WAVE-INDUCED MEAN MAGNITUDES IN PERMEABLE
SUBMERGED BREAKWATERS
By Fernando J. Méndez,1 Inigo J. Losada,2 Member, ASCE,
and Miguel A. Losada,3 Member, ASCE
ABSTRACT: The influence of wave reflection and energy dissipation by breaking and by porous flow induced
by a permeable submerged structure on second-order mean quantities such as mass flux, energy flux, radiation
stress, and mean water level is analyzed. For this purpose, analytical expressions for those mean quantities in
terms of the shape functions are obtained. The dependence of those quantities on the incident wave characteristics, structure geometry, and permeable material characteristics is modeled, extending the writers’ previous
work including wave breaking. Two models for regular waves are presented: a 2D model to be applied to
submerged trapezoidal breakwaters and a 3D model for submerged permeable rectangular breakwaters. Both
models are able to reproduce experimental wave height transformation as well as mean water level variations
along the wave flume with reasonable accuracy. Results give useful information for engineering applications of
wave height evolution and set-up and set-down evolution in the vicinity of the submerged permeable structure.
INTRODUCTION
Submerged breakwaters have been widely used for coastal
protection as wave energy dissipators and sediment transport
controllers. Energy dissipation is caused by two mechanisms:
wave breaking and friction on the surface and inside the permeable layer of the structure. Efficiency of the submerged
breakwater depends on the freeboard, the crest width, and the
permeable material characteristics. In regions of large tidal
range, reflection, transmission, and dissipation characteristics
may change considerably during a tidal cycle.
Several models to describe wave transformation over submerged breakwaters have been presented in the literature. Independently of the extensive existing work for impermeable
structures, the influence of structure permeability has also been
addressed in the last few years. In the following, some of that
work is addressed.
Rojanakamthorn et al. (1989, 1990) presented a mathematical model based on linear wave theory for a rectangular submerged breakwater and extended the solution in deriving a
modified mild-slope equation, including wave breaking, to
evaluate wave transformation over a trapezoidal porous breakwater. Irregular waves were analyzed using an individual wave
analysis technique to calculate the transformation of each wave
component. Gu and Wang (1992), using a boundary integral
element method (BIEM), developed a numerical model for
wave energy dissipation within porous submerged breakwaters
based on a linearized porous flow equation.
Losada et al. (1996a,b) assuming incompressible fluid and
irrotational motion inside and outside the permeable structure,
analyzed the interaction of submerged breakwaters with nonbreaking oblique incident regular waves and directional random waves. The influence of structure geometry, porous material properties, and wave characteristics on the kinematics
1
Postdoct. Res., Oc. and Coast. Res. Group, Universidad de Cantabria,
Dpto. de Ciencias y Técnicas del Agua y del Medio Ambiente, Av. de
los Castros s/n, 39005 Santander, Spain.
2
Prof., Oc. and Coast. Res. Group, Universidad de Cantabria, Dpto. de
Ciencias y Técnicas del Agua y del Medio Ambiente, Av. de los Castros
s/n, 39005 Santander, Spain.
3
Prof., Universidad de Granada, E.T.S.I. de Caminos, C.y.P. Campus
de la Cartuja s/n, 18071 Granada, Spain.
Note. Discussion open until July 1, 2001. To extend the closing date
one month, a written request must be filed with the ASCE Manager of
Journals. The manuscript for this paper was submitted for review and
possible publication on August 13, 1998. This paper is part of the Journal
of Waterway, Port, Coastal and Ocean Engineering, Vol. 127, No. 1,
January/February, 2001. 䉷ASCE, ISSN 0733-950X/01/0001-0007–0015/
$8.00 ⫹ $.50 per page. Paper No. 19011.
and dynamics over and inside the breakwaters is considered
in both papers.
Second-order theory was first presented for an impermeable
step by Massel (1983) where the growth of secondary harmonics was analyzed. Further research on nonlinear effects
can be found in Rey et al. (1992), Ohyama and Nadaoka
(1992), and Eldeberky and Battjes (1994) for impermeable
submerged structures. A limited number of papers can be
found where the nonlinear effects on permeable submerged
breakwaters are considered. Cruz et al. (1992, 1997) derived
a set of nonlinear vertically integrated equations similar to that
of Boussinesq to evaluate wave transformation induced by a
porous structure. Mizutani et al. (1998) developed a coupled
BIEM/finite-element method to study the nonlinear dynamic
interaction between waves and a wide crown submerged
breakwater without breaking. Losada et al. (1997) conducted
experimental work to analyze the influence of structure porosity on the generation of higher harmonics, comparing experimental results with linear models and defining an effective
water depth, hef , to evaluate the potential harmonic generation
by submerged permeable structures.
The literature review suggests that most of the nonlinear
work carried out has been concentrated on evaluating the wave
transformation induced by the presence of the permeable structure. However, little or no attention has been paid to the modeling of second-order averaged magnitudes such as mean water level, mass transport, radiation stress, etc., which are
obviously affected by the reflection and dissipation induced by
the structure. Several investigators are aware of the mean water level variations induced by submerged structures, as has
been pointed out in the experimental work of Gourlay
(1996a,b), Mory and Hamm (1997), and Loveless and Debski
(1998). However, at the moment, no extensive theoretical analysis has been carried out for submerged permeable breakwaters.
The main goal of this paper is to analyze the influence of
reflection and dissipation induced by a permeable submerged
breakwater on second-order mean quantities such as mean water level, mass flux, and radiation stress. The dependency of
these magnitudes on incident wave conditions, structure geometry, and permeable material characteristics are modeled extending the work in Losada et al. (1996a) to include wave
breaking and the evaluation of mean quantities. Regular waves
are considered. Results show that the models presented, especially the 2D model for an arbitrary geometry, are very useful to compute mean water level or energy flux variations
along submerged permeable breakwaters, giving accurate results based only on incident wave conditions, breakwater ge-
JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING / JANUARY/FEBRUARY 2001 / 7
ometry, and material without any further calibrating parameters. Therefore, those models may be a useful tool for
engineering applications.
This paper is organized as follows. After the introduction,
a short summary of the first-order solution is presented. In the
next section, expressions for the mean second-order quantities
are derived in terms of shape functions, including evanescent
modes. After a section on model calibration, the sensitivity of
the theoretical results to various breaking models and parameters are discussed. Next, the influence of incident wave conditions, breakwater geometry, and porous material characteristics on the second-order magnitudes is analyzed. Finally,
conclusions are given. Two appendices are also included.
The friction coefficient f depends on the seepage velocity and
on the following porous material characteristics: ε, porosity;
Kp, intrinsic permeability; Cf , turbulent friction coefficient; and
s, an inertia coefficient that takes into account the added mass,
generally taken to be one.
Once the system is solved, the potential in each region is
known and the instantaneous magnitudes (free surface, velocity and acceleration fields, and pressure) can be evaluated in
any domain point. Moreover, by using the analytical expressions given later in the paper, the mean or second-order flow
quantities can be determined.
THEORETICAL ANALYSIS
For a permeable submerged breakwater [Fig. 1(b)], Losada
et al. (1996b) presents a 2D solution based on an extension of
the mild-slope equation. This extension of the mild-slope
equation for wave propagation over a porous layer is derived
by multiplying the Laplace equation by their correspondent
vertical eigenfunctions in terms of the propagating mode only,
depending on the region where they apply and integrating over
depth. To solve the problem, the equation is discretized using
a finite-difference scheme defining a grid in the x-direction.
Using proper boundary conditions at the domain boundaries,
the complex amplitude of the water surface, ␸(x), can be calculated at each grid point. The velocity potentials inside and
outside the structure can be simply obtained by multiplying
␸(x) by the corresponding vertical eigenfunctions, Mo(z) and
Po(z). Again, once the potentials are known, the instantaneous
magnitudes can be evaluated easily.
In this paper, wave breaking is added to the 2D model following Rojanakamthorn et al. (1990). For this purpose a new
term is included in the mild slope equation such that
In the following, the theoretical basis for two models, a 3D
model to analyze flow conditions in the vicinity of rectangular
porous structures and a 2D model to be applied to trapezoidal
porous breakwaters, is introduced.
First-Order Solution
Rectangular Permeable Submerged Breakwater
For a rectangular permeable breakwater of width b and
height ␣h, submerged in a constant water depth h [Fig. 1(a)],
Losada et al. (1996a) developed a theoretical model for monochromatic small amplitude waves impinging obliquely on the
structure. With the assumption of irrotational motion and incompressible fluid inside and outside the porous structure, the
flow field is separated into four regions where Laplace’s equation holds. In each region, Laplace’s equation and the necessary boundary conditions determine a boundary value problem
which can be solved to obtain a velocity potential, ⌽(x, y, z,
t) in terms of unknown complex amplitudes. Each of the defined boundary value problems is solved by separation of variables resulting in Sturm-Liouville problems with associated
eigenvalues and orthogonal eigenfunctions. The eigenvalues
can be identified with the correspondent wavenumbers, which
inside and above the porous region are complex, the imaginary
part of the wavenumber representing the damping rate. Matching conditions at the interfaces guarantee the continuity of the
solution. Using the orthogonality of the eigenfunctions, a system of equations is finally obtained with the complex amplitudes as unknowns.
The effects of the porous material is considered in terms of
a friction coefficient, f, which can be calculated using Lorentz’s hypothesis of equivalent work (Sollitt and Cross 1972).
Trapezoidal Permeable Submerged Breakwater
ⵜh ⭈ (␤ⵜh ⭈ ␸) ⫹ (K 2o ␤ ⫺ i␴␤ fD)␸ = 0
where
␤=
冉 冊 冋冕
␴
ig
2
0
M 2o(z) dz ⫹ ε(s ⫺ if )
⫺h⫹␣h
冕
⫺h⫹␣h
(1)
册
P 2o(z) dz
⫺h
(2)
The term i␴␤ fD␸ has been added to account for wave breaking, where fD is an energy dissipation function; g = acceleration of gravity; i = imaginary unit; and ␴ = wave angular
frequency. Mo(z) and Po(z) are the vertical eigenfunctions outside and inside the permeable structure, respectively; and Ko
is the complex wavenumber for the first mode. The expressions of the vertical eigenfunctions, dispersion equations, and
complete potentials can be found in Appendix I.
The energy dissipation function due to wave breaking on a
submerged permeable breakwater takes into account the processes of wave decay and recovery and is expressed as (Rojanakamthorn et al. (1990)
fD =
KR
␣D tan ␨
␴cg
冑 冑
g
hef
␯ ⫺ ␯r
␯s ⫺ ␯r
(3)
where cg = group velocity; KR = real part of the most progressive wave number over the porous bed [(28)]; tan ␨ =
equivalent bottom slope at the breaking point, which is defined
as a mean slope in the distance 5hef ,b offshoreward from the
breaking point; and
␯=
兩␸兩
hef
(4)
␯s = 0.4(0.57 ⫹ 5.3 tan ␨)
FIG. 1. Definition Sketch: (a) Rectangular Breakwater; (b) Arbitrary Geometry
␯r = 0.4
8 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING / JANUARY/FEBRUARY 2001
冏
兩␸兩
hef 㛭 b
(5)
(6)
where the subscript b means the value at the breaking point.
hef is the effective water depth over the porous layer (Losada
et al. 1997). This term can be calculated using (17). The parameters ␯s and ␯r are expressed according to the results in the
paper of Watanabe and Dibajnia (1988). Therefore, although
the original value for ␣D is 2.5, a different ␣D has been found
in this work.
Instead of using the breaking criteria in Rojanakamthorn et
al. (1990), the following formulation from Iwata and Kiyono
(1985) for partial standing waves has been used to take into
account the reflection induced by the structure:
Hb
=
Lb
冋
0.218 ⫺ 0.076
冉
冊册
1 ⫺ 兩Ro兩
1 ⫹ 兩Ro兩
tanh kb hb
(7)
where Hb = wave height at the breaking point; kb = 2␲/L b; Lb
= wave length at the breaking point; hb = water depth above
the structure at the breaking point; and 兩Ro兩 = modulus of the
reflection coefficient.
Once the breaking condition has been found, fD is calculated
using the expression presented in Rojanakamthorn et al.
(1990). In order to arrive at a solution, an iterative procedure
has to be carried out.
Mean Quantities
Once the first-order solution is known, mass flux, energy
flux, and radiation stresses can be calculated. New expressions
taking into account the influence of wave reflection, transmission, and dissipation are obtained.
These magnitudes of the mean quantities are formulated in
terms of shape functions (see Appendix II), which present the
advantage of allowing a more compact expression.
Mass Transport
The total mass flux in the x-direction, Mx, is given by
冕
␳u dz = ␳u␩兩z=0
(8)
0
In terms of the shape functions, the mass flux can be written
as
Mx =
冏
1 2
a ␳ Re[H␩ H *]
u
2
(9)
z=0
where (*) stands for complex conjugate; ␳ = water density; a
= incident wave amplitude; and Re[ ] stands for real part of
the magnitude in brackets.
Radiation Stress
The presence of waves will result in an excess of momentum flux, which is defined as the radiation stress. The radiation
stress will be affected by the presence of reflected waves and
the dissipation induced by breaking waves above the structure
as well as by the flow through the porous material. The four
components of the radiation tensor can be expressed in terms
of the shape functions (Mei 1989) as
Sxx =
1 2
a
2
1
Syy = a2
2
冕
冕
0
␳(兩Hu兩2 ⫺ 兩Hw兩2)␰(z) dz ⫹
1
␳ga2兩H␩兩2 ⫹ S (4)
4
(10)
␳(兩H␯兩2 ⫺ 兩Hw兩2)␰(z) dz ⫹
1
␳ga2兩H␩兩2 ⫹ S (4)
4
(11)
⫺h
0
⫺h
Sxy =
1 2
a
2
冕
0
␳ Re[Hu H *]␰(z)
dz
␯
⫺h ⫹ ␣h < z < 0
⫺h < z < ⫺h ⫹ ␣h
1
sε
(13)
S (4) is a term that takes into account the modulation of the
wave field
S (4) =
⫹
冕 冋 冉冕
冉冕
0
1 2
a
2
⭸
⭸y
⫺h
⭸
⭸x
冊
0
␳ Re[Hu H *]␰(z⬘)
dz⬘
w
z
冊册
0
␳ Re[H␯ H *]␰(z⬘)
dz⬘
w
z
␰(z) dz
This term can be neglected for progressive waves, but it must
be included if there is wave energy reflection.
Energy Flux
The energy flux in the x-direction, Ᏺx, is given by
冕
0
Ᏺx =
( p ⫹ ␳gz)u dz
(14)
⫺h
Based on the linearized Bernoulli equation in a fluid region,
it can be obtained p ⫹ ␳gz = ⫺i␴␳⌽. For a porous medium,
the same equation (Sollitt and Cross 1972) yields p ⫹ ␳gz =
⫺␳␴⌽(si ⫹ f ).
Therefore, for a porous layer of height ␣h, the energy flux,
in terms of the shape functions, is defined as
Ᏺx =
1 2
a
2
冕
⫺h⫹␣h
␳ Re[Hu H *]ε
dz ⫹
p
⫺h
1 2
a
2
冕
0
␳ Re[Hu H *]
dz
p
⫺h⫹␣h
(15)
Expressions for other mean quantities including reflection and
dissipation, such as set-down, impulse, mean pressure or kinetic,
and potential mean energy can be found in Méndez (1997).
VALIDATION OF MODEL
␩
Mx =
再
␰(z) =
(12)
⫺h
In the fluid region, ␰(z) = 1, and in the porous region, ␰(z) is
Both models are validated using the experimental results
described in Rivero et al. (1998) and Tomé (1997). The set of
experimental tests was conducted at the large-scale wave flume
CIEM of LIM/UPC within the ‘‘Dynamics of Beaches’’ project. Free surface elevation and velocities were collected for
regular and irregular waves at 12 stations in a 100 m long and
3 m wide flume for a submerged breakwater on a 1:15 rigid
sloping bottom. The submerged breakwater, with 1:1.5 slopes
on both sides, has a crown width of 0.61 m and is constructed
of an impermeable core and an armor layer of quarrystones
with a mean weight of 25 Kg. The water depth at the toe of
the structure was 1.50 m.
The first order has been solved using (1) including wave
breaking. The numerical boundary condition at the end of the
flume has been modified to include the reflection induced by
the concrete sloping bottom. The result is given in terms of
the wave height envelope along the flume. To solve the problem, it is necessary to specify the mechanic characteristics of
the permeable material, ε, Kp, Cf, and s. These magnitudes
have been calculated using the empirical expressions in van
Gent (1995), which are based on the material D50, and the
resulting values are ε = 0.4, Kp = 2.5 ⭈ 10⫺5 m2, Cf = 0.32, and
s = 1.0. These characteristics are independent of the flow and
are therefore kept constant for all the cases considered in the
experimental work.
The mean water level variation is solved using the timeaveraged and depth-integrated momentum equation
⭸Sxx
⭸␩
= ⫺␳g(h ⫹ ␩)
⭸x
⭸x
(16)
where Sxx is expressed as in (10).
JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING / JANUARY/FEBRUARY 2001 / 9
FIG. 2. Wave Height H and Mean Water Level ␩ Evolution in Submerged Permeable Breakwater for Different Wave Conditions; Comparison between Experimental Data (Rivero et al. 1998) and Numerical Model Results
Fig. 2 shows the experimental results in Tomé (1997) versus
the numerical results obtained using the mild slope 2D model,
presented in this paper. In this figure, the wave height H and
the mean water level ␩ evolution are presented along the given
geometry for different incident wave conditions, showing the
experimental values and the numerical results calculated considering different approaches and different parameter values.
The influence of the breaking parameter ␣D is analyzed in Fig.
2(a). In Fig. 2(b), the influence of using h, hef , or h ⫺ ␣h in
(16) is shown. The theoretical results are calculated in Fig.
2(c) using the breaking model of Dally et al. (1985), denoted
by DDD’85, instead of the one presented in Rojanakamthorn
et al. (1990). Finally, the influence of the reflection at the
shoreline is analyzed in Fig. 2(d).
In general, there is good agreement between the experimental and numerical wave height transformation. The modulation of wave height induced by the reflection in front of
the breakwater is very well reproduced by the theory. However, the model tends to underpredict the experimental wave
height values above the crest.
The mean water level variations evaluated using the presented 2D model are also very satisfactory. The initial value
(x = ⫺6.0 m) for the theoretical calculations is imposed, since
the mean water level for deep water conditions is known from
the experiments. Results show a modulation of ␩ in front of
the breakwater induced by reflection. A set-down can be
clearly observed at the front slope or above the breakwater
crest just before the maximum wave height is reached before
breaking. After breaking a slightly modulated set-up is clearly
observed leewards the structure in the experimental results.
Influence of Breaking Parameter ␣D
In Fig. 2(a), three different lines have been plotted for three
values of ␣D, which controls the breaking process. It can be
observed that results are very sensitive to ␣D, especially after
breaking. In fact, ␣D could be used as a calibration parameter.
Furthermore, increasing ␣D results in a phase shift of the results in front of the structure. However, to be consistent, for
all the other cases considered in this paper, a constant value
10 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING / JANUARY/FEBRUARY 2001
of ␣D = 1.0 has been used so that no calibration parameter has
been necessary.
For ␣D = 0 (no breaking) there is still a set-up visible leewards the structure, in Fig. 2(a). This fact points out that the
set-up induced by a permeable submerged breakwater is due
to the radiation stress gradients induced by the dissipation associated with both breaking and friction.
hand, the mean water level with h ⫺ ␣h is higher than with
hef . In general, the use of neither h nor h ⫺ ␣h in (16) is valid
to represent the x-momentum equation for waves propagating
over a permeable layer, since both are asymptotic representations of the breakwater permeable characteristics (totally permeable or impermeable).
Discrepancies behind Structure
Breaking Model of Dally et al. (1985)
The analysis of the breaking has pointed out that the evaluation of the minimum water level is very much associated
with the breaking model as well as with the breaking limit.
The use of a breaking limit considering reflection (7), has improved the results.
A further analysis of wave breaking has been carried out
using the breaking model of Dally et al. (1985) and comparing
the theoretical results with those obtained via Rojanakamthorn
et al. (1990), as shown in Fig. 2(c). For the parameters selected, it can be seen that in front of the structure there is a
phase shift in the results. The breaking model by Dally et al.
(1985) with KD = 0.15 clearly underpredicts the experimental
results. However, with KD = 0.35, a very good estimate of the
mean water level variation behind the structure is obtained
even if the modulation is not present in the theoretical results.
Influence of Reflection at Shoreline
Although the model is capable of including reflection at the
shoreline, this effect has not been considered in an attempt to
avoid the use of any calibration parameter. However, in order
to show the importance of reflection, some theoretical calculations, including the modulus of reflection and the relative
phasing of incident and reflective waves at the shoreline, have
been plotted in Fig. 2(d). The best fit has been found with 兩R兩
= 0.07 and ␺ = 0⬚. The strong influence of the reflection in
the leeward region (differences of almost 1 cm in mean water
level) can be seen.
RESULTS
In this section the evolution of the mean quantities, mass
flux, energy flux, and radiation stress for a rectangular breakwater along the incident wave propagation direction are analyzed by applying the present methods. Next, some examples
that may occur in the engineering practice are considered and
the evolution of the wave height and mean water level for
different trapezoidal breakwater geometries and incident wave
characteristics is discussed. In all the results, the friction coefficient f for the porous medium has been obtained using Lorentz’s hypothesis of equivalent work [Sollitt and Cross
(1972); see (29) in Appendix I].
Rectangular Submerged Breakwater
Effective Water Depth hef
Although it is known that for a permeable layer, (16) is only
an approximation (Losada 1996), the equation has been integrated by means of a finite-difference technique obtaining the
mean water level variation on submerged breakwaters. In order
to take into account the effects of the permeability in the solution, an effective water depth hef has been used instead of h,
following Losada et al. (1997). Furthermore, (1) and (7) are
also solved using h = hef . The following example is given to
explain the purpose of hef . Consider a rectangular breakwater
with h = 10 m, ␣ = 0.4, a wave period T = 10 s, and varying
permeable material. For an extremely porous material ( f = 0,
ε = 1) the calculated water depth is hef = 10 m. For f → ⬁, ε
→ 0, the structure is almost impermeable and the wave feels
an effective water depth hef = 6 m. For an intermediate case,
say ( f = 1, ε = 0.5), the calculated value is hef = 7.1 m. Therefore, a wave propagating above this material is, in terms of
kh, equivalent to a wave propagating on an impermeable step
2.9 m high in a 10 m water depth. The procedure to calculate
hef is simple and can be found in Losada et al. (1997):
hef =
For all the cases considered, it can be seen that the 2D
model fails to reproduce the mean water level at the two
gauges immediately behind the structure and the wave height
above the crest. The reason for this disagreement could be due
to the neglect of several physical processes. The excess of
momentum flux due to turbulent fluctuations could be important due to the existence of eddies leewards the structure.
Moreover, the roller effect (mass flux and momentum flux) is
not considered in the breaking process. Additionally, flow separation, which is expected to occur seawards and leewards of
the structure (Losada et al. 1989; Tang and Chang 1998), reduces the effective water depth in their vicinity and may influence significantly the evolution of the wave height and of
the mean water level.
1
tanh⫺1
KR
冉 冊
␴2
gKR
(17)
where KR = real part of the most progressive wave number
over the porous bed, Ko, that can be calculated using (28).
In Fig. 2(b), the influence of using h (totally permeable
breakwater), h ⫺ ␣h (impermeable breakwater), or hef (permeable breakwater) in (16) is shown. It can be seen that h
underpredicts the set-up leewards the structure. On the other
Mass Flux
Using the model for the rectangular breakwater, the nondimensional mass flux evolution in the x-direction, Mx, for three
different geometries with b = 5 m, h = 1 m, and ␣ = 0.25, has
been evaluated and plotted in Fig. 3, versus nondimensional
distance x/b. E and c are defined as the mean energy and celerity of the incident wave, respectively. Incident wave conditions are Hi = 0.2 m, kh = 1.20 m, and ␪ = 30⬚. For each
geometry, three different materials have been considered,
whose characteristics can be found in Table 1. The solution
has been calculated using N = 20 evanescent modes.
Fig. 3(a) shows the mass flux evolution for a finite breakwater. In front of the structure, the mass flux is almost equal
for the three materials considered. This is a consequence of
the fact that the reflection induced by the structure is almost
identical for all three materials. On the top of the structure,
the mass flux is increased. The higher mass flux at x/b = 0
corresponds to material 1 and can be explained due to the fact
that for this material the effective water depth is the smallest
since its permeability is the lowest, thus resulting in a higher
increase in the velocities and wave height at the edge of the
breakwater crest. Material 3, with the highest porosity and
permeability, induces the smallest mass flux a the beginning
of the submerged structure. Along the breakwater the reduction
in mass flux can be clearly seen for all materials considered.
The dissipation rate is more important for material 2, which
owes the greatest imaginary part of the complex wavenumber.
Consequently, at the back face of the breakwater, the mass
JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING / JANUARY/FEBRUARY 2001 / 11
and thee different angles of incidence 0⬚, 30⬚, and 60⬚. Computations have been carried out with six evanescent modes,
and results are shown in Fig. 4.
In front of the structure and for a given breakwater height,
the energy flux is highest for normal incidence and decreases
for increasing incidence. The incidence is controlling the energy flux, since Ᏺx ⬃ ECg cos ␪(1 ⫺ 兩Ro兩2) and for the cases
considered Ro is small. For a given incidence, the structure
with ␣ = 0.3 has a higher energy flux, since it reflects less
than the one with ␣ = 0.7.
As expected, above the submerged breakwater the energy
flux decays more rapidly for ␣ = 0.7, since this structure induces a much higher dissipation than the one with ␣ = 0.3.
Therefore, it can be concluded that, under nonbreaking conditions and for a given breakwater geometry and porous material, increasing the freeboard results in higher energy flux
transmission. Moreover, if the freeboard is given, the energy
flux transmitted decreases with increasing oblique incidence.
Radiation Stress
Fig. 5 shows the evolution of the nondimensional radiation
stress components for a rectangular breakwater with ␣ = 0.6,
b = 0.5 m in h = 1 m and kh = 0.55. Incident wave height is
Hi = 0.2 m and five evanescent modes are used. Three different
angles of incidence have been used, namely, 15⬚, 30⬚, and 45⬚.
The components’ variation has been plotted along the structure
only.
FIG. 3. Nondimensional Mass Flux Evolution in Submerged
Breakwater for Different Leeward Boundary Conditions: Hi ⴝ 0.2
m; h ⴝ 1.0 m; kh ⴝ 1.20; ␪ ⴝ 30ⴗ; b ⴝ 5 m; ␣ ⴝ 0.25; N ⴝ 20
TABLE 1.
Permeable Material Characteristics
Material
(1)
D50
(2)
ε
(3)
Kp
(4)
Cf
(5)
1
2
3
5 cm
10 cm
15 cm
0.39
0.45
0.50
6.0 ⭈ 10⫺7 m2
3.0 ⭈ 10⫺6 m2
1.1 ⭈ 10⫺5 m2
0.15
0.11
0.10
flux for material 2 is smaller than for material 3. As region 3
has been considered to be semi-infinite in the x-direction, the
mass flux level remains constant along this region.
Fig. 3(b) presents the case of the geometry shown in 3(a)
but with a vertical impermeable backwall at x/b = 1. Here, the
backwall is controlling the mass flux. Material 1 with KI h =
⫺0.0234 is inducing the smallest dissipation and therefore the
reflection at the backwall is maximum, resulting in the lowest
mass transport in front of the submerged breakwater. For materials 2 and 3, the dissipation rate is higher resulting in a
higher gradient. For all three materials, the modulation due to
the reflection at the backwall is evident. As imposed, the mass
flux at the backwall is zero.
In Fig. 3(c), for a semi-infinite submerged breakwater, a
similar pattern can be seen for three materials, the only difference being apparent in the fact that the mass flux tends to
go to zero as x/b increases. This case can be considered analogous to that of a wave propagation on a porous layer of finite
width.
FIG. 4. Nondimensional Mass Flux Evolution in Finite Submerged Breakwater for Different Wave Incidence (␪) and Relative Breakwater Height (␣): Material 1; Hi ⴝ 0.4 m; h ⴝ 1.0 m; kh
ⴝ 0.72; b ⴝ 10 m; and N ⴝ 6
Energy Flux
Next the energy flux evolution along a submerged finite
rectangular step is analyzed. For material 1 and a breakwater
width b = 10 m and water depth h = 1 m, two different structure heights have been considered, ␣ = 0.3 and ␣ = 0.7. Incident wave conditions are given by Hi = 0.4 m, kh = 0.72,
FIG. 5. Nondimensional Radiation Stress Components Evolution in Finite Submerged Breakwater for Different Wave Incidence (␪): Material 2; Hi ⴝ 0.2 m; h ⴝ 1.0 m; kh ⴝ 0.55; b ⴝ 5 m;
␣ ⴝ 0.6; and N ⴝ 4
12 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING / JANUARY/FEBRUARY 2001
As expected, Sxx is maximum for normal incidence decreasing with wave incidence, since the x component of the momentum flux decreases with increasing incidence. On the contrary, Sxy and Syy increase with increasing ␪. Furthermore, for
any ␪, Sxx decreases along the breakwater, since the momentum
is being lost to dissipation induced by the permeable material.
Again, a slight modulation can be observed, induced by the
reflection of the structure’s backward face. A similar behavior
can be seen for Sxy, although the differences between the angles
of incidence considered are not as strong as for Sxx. Furthermore, the decay of Sxy is milder than for Sxx. The pattern of
Syy, even if for x/b < 0.5, is similar to that of the other components; for x/b > 0.5, there seems not to be much influence
of the wave incidence, the breakwater width being the variable
controlling its behavior.
given F, transmission is lowest for the widest breakwater due
to the dissipation generated by the permeable material.
In Fig. 6(b), the resultant mean water level is shown. In the
seaward region, ␩ is modulated. Stronger modulation is associated with higher reflection. At the breakwater toe, the setdown starts, reaching a maximum at the beginning of the crest.
Set-up starts to increase along the crest with a much higher
value for F = 1.5 m, where the set-up results from the friction
and breaking contribution. At the transition between the crest
and the backslope there is a change in the set-up slope associated with the change in water depth. For the cases considered, the set-up without breaking is approximately 2% of the
incident wave height Hi, while with breaking the set-up
reaches values up to 20% of Hi.
Mean Water Level Difference
Trapezoidal Submerged Breakwater
Wave Height and Mean Water Level
Wave height and mean water level evolution along a trapezoidal submerged permeable breakwater are determined for
different breakwater crest widths (b = 8.8 m, 27.2 m) and
freeboards (F = 3.5 m, 1.5 m). The structure has 1:2 slopes
on both sides. Computations have been carried out with null
reflection in the leeward region. The material characteristics
are given to be ε = 0.45, Kp = 2 ⭈ 10⫺4 m2, Cf = 0.15, and s =
1.0. Incident wave conditions are Hi = 3.0 m and T = 10 s.
Breakwater dimensions and results are plotted in Fig. 6.
Modulation of wave height in front of the breakwater is
clearly visible [Fig. 6(a)], with maxima positions slightly
shifted with respect to the breakwater crest starting position.
As expected, the lowest modulation is associated to F = 3.5
m and b = 27.2 m, since reflection is lowest for this case, 兩Ro兩
= 0.03.
For F = 3.5 m, dissipation is induced by friction only and
attenuation is low for both widths, resulting in higher transmission. For F = 1.5 m, wave breaking is controlling the propagation, resulting in a lower transmission. As expected, for a
FIG. 6. (a) Wave Height H Evolution and (b) Mean Water Level
␩ Evolution in Submerged Breakwater for Different Crown
Widths and Freeboards: Hi ⴝ 3.0 m; T ⴝ 10 s; Kp ⴝ 2 ⭈ 10⫺4 m2; Cf
ⴝ 0.15; s ⴝ 1, ␧ ⴝ 0.45
In Fig. 7, the nondimensional mean water level difference
(⌬␩/Hi) is shown for a trapezoidal submerged breakwater versus relative crown width b/L p for different relative freeboard
(F/Hi) values for two different incident wave heights Hi = 1.5
m and Hi = 3.0 m, and T = 10 s. L p is the wavelength at the
breakwater toe. ⌬␩ is the difference between the set-down in
deep water and the set-up in the leeward region at a distance
of L p /2 from the end of the breakwater crest. The breakwater
material characteristics are ε = 0.45, Kp = 2 ⭈ 10⫺4 m2, Cf =
0.15, and s = 1.0.
Decreasing F/Hi, i.e., reducing the freeboard, results in increasing (⌬␩/Hi), since breaking becomes more important.
Furthermore, for Hi = 3.0 m and in general, increasing the
incident wave height results in higher values of ⌬␩/Hi for a
given (F/Hi). In the figure, the transition line between breaking
and nonbreaking conditions is also plotted. Under nonbreaking
conditions and for a given relative width, the growth of
⌬␩/Hi with F/Hi is not very fast. However, above the breaking
limit, the increase in the mean water level difference increases
very rapidly, pointing out the nonlinear character of this magnitude.
For a given (F/Hi) increasing the relative width results in
higher mean water level difference, tending toward an asymptotic value. Taking into account that for a given (F/Hi), h at
the toe of the structure is fixed and T = 10 s, the previous
result indicates that increasing the breakwater width to an upper limit results in a higher set-up, since dissipation increases
with b. After that limit no further dissipation is obtained and
therefore no higher set-up. All the curves start close to 0, since
FIG. 7. Nondimensional ⌬␩/Hi Evolution in Trapezoidal Submerged Breakwater for Different Relative Crown Widths (b/L p)
and Relative Freeboards (F/Hi): T ⴝ 10 s; Kp ⴝ 2 ⭈ 10⫺4 m2; Cf ⴝ
0.15; s ⴝ 1; ␧ ⴝ 0.45
JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING / JANUARY/FEBRUARY 2001 / 13
b/L p → 0 implies no breakwater and therefore no dissipation.
For F/Hi = 0.33 and close to b/L p = 0.25, there is a resonant
condition, giving a maximum mean water level difference.
Under nonbreaking conditions and for Hi = 1.5 m, ⌬␩ varies
between 1 and 10% of the incident wave height and reaches
up to 25% of Hi under breaking conditions. For Hi = 3 m, the
percentage can reach 35%.
␾1(x, z) = aIo(z)e
⌽j (x, y, z, t) = Re[␾j (x, z)]e
j = 1, 2, 3, 4
(18)
where ␭ = ko sin ␪; ␪ = wave incidence angle; and ko = progressive wavenumber in region 1.
In regions 1 and 3, the velocity potentials are given by
In(z)Tn e ⫺iqn(x⫺b)
(20)
ig cosh kn(h ⫹ z)
␴
cosh kn h
(21)
Ro and To = reflection and transmission coefficients; Rn and Tn
= nondimensional coefficients of the evanescent mode n; and
kn = eigenvalue (wave number) that satisfies the dispersion
relationship
␴ 2 = gkn tanh kn h
(22)
Above and inside the breakwater, regions 2 and 4, the velocity potentials are given by
冘
冘
⬁
␾2(x, z) = a
Mn(z)[An e ⫺iQn x ⫹ Bn e iQn(x⫺b)]
(23)
Pn(z)[An e ⫺iQn x ⫹ Bn e iQn(x⫺b)]
(24)
n =0
⬁
␾4(x, z) = a
n =0
aAn and aBn = complex amplitudes of the waves propagating
above the breakwater; and Qn = 兹K 2n ⫺ ␭2. For normal incidence, Qn = Kn.
The depth dependency is now provided by the following
functions:
Mn(z) =
Pn(z) =
ig cosh Kn(z ⫹ h) ⫺ Fn sinh Kn(z ⫹ h)
␴
cosh Kn h ⫺ Fn sinh Kn h
ig [1 ⫺ Fn tanh Kn ␣h]
cosh Kn(z ⫹ h)
␴
(s ⫺ if )
cosh Kn h ⫺ Fn sinh Kn h
Fn =
冉
1⫺
冊
ε
(s ⫺ if )
tanh Kn ␣h
ε
1⫺
tanh2Kn ␣h
(s ⫺ if )
(25)
(26)
(27)
The complex wavenumber Kn can be determined using the
complex dispersion equation derived by Losada et al. (1996a)
for wave propagation over a porous medium
␴ 2 ⫺ gKn tanh Kn h = Fn[␴ 2 tanh Kn h ⫺ gKn]
(28)
The linearized friction coefficient, f, can be obtained using the
Lorentz equivalent work hypothesis (Sollitt and Cross 1972)
FIRST-ORDER SOLUTION
⫺i(␭y⫺␴t)
(19)
where In(z) = depth function associated to the n-mode
In(z) =
The boundary value problem of an incident wave train with
amplitude a and wave angular frequency ␴ interacting in constant depth h with a submerged dissipative medium of varying
nature of height ␣h and width b has been analyzed using eigenfunction expansions by several authors (e.g., Losada et al.
1996a; Méndez 1997). Considering a permeable breakwater,
the dissipative medium is defined in terms of its porosity ε,
an added mass factor s, and a linearized friction coefficient f.
Assuming an inviscid fluid and irrotational flow, a Laplace
equation in each of the regions may be defined. The boundary
value problem in each region j is solved if the velocity potential ⌽j (x, y, z, t) is known. For an obliquely propagating wave
onto the structure, the potential can be separated into
In(z)Rn e iqn x
n =0
n =0
冕 冕冉
T
APPENDIX I.
冘
⫹a
⬁
␾3(x, z) = a
CONCLUSIONS
In this paper two models, a 3D model for permeable rectangular breakwaters and a 2D model for permeable breakwaters of arbitrary geometry, are presented to evaluate secondorder magnitudes. The 2D model includes wave breaking
considering different breaking models, and once the incident
wave conditions, breakwater geometry, the porosity, and the
D50 of the porous material are known, the model is capable of
calculating the second-order magnitudes evolution without any
further calibrations. Both models are able to reproduce wave
height transformation as well as mean water level variations
along the breakwater with reasonable accuracy.
In front of the structure, second-order magnitudes are modulated due to the reflection induced by the structure. Under
nonbreaking conditions, second-order magnitudes are attenuated. The dissipation rate is dependent upon wave conditions,
breakwater geometry, and porous material characteristics. Under nonbreaking conditions and for a given breakwater geometry and porous material, increasing the freeboard yields
higher energy flux transmission. Furthermore, for a given freeboard, the energy flux transmitted decreases with increasing
oblique incidence. These conclusions are important especially
for the design of beach toe protection structures, providing the
models a tool to evaluate the energy flux transmitted.
Results point out that the set-up induced by a permeable
submerged breakwater is due to the radiation stress gradients
induced by the dissipation associated with both breaking and
friction. The set-up due to breaking is much more important
than the one due to dissipation induced by the porous material.
For the cases considered, the set-up without breaking is approximately 2–5% of the incident wave height Hi, while with
breaking the set-up reaches values up to 20% of Hi, or even
higher. These values are highly dependent on wave height.
The mean water level variations show a modulation in front
of the structure, a maximum set-down at the beginning of the
crest, and a progressive increase along the crest reaching its
maximum value approximately at the backslope. Roller effects, flow separation, and turbulent effects should be further
explored.
冘
⬁
⫺iqo x
f=
1
␴
0
V
冊
ε 2␯ 2
ε 3Cf
兩q兩 ⫹
兩q兩3
Kp
兹Kp
冕冕
dt dV
T
0
(29)
ε兩q兩 dt dV
2
V
where ␯ = kinematic fluid viscosity; Kp = intrinsic permeability; Cf = turbulent friction coefficient and q = seepage velocity
vector, q = (u, v, w).
APPENDIX II.
SHAPE FUNCTIONS
冘
⬁
e ⫺iqo x ⫹
冘
冘
Rn e iqn x
xⱕ0
n =0
⬁
H␩(x) =
[An e ⫺iQn x ⫹ Bn e iQn(x⫺b)]
n =0
⬁
Tn e ⫺iqn(x⫺b)
n =0
14 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING / JANUARY/FEBRUARY 2001
xⱖb
0ⱕxⱕb
(30)
for a rectangular submerged breakwater; and H␩ (x) = (1/a)␸(x)
for a submerged breakwater of arbitrary geometry:
(Hu(x, z), Hv(x, z), Hw(x, z))
=
冉
冊
1 ⭸␾(x, z)
i␭
1 ⭸␾(x, z)
, ⫺ ␾(x, z),
a
⭸x
a
a
⭸z
Hp(x, z) =
再
i␴
⫺ ␾(x, z) fluid region
a
(si ⫹ f )␴
⫺
␾(x, z) porous region
a
(31)
(32)
where ␾(x, z) has been defined in Appendix I for rectangular
breakwaters. For submerged breakwaters of arbitrary geometry, ␾(x, z) is
␾(x, z) =
再
Mo(z)␸(x)
⫺h ⫹ ␣h ⱕ z ⱕ 0
Po(z)␸(x)
⫺h ⱕ z ⱕ ⫺h ⫹ ␣h
(33)
ACKNOWLEDGMENTS
The writers acknowledge the funding provided by Spanish Comisión
Interministerial de Ciencia y Technologı́a (CICYT) under contracts
MAR96-1833 and MAR99-0653.
APPENDIX III.
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