Journal of Harbin Institute of Technology (New Series) DOI ... Recent... · intermediate Froude numbers, the evolution of free surface exhibits capillary wave pattern. Quasi-steady

Journal of Harbin Institute of Technology (New Series)

Received 2019-04-18.

Sponsored by the National Natural Science Foundation of China (Grant Nos. 51879159, 51490675, 11432009, and 51579145), Chang Jiang Scholars Program (Grant No. T2014099), Shanghai Excellent Academic Leaders Program (Grant No. 17XD1402300), Program for Professor of Special Appointment (Eastern Scholar)

at Shanghai Institutions of Higher Learning (Grant No. 2013022), and Innovative Special Project of Numerical Tank of Ministry of Industry and Information

Technology of China (2016-23/09).

*Corresponding author. Distinguished Professor of Yangtze River Scholar. E - mail: [email protected].

1

DOI:10.11916/j.issn.1005-9113.19036

Review: Recent Advancement of Experimental and Numerical

Investigations for Breaking Waves

Cheng Liu, Yiding Hu, Zheng Li and Decheng Wan*

(State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering,

Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai Jiao Tong University,

Shanghai 200240, China)

Abstract: Breaking wave is a complex physical phenomenon that takes place at the gas-fluid interface, which is

the chief reason for the generation of two-phase turbulence, wave energy dissipation, and mass transfer between air

and water. For marine hydrodynamics, the breaking bow wave of high speed vessels induces the bubble-mixed flow

travelling around the ship, eventually developing to be the turbulent wake which is easy to be detected by

photoelectric equipment. Besides, the flow-induced noise stemming from wave plunging may weaken the acoustic

stealth of water surface craft. In the oceanographic physics context, wave breaking accounts for the energy and mass

exchange of the ocean-atmosphere system, which has a great effect on the weather forecasts and global climate

predictions. Due to multi-scale properties of multiphase turbulent flows, a wide range of time and length scales

should be resolved, making it rather complicated for experimental and numerical investigations. In early reviews[1-4]

,

general mechanisms related to wave breaking problems are well-described. However, previous emphasis lies on the

phenomenological characteristics of breaking wave. Thus, this review summarizes the recent experimental and

numerical advances of the studies of air entrainment, bubble distribution, energy dissipation, capillary effect, and so

on.

Keywords: wave breaking; energy dissipation; air entrainment and void fraction; bubble and droplet size

distribution; capillary effects

CLC number: O353.2 Document code: A

1. Experimental Studies

In last few decades, a number of important

experiments have been performed and fully analyzed to

reveal the intrinsic physics of breaking waves. Different

measurement methods including the high-speed

cameras, bubble detection approach[5]

, fiber optical

probes[6-7]

, Laser Doppler Velocimer[8]

, Particle Image

Velocimetry (PIV) technologies[9]

, and acoustic

measurement[10]

have been adopted.

By applying a visualization technique, Bonmarin[11]

analyzed the steep wave breaking evolution. The wave

profile of a plunging crest, the related splash-up, and

the air-entrainment phenomenon were observed. Qiao

and Duncan[12]

performed detailed analysis for the

unsteady evolution of moderate spilling wave breakings.

Grue and Jensen[13]

measured the velocity field and

corresponding accelerations of the steep plunging

waves from experiment. They found that for the


2

horizontal and vertical accelerations over 1.1g (g

indicates the acceleration of gravity) and 1.5g, the wave

breaking phenomenon will be induced.

The experiment of breaking waves over an

underwater hydrofoil conducted by Duncan[14-15]

was

the first attempt to study the energy dissipation caused

by wave breaking. Rapp and Melville[8]

studied the

dispersion of the deep-water wave breakings by using

Laser Doppler measurement. They measured the loss of

excess momentum and energy flux of breaking waves,

and found over 10% flux loss for single spilling wave

breakers to as much as 25% flux loss for plunging wave

breakers. Chang and Liu[16]

investigated the turbulent

flow originated from wave breaking with limited water

depth. By applying the ensemble average approach, the

mean velocity, Reynolds stress, and strain rate were

obtained. It was concluded that the generation and

evolution of turbulence flows including its dissipation

are of equal importance, whereas the effect of

turbulence diffusion occupies a small portion.

Kimmoun and Branger[17]

performed the waves

breaking test in water tank by using a wedge body.

They calculated the spatial derivatives to acquire

adequate information for describing the flow structures.

Many existing experimental studies suggest that

the air fractions in wave-turbulence interaction process

can induce low-frequency-noise in the ocean. For a

better understanding of the air entrainment caused by

wave breaking, Lamarre and Melville[18]

exploited the

bubble plumes and found that the magnitude of

containing air fractions is many orders larger than

previous expectation. It was concluded that the

contribution of bubbles to sound propagation might be

underestimated to a very large extent if these plumes of

large bubbles are out of consideration. Later on,

Lamarre and Melville[19]

gave detailed measurements

for the evolution of the void-fraction field in bubble

plumes originated from large-scale deep water breakers.

Various parameters of air fractions were calculated and

comparison was made with previous 2-D laboratory

breaking waves. In the experiment performed by

Cartmill and Su[20]

, bubble radii in the bounds of 34 to

1200 μm were measured by an acoustic sensor at

different locations and deepness in a water tank, and the

scatters of bubble size are well conformed with -3

power-law from 10 to 1200 μm . Loewen et al.[10]

applied an image processing technology to measure the

size scatters of large bubbles generated by breaking

waves in fresh and salt water. They found that the

distributions of bubble size can be described by an

exponential (or power law) equation, and concluded

that the gas transportation resulted from large bubbles

should be taken into consideration even for the

low-speed wind. By measuring the bubble dimensions

for breaking waves in open water and laboratory

experiment, Deane and Stokes[5]

gave a quantitative

illustration of the bubble generation principles. They

found that for bubbles dimensions greater than about

1 mm, the bubble density conforms a -10/3 power law

correspondent to the bubble radius, in which the bubble

size distribution is dominated by turbulence intensity.

In terms of smaller bubbles stemming from droplet and

jet impact, a -3/2 power-law can be deduced. Air

entrainment of typical gravity wave breakers on a

slope-beach was studied experimentally by Mori and

Kakuno[21]

, in which the wave surface altitude, air

fractions, and bubble transportation were analyzed in

detail. Experimental results revealed a linear

dependence between the air portions and turbulent

intensity. Besides, the bubble distributions were

observed to follow a -3.4 power law against the bubble

diameters. It was noted that the bubble transportation

has no significant relations with the breaking positions

and free surface depth. Based on the experiment of

Blenkinsopp and Chaplin[22]

, further measurements

were performed in the water tank to study the

transportation of void fractions originated from the

wave breaking of fresh water, salted water, and sea


3

water. Experimental results suggested that the influence

of water type might be less significant compared with

the scale effect for investigating the evolution of bubble

plumes entrained due to breaking waves.

Andreas[23]

was the first to analyze the

transportation of seawater spray. In the study, previous

droplets creation functions were summarized, and new

formula for wind speeds over 32 ms−1

were proposed.

The droplet size distribution under strong wind

conditions were also studied experimentally by Veron et

al.[24]

by using high speed video camera. It showed that

the droplet dimension distribution follows -3 to -5

power laws when the droplets diameters vary between

196 and 5510 mm. The creation of droplets by two

kinds of plunging waves was studied systematically by

Towle[25]

, and results revealed that the wave breakers

induced by larger wave steepness generally lead to

larger numbers of droplets with greater radius.

By using high-image-density PIV, Lin and

Rockwell[9]

studied the evolution of quasi-steady

breaker in different stages, including the capillary wave

pattern and the fully developed breaking waves. For

high Froude number, surface tension plays an important

role to determine the breaker characteristics. In terms of

intermediate Froude numbers, the evolution of free

surface exhibits capillary wave pattern. Quasi-steady

breaking waves accompany with flow separation may

be generated by high Froude number at the breaker

zone.

Some of the representative experimental

investigations for breaking waves are summarized in

Table 1.

Table 1 Experimental and open ocean investigations for breaking waves

References Year Exp. Technique Breaking Wave Types Research Emphasize

Duncan[14-15]

1981

1983 Camera

Breaking waves due to submerged

hydrofoil

Surface profile, dimensions of

turbulent wake

Bonmarin[11]

1989 Camera Deep water wave breaking Splash phenomenon, air

entrainment

Rapp and Melville[8]

1990 LDV Deep water wave breaking Kinetic energy loss, air entrainment

and dissipation

Lamarre and

Melville[18-19]

1991

1994

Acoustic instruments,

camera Deep water breaking waves

Contribution of bubbles to sound

propagation

Cartmill and Su[20]

1993 Acoustic instruments Deep water breaking waves Bubble size distribution

Lin and Rockwell[9]

1995 PIV, camera Quasi-steady breaking wave Surface tension effect

Loewen et al.[10]

1996 Acoustic instruments,

camera

Breaking waves in freshwater and

saltwater Bubble size distribution

Chang and Liu[16]

1999 Camera Breaking waves of intermediate

depth Turbulence production, dissipation

Qiao and Duncan[12]

2001 PIV, Camera Spilling breakers Surface tension effect

Deane and Stokes[5]

2002 Bubble interface detection Breaking waves in laboratory and

open ocean Bubble density, size distributions

Grue and Jensen[13]

2006

PIV Deep water plunging breaking

Overturning jet and wave induced

acceleration

Rojas and Loewen[6]

2007 Fiber optical probes, camera Deep water breaking waves Peak, mean void fractions

Kimmoun and Branger[17]

2007 PIV Sloping beach wave breaking Full velocity field, void fraction of

the surf zone.

Blenkinsopp and

Chaplin[22]

2007 Fiber optical probes, camera

Breaking waves in fresh water,

salted water and sea water

Distribution of void fractions, effect

of scale


4

Mori and Kakuno[21]

2008 Camera Wave breaking in sloping beach

Free surface elevation, void

fraction, bubble distribution,

turbulent intensity

Blenkinsopp and

Chaplin[7]

2010

Fiber optical probes, camera Breaking waves induced by

submerged reef

Spatial, temporal evolution of the

bubble plumes

Veron et al.[24]

2012 Camera Breaking waves under strong wind Droplet size distribution under

strong wind conditions

Towle[25]

2014 Camera Plunging breaking waves Amounts, droplets size distribution

2. Numerical Studies

Experimental research can only be done for usual

wave conditions due to technical difficulties. Thus, it is

difficult to acquire an intensive understanding of the

multi-scale wave breaking phenomenon. Due to the

existence of complex hydrodynamic characteristics

(e.g., complex breaking patterns, coalescence and

separation of bubbles/droplets, and multiscale

liquid-gas turbulence), simulating the surf zone

breaking waves is still a challenge for traditional CFD.

With recent progresses of the numerical approaches, it

becomes feasible to resolve the two-phase flow

problems numerically even with the presence of the

above difficulties. Because of the recent advances in

computer architecture, detailed numerical studies of the

intermediate scale wave breaking problems that takes

account of water droplets and air bubbles become

possible. Scardovelli and Zaleski[26]

provided a

well-documented review on different numerical

approaches.

In early numerical studies, Lemoes[27]

; Takikawa

et al. [28]

; Lin and Liu[29-30]

; Bradford [31]

adopted the

Reynolds Averaged Navier-Stokes (RANS) models for

the analysis of breaking waves. Although the

understanding of the fundamental wave breaking

phenomenon arising in the surf zone can be improved

accordingly, the turbulent intensity was over predicted

at the breaking regions. Hieu et al.[32]

and Zhao et al.[33]

developed multi-scale method that can improve original

RANS predictions. The 2-D flow structures can be fully

resolved by RANS model, while the 3-D turbulence

interactions are represented by the specified eddy

viscosity model. Although air entrainment was not

involved, the numerical results are generally consistent

with experimental data.

Large eddy simulations (LES) and direct

numerical simulations (DNS) are two major branches

for the high-fidelity simulation of the turbulent wave

breaking. In LES, the large scales eddies are modeled

directly, while sub-grid viscosity model and

bubbly-flow models are utilized for representing small

scale eddy. Thus, Navier-Stokes equations can be

solved using the computational mesh with moderate

refinement. Shi et al.[34]

, Liang et al.[35-36]

, and Derakhti

and Kirby[37]

developed turbulent bubbly flow models

for the closure of governing equations. The numerical

work of Zhao and Tanimoto[38]

was the first attempt to

simulate breaking waves of a two-dimensional

configuration by using LES method, in which good

agreement was obtained compared with experimental

measurements. Watanabe and Saeki[39]

presented the

first numerical results of 3-D LES of a plunging

breaking wave. Similar 3-D numerical experiment of

plunging breakers were conducted by Mutsuda and

Yasuda[40]

, in which air entrainment and bubble

interactions were discussed in detail. Christensen and

Deigaard[41]

studied spilling breaking wave and

plunging breaking wave with different intensity.

Phenomenology study with a visualization of the

internal velocity field were given. 3-D vortex structures

created by the wave breaker in surf zone were

investigated by Watanabe et al.[42]

with the adoption of

single-phase model.

In contrast, as a promising technology, DNS is


5

becoming prevalent in solving the multi-phase flow

since no parametrizations are required. It is noted that

the mesh resolution should be sufficiently refined to be

taken into account of the small-scale flow structures.

The early-stage numerical models by Lin and Liu[29-30]

as well as Bradford[31]

failed in considering air

entrainment details until the studies of Lubin et al.[43]

and Biausser et al.[44]

DNS are often adopted to study

the evolution of wave breakings to provide numerical

data on wave dissipation and the splashing processes.

Periodic wave conditions with relatively small

wavelengths are usually adopted, and representative

research works can be found in Refs.[45-49]

. The physical

processes of wave breaking, velocity profiles, and

breaking intensities for plunging breakers were

investigated by Chen et al.[45]

Iafrati[50-51]

provided

quantitative statistics of the spray and air fraction in

breaking waves. Numerical investigation of plunging

breaking wave past an underwater bump were

performed by Kang et al.[52]

and Koo et al.[53]

, in which

the whole plunging wave breaking procedure including

the jet diving and secondary jet were analyzed.

Considering the computational cost for the 3-D

cases, the simulations mentioned above were mainly

conducted in 2-D domain. In addition, most of the DNS

studies aimed at low-Reynolds-number problems.

However, since 2-D simulations cannot capture the

flow structure of Langmuir revolution in wave breaking

proceedings[54]

, 3-D simulations are necessary to

fundamentally investigate the 3-D phenomenon

including bubble and spray formation. The differences

between 2-D and 3-D results were compared by Lubin

et al.[43]

, and numerical simulation revealed that the

energy dissipation after the wave breaking can be

enhanced by 3-D turbulent flow, while the enhancement

is less than 5% for half-wave period. To alleviate such

computational burdens of 3-D simulations, Sullivan et

al.[55-56]

developed an approach with intermediate

complexity. Different from the complete model of the

breaking waves, a stochastic term was added in

momentum equation with a linearizing free-surface

condition to represent the existence of wave breaking.

This model has been used to study the wave breaking

phenomenon considering Langmuir circulation.

However, forcing terms should be defined to keep

consistence with the experimental measurements. Thus

the stochastic model cannot capture the air entrainment,

and the density changes and buoyancy effect are not

involved in current stage (Sullivan et al.[56]

).

Recently, with the rapid development of numerical

schemes and computer hardware, more investigations

have been conducted to 3-D DNS focusing on the

vortex structures, energy dissipation, and air

entrainment, represented research works were given by

Christensen[57]

; Fuster et al.[58]

; Brucker et al.[59]

;

Lakehal and Liovic[60]

; Zhou et al.[61]

; Lubin and

Glockner [62]

, etc.

In the study of Brucker et al.[59]

, up to 134 million

grids were used and numerical tests showed that the

parallel scalability remain acceptable. More recent

numerical study presented by Lubin and Glockner[62]

used over 1 billion grids to investigate the vortex

filaments created at an early stage after the occurrence

of breaking. In the studies of Stern et al. [63]

and Wang

et al.[64]

, bow wave breakings adjacent to the wedge

shape obstacle and breaking wave past a underwater

bump were simulated using very large grids (up to 2.2

billion), which was the first endeavor to simulate

wave-breaking problems with the mesh resolution scale

around micrometers. Table 2 outlines the existing

simulation approaches for the fundamental analysis of

breaking waves.

Table 2 Numerical investigations for breaking waves (FD: finite difference; FV: finite volume; CIP:

constraint interpolation profile)


6

References Year Model Scheme Dim. Research Emphasize

Takikawa[28]

1997 RANS FD 2 Breaking wave transformation system on a slope

Lin and Liu[29-30]

1998 RANS FD 2 Evolution of wave sequence, wave breaking in the slope beach

region

Watanabe and Saeki[39]

1999 LES FD 3 Vorticity and velocity field during the wave breaking process

Chen et al.[45]

1999 DNS FD 2 Splash-up phenomenon of plunging breakers

Bradford[31]

2000 RANS FD 2 Spilling and plunging breaking waves past a sloping region

Song, Sirviente[46]

2004 DNS FD 2 Viscous and capillary effects in breaking wave

Sullivan et al.[55]

2004 DNS FD 3 Stochastic model and its application in breaking waves

simulation

Hieu et al.[32]

2004 RANS CIP, FD 2 Wave propagation in shallow water (wave shoaling, wave

reflection and air movement)

Zhao et al.[33]

2004 RANS FD 2 Multi-scale turbulence model for breaking wave

Watanabe et al.[42]

2005 LES - 3 Wave breaking and the breaking-induced turbulent two-phase

flow

Lubin et al.[65]

2006 LES - 3 Overturning, splash-up and air entrainment of plunging

breaking processes

Christensen[57]

2006 LES FV 3 Undertow, turbulence in breaking waves

Fuster et al.[58]

2009 DNS FV 3 Atomization of a liquid sheet, capillary effect

Iafrati[49]

2009 DNS - 2 Breaking waves of different initial steepness

Brucker et al.[59]

2010 DNS FD 3 Four types of deep-water breaking waves (weak, strong, and

very strong plunging)

Shi et al.[34]

2010 LES FD 2 Air fraction evolution predicted by two-phase flow model

Iafrati[48]

2011 DNS - 2 Energy dissipation in early stage breaking

Lakehal and Liovic[60]

2011 DNS FV 3 Breaking of steep water waves on a beach of constant bed slope

Liang et al.[36]

2012 LES - 2 Multi-size multi-gas bubble model to study evolution of

bubbles

Kang et al.[52]

, Koo et al.[53]

2012 DNS - 2 Plunging breakers for wave past a submerged bump

Wang et al.[64]

2016 DNS FD 3 Small-scale two-phase flow structures in a breaking wave

Derakhti and Kirby[37]

2014 LES - 3 LES model to incorporate entrained bubble populations

Zhou et al.[61]

2014 LES FV 3 Wave breaking-induced turbulence on seabed

Lubin and Glockner[62]

2015 DNS FD 3 Visualization of vortex filaments and its evolution for plunging

breaking waves

Deike et al.[49]

2015 DNS FV 3 Void fraction, bubble distribution, energy dissipation, surface

tension effect

3. Mechanism Analysis

The breaking process of complex multiphase

turbulent flows is a challenging problem for analysis.

General conclusions of the unsteady and statistical

characteristics of the energy dissipation, air entrainment,

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7

bubble cloud, aerated vortex filaments, and the

capillary effect are summarized and presented in this

section. Corresponding statistical methodologies and

important conclusions are also introduced and

discussed.

The typical breaking procedure can be described in

following aspects. As shown in Figs.1(a)-(c)

(reproduced from Wang et al.[64]

), a jet was first formed

in the wave forward direction (Fig.1(a)), and then it

impacted the water surface due to gravity (Fig.1(b)). In

the initial breaking stage, the wave dynamics exhibited

2-D flow pattern. Afterwards, the jet impacting and

entrapment of air cavity brought numerous bubbles and

droplets, and the flow became fully three dimensional

(Fig.1(c)). As summarized by Deane and Stokes[5]

as

well as Kiger and Duncan[66]

, there are two major

reasons for the generation of bubble clouds, i.e., jet

impacting the free surface creates small scale bubbles,

while the air-bubble collapsing accounts for the

formulation of larger bubbles. The influence of initial

slope for waves (𝑆) was analyzed numerically by Deike

et al.[67]

, and they found that the wave evolution was

close to the plunging breaking with initial slope

𝑆 > 0.42, while a spilling process was observed for

0.35 < 𝑆 < 0.42, which was also conformed with the

experiment done by Rojas and Loewen[68]

.

Fig.1 (a)-(c) are reproduced from Wang et al.[64]

, and (d)-(e) are reproduced from Lubin and Glockner[62]

;

(d)-(e) show the bottom view of the free surface for breaking waves, in which the air entrainment and cavity

break-up process can be demonstrated clearly. The wave impact that induces large air cavity is accompanied

by many small-size satellite bubbles. The thin filaments of air, which is an evident phenomenon in wave

breaking, are under numerical investigation by Lubin and Glockner[62]

.

3.1 Energy Dissipation

The total energy 𝐸 = 𝐸𝑘 + 𝐸𝑔 + 𝐸𝑠 for wave

breaking can be decomposed into three components

(the kinetic energy 𝐸𝑘 , the gravitational potential

energy 𝐸𝑔 , and the capillary potential energy 𝐸𝑠 ),

according to the formulations given by Chen et al.[45]

,

Iafrati[51]

, and Deike et al.[49]

as follows:

𝐸𝑘 =1

2∫ 𝜌𝒖2 d𝑥d𝑦 (1)

𝐸𝑔 = ∫ 𝜌𝑔𝑦 d𝑥d𝑦 (2)

𝐸𝑠 =𝛾(𝔏−1)

𝜌𝑤𝑔𝜆2 (3)

where 𝔏 represents the arc length. Due to the viscosity

of the fluid, the total wave energy exhibits descending

trend in time. In the following part, the methodologies

and preliminary conclusions for breaking energy

dissipation will be introduced briefly. It is noted that in

most of the theoretical and numerical studies, the

surface tension effect 𝐸𝑠 is ignored.

Fig.2(a) shows the time evolution of wave energy,


8

in which▽ denotes the gravitational potential energy,

◇ represents the capillary potential energy, ⭕

indicates the kinetic energy, and □ denotes the total

energy. Fig.2(b) shows the wave breaking factors as a

function of wave steepness, in which the

semi-empirical results are represented by solid line, ▼

indicates the DNS by Deike et al.[67]

, and other symbols

are from the experimental measurements of Banner and

Peirson[69]

, Drazen et al. [70]

and Grare et al. [71]

Fig.2(c) shows the total wave energy, kinetic energy,

and gravitational potential energy as a function of time.

(a) Time evolution of wave energy regenerated

from Deike et al.[67]

(b) Wave breaking factors as a function of wave

steepness regenerated from Deike et al.[67]

(c) Total wave energy, kinetic energy, and

gravitational potential energy as a function of time

regenerated from Wang et al.[64]

Figs.2 Numerically predicted energy dissipation

as a function of time.

Followed the theoretical study of Phillips[72]

, the

energy dissipation rate caused by breaking waves can

be described by 𝐸diss = ∫ 𝜖𝑙Λ(𝒄)d𝒄, where 𝒄 is the

phase velocity, 𝜖𝑙 indicates the dissipation rate of unit

width of wave crest, and Λ(𝒄) stands for the breaking

distribution, which can be measured in experiments

(Gemmrich et al.[73]

, Kleiss and Melville[74]

, and

Sutherland and Melville[75]

). The dissipation rate of

Duncan[15]

and Phillips[72]

can be adopted by defining

𝜖𝑙 = 𝑏𝜌𝑐5/𝑔 , where 𝑏 represents the

non-dimensional breaking parameter. The

semi-empirical relationships of the wave breaking

factors derived by Romero et al.[76]

is applied as

𝐸𝑘 = 𝑏 = 0.4(𝑆 − 0.08)2.5 (4)

where 𝑆 = 0.08 is an estimation of the threshold for

breaking. The accuracy of the above inertial model was

validated by numerous experimental studies (Deike et

al.[49]

, Romero et al.[76]

, Garrett et al.[77]

, Melville and

Pizzo[78]

, and Melville and Fedorov[79]

).

Deike et al.[49]

compared the evolution of the whole

wave energy and the kinetic and gravitational energy

per unit length of breaking wave, as shown in

Figs.2(a)-(b). The total wave energy shows an abrupt

decrease during the first wave period, which means the


9

majority of the wave energy is dissipated. They also

provided a fitted exponential function to predict energy

decay which is written as 𝐸 = 𝐸0𝑒−𝜁𝑡 , where 𝜁

represents the decay rate and 𝐸0 stands for the

primary wave energy per unit width of the wave crest.

The breaking influencing factor 𝑏 against the wave

slope is shown in Fig.2(b), in which the experimental

data of Romero et al.[76]

, Garrett et al.[77]

, and Drazen et

al.[70]

together with the 3-D DNS solutions (Deike et

al.[49]

) were given. Deike et al.[67]

inferred that the

initial setups for breaking cause the differences in

dissipation rate between DNS and experimental result

for 𝑆 varying from 0.35 to 0.4. The kinetic, gravity,

and potential energy components derived from 3-D

NDS were also given by Wang et al.[64]

, together with

the 2-D DNS solutions by Chen et al.[45]

as shown in

Fig.2(c). Although the time evolution of energy exhibits

similar trends for 2-D and 3-D simulations in general, a

higher jump after jet plunges was observed for 3-D case,

which is consistent with the numerical predictions by

Lubin et al.[43]

and Lubin and Glockner[62]

.

Deike et al.[49]

analyzed components of the

breaking waves energy for various factors 𝜖 and 𝐵𝑜

(Bond number, 𝐵𝑜 = 𝛥𝜌𝑔/(𝛾𝑘2) ), as shown in

Figs.3(a)-(d). Considering low surface tension

( 𝐵𝑜 = 1000) circumstance, the wave energy descends

exponentially with constant decaying rate for the

steepness below the breaking threshold. In the first few

wave periods, the wave energy is shown to be dispersed

very fast. Specifically, the gravitational energy is

dissipated completely, while the kinetic fraction

remains even in the immediate wave periods. They

found that during the wave propagation, the surface

tension potential energy could be neglected. Similar to

the investigation by Iafrati[50]

, the surface tension

energy occupies less than 8% of the total energy

dissipated.

Fig.3 Wave energy components (kinetic energy: ∗; surface tension potential energy:▽; gravitational potential

energy: ⭕; and the total energy: □) as functions of time, 𝑩𝒐 = 𝟏𝟎𝟎𝟎: (a) 𝝐 = 𝟎. 𝟑 and (b) 𝝐 = 𝟎. 𝟒𝟓;

𝑩𝒐 = 𝟓𝟎: (c) 𝝐 = 𝟎. 𝟑 and (d) 𝝐 = 𝟎. 𝟒𝟓 ((a)-(d) are reproduced from Deike et al.[49]

)

When the surface tension effect becomes dominated ( 𝐵𝑜 = 50, 100), it was concluded in Ref. [49]


10

that if the Bond number is relatively low, more energy

will be dissipated for non-breaking waves, while the

total energy dissipation shows less relevance to the

Bond number for the case of breaking wave.

The total energy dissipation affected by initial

steepness at different Bond numbers was also provided

in Ref.[49]

. For low surface tension, the energy

dissipation will be promoted for wave steepness larger

than the threshold. In terms of strong surface tension,

the existence of capillary effect will enhance the energy

dissipation rate. It was observed that a larger initial

steepness will further increase the total energy

dissipation.

3.2 Air Entrainment and Void Fraction

Air entrainment occurs when the jet re-contacts the

free surface with a large air pocket and bubbles

entrained. It also occurs when the droplets drop into the

water with high-speed, which was reviewed by Kiger

and Duncan[66]

. However, the investigation for

entrained air was not adequate in their study, so the

available corresponding studies were performed by

Deane and Stokes[5]

, Lamarre and Melville[18]

, and


.

The void fraction has been experimentally studied

by repeating measurements at various locations by

measuring electric conductivity[19]

and using optical

fiber probes[6,22]

. Deike et al.[49]

obtained void fraction

variation through 3-D DNS, in which the predicted

bubble flume evolution and the dynamics of void

fraction are consistent with the available experimental

measurements by Lamarre and Melville[19]

and


. Both plunging case

( 𝑆 = 0.55 ) and spilling breakers ( 𝑆 = 0.43 ) were

compared, and in the spilling case, bubble clouds

generated by entrained air are accumulated adjacent to

the free surface. In terms of plunging breaking waves,

larger bubbles rise above the water surface into the air,

while small bubbles remain in water phase for longer.

The entrained air evolving with time is presented

in Fig.4(a), which was reproduced from Deike et al.[49]

At 𝑡/𝑇 ≈ 1, the first peak is observed since the jet

impacts surface and a big pocket of air is entrapped.

Then the measured air volume decreases quickly due to

the collapse of the bubble and its re-contact into the

free surface. After that, bubbles are created and a

secondary air volume peak will be observed, which

corresponds to the situation where large amount of

bubbles are trapped beneath the water surface. Lamarre

and Melville[18]

described the fast decay as an

exponential function, 𝑉 = 𝑉0𝑒−𝒦𝑡/𝑇 , Deike et al.[67]

deduced that the 𝒦 factor varies from 2.5 to 4, and

approximately, it relies on the initial wave steepness.

𝒦 = 3.9 and 𝒦 = 5 were deduced respectively by


and Blenkinsopp and

Chaplin[22]

in their experiments.

(a) Time variation of the entrained air volume

obtained from the void fraction (- - -) and the closed

surfaces (▼) reproduced from Deike et al.[67]

(b) Volumetric entrained air �̅� against 𝝐𝒍𝑳𝒄 𝝆𝑾𝒈⁄

reproduced from Deike et al.[67]


11

(c) Normalized air volume against the initial wave

slope 𝑺. ▼ shows the total volume of air during the

active breaking process; ◇ represents the

experimental results of Deane and Stokes[5]

; ■

represents the experimental measurement of


; ◆ represents the

experimental results of Duncan[14]

Fig.4 Numerically predicted air volume as a function

of time.

Deike et al.

[67] further deduced the formulation for

predicting the time-averaged air entrainment volume

against the dissipation rate per unit width of wave crest

or the slope, given as, �̅� = 𝐵 𝜖𝑙𝐿𝑐 𝜌𝑊𝑔⁄ . Apply the

semi-empirical function for 𝜖𝑙 as (Romero et al.[76]

)

�̅� = 0.4𝐵 (𝑆 − 0.08)2.5𝐿𝑐𝑐5 𝑊𝑔2⁄ (5)

Fig.4(b) shows the total air volume �̅� against

𝜖𝑙𝐿𝑐 𝜌𝑊𝑔⁄ , in which a good agreement was obtained

between the theoretical and numerical scaling (Eq. (3)),

and the CFD solution is also consistent with available

experimental measurement by Deane and Stokes[5]

,

Duncan[14]

, and Lamarre and Melville[19]

. Total volume

of air against wave steepness 𝑆 is presented in Fig.4(c),

in which a reasonable match between the theoretical

model (Eq. (5)) and the DNS data is found.

3.3 Bubble and Droplet Size Distribution

The identification of the droplets and bubbles was

implemented with a neighbor searching algorithm

utilizing a non-recursive strategy. Herrmann[80]

proposed an efficient identification approach for

classifying droplets or bubbles shared by different

blocks[49,67]

.

Previous experimental investigation by Deane and

Stokes[5]

indicated that the size distribution of bubbles

is dependent with different power law scaling. A

theoretical scaling equation accounting for the turbulent

fragmentation was proposed by Garrett et al.[77]

, which

is written as

𝑁(𝑟) ∝ 𝑄𝜖−13𝑟−

103 （6）

where 𝑁(𝑟) stands for the bubble numbers in a unit

𝑚𝑚, 𝑄 and 𝜖 represent the air volume entrained per

volume and energy dissipation respectively, and 𝑟

indicates the bubble dimensions. It was found that when

the bubble dimension is smaller than Hinze scale, the

bubble shape can be held by surface tension effect. In

this situation, the influence of turbulence is expected to

be unimportant. Hinze scale was defined by Deane and

Stokes[5]

as 𝛼𝐻 = 2−8/5𝜖−2/5(𝜎𝑊𝑒𝑐/𝜌)3/5 , where

𝑊𝑒𝑐 denotes the critical Weber number and 𝜎 is the

surface tension coefficient. The scaling equation of

small bubbles was given by Deane and Stokes[5]

and is

expressed as

𝑁(𝑟) ∝ 𝑄(𝜎/𝜌)−3

2𝑣2𝑟−3

2 （7）

where 𝑣 indicates the jet falling speed. Soloviev and

Lukas[81]

derived a scaling equation for bubble size

distribution based on the analysis of buoyancy forces

dominated bubble fraction. Figs.5(a-c) illustrate that

bubbles created at the early stages of breaking waves

are relatively small while larger bubbles are formed in

the later stages of after breaking. Figs.5(a)-(b) present

the bubble size distributions for fine-grid simulations[64]

,

in which the bubble numbers (time-averaged) is close

to the experimental measurements (Deane and Stokes[5]

,

Mori and Kakuno[21]

, and Tavakolinejad[82]

). In the

simulation work of Wang et al.[64]

, the two-phase flow

is not as violent as those in the experiments, so the

slopes are slightly lower than the theoretical and

experimental results for large bubbles.


12

(a) Time-averaged bubble size distribution at

specified time for fine grid[64]

(b) Time-averaged results of different grid

resolution (coarse, medium, and fine grid) [64]

(c) Time-averaged bubble dimension distribution[67]

Fig.5 Numerically predicted bubble size distribution

In terms of droplet size distribution, a similar

scaling equation was given by Garrett et al.[77]

and

Deane and Stokes[5]

, which is written as

𝑁(𝑟) ∝ 𝑣−1(𝜎/𝜌)1

2𝑟−9

2 （8）

where 𝑣 represents the jet velocity of the spray. Fig.6

shows the theoretical and experimental results

(Lhuissier and Villermaux[83]

, Veron et al.[24]

, and

Towle[24]

), as well as the numerical predictions of Wang

et al.[64]

for time-averaged droplet size distribution. It is

noted that there are significant differences between the

experimental studies and simulations for analyzing the

spray formation mechanisms, (e.g., the plunging wave

breaking[26]

), the high wind speed generated spray

spume drops (Veron et al.[24]

), and bubble bursting

(Lhuissier and Villermaux[83]

), however, the distribution

of drop size also conforms a scaling power-law. Based

on the DNS results of Deike et al.[49]

, the

time-averaged bubble size distribution �̅�(𝑟) against

the wave breaking time is presented in Fig.6, in

which �̅�(𝑟) is found to obey a power law with

𝑁(𝑟) ∝ 𝑟𝑚 , and 𝑚 falls into the range of 3-3.5

according to the numerical results. The size

distribution predicted by the above formula is

consistent with many experimental measurements

(e.g., Deane and Stokes[5]

, Rojas and Loewen[6]

,


, and Terrill et al.[84]

).

Fig.6 Drop density distribution as a function of

radius, which is regenerated from Wang et al.[64]

3.4 Capillary Effects

Few researchers have paid attention to the

micro-mechanism inside the wave breaking processes,

considering the influence of capillary effects. The

surface tension effects were firstly studied by the


13

experiments of Liu and Duncan[85-87]

. They found that

the breaking kinematic energy dissipation is able to be

modified through adding surfactants. Capillary effects

on the wave breaking measured in experiment were

discussed extensively by Duncan[87]

. The numerical

investigation of capillary effects was accomplished in

the studies of Liu and Duncan[86]

and Deike et al.[49]

Deike et al.[49]

classified various small-scale wave

patterns into four groups, including the gravity induced

parasitic capillary waves, non-breaking waves, and

spilling and plunging breaking waves. By changing the

initial steepness of the wave and the Bond number, a

wave-state diagram was summarized, as shown in

Fig.7(a), where PB indicates plunging breakers, SB

represents spilling breakers, PCW represents parasitic

capillary waves, and NB stands for non-breaking

gravity waves. From Fig.7(b), the critical steepness 𝜖𝑐

for this case is around 0.32. Thus when 𝜖 < 𝜖𝑐 , no

wave breaking occurs. However, as pointed by Deike et

al.[49]

, since the critical steepness predicted by

numerical simulation only gives an ideal situation, 𝜖𝑐

in experiment or open sea is strongly corresponding to

the initial conditions, therefore the discrepancies of 𝜖𝑐

between numerical and experimental predictions are

unavoidable. Accurate evaluation of critical steepness is

challenging even in experimental studies, and the

corresponding experiments for critical steepness were

performed by Romero et al.[76]

, Drazen et al.[70]

, and

Perlin et al.[88]

As shown in Fig.7, when surface tension becomes

dominated, two types of wave breaking will be induced,

i.e., the gravity-capillary breakers and the spilling

breakers, and both are scaled as 𝜖 ∝ (1 + 𝐵𝑜−1/3),

which is consistent with the theoretical study of

Longuet-Higgins[89]

.

(a) Wave regime diagram derived from numerical

simulation for 𝑩𝒐 and 𝝐, in which ■ is the wave

breaking boundary, and ▼ is the spilling-plunging

boundary

(b) Wave state diagram derived from experiment

compares with the regimes obtained through the

simulation, in which ▼ is the plunging breakers, ● is

the spilling breakers, * is the breakers triggered by

modulation instability, and ◆ is the parasitic

capillary waves.

Fig.7 Numerically predicted wave regime diagram

3.5 Other Research Findings Related to Wave

Breaking

Recent progress in computational capabilities

offers the possibility to study multiscale phenomenon

existing in wave breaking. In this subsection, recent

development in the simulation of vortex filaments,

modulational instability, and wind turbulence over

breaking waves are introduced briefly.

The vortex filaments generated in the primary

stage of the wave breaking are often observed in

laboratories or open sea. Lubin and Glockner[62]

confirmed the existence of the unusual vorticial


14

structures under breaking waves and investigated the

mechanisms through 3-D DNS. From numerical results,

the streamwise vortex filaments were detected with an

upstream obliqueness (approximately 50∘ ) under

plunging breaking waves. However, Lubin and

Glockner[62]

pointed out that there is no contribution on

the dissipation process due to the presence of the vortex

filaments. The formation of large-scale vortex

structures in the air induced by the plunging wave and

its influence to the interaction between the ocean and

atmosphere were investigated numerically by Iafrati et

al.[90]

The energy dissipated by the wave breaker both in

fluid and gas was considered. It is particularly

important to note that the energy dissipation in air is

larger than that in water. Recently, the wind turbulence

over breaking waves was analyzed by Yang et al.[91]

in

detail. Effects of wave age and wave steepness were

investigated through statistics of turbulent airflow over

breaking waves.

4 Conclusion and Future Prospect

In this review, previous studies for wave breaking

phenomenon by laboratory experiments, numerical

simulations, and semi-empirical models are

summarized. The methodologies for analyzing

characteristic variables during wave breaking, including

the void fraction, the bubble size distribution, the

energy dissipation of the breaking wave, and the time

evolution of the total volume of air are illustrated.

In the last few decades, numerous experimental,

theoretical, and numerical methods have been proposed

for analyzing the breaking waves and their effects in

different scales. For experimental and open ocean

investigation, extensive measurements for the bubble

density, size distribution (Lamarre and Melville[18]

;

Cartmill and Su[20]

; Lin and Rockwell[9]

; Deane and

Stokes[5]

; Mori and Kakuno[21]

; Blenkinsopp and

Chaplin[7]

) under various initial wave steepness were

performed. Acoustic resonator, fiber optical probes, and

high speed camera were often used for those analyses.

Two experiments (Lin and Rockwell[9]

; Qiao and

Duncan[12]

) were designed aiming at analyzing the

surface tension effects, in which the PIV technology

was adopted for visualizing the velocity field of spilling

breakers. Kinetic energy dissipation and turbulence

intensity during the breaking procedure were well

studied by Rapp and Melville[8]

, Chang and Liu[16]

, and

Mori and Kakuno[21]

with PIV and high speed camera.

There were also quantitative statistics for droplets

distribution (Veron et al.[24]

; Towle[25]

) and air

entrainment (Bonmarin[11]

; Rapp and Melville[8]

) during

the violent breaking. Compared with experiment,

numerical simulations gave more details and deep

insights into the complex two-phase flow phenomenon.

Among the prevalent numerical methods mentioned in

this review, it was noticed that the RANS simulations

(e.g., Takikawa et al.[28]

; Lin and Liu[29-30]

; Bradford[31]

)

were only conducted in early years. Recently, the LES

(e.g., Watanabe et al.[42]

; Lubin et al.[62]

; Shi et al.[34]

),

especially the DNS (e.g., Lakehal and Liovic[60]

; Wang

et al.[64]

; Lubin and Glockner[62]

; Deike et al.[49]

) were

more preferred in the fundamental study of wave

breaking problems. The statistic of air entrainment

(Lubin et al.[43]

) and droplet/bubble distribution (Wang

et al.[64]

), as well as energy dissipation (Deike et al.[49]

)

were more simple and straightforward from a volume

of fluid (VOF) field than from experimental photograph.

In terms of semi-empirical method, Romero et al.[76]

proposed breaking parameter model, which can be used

for estimating dissipation rate. Deike et al.[67]

deduced

the formulation for predicting the volume of entrained

air. Garrett et al.[77]

developed bubble size scaling

equation accounting for the turbulent fragmentation.

Deane and Stokes[5]

further provided a scaling equation

of small bubbles and droplet size distribution.

Although existing theoretical, experimental, and

numerical tools are capable to provide majority

characteristics, such as the mean void fraction, and

droplet/bubble distribution, many fundamental physical


15

processes of breaking wave are lack of quantitative

analysis. For experimental studies, tracking and

recording evolution of single droplet/bubble are

challenging, especially for fully 3-D experiment. In

addition to the reliable approaches for accurate statistics

of air entrainment, droplets distributions need further

development, and existing PIV technologies cannot

provide adequate details for intensive investigations of

the turbulent two-phase flows, while the promising

X-ray micro-computed tomography technology[92]

is

considered to be extended for this purpose. For

numerical approaches, DNS provides an important

alternative in resolving wave breaking mechanism in

micro-scale. Due to the high requirement of the

computing resources in DNS, adaptive mesh (Liu and

Hu[93-96]

), which is capable to resolve the minimum

bubble/droplet by localized mesh refinement, shows

obvious advantage in large scale parallel computation.

Besides, the fast algorithm for two-phase flow

problems, which can greatly improve the computational

efficiency, can be considered in future work. In terms of

theoretical methods, up to now, all the semi-empirical

formulae mentioned in this review were validated by

model experiment and numerical simulations, while the

scaling effect is rarely considered. In the future, more

experimental and numerical investigations are required

to bridge the gap between model scale analysis and real

scale applications.

References

[1] Peregrine D H. Breaking waves on beaches.

Annual Review of Fluid Mechanics, 1983, 15(15):

149-178.DOI: 10.1146/annurev.fl.15.010183.001

053.

[2] Battjes J A. Surf-zone dynamics. Annual Review

of Fluid Mechanics, 1988, 20(1): 257-293. DOI:

10.1146/annurev.fl.20.010188.001353

[3] Svendsen I A, Putrevu U. Surf-zone

hydrodynamics. Advances in Coastal and Ocean

Engineering, 1996: 1-78. DOI: 10.1142/97898

12797575_0001.

[4] Christensen E D, Walstra D J, Emerat N. Vertical

variation of the flow across the surf zone. Coastal

Engineering, 2002, 45(3): 169-198. DOI: 10.

1016/S0378-3839(02)00033-9.

[5] Deane G B, Stokes M D. Scale dependence of

bubble creation mechanisms in breaking waves.

Nature (London), 2002, 418(6900): 839-844. DOI:

10.1038/nature00967.

[6] Rojas G, Loewen M R. Fiber-optic probe

measurements of void fraction and bubble size

distributions beneath breaking waves. Experiments

in Fluids, 2007, 43(6): 895-906.DOI: 10.1007/

s00348-007-0356-5.

[7] Blenkinsopp C E, Chaplin J R. Bubble size

measurements in breaking waves using optical

fiber phase detection probes. IEEE Journal of

Oceanic Engineering, 2010, 35(2): 388-401. DOI:

10.1109/joe.2010.2044940.

[8] Rapp R J, Melville W K. Laboratory

measurements of deep-water breaking waves.

Philosophical Transactions of the Royal Society A:

Mathematical, Physical and Engineering Sciences,

1990, 331(1622): 735-800. DOI: 10.1098/rsta.

1990.0098.

[9] Lin J C, Rockwell D. Evolution of a quasi-steady

breaking wave. Journal of Fluid Mechanics, 1995,

302(302): 29-44. DOI: 10.1017/S002211209500

3995.

[10] Loewen M R, O’Dor M A, Skafel M G. Bubbles

entrained by mechanically generated breaking

waves. Journal of Geophysical Research: Oceans,

1996, 101(C9). DOI: 10.1029/96jc01919.

[11] Bonmarin P. Geometric properties of deep-water

breaking waves. Journal of Fluid Mechanics, 1989,

209: 405-433. DOI: 10.1017/S0022112089003162.

[12] Qiao H, Duncan J H. Gentle spilling breakers:

Crest flow-field evolution. Journal of Fluid


16

Mechanics, 2001: 57-85. DOI: 10.1017/S002211

2001004207.

[13] Grue J, Jensen A. Experimental velocities and

accelerations in very steep wave events in deep

water. European Journal of Mechanics, B/Fluids,

2006, 25(5): 554-564. DOI: 10.1016/j.euromechflu.

2006.03.006.

[14] Duncan J H. An experimental investigation of

breaking waves produced by a towed hydrofoil.

Proceedings of the Royal Society A: Mathematical,

Physical and Engineering Sciences, 1981,

377(1770): 331-348. DOI: 10.1098/rspa.1981.

0127.

[15] Duncan J H. The breaking and non-breaking wave

resistance of a two-dimensional hydrofoil. Journal

of Fluid Mechanics, 1983, 126: 507-520. DOI:

10.1017/S0022112083000294.

[16] Chang K A, Liu L F. Experimental investigation of

turbulence generated by breaking waves in water

of intermediate depth. Physics of Fluids, 1999,

11(11): 3390-3400. DOI: 10.1063/1.870198.

[17] Kimmoun O, Branger H. A particle image

velocimetry investigation on laboratory surf-zone

breaking waves over a sloping beach. Journal of

Fluid Mechanics, 2007, 588: 353-397. DOI: 10.

1017/S0022112007007641.

[18] Lamarre E , Melville W K . Air entrainment and

dissipation in breaking waves[J]. Nature, 1991,

351,469-472.

[19] Lamarre E, Melville W K. Void-fraction

measurements and sound-speed fields in bubble

plumes generated by breaking waves. The Journal

of the Acoustical Society of America, 1994, 95(3):

1317-1328. DOI: 10.1121/1.408572.

[20] Cartmill J W, Su M Y. Bubble size distribution

under saltwater and freshwater breaking waves.

Dynamics of Atmospheres and Oceans, 1993,

20(1-2): 25-31. DOI: 10.1016/0377-0265(93)

90046-A.

[21] Mori N, Kakuno S. Aeration and bubble

measurements of coastal breaking waves. Fluid

Dynamics Research, 2008, 40(7-8): 616-626. DOI:

10.1016/j.fluiddyn.2007.12.013.

[22] Blenkinsopp C E, Chaplin J R. Void fraction

measurements in breaking waves. Proceedings of

the Royal Society A: Mathematical, Physical and

Engineering Sciences, 2007, 463(2088): 3151-

3170. DOI: 10.1098/rspa.2007.1901.

[23] Andreas E L. A new sea spray generation function

for wind speeds up to 32 m s−1. Journal of

Physical Oceanography, 1998, 28(11): 2175-2184.

[24] Veron F, Hopkins C, Harrison E L, et al. Sea spray

spume droplet production in high wind speeds.

Geophysical Research Letters, 2012, 39(39):

16602. DOI: 10.1029/2012GL052603.

[25] Towle D M. Spray droplet generation by breaking

water waves. College Park: University of

Maryland, 2014.

[26] Scardovelli R, Zaleski S. Direct numerical

simulation of free-surface and interfacial flow.


567-603. DOI: 10.1146/annurev.fluid.31.1.567.

[27] Lemos C M. Wave breaking: A numerical study.

Berlin: Springer-Verlag, 2013.

[28] Takikawa K. Internal characteristics and numerical

analysis of plunging breaker on a slope. Coastal

Engineering, 1997, 31(1-4): 143-161. DOI: 10.

1016/S0378-3839(97)00003-3.

[29] Lin P, Liu L F. A numerical study of breaking

waves in the surf zone. Journal of Fluid Mechanics,

1998, 359: 239-264. DOI: 10.1017/S002211209

700846X.

[30] Lin P, Liu L F. Turbulence transport, vorticity

dynamics, and solute mixing under plunging

breaking waves in surf zone. Journal of

Geophysical Research: Oceans, 1998, 103(C8).

DOI: 10.1029/98JC01360.

[31] Bradford S F. Numerical simulation of surf zone

dynamics. Journal of Waterway, Port, Coastal, and


17

Ocean Engineering, 2000, 126(1): 1-13. DOI: 10.

1061/(ASCE)0733-950X(2000)126:1(1).

[32] Hieu P D, Katsutoshi T, Ca V T. Numerical

simulation of breaking waves using a two-phase

flow model. Applied Mathematical Modelling,

2004, 28(11): 983-1005.DOI: 10.1016/j.apm.2004.

03.003.

[33] Zhao Q, Armfield S, Tanimoto K. Numerical

simulation of breaking waves by a multi-scale

turbulence model. Coastal Engineering, 2004,

51(1): 53-80.DOI: 10.1016/j.coastaleng.2003.12.

002.

[34] Shi F, Kirby J T, Ma G. Modeling quiescent phase

transport of air bubbles induced by breaking waves.

Ocean Modelling, 2010, 35(1-2): 105-117. DOI:

10.1016/j.ocemod.2010.07.002.

[35] Liang J H, McWilliams J C, Sullivan P P, et al.

Modeling bubbles and dissolved gases in the ocean.

Journal of Geophysical Research Oceans, 2011,

116(C3).DOI: 10.1029/2010JC006579.

[36] Liang J H, McWilliams J C, Sullivan P P, et al.

Large eddy simulation of the bubbly ocean: New

insights on subsurface bubble distribution and

bubble-mediated gas transfer. Journal of

Geophysical Research, 2012, 117(C4): C04002.

DOI: 10.1029/2011jc007766.

[37] Derakhti M, Kirby J T. Bubble entrainment and

liquid-bubble interaction under unsteady breaking

waves. Journal of Fluid Mechanics, 2014, 761:

464-506. DOI: 10.1017/jfm.2014.637.

[38] Zhao Q, Tanimoto K. Numerical simulation of

breaking waves by large eddy simulation and VOF

method. Coastal Engineering Proceedings, 1998,

1(26).

[39] Watanabe Y, Saeki H. Three-dimensional large

eddy simulation of breaking waves. Coastal

Engineering Journal, 1999, 41(3-4): 281-301. DOI:

10.1142/S0578563499000176

[40] Mutsuda H, Yasuda T. Numerical simulation of

turbulent air-water mixing layer within surf-zone.

Proceedings of the 27th International Conference

on Coastal Engineering. Sydney, Australia, 2000.

DOI: 10.1061/40549(276)59.

[41] Christensen E D, Deigaard R. Large eddy

simulation of breaking waves. Coastal Engineering,

2001, 42(1): 53-86. DOI: 10.1016/S0378-3839(00)

00049-1.

[42] Watanabe Y, Saeki H, Hosking R.

Three-dimensional vortex structures under

breaking waves. Journal of Fluid Mechanics, 2005,

545: 291-328. DOI: 10.1017/S0022112005006774.

[43] Lubin P, Vincent Stéphane, Caltagirone J P, et al.

Fully three-dimensional direct numerical

simulation of a plunging breaker. Comptes Rendus

Mécanique, 2003, 331(7): 495-501.DOI: 10.1016/

S1631-0721(03)00108-6.

[44] Biausser B, Fraunié P, Grilli T, et al. Numerical

analysis of the internal kinematics and dynamics

of three-dimensional breaking waves on slopes.

International Journal of Offshore and Polar

Engineering, 2004, 14(4): 247-256.

[45] Chen G, Kharif C, Zaleski S, et al.

Two-dimensional Navier-Stokes simulation of

breaking waves. Physics of Fluids, 1999, 1(1): 121.

DOI: 10.1063/1.869907.

[46] Song C, Sirviente A I. A numerical study of

breaking waves. Physics of Fluids, 2004, 16(7):

2649. DOI: 10.1063/1.1738417.

[47] Hendrickson K, Yue D K P. Navier-Stokes

simulations of unsteady small-scale breaking

waves at a coupled air-water interface.

Proceedings of the Twenty-Sixth Symposium on

Naval Hydrodynamics Office of Naval Research,

2006.

[48] Iafrati A. Energy dissipation mechanisms in wave

breaking processes: Spilling and highly aerated

plunging breaking events. Journal of Geophysical

Research: Oceans, 2011, 116(C7). DOI:


18

10.1029/2011JC007038.

[49] Deike L, Popinet S, Melville W K. Capillary

effects on wave breaking. Journal of Fluid

Mechanics, 2015, 769: 541-569. DOI:

10.1017/jfm.2015.103.

[50] Iafrati A. Numerical study of the effects of the

breaking intensity on wave breaking flows. Journal

of Fluid Mechanics, 2009, 622: 371-411.DOI:

10.1017/S0022112008005302.

[51] Iafrati A. Air-water interaction in breaking wave

events: quantitative estimates of drops and bubbles.

Proceedings of 28th Symposium on Naval

Hydrodynamics, Pasadena, California. 2010:

12-17.

[52] Kang D, Ghosh S, Reins G, et al. Impulsive

plunging wave breaking downstream of a bump in

a shallow water flume—Part I: Experimental

observations. Journal of Fluids and Structures,

2012, 32: 104-120. DOI: 10.1016/j.jfluidstructs.

2011.10.010.

[53] Koo B, Wang Z, Yang J, et al. Impulsive plunging

wave breaking downstream of a bump in a shallow

water flume—Part II: Numerical simulations.

Journal of Fluids and Structures, 2012, 32:

121-134. DOI: 10.1016/j.jfluidstructs.2011.10.011.

[54] Thorpe S A. Langmuir circulation. Annual Review

of Fluid Mechanics, 2004, 36(36): 55-79. DOI:

10.1146/annurev.fluid.36.052203.071431.

[55] Sullivan P P, McWilliams J C, Melville W K, et al.

The oceanic boundary layer driven by wave

breaking with stochastic variability. Part 1. Direct

numerical simulations. Journal of Fluid Mechanics,

2004, 507: 143-174. DOI: 10.1017/S00221120040

08882.

[56] Sullivan P P, McWilliams J C, Melville W K.

Surface gravity wave effects in the oceanic

boundary layer: Large-eddy simulation with vortex

force and stochastic breakers. Journal of Fluid

Mechanics, 2007, 593: 405-452.DOI: 10.1017/S0

02211200700897X.

[57] Christensen E D. Large eddy simulation of spilling

and plunging breakers. Coastal Engineering, 2006,

53(5-6): 463-485.DOI: 10.1016/j.coastaleng.2005.

11.001.

[58] Fuster D, Agbaglah G, Josserand C, et al.

Numerical simulation of droplets, bubbles and

waves: State of the art. Fluid Dynamics Research,

2009, 41(6): 065001. DOI: 10.1088/0169-5983/41/

6/065001.

[59] Brucker K A , O'Shea T T , Dommermuth D G , et

al. Three-Dimensional Simulations of Deep-Water

Breaking Waves[C]// The Proceedings of the 28th

Symposium on Naval Hydrodynamics. arXiv,

2010.

[60] Lakehal D, Liovic P. Turbulence structure and

interaction with steep breaking waves. Journal of

Fluid Mechanics, 2011, 674: 522-577. DOI:

10.1017/jfm.2011.3.

[61] Zhou Z, Sangermano J, Hsu T J, et al. A numerical

investigation of wave-breaking-induced turbulent

coherent structure under a solitary wave. Journal

of Geophysical Research: Oceans, 2014, 119(10):

6952-6973. DOI: 10.1002/2014jc009854.

[62] Lubin P, Glockner S. Numerical simulations of

three-dimensional plunging breaking waves:

Generation and evolution of aerated vortex

filaments. Journal of Fluid Mechanics, 2015, 767:

364-393. DOI: 10.1017/jfm.2015.62.

[63] Stern F, Wang Z, Yang J, et al. Recent progress in

CFD for naval architecture and ocean engineering.

Journal of Hydrodynamics, 2015, 27(1): 1-23. DOI:

10.1016/S1001-6058(15)60452-8.

[64] Wang Z, Yang J, Stern F. High-fidelity simulations

of bubble, droplet and spray formation in breaking


307-327. DOI: 10.1017/jfm.2016.87.

[65] Lubin P, Vincent S, Abadie S, et al.

Three-dimensional large eddy simulation of air


19

entrainment under plunging breaking waves.

Coastal Engineering, 2006, 53(8): 631-655.DOI:

10.1016/j.coastaleng.2006.01.001.

[66] Kiger K T, Duncan J H. Air-entrainment

mechanisms in plunging jets and breaking waves.


563-596.DOI: 10.1146/annurev-fluid-122109-160

724.

[67] Deike L, Melville W K, Popinet S. Air entrainment

and bubble statistics in breaking waves. Journal of

Fluid Mechanics, 2016, 801: 91-129. DOI:

10.1017/jfm.2016.372.

[68] Rojas G, Loewen M R. Void fraction

measurements beneath plunging and spilling

breaking waves. Journal of Geophysical Research:

Oceans, 2010, 115(C8): C08001. DOI:

10.1029/2009JC005614.

[69] Banner M L, Peirson W L. Wave breaking onset

and strength for two-dimensional deep-water wave

groups. Journal of Fluid Mechanics, 2007, 585:

93-115. DOI: 10.1017/S0022112007006568.

[70] Drazen D A, Melville W K, Lenain L. Inertial

scaling of dissipation in unsteady breaking waves.

Journal of Fluid Mechanics, 2008, 611: 307-332.

DOI: 10.1017/S0022112008002826

[71] Grare L, Peirson W L, Branger H, et al.

Growth and dissipation of wind-forced,

deep-water waves. Journal of Fluid Mechanics,

2013, 722:5-50. DOI: 10.1017/jfm.2013.88.

[72] Phillips O M. Spectral and statistical properties of

the equilibrium range in wind-generated gravity


505-531. DOI: 10.1017/S0022112085002221.

[73] Gemmrich J R, Banner M L, Garrett C, et al.

Spectrally resolved energy dissipation rate and

momentum flux of breaking waves. Journal of

Physical Oceanography, 2008, 38(6): 1296-1312.

DOI: 10.1175/2007jpo3762.1.

[74] Kleiss J M, Melville W K. Observations of wave

breaking kinematics in fetch-limited seas. Journal

of Physical Oceanography, 2010, 40(12):

2575-2604. DOI: 10.1175/2010jpo4383.1.

[75] Sutherland Peter, Melville W K, et al. Field

measurements and scaling of ocean surface

wave-breaking statistics. Geophysical Research

Letters, 2013, 40(12): 3074-3079. DOI:

10.1002/grl.50584.

[76] Romero L, Melville W K, Kleiss J M. Spectral

energy dissipation due to surface wave breaking.

Journal of Physical Oceanography, 2012, 42(9):

1421-1444.DOI: 10.1175/jpo-d-11-072.1.

[77] Garrett C, Li M, Farmer D. The connection

between bubble size spectra and energy dissipation

rates in the upper ocean. Journal of Physical

Oceanography, 2000, 30(9): 2163-2171.

DOI:10.1175/1520-0485(2000)030<2163:TCBBS

S>2.0.CO;2

[78] Pizzo N E, Melville W K. Vortex generation by

deep-water breaking waves. Journal of Fluid

Mechanics, 2013, 734:198-218. DOI: 10.1017/jfm.

2013.453.

[79] Melville W K, Fedorov A V. The equilibrium

dynamics and statistics of gravity-capillary waves.

Journal of Fluid Mechanics, 2015, 767: 449-466.

DOI: 10.1017/jfm.2014.740.

[80] Herrmann M. A parallel Eulerian interface

tracking/Lagrangian point particle multi-scale

coupling procedure. Journal of Computational

Physics, 2010, 229(3):745-759. DOI: 10.1016/j.

jcp.2009.10.009.

[81] Soloviev A, Lukas R. The near-surface layer of the

ocean: Structure, dynamics and applications.

Dordrecht: Springer, 2006. DOI: 10.1007/1-4020-

4053-9.

[82] Tavakolinejad M. Air bubble entrainment by

breaking bow waves simulated by a 2D+T

technique. College Park: University of Maryland,

2010.


20

[83] Lhuissier H, Villermaux E. Bursting bubble

aerosols. Journal of Fluid Mechanics, 2012, 696:

5-44. DOI: 10.1017/jfm.2011.418.

[84] Terrill E J, Melville W K, Stramski D. Bubble

entrainment by breaking waves and their influence

on optical scattering in the upper ocean. Journal of

Geophysical Research Oceans, 2001, 106(C8):

16815-16823. DOI: 10.1029/2000jc000496

[85] Liu X, Duncan J H. The effects of surfactants on

spilling breaking waves. Nature, 2003, 421(6922):

520-523. DOI: 10.1038/nature01357.

[86] Liu X, Duncan J H. An experimental study of

surfactant effects on spilling breakers. Journal of

Fluid Mechanics, 2006: 433-455.DOI: 10.1017/

S0022112006002011.

[87] Duncan J H. Spilling breaker. Annual Review of

Fluid Mechanics, 2001, 33(33):519-547. DOI:

10.1146/annurev.fluid.33.1.519.

[88] Perlin M, Choi W, Tian Z. Breaking waves in deep

and intermediate waters. Annual Review of Fluid

Mechanics, 2013, 45(1):115-145. DOI: 10.1146/

annurev-fluid-011212-140721.

[89] Longuet-Higgins M S. Capillary rollers and bores.

Journal of Fluid Mechanics, 1992, 240(1):

659-679. DOI: 10.1017/S0022112092000259.

[90] Iafrati A, Babanin A, Onorato M. Modulational

instability, wave breaking, and formation of

large-scale dipoles in the atmosphere. Physical

Review Letters, 2013, 110(18): 184504. DOI: 10.

1103/PhysRevLett.110.184504.

[91] Yang Z, Deng B, Shen L. Direct numerical

simulation of wind turbulence over breaking


120-155. DOI: 10.1017/jfm.2018.466.

[92] Mitroglou N, Lorenzi M, Santini M, et al.

Application of X-ray micro-computed tomography

on high-speed cavitating diesel fuel flows.

Experiments in Fluids, 2016, 57(11): 175. DOI:

10.1007/s00348-016-2256-z.

[93] Liu C, Hu C. An immersed boundary solver for

inviscid compressible flows. International Journal

for Numerical Methods in Fluids, 2017, 85(11):

619-640. DOI: 10.1002/fld.4399.

[94] Liu C, Hu C. An adaptive multi-moment FVM

approach for incompressible flows. Journal of

Computational Physics, 2018, 359: 239-262. DOI:

10.1016/j.jcp.2018.01.006.

[95] Liu C, Hu C. Adaptive THINC-GFM for

compressible multi-medium flows. Journal of

Computational Physics, 2017, 342: 43-65.DOI: 10.

1016/j.jcp.2017.04.032.

[96] Liu C, Hu C. Block-based adaptive mesh

refinement for fluid-structure interactions in

incompressible flows. Computer Physics

Communication, 2018, 232: 104-123. DOI: 10.

1016/j.cpc.2018.05.015.

https://doi.org/10.1146/annurev.fluid.33.1.519

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