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Proc. R. Soc. A (2007) 463, 3151–3170
doi:10.1098/rspa.2007.1901
Published online 25 September 2007
Void fraction measurements in breaking waves
B Y C. E. B LENKINSOPP
AND
J. R. C HAPLIN *
School of Civil Engineering and the Environment, University of Southampton,
Highfield, Southampton SO17 1BJ, UK
This paper describes detailed measurements and analysis of the time-varying
distribution of void fractions in three different breaking waves under laboratory
conditions. The measurements were made with highly sensitive optical fibre phase
detection probes and document the rapid spatial and temporal evolutions of both the
bubble plume generated beneath the free surface and the splashes above. Integral
properties of the measured void fraction fields reveal a remarkable degree of similarity
between characteristics of the two-phase flow in different breaker types as they evolve
with time. Depending on the breaker type, the energy expended in entraining air and
generating splash accounts for a minimum of between 6.5 and 14% of the total energy
dissipated during wave breaking.
Keywords: breaking waves; air entrainment; bubble plumes; splash; void fraction;
energy dissipation
1. Introduction
Wave breaking at sea is a violent and spectacular phenomenon that has an
important role in numerous environmental processes. Upon breaking, the impact
of the overturning jet with the water surface in the preceding trough generates a
large splash-up and the entrainment of a dense plume of air bubbles. This rapidly
evolves under the influence of wave-generated currents, turbulence, buoyancy,
dissolution and bubble coalescence. Thus, after wave breaking there exists a
continuum of time-dependent void fractions (defined as the probability that any
given point is in air and not water) varying from zero at large submergence to
unity at high elevations.
Entrainment of air by breaking waves has been shown to influence a number of
physical processes, including air–sea transfer of gases (Merlivat & Memery 1983;
Melville 1996), generation of sea-surface sound (Knudsen 1948; Deane 1997) and
the production of the sea-salt aerosol (Blanchard 1963; Cipriano & Blanchard
1981). The dominant factor for all of these processes is the spectrum of bubble
sizes, and numerous authors have attempted to make measurements of this
distribution (including Medwin (1970), Johnson & Cooke (1979), Cartmill & Su
(1993), Deane (1997), Farmer et al. (1998), Deane & Stokes (2002) and Leifer &
* Author for correspondence ([email protected]).
Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2007.1901 or via
http://www.journals.royalsoc.ac.uk.
Received 15 March 2007
Accepted 31 August 2007
3151
This journal is q 2007 The Royal Society
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C. E. Blenkinsopp and J. R. Chaplin
De Leeuw (2006)). However, it was suggested by Melville (1996) that for
accurate estimates of gas transfer it is also important to know the total volume of
air entrained in the water column and the subsequent evolution of this air
volume. In addition, it has been shown by Führböter (1970), Lamarre & Melville
(1991, 1994) and Hoque (2002) that the entrainment of large numbers of air
bubbles during breaking contributes significantly to the total energy dissipation.
Consequently, a detailed knowledge of the distribution of entrained air and the
behaviour of the entrained bubble plumes would contribute to a better
understanding of wave breaking in general as well as the influence of air
entrainment on these processes. However, largely due to the practical difficulties
of making measurements in a violent two-phase flow, the existing information
is limited.
Many previous authors have made estimates of void fraction in the bubbly
flow beneath breaking waves in either the laboratory or the field (Loewen et al.
1996; Deane 1997; Vagle & Farmer 1998; Kalvoda et al. 2003). However, these
measurements have generally been taken at a single often undefined location in
the flow, using instruments that average void fractions over regions and intervals
that are not small in comparison with the wave’s characteristic length and
time scales.
Papanicolau & Raichlen (1988), Bonmarin (1989) and Kalvoda et al. (2003)
examined the large-scale geometric properties of bubble plumes generated by
breaking waves using photographic and video techniques and provided
information about the variation in size, shape and position of typical bubble
plumes with time. However, they provided no quantitative data about the
concentration of air within the plumes.
Several researchers including Hwung et al. (1992), Hoque (2002), Cox & Shin
(2003) and Hoque & Aoki (2005) have carried out more detailed laboratory
investigations of the void fraction field in breaking waves, providing useful
information about the vertical and horizontal distributions of void fraction. But
by far the most comprehensive set of measurements was completed by Lamarre
(1993) and is also reported by Lamarre & Melville (1991, 1992, 1994). Lamarre
used a conductivity probe to produce contour plots of the time-varying void
fraction field beneath both two- and three-dimensional focused laboratory
breakers as well as making some further measurements in deep water ocean
waves. The results showed that the bubble plumes generated by breaking waves
undergo rapid transformations and lose 95% of their initially entrained air
volume during the first wave period. Lamarre & Melville (1991, 1994) also
concluded that the entrainment and subsequent submergence of large quantities
of air can account for between 30 and 50% of the total energy dissipated at
breaking. Measured void fraction results were used to calculate various integral
properties of the void fraction field and it was shown that these evolved as simple
functions of time and scaled reasonably well from small two-dimensional
laboratory waves to larger three-dimensional breaking waves.
These previous studies have made use of a variety of techniques including
various types of conductivity probe (Lamarre & Melville 1994; Hoque & Aoki
2005), acoustic techniques (Loewen et al. 1996) and laser methods (Hwung et al.
1992). In the present investigation we used optical fibre phase detection probes.
These are highly sensitive, low-profile instruments which rely on the change in
refractive index to detect the presence of air or water at the probe tip in a
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Breaking wave void fractions
2.0 m
6.0 m
9.0m
traverse
conductivity
probe
optical fibre
probes
0.7m
submerged
reef.
gradient = 1 : 10
absorbing
beach
0.62m
Figure 1. Experimental apparatus.
two-phase flow. Probes of this type were first used by Miller & Mitchie (1970)
and are described in detail by Cartellier & Achard (1991). Optical fibre probes
are commonly used for the investigation of two-phase flows, predominantly in the
chemical industry, and have been previously used to study a variety of flow
conditions including air-bubble columns (Chabot et al. 1992), hydraulic jumps
(Murzyn et al. 2004) and even liquid sprays (Hong et al. 2004). Serdula &
Loewen (1998) suggested that optical probes represent the most promising
technique for the examination of the high void fraction bubble plumes
entrained by breaking waves, but no such measurements seem to have been
published previously.
In this paper we report on a series of detailed measurements of the distribution
of time-dependent void fractions in the region of laboratory breaking waves using
a pair of highly sensitive phase detection probes. The experimental set-up and the
scope of the measurements are described in §2 along with a description of the
optical fibre probes used in this study. Then §3 outlines the method used to
compute the void fraction at each measurement location, and §4 discusses the
repeatability of the waves. A series of contour plots showing evolving void
fraction fields are presented in §5. These show that the chosen measurement
technique correctly captures the temporal and spatial evolutions of the two-phase
flow generated by breaking to a resolution that has not previously been achieved.
The void fraction data are further analysed in §6 to provide new information
about the characteristics of the bubble plume and the splash generated at
breaking. The effects of scale and of differences between freshwater and seawater
are discussed in §7, and the main findings of the study are summarized in §8.
2. Experimental arrangements
The experiments were performed in a glass-walled wave flume 17 m long, 0.44 m
wide filled with tap water to a depth of 0.7 m (figure 1). At one end, the flume is
equipped with an absorbing flap-type wave generator and at the other end a
vertical wedge of firm polyether foam for which the reflection coefficient was
between 2 and 8% for the wave conditions measured in the present experiments.
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C. E. Blenkinsopp and J. R. Chaplin
Table 1. Test cases for the void fraction measurements.
case
no.
f (Hz)
Ho (mm)
Hb (mm)
1
2
3
0.5
0.7
1.0
78.2
97.2
100.5
95
100
95
db (mm)
dissipation,
DE ( J)
vortex
area,
Av (m2)
breaker type
113
109
109
17.15
19.54
13.14
0.00299
0.00277
0.00188
strongly plunging
plunging
spilling/plunging
Regular waves were propagated over a submerged reef structure whose front
gradient was 1 : 10. The waves broke at its crest (0.082 m below still water level)
in depth-limited conditions and plunged into the deep water area behind,
allowing the bubble plume to become fully developed with no interaction
between the plume and the flume bed. This arrangement allowed large numbers
of repeatable breaking waves to be generated, enabling highly repetitive void
fraction measurements to be made.
Wave conditions on either side of the reef were measured with resistance-type
gauges. Collection of the void fraction data was triggered by the signal from a
fixed conductivity probe of the type used by Waniewski et al. (2001), which was
positioned just below the elevation of the wave crests at the break point where
part of the front face of the waves first became vertical.
Three cases were examined, from a strongly plunging wave to one that was on
the boundary between spilling and plunging. Details are shown in table 1, in
terms of the wave frequency f, deep water wave height Ho, breaker height Hb and
depth db, energy dissipated per unit crest length in each breaking wave DE and
vortex area Av. The energy dissipated was obtained from measurements of the
incident, transmitted and reflected waves as
DE Z ðEi K Et K Er ÞwL;
ð2:1Þ
where Ei, Et, Er are the incident, transmitted and reflected wave energies per unit
surface area, respectively; w is the width of the flume; and L is the wavelength in
a water depth of 0.7 m.
The cross-sectional area Av of the vortex enclosed beneath the underside of the
plunging jet as it strikes the undisturbed water in the wave trough was estimated
using video images recorded from the side of the wave flume.
Measurements of void fraction in both the bubble plume and the splash
generated by wave breaking were made with two-phase detection probes
mounted horizontally on an arm suspended from a traverse system. The probes
consisted of 62.5/125 mm multimode optical fibres with conical tips truncated at
a diameter of 20 mm. Except for the last 5 mm, the end of each fibre under water
was encased in a 0.8 mm diameter stainless steel tube. The conical end of the
fibre acts as a Descartes prism, reflecting injected light (at a wavelength of
854 nm) with an intensity that depends on the refractive index of the medium in
which it is immersed. At the other end of the fibre, an optical amplifier with a
time constant of less than 0.033 ms detects these variations, producing a high
output voltage when the tip is in air and a low output when the probe is
immersed in water. This output was sampled at 150 kHz.
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Breaking wave void fractions
z (mm)
100
0
–100
–200
–100
0
100
200
300
400
x (mm)
500
600
700
800
Figure 2. Grid for void fraction measurements, which typically covered the region K140 mm!x!
820 mm, K200 mm!z!120 mm.
The probes were positioned such that the tip of one was directly above the
other, with a vertical separation of 60 mm. Measurements were taken at between
770 and 970 positions on a 20!20 mm grid that is shown in figure 2. Data were
obtained for each wave case and at each grid point from series of between 200 and
400 waves.
3. Data processing
Data processing identified the passage of each air/water interface past each
probe, using a single threshold technique described by Cartellier & Achard
(1991) with the threshold level set at 10% of the difference between the water and
air levels. Approximations to the time-dependent void fractions a for each grid
point were obtained by partitioning the data from each wave period into 40 time
bins of equal duration and computing phase averages. The time origin of the first
bin coincided with the arrival of the trigger from the conductivity probe. A void
fraction was calculated for each grid point (i, j ) and each time bin (n) using the
following expression:
P
ta ði; j; nÞ
T ;
aði; j; nÞ Z
ð3:1Þ
w 40
where ta is the total residence time of air phase at the probe tip; w is the number
of waves in the run; and T is the wave period. A similar technique was used by
Chanson (2004) in studying open-channel surges.
4. Repeatability of the wave profile
The standard deviation of the wave height before breaking was less than 1.1% in
each of the three cases shown in table 1 while that of the wave period was less
than 0.6%. However, despite careful control of the generated wave profile, no two
breaking events were identical, particularly in the dynamic region close to the
overturning jet. To provide some evidence of the repeatability of the breaking
wave form, a video analysis was completed for all three wave cases and showed
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C. E. Blenkinsopp and J. R. Chaplin
wave 1
wave 2
wave 3
0 10 20 30 40
scale (mm)
Figure 3. Surface traces of three consecutive waves for the plunging wave case (wave case 1).
good results. As an example, the outlines of the upper water surface and the air
vortex for three consecutive breaking waves from case 1 at a time after the wave
has overturned are presented in figure 3. While there are some differences
between the waveforms, the overall shapes and positions relative to the reef in
the three cases are very similar. This suggests that, except in regions of very high
gradients, ensemble averages of measurements of void fractions in these
conditions will not be contaminated by variability in the waves.
5. Time-varying void fraction distributions
Figure 4 shows a sequence of contour plots of time-varying void fractions at
varying intervals computed as described above for wave case 2 alongside
photographs of the wave at corresponding times. As it is difficult to define the
free surface in a turbulent aerated flow, the 50% void fraction contour (shown in
yellow) is used to approximate the position of the free surface.
The void fraction colour scale in figure 4 is nonlinear in order to reveal
variations in void fractions towards the extremities of the range. But in some
regions this has the effect of highlighting small features that are not necessarily
fully resolved, and are not evident in the corresponding photograph. The
repeatability of void fraction measurements at different points within the bubble
plume and splash was assessed by completing multiple experimental runs at more
than five locations for each wave case. In regions where the void fraction
measured within a time bin was approximately 0.3%, the results differed by less
than 14%. However, for a given test duration, greater variability can be expected
in measured void fractions that are close to zero or 100%. Numerical simulations
were carried out to quantify this effect for the actual experimental conditions and
the results are summarized in figure 5. At a given mean void fraction a, 50% of
measurements of void fraction would fall within the inner shaded region, and
90% within the outer region. These results imply that the contours that are
identified with a void fraction of 0.1% in figure 4 have a 90% probability of
representing an actual void fraction of between 0.046 and 0.162%. At an indicated
void fraction of 1%, the corresponding range is 0.825–1.182%. (Similar ranges apply
at the other end of the scale, to departures from 100% in estimates of void fractions,
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Breaking wave void fractions
100
99.9
99.8
99.5
99
95
90
80
70
60
50
40
30
20
10
5
3
1
0.5
0.2
0.1
0
void
fraction
a (%)
7T/40
(a)
z (mm)
100
0
–100
–200
9T/40
(b)
z (mm)
100
0
–100
–200
11T/40
(c)
z (mm)
100
0
–100
–200
13T/40
z (mm)
100
(d)
0
–100
–200
15T/40
(e)
z (mm)
100
0
–100
–200
17T/40
(f)
z (mm)
100
0
–100
–200
19T/40
(g)
z (mm)
100
0
–100
–200
23T/40
(h)
z (mm)
100
0
–100
–200
29T/40
(i)
z (mm)
100
0
–100
–200
z (mm)
100
35T/40
0
–100
–200
( j)
0
200
400
x (mm)
600
800
0
200
400
x (mm)
600
800
Figure 4. (a–j ) Time-dependent void fraction distributions for wave case 2.
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C. E. Blenkinsopp and J. R. Chaplin
0.5
90%
a /a
0
50%
– 0.5
0.1
1
10
a (%)
Figure 5. Computations of the 50 and 90 percentile ranges of estimates of void fraction obtained
from numerical time domain simulations.
but they were approximately 16% wider because the mean residence time of droplets
measured by the probes was slightly larger than for the bubbles.)
It is clear from figure 4 that the optical fibre probe measurements capture the
primary features of the two-phase flow. There are rapid variations in void
fractions as well as the size and position of the bubble plume and splash-up, with
strong void fraction gradients close to the free surface and the breaking event.
The main features of the flow evident in figure 4 are described below.
Figure 4a (tZ7T/40)
— The breaking wave enters from the left. The shape of the crest as the wave
begins to overturn and the water entrainment in the vortex between the
plunging jet and the wave face are well captured by the measurements.
Figure 4b (tZ9T/40)
— The plunging jet strikes the free surface and begins to generate ‘spray’ which
initially appears to originate from the jet itself, i.e. the jet rebounds from the
free surface (Tallent et al. 1990).
Figure 4c to d (tZ11T/40 to 13T/40)
— The majority of air entrainment takes place between the times of figure 4c
and e. This period corresponds approximately to the formation phase
described by De Leeuw & Leifer (2002).
— As the overturning motion continues, the air trapped in the vortex beneath
the jet is driven down into the water column creating a region of high void
fractions. The bubble plume is very dense and consists predominantly of large
air cavities rather than small, defined bubbles.
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Breaking wave void fractions
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— At the same time, the plunging jet penetrates the free surface and pushes up a
triangle of un-aerated water in front of the wave (Basco 1985), forming an air
cavity between the upper surface of the jet and the distorted free surface in a
similar manner to that seen by Prosperetti & Oguz (1997) and Ledesma (2004)
for an unsteady plunging jet.
— The initial spray described above develops into a ‘core’ of splash (Tallent et al.
1990), which is ejected forwards from the region between the overturning jet and
the triangle of un-aerated water. This region can be clearly seen in figure 4d, where
the 50% contour line is distorted into a V-shape.
— The splash is ejected to an elevation above that of the breaking wave’s crest and its
leading edge is projected a horizontal distance of over 100 mm in front of the wave.
Figure 4e (tZ15T/40)
— The V-shape in the 50% contour becomes less well defined and the free surface
partially encloses the air cavity created between the upper surface of the
plunging jet and the triangle of un-aerated water.
— Between figure 4e and f, the water forming the primary splash-up falls back to
the surface, generating small amounts of spray and a secondary bubble plume
downstream. This remains close to the free surface and disperses as it is
carried downstream by the reformed wave.
Figure 4f to g (tZ17T/40 to 19T/40)
— This period corresponds to the injection phase described by De Leeuw & Leifer
(2002) where the air cavities entrained during the formation phase are driven
down into the water column. They are broken up into smaller bubbles
reaching a maximum penetration depth in figure 4g.
— The bubble plume is compressed, forcing air and water up through the free
surface. This creates a vertical spray of small droplets (seen in both the
photographs and void fraction measurements) which can reach elevations of
up to 5Hb. This can be clearly seen in both the photographs and void fraction
measurements. Similar spray was observed by Miller (1976).
— There is no evidence of a significant secondary splash-up generated by the
impact of the primary splash-up on the water surface as noted by previous
investigators (e.g. Peregrine 1983; Jansen 1986). This is probably a
consequence of the experimental set-up where the wave breaks by depthlimited conditions at the reef crest but then rapidly reforms into a nonbreaking wave in deep water.
Figure 4h to j (tZ23T/40 to 35T/40)
— This long period corresponds to the rise phase of De Leeuw & Leifer (2002).
During this period, the primary bubble plume slowly disperses as bubbles rise
to the surface and the plume spreads horizontally as it is carried downstream
by the motion of the reformed wave.
— Small amounts of spray are seen in the void fraction measurements close to the
free surface during this period. It is thought that this spray originates
predominantly from bubbles bursting at the surface.
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The full sequences of contour plots at intervals of T/40 for all three wave cases
listed in table 1 are provided as electronic supplementary material and
demonstrate the same general characteristics, although it is observed that, in
the spilling/plunging case, the overturning motion is significantly less violent and
there is a less well-defined active entrainment region.
6. Integral properties of the bubble plume and splash-up
To examine some of the characteristics of the bubble plume and splash-up, this
section discusses some integral properties of the void fraction field. Boundaries of
the bubble plume were taken as the 0.3 and 50% void fraction contours, and
those of the splash-up the 50 and 99.7% contours.
(a ) Total volume per unit width of air and splash
The total volume of air entrained beneath, and the volume of splash above, the
50% void fraction contour per unit crest width for each of the 40 phase average
bins were computed using the expressions
iZI jZj
X
X50
Vp ðnÞ Z
ai;j;n dz dx and
ð6:1Þ
iZ1 jZ1
Vs ðnÞ Z
jZJ
iZI X
X
ð1K ai;j;n Þdz dx;
ð6:2Þ
iZ1 jZj 50
where the instantaneous elevation of the 50% void fraction contour is at jZj50,
and dx and dz are the horizontal and vertical intervals of the measurement grid,
respectively.
Results are plotted as functions of time in figure 6. The volumes of air and
splash per unit width are normalized by the volume per unit width of the vortex
of air enclosed between the underside of the plunging jet and the wave face Vv.
In figure 6a, time is normalized by the time taken for a bubble with a diameter
of 2.5 mm (representative of those observed) to rise a distance equal to the
breaking wave height Hb. The bubble rise velocity u r was estimated from the
expression of Leifer et al. (2000) for clean air bubbles within the range rp!r!
4 mm where rp is the bubble radius above which bubbles begin to oscillate as
they rise through the bubble column (rpz0.67 mm at 208C).
u r Z ðu rm C j1 ðr K rc Þm1 Þexpðj2 Tðr K rc Þm 2 Þ;
ð6:3Þ
where the minimum velocity for an oscillating bubble, u rmZ22.16 cm sK1;
the critical radius below which bubbles do not oscillate at any temperature,
r c Z0.0584 cm; TZwater temperature (8C); j1 Z0.733; j 2Z4.792!10 K4 ;
m1ZK0.849; and m2ZK0.815.
The bubble rise time is a more satisfactory time scale for these processes than
the wave period, used in studies by Lamarre & Melville (1991), Cox & Shin
(2003), Kalvoda et al. (2003) and others. This is not surprising because, once
initiated, there is no strong physical dependence of the bubble plume and splashup on the wave properties and so there is no reason why they would be expected
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Breaking wave void fractions
(a)
(b)
1. strong plunging
2. plunging
3. spilling
exponential fit
1.6
1.2
Vs / Vv
1.2
Vp / Vv
1. strong plunging
2. plunging
3. spilling
1.6
0.8
0.8
0.4
0.4
0
2
4
t / (Hb / ur)
6
0
4
8

t / √ 2Hb / g
12
16
Figure 6. Volume per unit width of (a) the bubble plume, Vp/Vv, and (b) the splash-up, Vs/Vv,
both normalized by the volume of the vortex and presented as functions of time. The solid line in
(a) is the exponential fit ðVp =Vv ÞZ 7:29 expðK1:27ðt=ðHb =u r ÞÞÞ.
to evolve on the same time scale. For the splash results in figure 6b, the time axis
is normalized by the time taken for a splash droplet to fall a distance equal to the
breaking wave height Hb.
The volume of entrained air rises rapidly after the overturning jet strikes the
water surface, before reaching the peak value and decaying exponentially as
the plume degasses. The peak volume of entrained air Vp/Vv varies from 1.3 for
the strongly plunging wave, which has the largest vortex volume, to 1.6 in the
spilling/plunging case where the vortex volume is smallest. These are both
greater than the maximum of Vp/VvZ1 reported by Lamarre & Melville,
possibly owing to the greater sensitivity of the current measurement technique
and the optical probe’s ability to measure right up to the free surface, as well as
the differences in the nature of the waves. The splash volume (figure 6b) follows
a similar trend, reaching peak values in the same range as the bubble plume of
Vs/VvZ1.1–1.6.
(b ) Trajectories of the bubble plume and splash centroids
Horizontal positions of the centroids of the bubble plume and splash volume
are plotted in figure 7. Also shown is the linear water wave speed at the reef crest.
In all three cases, the bubble plume and splash volume centroids move at
approximately the wave speed during the period immediately after breaking,
while active bubble entrainment and splash generation are taking place. Once
the bubble plume begins to degas and disperse, it slows considerably. A similar
reduction in the splash centroid velocity is observed as the primary splash-up
initially generated by the plunging jet falls back to the water surface.
Elevations of the centroid and the base of the bubble plume are plotted in
figure 8. The maximum penetration of the bubble column is greater for the
plunging wave cases, reaching depths of approximately 1.8Hb, 2Hb and 1.4Hb for
the strongly plunging, plunging and spilling/plunging waves, respectively. These
values correspond to 2.4Ho, 2.1Ho and 1.3Ho which are rather greater than the
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(a)
(b) 0.6
0.4
horizontal distance
from crest, xs (m)
horizontal distance
from crest, xp (m)
0.5
0.3
0.2
1. strong plunging
2. plunging
3. spilling
wave celerity
0.1
0
2
4
0.4
0.2
6
0
1. strong plunging
2. plunging
3. spilling
wave celerity
4
8
12

t / √ 2Hb / g
t / (Hb /ur)
16
Figure 7. Horizontal displacement of (a) the bubble plume centroid, xp, and (b) the splash-up
centroid, xs, as functions of time.
(a)
(b)
0.4
1. strong plunging
2. plunging
3. spilling
ur (d = 2.5 mm)
1. strong plunging
2. plunging
3. spilling
ur (d = 2.5 mm)
0
z 0.3% /Hb
0
z p /Hb
1
–1
– 0.4
–2
– 0.8
0
2
4
t / (Hb /u r )
6
0
2
4
t / (Hb /u r )
6
Figure 8. Elevation of (a) the bubble plume centroid, z p, and (b) the base of the bubble plume
defined as the minimum elevation of the 0.3% void fraction contour, z 0.3%, as functions of time.
Straight lines represent the terminal velocity of a 2.5 mm diameter bubble.
depths of 0.5Ho to Ho measured by Chanson & Lee (1997) and 0.8Ho to 1.5Ho by
Yüksel et al. (1999) who both examined plunging waves using a similar
experimental set-up. A probable reason for the greater bubble penetration depth
measured in the current experiments is that the optical fibre probe measured
over at least 200 waves, while Chanson & Lee (1997) and Yuksel et al. (1999)
measured between three and five waves for each wave case using a video
technique. The subsequent rise velocities are compared with that of a 2.5 mm
diameter bubble, shown as a straight line. The bubble plume centroid rises more
slowly than individual bubbles contained within it, because over this interval
new bubbles are entrained and driven deeper, while those that were generated
earlier rise through the water column. By contrast, the base of the bubble plume
rises at approximately the same rate as a single bubble once the active
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Breaking wave void fractions
(a)
(b)
0.6
1. strong plunging
2. plunging
3. spilling
freefall
z99.7% /Hb
zs /Hb
0.4
0.2
0
1. strong plunging
2. plunging
3. spilling
freefall
1.2
0.8
0.4
0
–0.2
–0.4
–0.4
0
4
8
t / √2Hb /g
12
16
0
4
8
12
16
t / √2Hb /g
Figure 9. Elevation of (a) the splash-up centroid, z s, and (b) the crest of the splash-up defined as
the maximum elevation of the 99.7% void fraction contour, z 99.7%, as functions of time.
entrainment of bubbles has ceased and the base of the plume is no longer
refreshed with newly entrained bubbles.
Corresponding results for the centroid and upper surface of the splash-up are
plotted in figure 9. Although there is some scatter in the data, the results for all
three wave cases follow a similar general trend. The splash reaches a maximum
elevation of z 99.7%/HbZ1–1.2 over approximately the same period for all three
cases and as expected rises slightly higher in the two plunging cases than in the
spilling/plunging case (Yasuda et al. 1999). After reaching the maximum
elevation the splash then falls back to the water surface and it appears that the
splash duration is slightly shorter in the spilling case. The crest of the splash
returns to SWL only in the strongly plunging wave case, because, in the two
other cases, the wave period is sufficiently short for the results to be affected by
the presence of surface waves and small secondary splashes throughout a full
wave period. The highest elevation at which splashes were detected by the
optical fibre probes was approximately 2Hb for all cases, but individual droplets
were occasionally projected up to approximately 5Hb above the still water level
during the vertical splash phase discussed in §5 and shown in figure 4i–k.
(c ) Cross-sectional area of the bubble plume and splash-up
Bubble plume and splash-up areas are plotted as functions of time in figure 10.
The peak bubble plume area is approximately the same (Ab/AvZ12) for the two
plunging wave cases but reaches a higher value of Ab/AvZ16 in the spilling/
plunging wave case. Results presented in figure 10a display the same trend as
those of Papanicolau & Raichlen (1988), Lamarre & Melville (1991, 1994) and
Kalvoda et al. (2003).
The variation of the splash-up area with time (figure 10b) is similar for all
three wave cases. The splash is generated as the plunging jet strikes the free
surface and its area quickly increases to a maximum which varies in the range
8.2!As/Av!11.8. Once the peak value has been reached, the area of the splashup reduces very quickly as the spray falls back to the water surface.
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C. E. Blenkinsopp and J. R. Chaplin
(b) 12
1. strong plunging
2. plunging
3. spilling
16
splash area, As /Av
bubble plume area, Ap / Av
(a) 20
12
8
1. strong plunging
2. plunging
3. spilling
8
4
4
0
4
2
0
6
t / (Hb / ur)
4
8
12
16
t / (√2Hb /g)
Figure 10. Variation of (a) the bubble plume area, Ap/Av, and (b) the splash-up area, As/Av, as
functions of time.
(d ) Potential energy of air and splash volumes
Air entrainment and generation of splash contribute to the total energy
dissipated by wave breaking. Führböter (1970) argued that much of the energy is
initially transferred to the potential energy of bubbles driven into the water
column. They then rise to the surface, generating small-scale turbulence and heat
as their potential energy decreases. Upon reaching the surface, the bubbles burst,
producing small droplets as described by Blanchard (1989) and generating small,
high-frequency waves.
The total potential energy per unit width associated with air bubbles is
estimated by
iZI jZj
X
X50
Ep ðnÞ Z rg
ai;j;n zi;j;n dz dx;
ð6:4Þ
iZ1 jZ1
where zi;j;n is the instantaneous vertical distance from grid point (i, j ) to the 50%
void fraction contour which is considered to approximate the free surface.
Additional energy losses due to viscosity occur as the bubbles rise but these are
very small compared with the work required to entrain air against buoyancy.
It seems reasonable also to assume that the waves that are generated when the
splash-up falls back to the water surface (like those generated by the bursting
bubbles) do not contribute to the energy of the transmitted wave. Therefore, the
energy per unit width that is effectively lost through splash generation can be
estimated as
Es ðnÞ Z rg
iZI jZj
X
X50
ð1K ai;j;n Þ
z i;j;n dz dx:
ð6:5Þ
iZ1 jZ1
Potential energies of the bubble plume and splash volume normalized by the
wave energy dissipated during breaking are plotted in figure 11.
In all cases the peak energy expended in entraining air is greater than that
consumed in creating the splash. The energy lost in these processes accounts for
a higher percentage of the total in the plunging wave cases than for the
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3165
Breaking wave void fractions
(a) 0.10
0.08
1. strong plunging
2. plunging
3. spilling
0.08
0.06
Es /∆E
Ep / ∆E
(b) 0.10
1. strong plunging
2. plunging
3. spilling
0.04
0.02
0.06
0.04
0.02
0
2
4
t / (Hb / ur)
6
0
4
8
12
16
t / (√2Hb / g)
Figure 11. Potential energy of (a) the entrained bubble plume, Ep/DE, and (b) splash-up volume,
Es/DE, normalized by the wave energy dissipated during breaking as a function of time.
spilling/plunging wave. Results in figure 11 represent the potential energy of the
bubble plume and splash-up for each of the 40 phase averages. But none of them
corresponds to all of the energy in these processes, since individual bubbles and
droplets reach their maximum vertical displacement at different times. In other
words, kinetic energy has not been accounted for. Consequently, the total
contribution of air entrainment and splash cannot be estimated from the current
results. However, the peak values provide an estimate of the minimum energy
dissipation associated with these mechanisms.
The peak energies associated with air entrainment shown in figure 11a are
considerably smaller than estimates of between 20 (Hoque 2002) and 30–50%
(Lamarre & Melville 1991, 1994) of the energy dissipated at breaking. This large
discrepancy is surprising as it was noted in §6a that the volume of air measured in
the current experiment was greater than that measured by Lamarre & Melville.
Figure 11b suggests that the work required to raise the primary splash-up
accounts for at least 5% of the breaking energy dissipation for both of the
plunging wave cases, and 2.5% for the spilling/plunging wave case. These seem
to be the first estimates of the contribution of splash generation to the total
energy dissipated during wave breaking.
7. Interpretation of measurements made at small scale in freshwater
These experiments were conducted in a small-scale facility using freshwater and
the implications of this must be considered when applying the results to other
situations. A brief discussion of this is given below, while a more detailed analysis
is presented by Blenkinsopp (2007) and Blenkinsopp & Chaplin (2007).
(a ) Use of freshwater to model oceanic processes
It has been suggested that differences between freshwater and seawater affect
the nature of bubble plumes generated by breaking waves and plunging jets, but
there is conflicting evidence regarding the nature of any variations. For example,
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C. E. Blenkinsopp and J. R. Chaplin
several researchers, including Monahan & Zeitlow (1969), Haines & Johnson
(1995) and Chanson et al. (2006), present data indicating a greater number of
small, sub-millimetric bubbles in salt/seawater than in freshwater. However,
Loewen et al. (1996) and Wu (2000) conclude that no such difference exists.
Others including Monahan et al. (1994), Wu (2000) and Chanson et al. (2006)
have examined the void fraction distributions and the total volume of air
entrained by violent flows in freshwater, saltwater and seawater, without
reaching a consensus. To complicate matters further, Chanson et al. (2006) found
that air entrainment rates and bubble sizes generated by a steady jet in seawater
were significantly less than that in saltwater, pointing to the role of as yet
unidentified physical, chemical and biological properties. Air entrainment was
greatest in freshwater.
Blenkinsopp (2007) and Blenkinsopp & Chaplin (2007) present measurements
of the time-varying void fraction distributions in the bubble plumes generated by
laboratory-scale breaking waves in freshwater, artificial seawater created using a
commercial sea-salt mix and natural seawater. The results of these studies were
remarkably similar for all three water types. They suggested that there were no
significant differences in the distribution of entrained air or the temporal and
spatial evolutions of the bubble plume after initial entrainment. Observations
indicated that, in all three cases, the entrained bubbles were mostly of a similar
size and bubbles with diameters of the order of 1 mm or greater were visually
predominant. However, in the two seawater solutions, an additional population
of fine bubbles (d!0.3 mm) was evident during the later stages of the plume
evolution and, due to their small rise velocity, tended to accumulate over
repeated breaking events. In contrast to the results of Chanson et al. (2006),
there were no consistent differences in air entrainment rates and entrained
bubble sizes for the artificial and natural seawater tests. However, it is noted that
these studies examine cases with different flow and entrainment characteristics
and it is suggested that further work is required to fully understand the influence
of water type on air entrainment and bubble properties in breaking waves.
(b ) Effect of scale on air entrainment by breaking waves
As noted by Kobus & Koschitzky (1991), dynamic similarity cannot be
achieved when modelling air/water flows. The interpretation of the present
measurements therefore calls for an appreciation of scale effects.
Deane & Stokes (2002) observed two primary mechanisms responsible for air
entrainment in laboratory breaking waves; larger bubbles with a radius greater
than 1 mm are formed by the fragmentation of the air vortex, while smaller
bubbles are formed by the impact and subsequent splashing of the overturning
jet. It seems probable that the volume of air trapped within the vortex and
then entrained into the water column depends predominantly on the geometry
of the overturning wave. For wave heights of 50 mm or more, the effects of
surface tension and viscosity on this process can reasonably be neglected (Couriel
et al. 1998).
Regarding the second mechanism, Chanson et al. (2004) made measurements
of the two-phase flow generated by steady plunging jets with constant Froude
number at different scales and concluded that significant scale effects exist.
However, breaking waves are strongly time-dependent and it was suggested by
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Breaking wave void fractions
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Ledesma (2004) that they may be more suitably modelled by considering an
impacting water jet. Oguz et al. (1995) and Prosperetti & Oguz (1997) developed
potential flow models of an impacting planar jet and falling water mass,
respectively, and showed that, for strong impacts, the effects of surface tension
and viscosity were negligible and the process of air entrainment was dominated
by gravity and inertia effects. They suggested that the volume of air entrained by
a single jet impact was dependent on the jet Froude number. Based on these
studies it seems reasonable to assume that there should be no significant scale
effects associated with the plunging jet entrainment mechanism and that the
total volume of entrained air will scale geometrically. Therefore, the more
important effects of scale are probably linked to the subsequent processes of
bubble fragmentation, coalescence and rise.
The size distribution of bubbles at both laboratory and field scale has been
examined by a number of authors as noted in §1; however, there has been little
comment about how the results at these two scales compare. Deane & Stokes
(1999, 2002) made measurements of the distribution of bubble sizes in both
laboratory and ocean breakers and found evidence that the same bubble
formation mechanisms operate at both scales, leading to comparable bubble size
distributions. This conclusion was supported by Kobus & Koschitzky (1991),
who suggested that the bubble sizes generated by free-surface aeration always
exhibit the same absolute size. Consequently, the rise velocities of the bubbles
making up the bubble plume will remain constant, independent of the scale of the
breaking wave, and this will affect the detrainment of the bubble plume. This is
examined in detail by Blenkinsopp (2007) and Blenkinsopp & Chaplin (2007),
who developed a Lagrangian bubble tracking model to examine the temporal
evolution of entrained bubble plumes at different scales. It was found that at the
scale of the current model the bubble plume almost entirely dispersed within a
single wave period, while at prototype scale significant void fractions remained,
leading to an accumulation of bubbles over repeated waves. This must be taken
into account when interpreting the results presented in this paper.
8. Conclusions
With the aim of developing a better understanding of the two-phase flow in a
laboratory breaking wave, detailed measurements have been made of timedependent void fractions both above and below the free surface using optical fibre
phase detection probes. This was found to be a successful technique, capable of
capturing the fine spatial and temporal evolutions of the structure of the flow.
Measurements were made in spilling/plunging, plunging and strongly plunging
regular waves, breaking over the crest of a submerged reef. By taking moments of
the void fraction field about the free surface, it was shown that the
characteristics of the bubble plumes and splashes, such as the trajectories of
their centroids, were similar for the three breaker types. In addition it was
observed that these properties evolved as well-defined functions of time on a scale
that is not directly related to the wave period although it is noted that the
evolution of the bubble plume once it has been entrained is subject to significant
scale effects.
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C. E. Blenkinsopp and J. R. Chaplin
For all three wave cases, the peak volume of air entrained by the breaking
wave was approximately equal to that of the water in the splash-up. Both
exceeded the volume of the vortex of air enclosed between the underside of the
plunging jet and the wave face by between 30 and 60%. The work required to
entrain air against the effects of buoyancy in spilling/plunging and strongly
plunging waves accounts for at least 4 to 9%, respectively, of the total energy
dissipated. Similarly, the contribution to the energy loss due to the splash-up was
at least 2.5% for spilling/plunging waves and 5% for plunging waves.
Taken together, these results suggest that air entrainment and splash
generation account for at least 6.5–14% of the total energy dissipation, with
the percentage increasing with the intensity of the breaking wave.
C.E.B. acknowledges the receipt of an EPSRC Studentship. The authors thank Gilberto Brambilla
and Michael Roelens for their assistance in developing optical fibre phase detection probes.
References
Basco, D. R. 1985 A qualitative description of wave breaking. J. Waterw. Port Coastal Ocean Eng.
111, 171–188.
Blanchard, D. C. 1963 The electrification of the atmosphere by particles from bubbles in the sea.
Prog. Oceanogr. 1, 73–112. (doi:10.1016/0079-6611(63)90004-1)
Blanchard, D. C. 1989 The ejection of drops from the sea and their enrichment with bacteria and
other materials: a review. Estuaries 12, 127–137. (doi:10.2307/1351816)
Blenkinsopp, C. E. 2007 Air entrainment, splash and energy dissipation in breaking waves. PhD
thesis, School of Civil Engineering and Environment, University of Southampon.
Blenkinsopp, C. E. & Chaplin, J. R. 2007 Validity of small-scale physical models involving
breaking waves. In Proc. 22nd Int. Workshop on Water Waves and Floating Bodies, Plitvice,
Croatia.
Bonmarin, P. 1989 Geometric properties of deep-water breaking waves. J. Fluid Mech. 209,
405–433. (doi:10.1017/S0022112089003162)
Cartellier, A. & Achard, J. L. 1991 Local phase detection probes in fluid/fluid two-phase flows.
Rev. Sci. Inst. 62, 279–303. (doi:10.1063/1.1142117)
Cartmill, J. W. & Su, M.-Y. 1993 Bubble size distribution under saltwater and freshwater breaking
waves. Dyn. Atmos. Oceans 20, 25–31. (doi:10.1016/0377-0265(93)90046-A)
Chabot, J., Lee, S. L. P., Soria, A. & de Lasa, H. I. 1992 Interaction between bubbles and fiber
optic probes in a bubble column. Can. J. Chem. Eng. 70, 61–68.
Chanson, H. 2004 Unsteady air–water measurements in sudden open channel flows. Exp. Fluids 37,
899–909. (doi:10.1007/s00348-004-0882-3)
Chanson, H. & Lee, J.-F. 1997 Plunging jet characteristics of plunging breakers. Coast. Eng. 31,
125–141. (doi:10.1016/S0378-3839(96)00056-7)
Chanson, H., Aoki, S. & Hoque, A. 2004 Physical modelling and similitude of air bubble entrainment
at vertical plunging jets. Chem. Eng. Sci. 59, 747–758. (doi:10.1016/j.ces.2003.11.016)
Chanson, H., Aoki, S. & Hoque, A. 2006 Bubble entrainment and dispersion in plunging jet flows:
freshwater vs. seawater. J. Coast. Res. 22, 664–677. (doi:10.2112/03-0112.1)
Cipriano, R. J. & Blanchard, D. C. 1981 Bubble and aerosol spectra produced by a laboratory
breaking wave. J. Geophys. Res. 86, 8085–8092.
Couriel, E. D., Horton, P. R. & Cox, D. R. 1998 Supplementary 2-D physical modelling of breaking
wave characteristics. Technical Report 98114, Water Research Laboratory, Sydney, Australia:
University of New South Wales.
Cox, D. T. & Shin, S. 2003 Laboratory measurements of void fraction and turbulence in the bore
region of surf zone waves. J. Eng. Mech. 129, 1197–1205. (doi:10.1061/(ASCE)07339399(2003)129:10(1197))
Proc. R. Soc. A (2007)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
Breaking wave void fractions
3169
Deane, G. B. 1997 Sound generation and air entrainment by breaking waves in the surf zone.
J. Acoust. Soc. Am. 102, 2671–2689. (doi:10.1121/1.420321)
Deane, G. B. & Stokes, M. D. 1999 Air entrainment processes and bubble size distributions in the
surf zone. J. Phys. Oceanogr. 29, 1393–1403. (doi:10.1175/1520-0485(1999)029!1393:AEPABSO2.0.CO;2)
Deane, G. B. & Stokes, M. D. 2002 Scale dependence of bubble creation mechanisms in breaking
waves. Nature 418, 839–844. (doi:10.1038/nature00967)
De Leeuw, G. & Leifer, I. 2002 Bubbles outside the plume during the LUMINY wind-wave
experiment. In Gas transfer at water surfaces (ed. M. Donelan) Geophysics monograph series,
vol. 127, pp. 295–301. Washington, DC: American Geophysical Union.
Farmer, D. M., Vagle, S. & Booth, A. D. 1998 A free-flooding acoustical resonator for measurement
of bubble size distributions. J. Atmos. Oceangr. Tech. 15, 1132–1146. (doi:10.1175/15200426(1998)015!1132:AFFARFO2.0.CO;2)
Führböter, A. 1970 Air entrainment and energy dissipation in breakers. In Proc. 12th Int. Conf. on
Coastal Engineering (ASCE), Washington DC, vol. 1, pp. 391–398.
Haines, M. A. & Johnson, B. D. 1995 Injected bubble populations in seawater and freshwater
measured by a photographic method. J. Geophys. Res. 100, 7057–7068. (doi:10.1029/
94JC03226)
Hong, M., Cartellier, A. & Hopfinger, E. J. 2004 Characterisation of phase detection optical probes
for the measurement of the dispersed phase parameters in sprays. Int. J. Multiphase Flow. 30,
615–648. (doi:10.1016/j.ijmultiphaseflow.2004.04.004)
Hoque, A. 2002 Air bubble entrainment by breaking waves and associated energy dissipation. PhD
thesis, Department of Architecture and Civil Engineering. Toyohashi University of Technology,
Toyohashi, pp. 151.
Hoque, A. & Aoki, S. 2005 Distributions of void fraction under breaking waves in the surf zone.
Ocean Eng. 32, 1829–1840. (doi:10.1016/j.oceaneng.2004.11.013)
Hwung, H. H., Chyan, J. M. & Chung, Y. C. 1992 Energy dissipation and air bubbles mixing inside
surf zone. In 23rd Int. Conf. on Coastal Engineering (ASCE), Venice, Italy, vol. 1, pp. 308–321.
Jansen, P. C. M. 1986 Laboratory observations of the kinematics in the aerated region of breaking
waves. Coast. Eng. 9, 453–477. (doi:10.1016/0378-3839(86)90008-6)
Johnson, B. D. & Cooke, R. C. 1979 Bubble populations and spectra in coastal waters: a
photographic approach. J. Geophys. Res. 84, 3761–3766.
Kalvoda, P. M., Xu, L. & Wu, J. 2003 Macrobubble clouds produced by breaking wind waves: a
laboratory study. J. Geophys. Res. 108, 3207. (doi:10.1029/1999JC000265)
Knudsen, V. O., Alford, R. S. & Emling, J. W. 1948 Underwater amibient noise. Marine Res. 7,
410–429.
Lamarre, E. 1993 An experimental study of air entrainment by breaking waves. PhD thesis,
Department of Civil & Oceanographic Engineering. Cambridge, MA: MIT Press.
Lamarre, E. & Melville, W. K. 1991 Air entrainment and dissipation in breaking waves. Nature
351, 469–472. (doi:10.1038/351469a0)
Lamarre, E. & Melville, W. K. 1992 Instrumentation for the measurement of void-fraction in
breaking waves: laboratory and field results. IEEE J. Oceanic Eng. 17, 204–215. (doi:10.1109/
48.126977)
Lamarre, E. & Melville, W. K. 1994 Void-fraction measurements and sound-speed fields in bubble
plumes generated by breaking waves. J. Acoust. Soc. Am. 95, 1317–1328. (doi:10.1121/
1.408572)
Ledesma, R. G. 2004 An experimental investigation on the air entrainment by plunging jets. PhD
thesis, Department of Mechanical Engineering, University of Maryland, Maryland, pp. 192.
Leifer, I. & De Leeuw, G. 2006 Bubbes generated from wind-steepened breaking waves: 1. Bubble
plume bubbles. J. Geophys. Res. 111, C06020. (doi:10.1029/2004JC002673)
Leifer, I., Patro, R. K. & Bowyer, P. 2000 A study on the temperature variation of rise velocity for
large clean bubbles. J. Atmos. Ocean. Technol. 17, 1392–1402. (doi:10.1175/1520-0426(2000)
017!1392:ASOTTVO2.0.CO;2)
Proc. R. Soc. A (2007)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
3170
C. E. Blenkinsopp and J. R. Chaplin
Loewen, M. R., O’Dor, M. A. & Skafel, M. G. 1996 Bubbles entrained by mechanically generated
breaking waves. J. Geophys. Res. 101, 20 759–20 770. (doi:10.1029/96JC01919)
Kobus, H. & Koschitzky, H.-P. 1991 Local surface aeration at hydraulic structures. In Air
entrainment in free-surface flows, IAHR hydraulic structures design manual no. 4, hydraulic
design considerations (ed. I. R. Wood), p. 149. Rotterdam, The Netherlands: A. A. Balkema
Publications.
Medwin, H. 1970 In-situ acoustic measurements of bubble populations in coastal ocean waters.
J. Geophys. Res. 75, 599–611.
Melville, W. K. 1996 The role of surface-wave breaking in air–sea interaction. Annu. Rev. Fluid
Mech. 28, 279–321. (doi:10.1146/annurev.fl.28.010196.001431)
Merlivat, L. & Memery, L. 1983 Gas exchange across an air–water interface: experimental results
and modelling of bubble contribution to transfer. J. Geophys. Res. 88, 707–724.
Miller, R. L. 1976 Role of vortices in surf zone prediction: sedimentation and wave forces. In Beach
and nearshore sedimentation (eds R. A. Davis & R. L. Ethington) Society of economic
paleontologists and mineralogists, special publication no. 24, pp. 92–114. Tulsa, OK: Society of
Economic Paleontologists and Mineralogists.
Miller, N. & Mitchie, R. E. 1970 Measurement of local voidage in liquid/gas two-phase flow
systems. J. Br. Nuclear Energy Soc. 9, 94–100.
Monahan, E. C. & Zietlow, C. R. 1969 Laboratory comparisons of fresh-water and salt-water
whitecaps. J. Geophys. Res. 74, 6961–6966.
Monahan, E. C., Wang, Q., Wang, X. & Wilson, M. B. 1994 Air entrainment by breaking waves: a
laboratory assessment. Aer. Technol. 187, 21–26.
Murzyn, F., Mouaze, D. & Chaplin, J. R. 2004 Optical measurements of bubbly flow in hydraulic
jumps. Int. J. Multiphase Flow 31, 141–154. (doi:10.1016/j.ijmultiphaseflow.2004.09.004)
Oguz, H. N., Prosperetti, A. & Kolaini, A. R. 1995 Air entrainment by a falling water mass.
J. Fluid Mech. 294, 181–207. (doi:10.1017/S0022112095002850)
Papanicolaou, P. & Raichlen, F. 1988 Wave and bubble characteristics in the surf zone. In Sea
surface sound (ed. B. R. Kerman), pp. 97–109. Dordrecht, The Netherlands: Kluwer Academic
Publishers.
Peregrine, D. H. 1983 Breaking waves on beaches. Annu. Rev. Fluid Mech. 15, 149–178. (doi:10.
1146/annurev.fl.15.010183.001053)
Prosperetti, A. & Oguz, H. N. 1997 Air entrainment upon liquid impact. Phil. Trans. R. Soc. A
355, 491–506. (doi:10.1098/rsta.1997.0020)
Serdula, C. D. & Loewen, M. R. 1998 Experiments investigating the use of fibre-optic probes for
measuring bubble-size distributions. IEEE J. Oceanic Eng. 23, 385–399. (doi:10.1109/48.725233)
Tallent, J. R., Yamashita, T. & Tsuchiya, Y. 1990 Transformation characteristics of breaking
water waves. In Water wave kinematics (eds A. Torum & O. T. Gudmestad), pp. 509–524.
Dordrecht, The Netherlands: Kluwer Academic Publishers.
Vagle, S. & Farmer, D. M. 1998 A comparison of four methods for bubble size and void fraction
measurements. IEEE J. Oceanic Eng. 23, 211–222. (doi:10.1109/48.701193)
Waniewski, T. A., Brennen, C. E. & Raichlen, F. 2001 Measurements of air entrainment by bow
waves. J. Fluids Eng. 123, 57–63. (doi:10.1115/1.1340622)
Wu, J. 2000 Bubbles produced by breaking waves in fresh and salt waters: notes and correspondance.
J. Phys. Oceanogr. 30, 1809–1813. (doi:10.1175/1520-0485(2000)030!1809:BPBBWIO2.0.CO;2)
Yasuda, T., Mutsuda, H., Mizutani, N. & Matsuda, H. 1999 Relationships of plunging jet size to
kinematics of breaking waves with spray and entrained air bubbles. Coastal Eng. J. 41, 269–280.
(doi:10.1142/S0578563499000164)
Yüksel, Y., Bostan, T., Çevik, E., Çelikoglu, Y. & Günal, M. 1999 Two-phase flow structure in
breaking waves. In Proc. 9th Int. Offshore and Polar Eng. Conf., Brest, France, pp. 231–235.
Proc. R. Soc. A (2007)