34th Symposium on Naval Hydrodynamics Washington D.C. June 26 July 1 2022 The Dynamics of Drop Breakup in Breaking Waves

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34th Symposium on Naval Hydrodynamics
Washington, D.C., June 26 – July 1, 2022
The Dynamics of Drop Breakup in Breaking Waves
W.H.R. Chan (Center for Turbulence Research, Stanford University, USA;
Present Address: University of Colorado Boulder, USA)
ABSTRACT
Breaking surface waves generate drops of a broad
range of sizes that have a significant influence on
regional and global climates, as well as the
identification of ship movements. Characterizing these
phenomena requires a fundamental understanding of
the underlying mechanisms behind drop production.
The interscale nature of these mechanisms also
influences the development of models that enable cost-
effective computation of large-scale waves. Interscale
locality implies the universality of small scales and the
suitability of generic subgrid-scale (SGS) models,
while interscale nonlocality points to the potential
dependence of the small scales on larger-scale
geometry configurations and the corresponding need
for tailored SGS models instead. A recently developed
analysis toolkit combining theoretical population
balance models, multiphase numerical simulations,
and structure-tracking algorithms is used to probe the
nature of drop production and its corresponding
interscale mass-transfer characteristics above the
surface of breaking waves. The results from the
application of this toolkit suggest that while drop
breakup is a somewhat scale-nonlocal process, its
interscale transfer signature suggests that it is likely
capillary-dominated and thus sensitive not to the
specific nature of large-scale wave breaking, but rather
to the specific geometry of the parent drops.
INTRODUCTION
Breaking waves on the surfaces of oceans generate
drops of a broad range of scales (de Leeuw et al., 2011;
Veron, 2015; Wang et al., 2016; Erinin et al., 2019;
Deike, 2022; and references therein). These drops
have a significant influence on weather- and climate-
relevant ocean–atmosphere interactions, including the
enhancement of near-ocean-surface mass, momentum,
and energy transfer (Andreas, 1992; Andreas et al.,
1995; Veron, 2015; Deike, 2022), the distribution and
activation of organic material (Cunliffe et al., 2013;
Deike, 2022), as well as the production of nucleation
sites for cloud formation (Tao et al., 2012; Veron,
2015; Fan et al., 2016). The latter is especially relevant
in the naval context since ships have been observed to
generate trails of thunderclouds in their wake (Chang,
2017). Near-surface air–sea fluxes also modify the
atmospheric and oceanic boundary layers with
significant impact on ship operation and performance
(Andreas and Emanuel, 2001). Understanding the
generation of these ship trails and boundary-layer
modifications, and more generally the impact of drops
and their successors on our weather, climate, and
environment, require a thorough characterization of
the dynamics of drop formation and evolution.
The drop size distribution has been directly
characterized by a number of laboratory and field
experiments (Wu, 1979; Koga, 1981; Wu et al., 1984;
de Leeuw, 1986; Smith et al., 1993; Anguelova et al.,
1999; de Leeuw et al., 2000; Fairall et al., 2009; Veron
et al., 2012; Erinin et al., 2019, 2022; Mehta et al.,
2019), as well as recent numerical simulations (Wang
et al., 2016; Mostert et al., 2022). These results offer
a glimpse into the physical mechanisms behind drop
production. We have recently developed an analysis
toolkit that probes the causes behind the dynamic
evolution of the drop size distribution through a
combination of theoretical models based on
population balance analysis, numerical simulations
using an interface-capturing method, and structure-
tracking algorithms to identify and track individual
bubbles/drops and their associated breakup events
(Chan, 2020; Chan et al., 2021abc). This toolkit was
successfully used to probe the interscale nature of
turbulent bubble breakup in breaking waves, but is not
restricted to turbulent bubble fragmentation and may
be used to analyze a variety of multiphase flows in
different contexts. In this work, the interscale nature of
drop breakup above the surface of breaking waves is
investigated using the same analysis toolkit.
The objectives of this study are to utilize a
combination of theoretical analysis and numerical
simulations to probe the fundamental mechanisms
behind drop production in breaking waves by
analyzing the transfer of liquid mass within drops
between different drop sizes through the occurrence of
breakup events. The theoretical nature of the interscale
mass flux between drop sizes is specifically examined.
The interscale locality of this flux determines the
universality of the underlying drop breakup
mechanism and informs strategies for SGS model
development to enable cost-effective computation of
large-scale waves. A scale-local flux implies small-
scale universality and the feasible application of
universal SGS models, while a scale-nonlocal flux
suggests that models may have to be tailored to the
specific parameters of the flow since breakup is then
likely to be geometry dependent. The structure of this
paper is as follows: the employed methodology is
introduced, results obtained from application of the
analysis toolkit are discussed, and a summary and
outlook for future work are offered in the conclusions.
METHODOLOGY
The methodology employed in this work is discussed
in detail by Chan (2020) and Chan et al. (2021abc).
Key aspects of the methodology are recapitulated here.
Analysis toolkit workflow
Our analysis toolkit may be applied as follows: first,
an ensemble of numerical simulations of the flow of
interest is performed to resolve bubble and/or drop
breakup and/or coalescence processes and to obtain
sufficiently converged bubble and/or drop statistics in
the range of scales of interest. Second, structure-
tracking algorithms are applied to identify and track all
resolved bubbles and/or drops in the system. Bubble
and/or drop lineages are constructed and used to
identify breakup and/or coalescence events, which are
then used to evaluate the interscale bubble-mass
and/or droplet-mass flux. Third, the properties of the
measured flux are extracted and used to validate
existing theories or construct possible mechanisms for
bubble and/or drop formation. The cornerstone of this
analysis toolkit is the interscale mass flux, which will
be introduced next in the context of drop breakup.
The interscale droplet-mass flux
The interscale droplet-mass flux due to binary breakup
events,
, is defined as the rate of transfer of
liquid mass present in the form of drops that is lost by
all drops of sizes larger than a particular cutoff value
and gained by all drops of sizes smaller than . By
the principle of mass conservation, these two
quantities are equivalent. The breakup flux
at a particular time may be expressed as
 

 







where
and
are integration dummy variables
representing, respectively, the child and parent drop
sizes,


is the probability distribution of
sizes of child drops given a particular parent drop size
,

is the characteristic breakup rate of
drops of size
, and
 is the drop size
distribution of the droplet population.
Figure 1: Schematic of potential contributions to the
interscale droplet-mass flux as visualized in drop-size space.
The interscale flux
 is an
amalgamation of several possible contributions to the
rate of change of the drop size distribution .
Figure 1 illustrates the nature of these contributions.
The evolution of  may be expressed in terms of
the phenomenological population balance equation
 

 !"
 #$
% &'!(
)$
%*+,
-.
Here, we are neglecting the contributions of
nonbreakup events, such as coalescence events, to the
drop population. The effects of coalescence are
deferred to future work. Scale-nonlocal transport
carries droplet mass from one drop size to a distinct,
nonneighboring drop size and may be expressed as a
summation of discrete sources $
/01234
and sinks $
/567
.
The interscale flux due to scale-local transport,
80398
may be expressed as
80398
 :
;
which is the scale-local flux of
in drop-size space
(-space) carried at a characteristic speed :
. The
speed :
may be interpreted as the time  < =
taken for the droplet population to traverse a size range
> < . Observe that all the terms in Eq. (3) are
dependent only on (and ). This is a hallmark of
scale locality: the interscale flux across a certain cutoff
size is only dependent on quantities evaluated at
size , and no other parameters. In particular, it does
not depend on drop statistics at sizes much larger or
smaller than . If the actual interscale flux
were assessed to be highly local, then one may
reasonably approximate
? 
80398
@
and the effects of drop breakup in drop-size space can
be approximately described by a single quantity :
.
To analyze interscale locality, the interscale
flux
 may be decomposed into its differential
contributions from different parent drop sizes
, or
its differential contributions to different child drop
sizes
. The higher the degree of interscale locality in
each of the following metrics, the more concentrated
the differential contributions (integrands) are around
the cutoff size . The first metric, infrared locality,
may be expressed in terms of the following integrand
A

 
  





B
such that
  
A

- C
The interscale flux is highly infrared local if the
integrand A
decays as
increases away from . The
greater the decay rate, the higher the degree of infrared
locality. Figure 2 illustrates this behavior in a highly
infrared local process. In a completely infrared local
process, A
is a Dirac delta function centered at .
Figure 2: Schematic of infrared locality in drop-size space.
D
denotes the asymptotic decay exponent of A
.
The second metric, ultraviolet locality, may be
expressed in terms of the following integrand
A
  






E
such that
 
A
-F
The interscale flux is highly ultraviolet local if the
integrand A
decays as
decreases away from . The
greater the decay rate, the higher the degree of
ultraviolet locality. Figure 3 illustrates this behavior in
a highly ultraviolet local process. In a completely
ultraviolet local process, A
is a Dirac delta function
centered at . The analyses of infrared and ultraviolet
locality provide two distinct, but complementary,
ways of decomposing the interscale flux.
Figure 3: Schematic of ultraviolet locality in drop-size
space. D
denotes the asymptotic decay exponent of A
.
Analytical scaling laws for A
and A
may be
derived if the corresponding scaling laws for the
constituent quantities
,
, and are known. For
example, classical turbulence scaling laws for these
constituent quantities were used to derive the desired
scaling laws for turbulent bubble breakup. This is
currently not as feasible for the case of drop breakup
as the dominant underlying breakup mechanism
remains a subject of active research. In this work, we
focus on the measurement of A
and A
from numerical
simulations and their relation to the observed scaling
laws for
,
, and , and defer a more detailed
discussion of theoretical scaling laws to future work.
Breaking-wave simulation ensemble setup
Similar to the work described by Chan et al. (2018,
2019, 2021c), an ensemble of numerical simulations
of breaking waves was generated to obtain the desired
statistics. Details of the simulation parameters may be
found in the aforementioned references. In particular,
the simulations were initialized with a third-order
Stokes wave of steepness 0.55. The Reynolds and
Weber numbers of the wave, based on the wavelength
G and deep-water phase velocity H
I
JG=.K of the
fundamental mode, are respectively L(
I
 -FM
N
and O(
I
 -CM
, matching those of a 27-cm-
long water wave at atmospheric conditions. Note that
摘要：

34thSymposiumonNavalHydrodynamicsWashington,D.C.,June26–July1,2022TheDynamicsofDropBreakupinBreakingWavesW.H.R.Chan(CenterforTurbulenceResearch,StanfordUniversity,USA;PresentAddress:UniversityofColoradoBoulder,USA)ABSTRACTBreakingsurfacewavesgeneratedropsofabroadrangeofsizesthathaveasignificantinflu...

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