Numerical Simulation of solitary waves propagating on stepped slopes
beaches
FAYÇAL CHERGUI1 , MOHAMED BOUZIT2
1Marine Engineering Department ; 2Mecanical Engineering Department
University of Science and Technology of Oran Mohamed Boudiaf
El Mnaouar, BP 1505, Oran, Bir El Djir
ALGERIA
Abstract: - The objective of the current paper is to study the propagating and breaking of solitary waves on
stepped slopes beaches, to simulate the shoaling and breaking, specifically the location of breaking
point Xb, and solitary wave height at breaking Hb of solitary waves on the different stepped slopes.
Ansys Fluent is used to implement the simulation, a two-dimensional volume of fluid (VOF) which is based on
the Reynolds-Averaged Navier–Stokes (RANS) equations and the k–ε turbulence closure solver. The obtained
results were firstly validated with existing empirical formulas for solitary wave run-up on the slope without
stepped structure and are compared with the experimental and numerical results. The numerical computation
has been carried out for several, configurations of beach slopes with tan ß= 1:15, 1:20, 1:25, wave height H0=
0.04, 0.06, 0.08m, water depth h0= 0.15, 0.2, 0.25m, and step height Sh= 0.025, 0.05, 0.075m. A set of
numerical simulations were implemented to analyze shoaling and breaking of solitary waves, wave reflection,
wave transmission, and wave run-up with various parameters wave heights, water depth, beach slopes, and Sh
step height.
Key-Words: Solitary waves - Breaking point - Wave run-up - Stepped slope - Navier−Stokes equations
Received: May 25, 2021. Revised: April 13, 2022. Accepted: May 11, 2022. Published: June 28, 2022.
1 Introduction
Long waves such as tsunamis and waves
resulting from large displacements of water
caused by phenomena such as landslides and
earthquakes sometimes behave approximately
like solitary waves. The run-up, overturn, and
breaking of solitary waves are the most direct
ways to damage the constructions near the
offshore, the most famous are tsunamis. Indeed,
since the giant tsunami that occurred in the
Indian Ocean on 26 December 2004 caused
memorable losses [1], this phenomenon has
aroused particular interest, and it's necessary to
understand their dynamics more clearly to
predict them more efficiently. The first to start
theoretical studies of solitary waves [2,3,4,5] ,
more recent analyses of solitary waves have
been performed [6,7,8].
At present, numerical models for tsunami wave
simulations were mainly carried out. In most
cases, solitary waves were used to represent a
tsunami wave, in general, solitary waves were
used to represent a tsunami wave
[9,10,11,12,13]. The breaking factor plays an
important role in the calculation of wave
heights, some researchers have attempted to
define breaking factors for the run-up of a
solitary wave on plane slopes.
In the literature, the two-dimensional simulation
of wave induced free surface flows can carried
out by numerical models that integrate the two-
dimensional Navier- Stokes equations [14] , Ji
et al.[15] and Hieu et al.[16] Presented a
numerical two-phase flow model coupling with
the VOF method, including the processes of
wave shoaling, wave breaking, wave reflection,
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and air movement, compared the simulation of
wave breaking on a sloping bottom with the
conditions of the experiment by Ting et al.[17].
Dentale et al.[18] Evaluated the filtration of the
fluid within the interstices of a concrete blocks
breakwater by integrating the Reynolds
Averaged Navier-Stokes equations (RANS)
inside the voids. Khaware et al.[19] Investigates
the sensitivity of 5th order solitary wave models
are applied to shallow wave scenarios with high
relative heights, modeling from both an
analytical and a numerical perspective, using a
(VOF) method, coupled with the Open Channel
Flow module in ANSYS Fluent, it has been
shown that the Stokes wave theory can be
applied in the shallow regime and concluded
that the Explicit formulation is very sensitive
near breaking conditions requirement of very
small time step size. The turbulence model most
commonly used is the standard k–ε model,
Hsiao et al.[9] studied experimentally and
numerically of wave tsunami-like solitary
waves impinging and overtopping an
impermeable trapezoidal seawall on a 1:20
sloping beach, Reynolds-Averaged Navier–
Stokes (RANS) equations are used to describe
mean flow fields, and the modified k-ε closure
model is employed to examine turbulence
behaviors. Wu et al.[20] compared the tsunami
force on structure using different k-ε turbulence
models and pointed out that the k-ε RNG model
has higher calculation accuracy.
The research on the propagation, shoaling and
breaking of solitary waves were mainly carried
out. [21] has computed shoaling and breaking
of solitary waves on slopes, discussed the
characteristics of various breaking types and
made a clear classification of wave breaking,
they proposed a set of empirical formulae for
breaking criterion, as shown in Eq. (11), who S
is the breaking coefficient of the solitary wave,
which can be used to determine the wave
breaking type. According on Eq. (11), the value
of the breaking coefficient S for the cases in this
study can be calculated the results show 0.1 < S
< 0.26, that indicates that the wave breaking
type of all the cases in this study is plunging
breaking (PL). Hsiao et al.[22] simulated the
shoaling and breaking of solitary waves on a
beach , and re-examined some of a set of
empirical formulae of Grilli et al. [21] the
experimental results agree well. Akbari et
al.[23] Improved (SPH) method to study
solitary wave overtopping for different coastal
structures, and model are verified by simulating
solitary wave breaking over a sloping bed,
based on laboratory data presented by Grilli et
al. [21]. Gallerano et al.[24] propose a modified
turbulence model to represent the energy
dissipation due to the wave breaking.
Numerous studies are focusing on the run-up of
solitary waves, Wu et al.[25], Chou et al.[26]
and Lin et al.[27] indicated that the run-up
height increases with beach slope for breaking
solitary waves. Kuai et al.[28] Realized
experimental measurements demonstrate the
effect of the impact of the jet from a plunging
breaking solitary wave and the post breaking
bore formed on the resultant run-up showing
spatial snapshots in detail on 1:15 slope for
incident wave height H/ho = 0.40. [29]
Presented numerical investigations of the run-
up and hydrodynamic characteristics of solitary
waves influenced by beach slope, concludes the
run-up heights of breaking solitary waves
increase with beach slope. Yao et al.[29]
Investigated numerically the reduction of
tsunami-like solitary wave run-up by the pile
breakwater on a different sloping beach, the
adopted model was validated with existing
empirical formulas Synolakis et al. [30] and
Hsiao et al. [22] for solitary wave run-up on the
slope without the pile structure.
Other protection systems have been studied,
such as stepped form, which combines needs in
current coastal protection by increasing the
surface roughness of the coastal protection
structure to reduce wave run-up and wave
overtopping. A comprehensive literature review
by Kerpen et al.[31] detailed discusses previous
studies on stepped revetments as a coastal
protection measure. Experimental studies are
carried out by Kerpen et al.[32] for the
empirical derivation of a roughness reduction
coefficient for wave overtopping on stepped
revetments in relation to a smooth slope,
mentioned the importance of the dimensionless
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step height on the energy dissipation in the
wave run-up and, as a consequence, the
reduction of the mean overtopping discharge.
Shih et al.[33] investigated the energy
dissipation characteristics of stepped obstacles
in a physical wave flume, varying the angle of
the stepped surface with the slope of the
embankment surface, with various interaction
angles between the incident waves and the
rough stepped surfaces.
In the present study, the propagating solitary
waves on stepped slope beach was numerically
studied for the different wave height, beach
slopes, water depth, and step height. The
unsteady Reynolds averaged Navier-Stokes
(RANS) equations coupled with a k-ε
turbulence model are applied to simulate the
free surface elevation. The main purpose of this
study is to numerically simulate the shoaling
and breaking, specifically the location of
breaking point Xb, and solitary wave height at
breaking Hb of solitary waves on the different
stepped slope and investigate the run-up, the
wave reflection, and the wave transmission of
solitary waves influenced by the incident wave
height, beach slope, water depth, and step
height Sh.
2 Numerical model
2.1 Governing equations
Two dimensional Reynolds Averaged Navier
Stokes (RANS) are numerically solved in the
CFD solver FLUENT [34,35], to simulate the
unsteady and incompressible viscous fluids, in
which the continuity equation and the
momentum equation are, respectively:
1
2
Where i and j are the cyclic coordinates in an
orthogonal coordinate system whose values are
1 and 2, while ui are the time-averaged velocity
components; ρ is the density of the fluid; p is
the pressure.
In this study, the k-ε turbulence model proposed
by (Launder and Spalding., 1972) are selected
[20,34,36] , which are as following:
3
4
To identified interfaces between non-
penetrating fluids a Volume of Fluid Method
(VOF) is used by Khaware et al.[19] and Zhan
et al.[36]. The volume fraction of a specific
fluid (α) is defined as the ratio of the volume of
that fluid to total volume. Interfaces between
different fluids are identified by volume
fraction falling between 0 and 1.
Summation of volume fraction for all the fluids
should be equal to one
15
Volume fraction equation is given as,
.06
Total continuity equation for incompressible
fluid is given as,
.07
To solve this model, the PISO scheme is used
for pressure velocity coupling [19], second-
order upwind and compressive schemes are
used for momentum and volume fraction
respectively. First-order transient methods are
used allowed with Explicit formulation. Finally,
first-order upwind scheme was selected for the
discretization of the equations of turbulent
energy and dissipation.
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2.2 Wave generation and solitary wave
theory
Fluent formulates the Solitary wave theories
that are expressed using Jacobian and elliptic
functions based on the work by D. John Fenton
et al., 1998 [37]. These wave theories are valid
for high steepness finite amplitudes waves
operating in intermediate to deep liquid depth
range.
The generalized expression for wave profile for
5th order Stokes theory is given as.
,1
8
2
2
tanh
9
Where c is wave celerity and k is wave number,
bij ci , are complex expressions of kH.
Solitary wave theories are more widely used for
shallow depth regimes, are derived by assuming
that the waves have infinite wave length. 5th
order solitary wave expressions are complex
functions of relative height (H/h).
Wave profile for a shallow wave is defined as
,
10
where wave celerity and wave numbers are
given as
1
21
3
4
and x0 is the initial position of wave
3 Numerical setup
3.1 Numerical domain
The computational domain for the propagation
of solitary waves is shown in Fig. 1(a) two-
dimensional numerical wave flume is built, the
flume is with the total length of 15.5 m, height
of 0.4 m, the numerical waves were set to be
generated at 7 m seaside of the slope toe.
The free surface elevation along the flume was
measured by nine numerical wave gauges Fig.
1(a) with x=0 denotes the slope toe, which
(ng01) was placed -3m upstream of the slope,
(ng02) at the slope toe, (ng03) at 0.75m, (ng04)
at the middle of the stepped structure 1.5m
(stepped width), and (ng05) at end of the
stepped structure 3m, for the others are placed
downstream the stepped structure spaced 0.5m
until (ng09) where is placed at 4.5m
downstream the toe of the slope. H
0 is the
incident wave height; h0 is the still water depth
at the plane bed, ß is the angle of beach slope.
Fig. 1 (a) Schematic view of the numerical wave flume,
with x=0 denotes the slope toe;
(b) Schematic view of the stepped structure on slope.
The stepped structure are installed along of
slope just 3m in front of the toe Fig. 1(b), three
cases of stepped structure are studied
Sw=0.5m, 1m, 1.5m ( Sw is step width, Sh is
step height and B is stepped width fixed 3m )
for three different slopes ( tan = 0.15, 0.20,
0.25) , then we have a total of nine cases of
stepped structure.
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3.2 Boundary Conditions and mesh
description
The wave propagates from the left where
solitary waves are generated in the inlet
boundary to the right where no-slip wall
boundary condition is applied. The bottom and
stepped structure are considered a no-slip wall
and the top is considered a pressure outlet.
Fig. 2 (a) Layout of the numerical setup; (b) Numerical
grids of the computational domain.
Fig. 3 Variations of the normalized ɳ/
maximum free surface elevation of breaking point
and / maximum wave run-up on the slope with
different grid sizes.
To validate the model, the structured mesh is
applied Fig. 2(a). ANSYS ICEM is employed to
build a geometric model and generate grids, a
numerical convergence test was performed by
varying the grid size on slope using the typical
wave condition of h0 = 0.2 m and H0 = 0.06 m.
We also tested another four sizes (2, 6, 8 and 10
mm) Fig. (3) showing the results for the
normalized maximum free surface elevation
ɳ/ of breaking point and the normalized
maximum runup / on the slope. The
differences in the simulations were generally
6.5% when the grid size is 10 mm, 4% grid size
8mm and 2.5% when the grid size declined
from 4 mm to 2 mm.
In vertical direction, the grid size start from
4mm it was kept a constant all the way at 0.3m
and 4mm to 8mm to the domain top. In the
stream-wise direction, from the inlet of domain
to the slope toe, the grid size reduced gradually
from 8mm to 4mm, and were kept constant
4mm from the slope toe at a location 1m
downstream of stepped structure, to capture the
shoaling and breaking of solitary waves, and the
grid size increased gradually from 4 mm to 8
mm at the end of domain. After conducting
several simulations with various time step sizes,
a time step size of t= 0:001s was chosen to
ensure the solution's accuracy and convergence,
all the simulations have been carried out on a
Core i7-3770, 3.40 GHz, and RAM 16.0 GB
computer, the simulation was run for 12 s to
guarantee the completion of wave run-up and
rundown processes on the slope.
4 Results and Discussions
4.1 Model validation
Firstly, a numerical test was conducted without
stepped structure to verify the solitary waves
generated by the model at numerical wave
gauge (ng01), we compared the experimental
work and the numerical generated solitary wave
profile with the theoretic profile calculated by
the equation of Yao et al.[29] Fig. 4(a).
Numerical results are in good agreement with
theoretical solutions up to a relative height of
H0= 0.06m, except at the tail of solitary wave
deviates compared to the experimental result.
We then compared the run-up of the four
offshore solitary wave heights with a fixed
offshore water depth of h0= 0.2m. tested wave
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conditions (H0=0.02, 0.04, 0.06, 0.08m) with
the laboratory experiments and the numerical
simulations Fig. 4(b). It's observed that the
formula's Hsiao et al. [22] predicted slightly
less wave run-up than those predicted by
Synolakis et al. [30] for the current laboratory
slope of 1:20. The present result shows a slight
increase that those shown by Yao et al.[29].
When the wave height increases, the slightly
becomes more important. The maximum error
of run-up value compared to laboratory
experiments is approximately 2% for H0=
0.02m, 3%, for H0=0.04m, 6% for H0=
0.06m and 10% for H0=0.08m.
Fig. 4 (a) Comparison of the solitary wave profile of the
numerical present results with numerical and the theory
result; (b) Comparison of the maximum wave run-up R
on the slope between the simulations and the predictions
from the empirical formulas.
Similar experiments were also conducted by
Hsiao et al. [9], except that a trapezoidal
seawall was placed in the slope 1:20. A solitary
wave with a wave height of H0= 0.07m in the
water depth of h0= 0.2m was generated. Where
a reference wave gauge (wg01) was fixed at 1.1
m in front of the beach slope, another wave
gauge downstream of the slope, (wg03) placed
1.7m and (wg10) placed at 3.744m behind the
slope toe. The simulation results are also
compared, the comparison between the solitary
wave profile obtained by the experimental and
numerical simulation of (Hsiao et al., 2010),
shows that the present computed results are a
little different from the results Fig. 5.
Fig. 5 Comparison of the solitary wave profile of the
numerical present results with the experimental and
numerical.
A breaking wave is a wave whose amplitude
(Hb) reaches a maximum level (Breaking point
BP) at which process can start to occur, the
solitary breaking wave type can be categorized
by using a slope parameter, So, suggested by
Grilli et al. [21], as shown in Eq. (11).
1.521tan
√11
In which tan is the angle of beach slope, ε =
H0/h0 is the wave steepness, H0 is the wave
height, h0 is the water depth measured from the
flat flume bottom and Hb is solitary wave height
at breaking.
0.149
/.0.3 12
0.841exp6.421 13
The ranges of different breakers are
0.3<S0<0.37 for surging type (SU),
0.025<S0<0.30 for plunging type (PL) and
S0<0.025 for spilling type (SP) [21], the water
depth (hb) at breaking point can be estimated by
Eq.(12).
From the equation (12) we can extract the
theoretical values location of solitary wave
breaking point Xb, and compared with the
numerical value, as shown in Fig. 6, using the
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typical wave condition of h0 = 0.2 m and H0 =
0.02, 0.04, 0.06 and 0.08 m on 1:20 slope. it can
be observed that the difference between the
numerical and theoretical solution changes
according to the wave height, the maximum
difference is 0.2m.
Fig. 6 The location of breaking point between present
numerical solution and theoretical solution by Grilli et al.
[21]
It can be seen that the numerical results agree
well with the compared results, which
demonstrates that the present model is capable
of accurately simulating the propagation, the
shoaling and breaking of solitary waves on
slopes considerably well.
4.2 Wave surface elevations and breaking
wave
The wave surface elevation time histories at the
different gauges are presented Fig. 7, a
numerical reference gauge (ng01) is placed at
x=-3 m from the toe of the slope. It can be
noted that the wave height of the solitary wave
almost remains unchanged along the flat flume
bottom (ng01) for all cases study, indicating
that the numerical flume can simulate solitary
wave propagation considerably well, and a
slight increase observed (ng02).
It also can be shown for Sh=0 and Sh= 0.025m
that the wave height of the solitary wave
becomes large gradually (ng06), but this is not
the case for Sh=0.05m and Sh=0.075m the
wave height decreases abruptly because the
breaking point is closer behind the numerical
wave gauge, as the figure Fig. 7 shows very
well the breaking point phase of the wave, this
means that as the step height Sh rises, so does
the breaking point is further backward. The
solitary wave steepens because of shoaling
effect; and then the wave height decreases,
indicating the solitary wave breaks (ng08).
Fig. 7 Wave surface elevation time histories at the
different numerical gauges (h0=0.2m, H0= 0.06m, slope
1:20).
To show the effect of slope on the breaking
point, Fig. 8 present the elevation of the free
surface at the moments of the breaking point for
different slopes for water depth h0=0.2m, wave
height H
0= 0.06m, and stepped width Sw=1m,
the increase in slope results from an increase of
the stepped height Sh, for 1:15, 1:20, and 1:25,
Sh is 0.025, 0.05, and 0.075m respectively see
Fig. 1(b).
Fig. 8 Free surface elevation at the breaking point
moments for different slope (h0=0.2m, H0= 0.06m,
Sw=1m).
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It is possible to figure out the breaking point
location Xb, with the availability of detailed
wave surface distribution data, it is clearly
shown that milder slope caused the incipient
wave breaking further shoreward Fig. 8c, and
the rise of stepped height Sh caused an increase
of the solitary wave height at breaking Hb Fig.
8(a).
(b)
(c)
(d)
Fig. 9 Free surface elevation when the solitary wave
propagates on the 1:20 slope , H0= 0.06m , h0= 0.2m. (a)
Sh =0m , (b) Sh= 0.025m ; (c) Sh= 0.05m ; (d) Sh=
0.075m
Fig. 9. shows the changes in the wave surface
profile during the shoaling and breaking process
for the wave with H0= 0.06m at h0 = 0.2m on
1:20 slope for different step height Sh. The
wave surface becomes unsymmetrical about the
wave crest when the solitary wave arrives at the
toe of the slop, as shown in Fig. 9(a). As
propagating the further steepening of wave-
front causes the initiation of wave breaking
process, the wave-front face develops into
vertical and steeper with the decreasing local
water depth Fig. 9(b,c). The position of the
breaking point Xb decreases and the breaking
phenomenon is in advance when the stepped
height Sh increases Fig. 9(c).
After that, the wave crest continues to increase,
then rolls down, and finally plunges into the
water, as shown in Fig. 9(d,e). The obtained
geometric criterion of the breaking point form is
consistent with those of Chen et al.[11] ; Grilli
et al. [21] and Lin et al.[38] .
The breaking of the wave creates turbulence
and tumbling of the water flow, the crest
overturns with an ejecting water jet emanating,
during wave curling down it entraps a large
amount of air to form bubbles of different sizes,
and the surge continues to propagate with the
entrapped bubbles after the wave-front passes
through the still-water shoreline, it collapses
and the consequent run-up process commences,
the surging starts earlier when the stepped
height Sh rise, as shown in Fig. 9(f).
It is noted that the present model is capable of
capturing the plunging breaking, which means
that the numerical method in this study can
simulate wave breaking considerably well.
Fig. 10 Present numerical of the location of breaking
point Xb (a) ; and solitary wave height at breaking Hb (b)
for different stepped height Sh
The theatrical solitary wave height at breaking
from the equation (13) for h0= 0.2m, H0=
(a)
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0.06m, slope 1:20 is Hb (theatrical) = 0.09m
compared with the present numerical solitary
wave height at breaking without stepped
structure (Sh= 0m) is Hb (numerical)= 0.081m, the
maximum error of Hb value is 10%, and for Xb
is 3% Fig. 6. It is shown the presence of stepped
structures caused the reduction of the location
of breaking point Xb nearly 1m for Sh= 0.075m
Fig. 6a. However, a slight decrease of the wave
height at breaking Hb was also observed, where
an important portion of the incident wave could
be reflected due to an increase of stepped height
Sh Fig. 6(b).
4.3 Wave run-up, wave reflection, and wave
transmission
The maximum run-up height of the wave is
defined as the height from the still water level
to the highest position where the wave arrives
on the slope, we present numerical results for
run-up, and evaluate the influence of incident
wave height (H0= 0.04, 0.06, 0.08m), water
depth (h0=0.15, 0.2, 0.25m), beach slope (ß=
1:15, 1:20, 1:25), and step height (Sh= 0.025,
0.05, 0.075m) on the maximum run-up height of
the solitary wave.
Fig. 11. Variations of maximum run-up on the slope
with different: (a) wave heights; (b) water depths; (c)
beach slopes; and (d) and step height.
The maximum run-up increased with increasing
wave height Ho Fig. 11(a) also increased with
increasing water depth ho Fig. 11(b), due to the
incident wave energy rose with the increase of
wave height and water depth. The growth rate
of the maximum run-up is greater when the
beach slope is steeped and similarly when the
wave height increases.
Fig. 11(c) shows the influence of the step height
Sh (Sh= 0.025, 0.05, 0.075m) on the maximum
run-up height of the solitary wave. The run-up
decreases with increasing the step height Sh for
all wave height. For the influence of the beach
slope (ß= 1:15, 1:20, 1:25) Fig. 11(d). the run-
up increases with increasing beach slope tan ß
combined with increasing the step height Sh.
With a steeper beach slope, the breaking wave
cannot propagate on the slope, and run-up
decreases with the beach slope ( slightly step
height Sh).
Measurement points were chosen to analyze
wave reflection Hr at (ng01) and transmission
Ht at (ng08). It is visible (see Fig. 7) the wave
height at (ng08) decreases with increasing step
height Sh, moreover, the numerical wave gauge
(ng01) did not record any wave reflected for
plane slope without stepped structure Fig. 6(a),
however, the number and the height of reflected
wave depending on step height Sh.
The reflected wave slightly decreased with
increasing water depth Fig. 12(b), However, it
decreased with the increase of wave height Fig.
12(a). The transmitted wave height decreased
with increasing wave height, but it increased
rapidly with increasing water depth Fig. 12(a,b).
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Fig. 12. Variations of simulated reflected wave
height at (ng01) and transmitted wave height at (ng08)
with different: (a) wave heights; (b) water depths; (c)
beach slopes; and (d) and step height
For the effect of step height Sh and slope
beaches on reflected and transmitted wave, it is
clearly shown that increase of Sh cause slight
decrees of transmitted wave Fig. 11(c). It is the
same for slope beaches but decreasing rapidly
due to combined with an increase of step height
Sh Fig. 11(d). However, the reflected wave
increase with increasing of step height Sh and
slope beaches caused the incipient wave
breaking closer to the toe of the slope.
The reflection coefficient is inversely
proportional to the incident wave. Moreover,
Yao et al.[29] declared decreasing of wave
reflection with increasing beach slope was also
observed with the presence of pile structure in
slope, which is contrary to the case for wave
interaction with a slope without the pile
structure.
5 Conclusion
In this paper, solitary wave propagating on a
sloping beach are investigated, a two-
dimensional numerical model based on ANSYS
Fluent was established. The Navier–Stokes
equations with k-ε turbulence closure were
solved and the free surface was tracked by a
VOF method, a set of numerical simulations
were implemented to investigate the water
surface profile, shoaling and breaking of
solitary waves , wave reflection, the wave
transmission, and the wave run-up by varying
incident wave height, water height, beach slope,
and step height Sh.
- The results obtained from the Fluent solver
simulations with a multiphase model for both k-
ε are in corroboration with the empirical
formulas for solitary wave run-up on the slope,
also with numerical results and experimental
data, indicate that the numerical model can
accurately model the free surface elevation, the
breaking processes and the maximum run-up
for different slope beach.
- The breaking process dissipates energy in the
form of turbulence, the variation of the step
height Sh affects the location of breaking point
Xb back close to the toe of the slope, and
decrease of the wave height at breaking Hb, as
well as the maximum run-up, so the run-up
increases with increasing beach slope, wave
height, and water depth.
- The wave reflection decreased with the
increase of water depth, and wave height.
Otherwise, increased slightly with increasing
step height Sh and increased rapidly with beach
slope. As well as the wave transmitted
increased with the increase of water depth, and
decreased with increasing wave height, step
height, and beach slope.
References:
[01] A.C. Yalciner, A. Suppasri, E. Mas, N. Kalligeris,
O. Necmioglu, , F. Imamura, C. Ozer, , A. Zaytsev, , N.
M. Ozel, C. Synolakis, Field survey on the coastal
impacts of March 11, 2011 great east Japan tsunami ,
2012, Pure and Applied Geophysics (This volume).
[02] J. Boussinesq, Théorie de l’intumescence liquide
appelée onde solitaire ou de translation se propageant
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.10
Fayçal Chergui, Mohamed Bouzit
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106
Volume 17, 2022
dans un canal Rrectangulaire, Comptes Rendus Acad. Sci.
Paris, 1971, Vol 72, pp 755- 759.
[03] J. B. Keller, The Solitary wave and periodic waves
in shallow water, Commun. Appl. Math., 1948, Vol 1, pp
323-339.
[04] L. Rayleigh, On Waves, Phil. Mag.,1876, Vol 1, pp
257-279.
[05] W.H Munk, The Solitary wave theory and its
application to surf problems, Annals New York Acad.
Sci., 1949 , Vol 51, pp 376-423.
[06] J. D. Fenton, A ninth-order solution for solitary
waves, Jour. Fluid Mech., 1972 , Vol 53, pp 257-271.
[07] J. G. B. Byatt-Smith, M.S. Longuet- Higgins, On the
speed and profile of steep solitary waves, Proc. Roy. Soc.
London, Series A, 1976, Vol 350, pp 175-189.
[08] M. S. Longuet-Higgins, J. D. Fenton, On mass,
momentum, energy, and calculation of a solitary wave,
Proc. Roy. Soc. London, Series A, 1974 ,Vol 340, pp 471-
493.
[09] S.C. Hsiao, T.C. Lin, Tsunami-like solitary waves
impinging and overtopping an impermeable seawall:
Experiment and RANS modeling, Coastal Engineering
57 (2010) 1–18.
[10] C. Jiang, X. Liu, Y. Yao, B. Deng, J. Chen,
Numerical investigation of tsunami-like solitary wave
interaction with a seawall, Journal of Earthquake and
Tsunami Vol. 11, No. 1 (2017) 1740006.
[11] D. Cheng, X. Zhao, D. Zhang, Y. Chen, Numerical
study of dam-break induced tsunami-like bore with a
hump of different slopes, China Ocean Eng., 2017, Vol.
31, No. 6, P. 683–692.
[12] K. Shao, W. Liu, Y. Gao, Y. Ninga, The influence of
climate change on tsunami-like solitary wave inundation
over fringing reefs, Journal of Integrative Environmental
Sciences, 2019, VOL. 16, NO. 1, 71–88
[13] K. Qu, X.Y. Ren, S. Kraatz, Numerical investigation
of tsunami-like wave hydrodynamic characteristics and
its comparison with solitary wave, Applied Ocean
Research 63 (2017) 36–48.
[14] G. Cannata, L. Barsi, C. Petrelli, F. Gallerano,
Numerical investigation of wave fields and currents in a
coastal engineering case study, WSEAS Transactions on
Fluid Mechanics, Vol. 13, 2018, pp. 87–94.
[15] Q. Ji, X. Liu, Y. Wang, C. Xu, Q. Liu , Numerical
investigation of solitary waves interaction with an
emerged composite structure, Ocean Engineering 218
(2020) 108080.
[16] P.D. Hieu, T. Katsutoshi, V.T. Ca, Numerical
simulation of breaking waves using a two-phase flow
model, Applied Mathematical Modelling 28 (2004) 983–
1005.
[17] C.K. Ting Francis, T. Kirby James, Observation of
undertow and turbulence in a laboratory surf zone,
Coastal Engineering 24 (1994) 51-80.
[18] F. Dentale, G. Donnarumma, E.P. Carratelli, F.
Reale, A numerical method to analyze the interaction
between sea waves and rubble mound emerged
breakwaters, WSEAS Transactions on Fluid Mechanics,
Volume 10, 2015, pp. 105-115
[19] A. Khaware, V. Gupta, K. Srikanth, P. Sharkey,
Sensitivity analysis of non-linear steep waves using VOF
method, Tenth International Conference on
Computational Fluid Dynamics (ICCFD10), 2018,
Barcelona, Spain,
[20] L. Wu , Study on the tsunami force of the
superstructures of the medium and small span bridges.
(MA.Eng Dissertation) Southwest Jiaotong
University,2017, P.R. China
[21] S.T. Grilli, I.A. Svendsen, R. Subramanya, Breaking
criterion and characteristics for colitary waves on slopes,
J. Waterway, Port, Coastal, Ocean Eng. 1997.123:102-
112.
[22] S.C. Hsiao, T.W. Hsu, T.C. Lin, Y.H. Chang, On the
evolution and run-up of breaking solitary waves on a
mild sloping beach, Coastal Engineering, 2008, 55(12),
975–988.
[23] H. Akbari, . Simulation of wave overtopping using
an improved SPH method, Coastal Engineering, 2017,
126 (2017) 51–68.
[24] F. Gallerano , G. Cannata , L. Barsi , F. Palleschi ,
B. Iele, Simulation of wave motion and wave breaking
induced energy dissipation, WSEAS Transactions on
Fluid Mechanics, Volume 10, 2015, pp. 62-69.
[25] N.T. Wu, H. Oumeraci, W. Partenscky, Numerical
modelling of breaking wave impacts on a vertical wall,
1995, Coastal Engineering, pp 1672 - 1686.
[26] C.R. Chou, K.Ouyang, The deformation of solitary
waves on steep slopes, Journal of the Chinese Institute of
Engineers, 1999 , Vol. 22, No. 6, pp. 805-812.
[27] P. Lin, K. Chang, L.F. Philip, Liu, Run-up and
rundown of solitary waves on sloping beaches, J.
Waterway, Port, Coastal, Ocean Eng, 1999 ,125:247-255.
[28] Y. Kuai, M. Qi, J. Li, , Numerical study on the
propagation of solitary waves in the near-shore, Ocean
Engineering 165 (2018) 155–163.
[29] Y. Yao, J. Mei-juna, M. Da-weib, D. Zheng-zhic, L.
Xiao-jiana, A numerical investigation of the reduction of
solitary wave runup by A row of vertical slotted piles,
China Ocean Eng., 2020, Vol. 34, No. 1, P. 10–20
[30] C.E. Synolakis, The runup of solitary waves, Journal
of Fluid Mechanics, 1987, 185, 523–545.
[31] N.B. Kerpen, , T. Schlurmann, Stepped revetments
– Revisited, Proc. 6th Int. Conf. on the Application of
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.10
Fayçal Chergui, Mohamed Bouzit
E-ISSN: 2224-347X
107
Volume 17, 2022
Physical Modelling in Coastal and Port Eng. and Science
(Coastlab16), 2016, Ottawa, Canada, 1-10.
[32] N.B. Kerpen, T. Schoonees, T. Schlurmann, Wave
overtopping of stepped revetments, Water 2019, 11,
1035.
[33] R.S. Shih, W.K. Weng, C.Y Li, Characteristics of
wave attenuation due to roughness of stepped obstacles,
Ships and Offshore Structures, 2020,
10.1080/17445302.2019.1661652.
[34] W. Yang , Z. Wen, F. Li, Q. Li, Study on tsunami
force mitigation of the rear house protected by the front
house, Ocean Engineering 159 (2018) 268–279.
[35] P. Prabu, B.S. Murty, C. Abhijit, S.A. Sannasiraj,
Numerical investigations for mitigation of tsunami wave
impact on onshore buildings using sea dikes, Ocean
Engineering 187 (2019) 106159.
[36] J. Zhan, J. Zhang, Y. Gong, Numerical investigation
of air entrainment in skimming flow over stepped
spillways, Theoretical and Applied Mechanics
Letters,2016, S2095-0349(16)30009-5.
[37] D. John Fenton, The cnoidal theory of water waves,
Chapter 2 of Developments in Offshore Engineering, Ed.
J.B. Herbich, Gulf: Houston, 1998.
[38] Y. Li, F. Raichlen, Energy balance model for
breaking solitary wave run-up, J. Waterway, Port,
Coastal, Ocean Eng. 2003.129:47-59.
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