3.1. Shock–droplet interaction mechanism

The interaction of an incident shock wave with a droplet surface leads to different wave structures like reflected wave, transmitted wave, Mach-stem and diffracted wave, which affects the flow conditions around the droplet periphery. Although these features are physically present, some features like transmitted wave, external and internal flow field of a droplet are difficult to perceive using currently available experimental techniques. The transmitted wave cannot be visualised due to the limitation of shadowgraphy and/or schlieren imaging techniques and due to the spherical shape of the droplet which acts as a lens during shadowgraphy/schlieren imaging. The flow dynamics could not be obtained experimentally due to the requirement of a high-resolution and high-frame-rate particle image velocimetry system (minimum requirement of $20\,000$–$40\,000$ f.p.s.), which necessitates the usage of a high-power pulse-burst laser system (Beresh et al. Reference Beresh2015). In addition, there are several challenges associated with seeding particles in supersonic and shock tunnel flows (Havermann et al. Reference Havermann, Haertig, Rey and George2008). However, due to advancements in numerical methods and existing computational power, it is now possible to observe these features using numerical techniques as reported recently by several researchers (Chen Reference Chen2008; Sembian et al. Reference Sembian, Liverts, Tillmark and Apazidis2016; Sridharan et al. Reference Sridharan, Jackson, Zhang, Balachandar and Thakur2016; Guan et al. Reference Guan, Liu, Wen and Shen2018; Das & Udaykumar Reference Das and Udaykumar2020; Poplavski et al. Reference Poplavski, Minakov, Shebeleva and Boyko2020). Therefore, the discussion in the current subsection is based on the experimental observations of the present work and some flow features established in the above numerical studies. Figure 4 schematically shows the shock dynamics (figure 4a–g) and flow dynamics (figure 4h–o) realised during shock–droplet interaction. Figure 5 presents the schlieren images recorded at 0.42 M f.p.s. with a pixel resolution of $64\times 96$ pixels and corresponds to the features shown in figure 4.

Figure 4. Shock–droplet interaction mechanism. (a–g) Shock wave dynamics during different time instances; and (h–o) external (red lines) and internal (blue lines) flow streamlines for a droplet at different timeinstances.

Figure 5. Schlieren images recorded at 0.42 M f.p.s. with $64\times 96$ pixels resolution showing shock–droplet interaction mechanism. (a) Falling droplet approaches to the incident shock wave; (b) formation of reflected wave; (c) Mach reflection; (d) focusing of Mach waves at the leeward side of the droplet; (e) generation of primary vortical structure near leeward side of droplet; ( f) droplet deformation and vortex stripping; and (g) stripping of daughter droplets from the primary droplet periphery.

The incident shock interaction on the droplet surface creates a reflected wave, a transmitted wave and a diffracted wave. Depending on the acoustic impedance $(Z =\rho c)$ of the medium on either side of the interface, the reflected wave can be a shock wave or an expansion wave (Henderson Reference Henderson1989). It is interesting to note that if the medium through which the wave has travelled before reflection has a lower impedance than the medium on the other side of the reflecting surface, the reflected wave will be a shock wave and vice versa. For the current case, the reflected wave will be a shock wave $(Z_{water}> Z_{air})$ (figures 4b,i and 5b). The reference time $(t)$ is measured from the first instance of the shock interacting with the droplet surface. The reference time is non-dimensionalised using the inertial time scale $(\tau = D_i/V_i \sqrt {\rho _l/\rho _a})$ as $t^*=t/\tau$. The planar incident wave will undergo regular reflection until the incidence angle $(\alpha _i)$ is greater than the critical detachment angle $(\alpha _c)$ (Ben-Dor Reference Ben-Dor2007). If $\alpha _i < \alpha _c$, a regular reflection is not possible and transition to Mach reflection occurs (figures 4c,j and 5c). A triple point is formed at the intersection of the incident wave, reflected wave and Mach stem. A slip surface is also formed at the intersection of the Mach wave and droplet surface. A uniform stream of induced airflow follows the incoming incident shock wave (see figure 4h). The formed reflected shock wave deflects the uniform airflow as it pass through it (figure 4i–l). The induced flow remains subsonic in all the cases studied in this work; therefore, it gets deflected during its motion around the droplet surface without creating any other shock structures. The liquid motion inside the droplet and a high-pressure region near the droplet's front stagnation point prompted windward surface deformation of the droplet.

If the incident wave is a shock wave, the transmitted wave will always be a shock wave (Henderson Reference Henderson1989; Sembian et al. Reference Sembian, Liverts, Tillmark and Apazidis2016; Guan et al. Reference Guan, Liu, Wen and Shen2018; Das & Udaykumar Reference Das and Udaykumar2020) (figure 4b). The transmitted wave moves inside the droplet, resulting in a high-pressure region behind it and an induced fluid flow towards the leeward side of the droplet (figure 4i) (Guan et al. Reference Guan, Liu, Wen and Shen2018). As the pressure waves can travel faster in water in comparison with air, the transmitted wave gets detached from the incident wave and reaches the droplet's leeward side at an earlier instant (figure 4c,j) (Sembian et al. Reference Sembian, Liverts, Tillmark and Apazidis2016; Das & Udaykumar Reference Das and Udaykumar2020). On reaching the air–water interface at the leeward side, the transmitted wave reflects as an expansion wave ($Z_{air} < Z_{water}$) (Henderson Reference Henderson1989) which focuses at a point inside droplet due to the concave profile of reflecting surface (figure 4d). Following focusing, the expansion wave propagates towards the windward side, where it is again reflected from the interface. This reflection of the transmitted wave from the windward and leeward side of the droplet continues with some energy loss after every collision until it completely decays.

Diffraction of the Mach stem occurs after passing the equator point of the droplet. This contributes towards the lagging of the Mach wave near its contact point with the droplet surface (figure 4e,k). The Mach wave is formed axisymmetrically around the droplet surface and focuses at a point near the rear stagnation point (figures 4 f,l and 5d) which strengthens the shock and creates a local high-pressure region at that location. The internal flow direction remains unidirectional towards the leeward side of the droplet until the shock focuses at the droplet's rear stagnation point (figure 4i–l). A high-pressure region at the rear stagnation point deforms the droplet surface locally, triggering an upstream flow inside the droplet. The two counteracting flows, i.e. downstream flow from front stagnation point and upstream flow from rear stagnation point interact with each other at the local saddle point inside the droplet, which divert the two flows towards the droplet equator (figure 4m,n). During the development of external airflow around the droplet periphery, a toroidal vortex (primary vortex) is formed near the flow separation point (figure 4m), similar to the external flow around a solid sphere of similar Reynolds number. In later times, the primary vortex moves downstream and forms a recirculation flow region and effectuate a jet flow at the rear stagnation point. Two counter-rotating secondary vortices V2 and V3, that interact with the droplet surface are also formed between the flow separation point and the recirculating flow region. Subsequent to focusing at a point, Mach waves propagate on the leeward surface, thereby increasing the influence of the high-pressure region (figures 4m and 5d,e). The combined effect of recirculated jet flow and spreading of the high-pressure region flattens the leeward side of the droplet and deforms it into a cupcake shape (figures 4n and 5 f). The observed wave dynamics of droplet interaction with incident shock wave is in accordance with the previous work on shock–water-column interaction (Igra & Takayama Reference Igra and Takayama2003; Sembian et al. Reference Sembian, Liverts, Tillmark and Apazidis2016) and various two-dimensional numerical simulations using spherical dispersions (Mehta et al. Reference Mehta, Jackson, Zhang and Balachandar2016; Sridharan et al. Reference Sridharan, Jackson, Zhang, Balachandar and Thakur2016; Guan et al. Reference Guan, Liu, Wen and Shen2018; Das & Udaykumar Reference Das and Udaykumar2020), indicating the axisymmetric nature of the stage I interaction. The continuous deformation of the droplet increases the projected area for aerodynamic drag, which further enhances the flattening of the liquid droplet (figures 4o and 5g) and leads to droplet breakup in later time instants. Figure 6 represents the various droplet breakup morphologies observed during its interaction with a shock wave of different strengths. Panels (a,b) correspond to the bigger droplet size $(D_i \sim 2.5\ \textrm {mm})$ and panels (c,d) correspond to smaller droplet size $(D_i \sim 0.5\ \textrm {mm})$. The left-hand side images correspond to the low Weber number while the right-hand side corresponds to the high Weber number. Figure 6 essentially explains that two types of breakup modes (i.e. SIE and RTP) are observed based on Weber number and droplet sizes, and the interaction becomes more vigorous as the Weber number increases. The details of these mechanisms are provided in following subsections.

Figure 6. Droplet breakup morphologies for different shock Mach numbers and droplet sizes. (a) The SIE droplet breakup at $M_s = 1.12$, $We = 219$, $Re = 1.3*10^4$, $D_i = 2.5$. (b) The SIE droplet breakup at $M_s = 1.3$, $We = 1.6*10^3$, $Re = 4*10^4$, $D_i = 2.9$. (c) The RTP droplet breakup at $M_s = 1.12$, $We = 44$, $Re = 2.5*10^3$, $D_i = 0.5$. (d) The RTP droplet breakup at $M_s = 1.45$, $We = 795$, $Re = 1.2*10^4$, $D_i = 0.5$.

3.2. Global observation of shock–droplet interaction mechanism

Figure 7 shows the global observation of the shock–droplet interaction mechanism. The complete interaction dynamics can be divided into two stages: initial shock interaction (stage I) and droplet breakup dynamics (stage II). As discussed earlier, the initial shock interaction corresponds to the formation of different wave structures and a local change of ambient conditions around the droplet. These changes result in droplet deformation; but the droplet retains its coherent structure during the interaction stage I. The droplet deformation begins due to high-pressure regions at the front and rear stagnation points (due to external airflow near the front stagnation point, recirculating flow and shock focusing near the rear stagnation point), which overcomes the restoring surface tension and viscous forces acting on the droplet. A low-pressure region is formed near the equator of the droplet where the external flow velocity is comparatively higher, as in the case of external airflow over a cylinder/sphere of similar Reynolds number. Two high-pressure regions at the droplet's front and rear stagnation points and a low-pressure region near the equator direct the internal flow of liquid towards the droplet equator, resulting in the deformation of the spherical droplet into a cupcake shape. This pressure difference is further enhanced due to an increase in aerodynamic drag with deforming droplet shape. The shock–droplet interaction mechanism is universal up to this point for all cases of shock–droplet interaction. Beyond this (stage II interaction), depending on the relative dominance of pressure force or the shear force acting on the droplet surface, two modes of droplet breakup, i.e. RTP breakup (figure 7b) or SIE (figure 7c) breakup are observed.

Figure 7. Global observation of shock–droplet interaction. (a) Stage I corresponding to the initial shock interaction; (b) breakup dynamics (stage II) of a droplet ($D_i \sim 0.5$ mm) undergoing breakup through RTP mode; and (c) SIE mode of droplet breakup for $D_i \sim 2.5$ mm.

The RTP-based droplet breakup is characterised by complete deformation of the droplet into a flattened disc followed by the formation of single/multiple bags, which grow over time due to incoming airflow and undergoes rupture. The bag rupture is followed by the breakup of the central undeformed region and outer toroidal rim joining the bags (Jackiw & Ashgriz Reference Jackiw and Ashgriz2021). The shear-induced droplet breakup mode is characterised by the formation of KH-based surface waves (Theofanous et al. Reference Theofanous, Mitkin, Ng, Chang, Deng and Sushchikh2012; Guan et al. Reference Guan, Liu, Wen and Shen2018; Dorschner et al. Reference Dorschner, Biasiori-Poulanges, Schmidmayer, El-Rabii and Colonius2020) on the windward surface of the droplet. These waves undergo elongation and rupture, or the formed waves travel along the windward surface and merge to form a liquid sheet near the equator of the droplet. The external aerodynamic forces acting on the formed sheet lead to stretching in the cross-stream direction followed by rupture into fine mist (figure 7c). A detailed discussion on each droplet breakup mode is provided in the following subsections.

3.3. Shear-induced droplet breakup

The initial stages of shock–droplet interaction involving wave dynamics and droplet deformation were discussed in § 3.1. The present subsection strides towards understanding the breakup dynamics of a droplet through the SIE mode. Figure 8 (also supplementary movie 2) shows the time sequence images (from left to right) of the breakup dynamics of a $\sim$2.9 mm diameter droplet for different values of shock Mach numbers. As discussed earlier, the initial wave interaction and droplet deformation stages are similar for all experimental cases. The deformed droplet attains a cupcake shape, after which it undergoes a stripping type of droplet breakup. The stripping of daughter droplets near the equator of the primary droplet is triggered by the formation of KH waves (Theofanous et al. Reference Theofanous, Mitkin, Ng, Chang, Deng and Sushchikh2012; Liu et al. Reference Liu, Wang, Sun, Wang and Wang2018) near the front stagnation point of the droplet (see figure 8, third column). These waves are formed midway between the front stagnation point and droplet equator, along the droplet periphery, which is the region of highest shear (Liu et al. Reference Liu, Wang, Sun, Wang and Wang2018). The validation of formed surface waves with theoretical predictions is shown later. For a better understanding of the breakup dynamics, a high-resolution (8 $\mathrm {\mu }$m pixel$^{-1}$) and high-speed (at $50\,000$ f.p.s.) imaging of interaction phenomena was done in a pendant drop mode (see figure 9). This configuration will affect the breakup dynamics at the rear stagnation point; however, no distinguishable difference between pendant mode and free-falling configuration could be identified at the windward side of the droplet (see supplementary movie 3). This methodology for obtaining high-resolution and high-speed imaging was also recently employed by Jackiw & Ashgriz (Reference Jackiw and Ashgriz2021). Figure 9(a,b) corresponds to the shock Mach number of 1.12 and 1.47, respectively. Once formed, KH waves grow in amplitude (figure 9a $t^* = 0.37\text{--}0.4$ and figure 9b $t^* = 0.1\text{--}0.29$) and subsequently get deflected towards the flow direction (figure 9a $t^* = 0.47$ and figure 9b $t^* = 0.38$) when entrained by the external air flow (see figure 8, fourth column). The flow entrainment causes the surface waves to move towards the downstream direction along the windward surface and after reaching the flow separation point (at the droplet equator) the liquid carried with these waves gets accumulated as a thin sheet at this location (figure 9a $t^* = 0.47$ and figure 9b $t^* = 0.38$). The thin sheet undergoes rupture due to nucleation of multiple holes (shown later through top-view imaging) which expands in size in a temporal fashion leading to sheet rupture into numerous cylindrical ligaments (figure 9a $t^* = 0.52$ and figure 9b $t^* = 0.48$). The formed cylindrical ligaments further undergo secondary atomisation into smaller droplets through Rayleigh–Plateau modes of instability.

Figure 8. Droplet breakup dynamics at different Weber numbers for a droplet diameter ($D_i$) of $\sim 2.9$ mm. Time sequence images of droplet breakup from the instant of wire explosion to droplet breakup (from left to right). (a–f) The magnitude of non-dimensional numbers ($M_s , We, Re$) increases from (a) to ( f). See supplementary movie file 2 for evolution dynamics of cases in (a,c and e).

Figure 9. Zoomed-in view of shear-stripping breakup. (a) Time-sequence images of droplet breakup at $M_s =1.12$ and $D_i = 2.02$ mm; and (b) time-sequence images of droplet breakup at $M_s =1.47$ and $D_i = 2.24$ mm. For details see supplementary movie file 3.

In addition to the accumulation of droplet fluid near the equator due to KH waves (leading to sheet formation), the effect droplet deformation, which causes an internal flow directed towards the equator region (discussed in § 3.1) also augments the sheet formation at the equator (see figure 4m–o). Here, the fluid transport towards the equator through the droplet deformation is greater than micron-sized KH waves. Therefore, the sheet formation with the predominant KH wave mechanism will be thinner than the droplet deformation mechanism. Subsequently, the droplet sizes stripping from the rupture of the sheet will also be smaller for the KH-waves-based liquid accumulation mechanism. Therefore, during the early stages of droplet breakup (see figure 8, fourth and fifth columns), the extent of droplet deformation is less, and KH waves are the only dominant mechanism for sheet formation leading to a fine mist of daughter droplets.

Figures 8 and 9 and theoretical analysis shown later (§ 3.4) shows that the number of KH waves formed on the droplet surface increases with increase in Weber number. Moreover, the local incubation location of KH waves shifts towards the front stagnation point as the Weber number increases (see figure 9). The formed surface waves grow in amplitude and move towards the equator of the droplet, where they get accumulated at the stripping sheet and compensate for the liquid mass loss due to sheet rupture. The thin sheet formed due to KH waves experiences two driving forces leading to its motion: first, the entrainment force due to the shear acting on its surface by external airflow; and second, additional pull due to the induced internal motion by the movement of KH wave formed ahead of it (towards the equator). Therefore, the translational velocity of the wave formed near the front stagnation point is higher than the one formed near the equator. This results in merging of two surface waves as they translate towards the droplet equator, resulting in an increase of combined wave sheet thickness. The merging of surface waves can be seen in figure 9(b) ($t^* = 0.19\text{--}0.48$) and is shown more clearly in next subsection. The merged KH waves on reaching the droplet equator result in the formation of a thicker sheet; thereby the size of fragmenting daughter droplet increases (see figure 8, fifth column). The thickness of the stripping sheet is further augmented by the deformation of the primary droplet, which again increases the size of fragmenting droplets. Therefore, the size of daughter droplets monotonically increases from the beginning of breakup to the end during the SIE breakup regime (see figure 8). The breakup mechanism involving sheet formation at droplet equator due to the combined effect of accumulation of KH waves and droplet deformation occurs recurrently (discussed later) until the complete droplet disintegrates into daughter droplets or aerodynamic forces acting on the remaining sheet of the deformed droplet are unable to overcome the surface tension of the sheet. The stretched sheet in the latter case retracts back to form a single droplet.

More surface waves are formed at the higher Weber numbers (figure 8). This implies that the amount of liquid transported through these waves will also be higher. In addition to this, the extent of droplet deformation also decreases with increasing Weber number. Therefore, higher wavenumbers and lower droplet deformations result in the domination of KH-waves-based liquid transport mechanism at the stripping sheet, which therefore explains the formation of smaller droplet sizes with increasing shock Mach number (see figures 8 and 9). In contrast to this, at lower Mach numbers, the number of KH waves formed are lower, and the extent of droplet deformation is higher; therefore, the formed droplet sizes are bigger in such cases. The above explanation of SIE breakup based on the competing effect of KH waves and droplet-deformation-based liquid transmission towards the stripping sheet has been reported for the first time in this work. This explains both the temporal increase in the size of daughter droplets and reduction in daughter droplet sizes with increasing Weber number, observed during SIE-based breakup.

In addition to the surface waves formed at the windward surface of the droplet, a flattened disc and a lip are also formed on the leeward side of the droplet due to the internal flow induced inside the droplet by shock focusing and jet flow impinging on the rear stagnation point of the droplet (discussed in § 3.1). The formed lip interacts with the secondary vortices formed after external flow stabilisation around droplet (figure 4n). These formed vortices stretch and engulf the droplet lip sheet towards their core, as explained by Sharma, Singh & Basu (Reference Sharma, Singh and Basu2021), leading to its rupture. This mechanism also contributes to mist formation during SIE-based droplet breakup; however, it lasts for a short time interval and has an insignificant contribution in the total number of daughter droplet formations (figure 8).

3.4. Kelvin–Helmholtz-based surface instabilities

The previous subsection discusses the droplet breakup mechanism based on KH surface waves. Here, we present the theoretical analysis used to predict the wavelength of the formed surface waves. Existing models of KH instabilities based on zero and non-zero vorticity thickness (Marmottant & Villermaux Reference Marmottant and Villermaux2004; Jalaal & Mehravaran Reference Jalaal and Mehravaran2014) are used for the comparison of theoretical and experimental results. Figure 10 (also supplementary movie file 4) shows a zoomed-in view of the droplet windward surface which display the time sequence images of KH waves formation for Weber number values of $We = 2095$ (figure 10a) and $We = 91\,667$ (figure 10b), respectively. As can be seen from figure 10, instability waves are formed much before the droplet response time (i.e. the beginning of droplet deformation). As discussed earlier, after formation, these waves move towards the droplet equator and sometimes merge to form a combined wave, thereby increasing the thickness of the liquid sheet carried along with them. Figure 10 shows at $t^* =0.13\text{--}0.19$ and $t^* =0.46\text{--}0.52$, some KH-wave-induced liquid sheets are unable to sustain further elongation and undergo rupture before accumulating at the droplet equator. This is an additional mechanism responsible for mist formation during SIE-based droplet breakup.

Figure 10. Kelvin–Helmholtz instability near front stagnation point of droplet for $D_i \sim 2.5$ mm. Time sequence images of KH waves formation is shown for (a) $M_s \sim 1.16$ and (b) $M_s \sim 1.45$ from left to right. For details see supplementary movie file 4. Panels (c) and (d) show the schematic diagram for zero and non-zero vorticity thickness models for KH waves, respectively.

Considering an inviscid and incompressible airflow on the droplet surface, no gravity effects and negligible vorticity layer thickness (see figure 10c), one can apply the first-order linear perturbation analysis to obtain the following dispersion equation (Marmottant & Villermaux Reference Marmottant and Villermaux2004) for the instability growth rate $(\omega _{KH} )$:

(3.1)\begin{equation} \omega_{KH} = k_{KH} \frac{\rho_l V_l + \rho_a V_i}{\rho_l +\rho_a} \pm i \frac{k_{KH}}{\rho_a + \rho_l} \sqrt{ \rho_l \rho_a (V_i-V_l)^2 - (\rho_a + \rho_l)\sigma k_{KH}}. \end{equation}

Here, $k_{KH}$ is the wavenumber of the instability, $V_l$ is the velocity of the droplet fluid and subscript $KH$ refers to KH-based instability. In (3.1), surface tension will stabilise all shear instabilities above critical wavenumber $(k_{KH_c})$, where

(3.2)\begin{equation} k_{KH_c} > \frac{\rho_a \rho_l}{\rho_a+\rho_l}\frac{(V_i-V_l)^2}{\sigma}. \end{equation}

Now, considering $V_i \gg V_l$ and $\rho _l\gg \rho _a$, which is a valid assumption in our case, one can obtain the wavenumber ($k_{KH_{max}}$) corresponding to maximum growth rate ($\omega _{KH_{max}}$) as

(3.3a)$$\begin{gather} k_{KH_{max}} = \frac{2}{3}\frac{\rho_l {V_i}^2}{\sigma} = \frac{2}{3}\frac{We}{D_i}, \end{gather}$$

(3.3b)$$\begin{gather}\lambda_{KH} = \frac{2{\rm \pi}}{k_{KH_{max}}}= \frac{3{\rm \pi}}{We}{D_i}. \end{gather}$$

Figure 11 shows the comparison of experimental and theoretical prediction of KH-wave wavelength corresponding to maximum growth rate with different Weber numbers. As predicted by (3.3a), the wavelength of maximum growth rate decreases with an increase in Weber number. Although the zero-vorticity thickness model predicts the correct trend of experimental data, the absolute values were found to be $O(10\text{--}100)$ less than the experimental observation (see table 1, third column). Similar under-prediction of wavelength while using the assumption of zero-vorticity layer thickness was also reported by other researchers (Marmottant & Villermaux Reference Marmottant and Villermaux2004; Kim et al. Reference Kim, Desjardins, Herrmann and Moin2006; Jalaal & Mehravaran Reference Jalaal and Mehravaran2014). The impractical assumption of zero vorticity layer thickness between the external airflow and internal liquid flow led to this under-prediction of the theoretical model. Therefore, a modified model with a finite thickness of vorticity layer and linear variation of velocity profile (see figure 10d) based on the work of Lord Rayleigh (Reference Rayleigh1879) for two different phases in the vorticity layer were obtained by Villermaux (Reference Villermaux1998) as

(3.4)\begin{equation} \lambda_{KH} = \frac{2{\rm \pi}}{0.8}F\left(\frac{\rho_l}{\rho_a}\right)\delta_a, \end{equation}

where, $\delta _a$ is the boundary layer thickness of air phase (see figure 10d) and

(3.5)\begin{equation} F\left(\frac{\rho_l}{\rho_a}\right) = \frac{5}{6}-\frac{1}{6(\rho_l/\rho_a)}+ \frac{\sqrt{5+13(\rho_l/\rho_a)-37(\rho_l/\rho_a)^2+27 (\rho_l/\rho_a)^3}}{6\sqrt{2}(\rho_l/\rho_a)}. \end{equation}

The value of $\delta _a$ for the present work is obtained using boundary layer analysis, where

(3.6)\begin{equation} \delta_a \sim \frac{L_c}{\sqrt{Re_{L_c}}} = C_1\frac{L_c}{\sqrt{Re_{L_c}}}. \end{equation}

Here, $L_c$ is the characteristic length scale, which in the present case is the initial drop radius ($R_{i}$) and $C_1$ is constant replacing scaling analysis and its value is chosen as a fitting parameter for figure 11. For the limiting case of $(\rho _l/\rho _a)\gg 1$, (3.4) can be written as (Villermaux Reference Villermaux1998)

(3.7)\begin{equation} \lambda_{KH} = \frac{2{\rm \pi}}{1.5}\sqrt{\frac{\rho_l}{\rho_a}}\delta_a, \end{equation}

and the corresponding dispersion equation is obtained as (Marmottant & Villermaux Reference Marmottant and Villermaux2004)

(3.8)\begin{equation} \textrm{e}^{{-}2\kappa}=[1-\{2\varOmega^*+\kappa\}]\frac{1+0.5(\rho_l/\rho_a +1)(2\varOmega^*-\kappa)}{1+0.5(\rho_l/\rho_a -1)(2\varOmega^*-\kappa)}. \end{equation}

Here, $\kappa = k_{KH}\delta _a$ and $\varOmega ^* ={\omega _{KH}\delta _a}/{(V_i-V_l)}-2\kappa ({(V_i+V_l)}/{(V_i-V_l)})$. The ratio of predicted and experimental wavelength based on (3.7) and (3.4) is referred as ‘non-zero thickness ratio 1’ and ‘non-zero thickness ratio 2’ in table 1 and figure 11. The decreasing trend of KH wavelength ($\lambda _{KH}$) is well predicted by two models and in comparison with zero thickness model the predicted results based on non-zero vorticity thickness are of same order of magnitude of experimental data. The observed discrepancy in the absolute value of wavelength was also reported by Marmottant & Villermaux (Reference Marmottant and Villermaux2004), Kim et al. (Reference Kim, Desjardins, Herrmann and Moin2006) and Jalaal & Mehravaran (Reference Jalaal and Mehravaran2014), and is attributed to the linear velocity profile assumption and error in estimating $C_1$ in (3.6). Based on the experimental observations of the present work, no significant trend of KH wavelength variation along the droplet periphery could be identified, as the measured wavelengths were found to be distributed randomly. Qualitatively KH waves are generally formed at an azimuthal location midway between the front stagnation point and droplet equator (i.e. region of highest shear Liu et al. Reference Liu, Wang, Sun, Wang and Wang2018). The region near the front stagnation point remains smooth/unaffected. However, as the Weber number increases, the starting location of KH instabilities shift towards the front stagnation point, as can be seen from figure 9.

Figure 11. Wavelength variation of KH waves with shock Mach number ($M_s$) for droplet size ($D_i$) of ${\sim }2.5$ mm.

Table 1. Kelvin–Helmholtz instability comparison with theoretical models.

3.5. Top-view visualisation and recurrent breakup of liquid sheet

The high-speed shock–droplet interaction is essentially a three-dimensional phenomenon, which necessitates top-view imaging to acquire relatively more information. The time-sequence images showing the top view of droplet breakup dynamics is shown in figures 12(a) and 12(b) for Weber numbers values of 474 and 4133, respectively. As discussed earlier, the breakup dynamics are similar for both cases. However, the fragmentation of the primary droplet is more intense at a higher Weber number, leading to much finer daughter droplet sizes after the breakup. The ligaments at the droplet periphery are formed periodically along the azimuthal direction during the first breakup (figure 12a at $t^* = 0.65$ and figure 12b at $t^* = 0.48$). The periodic formation of fragmenting ligaments can be attributed to azimuthal modulation of the RT waves on the existing KH waves formed due to shear effects. In continuation of the work of Villermaux (Reference Villermaux1998) and Marmottant & Villermaux (Reference Marmottant and Villermaux2004) involving instability analysis of a liquid jet exposed to a coaxial airflow, Jalaal & Mehravaran (Reference Jalaal and Mehravaran2014) carried a similar analysis on a droplet exposed to external uniform flow and explained RTI-based azimuthal modulations on axisymmetric KH waves (formed due to shear) due to the acceleration of KH waves towards the droplet equator. The modulation of RT waves on the KH waves is shown in figure 13. Here figure 13(a) shows the cross-sectional location on which RT waves are shown in figure 13(b). The crest of RT waves get entrained to external airflow and result in ligament formation. The measurement of the RT wavelength was performed by measuring the distance between the stripping ligaments (as shown in figure 13c). An assumption is made that the formed RT waves stretch to form ligaments whose breakup can be seen through top-view imaging, as shown in the third column image of figures 12(a) and 12(b). The measurements are provided for a possible estimation in terms of order of magnitude basis. The exact values of RT wavelength could not be identified based on these images as randomness arises in the measurements due to the uncertain manner of ligament breakup under the influence of induced airflow. Therefore, values provided in table 2 must only be used for order-of-magnitude comparison. Similarly, no trend in the variation of these wavelengths could be explained with a change in shock Mach numbers using table 2.

Figure 12. Time sequence images showing the top view of shock–droplet interaction for droplet size ($D_i$) of $\sim 2.5$ mm at (a) $M_s \sim 1.3$ and (b) $M_s \sim 1.46$.

Figure 13. Azimuthal modulation of RT waves on KH waves. (a) Kelvin–Helmholtz waves formed near the front stagnation point of the droplet. Dashed lines show the cross-section of the KH wave on which RT waves are displayed. (b) Schematic diagram showing RT waves formed on droplet surface and their wavelength variation ($\lambda _T$) with Weber number ($We$). (c) Experimental image showing RT waves leading to droplet stripping during initial stages of droplet breakup.

Table 2. Rayleigh–Taylor modulation over KH waves. A comparison of experimental and theoretical wavelengths.

A comparison of the experimentally observed RT-waves wavelength is made with an existing model based on Marmottant & Villermaux (Reference Marmottant and Villermaux2004) analysis as

(3.9)\begin{equation} \lambda_{RT} \cong 2.8\delta_a {We_{\delta_{a}}}^{{-}1/3}\left(\frac{\rho_a}{\rho_l}\right)^{{-}1/3}. \end{equation}

Here, $We_{\delta _{a}}$ is a Weber number based on the vorticity layer thickness. The comparison of the theoretical prediction with experimental data is shown in table 2. Although the predicted results correlate with experimental data with the same order of magnitude, the absolute magnitudes of predicted wavelengths differ considerably (up to three times). The observed discrepancy could be due to the linear velocity profile assumption between the vorticity thickness layer and constant $C_1$, as discussed earlier, and similar discrepancies were also obtained in the numerical simulations of Jalaal & Mehravaran (Reference Jalaal and Mehravaran2014). The RT waves measurements were only done for the first instance of droplet breakup, corresponding to the KH waves formed nearest to the droplet equator. This limitation was due to merging KH waves and additional ligaments formation due to sheet rupture during later breakup instances.

Jarrahbashi & Sirignano (Reference Jarrahbashi and Sirignano2014) had studied through vorticity dynamics the instability mechanisms in round liquid jets. The azimuthal instability on the axisymmetric KH waves can arise through the baroclinic effect (RTI, as shown above and in Marmottant & Villermaux (Reference Marmottant and Villermaux2004)) and/or by streamwise vortex generation due to vortex tilting and stretching mechanisms (as in isodensity jets (Liepmann & Gharib Reference Liepmann and Gharib1992)) leading to lobe formation. The baroclinic effects are dominant during atmospheric gas pressures (i.e. at low gas–liquid density ratios), where the gas–liquid density ratio is $O (10^{-3})$. In such cases, RT instability is a dominant mode for lobe generation due to the high-density gradient across the interface (related to baroclinic vorticity). Also, the contribution from the vortex tilting and stretching mechanism is negligible because of lower gas inertia, thereby lower vortex strain near the liquid interface. However, at higher gas-density ratios, vortex–strain interactions are substantial and the contribution of the baroclinic effect decreases in streamwise vorticity generation. Therefore, stretching and tilting of vortices is a dominant mechanism in azimuthal modulation on KH waves for such cases (Jarrahbashi et al. Reference Jarrahbashi, Sirignano, Popov and Hussain2016). A similar observation was also obtained for a liquid sheet exposed to external airflows (Zandian, Sirignano & Hussain Reference Zandian, Sirignano and Hussain2017, Reference Zandian, Sirignano and Hussain2019a,Reference Zandian, Sirignano and Hussainb). For the shock Mach number cases considered in the present work, the gas–liquid density ratio is $O(10^{-3})$, and therefore the baroclinic effect (or RTI) mode of lobe generation will be predominant, as discussed earlier.

Further, formed lobes undergo a cascade process through three different mechanisms during ligaments/droplet formation. The involved mechanisms include: (i) lobe stretching leading to ligament formation; (ii) corrugation on the lobes leading to multiple ligament formations during stretching of corrugation; and (iii) perforation of holes on the lobe leading to bridge breakup and ligament formation (Zandian et al. Reference Zandian, Sirignano and Hussain2017, Reference Zandian, Sirignano and Hussain2019b). The mode of ligament formation was shown to be dependent on the modified Ohnesorge number $(Oh_m =\sqrt {(We_g )}/Re_l)$ and Reynolds number ($Re_l$) magnitude (Zandian et al. Reference Zandian, Sirignano and Hussain2017). In the present work, we have refrained from determining the exact mechanism of ligament formation as the time sequence evolution of ligament formation was beyond the spatial and temporal resolutions. The ligament formation is also obstructed during the top-view imaging. However, future experimental or numerical works are motivated to uncover the prevalent mechanism of ligament formation during lobe disintegration. The elongation of the lobes leading to ligament formation was shown to be a predominant mechanism during low gas–liquid density ratio by Jarrahbashi & Sirignano (Reference Jarrahbashi and Sirignano2014). Therefore, the same can be assumed to be prevalent in the present case while determining the wavelength of RT waves.

The phenomenon of SIE breakup of a droplet is recurrent, as shown in figure 14 and explained schematically using figure 14( f). Each breakup cycle involves the formation of a thin sheet at the droplet periphery. The elongation of the sheet due to its entrainment into the external airflow further decreases the sheet thickness, making it susceptible to surface instabilities. Any small perturbation on this sheet can result in the nucleation of holes on its surface (see figure 14a). After formation, these holes expand in all directions, leading to the sheet's rupture into cylindrical ligaments. The formed ligament further undergoes a secondary atomisation into a daughter droplet through RPI and, thus, completes one breakup cycle. The second cycles begin with the formation of the thin sheet due to the accumulation of liquid at the droplet periphery due to the combined effect of KH waves translation and droplet deformation (as explained earlier) and follow a similar breakup pattern as discussed above. The three cycles of SIE-based recurrent breakup of the droplet are shown in figure 14. A similar recurrent breakup of the liquid droplet was also shown by Dorschner et al. (Reference Dorschner, Biasiori-Poulanges, Schmidmayer, El-Rabii and Colonius2020) through numerical simulations.

Figure 14. Recurrent breakup of the liquid sheet: (a) first breakup; (b,c) second breakup; and (c–e) third breakup. The red and blue rectangular regions shows the time-sequence evolution of holes during second and third breakup, respectively. All images are of same scale. ( f) Schematic diagram showing recurrent breakup of a droplet.

3.6. A shift in droplet breakup dynamics during transition regime $(100< We<1000)$

Previously, the droplet breakup dynamics of low viscosity Newtonian fluids ($Oh<0.1$) were subdivided into five major regimes (i.e. vibrational breakup ($We < 11$), bag breakup ($11< We < 35$), multimode breakup ($35< We < 80$), sheet thinning breakup ($80< We < 350$) and catastrophic breakup ($We > 350$)) based on Weber number range (Guildenbecher et al. Reference Guildenbecher, López-Rivera and Sojka2009). These regimes were reclassified by Theofanous & Li (Reference Theofanous and Li2008) based on hydrodynamics instabilities, where previously identified bag and multibag breakup modes were included in the RTP regime (based on RTI), while sheet thinning and catastrophic regimes correspond to the SIE mechanism (based on KHI). The RTP regime ranges up to $We < 100$, the SIE regime corresponds to $We > 1000$, and a transition regime exists between these two regimes ($100< We<1000$). This reclassification is also validated in our experiments for RTP (figure 18a,b) and SIE (figure 8b–f) regime. In the transition regime, through figure 15 (also see supplementary movie files 5 and 6) we explain the dependence of the breakup mode on the size of the primary droplet. In the existing literature, a unidirectional breakup trend from bag breakup to catastrophic breakup is reported with increasing Weber number. The sheet thinning or catastrophic breakup are the only two breakup modes observed for $We > 80$. Here, using the ratio of initial drop diameter to the wavelength of KH wave, we explain the existence of multimode droplet breakup for Weber number values up to $\sim$795. Figure 15(a,b) shows the shear stripping mechanism, while figure 15(c, d) shows the multimode droplet breakup at a comparatively higher Weber numbers ($We \sim 347\,795$) values predicted from the previous studies ($We <80$). The important features in the evolution of multibag and shear stripping mode of droplet breakup are shown schematically in figure 15(e). The shear stripping breakup was discussed in detail in § 3.3 and multibag breakup will be discussed later in this section. The growth rate of KH waves and RT waves on a droplet surface can be obtained from (3.8) and (3.10), respectively. The results of instability growth rate from dispersion relation based on KH wave (3.8) and RT wave (discussed later) are plotted in figure 16 for two particular cases of low and high Weber numbers. It is clearly observed that the growth rate of KH waves is always higher than the RT waves of corresponding Weber number, indicating that the RT waves (which lead to a bag-type breakup mechanism) based droplet breakup can only occur during the inability of KH wave formation (Theofanous et al. Reference Theofanous, Mitkin, Ng, Chang, Deng and Sushchikh2012). Thus, finding the criterion for this transition becomes important.

Figure 15. A shift of droplet breakup mechanism from SIE to RTP with different droplet sizes and similar Weber numbers ($We$). (a,b) Time sequence images showing SIE mode of droplet breakup for $D_i \sim 2.5$ mm. (c,d) Time sequence images showing RTP mode of droplet breakup for $D_i \sim 0.5$ mm. For details see supplementary movie files 5 and 6. (e) Schematic diagram showing droplet breakup through RTP and SIE modes.

Figure 16. Rayleigh–Taylor and KH instabilities growth rate ($\omega$) versus wavenumber (k).

Figure 17 shows the ratio of droplet diameter with the predicted wavelength of KH wave based on (3.7). Green markers indicate the droplet sizes undergoing SIE-based breakup mode, while the red markers correspond to the RTP-based droplet breakup mechanism. It can be seen from figure 16 that KH waves are only formed if $D_i \gg \lambda _{KH}$ (at least by an order difference) while the RTP-based droplet breakup mode occurs when both $D_i$ and $\lambda _{KH}$ are of a similar order of magnitude. When $D_i \sim \lambda _{KH}$ the size of the fastest growing wavelength is similar or less than the size of the available area required for its formation, i.e. droplet windward surface length, which is of the order of the droplet radius. Therefore, KH waves cannot be formed under such cases (as predicted in figure 17), which gives way to the formation of much slower RT waves on the windward surface of the droplet after its complete deformation into a flattened disc. This explains the observation of multimode droplet breakup at Weber number values of $\sim$347 and $\sim$795. Thus, the present article indicates the importance of the $D_i/\lambda _{KH}$ ratio in addition to the Weber number for predicting the breakup mode of the droplet.

Figure 17. Transition of droplet breakup mode based on $D_i/\lambda _{KH}$ and $We$. Red markers indicates breakup thorough RTP mode while green markers correspond to the SIE mode. The orange region corresponds to RTP ($We < 100$), the green region corresponds to SIE ($We > 1000$) and the blue region corresponds to transition regime ($100< We<1000$).

Figure 18 shows the RTP-based droplet breakup mechanism (also see supplementary movie file 7). In these cases, the primary droplet undergoes complete deformation (unlike SIE) into a flattened disc. The formed disc becomes unstable due to the acceleration of low density induced airflow on its surface, resulting in RTI. The growth rate ($\omega _{RT}$) of RT instability can be expressed as (Drazin & Reid Reference Drazin and Reid2004; Zhao et al. Reference Zhao, Wu, Li, Xu and Liu2018)

(3.10)\begin{equation} {\omega_{RT}}^2 = \frac{k_{RT} [(\rho_l-\rho_a)a-{k_{RT}}^2\sigma]}{\rho_l+\rho_a}. \end{equation}

Here, $k_{RT}$ is the wavenumber of RT waves, $a$ is the acceleration of the droplet, which is defined as

(3.11)\begin{equation} a = \frac{F_a}{m_d} = \frac{3\rho_a V_i^2 C_d D_m^2}{4 D_i^3 \rho_l}. \end{equation}

Here, $C_d$ is the drag coefficient whose value is chosen as ${\sim }1.2$ (Zhao et al. Reference Zhao, Wu, Li, Xu and Liu2018) and $D_m$ is the maximum droplet diameter during flattened state before the bag formation. The variation of RT instability growth rate for different wavenumber is shown in figure 16. The wavelength corresponding to the maximum growth rate ($\lambda _{RT}$) is obtained as (Drazin & Reid Reference Drazin and Reid2004; Jain et al. Reference Jain, Prakash, Tomar and Ravikrishna2015)

(3.12)\begin{equation} \lambda_{RT} = 2{\rm \pi}\sqrt{\frac{3\sigma}{\rho_la}}. \end{equation}

Figure 18. Droplet breakup dynamics at different Weber numbers for a droplet of diameter $(D_i) \sim 0.4$ mm. Time sequence images of droplet breakup from the instant of wire explosion to droplet breakup (from left to right). (a–d) The magnitude of non-dimensional numbers ($M_s , We, Re$) increases from panels (a) to (d). For details see supplementary movie file 7.

The wavelength of formed instability determines the number of bags formed during the droplet breakup process. The mode of droplet breakup is classified by normalising the initial droplet diameter with RT wavelength ($N = D_i/\lambda _{RT}$) (Zhao et al. Reference Zhao, Liu, Li and Xu2010). In the present work, $N\sim 2\text{--}3$ for all considered cases (see supplementary table 1) which undergoes breakup through RTI. Therefore, only the multibag mode of droplet breakup is obtained (see figure 18) (Zhao et al. Reference Zhao, Liu, Li and Xu2010; Jain et al. Reference Jain, Prakash, Tomar and Ravikrishna2015). Several bags are formed during the multibag mode of droplet breakup on the flattened droplet between the central undeformed and peripheral rims. The formed bag grows in size along the streamwise direction, reducing the sheet thickness, making it vulnerable to rupture into secondary daughter droplets. This is then followed by the rupture of the peripheral rim, which disintegrates into larger droplet sizes compared with those formed by bag breakup. The central region of the deformed droplet takes the form of a stamen-shaped ligament, which is attached to the formed bag. After bag rupture, the stamen ligament either contracts back to form a single droplet (at low Weber numbers) or further undergoes atomisation in a manner similar to the primary droplet, depending on the external aerodynamic forces. The usage of a wide range of Weber numbers ($We \sim 30$–$15\,000$) in the present study essentially resulted in the observation of both RTP and SIE modes of droplet breakup and identifying the possible transition mechanism between two modes. Subsequently, the present experimental study illustrates the aspect ratio variation for different Weber numbers and droplet sizes of 2.5 mm and 0.5 mm (see figure 19). The results are shown up to the beginning of droplets stripping for the SIE breakup mode and the beginning of bag formation during the RTP mode. As can be seen from the figure, the extent of deformation for smaller droplet size (RTP breakup mode) is greater than the bigger droplet (SIE breakup). This data could be useful during the validation of future numerical models.

Figure 19. Aspect ratio variation with time: (a) schematic diagram; and (b,c) aspect ratio variation with time for droplet size of $\sim$2.5 mm and $\sim$0.5 mm, respectively.