Droplet size distribution in a swirl airstream using in-line holography technique

Raindrops reach earth in a variety of shapes and sizes due to their complex interactions with the atmosphere and accompanying microphysical processes, such as fragmentation (Villermaux & Bossa, 2009; Kostinski & Shaw, 2009), coalescence (Schlottke & Weigand, 2008; Chaitanya et al., 2021) and phase change (Schlottke & Weigand, 2008). These microphysical processes are further influenced by various factors, including meteorological conditions, cloud type from which raindrops originate, the topology of the earth, and air movement in the atmosphere. The distribution of shape and size of raindrops is one of the important factors in rainfall modelling. This was first noticed by Bentley (1904); von Lenard (1904). Subsequently, Marshall & Palmer (1948) established a relationship that correlates the average diameter of raindrops with rainfall rate, which has been used in rainfall modelling even today (Villermaux & Bossa, 2009). Although several researchers have studied the droplet fragmentation phenomenon due to its importance in a variety of industrial applications, such as combustion, surface coating, pharmaceutical manufacturing, and disease transmission modelling (Villermaux, 2007; Marmottant & Villermaux, 2004; Soni et al., 2020; Jackiw & Ashgriz, 2021; Kirar et al., 2022; Xu et al., 2022), there have been very few studies on the size distribution of satellite droplets after fragmentation of a large droplet (Srivastava, 1971; Villermaux & Bossa, 2009; Jackiw & Ashgriz, 2022; Gao et al., 2022). Moreover, to our knowledge, no one has previously investigated the droplet size distribution under swirl airstream, even though such a situation is frequently encountered during rainfall (Lewellen et al., 2000; Haan Jr et al., 2017) and in many industrial applications (Soni & Kolhe, 2021; Candel et al., 2014). This is the subject of the present investigation.

In a continuous airstream, a droplet undergoes morphological changes due to the development of the Rayleigh–Taylor instability during its early inflation stage and later encounters fragmentation due to the Rayleigh–Plateau capillary instability (Taylor, 1963), and nonlinear instability of liquid ligaments (Jackiw & Ashgriz, 2021, 2022). The droplet exhibits different breakup modes, such as vibrational, bag, bag-stamen, multi-bag, shear, and catastrophic breakup modes, depending on the speed and direction of the airstream (Pilch & Erdman, 1987; Dai & Faeth, 2001; Cao et al., 2007; Guildenbecher et al., 2009; Suryaprakash & Tomar, 2019; Soni et al., 2020). The Weber number, defined as ${\it We}\equiv\rho_{a}U^{2}d_{0}/\sigma$, wherein $\rho_{a}$, $\sigma$, $U$ and $d_{0}$ denote the density of the air, interfacial tension, average velocity of the airstream and equivalent spherical diameter of the droplet, is used to characterised the droplet breakup phenomenon. A droplet exhibits shape oscillations at a specific frequency for low Weber numbers and fragments into small droplets of comparable size. This is known as the vibrational breakup. As the Weber number is increased, the droplet evolves to form a single bag on the leeward side, encircled by a thick liquid rim. As a result of the bag and rim fragmentation, tiny and slightly bigger droplets are formed. This process is known as the bag breakup phenomenon. The bag-stamen and multi-bag morphologies are identical to the bag breakup mode, but with a stamen formation in the drop’s centre, resulting in a large additional drop during the breakup (bag-stamen fragmentation) and several bags formation (multi-bag mode fragmentation). In shear mode, the drop’s edge deflects downstream, fracturing the sheet into small droplets. When the Weber number is exceedingly high, the droplet explodes into a cluster of tiny fragments very quickly, resulting in catastrophic fragmentation. Taylor (1963) was the first to find the critical Weber number $({\it We}_{cr})$ at which the transition from the vibrational to the bag breakup occurs. Subsequently, several researchers have investigated droplet fragmentation subjected to an airstream in cross-flow configuration (when the droplet interacts with the airstream in a direction orthogonal to gravity) and in-line configurations, such as co-flow and oppose-flow, when the droplet interacts with the airstream flowing along and opposite to the direction of gravity, respectively. The critical Weber numbers in cross-flow (Pilch & Erdman, 1987; Guildenbecher et al., 2009; Krzeczkowski, 1980; Jain et al., 2015; Hsiang & Faeth, 1993; Wierzba, 1990; Kulkarni & Sojka, 2014; Wang et al., 2014) and oppose-flow (Villermaux & Bossa, 2009; Villermaux & Eloi, 2011) configurations were found to be about 12 and 6, respectively. Soni et al. (2020) showed that in the case of an oblique airstream, a freely falling droplet exhibits curvilinear motion while undergoing topological changes and that the critical Weber number decreases when the direction of the airstream changes from the cross-flow to the oppose-flow condition. The value of ${\it We}_{cr}$ was found to be about 12 in the cross-flow configuration, and its value approaches 6 when the inclination angle of the airstream with horizontal is greater than $60^{\circ}$. The initial droplet size, fluid properties, ejection height from the nozzle, and velocity profile/potential core region were also known to affect the critical Weber number (Hanson et al., 1963; Wierzba, 1990). All these investigations are for straight airflow in the no-swirl condition. The deformation and breakup of a water droplet falling in air have also been investigated (Szakall et al., 2009; Agrawal et al., 2017; Balla et al., 2019; Agrawal et al., 2020).

A few researchers (Merkle et al., 2003; Rajamanickam & Basu, 2017; Kumar & Sahu, 2019; Patil & Sahu, 2021; Soni & Kolhe, 2021) have examined the characteristics of swirl flow using high-speed imaging and particle image velocimetry (PIV) techniques. The droplet morphology and breakup phenomenon in a swirling airflow is more complex than in a straight airflow without a swirl. In the case of swirl airflow, the droplet encounters oppose-flow, cross-flow, and co-flow situations simultaneously, depending on its ejection position, airstream velocity, and swirl strength. Recently, Kirar et al. (2022) experimentally analysed the droplet fragmentation phenomenon under a swirling airstream using shadowgraphy and particle image velocimetry techniques. They discovered a new breakup mode, termed as ‘retracting bag breakup’, due to the differential flow field created by the wake of the swirler’s vanes and the central recirculation zone in swirl airflow. A theoretical analysis based on the Rayleigh-Taylor instability was also developed that uses a modified stretching factor than the straight airstream to predict the droplet deformation process under a swirl airstream. Apte et al. (2009) performed large-eddy simulations of spray atomization from a swirl injector. They used a Pratt and Whitney injector (a high-bypass turbofan engine family) to create a spray alteration/rotation. They found that the number probability density of the atomized droplets follows the Fokker-Planck equation. This partial differential equation provides the temporal evolution of the probability distribution of the velocity of a particle subjected to drag and random forces similar to that in Brownian motion. Apart from the studies mentioned above, Shanmugadas et al. (2018) experimentally investigated wall filming and atomization from a swirl cup injector. The injector uses a simplex nozzle and a primary swirler to create a liquid rim and a secondary swirler to create the ligaments and droplets from the rim by shearing action. Their study reveals that the atomization/fragmentation process is a strong function of the central recirculation zone of the primary swirl. Further, Shao et al. (2018) performed direct numerical simulations on the swirling sheet breakup process leading to the formation of sheets, ligaments, and droplets. They found that the size distribution of droplets in a swirling sheet breakup process approximately follows the log-normal distribution. As this discussion indicates, despite significant research on the various droplet breakup processes and flow characteristics in straight and swirling airstreams, the size distribution of satellite droplets following the fragmentation of a single droplet has not been investigated yet.

The size distribution of satellite droplets due to fragmentation is commonly analysed using Laser Diffraction (LD), Phase Doppler Particle Analyzer (PDPA), and in-line holography techniques (Gao et al., 2013b; Katz & Sheng, 2010; Kumar et al., 2019). In the LD technique, a radial sensor measures the light scattered by the satellite droplets in the forward direction. This is not an imaging-based approach and must be used in conjunction with a model-based inversion to estimate the size distribution function of the detected droplets. The PDPA is a single-point measurement technique where the phase difference of scattered light captured at multiple angles is utilized to obtain the size of the droplets. The LD and PDPA techniques are suitable for applications, such as spray atomization, where droplets are produced continuously. However, both methods suffer from a number of limitations, such as small sampling volume in the PDPA method and lack of spatial resolution in the LD technique. Moreover, while the PDPA is restricted to spherical particles, the LD technique is not sensitive to single droplet analysis.

In contrast, the digital in-line holography technique, which employs a deep-learning-based image processing method, has recently emerged as a powerful tool for capturing three-dimensional information about an object with a high spatial resolution (Shao et al. (2020)). This technique can also offer the spatial distribution of satellite droplets. Thus, it is a superior choice to be used in the current study to analyse the size distribution of satellite droplets resulting from the fragmentation of a water droplet subjected to a swirl airstream. A few researchers have employed in-line holography to determine the droplet’s size distribution in a straight airstream in no-swirl conditions. For instance, for an ethanol droplet in a cross-flowing airstream, Gao et al. (2013b); Guildenbecher et al. (2017) used the in-line holography technique to obtain the three-dimensional position and velocities of fragments in the bag breakup and shear-thinning regimes. Li et al. (2022) used this technique to analyse the size distribution of satellite droplets under an induced shock wave. The investigation by Jackiw & Ashgriz (2022) is one of the most relevant studies in the context of the present work. Jackiw & Ashgriz (2022) developed an analytical model for the combined multi-modal distribution resulting from the aerodynamic breakup of a droplet under no-swirl conditions. They demonstrated that their theoretical predictions agree well with the experimental results of Guildenbecher et al. (2017). Although Jackiw & Ashgriz (2022) performed fragmentation experiments for a water droplet under straight airflow without swirling using a shadowgraphy technique, they only analysed the size distributions of satellite ethanol droplets reported by Guildenbecher et al. (2017) using a digital in-line holography technique.

In the present work, we investigate the size distribution of satellite droplets resulting from the fragmentation of a freely falling water droplet under a swirl airstream of different strengths by employing both shadowgraphy and digital in-line holography techniques. We found that while the satellite droplets show mono-modal size distribution for the no-swirl airstream, the fragmentation exhibits bi-modal and multi-modal distributions in a swirl flow for the low and high swirl strengths, respectively. A theoretical analysis accounting for various mechanisms, such as the nodes, rim, and bag breakup mode, is also carried out to determine the size distribution of droplets for different swirl strengths. To the best of our knowledge, the present study is the first attempt to estimate the size distribution of satellite droplets under swirl airstreams, which are commonly observed in many natural and industrial applications.

The rest of the paper is organized as follows. In §2, the experimental set-up and procedure for both the shadowgraph and digital in-line holography are elaborated. In order to demonstrate the digital in-line holography technique used in our work, we have considered a typical bag fragmentation of a water droplet under a straight airstream without swirl and compared our result with the size distribution of an ethanol droplet considered by Guildenbecher et al. (2017). The influence of different swirl airstreams on the droplet fragmentation and the resulting size distribution are discussed in §3. A theoretical model is employed to predict the size distributions of satellite droplets obtained from our in-line holography technique. The concluding remarks are given in §4.

In the present study, while a shadowgraphy technique is employed to investigate the droplet deformation and breakup phenomena of a freely falling droplet under an imposed swirl airstream, a digital in-line holography technique is used to analyze the size distribution of satellite droplets after fragmentation of the primary drop. The experimental set-up consists of (i) an air nozzle (18 mm diameter) with a swirler, (ii) a droplet dispensing needle connected to a three-dimensional (3D) traverse system, (iii) a continuous wave laser, (iv) a spatial filter arrangement, (v) collimating optics which include concave and convex lenses, (vi) two high-speed cameras and (vii) a diffused backlit illumination. A schematic diagram of the complete experimental set-up is shown in figure 1. The inset at the bottom of figure 1 shows the back and side views of the swirler. To change the strength of the swirl airstream, two different types of eight-vane swirlers are used, one with a vane angle, $\theta=30^{\circ}$ and the other with $\theta=45^{\circ}$. The rest of the dimensions of the swirlers are the same: outer diameter ($D_{o}=22$ mm), inner diameter ($D_{i}=20$ mm), dome diameter ($D_{m}=12$ mm) and blade thickness ($=1$ mm). The swirl strength is characterized by the geometric Swirl number $(Sw)$, which is defined as (Beér, 1974)

such that when $\theta=30^{\circ}$ and $45^{\circ}$, we achieve $Sw=0.47$ (low swirl strength) and $Sw=0.82$ (high swirl strength), respectively. We have also calculated the flow-based swirl number $(Sw_{f})$ using the stereo-PIV measurements (Kirar et al., 2022). The flow swirl number $(Sw_{f})$ is given by

where $u_{y}$ and $u_{\theta}$ are the axial and tangential components of the velocity, and $R$ denotes the radius of the swirler (Candel et al., 2014). We found that the values of $Sw_{f}$ are about 0.42 and 0.58 for the corresponding geometric swirl numbers $Sw=0.47$ and 0.82, respectively.

The swirlers are fabricated with a tough resin using 3D printing technology and have provision for attaching at the exit of a metallic circular nozzle. In order to remove inlet airstream disturbances and straighten the flow, a honeycomb pallet is placed upstream of the nozzle. The air nozzle is connected to an ALICAT digital mass flow controller (model: MCR-500SLPM-D/CM, Make: Alicat Scientific, Inc., USA), which can regulate flow rates between 0 and 500 standard litres per minute. The accuracy of the flow metre is about 0.8% of the reading + 0.2% of the full scale. The mass flow controller is coupled to an air compressor for air supply. An air dryer and moisture remover are installed in the compressed airline to maintain dry air during the experiments.

A Cartesian coordinate system $(x,y,z)$ with its origin at the center of the swirler tip, as shown in figure 1, is used to analyze the dynamics. A syringe pump (model: HO-SPLF-2D, Make: Holmarc Opto-Mechatronics Pvt. Ltd., India) connected to a dispensing needle is used to create consistent-sized droplets of water (spherical diameter, $d_{0}=3.09\pm 0.07$ mm). We use a 20 Gauge needle to dispense the droplets. The outer and inner diameters of the needle are 0.908 mm and 0.603 mm, respectively, and the length of the needle is 25.4 mm. The dispensing needle is fixed to a three-dimensional traverse mechanism that allows it to change the location of its tip $(x_{d},y_{d},z_{d})$ and thus control the droplet’s location in the swirl airstream. A flow rate of 20 $\mu$l/s is maintained in the syringe pump to create a water droplet at the tip of a blunt needle. This flow rate is low enough that only gravity can detach the droplets from the needle. We ensure that no other droplets emerge from the needle during the droplet’s interaction with the swirl flow airstream.

For shadowgraphy, a high-speed camera (model: Phantom VEO 640L, make: Vision Research, USA) with a Nikkor lens of a focal length of 135 mm and a minimum aperture of $f/2$ is employed. It is designated as high-speed camera 1 in the schematic diagram (figure 1). The high-speed camera 1 is positioned at $x=180$ mm while maintaining an angle of $-30^{\circ}$ with the $x$ axis. To illuminate the background uniformly, a high-power light-emitting diode (model: MultiLED QT, Make: GSVITEC, Germany) is used along with a diffuser sheet as shown in figure 1. The images captured using the high-speed camera 1 have a resolution of $2048\times 1600$ pixels, and they are recorded at 1800 frames per second (fps) with an exposure duration of 1 $\mu$s and a spatial resolution of 29.88 $\mu$m/pixel. The droplet’s image sequence is stored in the internal memory of the camera and then transferred to a computer for further processing.

The digital in-line holography set-up consists of a laser, spatial filter, collimating lenses, and the high-speed camera 2 (model: Phantom VEO 640L; make: Vision Research, USA) with a Tokina lens (focal length of $100$ mm and a maximum aperture $f/2.8$, model: AT-X M100 PRO D Macro). The high-speed camera 2 is positioned at $x=180$ mm as shown in figure 1. A laser beam is generated using a continuous wave laser (model: SDL-532-100T, make: Shanghai Dream Lasers Technology Co. Ltd.) that produces 100 mW output power at a wavelength of 532 nm. The laser beam is passed through a spatial filter to produce a clean beam. The spatial filter consists of an infinity-corrected plan achromatic objective (20X magnification, make: Holmarc Opto-Mechatronics Ltd.) and a 15 $\mu$m pin-hole. The beam from the spatial filter is expanded using a plano-concave lens and then collimated using a plano-convex lens (make: Holmarc Opto-Mechatronics Ltd.) before it illuminates the droplet field of view. The diameter of the collimated beam is 50 mm. The interference patterns created due to the droplet’s disintegration are recorded using the high-speed camera 2 with a resolution of $2048\times 1600$ pixels at 1800 fps, at an exposure duration of 1 $\mu$s, and a spatial resolution of 19.27 $\mu$m/pixel. The high-speed camera 1 (used for the shadowgraphy) and high-speed camera 2 (employed for the digital in-line holography) are synchronized using a digital delay generator (model: 610036, Make: TSI, USA). While presenting the results of droplet size distributions, the error bars display the standard deviation of three repetitions for each set of parameters.

As discussed in the introduction, Kirar et al. (2022) employed shadowgraphy to illustrate different breakup modes for an ethanol droplet of diameter $d_{0}=2.7\pm 0.07$ mm and discovered a new breakup mode (retracting bag breakup mode) for a low value of the Swirl number $(Sw)$. To the best of our knowledge, the study by Kirar et al. (2022) is the first to explore the phenomenon of droplet breakup in a swirling airstream. In contrast to the present study, which focuses on droplet size distribution, their aim was to demonstrate a new breakup mode in a swirling flow using Raleigh-Taylor instability. Thus, before discussing the size distribution of droplets obtained using the digital in-line holography technique, we begin the presentation of our results by demonstrating the temporal evolution of the morphologies of a freely falling water droplet of diameter $d_{0}=3.09\pm 0.07$ mm interacting with an airstream of different Swirl strengths. Figure 5 depicts the temporal evolution of the droplet morphology for the no-swirl ($Sw=0$), low swirl ($Sw=0.47$), and high swirl ($Sw=0.82$) conditions for a fixed value of the Weber number (${\it We}=12.1$). The dimensionless locations of the tip of the dispensing needle $(x_{d}/D_{o},y_{d}/D_{o},z_{d}/D_{o})$ are $(0.0,0.01,0.80)$, $(0.12,0.01,0.80)$ and $(0.12,0.01,0.80)$ for $Sw=0$, $0.47$ and $0.82$, respectively. In the no-swirl case, the droplet is dispensed at the axis of symmetry, $x_{d}/D_{o}=0$, but in swirl flow cases, the droplet is dispensed at $x_{d}/D_{o}=0.12$, so that the droplet interacts with the swirl airstream in oppose/cross-flow conditions. The location of the dispensing needle for the no-swirl and swirl cases is shown in figure S2 of the supplementary information. These locations are chosen in accordance with the regime map presented in Kirar et al. (2022) (see, figure S4 in the supplementary material). The dimensionless time, $\tau\left(\equiv t/t_{d}\right)$ is mentioned at the top of each panel in figure 5. Here $t$ is the physical time and $t_{d}=d_{0}\sqrt{\rho/\rho_{a}}/U$ is the characteristic deformation time. In our experiments, $\tau=0$ represents the onset of the breakup of the droplet. In the case of bag breakup mode, it is the instant the inflated bag ruptures, but in the case of vibrational breakup mode, it is the instant the droplet starts to break into smaller droplets of comparable size. Thus, the value of $\tau$ before the fragmentation instant is a negative number.

It can be seen in figure 5 that the droplet undergoes a vibrational breakup mode in the no-swirl case (first row), but it exhibits a retracting bag breakup mode for $Sw=0.47$ (middle row) and a normal bag breakup mode for $Sw=0.82$ (bottom row). In the no-swirl case, as the spherical droplet enters the aerodynamic field, it deforms into a thick disk ($\tau=-0.96$) as a result of asymmetrical pressure distribution at the front and rear of the droplet. At this stage, the surface tension force opposes further deformation and try to bring the droplet to a spherical shape. Consequently, the droplet oscillates with an increasing amplitude due to the competition between the aerodynamic force favouring the deformation, and the viscous and surface tension forces opposing it (at $\tau=-0.38$ and $-0.19$). As the oscillations reach an amplitude comparable to the drop’s radius, it fragments into smaller satellite droplets.

For the low swirl strength ($Sw=0.47$), the droplet quickly changes its shape from a sphere to a slightly tilted disk due to the aerodynamic force in the direction of vane inclination. As the swirl is not strong enough, only the bottom part of the disk enters the low-velocity zone (schematically shown in figure 6) generated by the wake of the swirler. Consequently, the rest of the disk, which is in the high-shear region, undergoes inflation to form a bag. In the high shear zone, a negative pressure gets created that retracts the bag sheet in a direction opposite to the direction of bag growth ($\tau=-0.19$). Subsequent retraction of the bag exhibits capillary instability and causes the bag to rupture ($\tau=0$). When the bag ruptures, the surface tension pulls the liquid film surrounding it to the rim, causing it to break into satellite droplets ($\tau=0.48$).

For the high swirl strength ($Sw=0.82$), the droplet rapidly changes its topology from a sphere to a disk shape due to the unequal pressure distribution around its periphery ($\tau=-2.02$ to $-0.96$). The lower portion of the disk tilts as it is exposed to an opposed flow condition. In this case, the swirling strength is sufficient to retain the disk in the shear zone. In other words, the high swirl flow does not allow the disk to enter into the low-velocity zone created by the wake of the swirler (see figure 6). As a result, the entire disk adjusts vertically, making the breakup process more efficient ($\tau=-0.38$ to $-0.19$). The middle of the disk elongates, generating a thin liquid sheet in the centre and a thick rim on the outside ($\tau=0$). Finally, high shear velocity causes the bag and rim to break in the shear zone ($\tau=0.48$).

Next, we investigate the size distribution of a droplet after fragmentation under different swirl air strengths. Figure 5 demonstrates that the droplet disintegrates into a few large satellite droplets in the no-swirl situation for ${\it We}=12.1$ due to the vibrational breakup. Therefore, in the following, we mainly focus on the size distribution in the low and high swirl strengths that encounter retracting bag and normal bag breaking events. Figure 7 depicts the temporal evolution of the droplet size distribution for $Sw=0.47$ (low swirl strength) at ${\it We}=12.1$. In figure 7, the first, second and third columns show the shadowgraphy images, holograms, and the resultant size distributions of droplets at different instants, respectively. At $\tau=0$, the retracted bag ruptures due to the negative pressure gradient created by the wake of the swirler. It is interesting to note that, although the bag is ruptured, it does not produce any satellite drops at this stage ($\tau=0$). This phenomenon is distinct from the normal bag breakup observed in a straight airstream at high Weber numbers (see, for instance, Guildenbecher et al. (2009); Kulkarni & Sojka (2014); Soni et al. (2020)). For $\tau>0$, it can be seen in figure 7 that the bag sheet continues to retract and finally impinges on the rim. This leads to the fragmentation of the upper portion of the rim due to the capillary Rayleigh–Plateau instability. The fragmentation of the rim continues which leads to the generation of smaller secondary droplets of diameter upto 300 $\mu$m, as indicated by the histogram at $\tau=0.37$ in figure 7. During this process, the nodes formed on the rim due to the Rayleigh-Taylor instability also disintegrates, and this process continues till $\tau=0.56$ thereby producing more droplets in broader size range, $300~{}\mu\textrm{m}<d<1200~{}\mu\textrm{m}$. Finally, as the bottom of the disk is entrapped in the wake zone, it creates a low aerodynamic force that leads to thicker nodes. These thicker nodes (at $\tau=0.75$) break eventually because of the continuous swirl airflow, which in turn, creates larger satellite droplets ($d>1200$ $\mu\textrm{m}$). Note that the histogram is truncated at $d\approx 1600$ $\mu\textrm{m}$, and the distribution of larger droplets is not shown here. A volume-weighted distribution presented later in figure 11 provides the entire size distribution of the droplets.

Figure 8 illustrates the temporal evolution of the droplet size distribution for $Sw=0.82$ (high swirl strength) for the same Weber number (${\it We}=12.1$). In this case, the entire liquid bulk exposes to the shear zone (see the droplet position in figure 6) and changes its morphology from a disk to a thick toroidal rim with an elongated bag ($\tau=0$). The bursting of this bag ($\tau=0.18$) generates very fine satellite droplets $(d<300$ $\mu\textrm{m})$. Subsequently, the remaining portion of the bag liquid sheet propagates back towards the rim due to unbalanced tension and is finally absorbed in the rim. At this stage ($\tau=0.38$), the number of tiny satellite droplet increases, and fragmentation of rim initiates. The breakup of the rim advances and results in generation of intermediate-sized droplets ($300~{}\mu\textrm{m}<d<900~{}\mu\textrm{m}$) at $\tau=0.48$. Finally, due to the Rayleigh-Taylor instability, the nodes on the rim disintegrate and produce bigger satellite droplets ($d>900~{}\mu\textrm{m}$) at $\tau=0.66$. At this stage, the coalescence of smaller droplets also contributes to the size distribution.

Figure 9(a) shows temporal variation of normalised number-mean diameter for different Swirl numbers at ${\it We}=12.1$. The error bar at each data point represents the standard deviation of three repetitions. As time progresses, the relative velocity between airflow and liquid bulk decreases, thus producing bigger droplets leading to an increase in the number-mean diameter of satellite droplets. As discussed earlier, while the low swirl case ($Sw=0.47$) produces larger size droplets due to the fragmentation of rim and nodes (bag-film breakup does not contribute to the size distribution), in the high swirl case ($Sw=0.82$), the breakup of the bag, rim, and nodes contribute to the number-mean diameter and thereby producing smaller satellite droplets.

The Sauter mean diameter (commonly termed as SMD or $d_{32}$) is defined as surface area moment mean, which provides the mean size of a given droplet distribution. Thus, the reciprocal of the Sauter mean diameter is a direct measure of the surface area per unit volume of the satellite droplets, which is given by

In the present study, we use the ratio between the mean volume and mean surface area of the satellite droplets to calculate $d_{32}$.

To get a better insight into the size distribution of satellite droplets, we plot the temporal variation of normalised Sauter mean diameter ($d_{32}/d_{0}$) for different swirl numbers in figure 9(b). It can be seen that the Sauter mean diameter increases with time for both the low and high swirl strengths. However, in comparison to low swirl strength, high swirl strength produces secondary droplets with a smaller value of $d_{32}$. Therefore, for the same aerodynamic force, the high swirl strength generates more surface area (and hence more surface energy) than the low swirl strength after fragmentation. At a later time ($\tau=0.5$), the sudden jumps in the variation of $d_{32}/d_{0}$ is associated with the increase in the number of larger droplets.

The shape and size distribution of raindrops is one of the important factors in rainfall modelling, which is influenced by several microphysical processes, such as fragmentation, coalescence and phase change. In the present study, we examine the size distribution of satellite droplets produced by the interaction of a freely falling water droplet with a swirl airstream of different strengths using shadowgraphy and digital in-line holography techniques. The Weber number, which measures the aerodynamic force exerted on a droplet by the airstream, is considered to be low enough that the droplet undergoes vibrational breakup, producing only a few satellite droplets of comparable sizes in the no-swirl airstream. As we increase the swirl strength, the water droplet experiences the retracting bag breakup and normal breakup phenomena for $Sw=0.47$ (low swirl strength) and $Sw=0.82$ (high swirl strength), respectively. While Kirar et al. (2022) observed similar morphological changes for an ethanol droplet, they did not examine the size distribution of the satellite droplets following fragmentation, which is the subject of the current investigation. The digital in-line holography employed to obtain the size distributions of satellite droplets is a deep-learning-based image processing method that has recently emerged as a powerful tool for capturing three-dimensional information with high spatial resolution. Therefore, it is a more appropriate approach to be employed in the current study for the analysis of the size distribution of satellite droplets produced by fragmentation of a water droplet in a swirling airstream.

We observe that the number mean diameter of the satellite droplets increases over time due to the decrease in the relative velocity between the air and liquid bulk during the fragmentation process. In the high swirl case ($Sw=0.82$), the disintegration of the bag, rim, and nodes leads to smaller satellite droplets, whereas in the low swirl scenario ($Sw=0.47$), the fragmentation of the rim and nodes leads to larger satellite droplets. The bag rupture does not contribute to the size distribution for the low swirl case. The temporal variations of the Sauter mean diameter reveals that, for a given aerodynamic force, a high swirl strength creates more surface area (and thus more surface energy) than a low swirl strength.

A statistical theoretical analysis, similar to that of Villermaux & Bossa (2009) originally developed for straight airstream without swirl, is also carried out to determine the size distribution of droplets for various swirl strengths. Despite not accounting for all breakup modes, the statistical model predicts the experimental result for small droplets but differs from the results for large droplets when the swirl strength is low. In the case of low swirl number, additional mechanisms such as entrapment of disk in the wake zone of the swirler and absence of bag fragmentation results in larger droplets.

Furthermore, we have analytically evaluated the combined multi-modal distribution that accounts for nodes, rim, and bag breakup modes of a droplet in a swirl airflow. In sharp contrast to the earlier studies on the fragmentation of a droplet in a straight airstream without a swirl that observed a monomodal size distribution, we observed bimodal and multi-model distributions for low and high swirl strengths, respectively. We observe two distinct modes for the low swirl scenario $(Sw=0.47)$ due to the rim breakup driven by the capillary instability, which results in smaller droplets with a symmetric distribution, and the subsequent node breakup, which generates an asymmetrical size distribution. For $Sw=0.47$, we found that the deviations between experiments and the theoretical model predictions in the characteristic size of the droplets owing to the rim and node breakups are about 30% and 16%, respectively. In contrast, in the high swirl scenario $(Sw=0.82$), we observe three peaks in the size distribution due to the breakup of the bag, rim and nodes. Although the theoretically evaluated characteristic sizes are in reasonable agreement with the experimental results, the volume-weighted contributions of the bag and rim are over-predicted, whereas the contribution to node breakup is under-predicted. Quantitatively, for $Sw=0.82$, the deviations between experiments and the theoretical model predictions in the characteristic size of the droplets due to the bag, rim and node fragmentation are about 25%, 4% and 4%, respectively. Nevertheless, the present study is a first attempt to estimate the size distribution of satellite droplets under a swirl airstream, which is observed in many natural and industrial applications.