7/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
1/14
Laboratory study of solitary-wave transformation over bed-formroughness on fringing reefs
Pablo D. Quiroga 1, Kwok Fai Cheung
Department of Ocean and Resources Engineering, University of Hawaii at Manoa, Honolulu, HI, USA
a b s t r a c ta r t i c l e i n f o
Article history:
Received 16 August 2012
Received in revised form 1 May 2013Accepted 6 May 2013
Available online 14 June 2013
Keywords:
Bed-form roughness
Bores
Breaking waves
Fringing reefs
Macro roughness
Surf zone processes
This paper presents the formulation and implementation of a series of two-dimensional ume experiments
to investigate effects of bed-form roughness on coastal wave processes. The experiments were carried out
on a fringing reef model in a 104-m long and 4.6-m high ume at Oregon State University. The reef model
has a 1:12 face slope and a long at section for examination of wave shoaling and breaking as well as bore
formation and propagation. The model is 2.36 m tall and the water depth ranges from 2.36 to 2.66 m to pro-
duce dry and wet reef conditions. The bed-form roughness is modeled by timber beams placed across the
ume in four congurations by varying the height from 0.038 to 0.076 m and the spacing from 0.388 to
0.768 m to provide a range of pitch ratios from 5 to 20. The incident solitary wave height varies from 10 to
50% of the water depth to cover a range of breaking and non-breaking conditions. A series of wire and
sonic gauges measured the wave transformation along the ume and a digital camera recorded images of
the breaking waves on a background grid painted on a ume wall. The bed forms decrease the effective
depth for wave propagation and modify the structure of the free surface ow. In comparison to a control
experiment with plain concrete surface, the solitary wave breaks earlier and dissipates more energy on the
reef slope. The subsequent bore slows down with undulation over the shallow reef at, but speeds up for
more energetic ows in deeper water.
2013 Elsevier B.V. All rights reserved.
1. Introduction
Many tropical andsub-tropical islands in thePacic are susceptible
to ood hazards due to tsunamis, hurricanes, and high-surf events.
Accurate prediction of the near-shore wave conditions is important
in coastal structure design, land-use planning, and hazard assessment.
The presence of fringing reefs along these coastlines results in more
complex near-shore processes than those on gentle slopes and sandy
beaches in non-tropical environments (Gerritsen, 1981).Fig. 1shows
a cross section of the reef at Mokuleia on the north shore of Oahu,
Hawaii. The prole, which references to the mean sea level (MSL),
includes a fore reef and a shallow reefat typical of Pacic island envi-
ronments. The abrupt slope transition at the reef edge introduces ener-
getic breaking waves thatresult in bore formation andpropagation over
the shallow reefat (Roeber et al., 2010). The energy dissipation pro-
cesses are augmented by the irregular reef surface with an abundance
of coral heads and colonies of reef organisms (Hardy and Young,
1996; Lowe et al., 2005; Nelson, 1996).
Wave breaking and dissipation in fringing reef environments have
recently received attention in the research community. Nwogu and
Demirbilek (2010) reported a wave ume experiment on irregular
wave transformation over a scaled model of a fringing reef on Guam.
Roeber (2010) described two series of large-scale ume experiments
with 10 two-dimensional reefs modeled after cross-shore proles in
Hawaii, Guam, and American Samoa. Swigler (2009)conducted basin
experiments for solitary wave transformation over a three-dimensional
reef design. These studies provided data for validation and calibration
of numerical models and understanding of wave processes over reef
geometries (e.g., Bai and Cheung, 2012, 2013; Filipot and Cheung,
2012; Kazolea and Delis, 2013; Roeber and Cheung, 2012; Sheremet et
al., 2011; Shi et al., 2012; Tonelli and Petti, 2013). These experiments,
however, were performed on plexiglass or nished concrete surface
without the bed-form roughness commonly found in reef environments.
Lowe et al. (2005) concluded from a eld experiment at Kaneohe Bay on
the east shore of Oahu, Hawaii that bottom friction may dominate the
energy dissipation over the reefat.
The dissipation mechanism due to free surface ows over rough
beds has been a subject of intense investigation.Sleath (1987),Chen
et al. (2007),Dixen et al. (2008), andLowe et al. (2008)conducted
ume experiments to investigate dissipation over gravel, stone, and
coral beds. Parameterized roughness geometries consisted of regu-
larly placed pipes and triangles provide a systematic approach to
examine energy dissipation in unidirectional and oscillatory ows
Coastal Engineering 80 (2013) 3548
Corresponding author. Tel.: +1 808 956 3485; fax: +1 808 956 3498.
E-mail addresses:[email protected](P.D. Quiroga),[email protected]
(K.F. Cheung).1 Present address: ESA PWA, 550 Kearny Street, Ste 900, San Francisco, California
94108.
0378-3839/$ see front matter 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.coastaleng.2013.05.002
Contents lists available at SciVerse ScienceDirect
Coastal Engineering
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o a s t a l e n g
http://-/?-http://dx.doi.org/10.1016/j.coastaleng.2013.05.002http://dx.doi.org/10.1016/j.coastaleng.2013.05.002http://dx.doi.org/10.1016/j.coastaleng.2013.05.002mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.coastaleng.2013.05.002http://www.sciencedirect.com/science/journal/03783839http://www.sciencedirect.com/science/journal/03783839http://dx.doi.org/10.1016/j.coastaleng.2013.05.002mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.coastaleng.2013.05.002http://-/?-http://crossmark.crossref.org/dialog/?doi=10.1016/j.coastaleng.2013.05.002&domain=pdf7/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
2/14
(e.g., Mirfenderesk and Young, 2003; Ojha and Mazumder, 2010;
Suntoyo et al., 2008). Their results advance the understanding of
laminar and turbulent boundary layers over rough beds and provide
a good resource for validation of computational uid dynamic, wave
propagation, and circulation models (e.g.,Fuhrman et al., 2009; Lowe
et al., 2010; Suntoyo and Tanaka, 2009). These ume experiments use
a small roughness height compared to the water depth allowing
formation of laminar and turbulent layers well below the free surface
for parameterization of the dissipation processes with a wave fric-
tion factor.
Coral reef organismsform some of the most jagged surfaces in coast-
al waters. The roughness height, for example, is typically 0.51.0 m on
Oahu, Hawaii (Nunes and Pawlak, 2008). The reefat is quite shallow,
usually less than a couple of meters deep. The dissipation might depend
on the water depth in addition to the roughnessheight and spacing sug-
gested by Raudkivi (1988). For reef environments with sparse coral
communities, theow from surface waves detaches behinda roughness
element and re-attaches in front of the next. A recirculation region isformed adjacent to the roughness element with an internal boundary
layer. This process plays an important role on the macro-turbulence
structure and dissipation rate from bed-form roughness (Leonardi et
al., 2003). For shallow ows, the wakes behind the roughness elements
may extend to the water surface. Thecoupling between free surface and
bed-form induced ows is sporadic and not well understood with little
information in the technical literature.
Prior experiments on wave breaking over fringing reefs have been
performed over smooth or nished surfaces, while dissipation due to
bed-form roughness has been investigated with water depth much
greater than the roughness height. Due to coupling between bottom
friction and wave breaking over shallow reefs, it is necessary to combine
the two dissipation mechanisms in a single laboratory experiment to
characterize the physical processes. Since scaling is an issue for these
processes, physical experiments in a large ume are preferred.Quiroga
(2012) extended the large-scale experiments of Roeber (2010) by
including bed-form roughness on a fringing reef model. The laboratory
study provided measurements of solitary wave transformation over
roughness elements constructed of timber beams with the height and
spacing varied under a range of
ow conditions. The controlled labora-tory environment enables a systematic investigation of the bed-form
effects on wave shoaling, breaking, and bore propagation. This paper
provides a summary of the experiments and results from Quiroga
(2012)as well as further analysis and interpretation of the data.
2. Laboratory experiments
A seriesofumeexperimentswere carried out at theO.H. Hinsdale
Wave Research Laboratory, Oregon State University in 2009. The test
facility is a National Science Foundation designated site for tsunami
research within the Network for Earthquake Engineering Simulation.
Fig. 2 showsa schematicof theexperiments andthe pertinent physical
variables. The wave ume measures 104 m long, 3.66 m wide, and
4.6 m high. Prefabricated concrete slabs of 0.2 m thickness, 3.66 m
width, and 4.57 m length were mounted at bolt holes in the ume
walls to construct a fore reef with a 1:12 slope and a reefat 2.36 m
above the bottom. A wedge in front of the fore reef provides a smooth
transition between the 0.2-m slab and the oor of the ume. A
piston-type wavemaker with a programmable hydraulicactuator gen-
eratesthe incident solitary wave, which has been commonly used
in laboratory studies of wave transformation and runup (e.g., Briggs
et al., 1995; Grilli et al., 1994; Hsiao et al., 2008; Li and Raichlen,
2002; Roeber, 2010; Swigler, 2009; Synolakis, 1987). The use of soli-
tary waves allows precise measurements of wave transformation
and energy dissipation without interference from return ows, wave
setup, and end-wall reection. The laboratory experiments represent
a simplication of wind generated waves, which resembles a series
of solitary waves in shallow reef environments.
Coral reef roughness is inhomogeneous with varying length scalesand a broad spectral distribution (Nunes and Pawlak, 2008). For gen-
eralization and ease of installation, a geometrical representation of
the bed-form roughness was made using assemblies of 2 4 timber
beams at regular intervals across the reef model.This allows parameter-
ization of theroughnessin terms of theheight k and spacing. The pitch
ratio /kdenes the wake behind a roughness element and the overall
dissipation mechanism(Raudkivi, 1988).We focus on the k-typerough-
ness with /k 5 that exposes the recirculation vortices to the free
surface ow.Fig. 3illustrates the four bed-form congurations in the
experiments. BF1 and BF3 have the same pitch ratio of 10, but rough-
ness heights of 3.8 and 7.6 cm. BF2 and BF3 have the same roughness
Fig. 1.Fringing reef prole at Mokuleia, Hawaii.
Fig. 2.Schematic of wave ume experiments.
36 P.D. Quiroga, K.F. Cheung / Coastal Engineering 80 (2013) 3548
http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B2%807/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
3/14
height of 7.6 cm, but pitch ratios of 5 and 10. The friction factor varies
with the pitch ratio and becomes maximum around /k = 7 in the ab-
sence of a free surface (Leonardi et al., 2003). The four congurations
cover an important range of roughness conditions as well as a large
pitch ratio of 20 for BF4 that produces negligible interactions of the
turbulence between roughness elements. This parameter setting allows
examination of the roughness height and pitch ratio with a minimum
number of experiments.Fourteen resistance-typewave gauges(WG1 to WG14)with parallel
wires andsevensonicwave gauges(SW1 to SW7) measured thesurface
elevation along theume at a sampling rate of 50 Hz. Resistance-type
wire gauges give accurate readings of the surface elevation for non-
breaking waves. Since ultrasonic wave gauges can track sheet ows
over dry beds as well as turbulent bores with air entrainment and
spray, they are deployed on thereefat to provide redundant measure-
ments of the ow conditions. Each wave gauge was mounted at a
distance of 0.44 m from a side wall. Because of the relatively steep
fore reef, the wave amplitude increases rapidly and a steep wave front
develops just prior to breaking. The wave gauges might not capture
the breaking wave height. A video camera simultaneously recorded im-
ages of breaking waves at 30 fps over a 0.5 m 0.5 m grid painted on a
ume wall. Incipience of wavebreaking is dened at the moment whenthe wave front becomes near vertical just before spilling or jet forma-
tion at the crest (Grilli et al., 1997; Hsiao et al., 2008). Post-processing
of the video images identies the breaking wave and the breaker type.
Fig. 4 shows a video image of a test from which the breaking wave
heightHband depthhbcan be determined.
Experiments were conducted at four water depths,ho= 2.36, 2.46,
2.56, and 2.66 m, measured from the bottom of the ume. These corre-
spond to water depths ofhf = 0, 0.1, 0.2, and 0.3 m over the reefat for
examination of sheet ow and bore propagation. The wavemaker
generates the input solitary wave with dimensionless heights ofHo/ho= 0.1 to 0.5 at 0.05 increments. Despite formation of trailing
waves, the solitary wave remains stable before reaching the reef
slope.Fig. 5shows good repeatability of the measured wave proles
at WG1 over the range of test conditions (tdenotes time andgaccel-
eration due to gravity). The variation of the recorded wave heights
from each set of tests is less than 2% of the target value. The trailingwaves indicate adjustment of the solitary wave prole during
the generation process. The averaged recorded wave height is 2%
larger than the target value at Ho/ho= 0.5, but that increases to 6%
atHo/ho= 0.1 as the longer wave becomes less reproducible by the
wavemaker. The incident wave height is thus dened by measure-
ments from WG1 at 8.3 m in front of the reef slope. The roughness
elements were initially installed on the reef slope from WG2 to
WG9 for examination of wave shoaling and breaking. In the second
series of experiments, the roughness elements were transferred to
the reefat from WG9 to the end of the ume. The incident solitary
wave transforms over the plain reef slope into a bore on the reefat
for investigation of the roughness effects.
3. Data post-processing and analysis
The laboratory study included 32 series of tests for the four water
depths, four bed-form congurations, and two bed-form placements as
well as four series of control tests without the bed forms. Each test pro-
vides time series of the surface elevation at the gauges along the ume
and videos images of the breaking wave near the reef edge for the given
incident wave height. The quality of the recorded signals is a concern
for breaking waves and turbulent bores, which involve air entrainment
Fig. 3.Roughness element congurations and arrangements.
Fig. 4.Video capture of a breaking wave and its measurements.
37P.D. Quiroga, K.F. Cheung / Coastal Engineering 80 (2013) 3548
http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B4%80http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B3%807/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
4/14
and splashing. Additional post-processing with a weighted-average
lter was necessary to remove spikes and outliers from the wave
gauge signals. Very often these noises are severe enough to obscure
the signals of the breaking waves or turbulent bores. The redundant
measurements by the wire and sonic wave gauges on the reef at
allow cross validation and reconstruction of the wave signals. For qual-
ity assurance, each test was conducted at least twice and the measure-
ments with minimal or correctable anomalies were averaged forsubsequent analysis. This results in a total of 746 tests in the three-
month laboratory study.
The experiments provided a large volume of time series data for
post-processing and analysis. Fig. 6 shows, for example, the time
series of the free surface from the control and bed-form tests with
hf= 0.1 m and Ho/ho= 0.2. The recorded surface elevations with
BF2 on the reef slope and at are shown in the left and right columns,
respectively. The results from the control test in both columns are
identical. The input solitary wave shoals on the plain slope with
noticeable steepening. Reection from the reef slope was observed
during the laboratory experiments and recorded by WG5 and WG6
as an elongated tail of the wave prole. A near vertical front develops
at WG9 and a plunging breaker occurs across the reef edge. WG10 on
the reefat recorded two peaks in the control test corresponding to
the initial splash up from the plunger and the subsequent broken
wave. The ow transforms into a turbulent bore on the shallow
water over the at. The data at WG14 shows the incident bore and
its reection from the end wall. The presence of bed forms on the
fore reef slightly modies the shoaling processes, but does not seem
to increase reection. The wave breaks just before reaching WG9
resulting in a reduced height comparing to the control test data. The
bed forms on the at cause noticeable reection of the input bore at
WG9 that in turn extends the tail of the surface pro
les on the slope.The bed forms reduce the initial splash up at WG10 and increase the
subsequent surface undulation modifying both the bore height and
speed on the at.
The post-processed data denesthe time historiesof solitary wave
and bore propagation over the reef model for analysis of the bed-form
effects. Solitary waves produce net transport of water mass in the
direction of propagation. The horizontal water particle displacement
is of the order of the ow depth, which includes the water depth
and the wave height. An examination of the post-processed data
shows that the ow depth on the reef slope is greater than the
maximum roughness element spacing of 0.768 m for tests withHo/ho 0.2. This is true even at breaking near the reef edge as the
wave height is several times larger than the water depth as seen
inFig. 4. The subsequent surge on the reef slope has a much larger
horizontal length scale associated with the transition from subcrit-
ical to supercritical ow during the breaking process. The supercrit-
icalow in turn generates a bore downstream via a hydraulic jump
near the reef edge. Bore propagation on the reefat is analogous to
open channel ow that the water traverses across the roughness
elements regardless of the bore height. Because the water particle
displacement exceeds the roughness element spacing in most of
the tests, the results enable a comparative study of the effects
of roughness height and pitch ratio on wave propagation and
dissipation.
4. Results and discussion
The post-processed data is divided into three sets, which are evalu-
ated for the bed-form effects on wave shoaling, breaking, and borepropagation. The results from the control tests with plain beds provide
a reference to assess the change in wave properties and dissipation
rates. The primary ow associated with the solitary waves and bores
is unidirectional along the bed. Secondary ow features associated
with the bed forms might extend into the water column to inuence
the free-surface ow. However, the laboratory experiments were not
designed to collect velocity data over the water column for analysis of
the detailed ow structure. We utilize published results and established
theories for steady owsover bed-form roughness,albeit in theabsence
of a free surface, to assist interpretation of the laboratory data.
4.1. Wave shoaling
The bed forms were installed on reef slope during the rst phaseof the experiments to investigate their effects on wave shoaling and
breaking.Fig. 7shows, for example, the normalized surface envelopemax/Ho over the reef slope for the water depth hf= 0.1 m over the
reefat and the full range of input wave conditions from Ho/ho= 0.1
to 0.5. The initial wave transformation on the reef slope is primarily
shoaling. The smaller solitary waves have longer wavelengths and
thus shoal earlier, but do not break on the slope as observed for the
tests withHo/ho= 0.1. The wave height reduction between WG8 and
WG9 is due to reection by the reef slope and energy transmission
from the non-breaking wave to the reefat. The larger solitary waves
shoal later, but break earlier on the slope as observed in the tests with
Ho/ho = 0.2 to 0.5. The breaking wave height is not necessarily the
largest due to reection and transmission of energy. The reduction
of max/Ho at WG9 for the larger values of Ho/ho is due to a
Fig. 5.Solitary wave proles at WG1 from four separate test runs denoted by dark blue,
red, green, and light blue lines.
38 P.D. Quiroga, K.F. Cheung / Coastal Engineering 80 (2013) 3548
http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B5%807/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
5/14
combination of reection and breaking. The bed forms show clear
effects on the surface elevation as the wave shoals and breaks on
the slope. Since the slope is uniform, an increase in water depth sim-
ply shifts the shoaling process toward the reefat and vice versa. An
exception occurs at the reef edge, where the shoaling process is
interrupted and the wave height also depends on the water depth
over the reefat. For Ho/ho 0.2, gradual shoaling occurs in front
of WG3, the surface elevation increases rapidly from WG3 to WG7,
and wave breaking occur between WG7 and WG9. Since the solitarywave does not produce a trough during the transformation, the
maximum surface elevation recorded at any location is equivalent
to the local wave height.
The steady increase in wave height from WG3 and WG7 covers the
zone of rapid shoaling as dened bySynolakis and Skjelbreia (1993).
The measured data allows examination of the bed-form effects on the
shoaling process prior to wave breaking. Fig. 8plots the wave height
gradientH/xas a function of the normalized incident wave height
Ho/ho, where x is the distance between WG3 and WG7. The data
obtained with the four water depths, which represent different stages
of the shoaling process, show a similar pattern with the roughness
conguration. The control tests with a smooth concrete surface give
the largest wave height gradient and the scatter of the data about
the blue trend line provides an indication of laboratory errors. The
bed-form roughness increases the friction and dissipates wave energy
during the shoaling process. The wave height gradient decreases
appreciably from the control data beyond the margin of laboratory
errors. Both the pitch ratio and roughnessheight inuence the results.
Over the range of pitch ratios from 5 to 20, BF3 with a value of 10 re-
sults in the largest reduction. The green dotted lines tted through
the data shows a consistent pattern with the range of water depth
considered. This corroborates the nding ofLeonardi et al. (2003)
that the energy dissipation is maximum at an intermediate range of
pitch ratios. The energy dissipation is also a function of the roughnessheight. BF1 has the same pitch ratio of 10, but produces consistently
larger gradients because of having half of the roughness height in
comparison to BF3.
The celerity, which is the same as group velocity for long waves, is
a good indicator of the shoaling process.Fig. 9shows the normalized
average celerity C=
ffiffiffiffiffiffigh
q computed by timing the peaks between WG3
and WG7, wheregis acceleration due to gravity andh is the average
water depth between the two gauges. A larger water depth corre-
sponds to an earlier stage of the shoaling process that results in a
more gradual increase of the celerity with the normalized incident
wave height Ho/ho. The data does not show a clear pattern of the rough-
ness effects on the celerity. This can be explained by the nonlinear
long-wave equations, in which the bottom friction is an external force
with no effects on the characteristic lines. However, the scatter appears
Fig. 6.Time series of surface elevation at wave gauges along the ume forHo/ho= 0.2 and hf= 0.1 m. Blue and black lines denote gauge data from the control and BF2 tests.
39P.D. Quiroga, K.F. Cheung / Coastal Engineering 80 (2013) 3548
http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B6%807/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
6/14
to increase with the roughness height and is beyond the range of labo-
ratory errors as inferred from thet of the control data. Forsteady ows
over bed forms, Jimnez (2004) approximated the roughness layer
thickness as
kR k min 1 =k; 5 : 1
This gives a value of 0.38 m for the largest roughness height ofk=
7.6 cm. The average water depth between WG3 and WG7 ranges from1 to 1.3 m for the four experiments. The turbulence introduced by the
bed forms probably does not extend far enough into the water column
to inuence the momentum balance in the free-surface ow and the
resulting wave properties. The green dotted line tted through the
BF3 data indicates a slight reduction of the celerity probably associated
with the smaller wave height in comparison to the control tests. The
bed-form roughness dissipates energy and decreases the wave height
through bottom friction, but does not signicantly modify the celerity
during the shoaling process.
4.2. Wave breaking
Wave breaking occurs between WG7 and WG9 for most of thetests.
The wave height shows themost rapid decrease across a narrow region
as noted by Synolakis and Skjelbreia (1993). Despite reection from the
steep slope and transmission to the at, the wave height gradient be-
tween these two gauges provides a general indication of the bed-form
effects on the breaking process especially for the larger wave heights
and the tests with shallower water.Fig. 10shows the wave height gra-
dient H/x as a function of the normalized incident wave height Ho/hofor thefourseries of tests with the waterdepth hf= 0 to0.3 mover the
reefat. Thewave height gradient, in general, shows a downward trend
with Ho/ho as energetic breaking of the largerwaves dissipates more en-ergy over a short distance. An increase in water depth effectively shifts
thesmaller breakingwaves toward thereef edge at WG9 and enhances
the effects of the standing water over the reefat on the breaking pro-
cess. In some cases, the incident wave does not break on the reef slope
but transforms into a bore immediately after propagating over the reef
edge. This results in less energy dissipation and even some effects of
shoaling between WG7 and WG9 that accounts for the initial upward
trend and positive values of the gradient for hf= 0.2 and 0.3 m.
The results show a clear inuence of the bed forms on the wave
breaking process. The control tests with a smooth bed yield the highest
gradient or lowest energy dissipation rate as indicated by the blue trend
line. The energy dissipation shows much stronger dependence on the
pitch ratio than theroughnessheight. Thegreendash linetted through
the BF2 data with the lowest pitch ratio of 5 shows the highest
Fig. 7.Maximum surface elevation along thereef slope for hf= 0.1 m. Bluecrosses connected by bluelines denote gauge data andback circles denote video dataof breaking waves from
the control tests; magenta squares, green diamonds, green triangles, and magenta circles denote gauge data from the BF1, BF2, BF3, and BF4 tests.
40 P.D. Quiroga, K.F. Cheung / Coastal Engineering 80 (2013) 3548
http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B7%807/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
7/14
dissipation rate among the four bed-form congurations, while BF4
with the largest pitch ratio of 20 produce results close to the control
tests. BF1 and BF3, which have the same pitch ratio of 10, produce
very similar results despite their roughness heights of 3.8 and 7.6 cm.
In contrast to wave shoaling, a smaller pitch ratio of 5 produces the
highest dissipation rate in the surf zone. Bed friction due to roughness
is not the primary mechanism for energy dissipation during wave
breaking. The average water depth between WG7 and WG9 rangesfrom 0.25 to 0.55 m for the fourexperiments. Secondaryows generat-
ed by the bed forms probably extend far enough into the water column
to modify the momentum balance and the subsequent wave breaking
process. It was observed during the experiments that the bed forms ini-
tiate more energetic, plunging breakers in deeper water and increase
the subsequent splashing and air entrainment in the surf zone.
Analysis of the videos taken during the experiments pinpoints the
breaking wave location and height. Only the data with incident wave
heights ofHo/ho 0.2 is considered for their well-dened breaking
incipience up to the reef edge. Fig. 11shows strong correlation be-
tween the normalized breaking depth hb/ho with the four bed-form
congurations. The results fromhf= 0 and 0.1 m, which both corre-
spond to plunging breakers on the reef slope, give similar relations for
cross validation. As the water depth increases to hf= 0.2 and 0.3 m,
the smaller waves develop into bores at the reef edge resulting in a
constant breaking depth. Other than that the trend lines through
the control and BF2 data highlight the increase in breaking depth
with the roughness element height and density. The increase in
breaking depth varies with the incident wave height and reaches an
average value ofhb/ho= 0.034 or hb= 9.2 cm for Ho/ho 0.45.
The roughness elements introduce a displacement height at the bot-
tom that decreases the effective depth for the ow and forces thewaves to break earlier. Jackson (1981) estimated the displacement
height at 0.7 k from a large range of bed-form roughness in steady
ow. This suggests a displacement height of 5.3 cm for BF2 that is
considerably smaller than the increase in breaking depth. A second
mechanism, in which the roughness elements redirect the predomi-
nantly horizontal ow upward, might explain the change in the
free-surface ow. Numerical model results of solitary wave propaga-
tion on bed forms show transfer of momentum from the bottom to
the free surface causing acceleration of the wave crest to produce
more energetic breaking at deeper water (Sambe et al., 2011).
Fig. 12plots the breaking wave heightHbas a function of the inci-
dent wave heightHo, both normalized by the water depthho. The four
series of tests with hf = 0 to 0.3 m show very similar trend lines
through the control and BF2 data. Despite producing a larger breaking
Fig. 8.Wave height gradient on the reef slope between WG3 and WG7 as a function of the solitary wave height at WG1. Blue crosses, magenta squares, green diamonds, green tri-
angles, and magenta circles denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green dotted trend lines pass through the control and BF3 data.
41P.D. Quiroga, K.F. Cheung / Coastal Engineering 80 (2013) 3548
http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B8%807/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
8/14
depth, the bed forms do not have signicant effects on the breaking
wave height. The larger breaking depth appears to offset the upward
momentum transfer in maintaining the wave height. The breaking,
however, is more energetic contributing to larger wave height reduc-
tions as seen inFig. 10. The breaking index, which is dened as the
ratio of the breaking wave height and depth, is an important param-
eter in surf-zone processes.Fig. 13shows the breaking index Hb/hbas
a function of the normalized incident wave height Ho/ho. Because of
the steep 1:12 slope, the waves break in shallower water and produce
relatively high breaking indices. In comparison, Grilli et al. (1997)reported breaking indices of 1.36 to 1.47 with the same criterion for
numerically generated waves on a gentle beach slope of 1:35. The
breaking index decreases with the roughness height and element
density since the waves break in deeper water while the breaking
height remains essentially unchanged. An exception to this general
pattern arises from the breaking waves at the reef edge in which
the standing water over the at denes the breaking depth of the
smaller waves resulting in drastically different trend lines for hf=
0.2 and 0.3 m.
4.3. Bore propagation
Bore propagation and decay have been studied extensively through
the nonlinear shallow-water equations since the seminal works of
Stoker (1957)and Whitham (1958). The shock-capturing techniques
approximate breaking and broken waves as bores and conserves mo-
mentum across the discontinuities to account for energy dissipation.
Recent Boussinesq and non-hydrostatic models have incorporated
these techniques to describe bore propagation and energy dissipation
in the surf zone with encouraging results (e.g.,Bai and Cheung, 2012,
2013; Kazolea and Delis, 2013; Roeber and Cheung, 2012; Shi et al.,
2012; Tonelli and Petti, 2013; Zijlema et al., 2011). Here, we present a
laboratory dataset that elucidates these processes in the presence of
bed-form roughness.The bed-form roughness was transferred from the reef slope to the
at in the second phase of the laboratory study. The experiments
were repeated with smooth concrete surface on the reef slope to pro-
vide the input wave at the reef edge. The broken or breaking wave
transforms into a bore over the bed forms on the reef at. Fig. 14
plots, for example, the surface elevation envelope from WG9 to WG14
for hf= 0.1 m. The input wave height at WG9 is generally well de-
ned, but might include effects of reection and ow-backup from
the bed forms. Turbulence and air entrainment from the breaking
wave produce modulations of the surface elevation as far as WG12.
The bore height decreases along the at due to reection by the
bed forms, dissipation from turbulence and bottom friction, and at-
tenuation of the input ow at the reef edge over time. The bores typ-
ically stabilize between WG13 and WG14 to dene the output
Fig. 9.Average celerity on the reef slope between WG3 and WG7 as a function of the solitary wave height at WG1. Blue crosses, magenta squares, green diamonds, green triangles,
and magenta circles denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green dotted trend lines pass through the control and BF3 data.
42 P.D. Quiroga, K.F. Cheung / Coastal Engineering 80 (2013) 3548
http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B9%807/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
9/14
condition of the transformation process. The control tests with a
smooth bed give the largest surface elevation. The bed-form rough-
ness increases the turbulence in theow and the dissipation causes
signicant reduction of the bore height. For the dry bed case, the
bore completely collapses and turns into a sheet ow on the reef
at conforming to the theoretical analysis and numerical results of
Hibberd and Peregrine (1979). The bed forms create extensive local
splashing that is not amenable to either the wire or sonic gauges.
We only consider the results obtained from the water depth hf=
0.1, 0.2, and 0.3 m in the subsequent analysis.Fig. 15 plots the wave height gradient H/x between WG9 and
WG14 as a function of the normalized wave height H/hf, where H is
the average between the two gauges. The use ofHinstead of the input
wave height at WG9 reduces the data scatter and improves the correla-
tion. The gradientshows a clearrelation with thewaveheight,roughness
height, and water depth. The dissipation rate increases with the wave
height due to more energetic wave breaking and turbulence at the
front. The shorter wavelength associated withthe larger incidentsolitary
wave also contributes to the attenuation of the wave height along the
reefat. The results show strong dependence on the roughness height,
because of the direct relation between the form drag on the roughness
element and the bottom friction. A higher pitch ratio for the same
roughness height contributes to slight reduction of the dissipation rate
within the range of laboratory errors. Since the roughness layer
thickness is of the same order as the water depth, the wake extends
to the downstream free surface regardless of the element spacing. The
wave height reduction due to the bed forms also depends on the
water depth. A smaller water depth amplies the effects of the bed
forms in dissipating the wave energy. These ndings have important
implications for numerical modeling of wave transformation over fring-
ing reefs. The pitch ratio, which is often poorly dened and highly var-
iable,plays a secondary rolein energydissipation. Theroughnessheight
and water depthare key parameters that dene the wave friction factor
over the reefat.The bore speed is an important characteristic parameter in shock
propagation that can be inuenced by the turbulence generated
from the bed forms. Fig. 16 shows the normalized average celerityC=
ffiffiffiffiffiffiffiffighf
q between WG9 and WG14 as a function of the dimensionless
wave heightH/hf. The results show that the bed-form roughness plays a
signicant role in borepropagation onthe reefat.When thewave height
is small, the bed forms tend to decrease the bore speed. Observations
during the experiments show that the roughness elements generate an
undulating bore, which has a smaller propagation speed. As the ow
depth increases with the wave height, the undulation over the bed
forms diminishes. The increase in propagation speed can be attributed
in part to reduction of the effective depth. The bed forms also enhance
vertical advection that increases the ow speed near the surface as
shown in the numerical results ofSambe et al. (2011). A higher pitch
Fig. 10.Wave height gradient across the wave breaker on the reef slope between WG7 and WG9 as a function of the solitary wave height at WG1. Blue crosses, magenta squares,
green diamonds, green triangles, and magenta circles denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green dash trend lines pass through the control and BF2
data.
43P.D. Quiroga, K.F. Cheung / Coastal Engineering 80 (2013) 3548
http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B1%B07/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
10/14
ratio with more widely spaced roughness elements slightly reduces
their effect on the free surface ow. The overall effect of the bed forms
on the bore speed diminishes with increasing water depth. The rough-
ness layers from all four bed forms likely extend to the water surface
to inuence the free surface ow forhf= 0.1 m. The results from BF1
and BF4 with an estimated 0.18-m roughness layer from Eq.(1)show
convergence to the control test data athf= 0.2 m, while BF2 and BF3
with an estimated 0.36-m roughness layer still have inuence on the
bore speed athf= 0.3 m.
5. Conclusions and recommendations
A series of large-scalelaboratory experimentshave provideda unique
dataset to elucidate wave shoaling, breaking, and bore propagation over
bed-form roughness on an idealized fringing reef. The use of solitary
waves in the experiments allows precise measurements of wave trans-
formation across the reef without interference from return ows, wave
setup, and end-wall reection. Construction of the bed forms by timber
beams facilitates a systematic approach to investigate their effects
in terms of the roughness height and pitch ratio. The experiments
utilized the k-type roughness, which generates turbulence into the
water column, to examine bed-form effects on wave propagation and
dissipation. A set of control tests with smooth concrete surface on the
reef provides a reference dataset for comparison. The wave transforma-
tion processes were recorded by resistance and sonic wave gauges
along the ume as well as a video system focusing on the wave breaker.
The bed-form roughness increasesthe bottomfriction and dissipates
wave energy during the shoaling process. Since the roughness height
is small compared to the water depth, the bed forms do not modify
the wave propagation characteristics appreciably. The measurements
reafrm a previous nding that the highest dissipation rate occurs atan intermediate range of pitch ratios. As the water depth decreases
relative to the roughness height, a smaller pitch ratio produces the
highest dissipation rate at wave breaking. The bed forms reduce the
effective water depth for wave propagation and increase the vertical
advection. The breaking depth increases with the roughness height
and density, while the breaking wave height remainslargely unaffected.
The roughness height and water depth, instead of the pitch ratio,
dene the bore propagation and dissipation rate on the reef at. The
celerity of the smaller waves decreases because of formation of undular
bores over the bed forms, while the larger bores speed up due to reduc-
tion in the effective depth and enhancement of vertical advection.
The experiments also produced a dataset to validate and calibrate
numerical free-surface ow models for tropical coastal environments.
The recorded dissipation on the reef slope and at can be readily
Fig. 11. Wave breaking depth on the reef slope as a function of the solitary wave heightat WG1. Blue crosses, magenta squares, green diamonds, green triangles, and magenta circles
denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green dash trend lines pass through the control and BF2 data.
44 P.D. Quiroga, K.F. Cheung / Coastal Engineering 80 (2013) 3548
http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B1%B17/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
11/14
used to calibrate the wave friction factor and the eddy viscosity for
phase-averaged and phase-resolving models. Existing parameteriza-
tions, however, are insufcient to account for the bed form effects on
wave propagation in shallow water. These sub-grid features modify
intrinsicwave properties in addition to thedissipationrate.It is neces-
sary to develop new algorithms that enhance vertical advection in a
numerical wave model to describe acceleration of the surface ow,
formation of undular bore, and modication of the bore speed. Futurelaboratory studies might try to model more realistic wave conditions.
Additional experiments with a wider range of roughness heights and
pitch ratios will certainly help produce a comprehensive dataset for
development of numerical models.
Acknowledgment
This study was funded in part by the National Science Foundation
Grant No. 0530759 through the Network for Earthquake Engineering
Simulation. The National Tsunami Hazard Mitigation Program provid-
ed additional support via Hawaii State Civil Defense through Grant
No. NA09NWS4670016. The authors would like to thank Dan Cox,
Jason Killian, Tim Maddox, Ian Robertson, Abdulla Mohamed, Volker
Roeber, and Kim Quesnel for the assistance with the laboratory
experiments as well as the two anonymous reviewers, whose com-
ments and suggestions have greatly improved this paper. SOEST Contri-
bution No. 8770.
References
Bai, Y., Cheung, K.F., 2012. Depth-integrated free-surface ow with a two-layer non-hydrostatic formulation. International Journal for Numerical Methods in Fluids69 (2), 411429.
Bai, Y., Cheung, K.F., 2013.Depth-integrated free surface ow with parameterized non-hydrostatic pressure. International Journal for Numerical Methods in Fluids 71 (4),403421.
Briggs, M.J., Synolakis,C.E., Harkins,G.S., Green, D.R.,1995. Laboratoryexperiments oftsunami runup on a circular island. Pure and Applied Geophysics 144 (3/4),569593.
Chen, D., Chen, C., Tang, F.-E., Stansby, P., Li, M., 2007. Boundary layer structure ofoscillatory open-channel shallow ows over smooth and rough beds. Experimentsin Fluids 42 (5), 719736.
Dixen, M., Hatipoglu, F., Sumer, B.M., Fredse, J., 2008. Wave boundary layer over astone-covered bed. Coastal Engineering 55 (1), 120.
Filipot, J.F., Cheung, K.F., 2012. Spectral wave modeling in fringing reef environments.Coastal Engineering 67, 6779.
Fuhrman, D.R., Fredse, J., Sumer, B.M., 2009. Bed slope effects on turbulent waveboundary layers: 1. Model validation and quantication of rough-turbulent re-sults. Journal of Geophysical Research 114, C03024. http://dx.doi.org/10.1029/
2008JC005045 .
Fig. 12.Breaking wave height on thereef slope as a function of thesolitarywave height at WG1. Blue crosses, magenta squares, green diamonds,greentriangles, andmagenta circles
denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green dash trend lines pass through the control and BF2 data.
45P.D. Quiroga, K.F. Cheung / Coastal Engineering 80 (2013) 3548
http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0005http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0005http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0005http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0005http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0005http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0005http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0005http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0010http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0010http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0010http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0010http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0010http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0010http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0010http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0015http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0015http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0015http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0015http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0015http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0020http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0020http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0020http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0020http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0020http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0020http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0020http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0025http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0025http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0025http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0025http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0030http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0030http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0030http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0030http://dx.doi.org/10.1029/2008JC005045http://dx.doi.org/10.1029/2008JC005045http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B1%B2http://dx.doi.org/10.1029/2008JC005045http://dx.doi.org/10.1029/2008JC005045http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0030http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0030http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0025http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0025http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0020http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0020http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0020http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0015http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0015http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0015http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0010http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0010http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0010http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0005http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0005http://refhub.elsevier.com/S0378-3839(13)00093-8/rf00057/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
12/14
Gerritsen, F., 1981.Wave Attenuationand Wave Set-upon a CoastalReef.PhD DissertationUniversity of Trondheim, Norway.
Grilli, S.T., Subramanya, R., Svendsen, I.A., Veeramony, J., 1994. Shoaling of solitarywaves on plane beaches. Journal of Waterway, Port, Coastal and Ocean Engineering120 (6), 609628.
Grilli, S.T., Svendsen, I.A., Subramanya, R., 1997.Breaking criterion and characteristics forsolitary waves on slopes. Journal of Waterway, Port, Coastal and Ocean Engineering123 (3), 102112.
Hardy, T.A., Young, I.R., 1996.Field study of wave attenuation on an offshore coral reef.Journal of Geophysical Research 101 (C6), 1431114326.
Hibberd, S., Peregrine, D.H., 1979.Surf and run-up on a beach: a uniform bore. Journalof Fluid Mechanics 95 (2), 323345.
Hsiao, S.-C., Hsu,T.-W.,Lin, T.-C., Chang, Y.-H., 2008. On theevolutionand run-up of breakingsolitary waves on a mild sloping beach. Coastal Engineering 55 (12), 975988.
Jackson, P.S., 1981.On the displacement height in the logarithmic velocity prole. Journalof Fluid Mechanics 111, 1525.
Jimnez, J., 2004. Turbulent ows over rough walls. Annual Review of Fluid Mechanics36, 173196.
Kazolea, M., Delis, A.I., 2013. A well-balanced shock-capturing hybrid nite volume-nite difference numerical scheme for extended 1D Boussinesq models. AppliedNumerical Mathematics 67, 167186.
Leonardi, S., Orlandi, P., Smalley, R.J., Djenidi, L., Antonia, R.A., 2003. Direct numericalsimulations of turbulent channel ow with transverse squarebars on one wall. Journalof Fluid Mechanics 491, 229238.
Li, Y., Raichlen, F., 2002. Non-breaking and breaking solitary wave run-up. Journal ofFluid Mechanics 456, 295318.
Lowe, R.J., Falter, J.L., Bandet, M.D., Pawlak, G., Atkinson, M.J., Monismith, S.G., Koseff,J.R., 2005. Spectral wave dissipatio n over a barrier reef. Journal of Geophysi cal
Research 110, C04001.http://dx.doi.org/10.1029/2004JC002711 .
Lowe, R.J., Shavit, U., Falter, J.L., Koseff, J.R., Monismith, S.G., 2008.Modelingow in coralcommunities with and without waves: a synthesis of porous media and canopyow approaches. Limnology and Oceanography 53 (6), 26682680.
Lowe, R.J., Hart, C., Pattiaratchi, C.B., 2010. Morphological constraints to wave-drivencirculation in coastal reeflagoon systems: a numerical study. Journal of GeophysicalResearch 115, C09021.http://dx.doi.org/10.1029/2009JC005753.
Mirfenderesk, H., Young, I.R., 2003.Direct measurements of the bottom friction factorbeneath surface gravity waves. Applied Ocean Research 25 (5), 269287.
Nelson, R.C., 1996.Hydraulic roughness of coralreef platforms. Applied Ocean Research18 (5), 265274.
Nunes, V., Pawlak, G., 2008. Observations of bed roughness of a coral reef. Journal ofCoastal Research 24 (2B), 3950.
Nwogu, O., Demirbilek, Z., 2010. Infragravitywave motionsand runup overshallowfring-ingreefs.Journal of Waterway,Port,Coastal andOceanEngineering 136(6), 295305.
Ojha, S.P., Mazumder, B.S., 2010. Turbulence characteristics ofow over a series of 2-Dbed forms in the presence of surface waves. Journal of Geophysical Research 115,F04016.http://dx.doi.org/10.1029/2008JF001203.
Quiroga, P.D., 2012.Effects of Bed-form Roughness on Long-wave Transformation overFringing Reefs. MS Thesis University of Hawaii, Honolulu.
Raudkivi, A.J., 1988.The roughness height under waves. Journal of Hydraulic Research26 (5), 569584.
Roeber, V., 2010. Boussinesq-Type Model for Nearshore Wave Processes in FringingReef Environment. PhD Dissertation University of Hawaii, Honolulu.
Roeber, V., Cheung, K.F., 2012.Boussinesq-type model for energetic breaking waves infringing reef environment. Coastal Engineering 70, 120.
Roeber, V., Cheung, K.F., Kobayashi, M.H., 2010. Shock-capturing Boussinesq-typemodel for nearshore wave processes. Coastal Engineering 57 (4), 407423.
Sambe, A.N., Sous, D., Golay, F., Fraunie, P., Marcer, R., 2011. Numerical wave breaking
with macro-roughness. European Journal of Mechanics B/Fluids 30 (6), 577
588.
Fig. 13.Wave breaking index on the reef slope as a function of the solitary wave height at WG1. Blue crosses, magenta squares, green diamonds, greentriangles, and magenta circles
denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green dash trend lines pass through the control and BF2 data.
46 P.D. Quiroga, K.F. Cheung / Coastal Engineering 80 (2013) 3548
http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0035http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0035http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0040http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0040http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0040http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0040http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0040http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0045http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0045http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0045http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0045http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0045http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0050http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0050http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0050http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0050http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0055http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0055http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0055http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0055http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0060http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0060http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0060http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0060http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0065http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0065http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0065http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0065http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0065http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0065http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0070http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0070http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0070http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0070http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0070http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0070http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0210http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0210http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0210http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0210http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0210http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0210http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0210http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0210http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0075http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0075http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0075http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0075http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0075http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0075http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0075http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0080http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0080http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0080http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0080http://dx.doi.org/10.1029/2004JC002711http://dx.doi.org/10.1029/2004JC002711http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0095http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0095http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0095http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0095http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0095http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0095http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0095http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0095http://dx.doi.org/10.1029/2009JC005753http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0100http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0100http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0100http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0100http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0105http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0105http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0105http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0105http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0110http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0110http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0110http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0110http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0115http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0115http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0115http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0115http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0115http://dx.doi.org/10.1029/2008JF001203http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0125http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0125http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0130http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0130http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0130http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0130http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0135http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0135http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0140http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0140http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0140http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0140http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0145http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0145http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0145http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0145http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0155http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0155http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0155http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0155http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B1%B3http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0155http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0155http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0145http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0145http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0140http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0140http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0135http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0135http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0130http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0130http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0125http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0125http://dx.doi.org/10.1029/2008JF001203http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0115http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0115http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0110http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0110http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0105http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0105http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0100http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0100http://dx.doi.org/10.1029/2009JC005753http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0095http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0095http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0095http://dx.doi.org/10.1029/2004JC002711http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0080http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0080http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0075http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0075http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0075http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0210http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0210http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0210http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0070http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0070http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0065http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0065http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0060http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0060http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0055http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0055http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0050http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0050http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0045http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0045http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0045http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0040http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0040http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0040http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0035http://refhub.elsevier.com/S0378-3839(13)00093-8/rf00357/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
13/14
Sheremet, A., Kaihatu, J.M., Su, S.-F., Smith, E.R., Smith, J.M., 2011. Modeling ofnonlinear wave propagation over fringing reefs. Coastal Engineering 58 (12),11251137.
Shi, F., Kirby, J.T., Harris, J.C., Geiman, J.D., Grilli, S.T., 2012.A high-order adaptive time-stepping TVD solver for Boussinesq modeling of breaking waves and coastal inun-dation. Ocean Modelling 4344, 3651.
Sleath, J.F.A., 1987.Turbulent oscillatory ow overrough beds. Journal of Fluid Mechanics182, 369409.
Stoker, J.J., 1957.Water Waves: The Mathematical Theory with Applications. IntersciencePublishers, New York (567 pp.).
Suntoyo, Tanaka, H., 2009. Numerical modeling of boundary layer ows for a solitarywave. Journal of Hydro-environment Research 3 (3), 129137.
Suntoyo, Tanaka, H., Sana, A., 2008. Characteristics of turbulent boundary layers over arough bed under saw-tooth waves and its application to sediment transport. CoastalEngineering 55 (12), 11021112.
Swigler, D.T., 2009.Laboratory Study Investigating the Three-Dimensional Turbulenceand Kinematic Properties Associated with a Breaking Solitary Wave. MS ThesisTexas A&M University, College Station.
Synolakis, C.E., 1987.Therunup of solitary wave. Journal of Fluid Mechanics 185, 523545.Synolakis, C.E., Skjelbreia, J.E., 1993.Evolution of maximum amplitude of solitary waves
on plane beaches. Journal of Waterway, Port, Coastal and Ocean Engineering 119(3), 323342.
Tonelli, M., Petti, M., 2013. Numerical simulation of wave overtopping at coastal dikesand low-crested structures by means of a shock-capturing Boussinesq model. CoastalEngineering 79, 7588.
Whitham, G.B., 1958. On the propagation of shock waves throughregionsof non-uniformarea or ow. Journal of Fluid Mechanics 4 (4), 337360.
Zijlema, M., Stelling, G.S., Smit, P., 2011.SWASH: an operational public domain code forsimulating waveeldsand rapidly variedowsin coastalwaters. CoastalEngineering58, 9921012.
List of symbols
BF: bed formC: celerityhb: wave breaking depth determined from video imagesho: water depth in the wave umehf: water depth over the reefat
Hb: breaking wave height determined from video imagesHo: incident wave height recorded at WG1h: average water depth between WG7 and WG9H: average bore height between WG9 and WG14k: roughness heightSG: sonic gaugeWG: wire gaugeH: wave height difference between WG3 and WG7 for wave shoaling, WG7 and WG9for wave breaking, and WG9 and WG14 for bore propagationhb: increase in breaking depth due to bed formsx: distance between WG3 and WG7 for wave shoaling, WG7 and WG9 for wavebreaking, and WG9 and WG14 for bore propagation: roughness element spacing: surface elevation from the still water levelmax: maximum surface elevation from the still water level
Fig. 14.Maximum surface elevation along the reefat for hf= 0.1 m. Blue crosses, magenta squares, green diamonds, green triangles, and magenta circles denote data from the
control, BF1, BF2, BF3, and BF4 tests; the control data is connected by blue lines.
47P.D. Quiroga, K.F. Cheung / Coastal Engineering 80 (2013) 3548
http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0160http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0160http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0160http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0160http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0160http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0165http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0165http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0165http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0165http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0165http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0165http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0165http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0170http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0170http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0170http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0170http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0170http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0170http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0150http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0150http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0175http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0175http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0175http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0175http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0175http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0175http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0180http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0180http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0180http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0180http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0180http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0215http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0215http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0215http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0185http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0185http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0185http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0190http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0190http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0190http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0190http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0190http://refhub.elsevier.com/S0378-3839(13)00093-8/rf9000http://refhub.elsevier.com/S0378-3839(13)00093-8/rf9000http://refhub.elsevier.com/S0378-3839(13)00093-8/rf9000http://refhub.elsevier.com/S0378-3839(13)00093-8/rf9000http://refhub.elsevier.com/S0378-3839(13)00093-8/rf9000http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0195http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0195http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0195http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0195http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0195http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0195http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0200http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0200http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0200http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0200http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0200http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0200http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0200http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0200http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0200http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B1%B4http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0200http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0200http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0200http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0195http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0195http://refhub.elsevier.com/S0378-3839(13)00093-8/rf9000http://refhub.elsevier.com/S0378-3839(13)00093-8/rf9000http://refhub.elsevier.com/S0378-3839(13)00093-8/rf9000http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0190http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0190http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0190http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0185http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0215http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0215http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0215http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0180http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0180http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0180http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0175http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0175http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0150http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0150http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0170http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0170http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0165http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0165http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0165http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0160http://refhub.elsevier.com/S0378-3839(13)00093-8/rf0160http://refhub.elsevier.com/S0378-3839(13)00093-8/rf01607/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs
14/14
Fig. 16.Average celerity on the reefat between WG9 and WG14 as a function of the
average wave height. Blue crosses, magenta squares, green diamonds, green triangles, and
magenta circles denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green
dash trendlinespassthrough thecontrol andBF2 data.Note that thevertical andhorizontal
axes of the three panels have different ranges for presentation of the data.
Fig. 15.Waveheight gradienton thereefat between WG9 andWG14as a functionof the
average wave height. Blue crosses, magenta squares, green diamonds, green triangles, and
magenta circles denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green
dash trend lines pass through the control and BF2 data. Note that the horizontal axes of the
three panels have different ranges for presentation of the data.
48 P.D. Quiroga, K.F. Cheung / Coastal Engineering 80 (2013) 3548
http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B1%B5http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B1%B6