t)$. In swirl cases, the growth rate of $\alpha (t)$ remains exponential, but the slope of the evolution process has increased as the swirl strength has increased, indicating that the swirl flow has expedited the drop deformation process. These findings are also consistent with our analysis of RT instability in the previous section, which demonstrated that the number of fingers increases as the swirl strength increases (see figure 11). Also, Soni et al. (Reference Soni, Kirar, Kolhe and Sahu2020) showed that as the direction of the straight airstream changes from cross-flow to oppose-flow (i.e. by changing the orientation of the interaction of the droplet with airstream), the critical Weber number requirement for bag breakup decreases. It can be seen in figure 13 that the theoretical predictions with $f = 3.01$ and 3.41 agree well with the experimental observations for $Sw=0.47$ and 0.82 at $z_d=-0.13$ (cross-flow configuration), respectively. Thus, it is reasonable to argue that increasing the vane angle (increasing the swirl strength) increases the stretching rate factor and it approaches the value $(f = 4)$ for the perfect oppose-flow configuration (Villermaux & Bossa Reference Villermaux and Bossa2009). It is also worth noting that the values of the stretching factor $f$ used in the theoretical predictions in figure 13 are fitted values, but the variation of the stretching factor with an increase in the swirl strength is consistent with the velocity fields (figure 8) obtained from the stereo-PIV measurements. Figure 19 (in the Appendix) depicts the variation of $U_r/U_x$ at the droplet suspension location ($\bar x_d = 0.01,\ \bar y_d = 0.89$) with $f$, demonstrating that increasing the swirl strength increases the velocity in the transverse direction, allowing for easier droplet fragmentation.

Figure 13. Evolution of $\alpha (t)=R(t)/(d_0/2)$ with dimensionless time $\tau$ for different values of the stretching factor $(f)$ and swirl conditions. Here, $We=15.77$. Symbols represent the experimental results, with an error bar generated from five repetitions. The dashed lines represent the theoretical results.

Next, we model the growth of the bag and develop its correlation with Weber number $We$ and stretching rate $f$. The bag's growth can be described by its elongation, $l(t)$, as shown in figure 12. For sufficiently large dimensionless time $\tau$, an asymptotic solution of $\alpha = \exp ({\tau \sqrt {{f^2}/{4} - {24}/{We}}})$ is used to solve the bag evolution process $(\beta (t) = { l(t) }/{ d_0/2 } )$. The differential equation for the elongation of the bag can be obtained by balancing the forces at the tip of the bag (Kulkarni & Sojka Reference Kulkarni and Sojka2014), which is given by

(3.12)\begin{equation} \frac{{\rm d}^2 \beta}{{\rm d} \tau^2} - \frac{24}{We} \beta - 2 \alpha^2 = 0, \end{equation}

and solved using the initial conditions $\beta (0) = 0$ and $\beta '(0) = 0$, which signify that there was no bulging in the disk and no progression of the disk to bulging at the onset of the bag formation. Thus, we get

(3.13)\begin{equation} \beta ={-} \frac{1}{\gamma} \left ( \frac{\sqrt{m}}{\sqrt{n}} + 1 \right ) {\rm e}^{\tau\sqrt{n}} + \frac{1}{\gamma} \left ( \frac{\sqrt{m}}{\sqrt{n}} - 1 \right ) {\rm e}^{-\tau\sqrt{n}} + \frac{2}{\gamma} e^{\tau \sqrt{m}}, \end{equation}

where $\gamma = f^2 - 120/We$, $m = f^2 - 96/We$ and $n = 24/We$. In figure 14, we plot the temporal evolution of $\beta$ obtained from our experiments for different swirl conditions. The theoretical predictions obtained using (3.13) are also shown in figure 14 for the corresponding values of the stretching factor $f$ (as discussed in relation to figure 13). The experimental results reveal the bag development process displays exponential growth, and match the theoretical predictions quite well. It can also be seen that increasing the swirl strength increases the growth rate of the bag development. We confirm that the theoretical growth rate predictions using $f = 4$ and $3.41$ follow nearly identical trends, as an oblique airstream with obliquity $\ge 45^{\circ}$ (towards the oppose-flow configuration) produces a critical Weber number $\approx 6$ (Soni et al. Reference Soni, Kirar, Kolhe and Sahu2020). This indicates that a swirling airstream interacting with a droplet at an angle greater than $45^{\circ}$ behaves similar to the oppose-flow configuration. As a consequence, $f = 4$ and $3.41$ perfectly match the results of Villermaux & Bossa (Reference Villermaux and Bossa2009) (oppose-flow) and $\,f = 2\sqrt 2$ matches the results of Kulkarni & Sojka (Reference Kulkarni and Sojka2014) (cross-flow) as seen in figure 14.

Figure 14. Evolution of $\beta (t)=l(t)/(d_0/2)$ with dimensionless time $\tau$ for different values of the stretching factor $(f)$ and swirl conditions. Here, $We=15.77$. Symbols represent the experimental results, with an error bar generated from five repetitions. The dashed lines represent the theoretical results. The results reported by Kulkarni & Sojka (Reference Kulkarni and Sojka2014) (for cross-flow) and Villermaux & Bossa (Reference Villermaux and Bossa2009) (for oppose-flow) in the case of straight airflow are also shown.