AA 2023

# Deformation and breakup of bubbles and drops in turbulence

Rui Ni1 1Department of Mechanical Engineering, The Johns Hopkins University,
Baltimore, MD 21218, USA; email<EMAIL_ADDRESS>
###### Abstract

Fragmentation of bubbles and droplets in turbulence produces a dispersed phase
spanning a broad range of scales, encompassing everything from droplets in
nanoemulsions to centimeter-sized bubbles entrained in breaking waves. Along
with deformation, fragmentation plays a crucial role in enhancing interfacial
area, with far-reaching implications across various industries, including
food, pharmaceuticals, and ocean engineering. However, understanding and
modeling these processes is challenging due to the complexity of anisotropic
and inhomogeneous turbulence typically involved, the unknown residence time in
regions with different turbulence intensities, and difficulties arising from
the density and viscosity ratios. Despite these challenges, recent advances
have provided new insights into the underlying physics of deformation and
fragmentation in turbulence. This review summarizes existing works in various
fields, highlighting key results and uncertainties, and examining the impact
on turbulence modulation, drag reduction, and heat and mass transfer.

###### doi:

10.1146/((please add article doi))

###### keywords:

turbulent multiphase flow, deformation and breakup/fragmentation, emulsion,
polydispersed droplets and bubbles, lift and drag, heat and mass transfer

††journal: Xxxx. Xxx. Xxx. Xxx.

##  1 Introduction

Mixing two immiscible fluids (gas-liquid or liquid-liquid) in turbulence
produces polydispersed droplets or bubbles that can freely deform, break, and
coalesce while interacting with the surrounding turbulence. These processes
are fundamentally important and practically relevant to multiple fields,
including bubble-mediated air-sea mass exchange (Villermaux et al., 2022),
chemical emulsions, food science, nuclear thermal hydraulics, and two-phase
heat transfer. In contrast to the deformation and breakup of droplets in low-
Reynolds-number viscous flows (Stone, 1994), in turbulence, these dynamics are
intimately linked to multiple length and time scales associated with the
background turbulent eddies.

A wide range of drop/bubble sizes can therefore be achieved via the adjustment
of the turbulence characteristics. For example, turbulence generated by a
simple batch stirrer system can break an oil-water mixture into macroemulsions
with the size of the dispersed oil droplets at $\mathcal{O}(1$–$100$ $\mu$m).
But if nanoemulsions with droplets of $\mathcal{O}(10$–$100$) nm are desired,
the turbulent scales have to be much smaller, requiring an energy-intensive
high-pressure homogenizer (HPH) method (Schultz et al., 2004, Håkansson,
2019).

Despite the wide range of scales involved, many key concepts crucial to
understanding deformation and breakup in turbulence can be traced back to the
seminal works by Kolmogorov (1949) and Hinze (1955), i.e. the Kolmogorov-Hinze
(KH) framework. The KH framework has gained widespread acceptance in various
fields, however, it is crucial to acknowledge that it contains a number of
assumptions and hypotheses. The purpose of this review is to gather studies
from various disciplines that investigate the deformation and breakup of both
droplets and bubbles in turbulence, in order to determine the regimes in which
the KH framework is applicable and more importantly, where it may fall short
and new challenges and opportunities await.

[h]

##  2 Key hypotheses and assumptions in the Kolmogorov-Hinze framework

(a) Turbulence was assumed to be homogeneous and isotropic. (b) The drop size
was assumed to be in the inertial range of turbulence. (c) Droplets were
assumed to be neutrally buoyant, with buoyancy and density ratio disregarded.
(d) It was hypothesized that the breakup is driven by the dynamic pressure
caused by changes in velocity over distances at the most equal to the drop
diameter. (e) The framework assumes that the interaction between drops and
turbulence is one-way, with droplets having no effect on the turbulent
dynamics. (f) While Kolmogorov took into account the kinematic viscosity ratio
between the two phases to separate different regimes, Hinze proposed to use
the Ohnesorge number (defined in Section 2) to measure the importance of the
inner viscosity.

This review provides an overview of the dynamics of deformation and breakup
and their impacts on momentum, mass, and heat transfer, with a particular
focus on experimental methods and results and a limited survey of simulation
findings. For in-depth coverage of numerical methods for resolving deformable
interfaces in turbulence, readers are referred to the recent reviews by
Tryggvason et al. (2013) and Elghobashi (2019). The subject is closely related
to the broader realm of particle-laden turbulence, including spherical
(Balachandar & Eaton, 2010, Brandt & Coletti, 2022), non-spherical (Voth &
Soldati, 2017), and buoyant particles (Mathai et al., 2020), but with a
particular emphasis on deformability. This review also complements other
comprehensive reviews on fragmentation (Villermaux, 2007), bubble dynamics
(Magnaudet & Eames, 2000, Risso, 2018, Lohse, 2018), and the complexity
introduced by surfactant (Takagi & Matsumoto, 2011), phase inversion (Bakhuis
et al., 2021), and non-Newtonian liquids (Zenit & Feng, 2018).

The problem being considered involves bubbles and droplets of a specific
diameter, denoted as $D$, being deformed and fragmented by surrounding
turbulence characterized by parameters, such as energy dissipation rate
($\epsilon$), fluctuation velocity ($u^{\prime}$), integral scale ($L$), and
the Kolmogorov scale ($\eta$). The density, dynamic and kinematic viscosities
are denoted by $\rho$, $\mu$, and $\nu$, respectively. The fluid properties of
the carrier phase and dispersed phase can be differentiated using subscripts
$c$ and $d$, respectively. The interfacial tension between the two phases is
represented by $\sigma$.

The problem at hand is characterized by a multitude of parameters, and as a
result, the relevant dimensionless groups are also vast. However, by making
some key assumptions and hypotheses, as outlined in Textbox 1, Kolmogorov
(1949) was able to simplify the problem. He proposed that, for the deformation
and breakup of large bubbles/droplets ($\eta\ll D\ll L$), the most important
dimensionless number is the Weber number, which is a measure of the ratio
between the inertial forces to surface tension forces.

$We_{t}=\frac{\rho_{c}u_{D}^{2}D}{\sigma}$ (1)

where $u_{D}$ is the eddy velocity of size $D$ and $u_{D}^{2}=C_{2}(\epsilon
D)^{2/3}$ is the estimation using the second-order structure function in the
inertial range in homogeneous and isotropic turbulence (HIT), where
$C_{2}\approx 2.3$ is the Kolmogorov constant. Furthermore, it was postulated
that, if the Weber number is the only dimensionless number that affects the
breakup problem, there must exist a critical Weber number ($We_{t}^{c}$) that
corresponds to the critical diameter($D^{c}$), below which the droplets remain
stable for a prolonged period in turbulence.

$D^{c}=\left(\frac{We_{t}^{c}\sigma}{\rho_{c}C_{2}\epsilon^{2/3}}\right)^{3/5}$
(2)

The idea of a critical Weber number implies an abrupt shift from a finite
breakup probability to zero at $We_{t}^{c}$, a simplistic view which does not
account for turbulent fluctuations. Although the likelihood of eddies with
local energy dissipation rates significantly higher than the mean is low, it
is not zero. Therefore, even if the mean Weber number is below $We_{t}^{c}$,
the occasional high-energy eddies can still break bubbles or droplets.
Additionally, while $We_{t}$ captures the contribution of turbulence,
persistent large-scale forcing, such as shear or buoyancy, can aid and even
dominate deformation and breakup. To understand the fundamental breakup
mechanisms, their contributions must be distinguished from those of
turbulence. Lastly, incorporating the effects of viscosity poses significant
challenges, requiring a systematic review of existing experimental data. To
this end, this review is structured as follows. In Section 2, different
regimes of deformation and breakup driven by turbulence, including the effects
of large-scale forcing and viscous damping, are reviewed. Section 3 provides
an overview of the key results and models of breakup frequency. In Section 4,
the findings on how deformation and breakup influence the momentum, heat, and
mass transfer between phases are summarized.

##  3 Various breakup Regimes

Figure 1 illustrates the relevant regimes that have been studied and the
typical deformation and breakup morphology that has been observed. Figure 1a
emphasizes the problems dominated by inertia but separately considers the
effects of small-scale turbulence ($We_{t}$) and large-scale forcing. The
large-scale forcing can arise in various forms, including a persistent mean
shear with the shear rate denoted as $\mathcal{S}$ and a pressure gradient
induced by buoyancy-driven migration, with their roles in deformation measured
by the shear Weber number
$We_{\mathcal{S}}=\rho_{c}\mathcal{S}^{2}D^{3}/\sigma$ and the Eötvös or Bond
number $Eo=\Delta\rho gD^{2}/\sigma$, respectively.

Figure 1b emphasizes the transition from an inertia-dominated to a viscous-
dominated regime when $D$ crosses the Kolmogorov length scale
($\eta=(\nu_{c}^{3}/\epsilon)^{1/4}$) and the viscous effect becomes more
pronounced. In the viscous regime, the crucial dimensionless number is the
Capillary number, i.e. $Ca_{t}=\sqrt{\mu_{c}\rho_{c}\epsilon}D/\sigma$. As the
viscous effect of the outer fluid becomes relevant, it is also necessary to
consider the regimes when the inner viscosity matters as well. As a result,
another key dimensionless number, i.e. the Ohnesorge number
($Oh=\mu_{d}/\sqrt{\rho_{d}\sigma D}$), is considered to measure the relative
significance of $\mu_{d}$ in resisting and damping deformation.

Figure 1: A parameter space of deformation and breakup of bubbles/droplets in
turbulence characterized by (a) the Weber number defined based on the small-
scale turbulence ($We_{t}$) versus large-scale persistent forcing measured by
either shear ($We_{\mathcal{S}}$) or buoyancy ($Eo$), and (b) the Ohnesorge
number ($Oh$) and the size of the bubbles/drops ($D$) relative to the
Kolmogorov scale ($\eta$). The inset panels are adapted with permission from
RF98 (Risso & Fabre, 1998), VVSL16 (Verschoof et al., 2016), RCR2011 (Ravelet
et al., 2011), MWMCG06 (Mason et al., 2006), EAL04 (Eastwood et al., 2004),
and QMN20 (Qi et al., 2020). The placement of these insets in the parameter
space only indicates the general regimes they correspond to, not their exact
parameters.

###  3.1 Inertia-dominated regime ($D>\eta$)

####  3.1.1 Intense homogeneous and isotropic turbulence ($We_{t}>Eo$ and
$We_{t}>We_{\mathcal{S}}$)

In this regime, the classical KH framework is most applicable. However, it can
be difficult to achieve these ideal conditions in experiments. In closed
systems, HIT can be generated by forcing flows from multiple symmetrical
locations. As these flows merge, HIT can be produced near the center, where it
is farthest from the momentum sources and therefore has the lowest energy
dissipation rate. This location is also where measurements were typically
taken. As a result, the probability of breakup is much higher outside the
measurement volume than inside. Bubbles and drops that are likely to break
would have already been broken before entering the measurement volume, making
it challenging to study their behavior in classical HIT systems.

One solution is to use HIT that decays along one direction and guide bubbles
or droplets through turbulence along the opposite direction. In this way, the
energy dissipation rate that bubbles/drops encounter continues to increase and
the measurement volume can be set at a location where the energy dissipation
rate is the highest but the flow is still HIT. Masuk et al. (2019b) designed a
vertical water tunnel with a jet array located at the top of the test section
and firing jets co-axially with the mean flow downward into the test section.
The facility and its key dimensions are shown in Figure 2a. Tan et al. (2023)
showed that, in this facility, the flow becomes HIT at around six nozzle
spacings below the jet array, and such HIT continues to decay. The decay was
found to scale with the nozzle diameter ($d_{n}$) and the jet velocity at the
nozzle exit ($v_{j}$). In particular, the fluctuation velocity follows
$u^{\prime}/v_{j}=(x/d_{n})^{-1}$, and the energy dissipation rate decays as
$\epsilon/(v_{j}^{3}/d)=0.76(x/d)^{-7/2}$. In this setup, the bubbles were
injected at the bottom of the test section where the energy dissipation rate
is the weakest. As they rise, the turbulence intensity grows, and eventually
reaches a point where it is sufficient to cause bubbles to deform and break.
Turbulence at this location, where the measurement volume is also placed,
features large energy dissipation rates of $\mathcal{O}(1)$ m2/s3, which is
sufficient for bubbles of size $\mathcal{O}(1)$ mm to reach the condition of
$We_{t}>Eo$.

Figure 2: (a) Schematic of the side view of a vertical water tunnel that uses
a jet array to produce intense HIT to study deformation and breakup (Masuk et
al., 2019b, Qi et al., 2020). (b) The top view of the octagonal test section
along with six cameras that were used to measure the shape of deformed bubbles
simultaneously with the nearby 3D turbulence (Masuk et al., 2019a, Qi et al.,
2020). (c) Examples of one strongly-deformed bubble captured by three
different cameras. (d) The distribution of the Weber number defined based on
different velocities. The red line indicates the log-normal distribution
predicted based on the distribution of the instantaneous energy dissipation
rate (Masuk et al., 2021c).

Apart from increasing turbulence intensity, another approach to reach
$We_{t}>Eo$ is to weaken the buoyancy effect. Risso & Fabre (1998) conducted
experiments in a parabolic flight, reducing the gravitational constant to
$g=0.4$ m/s2 and effectively reducing $Eo$ by 25 times. In this experiment,
turbulence was generated via an axisymmetrical momentum jet close to the
bottom of the device (Risso & Fabre, 1997). The average size of bubbles
investigated is about 18 mm, which would have broken due to buoyancy alone if
the experiments were conducted under the Earth’s gravity (Tripathi et al.,
2015), but under microgravity, the buoyancy effect was much weaker, and
breakup was dominated by turbulence. One sequence of images for strongly-
deformed bubble in such an environment is shown in an inset of figure 1a. The
critical Weber number averaged over all the cases was reported to be around
4.5. A numerical version of the same experiment was conducted by Qian et al.
(2006) using the lattice Boltzmann method with bubbles being fragmented in
homogeneous turbulence in a three-dimensional periodic box. The reported
critical Weber number was around 3.

Slip velocitythe instantaneous velocity difference between the two phases.
This quantity is different from the mean slip velocity used in atomization.

Although microgravity helps reduce the impact of the relative motion between
the two phases (slip velocity) driven by buoyancy, this rising motion is not
the sole source of the additional pressure gradient that could drive
deformation. Even with a neutrally-buoyant dispersed phase, as assumed in the
KH framework, slip velocity ($u_{\text{slip}}$) cannot be fully eliminated due
to finite size effects (Homann & Bec, 2010, Bellani & Variano, 2012). To
determine the extent to which deformation is driven by slip velocity, Masuk et
al. (2021c) conducted an experiment to measure the shape of deforming bubbles
simultaneously with their surrounding turbulence in 3D. This challenging
experiment was accomplished using a diagnostic setup, which included six
cameras positioned around the test section (figure 2b). Typical images of a
deformed bubble, along with the nearby tracers, are shown in figure 2c. The
shadows of high concentration of tracers were tracked using the openLPT method
(Tan et al., 2020), while the bubble geometry was reconstructed using a
technique that employs surface tension as an additional physical constraint,
resulting in improved reconstruction quality (Masuk et al., 2019a).

These simultaneous measurements provide insight into the Lagrangian evolution
of the bubble Weber number and the shape of individual bubbles (Masuk et al.,
2021c). The distributions of the Weber number based on different velocity
scales are shown in figure 2d.
$We_{t}=\rho(\widetilde{\lambda_{3}}D)^{2}D/\sigma$, in this case, was
determined by using the eigenvalue, $\lambda_{3}$, that corresponds to the
most compressive direction ($\hat{\mathbf{e}}_{3}$), of the strain rate tensor
coarse-grained at the bubble scale. The ensemble average of this definition of
$We_{t}$ should be equivalent to the one proposed in the KH framework, but it
provides a more accurate representation of the relevant instantaneous Weber
number, which was confirmed by the fact that the semi-minor axis of the bubble
preferentially aligns with $\hat{\mathbf{e}}_{3}$. This alignment suggests
that it is the converging flow near the bubble and the resulting pressure rise
on the interface that leads to compression and deformation. The PDF of
$We_{t}$ can be captured through a log-normal distribution (red line in figure
2d), calculated by following the definition of $We_{t}$ and the distribution
of local energy dissipation rate that is described by the refined Kolmogorov
theory (Kolmogorov, 1962) and the multi-fractal spectrum (Meneveau &
Sreenivasan, 1991).

In addition to the strain rate, the instantaneous slip velocity was also
calculated and divided into the horizontal ($x$) and vertical components
($z$). Their respective Weber numbers, i.e. $We_{\text{slip,x}}$ and
$We_{\text{slip,z}}$, can be calculated by using the slip velocity as the
velocity scale. The PDF of $We_{\text{slip,z}}$ contains the contribution by
the buoyancy-driven deformation, but the overall shapes of
$We_{\text{slip,x}}$ and $We_{\text{slip,z}}$ remain similar to each other.
The difference between $We_{\text{slip,x}}$ and $We_{t}$ (figure 2d), on the
other hand, is significant, underscoring the difference between two
deformation mechanisms brought by turbulence and finite-sized bubbles.

Through simultaneous measurements, a simple relationship between $We_{t}$ and
aspect ratio of the bubble $\alpha$, following $\alpha=2We_{t}^{2/3}/5+1.2$,
was determined by minimizing the difference between the PDF of $\alpha$
obtained from direct shape measurement and that calculated from $We_{t}$. This
relationship provides a way to describe bubble deformation in turbulence,
which complements studies that investigated the deformation of gas bubbles
rising in quiescent liquids (Legendre et al., 2012). In turbulence, while it
was noted that the fit against $We_{t}$ is slightly better than against
$We_{\text{slip}}$, the difference was not substantial, suggesting that both
are important in bubble deformation. However, the orientation analysis by
Masuk et al. (2021b) indicates that the bubble semi-minor axis aligns
signficantly better with the slip velocity than with $\hat{\mathbf{e}}_{3}$,
emphasizing the crucial roles played by slip velocity in bubble deformation in
turbulence.

####  3.1.2 Shear or buoyancy dominated deformation ($We_{t}<Eo$ or
$We_{t}<We_{\mathcal{S}}$)

#####  3.1.2.1 Buoyancy dominated regime ($We_{t}<Eo$)

In this regime, deformation is predominantly driven by buoyancy and the
turbulence effect is minimal. Sevik & Park (1973) conducted an experiment on
the breakup of bubbles in a turbulent jet. Bubbles with diameter varying from
4.0 mm to 5.8 mm were injected along the centerline of the jet. $Eo$ ranges
from 2.1 to 4.5, and the critical Weber number determined was about 1.3. This
critical $We_{t}$ is much smaller than 4.5 reported by Risso & Fabre (1998)
based on the experiments conducted in microgravity, indicating less stress
from turbulence was needed to break bubbles thanks to the extra help from
buoyancy.

Ravelet et al. (2011) conducted an experiment of large bubbles rising in weak
turbulence and reported two different Weber numbers, one based on the bubble’s
typical rise velocity and the other on the velocity gradient across the
bubble. The fact that the former Weber number was close to 11.6 indicates that
bubbles were strongly deformed due to buoyancy. The latter one, $We_{t}$ as
defined in equation 1, was around 1.8, which was about an order of magnitude
smaller. Figure 1a displays snapshots of a deforming bubble, which show
resemblance to bubbles rising in a still medium (Mougin & Magnaudet, 2001),
with the short axis of the bubble preferentially tilted towards the vertical
direction and exhibiting periodic motions. These similarities were expected
since bubbles were still primarily compressed in the vertical direction and
the same wake instability occurred (Zenit & Magnaudet, 2008). Despite these
similarities, the time series of bubble deformation in turbulence were more
chaotic and the decorrelation timescale was associated with the mode-2 natural
frequency of the small-amplitude bubble oscillation, i.e.
$f_{2}=\sqrt{96\sigma/\rho_{c}D^{3}}$. The natural oscillation timescale was
proposed as an important timescale by Sevik & Park (1973) and Risso & Fabre
(1998) based on the physical picture of a bubble resonating with turbulent
perturbations at its natural frequency. However, with strong buoyancy, Ravelet
et al. (2011) suggested that the preferential sliding motion between the two
phases significantly changes the deformation dynamics, leading to breakup
driven by single intense eddies rather than the stochastic resonance observed
under microgravity ($We_{t}>Eo$). This work implied that persistent
deformation in one direction could alter the deformation dynamics driven by
turbulence more than just adding to the stress.

#####  3.1.2.2 Shear dominated regime ($We_{t}<We_{\mathcal{S}}$)

Levich (1962) considered the breakup of small drops immersed in the
logarithmic sub-layer of a turbulent boundary layer (TBL). The mean velocity
parallel to the wall, $\langle U\rangle$, in the wall normal direction ($y$)
is described by $\langle U\rangle=U_{\tau}\ln(y/y_{0})/\kappa$, where
$U_{\tau}=\sqrt{\tau_{w}/\rho_{c}}$ is the friction velocity and is determined
by the wall shear stress ($\tau_{w}$). The characteristic length scale is
expressed as $\delta_{0}=\nu_{c}/U_{\tau}$ or $y_{0}=\delta_{0}/9$.
$\kappa\approx 0.4$ is the von-Kármán constant. Levich (1962) argued that the
pressure gradient that drives the drop deformation is dominated by the
persistent large-scale shear across the drop size $D$ from $y$ to $y+D$.
Assuming $d\ll y$, the Weber number ($We_{\mathcal{S}}$) can be expressed as a
function of $y$

$We_{\mathcal{S}}=\frac{2\rho_{c}U_{\tau}^{2}D^{2}}{\kappa^{2}y\sigma}\ln\frac{y}{y_{0}}\quad\mathrm{and}\quad
We_{\mathcal{S,\text{max}}}=\frac{\ln(180)\rho_{c}U_{\tau}^{3}D^{2}}{10\kappa^{2}\nu_{c}\sigma}\approx\frac{3\rho_{c}U_{\tau}^{3}D^{2}}{\kappa^{2}\nu_{c}\sigma}$
(3)

where $We_{\mathcal{S,\text{max}}}$ is the largest value of $We_{\mathcal{S}}$
that can be reached near the bottom end of the log layer ($y\approx
20\delta_{0}$). Equation 3 is slightly different from the original work by
Levich (1962) after I corrected some issues, e.g. the assumption of the bottom
of the log layer at $y\approx e\delta_{0}$. Although it was not explicitly
mentioned in the work by Levich (1962), $We_{\mathcal{S,\text{max}}}$ can be
re-written as $We_{\mathcal{S,\text{max}}}=3Re_{d}\tau_{w}D/\kappa^{2}\sigma$,
which is essentially the Weber number based on the wall shear stress
($We_{\tau}=\tau_{w}D/\sigma$) multiplied by the droplet Reynolds number
$Re_{c}=U_{\tau}D/\nu_{c}$ based on $U_{\tau}$ and the carrier-phase fluid
properties.

Yi et al. (2021) studied the behavior of an oil-water emulsion in a Taylor-
Couette (TC) system, which consists of a fluid layer between two counter-
rotating cylinders. The resulting flows featured two thin turbulent boundary
layers (TBLs) near the surfaces of the inner and outer cylinders, leaving a
larger bulk region with more homogeneous and isotropic turbulence. Yi et al.
(2021) first employed the KH framework, assuming that most of the droplet
breakup occurred within the bulk region. Although the scaling law was
verified, as the results were reanalyzed (Yi et al., 2022, 2023), it became
evident that the value of the critical Weber number is much smaller than 1,
from 0.013 to 0.018 using the bulk turbulence, implying that the turbulent
stresses were not sufficient to overcome the interfacial tension in the bulk.
This led them to conclude that the majority of droplets stayed in the bulk but
most breakup occurred in the TBLs. When using Levich’s definition of Weber
number, Yi et al. (2022) found that the critical Weber number is close to 5,
which is order unity and more reasonable. This finding suggests that the
breakup was indeed driven primarily by the mean shear in the TBL, whose
thickness was around 5 times the average droplet diameter.

Bubble breakup also occurs when they are directly injected into the near-wall
region of a TBL. Madavan et al. (1985) found that the bubble size in the TBL
is determined by the free-stream velocity and gas flow rate, and is not
affected by the method of gas injection. This finding implies that the size
distribution of bubbles is primarily controlled by breakup and coalescence.
Rather than following equation 3, Pal et al. (1988) proposed a new way to
calculate the Weber number by estimating the local energy dissipation rate
$\epsilon\sim U_{\tau}^{3}/\theta$ ($\theta$ is the momentum thickness)
experienced by bubbles based on the turbulence within the boundary layer.
Sanders et al. (2006) revised this definition, replacing the momentum
thickness $\theta$ with $\kappa y$, resulting in a new definition of the
turbulent Weber number.

$We_{t}=\frac{2\rho_{c}U_{\tau}^{2}D^{5/3}}{(\kappa y)^{2/3}\sigma}$ (4)

Bubbles located approximately $y=$1 mm ($y\approx 370\delta_{0}$) away from
the wall in flows with a friction velocity of $U_{\tau}=0.37$ m/s have been
observed to have a size distribution of 320$\pm$130 $\mu$m (Sanders et al.,
2006). This size distribution can be explained by assuming $We_{t}$ in
equation 4 has a critical value. But it is important to note that, for a
critical $We_{t}$ near unity, $We_{\mathcal{S}}$ using equation 3 is roughly
19.

####  3.1.3 Mixed regime ($We_{t}\approx We_{\mathcal{S}}$)

Most turbulent flows, e.g. homogeneous shear flow (Rosti et al., 2019,
Trefftz-Posada et al., 2023), turbulent pipe or channel flows (Angeli &
Hewitt, 2000, Scarbolo et al., 2015, Mangani et al., 2022), breaking waves
(Garrett et al., 2000, Deane & Stokes, 2002), turbulent jets (Martínez-Bazán
et al., 1999), von Kármán swirling flow (Ravichandar et al., 2022), or stirred
tanks (Shinnar, 1961), typically involve both large-scale flows and
turbulence.

Efforts have been made to minimize the impact of large-scale flows by
injecting bubbles or droplets in regions that are closer to HIT, such as the
centerline of jets (Martínez-Bazán et al., 1999) or the center of the von
Kármán swirling flow (Ravichandar et al., 2022). However, as the injected
bubbles or droplets are carried away from the injection point, the influence
of large-scale flows may still be present.

In chemical or petroleum engineering, the size of oil droplets broken by
turbulence in pipes or stirred tanks is often studied using the critical Weber
number defined based on the global energy dissipation rate
$\langle\epsilon\rangle$, where $\langle...\rangle$ represents the ensemble
average over the entire device. For batch rotor-stator systems and
conventional stirred tanks, $\langle\epsilon\rangle$ scales with the rotor
speed $N$ and the rotor diameter $L$, as expressed by
$\langle\epsilon\rangle\sim N^{3}L^{2}$ (Rushton, 1950, Chen & Middleman,
1967). This $\langle\epsilon\rangle$ leads to the critical Weber number being
defined as $We^{c}_{t}=\rho_{c}N^{2}L^{4/3}D^{5/3}/\sigma$, from which the
critical drop size can then be determined.

In pipe flows, the global energy dissipation rate is expressed as
$\langle\epsilon\rangle=fU_{c}^{3}/2D_{p}$, where $f$ is the friction factor,
$U_{c}$ is the mean axial velocity of the continuous phase, and $D_{p}$ is the
pipe diameter. This leads to a critical Weber number that scales with
$Df^{2/3}$. However, Kubie & Gardner (1977) argued that the velocity scale
should be the fluctuation velocity, not the centerline velocity. The
fluctuation velocity is approximately equal to 1.3$U_{\tau}$, where
$U_{\tau}=(f/8)^{1/2}U_{c}$. Based on this, the critical Weber number can be
rewritten as $We^{c}_{t}=f\rho_{c}DU_{c}^{2}/\sigma$, which scales with $f$
instead of $f^{2/3}$. However, experiments conducted by Angeli & Hewitt (2000)
found that the critical drop size scales with $f^{-3}$, albeit from a very
narrow range of $f$. Thus, more experiments are needed to fully resolve the
debate and determine the appropriate relationship between the critical drop
size and friction factor in pipe flows.

####  3.1.4 Viscous effects in the inertia-dominated regime

For droplets with size $D\gg\eta$, the deformation is dominated by flow
inertia and the viscosity of the carrier phase can be neglected. However, the
inner viscous damping may still play a significant role in deformation
dynamics, comparable or even exceeding the impact of surface tension, as
quantified by the dimensionless number, $Oh$.

Davies (1985) and Calabrese et al. (1986) considered this problem and assumed
a total balance of stresses between the external forcing by turbulence and
internal damping: $\rho_{c}\left(\varepsilon
D\right)^{2/3}\sim\sigma/D+\mu_{\mathrm{d}}\left(\varepsilon
D\right)^{1/3}\sqrt{\rho_{c}/\rho_{d}}/D$. It can also be expressed in the
dimensionless form, $We_{t}\sim
c_{1}+c_{2}Oh^{2}Re_{d}\sqrt{\rho_{c}/\rho_{d}}$, where
$Re_{d}=\rho_{d}(\epsilon D)^{1/3}D/\mu_{d}$ is the droplet Reynolds number
based on the eddy velocity and inner fluid properties, and $c_{1}$ and $c_{2}$
are two fitting constants. Droplets will deform if the left side is larger
than the right side, which implies that quantities of interest $\mathcal{Q}$
(e.g. aspect ratio or breakup frequency) should be a function of the new
dimensionless number following

$\mathcal{Q}=f\left(\frac{We_{t}}{c_{1}+c_{2}Oh^{2}Re_{d}\sqrt{\rho_{c}/\rho_{d}}}\right)$
(5)

Equation 5 indicates that, for deformation and breakup, the primary
dimensionless number is $We_{t}$ when the inner viscous damping is negligible,
and $We_{t}/Oh^{2}Re_{d}\sqrt{\rho_{c}/\rho_{d}}$ when it is important to
consider.

To investigate the viscous effect, Eastwood et al. (2004) injected oil
droplets in a turbulent jet along the centerline, using the same setup as
Martínez-Bazán et al. (1999) in their investigation of bubble breakup. The
values of $Oh$ in the experiment ranged from $\mathcal{O}(10^{-2})$ to
$\mathcal{O}(10^{-1})$, and $We_{t}$ was roughly $\mathcal{O}(10)$. As
illustrated in figure 1b, the experiment revealed a clear long filament,
indicating a significant deformation preceding breakup. The extent of
stretching increased with an increase in droplet viscosity, and droplets
within the inertial sub-range stretched to lengths comparable to the local
integral scale before fragmentation. This long filament was confirmed by other
experiments (Andersson & Andersson, 2006, Solsvik & Jakobsen, 2015) and
simulations (Håkansson et al., 2022), and it was found to be connected to the
large number of daughter droplets generated from droplet breakup.

Recognizing the importance of this process, Maaß & Kraume (2012) adopted the
idea originally proposed by Janssen & Meijer (1993) to describe a drop
elongating in one dimension and thinning in the other two exponentially over
time, driven by a constant straining flow. Given a stretching rate, the
breakup time could be estimated once a critical diameter of the neck was
determined, which was assumed to be related to the critical Capillary number
based on $\mu_{d}$ and the stretching rate. However, this model did not
account for the instability of the filament itself (Ruth et al., 2022) or the
possible interruption by small-scale eddies, as it assumed a persistent
elongation at the scale of the filament.

Vankova et al. (2007) investigated the size of emulsion droplets produced
using a HPH with various oils, resulting in a range of $Oh$ from
$\mathcal{O}(10^{-1})$ to $\mathcal{O}(10)$. The authors adopted equation 5
and adjusted two constants, $c_{1}$ and $c_{2}$, to fit their experimental
results. The obtained values were $0.78$ and $0.37$ for $c_{1}$ and $c_{2}$
respectively. However, subsequent analysis by Zhong & Ni (2023) questioned the
validity of the linear combination of the restoring and dissipative terms in
equation 5 and proposed a new equation to better collapse all the data.

$\mathcal{Q}=f\left(\frac{We_{t}}{1+Oh}\right)$ (6)

$\mathcal{Q}$, in this case, is the non-dimensionalized breakup frequency.
This quantity will be further discussed in detail in Section 4. Note that this
relationship was established based on limited experimental data. In order to
further examine the validity of this relationship, it is possible to use
simulation databases, such as the one by Mangani et al. (2022), with well-
controlled characteristics that cover a broad range of density and viscosity
ratios in turbulence.

###  3.2 Viscous-dominated regime ($D<\eta$)

####  3.2.1 Experimental Methods

There are three main experimental methods for producing droplets with
$D<\eta$: (a) generating turbulence with high $\epsilon$; the typical value of
$\epsilon$ can be calculated by satisfying two criteria
$Ca_{t}=\sqrt{\rho_{c}\mu_{c}\epsilon}D/\sigma>Ca_{t}^{c}$ and
$D<(\nu_{c}^{3}/\epsilon)^{1/4}$. (b) reducing surface tension by adding
surfactants, and (c) increasing the viscosity of the carrier phase. In food
processing industries, including dairy, breaking droplets into nanometer sizes
is important for the desired texture, color, and stability for storage. Option
(b) or (c) is less ideal due to the required food-grade chemical additives,
leaving option (a) as the primary method. The HPH is the key technique for
this purpose (Håkansson, 2019). It uses a high-pressure piston pump (50–200
MPa) and a narrow gap ($\mathcal{O}(100)$ $\mu$m) to accelerate emulsions to
velocities of up to $\mathcal{O}(100)$ m/s, creating a localized turbulent jet
(Bisten & Schuchmann, 2016) with $\epsilon$ in the range of
$\mathcal{O}(10^{8})$ to $\mathcal{O}(10^{9})$ m2/s3 and fragmenting droplets
to sizes of $\mathcal{O}(10$–$100)$ nm.

The high-speed colloid mill (rotor-stator) system, is another commonly used
method for emulsion preparation. This type of system is similar to the high-
Reynolds-number TC system (van Gils et al., 2011, Grossmann et al., 2016), but
with a small gap size ($h=\mathcal{O}(100$) $\mu$m) and high rotor spin rate
(inner cylinder with the radius of $r_{i}=\mathcal{O}(10$) cm) at
$\omega_{i}=\mathcal{O}(10^{4}$) revolutions per minute (RPM), which results
in a moderate Reynolds number ($Re=\omega_{i}r_{i}h/\nu_{c}$) at around
$\mathcal{O}(10^{4}$) but a significant mean shear and turbulent energy
dissipation rate (Schuster et al., 2012).

####  3.2.2 Negligible inner viscosity $Oh\ll 1$

By systematically increasing the viscosity of the continuous oil phase
($\mu_{c}$) by two orders of magnitude while keeping the dispersed aqueous
phase constant (option c in the previous section), Boxall et al. (2012)
studied the transition of the dynamics of droplet breakup from the inertia-
dominated to the viscous-dominated regimes. The droplets were fragmented by
turbulence in a customized mixing cell driven by a six-blade impeller. The
droplet size was determined using the focused beam reflectance method, and the
average droplet size was calculated only after the steady state was reached,
which took approximately three hours.

If $Oh$ is negligibly small, the only dimensionless number that matters to the
problem is the capillary number ($Ca_{t}$). Assuming that a critical capillary
number ($Ca_{t}^{c}$) exists, Shinnar (1961) suggested that the critical
droplet size ($D^{c}$) can be determined as follows

$D^{c}=\frac{Ca_{t}^{c}\sigma}{\sqrt{\mu_{c}\rho_{c}\epsilon}}$ (7)

In the experiments conducted by Boxall et al. (2012), the impeller speed ($N$)
and diameter ($L$) were kept almost constant, so the energy dissipation rate
($\epsilon\sim N^{3}L^{2}$) did not vary significantly. As $\mu_{c}$
increased, it was shown that the droplet size remained unchanged for low
values of $\mu_{c}$, and scaled with $\mu_{c}^{-1/2}$ in the viscous-dominated
regime at high $\mu_{c}$. This finding provides direct support for Equation 7.

Figure 3: (a) The critical Capillary number measured from nanoemulsions made
with a homogenizer as a function of $Oh$. The black solid line shows the 2/5
scaling, with the blue shaded area representing the $\pm 0.1$ uncertainty in
the scaling exponent. (b) The critical drop size $D^{c}$ versus the viscosity
of the carrier phase for nanoemulsions made with two different types of
surfactants. Figures are adapted with permission from Wooster et al. (2008)
and Gupta et al. (2016a)

####  3.2.3 Large inner viscosity $Oh\gtrsim 1$

In the previous section, a water-in-oil emulsion was considered and the
viscosity of the dispersed phase was negligible in comparison to the
continuous phase. For other types of emulsions, such as oil-in-water,
$\mu_{d}$ is large and the viscous damping by the inner fluid cannot be
neglected, and it is likely that $Ca_{t}^{c}$, if it exists, is dependent on
the value of $Oh$.

To model this dependence, Gupta et al. (2016b) proposed a model based on the
physical picture of a part of the droplet, with the size of the instability
length scale, being extruded from the parent droplet due to the surrounding
turbulence. By assuming that the propagation timescale of the instability is
dominated by the viscous diffusion of eddy momentum into the droplet, a new
formulation was derived, $Ca^{c}_{t}\sim Oh^{2/5}$, indicating that the
critical Capillary number is not a constant, but a function of $Oh^{2/5}$.
Figure 3a shows the measured $Ca^{c}_{t}$ over a range of $Oh$ by
systematically varying the types of oils used in experiments. The oil droplet
size obtained from HPH was substituted into the definition of $Ca_{t}$ to
obtain its value. The proposed scaling seems to capture the scaling between
$Ca^{c}_{t}$ and $Oh$ well.

This model proposed by Gupta et al. (2016b) further predicts the critical
droplet size relative to all other fluid properties as follows:
$D^{c}=C_{1}(\sigma^{5/6}\mu_{d}^{1/3})/[(\rho_{d}\sigma)^{1/5}(\mu_{c}\rho_{c}\epsilon)^{5/12}]$.
Specifically, it implies that $D$ scales with $\mu_{d}^{1/3}$,
$\mu_{c}^{-5/12}$, and $\epsilon^{-5/12}$. The scaling was compared to
experimental data obtained by Wooster et al. (2008), who created an oil-in-
water emulsion with varying $\mu_{c}$ by adding various concentrations of
polyethylene glycol into water. The comparison is shown in figure 3b. Although
the proposed scaling of $D\sim\mu_{c}^{-5/12}$ by Gupta et al. (2016a) agrees
well with the data, it is difficult to distinguish it from the one proposed by
Shinnar (1961) ($D\sim\mu_{c}^{-1/2}$), which did not account for $\mu_{d}$
and $Oh$.

In addition, Wooster et al. (2008) actually reported two datasets using the
same emulsions with the only difference being the types of surfactant added.
The data adopted by Gupta et al. (2016a) is the one with 5.6 wt % sodium
dodecylsulphate (SDS, 98.5%). The data using 5.6 wt % polysorbate 80 (Tween
80, 98%) is also shown in figure 3b (triangles), which deviates noticeably
from the proposed -5/12 scaling, agreeing better with $D\sim\mu_{c}^{-1/6}$.
This difference implies the possible complexity introduced by surfactant and
coalescence.

Nevertheless, assuming the scaling proposed by Gupta et al. (2016a) is correct
and combining the two regimes, i.e. with or without significant viscous
effects, one can express quantities of interest in the problem of viscous
deformation and breakup using an equation similar to equation 5

$\mathcal{Q}=f\left(\frac{Ca_{t}}{c_{3}+c_{4}Oh^{2/5}}\right)$ (8)

where $c_{3}$ and $c_{4}$ are constants that are yet to be determined to
understand the critical Capillary number and the transitional $Oh$ from the
regime where the inner viscous damping is important to where it is not.

##  4 Deformation and breakup: timescales and dynamics

###  4.1 Characteristic timescales

The prediction of the evolution of the drop and bubble size distribution over
space and time can be captured by solving the population balance equation,
which is a Boltzmann-type equation. This approach has been widely implemented
in many simulation methods to predict the dynamics of polydispersed particles,
bubbles, and drops that constantly coalesce or break (Marchisio & Fox, 2013,
Shiea et al., 2020). In the population balance equation, there are three key
quantities to describe breakup, the breakup frequency, the daughter
bubble/droplet size distribution, and the number of daughters.

For breakup frequency, selecting the right timescale to non-dimensionalize it
is the first challenge. The discussions of the characteristic timescale of
deformation and breakup can be traced back to Section 127 of the book by
Levich (1962), who considered four different breakup timescales based on the
inner viscosity and interface velocity. These regimes can be determined by
estimating the magnitude of the three terms, the pressure gradient $\nabla
p/\rho_{d}$, unsteady term $\partial u_{d}/\partial t$, and viscous term
$\nu_{d}\nabla^{2}u_{d}$, in the wave equation that describes the inner fluid
motion during breakup. Four timescales have been proposed, as expressed in the
following equations.

$\frac{\partial u_{d}}{\partial t}\sim\frac{\nabla
p}{\rho_{d}}\sim\frac{p}{D\rho_{d}};\frac{\partial u_{d}}{\partial
t}\sim\frac{D}{\tau^{2}};p\sim\frac{\sigma}{D}\Rightarrow\tau\sim\sqrt{\frac{\rho_{d}D^{3}}{\sigma}}~{}(\textrm{low
viscosity, low speed})$ (9) $\frac{\nabla
p}{\rho_{d}}\sim\nu_{d}\nabla^{2}u_{d}\sim\frac{\nu_{d}}{D\tau};\frac{\nabla
p}{\rho_{d}}\sim\frac{p}{D\rho_{d}};p\sim\frac{\sigma}{D}\Rightarrow\tau\sim\frac{\mu_{d}D}{\sigma}~{}(\textrm{high
viscosity, low speed})$ (10) $\frac{\partial u_{d}}{\partial
t}\sim\frac{\nabla p}{\rho_{d}}\sim\frac{p}{D\rho_{d}};\frac{\partial
u_{d}}{\partial
t}\sim\frac{D}{\tau^{2}};p\sim\rho_{c}u_{c}^{2}\Rightarrow\tau\sim\frac{D}{u_{c}}\sqrt{\frac{\rho_{d}}{\rho_{c}}}~{}~{}(\textrm{low
viscosity, high speed})$ (11) $\frac{\nabla
p}{\rho_{d}}\sim\nu_{d}\nabla^{2}u_{d}\sim\frac{\nu_{d}}{D\tau};\frac{\nabla
p}{\rho_{d}}\sim\frac{p}{D\rho_{d}};p\sim\rho_{c}u_{c}^{2}\Rightarrow\tau\sim\frac{\mu_{d}}{\rho_{c}u_{c}^{2}}~{}~{}(\textrm{high
viscosity, high speed})$ (12)

where $\tau$ is the characteristic breakup timescale. $u_{d}$ is the
characteristic inner fluid velocity, which does not show up in the final
estimation of $\tau$ because $u_{d}$ scales roughly with $D/\tau$.

The eddy turnover time has been proposed as another characteristic timescale,
$t_{D}=\epsilon^{1/3}D^{-2/3}$, for describing bubble fragmentation in
breaking waves (Garrett et al., 2000, Deane & Stokes, 2002, Chan et al., 2021,
Gao et al., 2021). In particular, the scaling between $t_{D}$ and $D$ directly
results in the steady-state bubble size distribution scaling with $D^{-10/3}$,
which was also observed in droplet breakup in turbulence (Soligo et al., 2019,
Crialesi-Esposito et al., 2023a). Note that the eddy turnover time is, in
fact, in line with equation 11 given by Levich (1962), if the characteristic
velocity scale ($u_{c}$) of the outer flow is set as the eddy velocity at the
bubble size $u_{D}=(\epsilon D)^{1/3}$, as proposed in the KH framework. The
only difference left is that $t_{D}$ does not account for the density ratio
between the two phases.

Another proposed timescale is the natural oscillation frequency, $f_{2}$,
which is associated with the second eigenmode of weak-amplitude oscillations
(Lamb, 1879). Assuming inviscid fluids,
$1/f_{2}=\sqrt{(3\rho_{d}+2\rho_{c})D^{3}/30\sigma}$. Although similar to the
timescale listed in equation 9, there is an important distinction to note:
Levich’s model only accounted for the density of the inner fluid, whereas a
more complicated relationship with both $\rho_{d}$ and $\rho_{c}$ is required
for $1/f_{2}$.

###  4.2 Experimental results

Figure 4: Breakup frequency of (a) bubbles and (b) droplets normalized by the
eddy turnover frequency, $f_{e}=(\epsilon^{1/3}D^{-2/3})$, as a function of
the key dimensionless number. The datasets that were compiled include VZS18
(Vejražka et al., 2018), MML99 (Martínez-Bazán et al., 1999), SJ15 (Solsvik &
Jakobsen, 2015), VT07(Vankova et al., 2007), EAL04 (Eastwood et al., 2004),
VMA22 (Vela-Martín & Avila, 2022), HFSJ20 (Herø et al., 2020). Models include
the ones by CT77 (Coulaloglou & Tavlarides, 1977), QTN22 (Qi et al.,
2022),QMN20 (Qi et al., 2020), and WWJ03 (Wang et al., 2003).

Zhong & Ni (2023) compiled experimental results on the breakup frequency of
bubbles and oil droplets with sizes $D>\eta$ based on the recommendation made
by Håkansson (2020). The eddy turnover time of size $D$ was used as the
characteristic timescale, chosen from a list of options mentioned in the
previous section. Figure 4 clearly shows that the breakup frequency drops
sharply as $We_{t}$ decreases, indicating droplets or bubbles with smaller
$We_{t}$ take longer to break. Thus, it is evident that the definition of a
critical Weber number depends on the observation time (Vela-Martín & Avila,
2022). If one waits longer during an experiment, smaller droplets or bubbles
can be obtained for a given level of turbulence.

Although the data for bubbles (figure 4a) showed better agreement,
discrepancies were still noticeable at high $We_{t}$. Specifically, Martínez-
Bazán et al. (1999) reported a plateau in breakup frequency ($g$) close to 1,
while Vejražka et al. (2018) claimed that $g$ increased towards 10 without
reaching a plateau. This disparity could be due to the different experimental
conditions employed: the former was conducted in turbulence closer to HIT,
while the latter involved injecting bubbles along the centerline of a
turbulent jet. As the bubbles spread away from the centerline as they migrate
downstream, they may experience mean shear.

Breakup frequency ($g$)the fraction of bubbles/droplets that break per unit
time in turbulence

The inverse of the mean shear rate ($\mathcal{S}$) was suggested by Zhong & Ni
(2023) as another potential timescale, estimated by dividing the centerline
velocity of the turbulent jet by its width, following the canonical turbulent
jet. The estimated shear rate is plotted as a black solid line in figure 4a,
matching the measured breakup frequency quite well and providing an
alternative timescale to consider for breakup frequency in inhomogeneous and
anisotropic turbulence.

For droplet data in figure 4b, equation 6 mentioned in Section 3.1.4, proposed
by Zhong & Ni (2023), was used to compile different datasets with different
$Oh$ . The datasets provided by Vankova et al. (2007), with $Oh$ ranging from
$\mathcal{O}(10^{-2})$ to $\mathcal{O}(10)$, collapsed with one another well
using equation 6. However, when including all the datasets, a nearly two-
orders-of-magnitude variation in the breakup frequency was observed. This
difference primarily arises from the experiments conducted by Vankova et al.
(2007) in a homogenizer, where the drop size was in the order of
$\mathcal{O}(10^{-5})$ m, as compared to other experiments that involved much
larger drops of $\mathcal{O}(10^{-3})$ m, implying either large systematic
uncertainties between large- and small-scale experiments or a potential hidden
size dependence that was not accounted for in the current selection of
dimensionless groups.

The deformation and breakup of bubbles and droplets could potentially be
understood under a unified framework if a suitable set of dimensionless
numbers is chosen to collapse all available data. In an attempt to achieve
this, Zhong & Ni (2023) selected two models that represent the upper (Wang et
al., 2003) and lower (Qi et al., 2020) bounds of bubble experiments, as
illustrated by the shaded area in Figure 4a. The same shaded area was overlaid
twice on top of the droplet datasets in Figure 4b, once with (lower shaded
area with dashed lines as bounds) and once without (upper shaded area with
solid lines as bounds) the density ratio, $\sqrt{\rho_{d}/\rho_{c}}$, as
suggested by Levich’s timescales (Equation 11), to show how well the bubble
and droplet data collapse. The inclusion of $\sqrt{\rho_{d}/\rho_{c}}$
resulted in the collapse of most of the available bubble and droplet data,
except for the dataset by Vankova et al. (2007). In contrast, when the density
ratio was not considered, the bubble data showed better agreement with the
results by Vankova et al. (2007). This finding suggests that the existing
droplet data exhibits too much disparity to draw a definitive conclusion
regarding the effectiveness of including the density ratio term for
characterizing the breakup timescale.

###  4.3 Numerical simulations

In addition to experiments, with the development of more advanced direct
numerical simulation (DNS) algorithms for two-phase flows (Elghobashi, 2019)
and Graphics Processing Unit (GPU) based codes (Crialesi-Esposito et al.,
2023b), it is possible to conduct a large number of simulations of breakup
events to collect statistics. For example, Liu et al. (2021) implemented an
efficient simulation scheme for the phase-field method to simulate the breakup
of a large drop and the coalescence of $\mathcal{O}(10^{3})$ drops.

In addition to the simulation schemes, in general, two strategies have been
adopted so far. The first one involves a larger simulation domain with many
drops and a limited number of selected dimensionless numbers, and drops are
allowed to break and coalesce (Dodd & Ferrante, 2016, Roccon et al., 2017,
Scarbolo et al., 2015, Mangani et al., 2022, Crialesi-Esposito et al., 2023a).
This strategy is particularly suitable for investigating breakup at high
concentrations in complex environments that are relevant to many applications,
such as emulsions and breaking waves. Since it simulates both breakup and
coalescence in turbulence, it also helps illustrate the energy transferred
between the two phases (Dodd & Ferrante, 2016, Crialesi-Esposito et al.,
2023a).

The second method relies on a smaller domain with only one drop but many more
runs, ranging from hundreds (Rivière et al., 2021) to over 30,000 (Vela-Martín
& Avila, 2022), to cover a wider parameter space. The advantage of this method
is the isolation of the breakup events without the complication of
coalescence. This approach is particularly useful for investigating parameters
under which breakup takes a long time, a regime where experiments suffer from
large uncertainty and finite residence time.

###  4.4 Deformation and breakup models

If reliable models for describing deformation and breakup can be developed, it
is much more computationally efficient to integrate these models along the
Lagrangian trajectories of point bubbles/droplets in turbulence to study their
breakup frequency and probability. In the following, we review some of the
deformation and breakup models.

For viscous fluids, Maffettone & Minale (1998) developed a model (M&M) to
describe the evolution of both shape and orientation of neutrally-buoyant
spheroidal droplets in a linear velocity gradient. This model was validated
against several experimental studies in low Reynolds number. Recently, the
model has been applied to simulating the deformation of many sub-Kolmogorov-
scale neutrally-buoyant droplets ($D\ll\eta$) in turbulence by integrating the
M&M equation numerically along their Lagrangian trajectories (Biferale et al.,
2014, Spandan et al., 2016).

For inertia-dominated deformation and breakup, it is much more challenging to
develop a deformation model. One model simplified the problem by ignoring the
orientation and proposed to describe a droplet as a linear damped oscillator
that is forced by the instantaneous turbulent fluctuations at the drop scale
(Risso & Fabre, 1998, Lalanne et al., 2019). The equation can be written in a
dimensionless form as follows,

$\frac{d^{2}\hat{a}}{d\hat{t}^{2}}+2\xi\frac{d\hat{a}}{d\hat{t}}+\hat{a}=K^{\prime}We_{t}(t)$
(13)

where $\hat{a}$ represents the difference between the semi-major axis of the
deformed geometry and the spherical-equivalent radius divided by $D/2$. The
damping coefficient is given by $\xi=1/2\pi\tau_{d}f_{2}$, where $\tau_{d}$ is
the damping time scale defined as $\tau_{d}=D^{2}/80\nu_{c}$ for bubbles
(Risso & Fabre, 1998) but a much more complicated implicit solution for
droplets (Lalanne et al., 2019). In contrast to previous models that assumed
an additive relationship between viscous stress and surface tension (Davies,
1985, Calabrese et al., 1986), this model correctly incorporated the
dissipative nature of viscous damping, and it has been successfully compared
to experimental data on breakup statistics, even in inhomogeneous turbulence
(Galinat et al., 2007, Maniero et al., 2012).

To account for the orientation and add multiple deformation mechanisms, Masuk
et al. (2021a) adapted the MnM model into the inertia-dominated regime by
making three important modifications: (a) Velocity gradients were coarse-
grained at the size of the bubble; (b) Deformation due to slip velocity was
accounted for by using a pseudo-strain-rate tensor; and (c) A pseudo-rotation
tensor was added to model the wake-induced bubble rotation. The modified model
has been successfully used to predict deformation and orientation for bubbles
in both turbulence and quiescent media, with the predicted statistics agreeing
well with experimental data. These findings suggest that the modified model
effectively captures the key mechanisms responsible for inertial deformation
and breakup.

###  4.5 Recent models for breakup frequency

Recent advances in modeling breakup mechanisms have highlighted the importance
of several previously neglected factors, including gas density, eddies of
different sizes, and turbulence intermittency, which we will summarize here.
In the past, air bubbles were often modeled as having negligible density and
viscosity. However, it has been shown that bubbles made of heavier gases can
break more frequently (Wilkinson et al., 1993). This phenomenon was explained
by Andersson & Andersson (2006), who pointed out that deformation typically
results in a dumbbell shape with two uneven ends. As the smaller end retracts
due to surface tension, air flow accelerates through the neck, which reduces
the local pressure and speeds up the breakup process. Larger gas density tends
to lower the local pressure and shorten the breakup time even further. These
observations and proposed mechanisms have inspired new models developed by
Xing et al. (2015) and Zhang et al. (2020), which incorporate the effect of
backflow and gas density.

In addition to the density effect, to accurately model bubble-eddy collision,
it is crucial to determine which eddy scales should be considered. The KH
framework assumes that only the drop-scale eddy is significant, with both
larger and smaller scales being negligible. Conversely, some models consider
eddies of all length scales from the turbulent spectrum (Karimi & Andersson,
2018, Castellano et al., 2019). However, recent research by Vela-Martín &
Avila (2021), which employed direct numerical simulation of a single drop
being deformed in turbulence, found that the impact of eddies with different
length scales on the variation of surface free energy is not equal. Turbulent
fluctuations at scales smaller than the drop diameter cause the majority of
surface deformation, while the contribution of scales close to or larger than
$D$ is relatively minor.

Qi et al. (2022) designed an experiment using the head-on collision between
two vortex rings to isolate the turbulent scales. During the early stage
before the collision, only intact large-scale vortices were accessible, while
the post-collision late stage was filled with many small eddies. Despite a
lower overall $We_{t}$ in the late stage, bubbles were found to break up in a
more violent and faster manner due to the presence of small eddies. Building
on this finding, the authors developed a new model that considers not only the
stress criterion, which requires the incoming eddy to exert sufficient stress
to overcome the restoring surface tension, but also the time scale. The
breakup must occur within the time before the bubble relaxes. This key idea
emphasizes that, instead of being gradually and consistently stretched by
flows at their own length scales, bubbles are fragmented by small eddies,
resulting in a sudden and intense local deformation over a short period of
time. The predicted breakup frequency as a function of $We_{t}$ is shown as
the purple solid line in figure 4a, which agrees with the experimental data by
Vejražka et al. (2018).

Numerous studies have investigated the effect of turbulence intermittency on
bubble breakup. Recent models have examined the impact of intermittency on the
turbulent energy spectrum, as noted by Bagkeris et al. (2021) and Solsvik &
Jakobsen (2016). However, the effect of intermittency on the distribution of
$\epsilon$, which can be derived from the multi-fractal model and described by
a log-normal distribution (Meneveau & Sreenivasan, 1991), is more pronounced
than that on the energy spectrum. This distribution can be incorporated into
modeling quantities such as the breakup probability (Masuk et al., 2021c),
eddy velocity (Qi et al., 2022), and breakup frequency (Qi et al., 2020).

In particular, Qi et al. (2020) modified the model originally proposed by
Martínez-Bazán et al. (1999) to account for the non-negligible breakup
frequencies for small bubbles when exposed to intermittent turbulent eddies.
The model prediction is shown as the blue solid line in figure 4a and the
lower bounds for the two shaded areas in figure 4b. The classical model by
Coulaloglou & Tavlarides (1977) (black dashed line) fits the data by Vankova
et al. (2007) well. However, for most other datasets, a slower decay of the
breakup frequency $g$ as $We_{t}$ decreases is observed, which is better
predicted by Qi et al. (2020).

##  5 Modulation of mass, momentum, and heat transfer by deformation

###  5.1 Deformation affecting effective bubble forces

The motion of large bubbles and droplets in turbulence can be characterized by
the combined effect of multiple hydrodynamic forces, such as buoyancy, drag,
lift, added mass, Basset history, and pressure forces (Magnaudet & Eames,
2000, Sridhar & Katz, 1995). As the majority of these forces are shape-
dependent, it is not surprising that bubble and droplet deformability can
significantly impact their translational motion and local concentration in
turbulence.

Most research on forces experienced by bubbles has focused on their behavior
in laminar shear (Legendre & Magnaudet, 1998, Tomiyama et al., 2002, Lu &
Tryggvason, 2008, Dijkhuizen et al., 2010, Hessenkemper et al., 2020). Bubble
deformation, driven primarily by buoyancy, is measured by $Eo$. As $Eo$
increases, the lift force undergoes a transition from positive to negative
values. This shift in direction is attributed to the stretching and tilting of
vorticity generated at the bubble surface, which transforms into a pair of
counter-rotating streamwise vortices in the bubble wake. These vortices have
the opposite sign compared to those produced around a spherical bubble,
resulting in a negative lift force. In addition to vorticity production,
direct asymmetric deformation caused by external shear can also lead to
negative lift (Zhang et al., 2021, Hidman et al., 2022).

In turbulence, Sugrue (2017) proposed a new dimensionless number taking the
product of $Eo$ and the ratio between the local turbulent kinetic energy and
the squared relative velocity between the two phases. This new number is
linked to the Weber number based on the fluctuation velocity. The authors
carried out extensive simulations, varying lift coefficients, and minimizing
the differences between experimental and simulated results. This allowed them
to extract lift coefficients for different flow conditions. The results showed
that the lift coefficients exhibited a similar inversion to those observed in
laminar shear flow. However, two key differences were noted. Firstly, the
magnitude of the coefficients was much smaller, and secondly, the inversion
diameter was smaller for turbulence-driven cases.

To measure the lift coefficient in turbulence, Salibindla et al. (2020)
conducted an experiment in nearly HIT. Although the flow does not have a mean
shear, the bubbles were constantly subjected to local shear and vorticity. The
transition of bubble rising velocity in turbulence from lower to faster than
its counterpart in an otherwise-quiescent medium was found as the bubble size
increased. Based on this finding and the access to the statistics of both
phases, the lift and drag forces experienced by bubbles with different sizes
were determined, and the lift inversion at smaller bubble size was observed
experimentally. The lift inversion was correlated to the turbulence-induced
deformation measured by $We_{t}$, which is close to 1 as the inversion occurs,
suggesting that turbulence-induced bubble deformation becomes essential. The
transition of the bubble’s rising velocity was linked to the preferential
sampling of different regions (upward or downward) in turbulence. This work
also supports the mechanism proposed by Spelt & Biesheuvel (1997) that small
spherical bubbles tend to preferentially sample the downward flows in
turbulence instead of being trapped in the vortex cores (Wang & Maxey, 1993)
and also quantitatively explains other previous experiments (Poorte &
Biesheuvel, 2002, Aliseda & Lasheras, 2011, Prakash et al., 2012).

It is worth noting that the transition of bubble rising velocity was not
observed in another work conducted by Ruth et al. (2021). In this study, the
change in rise velocity was attributed mainly to drag rather than lift,
highlighting the need for further investigation into how deformable bubbles
modify lift and drag forces in intense turbulence where deformation is driven
by local turbulence instead of buoyancy. Nevertheless, once lift and drag
forces are determined, the added mass force can also be evaluated
experimentally. Recent work by Salibindla et al. (2021) showed that the added
mass force experienced by bubbles in turbulence can be accurately modeled
using the solid spheroid approximations (Lamb, 1879). These findings suggest
that the instantaneous added-mass force experienced by deformable bubbles can
be approximated by appropriately oriented spheroids with the correct
instantaneous aspect ratios.

###  5.2 Turbulent drag reduction

Figure 5: The drag of turbulent Taylor–Couette flow during its transition from
non-boiling (grey shaded areas) to boiling at $t=t_{boil}$ and the key
quantities, including (a) Liquid temperature $T_{TC}$, (b) volume fraction
$\alpha$, (c) drag reduction (DR) as a function of time; Two time steps
correspond to the photographs shown in (d) and (e). Figures are adapted with
permission from Ezeta et al. (2019).

The deformation and breakup of bubbles and drops in turbulent boundary layers
have been extensively studied in the context of drag reduction (Ceccio, 2010,
Murai, 2014) in various configurations, including turbulent TC (van Gils et
al., 2013), flat plates (Sanders et al., 2006), channel flows (Murai et al.,
2007, Tanaka et al., 2021), and even under the model ship hull (Tanaka et al.,
2022). Several mechanisms have been proposed to explain the origin of the
bubble-induced drag reduction effect (Ferrante & Elghobashi, 2004, Lu et al.,
2005, Lu & Tryggvason, 2008, van den Berg et al., 2007). The successful drag
reduction experiments have been summarized by Murai (2014) in two regimes:
relatively small bubbles in high-speed flows or large bubbles in low-speed
flows. A recent overview of this topic was presented by Lohse (2018), who
highlights the difference between bubble-induced drag reduction at small
Reynolds numbers and large Reynolds numbers, attributed to the effects of
Froude number and Weber number, respectively.

While most experiments on bubble-mediated drag reduction focused on large-
scale averaged skin friction, a few studies measured the couplings between the
two phases. Kitagawa et al. (2005) conducted 2D simultaneous measurements of
both phases in a horizontal turbulent channel to determine the mechanism of
drag reduction caused by bubbles. The bubbles, which were roughly 530 $\mu$m
and deformable close to the wall, were shown to cause a drop in the Reynolds
stress of the carrier phase. The reduction ratio was almost the same as that
of the skin friction coefficient. Murai et al. (2007) measured the fluctuation
of the wall shear stress due to the passage of individual deformable bubbles
with size comparable to the boundary layer thickness using a shear transducer.
They found that the bubbles considerably reduced the local wall shear stress,
which was induced by the two roll vortices upstream and downstream of the
bubble that modified the local turbulent shear stress (Oishi & Murai, 2014).

Drag reduction has been a topic of extensive study in the context of high-
Reynolds-number TC systems because it is a closed system where frictional drag
can be measured as the global torque. van Gils et al. (2013) showed that the
system transitions from moderate drag reduction of 7% to a more significant
one with nearly 40% drag reduction at Reynolds number ($Re$) above $10^{6}$
and gas void fraction of 4%. This transition was observed as the Weber number
crosses one and the bubble becomes more deformable, even as the size becomes
smaller as $Re$ increases. As $Re$ increases and drag decreases, a larger
bubble aspect ratio was observed (as shown in figure 1a), signaling the
connection between deformation and drag reduction. This point was further
supported by Verschoof et al. (2016), who showed that the large drag reduction
(40%) could be ‘turned off’ by adding some surfactant. The surfactant reduces
surface tension and hinders coalescence, which leads to much smaller bubbles
with smaller $We_{t}$. Similar levels of drag reduction were also observed in
boiling TC driven by vapor bubbles, again due to their deformation in
turbulence, as shown in figure 5. As the vapor bubble volume fraction $\alpha$
increases, the probability of finding a larger value of $We_{t}$ increases,
probably due to the presence of more-deformable larger bubbles formed by
coalescence. Finally, a recent study by Wang et al. (2022) investigated how
viscosity ratios between the two phases affect drag in TC and found that the
drag coefficient increases as the inner viscosity increases and drop
deformability weakens, further reaffirming the importance of deformation in
turbulent drag reduction.

Extensive simulations have been also conducted to explore the potential
mechanism of drag reduction driven by deformable bubbles/droplets. Iwasaki et
al. (2001) demonstrated that the droplets can attenuate near-wall streamwise
vortices via deformation. Lu et al. (2005) found that large deformable bubbles
can lead to significant drag reduction by suppressing streamwise vorticity
near the wall, whereas less-deformed bubbles tend to bring additional shear
rate near the viscous sublayer to increase drag. Spandan et al. (2018)
reported that deformable bubbles can reduce drag in TC flows by modulating
dissipation in their wakes, regardless of whether the carrier fluid is weakly
or highly turbulent. Overall, these studies underscore different mechanisms at
play in bubble/droplet-mediated turbulent drag reduction through a deformable
interface.

###  5.3 Turbulence modulation

Dodd & Ferrante (2016) performed direct numerical simulations to investigate
the behavior of finite-sized drops in decaying isotropic turbulence, exploring
a range of Weber numbers, density ratios, and viscosity ratios between the two
phases. In this work, the turbulence kinetic energy (TKE) equations were
derived to capture the energy transfer between two phases, with a particular
focus on the role of interfacial energy. It was shown that, while the presence
of droplets always enhances the dissipation rate near the droplet interface,
the initial turbulence decay rate is faster in the presence of more deformable
drops (i.e., larger Weber numbers). However, the decay rate becomes
independent of the Weber number later on, likely because the turbulence has
decayed to a point where it is no longer strong enough to deform or break any
drops. The study also demonstrated that droplet coalescence acts as a source
of TKE through the power of surface tension, while breakup serves as a sink of
TKE.

In their investigation of turbulence modulation at different scales, Freund &
Ferrante (2019) analyzed the same data generated by Dodd & Ferrante (2016)
using wavelet transforms instead of Fourier transforms because wavelet
restricts the effects of non-smoothness locally while preserving spatial
information. At a distance larger than 5$\eta$ or $D/4$, the carrier-phase
spectra remained nearly unaffected, but the energy at high wavenumbers
increased close to the drop interface due to enhanced local velocity
gradients. They also observed that drops with larger density ratios reduced
the energy at low wavenumbers compared to neutrally-buoyant drops. In a
separate study, Scarbolo et al. (2013) examined the interaction between
turbulence and one large deformable droplet and showed that the presence of
the interface results in vorticity generation and turbulence damping near the
interface, and the distance from the interface where these effects are present
depends on the surface tension.

The spectrum analysis of turbulence modulation by drops in HIT was also
studied by Mukherjee et al. (2019). They showed that the presence of dispersed
drops leads to a transfer of energy from large scales to small scales, as the
drops subtract energy from the former and inject it into the latter. This
transfer of energy is reflected in the energy spectra, which cross the spectra
of the single-phase turbulence at a length scale close to the Kolmogorov-Hinze
scale, as initially proposed by Perlekar et al. (2014) for a different system.
Crialesi-Esposito et al. (2022) provided further insights into the mechanisms
behind this phenomenon, showing that surface tension forces play a key role in
absorbing energy from large scales and reducing its transfer through advection
terms. Eventually, this energy is transferred to small scales by surface
tension. They also noted that the modulation of turbulence spectra is more
sensitive to the viscosity ratio, while the scale-by-scale energy budget
depends more on the volume fractions.

Bubble-induced turbulence in a swarm of rising bubbles in an otherwise
quiescent fluid has been extensively studied in recent years. The phenomenon
has been investigated (Riboux et al., 2010, Innocenti et al., 2021, Pandey et
al., 2022), modelled (Ma et al., 2017, Du Cluzeau et al., 2019), and also
recently reviewed by Risso (2018) and Mathai et al. (2020). Mercado et al.
(2010) measured the energy spectrum of the carrier phase using a phase-
sensitive anemometry and found that the energy decays with wave number
following a power law of $-3.2$, which is close to the $-3$ scaling proposed
by Lance & Bataille (1991). They also observed that even at a small gas volume
fraction, typically from 0.28% to 0.74%, deformable bubbles tend to cluster
along the vertical direction at both small and large scales, which was
attributed to deformation (Bunner & Tryggvason, 2003).

###  5.4 Heat and mass transfer

The study of heat and mass transfer in turbulent multiphase flows is a complex
and multifaceted topic that deserves a dedicated review because a wide range
of relevant interfacial transfer phenomena, including boiling/condensation
(Russo et al., 2014), dissolution (Mac Huang et al., 2015, Farsoiya et al.,
2023), melting (Machicoane et al., 2013), evaporation (Birouk & Gökalp, 2006,
Duret et al., 2012, Marié et al., 2014, Méès et al., 2020), and the induced
Stefan flow, can be potentially modulated by turbulence and a deformable
interface.

Deformation and breakup have been shown to affect heat and mass transfer.
However, the extent of their influence on these processes is not yet fully
understood beyond their effect on the size distribution and interfacial area.
Recently, Albernaz et al. (2017) investigated the deformation and heat
transfer of a single drop in HIT and found a negative correlation between
local curvature and temperature on the droplet surface. Wang et al. (2019)
found that the kinematics of deformable bubbles and droplets could
significantly enhance the heat transfer in turbulent convection, and revealed
that the emergent size distribution of the bubbles and droplets in the system
governed the degree of heat transfer enhancement achievable. Dodd et al.
(2021) used direct numerical simulation (DNS) to study finite-size,
deformable, and evaporating droplets in HIT, and they showed that higher
surface curvature induced by deformation and breakup leads to higher
evaporation rates, especially for cases with large Weber numbers. Shao et al.
(2022) demonstrated that the Stefan flow induced by evaporation reduces the
coalescence rate and attenuates the turbulence kinetic energy. Scapin et al.
(2022) extended the problem to homogeneous shear flow and found that the
larger surface area due to deformation leads to an overall larger mass
transfer rate for drops with higher Weber numbers in persistent mean shear.
They also observed a weak correlation between the interfacial mass flux and
curvature at high temperature and a positive correlation at large Weber
number, low ambient temperature, and slower evaporation. Boyd & Ling (2023)
simulated the aerodynamic breakup of an acetone drop in a high-speed and high-
temperature vapor stream and showed that as the drop deforms, the increase of
frontal surface area results in a significantly increased rate of evaporation
and a nonlinear decrease in drop volume over time.

[SUMMARY POINTS]

  1. 1.

New experiments capable of measuring the shape of deforming bubbles and drops
simultaneously with the detailed surrounding turbulence in 3D have been made
possible with the advancement of diagnostic methods and new facilities that
can generate controlled turbulence.

  2. 2.

Turbulence in many applications is usually inhomogeneous and anisotropic.
Deformation and breakup in these systems are often subjected to both a non-
uniform distribution of turbulence intensity and a persistent large-scale
shear.

  3. 3.

The primary dimensionless parameter for breakup driven by the flow inertia of
the carrier phase ($D\gg\eta$) is $We/(1+Oh)$, while for breakup driven by the
viscous stress of the carrier phase ($D\ll\eta$), it is
$Ca/(c_{3}+c_{4}Oh^{2/5})$. These relationships were established based on
limited data, and further studies are required to validate them.

  4. 4.

Deformation is driven by large-scale eddies and breakup is accelerated by
small-scale eddies. The multiscale nature of breakup is a key component in
understanding the inertia-dominated breakup.

  5. 5.

Existing works showed the intricacies of the interplay between the local
interface curvature, local interfacial mass flux, and the induced Stefan flow
for two-phase heat and mass transfer with a deformable interface.

[FUTURE ISSUES]

  1. 1.

Understanding the potential synergies between large-scale shear and local
turbulence in deforming and breaking bubbles/droplets requires additional
research.

  2. 2.

The impact of density ratio, Reynolds number, and additives such as salt and
surfactant on deformation and breakup should be further investigated, as their
roles remain poorly understood despite previous research.

  3. 3.

Additional research is needed to better understand how the time history of
bubbles or droplets affects their deformation and breakup processes as they
traverse through inhomogeneous turbulence with varying local intensities.

## DISCLOSURE STATEMENT

The author is not aware of any affiliations, memberships, funding, or
financial holdings that might be perceived as affecting the objectivity of
this review.

## ACKNOWLEDGMENTS

The author thanks colleagues who provided advice and suggestions on the
manuscript. The author also acknowledges support from NSF grant CBET 1854475
and CAREER-1905103 and Office of Naval Research grant N00014-21-1-2083 and
N00014-21-1-2123.

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