Self-similar Features in Secondary Breakup of a Droplet and Ligament Mediated Fragmentation under Extreme Conditions

TL;DR

The paper addresses predicting droplet fragmentation under extreme aerodynamic forcing by resolving a multiscale deformation cascade in shock-induced aerobreakup. It combines high-speed visualization with a local-Weber-number framework, revealing self-similar sub-secondary breakups that connect global breakup to ligament-mediated fragmentation. The study finds a universal droplet-size distribution describable by a compound Gamma form with a corrugation limit near $n\approx 4$ and establishes a scaling $\langle d\rangle\sim We^{-1/3}$, augmented by a time-integrated distribution approach. Together, these results provide a unified, multiscale framework for predicting fragment statistics in extreme flows and offer potential applicability to primary atomization and ocean spray phenomena.

Abstract

Droplet formation is relevant in many applications spanning natural and artificial settings. A physical understanding of aerobreakup or air-assisted secondary atomization and predicting size distributions in these applications is non-trivial. We show that extreme airflow speeds induce catastrophic breakup, which, although chaotic and seemingly obscure, is not hopelessly unstructured. In the present study, through shockwave-induced breakups, we investigate the associated intermediate processes at smaller spatiotemporal scales at very high Weber numbers ($We \sim 10^3-10^4$). We demonstrate that microscale protrusions decorate the disintegrating droplet interface and eventually fragment, resulting in the generation of daughter droplets. We discover these undulations to follow breakup patterns (sub-secondary breakup) that resemble a scaled-down version of secondary atomization. The consistent topology across a vast range of scales $(10^{-6}m-10^{-2}m)$ suggests a self-similar mechanism bridged by local Weber number. The normalized size distribution of the resultant droplets exhibits universality and $We$ invariance at all extreme conditions, including transient statistics for subsequent time periods. This conforms to a universal modified gamma distribution characterized by ligament shape factors, which tend toward the limiting behavior associated with the maximum corrugations physically possible. Scaling laws based on the $We$ are derived for the averaged diameter as $\sim We^{-1/3}$, using a high-energy aerodynamic breakup mechanism and subsequently used to derive a time-integrated distribution. These observations reinforce the idea of a self-similar mechanism for the catastrophic droplet breakups, encompassing multiscale deformation cascades, self-similar sub-secondary breakups, maximally corrugated ligaments, and universal droplet size distributions.

Figures (13)

• Figure 1: (a) Shadow images of a shock induced breakup of a droplet at $We\approx2000$, depicting an SIE mode. Schematic illustrating (b) Rayleigh-Taylor Piercing (RTP) (c) Shear-Induced Entrainment (SIE) (d) An undulation fragmentation through sub-secondary breakup process (e) Shadow images of bag breakup of a droplet at $We\approx15$, depicting an RTP mode. This figure adapted from chandraAerodynamicBagBreakup2024 (f) Shadow images of a sub-secondary breakup process of a droplet, showing the bag mode at much smaller length scales.
• Figure 2: (a) Experimental setup illustrating a wire-explosion based shock tube, actuated by a pulsed high voltage source. An acoustic levitator is used to place the droplet at the opening, and a shadowgraphy system is deployed for imaging. (b) A droplet interacting with the high-speed flow emerging from the shock tube, illustrating the shock wave and a starting jet.
• Figure 3: The deformation cascade happening at a range of spatiotemporal scales, illustrating three prominent stages: global deformations, sub-secondary breakups, and ligament-mediated droplet generation.
• Figure 4: Evolution of an undulation over droplet surface undergoing sub-secondary breakup processes (a) ligament mode (b) bag mode. The scale bar represents $100\mu m$ and consecutive frames are separated by an interval of $\Delta t=13.33\mu s$. Schematic illustrating sub-secondary breakup processes. The early stage KHI initiates an undulation with characteristic scale $\xi\sim \lambda_{\rm KH}$ with local effective air speed $u_a'$. Beyond a particular finite amplitude, this undergoes destabilization through sub-secondary breakup processes (a) the ligament mode, characterized by transverse acceleration-induced azimuthal RTI, and (b) the bag mode, characterized by longitudinal acceleration-induced azimuthal RTI. Both these modes eventually create terminal ligaments that generate droplets through RPI. $\lambda_{\rm KH}$, $\lambda_{\rm RT}$ and $\lambda_{\rm RP}$ represents wavelength for KHI, RTI and RPI respectively.
• Figure 5: Striping process in catastrophic SIE as a recurrent sub-secondary breakup mechanism. (a) Schematic depicting the continuous formation of undulations indexed '$i$', and their recurrent breakup generating droplets with distributions $p^i(d)$. (b) Experimental images of the stripping process with recurrent undulation breakup. The scale bar represents $100\mu m$ and consecutive frames are separated by an interval of $\Delta t=13.33\mu s$.