# Bubble pinch-off in turbulence

^{a}Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544;^{b}Département de Physique, Ecole Normale Supérieure, PSL (Paris Sorbonne Lettres) Research University, 75005 Paris, France;^{c}Princeton Environmental Institute, Princeton University, Princeton, NJ 08544

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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved November 3, 2019 (received for review June 7, 2019)

## Significance

As a bubble breaks apart, the final pinching culminates in a singularity. We investigate the pinch-off of a bubble in turbulence and demonstrate that the turbulent flow field freezes during the pinching process, opening the route for a self-similar collapse close to the one predicted for unperturbed configuration. The role of the turbulent flow field is, therefore, to set the complex initial conditions, which can lead to oscillations of the neck shape during the collapse and the eventual escape from self-similarity with the appearance of a kink-like interfacial structure. This work can be seen as a prototype for understanding the route to finite-time singularities in realistic multiscale systems where random perturbations are present, with both fundamental and practical implications.

## Abstract

Although bubble pinch-off is an archetype of a dynamical system evolving toward a singularity, it has always been described in idealized theoretical and experimental conditions. Here, we consider bubble pinch-off in a turbulent flow representative of natural conditions in the presence of strong and random perturbations, combining laboratory experiments, numerical simulations, and theoretical modeling. We show that the turbulence sets the initial conditions for pinch-off, namely the initial bubble shape and flow field, but after the pinch-off starts, the turbulent time at the neck scale becomes much slower than the pinching dynamics: The turbulence freezes. We show that the average neck size,

The breakup dynamics of bubbles and droplets are central to many natural and engineering processes, playing a role in heat, mass, and momentum transfer at the ocean–atmosphere surface (1, 2), rain drops falling on solids and liquids (3), and industrial liquid atomization and fragmentation (4). In such configurations, the gas–liquid interface is often surrounded by a violent turbulent flow characterized by perturbations of various strengths at various scales. As a result, the external forcing can strongly deform the bubble or droplet interface, leading to a complex geometrical shape and various breaking scenarios (5, 6). During the final instant of the collapse, the pinching dynamics approach a finite-time singularity, which has been studied extensively in idealized configurations in an attempt to identify such singularities and their regularization by viscous forces (7, 8). For these practical and fundamental reasons, the final pinching of a gas bubble or liquid droplet continues to draw extensive research interest.

In a quiescent fluid, starting from a purely axisymmetric fluid neck, the dynamic eventually becomes independent of the initial condition as it approaches the finite-time singularity corresponding to the time of pinch-off. Experiments and theories for the breakup of air bubbles or liquid droplets (8⇓⇓⇓⇓⇓–14) have described the interface neck diameter, d, thinning toward pinch-off by a self-similar dynamic

A central assumption of the above description is the circular shape of the neck at the start of pinch-off and the axisymmetry of the surrounding flow field. Memory effects in bubble pinch-off, evidenced by initial perturbations to the neck shape yielding shape oscillations throughout the collapse, have been identified and described (18⇓–20) as have misalignments of the neck with respect to gravity and the rapid injection of gas, which alter the singularity (19, 21). These studies have extended our understanding of pinch-off to controlled asymmetric situations, but they are limited to idealized configurations in which background perturbations are of a high degree of symmetry. Here, we investigate the final pinch-off dynamics of a bubble in a fully developed turbulent flow. This study can be seen as a prototype for understanding the route to singularities in realistic multiscale systems where perturbations at multiple scales are inherently present.

## Frozen Turbulence during Pinch-Off

Fig. 1 shows a schematic of the bubble deformation and pinch-off scenario in a turbulent flow. We blow air through needles with inside diameter *Materials and Methods*). The turbulence dissipation rate ϵ is between 600 and 5,400 cm^{2}/s^{3}. The growth of the air bubble lasts from

During the collapse in a quiescent fluid, the flow space and timescales shrink to 0 with accelerating dynamics up to the pinch-off singularity time

## Experimental Collapse with a Turbulent Background

Despite the divergence of the turbulence and pinching scales, individual observations of pinch-off in a turbulent background reveal rich dynamics absent with quiescent backgrounds. Among the

Evidence of these memory effects in turbulent pinch-off is given for an experimental observation in Fig. 2. We record the bubble pinching off from a needle with 2 orthogonal high-speed cameras, yielding the views of the neck region in the laboratory *A* and *B*. An image processing algorithm is used to infer the orientation of the neck axis *Materials and Methods* and Fig. 2 *B* and *C*). The blue and green curves in Fig. 2*C* show the neck sizes *C*, indicating that the neck shape is an oscillating ellipse with average size that follows closely the unperturbed results described in the literature, *D*. This oscillation is characteristic of the Bell–Plesset collapse, which describes an interface undergoing a prescribed acceleration. This phenomenon was treated analytically by Bell (23) and Plesset (24); more recently, the temporal evolution of this collapse was derived independently (18, 25) in the context of a collapsing bubble neck subject to small-amplitude azimuthal shape perturbations. A neck with a mode n perturbation has a cross-section described by *C* and *D* shows, as dashed lines, results from the azimuthal perturbation model, where an **1** no longer holds. Moreover, Fig. 2*F* shows that the oscillation phase is set by the local neck size over a vertical range near the neck minimum. This suggests that the perturbation from the cylindrical shape does not vary significantly along the neck axis. Other examples of similar oscillatory dynamics are shown in *SI Appendix*. In many cases, oscillations occur about *Materials and Methods*). We initialize a bubble in a quiescent background with a neck cross-section that is initially slightly elliptic (mode *SI Appendix*).

## Escape from Self-Similarity Close to Singularity

Next, we address the cases where the initial dynamics of the average neck size are still described by the self-similar route to finite-time singularity, but the neck thinning eventually escapes self-similarity at some point before pinch-off. We observe that such events arise more often for larger initial neck sizes and stronger turbulence and that the escape from self-similarity can be preceded by shape oscillations, such as those in Fig. 2, or displacement of the thinnest portion of the neck in either the vertical or horizontal direction, which indicates more complex 3D dynamics. A combination of the 2 perturbations has also been observed. The final escape from self-similarity is accompanied by a kink structure appearing in the interface near the neck minimum. We record the size of the neck,

Fig. 3*A* shows an example of the appearance of a kink associated with a vertical displacement of the thinnest neck position. In this case, the kink structure first appears on the left side of the *A*) and corresponds to the first inflection point in *A*, *Inset*). The kink next appears in the 2 sides of the neck viewed in the *B* shows that the axial position of the thinnest point of the neck changes as the kink develops so that the final pinch-off point is clearly distinct from the initial neck. As evidence for the complex 3D structure of the perturbation, the axial locations of the minimum *B*: these oscillate about the average. Plotting the oscillation signal *B* as a function of the local neck thickness, we see that various slices of the neck no longer share the same oscillation characteristics as had been the case in Fig. 2. In other cases, shape oscillations are not observed, but there is a persistent horizontal displacement of the thinnest part of the neck related to a large asymmetry, leading to different sides of the neck collapsing at consistently different rates. Such configurations, together with additional cases similar to Fig. 3 *A* and *B*, are shown in *SI Appendix*, all of which display a kink that forms and causes the collapse to escape self-similarity in its final stages.

To show that this 3D behavior responsible for escaping self-similarity can be induced only by the initial bubble shape itself, we perform direct numerical simulations of an asymmetric bubble collapsing due to surface tension. We initialize an asymmetric bubble shape resembling a dumbbell with an off-center circular axis, shown in Fig. 3*C*, and a quiescent initial condition. The result is an elliptical neck shape with asymmetry that persists through the pinch-off without oscillation, evidenced by the plots of *C*. The asymmetry is further understood by considering the motion of 4 separate points on the neck interface, the collapse velocities of which are shown in Fig. 3*C*, *Insets*: one side of the neck consistently moves away from the neck center during the collapse, as the side opposite to it collapses much more quickly. Eventually, a kinked structure appears in the neck region. Given the initially quiescent state of the simulation, the complex pinching dynamics can be attributed to the bubble’s initial asymmetric shape. Fig. 3*D* shows, for the experimental and numerical cases in which a kink is observed, the size of the neck at kink formation *SI Appendix*.

As shown in Figs. 2 and 3, the averaged neck thinning in turbulence initially follows the unperturbed results described in the literature, **1** does not capture the neck evolution. In some cases, these dynamics are followed until the end of the observable time window (Fig. 2), while in other cases, the collapse escapes self-similarity (Fig. 3). Fig. 4*A* shows the normalized ensemble-averaged mean neck diameter, *A*, *Inset*, is also centered around

## Control of Kink Formation by the Turbulence

In Fig. 3, we showed that the escape from self-similarity by kinking is controlled by the asymmetry in the neck shape induced by the pinching dynamics. Here, we relate this asymmetry to the turbulent initial conditions by considering the competition between turbulent stresses and surface tension, which sets the initial bubble shape. This is parameterized by the ratio of the needle size, *B* shows the likelihood of a kink being resolved (up until the experimental limit of *C*, *D*. At small *SI Appendix*), suggesting that capillarity governs the kink development.

The kink dynamics presented are very unlikely to be caused by turbulence intermittency given the high velocity of the kink’s development (which is faster than velocities measured in the tails of velocity fluctuation distribution as shown in *SI Appendix*). Previous literature suggests that similar kinked structures are indicative of an **1** with

## Conclusion

We present experiments on bubble pinch-off in turbulence and demonstrate that the effect of the turbulent background flow can be reduced to its role in setting the initial conditions. We show that the turbulent fluctuations freeze after the pinch-off process starts (i.e., the turbulent timescale at the scale of the neck is much slower than the pinching dynamic). The frozen turbulence sets the stage for complex behavior, such as neck shape oscillations, during the collapse. We identify a controlling nondimensional parameter,

## Materials and Methods

### Experimental Setup and Analysis.

Turbulence in the water is created by 8 submerged pumps with outlets that are attached via flexible tubing to nozzles arranged at the vertices of a 28-cm cube and pointed toward the cube’s center. A sketch of the experimental setup is provided in *SI Appendix*. Inspired by refs. 30 and 31, this setup induces a largely homogeneous, isotropic flow in the cube center. A needle in the range ^{2}/s^{3} (*SI Appendix*. For the 3 increasing values of non-0 ϵ, the Kolmogorov microscale is 58, 47, and 34 μm, respectively, and the Taylor microscale is 1.7, 1.7, and 1.2 mm, respectively. This is comparable with the initial size of the neck as pinch-off begins so that the bubble is initially within the turbulence inertial subrange. The integral length scale is estimated to be

The imaging of the pinch-off is done with a 2-camera setup similar to that in refs. 18 and 21, which provides a measure of the neck’s asymmetry with views from 2 orthogonal angles. One camera (Phantom v2012) films at *SI Appendix*.

Approximately 300 cases of turbulent bubble pinch-off were imaged with the 2-camera setup, and a further **1**, and finding the initial conditions *A* shows the ensemble average of *B* is obtained by choosing the time range over which the power law fit is performed independently for each case to maximize the fit coefficient of determination. Here, we enforce that the chosen range of *B* is computed for each combination of needle size and turbulence dissipation rate, which defines the ratio

### Direct Numerical Simulations.

We perform direct numerical simulations with the open source package Basilisk (26, 27), solving the 3D 2-phase Navier–Stokes equations with surface tension using an air-water density ratio of 850 and a viscosity ratio of 5.12 on an adaptive octree mesh. This solver has been extensively validated for complex multiphase flow processes with interface topological changes and reconnection (1, 33⇓–35). To validate our configuration, we perform a 3D simulation of axisymmetric bubble pinch-off and recover the pinching dynamics expected from the theoretical and experimental results, the neck size following *SI Appendix*.

### Data Availability.

Data and code to reproduce plots are available at http://arks.princeton.edu/ark:/88435/dsp014f16c5691 (36).

## Acknowledgments

This work was supported by NSF Faculty Early Career Development (CAREER) Program Award Chemical, Bioengineering, Environmental and Transport Systems (CBET) 1844932 and American Chemical Society Petroleum Research Fund Grant 59697-DNI9 (to L.D.). Computations were partially performed using allocation TG-OCE140023 (to L.D.) from the Extreme Science and Engineering Discovery Environment, which is supported by NSF Grant ACI-1053575.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: ldeike{at}princeton.edu.

Author contributions: D.J.R. and L.D. designed research; D.J.R., W.M., S.P., and L.D. performed research; D.J.R., W.M., and L.D. analyzed data; and D.J.R., W.M., S.P., and L.D. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

Data deposition: Data and code to reproduce plots are available at http://arks.princeton.edu/ark:/88435/dsp014f16c5691.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1909842116/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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