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Wave reflection from perforated-wall caisson breakwaters
COASTAL ENGINEERING ELSEVIER
Coastal Engineering 26 ( 1995) 177-193
Wave reflection from perforated-wall caisson breakwaters Kyung Duck Suh, Woo Sun Park Korea Ocean Research & Development
Institute, 1270 Sa-dong, Ansan, Kyonggi-do 425-l 70, South Korea
Received 8 August 1994; accepted 18 September 1995
Abstract Analytical models for predicting wave reflection from a perforated-wall caisson breakwater have been developed. Most of the existing models deal with the case in which the waves are normally incident to the caisson lying on a flat sea bottom. In the present paper, using the Galerkin-eigenfunction method, an analytical model is developed that can predict the reflection coefficient of a perforated-
wall caisson mounted on a rubble mound foundation when waves are obliquely incident to the breakwater at an arbitrary angle. The developed model is compared with other theoretical results and hydraulic experimental data.
1. Introduction Composite breakwaters have been widely used at various locations, which consists of the lower rubble foundation and the upper upright structure made of concrete blocks or caissons. One of the drawbacks of a conventional composite breakwater compared to a rubble mound breakwater is that the increased agitation on the sea side of the breakwater due to severe wave reflection from the upright structure can originate difficulties in navigation or ship operation (see McBride et al., 1994, for example). As a countermeasure to overcome this drawback, a mound of concrete blocks of energy dissipating type is placed in front of the breakwater. In this case, however, wave transmission due to overtopping may increase so that the tranquillity inside the harbour may deteriorate (see Lee et al., 1994, for example). As another means for reducing wave reflection, a perforated-wall caisson is often adopted, which reduces the wave reflection by energy dissipation due to turbulence generated when the waves intrude into the wave chamber through the perforated wall. In order to examine the reflection characteristics of a perforated-wall caisson breakwater, hydraulic model tests have been used (Jarlan, 1961; Marks and Jarlan, 1968; Terret et al., 1968; Tanimoto et al., 1976, among others]. Efforts towards developing analytical models 0378.3839/95/$09.50
0 1995 Elsevier Science B.V. All rights reserved
.SSDIO378-3839(95)00027-5
178
K. D. Sub, W.S. Park/Coastal
Engineering 26 (1995) 177-193
for predicting the reflection coefficient have also been made. Based upon linearized shallow water wave theory, Kondo ( 1979) developed an analytical model for calculating reflection coefficient of a perforated-wall caisson breakwater having one or two perforated walls. Kondo concluded that the breakwater having two perforated walls could give much lower reflection coefficient than that having one perforated wall. Recently, using the method of matched asymptotic expansions, Kakuno et al. ( 1992) developed an analytical model for predicting wave reflection from a single-chamber perforated-wall caisson. A similar model and comparisons with large-scale laboratory data were given by Bennett et al. ( 1992). On the other hand, Fugazza and Natale ( 1992) proposed a closed-form solution for wave reflection from a multi-chamber perforated-wall caisson. Fugazza and Natale showed that the reflection is minimized when the wave chamber width is about one quarter of the wave length. In contrast to the conclusion made by Kondo ( 1979), Fugazza and Natale showed that the perforated-wall breakwater with a single wave chamber could give the largest reduction of wave reflection in the range of practical applications. The aforementioned analytical approaches deal with the case in which the waves are normally incident to the perforated-wall caisson lying on a flat sea bottom. In the present study, we develop an analytical model that can predict the wave reflection from a singlechamber perforated-wall caisson mounted on a rubble mound foundation when waves are obliquely incident to the breakwater at an arbitrary angle. The method of solution adopted in this paper consists of solving the wave potential on the rubble foundation with the matching conditions at the seaward toe of the rubble foundation and at the location of the perforated wall. The methodology follows the Galerkin-eigenfunction method used by Masse1 ( 1993) in developing the extended refractiondiffraction equation. This equation includes higher order terms of the bottom slope and the term proportional to the bottom curvature which were neglected in the mild-slope equation so that it can be applied to wave propagation over a bed consisting of substantial variations in water depth. In the design of a composite breakwater, the dimension and shape of the rubble mound are usually determined from various safety considerations such as wave pressure, bottom foundation, and so on, for an extreme wave condition (e.g., design wave condition of 50 or 100 years’ return period). For example, Tanimoto et al. ( 1987) have investigated the effect of rubble mound configuration upon the wave force acting on the caisson. From the viewpoint of wave reflection, however, the ordinary waves are of more interest than the severe storm waves because during the severe wave condition most ships seek refuge into harbors. Thus, in the present study, we deal with the reflection of ordinary nonbreaking waves. The present model is compared with the analytic solution of Fugazza and Natale ( 1992) for a perforated-wall caisson lying on a flat sea bottom. For the cases in which the caisson is mounted on a rubble mound foundation, the model is compared with the experimental data reported by Tanimoto et al. ( 1976). To examine the performance of the model for oblique incidence of waves, the model is tested against the experimental data obtained by Ijima et al. ( 1978) in a wave basin. In the following section, the analytical model for predicting the reflection coefficient of a perforated-wall caisson breakwater is derived. The next section includes comparison of the developed model with other theoretical and experimental results. The major conclusions then follow.
K.D. Sub, W.S. Park/Coastal
Engineering 26 (1995) 177-193
179
2. Theoretical analysis Let us consider the perforated-wall caisson breakwater sketched in Fig. 1, in which 13~is the incident wave angle, B is the wave chamber width of the perforated-wall caisson. The x-axis and y-axis are taken to be normal and parallel, respectively, to the breakwater crest line, and the water depth is assumed to be constant in they-direction. The vertical coordinate z is measured vertically upwards from the still water line. In Region 2 ( - b 5 x 5 0) the water depth h(x) is a varying function of x. For x I - b (Region 1) and 0 2 x I B (Region 3), the water depth is constant and equal to hi and h,, respectively. The rubble mound foundation is assumed to be impervious. Assuming inviscid irrotational flow, the velocity potential @(x,y,z,t) for the monochromatic wave propagating over the water depth h(x) with the angular frequency w and wave height H can be expressed as @(x,y,z,t)
- igH
= Re
20
Kwz)
exp( - id
>
in which i = \/- 1; g = gravitational acceleration, and the symbol Re represents the real part of a complex value. The wave number k must satisfy the dispersion relationship: w2=gktanh(kh)
Linearizing the free-surface boundary for the potential +(x,y,z) is obtained:
a4J ,,-A4=0 a+
a+dh
%f---=O
conditions,
the following
boundary
value problem
at z=O
(4)
at z= -h(x)
Incident
Reflected
Incident
‘mve
wcwe \
wave
Fig. 1. Schematic
:*I
I z=-b
diagram and coordinate
I
I
2=0
z=B
system for calculation
-z
-
of wave reflection.
180
K.D. Suh, W.S. Park/ Coastal Engineering 26 (1995) 177-193
in which V represents the horizontal gradient operator, and h = w2/g is a wave number in deep water. The Gale&in-eigenfunction method is used to formulate the problem. The Galerkin approach assumes that the function $(x,y,z) can be expanded in terms of N+ 1 depthdependent functions 2, (x,2) :
(6) The functions Z,(x,z)
Z,*(w) =
are taken as
cos[(Y,(z+h)l
(7)
cos( a,h)
so as to form a complete orthogonal set of eigenfunctions in the domain wave numbers (Y, are the solution of the following dispersion relation: h+a,tan(cr&)
=0
[ - h(x) ,O] . The
(8)
which has an infinite discrete set of real roots + (Y,,and a pair of imaginary roots cu, = f ik. Therefore, the function Z,(x,z) represents the free propagating wave mode, while the functions Z,(x,z) (n r 1) correspond to the non-propagating evanescent wave modes. The functions Z,(x,z) satisfy the free surface boundary condition (4) and do not satisfy the bottom boundary condition (5) individually. However, the global set of orthogonal functions should satisfy this condition. This is known as a tau method (Canuto et al., 1988). In the tau method, a sufficient number of the functions cp,(x,y) in the approximated solution (6) is chosen to ensure exact satisfaction of the bottom boundary condition. Turning back to the breakwater problem sketched in Fig. 1, the solution of the boundary value problem given by (3)-( 5) may be constructed from the particular solutions in each region of the fluid domain: +,(x,y,z)
= [exp[ik,(x+b)cos&,]
coshk,(z+h,)
cash W
-exp[
-ik,(x+b)cos&]}exp(i~y)
+ CR,exp[P,,,(x+b)lexp(iXy)Z,,(h,,z) a, n
422(w,z)= C~n,,(x)exp(iXv)Z2,,(h2,z>
(9)
(10)
4.n
Mwz)
= CT,exp(-P3.~)exp(i~)Z,.,(h,,z)
(11)
4.”
in which &=
\/o&+x’,
If II = 0 (propagating A = kj tanh( kjhj),
x=kj mode),
sinf?j=k,sinO,
=constant
(j= 1,2,3)
(12)
(8) and ( 12) yield
/I,,0 = f ikj cosOj
(13)
For /3i,o we take the negative sign for the reflected wave in Region 1, while we need both positive and negative for the waves inside the wave chamber (i.e., Region 3).
K.D. Sub, W.S. Park/ Coastal Engineering 26 (1995) 177-193
181
The potential ~j(X,Y,Z) must satisfy the matching conditions which provide continuity of pressure and horizontal velocity, normal to the vertical planes separating the fluid regions and no-flux condition at the wall on the downwave side of the wave chamber, i.e. +I=&,
z=fj$
(X’
-b,
-h,1z
(t+$ 1!f$ !!b=s
g)3=42+
ax
a43 ,,=o
(x=0,
ax
(14)
-h,_
(15)
-h,_
(x=B,
(16)
In first equation of ( 15)) the second term on the right side represents the energy loss at the perforated wall (Kondo, 1979; Fugazza and Natale, 1992). In this equation, 4 is the length of the jet flowing through the perforated wall, which is usually taken as the thickness of the wall, and y is the linearized dissipation coefficient given by Fugazza and Natale (1992) as W ,/W2(R
5 + cosh2k,h,
in which H, = incident wave height at the perforated C = 1 - PW, P = t?k,, r = porosity of the wall, and
(
(y=p
1
i-cost&c,
2
(17)
+ 1)2+ ~‘2 2k3h3 + sinh2k,h,
1
-1
wall, W= tan(k3B),
R = ykJo,
(18)
is the energy loss coefficient at the perforated wall, which is a modification of the head loss coefficient for the plate orifice formula (see Mei, 1983, p. 257). In the preceding equation, r cos0, denotes the effective ratio of the opening of the porous wall taking into account the oblique incidence of the waves to the wall. For normal incidence, this reduces to r as in Fugazza and Natale ( 1992). C, is the empirical discharge coefficient at the perforated wall. Hattori ( 1972) concluded that the discharge coefficient ranged from 0.4 to 0.75 and Fugazza and Natale ( 1992) showed that the use of C, = 0.55 provided good agreement with experimental data for perforated breakwaters. Thus C,=O.55 was used in the present study. Rearranging of ( 17) gives a quartic polynomial of 7, which can be solved by the eigenvalue method (see Press et al., 1992, p. 368). As shown in (6), the Gale&in solution consists of a free propagating wave mode and non-propagating evanescent modes. The evanescent wave modes would be of importance in the region near the breakwater. However, since we are interested in the reflected wave far from the breakwater, we focus at the solution for the propagating mode. Masse1 ( 1993) showed that in Region 2 the function &(x) satisfies the following ordinary differential equation:
d240 -+D(x) dx?
in which
2+E(x)&=O
(19)
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K. D. Sub, W. S. Park/Coastal
G(M) D(X) =hz
Engineering 26 (1995) 177-l 93
dh
(20) (21)
In these equations,
(22)
p=i(l+&) G(kh) =
kh
l-3?+
T+kh(l-?)
1
27 T+kh(l-?)
(23)
in which T= tanh( kh) . R$) and R$’ are given by Rg)
1
=
cosh’( kh)
( WJ, + WJ, + w,z, + W,Z, + W& + W,)
(24)
(UJ,
(25)
and R$.’ =
cosh:(kh)
+
u212+
U34)
The expressions for W,,, U,, and Z,,are given in Appendix A. In order to solve ( 19) in Region 2, we need the boundary conditions at x = - b and x = 0. For the sake of readability of the text, the derivation of the boundary conditions is separated into Appendix B. After some algebraic manipulation, the boundary conditions in (42) of Appendix B are reduced to the following two equations:
d4d -b) dx eo(o)
=
=z[2-&( Iexp(
-b)]k, -Pu@)
P3,0~exp( -&$)
cos 0, +exp(L%d) -expU%,$)
1
@do)
_[_9 w
dx
(27)
The differential equation ( 19) with the preceding two boundary conditions can be solved using the finite-difference method. Using the forward-differencing for d@,,( - b) /dx, backward-differencing for d&,( 0) ldx, and central-differencing for the derivatives in ( 19)) the boundary value problem ( 19)) (26) and (27) was approximated by a system of linear equations, AY = B, where A is a tridiagonal band type matrix, Y is a column vector and B is also a column vector. The subroutines given in the book of Press et al. ( 1992) were used to solve this matrix equation. In particular we are interested in &( -b), from which the reflection coefficient K, is calculated using (33) of Appendix B.
3. Comparison
with other theoretical and experimental
results
For a perforated-wall caisson lying on a flat sea bottom, Fugazza and Natale ( 1992) showed that the reflection of waves normally incident to the caisson is at its minimum when
K.D. Sub, W.S. Park/Coastal
Engineering 26 (1995) 177-193
183
B/L Fig. 2. Reflection coefficient K, versus normalized chamber width B/L for a perforated-wall caisson lying on a flat sea bottom; Comparison between present theory and the solution of Fugazza and Natale ( 1992).
Bf L = 0.25 in which L is the wave length. The reason is that when B/L = 0.25 the reflection of the porous wall and the reflection of the impermeable wall are in opposite phases. Fig. 2 shows the variation of reflection coefficients with respect to B/L calculated by the formula of Fugazza and Natale ( 1992) and by the present theory. Water depth h = 3 m, wave height H = 1 m, wave chamber width B = 4 m, and porosity of the wall r = 0.3 and 0.6 were used. The two different theories give almost identical results as shown in Fig. 2, indicating minimum reflection occurred at B/L = 0.25. This result is not surprising because the same expression for the perforated-wall dissipation coefficient (i.e., Eq. 17) was used in both theories. In order to examine the effect of rubble mound foundation upon wave reflection, the reflection coefficients were calculated by changing the height and slope of the rubble mound. Total water depth in front of the breakwater hi = 10 m, wave period T= 8 s, incident wave height H = 2 m, thickness of the perforated wall 1= 0.6 m, porosity of the wall r= 0.3 and wave chamber width B = 10 m were used. The waves were assumed to be normally incident to the breakwater. The perforated wall of the caisson was assumed to be located at the upper end of the mound slope so that there was no berm in front of the caisson. The mound height ranged from 0.1 to 0.5 times the total water depth and the mound slope ranged from 1 : 4 to 1 : 1 for each mound height. For the solution of Fugazza and Natale ( 1992) the wave height at the perforated wall, H,, was calculated by linear shoaling, i.e., H, = H( C,,/C,,) ” 2. After the reflection coefficient at the top of the rubble mound was calculated using this wave height, the reflected wave height on the flat sea bottom was calculated by applying the shoaling formula in the inverse manner. For the present solution, the incident wave height at the location of the perforated wall was calculated by assuming the absence of the caisson but including the partial reflection from the mound slope as done in Masse1 ( 1993 ) and it was used to calculate the dissipation coefficient, ‘y, in ( 17). Then ( 19) was solved on the mound slope to calculate the reflection coefficient. Fig. 3 shows the ratio of the reflection coefficient of the present solution to the solution of Fugazza and Natale ( 1992) for the variation of mound height and slope. Except for very high mound with very mild slope, the reflection coefficient calculated by the present theory is slightly higher than that by Fugazza and Natale ( 1992). The difference becomes significant as the mound height increases and the mound slope becomes steeper. Note that the wave reflection from the mound slope is included in the present model but not in the solution of Fugazza and Natale incorporated with the linear shoaling analysis. A portion of incident
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K.D. Sub, W.S. Park / Coastal Engineering 26 (1995) 177-I 93
Fig. 3. Contours of the ratio of the reflection coefficient of the present model versus the solution of Fugazza and Natale ( 1992) as a function of mound height and slope.
wave energy will be reflected seaward by the rubble mound. This causes the reduction of incident wave height in front of the caisson. Therefore the less energy dissipation due to the perforated wall increases the height of the reflected wave. These two effects make the present theory predict higher wave reflection than that of Fugazza and Natale. Fig. 3 indicates that in the practical range of mound height and slope values the difference between the two theories is negligible. Thus the formula developed by Fugazza and Natale ( 1992)) along with the linear shoaling analysis over the mound slope, could be used for most practical situations if the waves are normally incident to the breakwater. In the preceding analysis, we have investigated the effect of rubble mound analytically. It may be useful to compare the model with experimental data for verifying the predictability of the model. Tanimoto et al. ( 1976) reported hydraulic model experiments for wave reflection from a perforated-wall caisson mounted on a rubble foundation. They used irregular waves of the Bretschneider-Mitsuyasu spectrum (see Goda, 1985, p. 25). Even though the present theory is developed for regular waves, experimental data using regular waves are rare. Thus the present theory is compared with the Tanimoto et al.‘s irregular wave experimental data. For the present analysis the wave period and height were taken as the significant wave period T, ,3 and the root-mean-square wave height H,,,( = H, ,3/ respectively. The experiment of Tanimoto et al. ( 1976) includes a total of 102 cases of various combinations of different wave conditions and structural dimensions of the break water. For all the cases the following parameters were constant: h, =42 cm, mound slope= 1 : 2, and r=0.236. The perforated wall included a number of circular holes of diameter of 2.5 cm. The ranges of other wave and structural parameters were as follows: T, ,3 = 1.O - 2.22 s, HI,, - 3.97 - 20.6 cm, h3 = 27 and 32 cm, e = 3 and 6 cm, crest height of breakwater h, = 10 and 16 cm, mound berm width et, = 20 and 2.5 cm, and wave chamber width including perforated-wall thickness B = 20 - 53 cm. The comparison of the measured and calculated reflection coefficients is shown in Fig. 4. For several points near the calculated reflection coefficient of about 0.1 significant underprediction is observed. Careful investigation was made for the experimental conditions of these data, but any particular reason could not be found. Statistically these data seem to be outliers which lie far beyond the scatter of the remaining data points (cf. Neter et al., 1985, pp. 114-l 16).
fi) ,
K.D. Sub, W.S. Park/Coastal
Fig. 4. Comparison al. ( 1976).
of the reflection coefficients
Engineering 26 (1995) 177-193
between present theory and the experimental
185
data of Tanimoto et
Fig. 4 shows that the model is not in very good agreement with the experimental data even without the outliers though the overall agreement is acceptable. Probably the behavior of an irregular wave in the breakwater could not be fully synthesized in parameters T, ,3 and H,,, as the energy dissipation phenomenon at the perforated wall is intrinsically nonlinear. In order to examine the effect of nonlinearity, the error ( = calculation/measurement) is plotted in Fig. 5 in terms of H,f gT2 in which H,, is the deepwater wave height corresponding to Hi/, and the wave period T is equal to T,,,. The data of the outliers were not included. The model tends to overestimate the reflection coefficient as the wave steepness increases probably because for steeper waves additional energy dissipation due to wave breaking may occur in the experiment. The error shown in Fig. 5 ranges approximately between 0.6 and 1.5. As for the error of a model, Goda ( 1994) proposed a method of judging the order-ofmagnitude accuracy and a concept of research efficiency with regards to the final accuracy. According to his judgement, the results in Fig. 5 are in the status between “start of engineering” and “reassessment of research needs”. Therefore, for the present model to be used as a practical engineering tool further refinement of the model may be necessary. Finally, to examine the performance of the model for the cases in which waves are obliquely incident to the breakwater, the model was compared against the experimental data reported by Ijima et al. ( 1978). They carried out hydraulic model experiments in a wave basin 20 m long, 9 m wide, and 0.6 m high. Regular waves were generated by a wave paddle located at one end of the basin, and on the downwave side the model breakwater was
Fig. 5. Error ( = calculation/measurement) et al. ( 1976)
of the model in terms of wave steepness for the experiment of Tanimoto
K.D. Sub, W.S. Park/ Coastal Engineering 26 (1995) 177-193
186 Table 1 Test conditions
and measured and calculated
reflection coefficients
for experiment of Ijima et al. (1978)
T(s)
@(“)
B (cm)
B COSOIL
(Kr)mea,
(Kr)ca,,
1.314
40
20 30 40 50
0.077 0.115 0.154 0.192
0.774 0.720 0.473 0.326
0.637 0.411 0.245 0.124
50
20 30 40 50
0.065 0.097 0.129 0.162
0.744 0.652 0.417 0.339
0.711 0.510 0.354 0.233
60
20 30 40 50
0.050 0.075 0.101 0.126
0.820 0.729 0.488 0.386
0.798 0.635 0.494 0.378
40
20 30 40 50
0.063 0.094 0.126 0.157
0.802 0.727 0.552 0.344
0.744 0.539 0.375 0.247
50
20 30 40 50
0.053 0.079 0.105 0.132
0.765 0.562 0.539 0.356
0.799 0.625 0.475 0.353
60
20 30 40 50
0.041 0.061 0.082 0.102
0.69 1 0.645 0.608 0.497
0.861 0.729 0.602 0.49 1
1.554
installed obliquely to the wave paddle so that the waves would be obliquely incident to the breakwater. Along the side walls of the basin wave absorber was installed to prevent wave reflection from the walls. Two different kinds of model caisson were used: one having vertical partition walls inside the wave chamber and the other without partition walls. In the present analysis only the data for the caisson without partition walls are used. Tests were made for 8= 40”, 50”, and 60” for which rereflection of the waves from the wave paddle did not occur. For each wave angle, two different wave periods T= 1.3 14 and 1.554 s and four different chamber widths B = 20, 30, 40, and 50 cm were tested, so the total number of cases was 24. For all the cases the following parameters were constant: h = 30 cm, H= 3 cm, e =7 cm, and r=0.25. The perforated-wall caisson was installed on a flat bed without a mound. The test conditions and the measured and calculated reflection coefficients are presented in Table 1, and the comparison between the measured and calculated reflection coefficients is shown in Fig. 6. The open and solid symbols in the figure denote the data for T= 1.314 and 1.554 s, respectively, and the different shape of symbols denotes a different wave angle. For most cases the reflection coefficient is larger for longer wave period, as expected. The
K.D. Sub, W.S. Park/Coastal
00
02
Engineering 26 (1995) 177-193
04
06
08
187
10
(K&m Fig. 6. Comparison (1978).
of the reflection coefficients
between present theory and the experimental
data of Ijima et al.
model tends to underpredict the reflection coefficient for smaller wave angle, but as a whole the agreement between measurement and calculation is acceptable. As mentioned at the beginning of this section, when waves are normally incident to the breakwater, the wave reflection is minimized when the reflection of the porous wall and the reflection of the impermeable wall are in opposite phases (i.e., B/L=0.25). When the waves are obliquely incident to the breakwater, this would happen when B cos0IL = 0.25. The values of B case/L in the experiment of Ijima et al. ( 1978) are presented in Table 1. For all the test conditions in the experiment, the values of B costIlL are less than 0.25, and for the same angle of incidence both measured and calculated reflection coefficients decrease as the wave chamber width increases, as expected. For the same chamber width, the theoretical result shows that the reflection coefficient increases with the angle of incidence. Such a trend is also observed in the measurement, even though the reflection coefficient for 8= 50” is slightly less than that for 8=40” in many cases. In order to see in detail the effect of angle of incidence and wave chamber width upon the wave reflection, the reflection coefficients were calculated by changing the wave angle from 0 to 80” and B costIlL from 0.1 to 0.5. The wave period T= 1.554 s was used and the
Wave
Angle
(degree)
Fig. 7. Contours of the reflection coefficient as a function of wave angle and B cosB/L; the lines of B/L= zo.5. 1.0 and 1.5 are also included.
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K.D. Suh, W.S. Park/Coastal
Engineering 26 (1995) 177-l 93
breakwater parameters were the same as those in the experiment of Ijima et al. ( 1978) except that the perforated-wall thickness was taken to be zero. The calculated results are shown in Fig. 7. For normal incidence of waves, as the same as shown in the previous analysis (see Fig. 2)) the wave reflection is minimum at B cosO/L=0.25 and increases as getting away from this value. The same trend is kept for oblique incidence of waves, but the solution exhibits singular behavior (i.e., the reflection coefficient shows a sharp peak) at B/L = 0.5, 1.O, 1.5, and so on, as shown in Fig. 7. In the practical design of a perforatedwall caisson, the value of B/L is usually less than 0.5. Thus, when the waves are obliquely incident to the breakwater, to minimize the wave reflection, the value of B cosd/Lshould be around 0.25 but the condition B/L - 0.5should be avoided. The experimental data of Ijima et al. (1978) covers the range of B cod/L approximately less than 0.2. In order to fully validate the results discussed above, experimental data in a wide range of B costIlL may be needed.
4. Conclusions Using the Gale&in-eigenfunction method, an analytical model was developed that can predict the reflection coefficient of a perforated-wall caisson mounted on a rubble foundation when the waves are obliquely incident to the breakwater at an arbitrary angle. In the case that the waves are normally incident to a perforated-wall caisson lying on a flat sea bottom, the result of the proposed model was shown to exactly agree with the solution by Fugazza and Natale ( 1992). In order to examine the predictability of the model for the cases in which the caisson is laid on a rubble mound or the waves are obliquely incident to the breakwater, the developed model was tested against the hydraulic experimental data reported by Tanimoto et al. ( 1976) and by Ijima et al. ( 1978)) respectively. The results showed that the developed model was in reasonable agreement with observations even though it was not so accurate as to be used for a practical engineering purpose. It was shown, by incorporating the linear shoaling analysis over the mound slope, that the formula of Fugazza and Natale ( 1992) originally developed for a caisson lying on a flat sea bottom could be used for the case in which the caisson is mounted on a rubble mound if the waves are normally incident to the breakwater. However, the solution of Fugazza and Natale ( 1992) with the linear shoaling analysis, which does not take into account the partial reflection from the mound slope, gives slightly lower wave reflection than the present theory. The method proposed in the present paper generalizes the formula by Fugazza and Natale ( 1992), of which it maintains the linearized expression for the perforated-wall dissipation coefficient (i.e., Eq. 17). As this linearization influences in a non-negligible manner the analytic solution, only a comparison between theoretical results and the (numeric) solution of the nonlinear problem would explain the influence of the simplification that were introduced. When the waves are normally incident to a perforated-wall caisson breakwater, it has been known that the wave reflection is minimized when the nondimensionalized wave chamber width B/L is about 0.25. In the present study, it was shown analytically that when the waves are obliquely incident to the breakwater, the minimum reflection would occur at B cosf?/L = 0.25and the wave reflection would increase sharply at B/L = 0.5. This phenom-
K.D. Sub, W.S. Park / Coastal Engineering 26 (1995) 177-193
189
enon was partially verified by the experimental data of Ijima et al. ( 1978) _In order to fully validate this phenomenon, however, experimental data in a wide range of B costIlL are needed. The present theory assumes that the wave chamber is infinitely long in the direction parallel to the breakwater crest line so that there is no partition wall in the wave chamber. In the practical design of a perforated-wall caisson, however, partition walls are often used for the purpose of reinforcement of the wave chamber. In this case, when the waves are obliquely incident to the breakwater, standing waves in the lateral direction will be generated inside the wave chamber, which may somehow affect the wave reflection and need more study in the future. The interaction between the permeable structure and waves is still an attractive subject for coastal researchers and engineers. The approach adopted in the present study may be applicable to the study of the hydraulic characteristics of other types of permeable coastal structure.
Acknowledgements This work was funded by the Korea Ocean Research and Development Institute and the Korea Ministry of Science and Technology under Project No. BSPE 00468 and BSPN 0026 1, respectively. The writers would like to thank Prof. Natale of the University of Pavia for suggesting the use of the effective opening ratio of the porous wall in the calculation of the energy loss coefficient for oblique incidence of waves.
Appendix
A. Components
in R&’ and R$$
190
K.D. Sub, W.S. Park/Coastal
V,=
-ik
u,=
-f6”
I, = $
(2kh + sinh2kh)
1,=&
(cosh2kh-
Engineering 26 (1995) 177-193
1)
Z3= & kh cosh2kh - 1 sinh2kh ‘( Z4= $
Z5= &
"I'= -sinh2~~~~ +-Ah f&"= -sinh2ckhj +Ah Af6”
Appendix
B. Derivation
1
2-ifb1T-ksinh(2kh)]
of boundary
condition
at x= -b and x=0
The boundary condition at x = - b is the same as that obtained by Masse1 ( 1993) for waves propagating on straight and parallel bottom contours. For the sake of completeness the boundary condition is rederived here. To obtain the boundary conditions it is convenient to use the orthogonality of the functions in (7). At x = - b, the potential 4, ( - b,y,z) is +I( -b,y,z)
= C R,exp(ir(y)Z,,n(h,,z) a1.n
= c “l,“c~~~~D) I,n I a1.n
cosar;(zi-hr) 1.n
(28)
in which 0
B:,,= Functions
I -hl
cos2a,.,(lth,)dz=~
sin(2q,hI) 2%&r
1
(29)
[cos~yr,,(z + h,) I /B,,, form a complete orthogonal set in domain [ - h,,O]. Thus
K.D. Sub, W.S. Park/Coastal
R,=
A( -hY,Z)
Engineering 26 (1995) 177-l 93
COS~If + h, )
191
dz
(30)
1-n
By invoking
the pressure continuity
at x = - b and using h2( - b) = h,, we get
0
COSQLnh
R,= 6,
exp(W)
I 44
_-h,
-b,y,z)
cosc-u,,,(z+h,) B 1.n
dz
The second integral in the preceding equation vanishes if m # n and is equal to B:., if m = n; thus (32)
&=&(-b) Particularly, R,=
for n = 0, we obtain
1 +K,.=qo(
-b)
(33)
in which K, is the reflection coefficient. The velocity continuity i( 1 - K,)k,cosf$
=
-b)
d@o(
(34)
dr
in which deo( -b) ldx denotes d&/dx at x= -b. The boundary conditions at x = 0 can be evaluated 4S( O,y,z) at x = 0 is given by 43(09~z)
=
at x = - b gives, for n = 0,
C T, exp(iXy)S,,,(h3,z)
=
C 4.n
4.”
in a similar
way. The potential
B3,Jnexp(b) cosa,,Az+h,) cost
h
3,n
3
B 3.n
(35)
in which sin(2a3,,h3) 2a3.A
1
(36)
Following the procedure similar to (30) through (32) and using the pressure continuity x = 0, we obtain for n = 0
at
(37) The velocity continuity fJoTo
=
at x = 0 with the use of h2( 0) = h3 and (Y*,~= LY~,~ at x = 0 gives
-y
Finally the no-flux condition
(38) at x = B gives
192
K.D. Sub, W.S. Park/Coastal
Engineering 26 (I 995) 177-193
(39)
P3,0T0ev ( - A,$) = 0 The complex-valued amplitude To inside the wave chamber both forward-propagating and backscattered waves, i.e.,
(i.e., Region 3) includes
in which T,, and T,, denote the amplitude of a forward-propagating wave, respectively. The corresponding value of &, is given by
wave and a backscattered
&,(=
ik,cos&,
P3,& = - ik,cos&
After substituting (40) and (41) into (37) - (39), the boundary conditions x = 0 for the progressive waves ( IZ= 0) can be summarized as follows:
(41) at x = - b and
c&-b)=l+K, i( 1 -K,) k,cos& =
dQ%( -b) dx
T,+T,=&,(O)+ p3,e(T,,-T,) TGXP( -P&U
(42) = -v = TO,exp(P&U
References Bennett, G.S., McIver, P. and Smallman, J.V., 1992. A mathematical model of a slotted wavescreen breakwater. Coastal Eng., 18: 23 l-249. Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. 1988. Spectral Methods in Fluid Dynamics. SpringerVerlag, New York, 557 pp. Fugazza, M. and Natale, L., 1992. Hydraulic design of perforated breakwaters. J. Waterw. Port Coastal Ocean Eng., 118: 1-14. Goda, Y., 1985. Random Seas and Design of Maritime Structures. University of Tokyo Press, 323 pp. Goda, Y., 1994. A plea for engineering-minded research efforts in harbor and coastal engineering. In: Proc. HydroPort’94. Port and Harbour Research Institute, Yokosuka, Vol. 1, pp. 1-21. Hattori, M., 1972. Transmission of waves through perforated wall. Coastal Eng. Jpn., 15: 69-79. Ijima, T., Okuzono, H. and Ushifusa, Y., 1978. The reflection coefficients of permeable quaywall with reservoir against obliquely incident waves. Rep. Coll. Eng., Kyushu Univ., 5 1: 245-250 (in Japanese). Jarlan, G.E., 1961. A perforated vertical wall breakwater. Dock Harbour Auth., X11(486): 394-398. Kakuno, S., Oda, K. and Liu, P.L.-F., 1992. Scattering of water waves by vertical cylinders with a backwall. In: Proc. 23rd Coastal Eng. Conf., Venice, Vol. 2, pp. 1258-1271. Kondo, H., 1979. Analysis of breakwaters having two porous walls, In: Proc. Coastal Structures ‘79, Vol. 2, pp. 962-977. Lee, D.S., Park, W.S., Suh, K.D. and Oh, Y.M., 1994. Hydraulic experiments for basic design of offshore breakwater for protecting artificial island. In: Proc. Hydro-Port’94. Port and Harbour Research Institute, Yokosuka, Vol. 1, pp. 563-580. Marks, M. and Jarlan, G.E., 1968. Experimental study on a fixed perforated breakwater. In: Proc. 1 lth Coastal Eng. Conf., London, Vol. 3, pp. 1121-l 140.
K.D. Sub, W.S. Park/Coastal Engineering 26 (1995) 177-193
193
Massel, S.R., 1993. Extended refractiondiffraction equation for surface waves. Coastal Eng., 19: 97-126. McBride, M.W., Smallman, J.V. and Allsop, N.W.H., 1994. Design of harbour entrances: breakwater design and vessel safety. In: Proc. Hydro-Port’94. Port and Harbour Research Institute, Yokosuka, Vol. 1, pp. 525-541. Mei, CC. 1983. The Applied Dynamics of Ocean Surface Waves. Wiley, New York, 740 pp. Neter, J., Wasserman, W. and Kutner, M.H., 1985. Applied Linear Statistical Models, 2nd ed. Richard D. Irwin, Inc., Homewood, IL, 1127 pp. Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. 1992. Numerical Recipes in FORTRAN: the Art of Scientific Computing. Cambridge University Press, 963 pp. Tanimoto, K., Haranaka, S., Takahashi, S., Komatsu, K., Todoroki, M. and Osato, M., 1976. An experimental investigation of wave reflection, overtopping and wave forces for several types of breakwaters and sea walls. Tech. Note of Port and Harbour Res. Inst., Ministry of Transport, Japan, No. 246, 38 pp. (in Japanese, with English abstract). Tanimoto, K., Takahashi, S. and Kimura, K., 1987. Structures and hydraulic characteristics of breakwaters - the state of the art of breakwater design in Japan. Rep. of Port and Harbour Res. Inst., Japan, Vol. 26, No. 5, 55 PP. Terret, F.L., Osorio, J.D.C. and Lean, G.H., 1968. Model studies of a perforated breakwater. Eng. Conf., London, Vol. 3, pp. 1104-l 120.
In: Proc. 1 lth Coastal
Coastal Engineering 26 ( 1995) 177-193
Wave reflection from perforated-wall caisson breakwaters Kyung Duck Suh, Woo Sun Park Korea Ocean Research & Development
Institute, 1270 Sa-dong, Ansan, Kyonggi-do 425-l 70, South Korea
Received 8 August 1994; accepted 18 September 1995
Abstract Analytical models for predicting wave reflection from a perforated-wall caisson breakwater have been developed. Most of the existing models deal with the case in which the waves are normally incident to the caisson lying on a flat sea bottom. In the present paper, using the Galerkin-eigenfunction method, an analytical model is developed that can predict the reflection coefficient of a perforated-
wall caisson mounted on a rubble mound foundation when waves are obliquely incident to the breakwater at an arbitrary angle. The developed model is compared with other theoretical results and hydraulic experimental data.
1. Introduction Composite breakwaters have been widely used at various locations, which consists of the lower rubble foundation and the upper upright structure made of concrete blocks or caissons. One of the drawbacks of a conventional composite breakwater compared to a rubble mound breakwater is that the increased agitation on the sea side of the breakwater due to severe wave reflection from the upright structure can originate difficulties in navigation or ship operation (see McBride et al., 1994, for example). As a countermeasure to overcome this drawback, a mound of concrete blocks of energy dissipating type is placed in front of the breakwater. In this case, however, wave transmission due to overtopping may increase so that the tranquillity inside the harbour may deteriorate (see Lee et al., 1994, for example). As another means for reducing wave reflection, a perforated-wall caisson is often adopted, which reduces the wave reflection by energy dissipation due to turbulence generated when the waves intrude into the wave chamber through the perforated wall. In order to examine the reflection characteristics of a perforated-wall caisson breakwater, hydraulic model tests have been used (Jarlan, 1961; Marks and Jarlan, 1968; Terret et al., 1968; Tanimoto et al., 1976, among others]. Efforts towards developing analytical models 0378.3839/95/$09.50
0 1995 Elsevier Science B.V. All rights reserved
.SSDIO378-3839(95)00027-5
178
K. D. Sub, W.S. Park/Coastal
Engineering 26 (1995) 177-193
for predicting the reflection coefficient have also been made. Based upon linearized shallow water wave theory, Kondo ( 1979) developed an analytical model for calculating reflection coefficient of a perforated-wall caisson breakwater having one or two perforated walls. Kondo concluded that the breakwater having two perforated walls could give much lower reflection coefficient than that having one perforated wall. Recently, using the method of matched asymptotic expansions, Kakuno et al. ( 1992) developed an analytical model for predicting wave reflection from a single-chamber perforated-wall caisson. A similar model and comparisons with large-scale laboratory data were given by Bennett et al. ( 1992). On the other hand, Fugazza and Natale ( 1992) proposed a closed-form solution for wave reflection from a multi-chamber perforated-wall caisson. Fugazza and Natale showed that the reflection is minimized when the wave chamber width is about one quarter of the wave length. In contrast to the conclusion made by Kondo ( 1979), Fugazza and Natale showed that the perforated-wall breakwater with a single wave chamber could give the largest reduction of wave reflection in the range of practical applications. The aforementioned analytical approaches deal with the case in which the waves are normally incident to the perforated-wall caisson lying on a flat sea bottom. In the present study, we develop an analytical model that can predict the wave reflection from a singlechamber perforated-wall caisson mounted on a rubble mound foundation when waves are obliquely incident to the breakwater at an arbitrary angle. The method of solution adopted in this paper consists of solving the wave potential on the rubble foundation with the matching conditions at the seaward toe of the rubble foundation and at the location of the perforated wall. The methodology follows the Galerkin-eigenfunction method used by Masse1 ( 1993) in developing the extended refractiondiffraction equation. This equation includes higher order terms of the bottom slope and the term proportional to the bottom curvature which were neglected in the mild-slope equation so that it can be applied to wave propagation over a bed consisting of substantial variations in water depth. In the design of a composite breakwater, the dimension and shape of the rubble mound are usually determined from various safety considerations such as wave pressure, bottom foundation, and so on, for an extreme wave condition (e.g., design wave condition of 50 or 100 years’ return period). For example, Tanimoto et al. ( 1987) have investigated the effect of rubble mound configuration upon the wave force acting on the caisson. From the viewpoint of wave reflection, however, the ordinary waves are of more interest than the severe storm waves because during the severe wave condition most ships seek refuge into harbors. Thus, in the present study, we deal with the reflection of ordinary nonbreaking waves. The present model is compared with the analytic solution of Fugazza and Natale ( 1992) for a perforated-wall caisson lying on a flat sea bottom. For the cases in which the caisson is mounted on a rubble mound foundation, the model is compared with the experimental data reported by Tanimoto et al. ( 1976). To examine the performance of the model for oblique incidence of waves, the model is tested against the experimental data obtained by Ijima et al. ( 1978) in a wave basin. In the following section, the analytical model for predicting the reflection coefficient of a perforated-wall caisson breakwater is derived. The next section includes comparison of the developed model with other theoretical and experimental results. The major conclusions then follow.
K.D. Sub, W.S. Park/Coastal
Engineering 26 (1995) 177-193
179
2. Theoretical analysis Let us consider the perforated-wall caisson breakwater sketched in Fig. 1, in which 13~is the incident wave angle, B is the wave chamber width of the perforated-wall caisson. The x-axis and y-axis are taken to be normal and parallel, respectively, to the breakwater crest line, and the water depth is assumed to be constant in they-direction. The vertical coordinate z is measured vertically upwards from the still water line. In Region 2 ( - b 5 x 5 0) the water depth h(x) is a varying function of x. For x I - b (Region 1) and 0 2 x I B (Region 3), the water depth is constant and equal to hi and h,, respectively. The rubble mound foundation is assumed to be impervious. Assuming inviscid irrotational flow, the velocity potential @(x,y,z,t) for the monochromatic wave propagating over the water depth h(x) with the angular frequency w and wave height H can be expressed as @(x,y,z,t)
- igH
= Re
20
Kwz)
exp( - id
>
in which i = \/- 1; g = gravitational acceleration, and the symbol Re represents the real part of a complex value. The wave number k must satisfy the dispersion relationship: w2=gktanh(kh)
Linearizing the free-surface boundary for the potential +(x,y,z) is obtained:
a4J ,,-A4=0 a+
a+dh
%f---=O
conditions,
the following
boundary
value problem
at z=O
(4)
at z= -h(x)
Incident
Reflected
Incident
‘mve
wcwe \
wave
Fig. 1. Schematic
:*I
I z=-b
diagram and coordinate
I
I
2=0
z=B
system for calculation
-z
-
of wave reflection.
180
K.D. Suh, W.S. Park/ Coastal Engineering 26 (1995) 177-193
in which V represents the horizontal gradient operator, and h = w2/g is a wave number in deep water. The Gale&in-eigenfunction method is used to formulate the problem. The Galerkin approach assumes that the function $(x,y,z) can be expanded in terms of N+ 1 depthdependent functions 2, (x,2) :
(6) The functions Z,(x,z)
Z,*(w) =
are taken as
cos[(Y,(z+h)l
(7)
cos( a,h)
so as to form a complete orthogonal set of eigenfunctions in the domain wave numbers (Y, are the solution of the following dispersion relation: h+a,tan(cr&)
=0
[ - h(x) ,O] . The
(8)
which has an infinite discrete set of real roots + (Y,,and a pair of imaginary roots cu, = f ik. Therefore, the function Z,(x,z) represents the free propagating wave mode, while the functions Z,(x,z) (n r 1) correspond to the non-propagating evanescent wave modes. The functions Z,(x,z) satisfy the free surface boundary condition (4) and do not satisfy the bottom boundary condition (5) individually. However, the global set of orthogonal functions should satisfy this condition. This is known as a tau method (Canuto et al., 1988). In the tau method, a sufficient number of the functions cp,(x,y) in the approximated solution (6) is chosen to ensure exact satisfaction of the bottom boundary condition. Turning back to the breakwater problem sketched in Fig. 1, the solution of the boundary value problem given by (3)-( 5) may be constructed from the particular solutions in each region of the fluid domain: +,(x,y,z)
= [exp[ik,(x+b)cos&,]
coshk,(z+h,)
cash W
-exp[
-ik,(x+b)cos&]}exp(i~y)
+ CR,exp[P,,,(x+b)lexp(iXy)Z,,(h,,z) a, n
422(w,z)= C~n,,(x)exp(iXv)Z2,,(h2,z>
(9)
(10)
4.n
Mwz)
= CT,exp(-P3.~)exp(i~)Z,.,(h,,z)
(11)
4.”
in which &=
\/o&+x’,
If II = 0 (propagating A = kj tanh( kjhj),
x=kj mode),
sinf?j=k,sinO,
=constant
(j= 1,2,3)
(12)
(8) and ( 12) yield
/I,,0 = f ikj cosOj
(13)
For /3i,o we take the negative sign for the reflected wave in Region 1, while we need both positive and negative for the waves inside the wave chamber (i.e., Region 3).
K.D. Sub, W.S. Park/ Coastal Engineering 26 (1995) 177-193
181
The potential ~j(X,Y,Z) must satisfy the matching conditions which provide continuity of pressure and horizontal velocity, normal to the vertical planes separating the fluid regions and no-flux condition at the wall on the downwave side of the wave chamber, i.e. +I=&,
z=fj$
(X’
-b,
-h,1z
(t+$ 1!f$ !!b=s
g)3=42+
ax
a43 ,,=o
(x=0,
ax
(14)
-h,_
(15)
-h,_
(x=B,
(16)
In first equation of ( 15)) the second term on the right side represents the energy loss at the perforated wall (Kondo, 1979; Fugazza and Natale, 1992). In this equation, 4 is the length of the jet flowing through the perforated wall, which is usually taken as the thickness of the wall, and y is the linearized dissipation coefficient given by Fugazza and Natale (1992) as W ,/W2(R
5 + cosh2k,h,
in which H, = incident wave height at the perforated C = 1 - PW, P = t?k,, r = porosity of the wall, and
(
(y=p
1
i-cost&c,
2
(17)
+ 1)2+ ~‘2 2k3h3 + sinh2k,h,
1
-1
wall, W= tan(k3B),
R = ykJo,
(18)
is the energy loss coefficient at the perforated wall, which is a modification of the head loss coefficient for the plate orifice formula (see Mei, 1983, p. 257). In the preceding equation, r cos0, denotes the effective ratio of the opening of the porous wall taking into account the oblique incidence of the waves to the wall. For normal incidence, this reduces to r as in Fugazza and Natale ( 1992). C, is the empirical discharge coefficient at the perforated wall. Hattori ( 1972) concluded that the discharge coefficient ranged from 0.4 to 0.75 and Fugazza and Natale ( 1992) showed that the use of C, = 0.55 provided good agreement with experimental data for perforated breakwaters. Thus C,=O.55 was used in the present study. Rearranging of ( 17) gives a quartic polynomial of 7, which can be solved by the eigenvalue method (see Press et al., 1992, p. 368). As shown in (6), the Gale&in solution consists of a free propagating wave mode and non-propagating evanescent modes. The evanescent wave modes would be of importance in the region near the breakwater. However, since we are interested in the reflected wave far from the breakwater, we focus at the solution for the propagating mode. Masse1 ( 1993) showed that in Region 2 the function &(x) satisfies the following ordinary differential equation:
d240 -+D(x) dx?
in which
2+E(x)&=O
(19)
182
K. D. Sub, W. S. Park/Coastal
G(M) D(X) =hz
Engineering 26 (1995) 177-l 93
dh
(20) (21)
In these equations,
(22)
p=i(l+&) G(kh) =
kh
l-3?+
T+kh(l-?)
1
27 T+kh(l-?)
(23)
in which T= tanh( kh) . R$) and R$’ are given by Rg)
1
=
cosh’( kh)
( WJ, + WJ, + w,z, + W,Z, + W& + W,)
(24)
(UJ,
(25)
and R$.’ =
cosh:(kh)
+
u212+
U34)
The expressions for W,,, U,, and Z,,are given in Appendix A. In order to solve ( 19) in Region 2, we need the boundary conditions at x = - b and x = 0. For the sake of readability of the text, the derivation of the boundary conditions is separated into Appendix B. After some algebraic manipulation, the boundary conditions in (42) of Appendix B are reduced to the following two equations:
d4d -b) dx eo(o)
=
=z[2-&( Iexp(
-b)]k, -Pu@)
P3,0~exp( -&$)
cos 0, +exp(L%d) -expU%,$)
1
@do)
_[_9 w
dx
(27)
The differential equation ( 19) with the preceding two boundary conditions can be solved using the finite-difference method. Using the forward-differencing for d@,,( - b) /dx, backward-differencing for d&,( 0) ldx, and central-differencing for the derivatives in ( 19)) the boundary value problem ( 19)) (26) and (27) was approximated by a system of linear equations, AY = B, where A is a tridiagonal band type matrix, Y is a column vector and B is also a column vector. The subroutines given in the book of Press et al. ( 1992) were used to solve this matrix equation. In particular we are interested in &( -b), from which the reflection coefficient K, is calculated using (33) of Appendix B.
3. Comparison
with other theoretical and experimental
results
For a perforated-wall caisson lying on a flat sea bottom, Fugazza and Natale ( 1992) showed that the reflection of waves normally incident to the caisson is at its minimum when
K.D. Sub, W.S. Park/Coastal
Engineering 26 (1995) 177-193
183
B/L Fig. 2. Reflection coefficient K, versus normalized chamber width B/L for a perforated-wall caisson lying on a flat sea bottom; Comparison between present theory and the solution of Fugazza and Natale ( 1992).
Bf L = 0.25 in which L is the wave length. The reason is that when B/L = 0.25 the reflection of the porous wall and the reflection of the impermeable wall are in opposite phases. Fig. 2 shows the variation of reflection coefficients with respect to B/L calculated by the formula of Fugazza and Natale ( 1992) and by the present theory. Water depth h = 3 m, wave height H = 1 m, wave chamber width B = 4 m, and porosity of the wall r = 0.3 and 0.6 were used. The two different theories give almost identical results as shown in Fig. 2, indicating minimum reflection occurred at B/L = 0.25. This result is not surprising because the same expression for the perforated-wall dissipation coefficient (i.e., Eq. 17) was used in both theories. In order to examine the effect of rubble mound foundation upon wave reflection, the reflection coefficients were calculated by changing the height and slope of the rubble mound. Total water depth in front of the breakwater hi = 10 m, wave period T= 8 s, incident wave height H = 2 m, thickness of the perforated wall 1= 0.6 m, porosity of the wall r= 0.3 and wave chamber width B = 10 m were used. The waves were assumed to be normally incident to the breakwater. The perforated wall of the caisson was assumed to be located at the upper end of the mound slope so that there was no berm in front of the caisson. The mound height ranged from 0.1 to 0.5 times the total water depth and the mound slope ranged from 1 : 4 to 1 : 1 for each mound height. For the solution of Fugazza and Natale ( 1992) the wave height at the perforated wall, H,, was calculated by linear shoaling, i.e., H, = H( C,,/C,,) ” 2. After the reflection coefficient at the top of the rubble mound was calculated using this wave height, the reflected wave height on the flat sea bottom was calculated by applying the shoaling formula in the inverse manner. For the present solution, the incident wave height at the location of the perforated wall was calculated by assuming the absence of the caisson but including the partial reflection from the mound slope as done in Masse1 ( 1993 ) and it was used to calculate the dissipation coefficient, ‘y, in ( 17). Then ( 19) was solved on the mound slope to calculate the reflection coefficient. Fig. 3 shows the ratio of the reflection coefficient of the present solution to the solution of Fugazza and Natale ( 1992) for the variation of mound height and slope. Except for very high mound with very mild slope, the reflection coefficient calculated by the present theory is slightly higher than that by Fugazza and Natale ( 1992). The difference becomes significant as the mound height increases and the mound slope becomes steeper. Note that the wave reflection from the mound slope is included in the present model but not in the solution of Fugazza and Natale incorporated with the linear shoaling analysis. A portion of incident
184
K.D. Sub, W.S. Park / Coastal Engineering 26 (1995) 177-I 93
Fig. 3. Contours of the ratio of the reflection coefficient of the present model versus the solution of Fugazza and Natale ( 1992) as a function of mound height and slope.
wave energy will be reflected seaward by the rubble mound. This causes the reduction of incident wave height in front of the caisson. Therefore the less energy dissipation due to the perforated wall increases the height of the reflected wave. These two effects make the present theory predict higher wave reflection than that of Fugazza and Natale. Fig. 3 indicates that in the practical range of mound height and slope values the difference between the two theories is negligible. Thus the formula developed by Fugazza and Natale ( 1992)) along with the linear shoaling analysis over the mound slope, could be used for most practical situations if the waves are normally incident to the breakwater. In the preceding analysis, we have investigated the effect of rubble mound analytically. It may be useful to compare the model with experimental data for verifying the predictability of the model. Tanimoto et al. ( 1976) reported hydraulic model experiments for wave reflection from a perforated-wall caisson mounted on a rubble foundation. They used irregular waves of the Bretschneider-Mitsuyasu spectrum (see Goda, 1985, p. 25). Even though the present theory is developed for regular waves, experimental data using regular waves are rare. Thus the present theory is compared with the Tanimoto et al.‘s irregular wave experimental data. For the present analysis the wave period and height were taken as the significant wave period T, ,3 and the root-mean-square wave height H,,,( = H, ,3/ respectively. The experiment of Tanimoto et al. ( 1976) includes a total of 102 cases of various combinations of different wave conditions and structural dimensions of the break water. For all the cases the following parameters were constant: h, =42 cm, mound slope= 1 : 2, and r=0.236. The perforated wall included a number of circular holes of diameter of 2.5 cm. The ranges of other wave and structural parameters were as follows: T, ,3 = 1.O - 2.22 s, HI,, - 3.97 - 20.6 cm, h3 = 27 and 32 cm, e = 3 and 6 cm, crest height of breakwater h, = 10 and 16 cm, mound berm width et, = 20 and 2.5 cm, and wave chamber width including perforated-wall thickness B = 20 - 53 cm. The comparison of the measured and calculated reflection coefficients is shown in Fig. 4. For several points near the calculated reflection coefficient of about 0.1 significant underprediction is observed. Careful investigation was made for the experimental conditions of these data, but any particular reason could not be found. Statistically these data seem to be outliers which lie far beyond the scatter of the remaining data points (cf. Neter et al., 1985, pp. 114-l 16).
fi) ,
K.D. Sub, W.S. Park/Coastal
Fig. 4. Comparison al. ( 1976).
of the reflection coefficients
Engineering 26 (1995) 177-193
between present theory and the experimental
185
data of Tanimoto et
Fig. 4 shows that the model is not in very good agreement with the experimental data even without the outliers though the overall agreement is acceptable. Probably the behavior of an irregular wave in the breakwater could not be fully synthesized in parameters T, ,3 and H,,, as the energy dissipation phenomenon at the perforated wall is intrinsically nonlinear. In order to examine the effect of nonlinearity, the error ( = calculation/measurement) is plotted in Fig. 5 in terms of H,f gT2 in which H,, is the deepwater wave height corresponding to Hi/, and the wave period T is equal to T,,,. The data of the outliers were not included. The model tends to overestimate the reflection coefficient as the wave steepness increases probably because for steeper waves additional energy dissipation due to wave breaking may occur in the experiment. The error shown in Fig. 5 ranges approximately between 0.6 and 1.5. As for the error of a model, Goda ( 1994) proposed a method of judging the order-ofmagnitude accuracy and a concept of research efficiency with regards to the final accuracy. According to his judgement, the results in Fig. 5 are in the status between “start of engineering” and “reassessment of research needs”. Therefore, for the present model to be used as a practical engineering tool further refinement of the model may be necessary. Finally, to examine the performance of the model for the cases in which waves are obliquely incident to the breakwater, the model was compared against the experimental data reported by Ijima et al. ( 1978). They carried out hydraulic model experiments in a wave basin 20 m long, 9 m wide, and 0.6 m high. Regular waves were generated by a wave paddle located at one end of the basin, and on the downwave side the model breakwater was
Fig. 5. Error ( = calculation/measurement) et al. ( 1976)
of the model in terms of wave steepness for the experiment of Tanimoto
K.D. Sub, W.S. Park/ Coastal Engineering 26 (1995) 177-193
186 Table 1 Test conditions
and measured and calculated
reflection coefficients
for experiment of Ijima et al. (1978)
T(s)
@(“)
B (cm)
B COSOIL
(Kr)mea,
(Kr)ca,,
1.314
40
20 30 40 50
0.077 0.115 0.154 0.192
0.774 0.720 0.473 0.326
0.637 0.411 0.245 0.124
50
20 30 40 50
0.065 0.097 0.129 0.162
0.744 0.652 0.417 0.339
0.711 0.510 0.354 0.233
60
20 30 40 50
0.050 0.075 0.101 0.126
0.820 0.729 0.488 0.386
0.798 0.635 0.494 0.378
40
20 30 40 50
0.063 0.094 0.126 0.157
0.802 0.727 0.552 0.344
0.744 0.539 0.375 0.247
50
20 30 40 50
0.053 0.079 0.105 0.132
0.765 0.562 0.539 0.356
0.799 0.625 0.475 0.353
60
20 30 40 50
0.041 0.061 0.082 0.102
0.69 1 0.645 0.608 0.497
0.861 0.729 0.602 0.49 1
1.554
installed obliquely to the wave paddle so that the waves would be obliquely incident to the breakwater. Along the side walls of the basin wave absorber was installed to prevent wave reflection from the walls. Two different kinds of model caisson were used: one having vertical partition walls inside the wave chamber and the other without partition walls. In the present analysis only the data for the caisson without partition walls are used. Tests were made for 8= 40”, 50”, and 60” for which rereflection of the waves from the wave paddle did not occur. For each wave angle, two different wave periods T= 1.3 14 and 1.554 s and four different chamber widths B = 20, 30, 40, and 50 cm were tested, so the total number of cases was 24. For all the cases the following parameters were constant: h = 30 cm, H= 3 cm, e =7 cm, and r=0.25. The perforated-wall caisson was installed on a flat bed without a mound. The test conditions and the measured and calculated reflection coefficients are presented in Table 1, and the comparison between the measured and calculated reflection coefficients is shown in Fig. 6. The open and solid symbols in the figure denote the data for T= 1.314 and 1.554 s, respectively, and the different shape of symbols denotes a different wave angle. For most cases the reflection coefficient is larger for longer wave period, as expected. The
K.D. Sub, W.S. Park/Coastal
00
02
Engineering 26 (1995) 177-193
04
06
08
187
10
(K&m Fig. 6. Comparison (1978).
of the reflection coefficients
between present theory and the experimental
data of Ijima et al.
model tends to underpredict the reflection coefficient for smaller wave angle, but as a whole the agreement between measurement and calculation is acceptable. As mentioned at the beginning of this section, when waves are normally incident to the breakwater, the wave reflection is minimized when the reflection of the porous wall and the reflection of the impermeable wall are in opposite phases (i.e., B/L=0.25). When the waves are obliquely incident to the breakwater, this would happen when B cos0IL = 0.25. The values of B case/L in the experiment of Ijima et al. ( 1978) are presented in Table 1. For all the test conditions in the experiment, the values of B costIlL are less than 0.25, and for the same angle of incidence both measured and calculated reflection coefficients decrease as the wave chamber width increases, as expected. For the same chamber width, the theoretical result shows that the reflection coefficient increases with the angle of incidence. Such a trend is also observed in the measurement, even though the reflection coefficient for 8= 50” is slightly less than that for 8=40” in many cases. In order to see in detail the effect of angle of incidence and wave chamber width upon the wave reflection, the reflection coefficients were calculated by changing the wave angle from 0 to 80” and B costIlL from 0.1 to 0.5. The wave period T= 1.554 s was used and the
Wave
Angle
(degree)
Fig. 7. Contours of the reflection coefficient as a function of wave angle and B cosB/L; the lines of B/L= zo.5. 1.0 and 1.5 are also included.
188
K.D. Suh, W.S. Park/Coastal
Engineering 26 (1995) 177-l 93
breakwater parameters were the same as those in the experiment of Ijima et al. ( 1978) except that the perforated-wall thickness was taken to be zero. The calculated results are shown in Fig. 7. For normal incidence of waves, as the same as shown in the previous analysis (see Fig. 2)) the wave reflection is minimum at B cosO/L=0.25 and increases as getting away from this value. The same trend is kept for oblique incidence of waves, but the solution exhibits singular behavior (i.e., the reflection coefficient shows a sharp peak) at B/L = 0.5, 1.O, 1.5, and so on, as shown in Fig. 7. In the practical design of a perforatedwall caisson, the value of B/L is usually less than 0.5. Thus, when the waves are obliquely incident to the breakwater, to minimize the wave reflection, the value of B cosd/Lshould be around 0.25 but the condition B/L - 0.5should be avoided. The experimental data of Ijima et al. (1978) covers the range of B cod/L approximately less than 0.2. In order to fully validate the results discussed above, experimental data in a wide range of B costIlL may be needed.
4. Conclusions Using the Gale&in-eigenfunction method, an analytical model was developed that can predict the reflection coefficient of a perforated-wall caisson mounted on a rubble foundation when the waves are obliquely incident to the breakwater at an arbitrary angle. In the case that the waves are normally incident to a perforated-wall caisson lying on a flat sea bottom, the result of the proposed model was shown to exactly agree with the solution by Fugazza and Natale ( 1992). In order to examine the predictability of the model for the cases in which the caisson is laid on a rubble mound or the waves are obliquely incident to the breakwater, the developed model was tested against the hydraulic experimental data reported by Tanimoto et al. ( 1976) and by Ijima et al. ( 1978)) respectively. The results showed that the developed model was in reasonable agreement with observations even though it was not so accurate as to be used for a practical engineering purpose. It was shown, by incorporating the linear shoaling analysis over the mound slope, that the formula of Fugazza and Natale ( 1992) originally developed for a caisson lying on a flat sea bottom could be used for the case in which the caisson is mounted on a rubble mound if the waves are normally incident to the breakwater. However, the solution of Fugazza and Natale ( 1992) with the linear shoaling analysis, which does not take into account the partial reflection from the mound slope, gives slightly lower wave reflection than the present theory. The method proposed in the present paper generalizes the formula by Fugazza and Natale ( 1992), of which it maintains the linearized expression for the perforated-wall dissipation coefficient (i.e., Eq. 17). As this linearization influences in a non-negligible manner the analytic solution, only a comparison between theoretical results and the (numeric) solution of the nonlinear problem would explain the influence of the simplification that were introduced. When the waves are normally incident to a perforated-wall caisson breakwater, it has been known that the wave reflection is minimized when the nondimensionalized wave chamber width B/L is about 0.25. In the present study, it was shown analytically that when the waves are obliquely incident to the breakwater, the minimum reflection would occur at B cosf?/L = 0.25and the wave reflection would increase sharply at B/L = 0.5. This phenom-
K.D. Sub, W.S. Park / Coastal Engineering 26 (1995) 177-193
189
enon was partially verified by the experimental data of Ijima et al. ( 1978) _In order to fully validate this phenomenon, however, experimental data in a wide range of B costIlL are needed. The present theory assumes that the wave chamber is infinitely long in the direction parallel to the breakwater crest line so that there is no partition wall in the wave chamber. In the practical design of a perforated-wall caisson, however, partition walls are often used for the purpose of reinforcement of the wave chamber. In this case, when the waves are obliquely incident to the breakwater, standing waves in the lateral direction will be generated inside the wave chamber, which may somehow affect the wave reflection and need more study in the future. The interaction between the permeable structure and waves is still an attractive subject for coastal researchers and engineers. The approach adopted in the present study may be applicable to the study of the hydraulic characteristics of other types of permeable coastal structure.
Acknowledgements This work was funded by the Korea Ocean Research and Development Institute and the Korea Ministry of Science and Technology under Project No. BSPE 00468 and BSPN 0026 1, respectively. The writers would like to thank Prof. Natale of the University of Pavia for suggesting the use of the effective opening ratio of the porous wall in the calculation of the energy loss coefficient for oblique incidence of waves.
Appendix
A. Components
in R&’ and R$$
190
K.D. Sub, W.S. Park/Coastal
V,=
-ik
u,=
-f6”
I, = $
(2kh + sinh2kh)
1,=&
(cosh2kh-
Engineering 26 (1995) 177-193
1)
Z3= & kh cosh2kh - 1 sinh2kh ‘( Z4= $
Z5= &
"I'= -sinh2~~~~ +-Ah f&"= -sinh2ckhj +Ah Af6”
Appendix
B. Derivation
1
2-ifb1T-ksinh(2kh)]
of boundary
condition
at x= -b and x=0
The boundary condition at x = - b is the same as that obtained by Masse1 ( 1993) for waves propagating on straight and parallel bottom contours. For the sake of completeness the boundary condition is rederived here. To obtain the boundary conditions it is convenient to use the orthogonality of the functions in (7). At x = - b, the potential 4, ( - b,y,z) is +I( -b,y,z)
= C R,exp(ir(y)Z,,n(h,,z) a1.n
= c “l,“c~~~~D) I,n I a1.n
cosar;(zi-hr) 1.n
(28)
in which 0
B:,,= Functions
I -hl
cos2a,.,(lth,)dz=~
sin(2q,hI) 2%&r
1
(29)
[cos~yr,,(z + h,) I /B,,, form a complete orthogonal set in domain [ - h,,O]. Thus
K.D. Sub, W.S. Park/Coastal
R,=
A( -hY,Z)
Engineering 26 (1995) 177-l 93
COS~If + h, )
191
dz
(30)
1-n
By invoking
the pressure continuity
at x = - b and using h2( - b) = h,, we get
0
COSQLnh
R,= 6,
exp(W)
I 44
_-h,
-b,y,z)
cosc-u,,,(z+h,) B 1.n
dz
The second integral in the preceding equation vanishes if m # n and is equal to B:., if m = n; thus (32)
&=&(-b) Particularly, R,=
for n = 0, we obtain
1 +K,.=qo(
-b)
(33)
in which K, is the reflection coefficient. The velocity continuity i( 1 - K,)k,cosf$
=
-b)
d@o(
(34)
dr
in which deo( -b) ldx denotes d&/dx at x= -b. The boundary conditions at x = 0 can be evaluated 4S( O,y,z) at x = 0 is given by 43(09~z)
=
at x = - b gives, for n = 0,
C T, exp(iXy)S,,,(h3,z)
=
C 4.n
4.”
in a similar
way. The potential
B3,Jnexp(b) cosa,,Az+h,) cost
h
3,n
3
B 3.n
(35)
in which sin(2a3,,h3) 2a3.A
1
(36)
Following the procedure similar to (30) through (32) and using the pressure continuity x = 0, we obtain for n = 0
at
(37) The velocity continuity fJoTo
=
at x = 0 with the use of h2( 0) = h3 and (Y*,~= LY~,~ at x = 0 gives
-y
Finally the no-flux condition
(38) at x = B gives
192
K.D. Sub, W.S. Park/Coastal
Engineering 26 (I 995) 177-193
(39)
P3,0T0ev ( - A,$) = 0 The complex-valued amplitude To inside the wave chamber both forward-propagating and backscattered waves, i.e.,
(i.e., Region 3) includes
in which T,, and T,, denote the amplitude of a forward-propagating wave, respectively. The corresponding value of &, is given by
wave and a backscattered
&,(=
ik,cos&,
P3,& = - ik,cos&
After substituting (40) and (41) into (37) - (39), the boundary conditions x = 0 for the progressive waves ( IZ= 0) can be summarized as follows:
(41) at x = - b and
c&-b)=l+K, i( 1 -K,) k,cos& =
dQ%( -b) dx
T,+T,=&,(O)+ p3,e(T,,-T,) TGXP( -P&U
(42) = -v = TO,exp(P&U
References Bennett, G.S., McIver, P. and Smallman, J.V., 1992. A mathematical model of a slotted wavescreen breakwater. Coastal Eng., 18: 23 l-249. Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. 1988. Spectral Methods in Fluid Dynamics. SpringerVerlag, New York, 557 pp. Fugazza, M. and Natale, L., 1992. Hydraulic design of perforated breakwaters. J. Waterw. Port Coastal Ocean Eng., 118: 1-14. Goda, Y., 1985. Random Seas and Design of Maritime Structures. University of Tokyo Press, 323 pp. Goda, Y., 1994. A plea for engineering-minded research efforts in harbor and coastal engineering. In: Proc. HydroPort’94. Port and Harbour Research Institute, Yokosuka, Vol. 1, pp. 1-21. Hattori, M., 1972. Transmission of waves through perforated wall. Coastal Eng. Jpn., 15: 69-79. Ijima, T., Okuzono, H. and Ushifusa, Y., 1978. The reflection coefficients of permeable quaywall with reservoir against obliquely incident waves. Rep. Coll. Eng., Kyushu Univ., 5 1: 245-250 (in Japanese). Jarlan, G.E., 1961. A perforated vertical wall breakwater. Dock Harbour Auth., X11(486): 394-398. Kakuno, S., Oda, K. and Liu, P.L.-F., 1992. Scattering of water waves by vertical cylinders with a backwall. In: Proc. 23rd Coastal Eng. Conf., Venice, Vol. 2, pp. 1258-1271. Kondo, H., 1979. Analysis of breakwaters having two porous walls, In: Proc. Coastal Structures ‘79, Vol. 2, pp. 962-977. Lee, D.S., Park, W.S., Suh, K.D. and Oh, Y.M., 1994. Hydraulic experiments for basic design of offshore breakwater for protecting artificial island. In: Proc. Hydro-Port’94. Port and Harbour Research Institute, Yokosuka, Vol. 1, pp. 563-580. Marks, M. and Jarlan, G.E., 1968. Experimental study on a fixed perforated breakwater. In: Proc. 1 lth Coastal Eng. Conf., London, Vol. 3, pp. 1121-l 140.
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Massel, S.R., 1993. Extended refractiondiffraction equation for surface waves. Coastal Eng., 19: 97-126. McBride, M.W., Smallman, J.V. and Allsop, N.W.H., 1994. Design of harbour entrances: breakwater design and vessel safety. In: Proc. Hydro-Port’94. Port and Harbour Research Institute, Yokosuka, Vol. 1, pp. 525-541. Mei, CC. 1983. The Applied Dynamics of Ocean Surface Waves. Wiley, New York, 740 pp. Neter, J., Wasserman, W. and Kutner, M.H., 1985. Applied Linear Statistical Models, 2nd ed. Richard D. Irwin, Inc., Homewood, IL, 1127 pp. Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. 1992. Numerical Recipes in FORTRAN: the Art of Scientific Computing. Cambridge University Press, 963 pp. Tanimoto, K., Haranaka, S., Takahashi, S., Komatsu, K., Todoroki, M. and Osato, M., 1976. An experimental investigation of wave reflection, overtopping and wave forces for several types of breakwaters and sea walls. Tech. Note of Port and Harbour Res. Inst., Ministry of Transport, Japan, No. 246, 38 pp. (in Japanese, with English abstract). Tanimoto, K., Takahashi, S. and Kimura, K., 1987. Structures and hydraulic characteristics of breakwaters - the state of the art of breakwater design in Japan. Rep. of Port and Harbour Res. Inst., Japan, Vol. 26, No. 5, 55 PP. Terret, F.L., Osorio, J.D.C. and Lean, G.H., 1968. Model studies of a perforated breakwater. Eng. Conf., London, Vol. 3, pp. 1104-l 120.
In: Proc. 1 lth Coastal