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Analysis of submerged platform breakwater by Eigenfunction expansion method
Ckeon Engng, Vol. 23, No. 8. pp. 649-566. 1996 Copyright 0 1996 Elsevm Science Ltd Printed in Great Britain. All riehtr reserved 0029-8018/96 $k.OO + 0.00
Pergamon
ANALYSIS OF SUBMERGED PLATFORM BREAKWATER EIGENFUNCTION EXPANSION METHOD
BY
Hin-Fatt Cheong, N. Jothi Shankar and S. Nallayarasu Department Civil Engineering, National University of Singapore, Singapore, 05 1I, Singapore (Received 15 July 1995; accepted in jfinal,fonn 23 August 1995)
1.
INTRODUCTION
There have been many theoretical and numerical studies on the hydrodynamic behaviour of submerged platform breakwaters. A brief review of the important contributions is presented below. Heins (1948, 1950) presented a solution procedure for the diffraction of water waves by a thin semi-infinite surface horizontal plate in water of finite depth and extended it to the case of a submerged plate in water of finite depth. Using the Wiener-Hopf technique, Greene and Heins (1953) extended these solutions to the case of a thin surface plate in water of infinite depth. Stoker (1957) obtained simple expressions for the computation of the transmission and reflection coefficients by matching the horizontal velocity and pressures between adjacent regions for the submerged plate. Burke (1964) extended the WeinerHopf technique to the case of a thin submerged plate in deep water. The solution procedure is complicated due to the iterative nature of the computation of the coefficients. Though the above methods have the ability to compute transmission and reflection coefficients, the formulations do not account for the thickness of the plate and the solutions are applicable to only certain wave regimes as mentioned in Table 1. Ijima et al. (1971) presented a method using Eigenfunction Expansions of the wave dispersion relation for the case of a surface plate in water of finite depth. In this method, the fluid domain is divided into three regions namely the upstream region, the downstream region and the region under the plate. In each region, the velocity potentials are expressed in terms of eigen-function expansions of the wave dispersion relation which satisfy the Laplace equation and the respective boundary conditions. The solution for the transmission and reflection coefficients was then obtained by matching the horizontal velocity and the velocity potentials at the adjoining regions. However, Ijima’s solution does not consider the thickness of the plate. The method due to Ijima et al. (1971) was extended by Liu and Iskandarani (1989) to the case of a submerged plate of finite thickness in finite water depth. However this solution becomes unstable in shallow water conditions, The method of matched asymptotic expansions was applied to a surface plate in deep 649
Hin-Fatt Cheong et al.
650
Table 1. Summary of past investigations Researchers
Water depth
Heins (1948) Heins (1950) Greene & Heins (1953) Burke (1964) Ijima (1971) Durgin (1972)
Type of plate
Finite Finite Infinite Infinite Finite Finite
Surface and Thin Submerged and Thin Surface and Thin Submerged and Thin Surface and Thin Surface, Submerged and Thin Surface and Thin
Siew (1977)
Finite
Submerged and Thin
Patarapanich (1978)
Finite
Liu (1989)
Finite
Surface, Submerged and Thin Surface, Submerged and Thick
Leppington (1972)
Solution method used Diffraction equations Diffraction equations Wiener-Hopf technique Wiener-Hopf technique Eigenfunction expansions Experimental Studies and Airfoil theory Matched asymptotic expansions Matched asymptotic expansions Finite Element Methods Eigenfunction expansions
water by Leppington (1972) and was extended to the case of a submerged plate in shallow water by Siew and Hurley (1977). Bai (1975), presented a numerical method using finite elements for solving the linearized water-wave diffraction problem due to submerged objects. Later, Patarapanich (1978), applied this method to the case of a submerged horizontal plate in finite water depth. Nallayarasu et al. (1992), applied the same method to the case of a fixed submerged inclined plate in finite water depth. Based on the above critical review of past investigations on a submerged horizontal plate using Eigenfunction Expansion Method, it is observed that the proposed solutions are not applicable to all wave regimes covering shallow water to deep water regions. The objective of this paper is to extend the applicability of the Eigenfunction Expansion Method to the case of shallow water waves and to compare the solutions by the Eigenfunction expansion method for the complete range of water depths with those obtained using finite element method and the matched asymptotic expansion method. 2.
FORMULATION
OF THE PROBLEM
The theoretical formulation is based on the diffraction of water waves by a submerged horizontal plate which is governed by Laplace’s equation with a mixed type free surface boundary condition, a homogeneous Neumann condition on the plate and the bed, and a radiation condition at infinity as shown in Fig. 1. The time variable can be suppressed by assuming the formulation to be valid only for sinusoidal motion so that the governing equation in the fluid domain Cl and boundary conditions become: Governing eqn. in fi, V2ed = 0 Free Surface Boundary FF
ah --ko$d=O ay
Bed Boundary FBed
aty=O
Analysis of submerged platform breakwater
651
&jd(y=-d) Fig. 1. Definition diagram for diffraction of waves by a submerged body.
a& x=Oaty=-d
(3)
Radiation Boundary rR
39, _ Body Boundary Ts a@, -= an
-- 344
(5)
an
where &, is the complex diffracted potential and $I is the complex incident wave potential which is defined as,
and o and a are the circular frequency and amplitude of the incident wave respectively and n is the outward normal to the body boundary. k and k,, are wave numbers defined by the dispersion relation,
In the following sections, the solution of the above problem using EFE and the modification required in the shallow water region will be described briefly with a short note on the FEM solution and LWS by the method of matched aymptotic expansion. 3.
3.1.
SOLUTION
METHODOLOGY
Eigenfunction expansion method
In this paper, the methodology adopted by Liu and Iskandarani (1989) is used to express the velocity potential in various regions (see Fig. 2). The fluid domain is divided into four
Hin-Fatt Cheong er al.
652
Y
-1
‘L
3
d’
1 upstream
I
x
1
Above the Plate
t’
T II d
B
.
*;
4 Below the Plate
D
uownstream
;
!
Fig. 2. Definition diagram for solution by eigenfunction expansion method.
regions as shown in Fig. 3. In each region the velocity potentials which satisfy the Laplace equation and the corresponding boundary conditions are expressed in terms of the Eigenfunction Expansions of wave dispersion relation. The solution to the velocity potential in each region is obtained by matching the velocity and pressure along the common boundaries between various regions. 3.2.
Gene&
solution
The general solution (Liu and Iskandarani, 1989) for the velocity potential satisfying the required boundary conditions in each region is expressed as follows with the subscript separating the region. The first term in each expression represents the propagating mode and the term inside the infinite sum is the evanescent mode. m
coshkb +4 $1=a coshkd
(eik(x
+ I) +
R
e-jkcx +“)) +
coskJy + d> ekm(x + I) A,,, cosk,d I Wl=l -d xc-1 (8)
2
Y t
x
* d’ 1 Upstream
-
/
+ -
4 Above the Plate 2
s B
Downstream
d
3 Below the Plate . Fig. 3. Definition diagram for shallow water solution.
653
Analysis of submerged platform breakwater coshk(J $2 =
+ 6)
TeiMx-O +
coshkd
cosk,,,(y + d) Cm
m=l
cosk,,,d
e -d
03 = a
coshk’b + d’) coshk’d’
C,,cos(k’x)
O”cosk’,(y + d’) (C,cosh(kix) + Dosin(k’x) + 2 cosK’ d, m m=l -d'
(9)
+ D,sinh(kj&),
(10)
> x
where the eigenvalues k,, k’ and k’,, can be obtained from the following equations using a numerical method such as the Newton-Raphson method. co2= -gk&n(k,d)
= gk’tanh(k’d’) = -gk’,tan(k’,d’)
(11)
And under the plate, no free surface exists, and the solution for the velocity potential is (p4= a(Eo + F&l
+
c E ,cosh(hd)
+ F,sinh(h,&cosh,(y
+ d)}
??I=1 -d
+ t’),-I
> 61
(12)
where L
m3r = (d_tt_d’)’
a = -
i
iga w
1
trot
(13)
The matching conditions are applied together with the orthogonality of the eigenfunction which leads to a system of (6N+6) simultaneous equations to solve for the same number of unknowns. This system of equations are solved in complex arithmetic using the GaussJordan method to give the coefficients R,T,C,,D,,E,,F,,A,,B,,D,,E,, and F,, m =1,2,3 ..... w, where R and T are the complex reflection and transmission coefficients. The infinite series in the expressions are truncated to N terms. The convergence tests show that 16 terms are sufficient to take care of standing waves. 3.3.
Modified eigenfinction
expansion method
The expressions for the velocity potential for the regions 3 and 4 are derived from equations (10) and (12) by neglecting the evanescent modes which do not transmit mass or momentum flux into other regions through the interface which also avoids the computation of the sine/cosine hyperbolic series. However this approximation is valid only in the shallow water region in which the assumption of a uniform horizontal velocity is valid. The velocity potential in region 4 is a combination of a constant potential and a linear potential term varying in the x direction and is sufficient enough to approximate the behaviour of the potential in this region. The previous studies (Drimer et al. (1991)) have shown that the variation of velocity below the plate is linear so that the approximate solution could be used. An alternative method to check these solutions is to compute the horizontal velocity below the plate using the linear wave theory and assess its uniformity. The general solutions for regions 3 and 4 are approximated by the shallow water approximation which provide the following expressions:
654
Hin-Fatt Cheong et al. ($13=
acoF;;;,;,d’ ) x) + D,sin(kic)), (COcos(k’
$4 = a&
+ F&Z),-d
+ t’),
-d’
> x
x-cl
(14)
(15)
The matching conditions are very similar to the previous section except that the number of unknown coefficients is reduced to (4N+4) with the same number of equations. 4.
PRESSURES,
FORCES AND TRANSMISSION
CHARACTERISTICS
Once the complex velocity potential is obtained, the normalised pressures, forces and the transmission and reflection coefficients can be computed as follows: P = -
P
z =-
P
&(aI + Cad)= iWp($, + @exp(
- iof)
where @, = Qxp( - iwt) and @‘d= $+exp( - iwt) (17) The horizontal and vertical forces on the plate can be found by integrating the pressures along the body boundary TB using the trapezoidal rule. These forces are periodic in time and the amplitude of the variation can be expressed as F, =
pn&, I l-E
Fy =
pnyds I l-l?
(18)
These horizontal and vertical forces are normalised with respect to (pgat’) and (pgaB) respectively in which t’ and B are the thickness and width of the submerged horizontal plate respectively. (19) Also n, and n,, are the x and y components of the outward normal vector from the body boundary. The free surface spatial wave amplitude can be computed from IJ(J = 0) = -
la+
gat
=
-
$$@, +ad) = - T ((I, + &J exp (-iwt)
(20)
The reflection IR] and transmission 14 coefficients are computed from the free surface elevation as follows:
(21) where a- and a+ are the complex wave amplitudes at the seaward and landward boundaries respectively and a is the incident wave amplitude.
Analysis of submerged platform breakwater 5. 5.1.
NUMERICAL
655
CONSIDERATIONS
Finite element method
In the Finite Element Method (see appendix B), the ratio of element size to the incident wave length (L) is an important parameter influencing the rate of convergence. The convergence test is carried out for various d/L and B/L ratios by changing the incident wave period and it is found that the element size ratio (ELR) for d/L ratio greater than 0.1 and less than 0.5 is l/25-1/30 and for d/L ratio less than 0.1, ELR is about l/60 and for d/L ratio greater than 0.5, ELR of l/20 is sufficient for the solution to converge. The truncation of the radiation boundary at 4d is based on the convergence test, by varying the distance from 2d-5d and plotting the diffracted potential along the free surface for each of the tests. It was observed that at 4d the local disturbance due to the presence of the body was insignificant and can be ignored see also Fig. 4. 5.2.
Eigenfinction
expansion method
in the Eigenfunction Expansion Method, the infinite series in the expansions is truncated at N terms and a convergence study has been carried out by increasing N from 1 to 18. The reflection and transmission coefficients converge very fast for d/L ratios greater than 0.1 with N=16 up to a four decimal accuracy. The computation is carried out with double precision complex arithmetic in DEC ULTRIX with 32 bit processors. But for d/L ratios less than 0.1, the sinh&) and cosh&) terms in the expansion of @, create numerical instability in the computation, i.e., with B/d >2, for n >lO, the above terms exceed the limit of the machine (l.OE f308) and hence for B/d >>2, the Eigenfunction Expansion becomes numerically unstable. Hence the solutions for regions 3 and 4 are modified using only the propagating modes to compute the approximate solution. The solution for eigenvalues from equation (16) are obtained by second order Newton-Raphson iterative algorithm as these eigenvalues play an important role in the solution of the equations. The eigenvalues are calculated up to an accuracy of 16 decimal digits using the starting approximation described in Newman (1990). 6.
RESULTS
AND DISCUSSION
The Finite Element Method (FEM), the method of Long Wave Solution (LWS) and the method of Eigenfunction Expansions (EFE) are applied to the case of a submerged hori-
Fig. 4. Definition diagram for finite element method.
Hin-Fatt Cheong er al.
0.00
0.25
0.50
0.75
1.00
dn Fig. 5(a). Variation of reflection coefficient IRI with d/L (B/d =l.O, d’/d =0.3, f/d =0.025).
EFE
0.00
0.25
0.50
0
0
0
0
FEM
v
.
.
.
LWS
0.75
1.00
d/L Fig. 5(b). Variation of Transmission Coefficient 111with d/L (B/d =I .O, d’/d =0.3, t’/d =0.025).
zontal plate breakwater with B/d =I.O, d’/d =0.3 and t’/d=0.025. To provide an overall picture of the results obtained with the three methods, the variation of the reflection and transmission coefficients with d/L is shown in Figure 5a and b respectively. The FEM results are in good agreement with those of EFE but the LWS (see appendix A) shows a deviation of lo-20% in the intermediate water region for both reflection and transmission coefficients and about 80-90% in the deep water region for reflection coefficient and 5060% for transmission coefficient which shows that the LWS is not suitable in this region. Fig. 5c and d depict the variation of dynamic horizontal and vertical force coefficients with d/L. In Fig. 5c, the LWS results are in good agreement with FEM and EFE results upto dkO.125, and for d/L greater than 0.125, the variation is very large. The peak
657
Analysis of submerged platform breakwater 2.00
1.50 2
Y 1.00
0.50
0.00 0.00
Fig. 5(c). Variation of Horizontal Force Coefficient IFx( with &L (B/d =l.O, d’/d =0.3, t’/d =0.025). 1.50
1.00 5.
Y
0.50
0.00 0.00
0.25
0.50
0.75
1.00
d/L Fig. 5(d). Variation of Vertical Force Coefficient /Fyi with d/L (B/d =1 .O, d’/d =0.3, F/d =0.025).
horizontal and vertical force coefficients occur at tikO.26 and 0.25 respectively, where as the peak reflection and transmission coefficients occur in the proximity of d/L =0.25. Hence, for a particular configuration of a submerged plate, both EFE and FEM show a slight phase shift in the horizontal and vertical force coefficients, and this could be due to the singularity of the solution near the edges of the plate. 6.1.
Shallow water region
In shallow water depths, the coefficients obtained by Modified EFE are in good agreement with those obtained by FEM and LWS. Fig. 6a, b, c and d depict the variation of reflection,
658
Hin-Fatt Cheong et al.
O.ooO
0.004
0.008
0.012
0.016
0.020
B/gT2
Fig. 6(a). Variation of Reflection Coefficient (RIwith B/(gT*) (d/L =0.0016, d’/d =0.3, C/d =0.025).
E
0.008
0.012
0.016
0.020
B/gT2
Fig. 6(b). Variation of Transmission Coefficient 14 with B/(gT* (d/L =0.016, d’/d =0.3, t’/d =0.025).
transmission, horizontal force and vertical force coefficients respectively with B/gT* for ~YikO.016, d’/d=O.3 and t’/d=O.025. Even though the modified EFE is applicable in shallow water, a comparison made in the intermediate water region (d/L =0.16)shows that this modified EFE can also be used in intermediate water regions. The maximum deviation is found to be within 2% from the complete EFE solution (Fig. 6e-h). The computational CPU time for each case is noted and it is found that the Modified EFE takes about 1140th of the time taken by FEM. Hence in this region, the computation of IRI and IZj and the force coefficients IF,1 and IF,1 can be carried out using the modified EFE rather than FEM.
659
Analysis of submerged platform breakwater 2.m
Q 0
0
0
4444
1
EFE FEhi
1.875
0.000
0.004
0.008 B/gT
0.020
0.016
0.012 2
Fig. 6(c). Variation of Horizontal Force Coefficient IFxl with B/(gT*) (d/L =0.016, d’/d =0.3, f/d =0.025).
0
4
0
o
4444
O.OiiO
0.004
0.012
0.008
WgT
EFE FEhl
0.016
0.020
2
Fig. 6(d). Variation of Vertical Force Coefficient lFy/ with B/(gT’) (d/L =0.016, d’/d =0.3, f/d =0.025).
6.2.
Intemediate
water region
Fig. 7a, b, c and d depict the variation of reflection, transmission, horizontal force and vertical force coefficients respectively with B/gTz for d/L = 0.192, d’ld = 0.3, and t’/d=O.025 in the intermediate water region. It be can seen from these figures that the agreement between FEM and EFE solutions is generally good except at peaks, where the deviation is about 3-4% for the horizontal and vertical force coefficients around B/gP = 0.05. In the EFE computations for intermediate water regions the convergence is faster with 16 evanescent modes in the expansion.
660
Hin-Fatt Cheong et al.
Full EFE 0
O.OOU
0.025
Q
0
0
0.050
modified
EFE
0.075
0.100
B/gT*
Fig. 6(e). Variation of Reflection Coefficient lRI with B/(gT’) (d/L =0.16, d’/d =0.3, f/d =0.025).
0.62
0.50 O.OCKl
0.025
0.050
0.075
0.100
B/gT* Fig. 6(f). Variation of Transmission Coefficient 14 with B/(gT*) (d/L =0.16, d’/d =0.3, f/d =0.025).
6.3.
Deep water region
Fig. ga, b, c and d depict the variation of reflection, transmission, horizontal force and vertical force coefficients respectively with B/gT* for d/L = 0.6, d’ld = 0.3, and t’ld = 0.025 for deep water region, It observed from from these figures that the agreement between FEM and EFE solutions is good except at peaks, where the deviation is about 6-7% for horizontal and vertical force coefficients around B/gT* =0.05. In the EFE computations for deep water regions the convergence is faster with 16 evanescent modes in the expansion.
661
Analysis of submerged platform breakwater
0.025
0.075
0.050
WgT
0.100
2
Fig. 6(g). Variation of Horizontal Force Coefficient (Fx] with B/(gT*) (OX. =0.16, d’/d =0.3, t’/d =0.025).
0.00
1 O.OCQ
I
I
I
0.025
0.050
0.075
WgT
0.100
2
Fig. 6(h). Variation of Vertical Force Coefficient lFy( with B/(gT*) (OX =0.16, d’/d =0.3, t’/d =0.025).
7.
CONCLUSIONS
Based on the analysis of interaction of regular waves with a submerged horizontal plate, it is found that in general the EFE solution is in good agreement with the numerical solution obtained by FEM for intermediate and deep water regions and in shallow water regions, FEM results are in good agreement with the LWS. However in shallow water regions, the computation of hyperbolic series in EFJZ becomes difficult due to numerical overflow. Hence it can be deduced that El% analytical solution can only be obtained for deep and intermediate water regions and for shallow water regions, if the B/d ratio is less than 2.0. The modified EFE solution is in good agreement with FEM and LWS in shallow
662
HimFatt Cheong et al. 1.00
Em FEh4
4444
0.75
0.50
0.25
0.00 O.OCO
0.020
0.040
WgT
0.060
0.080
0.100
2
Fig. 7(a). Reflection Coefficient 1R)with B/(gT’) (d/L =0.192, d’/d =0.3, f/d =0.025)
1.00
0.88
E
635
0.62
0.50
1
O.ooO
0.020
I
I
0.060
0.040
WgT
1
0.080
0.100
2
Fig. 7(b). Variation of Transmission Coefficient 14 Ifi with B/(gT*) (d/L =0.192, d’/d =0.3, f/d =0.025).
water as well as in deep and intermediate water regions. In deep and intermediate water regions, the horizontal and vertical force coefficients computed using EFE solution have 2-3% deviation from FEM solution. In respect of computational CPU time it is found that both EFE and LWS take approximately 1/2Oth and 1/4Oth of the time taken by FEM. REFERENCES Bai, K.J. 1975. Diffraction of oblique waves by an infinite cylinder.. J. of Fluid Mechanics. 68, 513-535. Burke, J.E. 1964. Scattering of surface waves on an Infinitely Deep Fluid.. J. of Mathematical Physics 5(6), 805-819.
663
Analysis of submerged platform breakwater 2.00
1.50
1.00 z-
050
0.00 O.OlM
0.020
0.040
0.060
0.080
0.100
B/gT2 Fig. 7(c). Variation of Horizontal Force Coefficient /Fxl with B/(gT*) (d/L =0.192, d’/d =0.3, t’/d =0.025)
1.12
-
0.75
-
038
-
-2
0.00
-L% 0.000
0.020
0.040
OMll
B/gT2
0.080
0.100
-
Fig. 7(d). Variation of Vertical Force Coefficient ]Fyl with B/(gT’) (&L =0.192, d’/d =0.3, t’/d =0.025). Drimer, N., Agnon, Y. and Stiassnie, M. 1991. A simplified analytical model for a floating breakwater in water of finite depth. Jourrnal of applied ocean research 14, 33-41. Greene, T.R. and Heins, A.E. 1953. Water Waves over a channel of Infinite Depth. Quarl. of Applied Mathematics 11(2), 201-214. Heins, A.E. 1948. Water Waves over a channel of Finite depth with a Dock.. American .I. of Mathematics 70, 730-748. Heins, A.E. 1950. Water Waves over a channel of Finite depth with a submerged plane barrier.. Canadian J. of Mathematics 2, 210-222. Ijima, T., Ozaki, S., Eguchi, Y. and Kobayashi, A. 1971. Breakwater and Quay wall by Horizontal plates.. Proc. of Coastal Engg. Con& 3, 1537-1556. Leppington, F.C. 1972. On the Radiation and Scattering of Short Surface Waves.. J. of Fluid Mechanics 56(l), 101-l 19.
664
Hin-Fatt Cheong et nl. 0.50 EW QQ44
O.ooO
0.050
0.100
0.150 WgT
FEU
0.200
02.50
2
Fig. 8(a). Reflection Coefficient IRI with B/(gT’) (d/L =0.6, d’/d =0.3, t’/d =0.025). 1.05 EFE 4444
O.OCWl
0.050
0.100
0.150 WgT
FEh.9
0.200
0.250
2
Fig. 8(b). Variation of Transmission Coefficient Iq with B/(gT2) (d/L =0.6, d’/d =0.3, F/d =0.025). Liu, P.L.F and Iskandarani, M. 1989. “Hydodynamic Wave forces on submerged Horizontal plate.“, Proc. of 23rd Cow., IAHR, C51-C64. Nallayarasu, s., Cheong, H.F. and Shankar, N.J. 1992. “Wave induced dynamic pressures and forces on a submerged inclined plate by Finite Element Method.” Proc. of Znr. Conf: on Numerical methods in Engg., Singapore, 1, 113-118. Newman, J.N. 1990. Numerical solutions of the water-wave dispersion relation. Journal of applied ocean research
12(l),
14-18.
Patarapanich, M. 1978. “Wave Reflection from a Fixed Horizontal plate.“, Proc. of the Inr. conj on Water Resources Engg., Asian Institute of Technology, Bangkok, 427-446. Siew, P.F. and Hurley, D.G. 1977. Long Surface Waves Incident on a Submerged Horizontal Plate.. J. of Fluid Mechanics 83, 141-151. Stoker, J.J. 1957. Water waves, Interscience, New York.
665
Analysis of submerged platform breakwater
\
O.OCiI
0.050
0.100
0.150
0.200
0.250
B/gT2 Fig. 8(c). Variation of Horizontal Force Coefficient ]Fxl with B/(gT*) (d/L =0.6, d’/d =0.3, t’/d =0.025).
--I O.ooO
0.050
0.100
0.150
0.200
0.250
WgT2
Fig. 8(d). Variation of Vertical Force Coefficient IFyl with B/(gT*) (d/L =0.6, d’/d =0.3, t’/d =0.025). APPENDIX
A
Matched Asymptotic Expansion Method (LWS) In this method (Siew and Hurley, 1977), the problem domain in consideration is divided into four outer regions as shown in Fig. 2. The solution to the velocity potentials is determined after normalising the governing equations and boundary conditions with an appropriate length scale. Close to the edges of the plate, Schwarz-Christoffel mapping is applied to obtain the solution for the inner region. The transmission (7) and reflection (R) coefficients are obtained by expanding the inner solution to the outer regions and matching them with the outer solutions. The zero order reflection and transmission coefficients and the velocity potentials in various regions are expressed in closed form and are summarised as follows:
666
Hin-Fatt Cheong et al. al = [eik(~+ 0 + Re@+ a2 = [T&M-O
+ O],-iCOl
(22)
le(-iw)
1 a4 =[ueikx +v-ikyiot )I $=
pf+Q
emiot
[
R = $ [kBsin(k’B)-2p( 1 -cos(k’B)]
(23) (24) (25) (26) (27)
where (28)
(29)
(30)
(31) where X is defined as X = 2p(l -cos(k’B))
KBd + (1 + p2) m sin(k’B) + 2i
(32)
and p* is the submergence ratio (d’/d) and 1 is the half width of the plate (B/2). k’ and k are the wave numbers in depth d’and d respectively which can be computed using dispersion relationship given by equation (11). The complex constants R, T, P, Q, U and V can be obtained using the equations (26)-(32) by expressing them in matrix form and the velocity potential in each region can be computed using equations (22)-(25). APPENDIX
B
Finite Element Method (FEM) In this method, the fluid domain is subdivided into 8 noded isoparametric rectangular elements within which the function $ is continuous and bounded. The free surface boundary, and the body boundaries are discretized using 3 noded quadratic line element and the radiation boundary using boundary damper 3 noded quadratic line element. The problem is solved for the complex diffracted potential using variational principle resulting in a set of linear simultaneous equations. A detailed discussion on variational formulation can be found elsewhere Bai (1975). A finite element program based on linear diffraction theory is developed to solve this problem for various wave parameters and plate dimensions.
Pergamon
ANALYSIS OF SUBMERGED PLATFORM BREAKWATER EIGENFUNCTION EXPANSION METHOD
BY
Hin-Fatt Cheong, N. Jothi Shankar and S. Nallayarasu Department Civil Engineering, National University of Singapore, Singapore, 05 1I, Singapore (Received 15 July 1995; accepted in jfinal,fonn 23 August 1995)
1.
INTRODUCTION
There have been many theoretical and numerical studies on the hydrodynamic behaviour of submerged platform breakwaters. A brief review of the important contributions is presented below. Heins (1948, 1950) presented a solution procedure for the diffraction of water waves by a thin semi-infinite surface horizontal plate in water of finite depth and extended it to the case of a submerged plate in water of finite depth. Using the Wiener-Hopf technique, Greene and Heins (1953) extended these solutions to the case of a thin surface plate in water of infinite depth. Stoker (1957) obtained simple expressions for the computation of the transmission and reflection coefficients by matching the horizontal velocity and pressures between adjacent regions for the submerged plate. Burke (1964) extended the WeinerHopf technique to the case of a thin submerged plate in deep water. The solution procedure is complicated due to the iterative nature of the computation of the coefficients. Though the above methods have the ability to compute transmission and reflection coefficients, the formulations do not account for the thickness of the plate and the solutions are applicable to only certain wave regimes as mentioned in Table 1. Ijima et al. (1971) presented a method using Eigenfunction Expansions of the wave dispersion relation for the case of a surface plate in water of finite depth. In this method, the fluid domain is divided into three regions namely the upstream region, the downstream region and the region under the plate. In each region, the velocity potentials are expressed in terms of eigen-function expansions of the wave dispersion relation which satisfy the Laplace equation and the respective boundary conditions. The solution for the transmission and reflection coefficients was then obtained by matching the horizontal velocity and the velocity potentials at the adjoining regions. However, Ijima’s solution does not consider the thickness of the plate. The method due to Ijima et al. (1971) was extended by Liu and Iskandarani (1989) to the case of a submerged plate of finite thickness in finite water depth. However this solution becomes unstable in shallow water conditions, The method of matched asymptotic expansions was applied to a surface plate in deep 649
Hin-Fatt Cheong et al.
650
Table 1. Summary of past investigations Researchers
Water depth
Heins (1948) Heins (1950) Greene & Heins (1953) Burke (1964) Ijima (1971) Durgin (1972)
Type of plate
Finite Finite Infinite Infinite Finite Finite
Surface and Thin Submerged and Thin Surface and Thin Submerged and Thin Surface and Thin Surface, Submerged and Thin Surface and Thin
Siew (1977)
Finite
Submerged and Thin
Patarapanich (1978)
Finite
Liu (1989)
Finite
Surface, Submerged and Thin Surface, Submerged and Thick
Leppington (1972)
Solution method used Diffraction equations Diffraction equations Wiener-Hopf technique Wiener-Hopf technique Eigenfunction expansions Experimental Studies and Airfoil theory Matched asymptotic expansions Matched asymptotic expansions Finite Element Methods Eigenfunction expansions
water by Leppington (1972) and was extended to the case of a submerged plate in shallow water by Siew and Hurley (1977). Bai (1975), presented a numerical method using finite elements for solving the linearized water-wave diffraction problem due to submerged objects. Later, Patarapanich (1978), applied this method to the case of a submerged horizontal plate in finite water depth. Nallayarasu et al. (1992), applied the same method to the case of a fixed submerged inclined plate in finite water depth. Based on the above critical review of past investigations on a submerged horizontal plate using Eigenfunction Expansion Method, it is observed that the proposed solutions are not applicable to all wave regimes covering shallow water to deep water regions. The objective of this paper is to extend the applicability of the Eigenfunction Expansion Method to the case of shallow water waves and to compare the solutions by the Eigenfunction expansion method for the complete range of water depths with those obtained using finite element method and the matched asymptotic expansion method. 2.
FORMULATION
OF THE PROBLEM
The theoretical formulation is based on the diffraction of water waves by a submerged horizontal plate which is governed by Laplace’s equation with a mixed type free surface boundary condition, a homogeneous Neumann condition on the plate and the bed, and a radiation condition at infinity as shown in Fig. 1. The time variable can be suppressed by assuming the formulation to be valid only for sinusoidal motion so that the governing equation in the fluid domain Cl and boundary conditions become: Governing eqn. in fi, V2ed = 0 Free Surface Boundary FF
ah --ko$d=O ay
Bed Boundary FBed
aty=O
Analysis of submerged platform breakwater
651
&jd(y=-d) Fig. 1. Definition diagram for diffraction of waves by a submerged body.
a& x=Oaty=-d
(3)
Radiation Boundary rR
39, _ Body Boundary Ts a@, -= an
-- 344
(5)
an
where &, is the complex diffracted potential and $I is the complex incident wave potential which is defined as,
and o and a are the circular frequency and amplitude of the incident wave respectively and n is the outward normal to the body boundary. k and k,, are wave numbers defined by the dispersion relation,
In the following sections, the solution of the above problem using EFE and the modification required in the shallow water region will be described briefly with a short note on the FEM solution and LWS by the method of matched aymptotic expansion. 3.
3.1.
SOLUTION
METHODOLOGY
Eigenfunction expansion method
In this paper, the methodology adopted by Liu and Iskandarani (1989) is used to express the velocity potential in various regions (see Fig. 2). The fluid domain is divided into four
Hin-Fatt Cheong er al.
652
Y
-1
‘L
3
d’
1 upstream
I
x
1
Above the Plate
t’
T II d
B
.
*;
4 Below the Plate
D
uownstream
;
!
Fig. 2. Definition diagram for solution by eigenfunction expansion method.
regions as shown in Fig. 3. In each region the velocity potentials which satisfy the Laplace equation and the corresponding boundary conditions are expressed in terms of the Eigenfunction Expansions of wave dispersion relation. The solution to the velocity potential in each region is obtained by matching the velocity and pressure along the common boundaries between various regions. 3.2.
Gene&
solution
The general solution (Liu and Iskandarani, 1989) for the velocity potential satisfying the required boundary conditions in each region is expressed as follows with the subscript separating the region. The first term in each expression represents the propagating mode and the term inside the infinite sum is the evanescent mode. m
coshkb +4 $1=a coshkd
(eik(x
+ I) +
R
e-jkcx +“)) +
coskJy + d> ekm(x + I) A,,, cosk,d I Wl=l -d
2
Y t
x
* d’ 1 Upstream
-
/
+ -
4 Above the Plate 2
s B
Downstream
d
3 Below the Plate . Fig. 3. Definition diagram for shallow water solution.
653
Analysis of submerged platform breakwater coshk(J $2 =
+ 6)
TeiMx-O +
coshkd
cosk,,,(y + d) Cm
m=l
cosk,,,d
e -d
03 = a
coshk’b + d’) coshk’d’
C,,cos(k’x)
O”cosk’,(y + d’) (C,cosh(kix) + Dosin(k’x) + 2 cosK’ d, m m=l -d'
(9)
+ D,sinh(kj&),
(10)
> x
where the eigenvalues k,, k’ and k’,, can be obtained from the following equations using a numerical method such as the Newton-Raphson method. co2= -gk&n(k,d)
= gk’tanh(k’d’) = -gk’,tan(k’,d’)
(11)
And under the plate, no free surface exists, and the solution for the velocity potential is (p4= a(Eo + F&l
+
c E ,cosh(hd)
+ F,sinh(h,&cosh,(y
+ d)}
??I=1 -d
+ t’),-I
> 61
(12)
where L
m3r = (d_tt_d’)’
a = -
i
iga w
1
trot
(13)
The matching conditions are applied together with the orthogonality of the eigenfunction which leads to a system of (6N+6) simultaneous equations to solve for the same number of unknowns. This system of equations are solved in complex arithmetic using the GaussJordan method to give the coefficients R,T,C,,D,,E,,F,,A,,B,,D,,E,, and F,, m =1,2,3 ..... w, where R and T are the complex reflection and transmission coefficients. The infinite series in the expressions are truncated to N terms. The convergence tests show that 16 terms are sufficient to take care of standing waves. 3.3.
Modified eigenfinction
expansion method
The expressions for the velocity potential for the regions 3 and 4 are derived from equations (10) and (12) by neglecting the evanescent modes which do not transmit mass or momentum flux into other regions through the interface which also avoids the computation of the sine/cosine hyperbolic series. However this approximation is valid only in the shallow water region in which the assumption of a uniform horizontal velocity is valid. The velocity potential in region 4 is a combination of a constant potential and a linear potential term varying in the x direction and is sufficient enough to approximate the behaviour of the potential in this region. The previous studies (Drimer et al. (1991)) have shown that the variation of velocity below the plate is linear so that the approximate solution could be used. An alternative method to check these solutions is to compute the horizontal velocity below the plate using the linear wave theory and assess its uniformity. The general solutions for regions 3 and 4 are approximated by the shallow water approximation which provide the following expressions:
654
Hin-Fatt Cheong et al. ($13=
acoF;;;,;,d’ ) x) + D,sin(kic)), (COcos(k’
$4 = a&
+ F&Z),-d
+ t’),
-d’
> x
x-cl
(14)
(15)
The matching conditions are very similar to the previous section except that the number of unknown coefficients is reduced to (4N+4) with the same number of equations. 4.
PRESSURES,
FORCES AND TRANSMISSION
CHARACTERISTICS
Once the complex velocity potential is obtained, the normalised pressures, forces and the transmission and reflection coefficients can be computed as follows: P = -
P
z =-
P
&(aI + Cad)= iWp($, + @exp(
- iof)
where @, = Qxp( - iwt) and @‘d= $+exp( - iwt) (17) The horizontal and vertical forces on the plate can be found by integrating the pressures along the body boundary TB using the trapezoidal rule. These forces are periodic in time and the amplitude of the variation can be expressed as F, =
pn&, I l-E
Fy =
pnyds I l-l?
(18)
These horizontal and vertical forces are normalised with respect to (pgat’) and (pgaB) respectively in which t’ and B are the thickness and width of the submerged horizontal plate respectively. (19) Also n, and n,, are the x and y components of the outward normal vector from the body boundary. The free surface spatial wave amplitude can be computed from IJ(J = 0) = -
la+
gat
=
-
$$@, +ad) = - T ((I, + &J exp (-iwt)
(20)
The reflection IR] and transmission 14 coefficients are computed from the free surface elevation as follows:
(21) where a- and a+ are the complex wave amplitudes at the seaward and landward boundaries respectively and a is the incident wave amplitude.
Analysis of submerged platform breakwater 5. 5.1.
NUMERICAL
655
CONSIDERATIONS
Finite element method
In the Finite Element Method (see appendix B), the ratio of element size to the incident wave length (L) is an important parameter influencing the rate of convergence. The convergence test is carried out for various d/L and B/L ratios by changing the incident wave period and it is found that the element size ratio (ELR) for d/L ratio greater than 0.1 and less than 0.5 is l/25-1/30 and for d/L ratio less than 0.1, ELR is about l/60 and for d/L ratio greater than 0.5, ELR of l/20 is sufficient for the solution to converge. The truncation of the radiation boundary at 4d is based on the convergence test, by varying the distance from 2d-5d and plotting the diffracted potential along the free surface for each of the tests. It was observed that at 4d the local disturbance due to the presence of the body was insignificant and can be ignored see also Fig. 4. 5.2.
Eigenfinction
expansion method
in the Eigenfunction Expansion Method, the infinite series in the expansions is truncated at N terms and a convergence study has been carried out by increasing N from 1 to 18. The reflection and transmission coefficients converge very fast for d/L ratios greater than 0.1 with N=16 up to a four decimal accuracy. The computation is carried out with double precision complex arithmetic in DEC ULTRIX with 32 bit processors. But for d/L ratios less than 0.1, the sinh&) and cosh&) terms in the expansion of @, create numerical instability in the computation, i.e., with B/d >2, for n >lO, the above terms exceed the limit of the machine (l.OE f308) and hence for B/d >>2, the Eigenfunction Expansion becomes numerically unstable. Hence the solutions for regions 3 and 4 are modified using only the propagating modes to compute the approximate solution. The solution for eigenvalues from equation (16) are obtained by second order Newton-Raphson iterative algorithm as these eigenvalues play an important role in the solution of the equations. The eigenvalues are calculated up to an accuracy of 16 decimal digits using the starting approximation described in Newman (1990). 6.
RESULTS
AND DISCUSSION
The Finite Element Method (FEM), the method of Long Wave Solution (LWS) and the method of Eigenfunction Expansions (EFE) are applied to the case of a submerged hori-
Fig. 4. Definition diagram for finite element method.
Hin-Fatt Cheong er al.
0.00
0.25
0.50
0.75
1.00
dn Fig. 5(a). Variation of reflection coefficient IRI with d/L (B/d =l.O, d’/d =0.3, f/d =0.025).
EFE
0.00
0.25
0.50
0
0
0
0
FEM
v
.
.
.
LWS
0.75
1.00
d/L Fig. 5(b). Variation of Transmission Coefficient 111with d/L (B/d =I .O, d’/d =0.3, t’/d =0.025).
zontal plate breakwater with B/d =I.O, d’/d =0.3 and t’/d=0.025. To provide an overall picture of the results obtained with the three methods, the variation of the reflection and transmission coefficients with d/L is shown in Figure 5a and b respectively. The FEM results are in good agreement with those of EFE but the LWS (see appendix A) shows a deviation of lo-20% in the intermediate water region for both reflection and transmission coefficients and about 80-90% in the deep water region for reflection coefficient and 5060% for transmission coefficient which shows that the LWS is not suitable in this region. Fig. 5c and d depict the variation of dynamic horizontal and vertical force coefficients with d/L. In Fig. 5c, the LWS results are in good agreement with FEM and EFE results upto dkO.125, and for d/L greater than 0.125, the variation is very large. The peak
657
Analysis of submerged platform breakwater 2.00
1.50 2
Y 1.00
0.50
0.00 0.00
Fig. 5(c). Variation of Horizontal Force Coefficient IFx( with &L (B/d =l.O, d’/d =0.3, t’/d =0.025). 1.50
1.00 5.
Y
0.50
0.00 0.00
0.25
0.50
0.75
1.00
d/L Fig. 5(d). Variation of Vertical Force Coefficient /Fyi with d/L (B/d =1 .O, d’/d =0.3, F/d =0.025).
horizontal and vertical force coefficients occur at tikO.26 and 0.25 respectively, where as the peak reflection and transmission coefficients occur in the proximity of d/L =0.25. Hence, for a particular configuration of a submerged plate, both EFE and FEM show a slight phase shift in the horizontal and vertical force coefficients, and this could be due to the singularity of the solution near the edges of the plate. 6.1.
Shallow water region
In shallow water depths, the coefficients obtained by Modified EFE are in good agreement with those obtained by FEM and LWS. Fig. 6a, b, c and d depict the variation of reflection,
658
Hin-Fatt Cheong et al.
O.ooO
0.004
0.008
0.012
0.016
0.020
B/gT2
Fig. 6(a). Variation of Reflection Coefficient (RIwith B/(gT*) (d/L =0.0016, d’/d =0.3, C/d =0.025).
E
0.008
0.012
0.016
0.020
B/gT2
Fig. 6(b). Variation of Transmission Coefficient 14 with B/(gT* (d/L =0.016, d’/d =0.3, t’/d =0.025).
transmission, horizontal force and vertical force coefficients respectively with B/gT* for ~YikO.016, d’/d=O.3 and t’/d=O.025. Even though the modified EFE is applicable in shallow water, a comparison made in the intermediate water region (d/L =0.16)shows that this modified EFE can also be used in intermediate water regions. The maximum deviation is found to be within 2% from the complete EFE solution (Fig. 6e-h). The computational CPU time for each case is noted and it is found that the Modified EFE takes about 1140th of the time taken by FEM. Hence in this region, the computation of IRI and IZj and the force coefficients IF,1 and IF,1 can be carried out using the modified EFE rather than FEM.
659
Analysis of submerged platform breakwater 2.m
Q 0
0
0
4444
1
EFE FEhi
1.875
0.000
0.004
0.008 B/gT
0.020
0.016
0.012 2
Fig. 6(c). Variation of Horizontal Force Coefficient IFxl with B/(gT*) (d/L =0.016, d’/d =0.3, f/d =0.025).
0
4
0
o
4444
O.OiiO
0.004
0.012
0.008
WgT
EFE FEhl
0.016
0.020
2
Fig. 6(d). Variation of Vertical Force Coefficient lFy/ with B/(gT’) (d/L =0.016, d’/d =0.3, f/d =0.025).
6.2.
Intemediate
water region
Fig. 7a, b, c and d depict the variation of reflection, transmission, horizontal force and vertical force coefficients respectively with B/gTz for d/L = 0.192, d’ld = 0.3, and t’/d=O.025 in the intermediate water region. It be can seen from these figures that the agreement between FEM and EFE solutions is generally good except at peaks, where the deviation is about 3-4% for the horizontal and vertical force coefficients around B/gP = 0.05. In the EFE computations for intermediate water regions the convergence is faster with 16 evanescent modes in the expansion.
660
Hin-Fatt Cheong et al.
Full EFE 0
O.OOU
0.025
Q
0
0
0.050
modified
EFE
0.075
0.100
B/gT*
Fig. 6(e). Variation of Reflection Coefficient lRI with B/(gT’) (d/L =0.16, d’/d =0.3, f/d =0.025).
0.62
0.50 O.OCKl
0.025
0.050
0.075
0.100
B/gT* Fig. 6(f). Variation of Transmission Coefficient 14 with B/(gT*) (d/L =0.16, d’/d =0.3, f/d =0.025).
6.3.
Deep water region
Fig. ga, b, c and d depict the variation of reflection, transmission, horizontal force and vertical force coefficients respectively with B/gT* for d/L = 0.6, d’ld = 0.3, and t’ld = 0.025 for deep water region, It observed from from these figures that the agreement between FEM and EFE solutions is good except at peaks, where the deviation is about 6-7% for horizontal and vertical force coefficients around B/gT* =0.05. In the EFE computations for deep water regions the convergence is faster with 16 evanescent modes in the expansion.
661
Analysis of submerged platform breakwater
0.025
0.075
0.050
WgT
0.100
2
Fig. 6(g). Variation of Horizontal Force Coefficient (Fx] with B/(gT*) (OX. =0.16, d’/d =0.3, t’/d =0.025).
0.00
1 O.OCQ
I
I
I
0.025
0.050
0.075
WgT
0.100
2
Fig. 6(h). Variation of Vertical Force Coefficient lFy( with B/(gT*) (OX =0.16, d’/d =0.3, t’/d =0.025).
7.
CONCLUSIONS
Based on the analysis of interaction of regular waves with a submerged horizontal plate, it is found that in general the EFE solution is in good agreement with the numerical solution obtained by FEM for intermediate and deep water regions and in shallow water regions, FEM results are in good agreement with the LWS. However in shallow water regions, the computation of hyperbolic series in EFJZ becomes difficult due to numerical overflow. Hence it can be deduced that El% analytical solution can only be obtained for deep and intermediate water regions and for shallow water regions, if the B/d ratio is less than 2.0. The modified EFE solution is in good agreement with FEM and LWS in shallow
662
HimFatt Cheong et al. 1.00
Em FEh4
4444
0.75
0.50
0.25
0.00 O.OCO
0.020
0.040
WgT
0.060
0.080
0.100
2
Fig. 7(a). Reflection Coefficient 1R)with B/(gT’) (d/L =0.192, d’/d =0.3, f/d =0.025)
1.00
0.88
E
635
0.62
0.50
1
O.ooO
0.020
I
I
0.060
0.040
WgT
1
0.080
0.100
2
Fig. 7(b). Variation of Transmission Coefficient 14 Ifi with B/(gT*) (d/L =0.192, d’/d =0.3, f/d =0.025).
water as well as in deep and intermediate water regions. In deep and intermediate water regions, the horizontal and vertical force coefficients computed using EFE solution have 2-3% deviation from FEM solution. In respect of computational CPU time it is found that both EFE and LWS take approximately 1/2Oth and 1/4Oth of the time taken by FEM. REFERENCES Bai, K.J. 1975. Diffraction of oblique waves by an infinite cylinder.. J. of Fluid Mechanics. 68, 513-535. Burke, J.E. 1964. Scattering of surface waves on an Infinitely Deep Fluid.. J. of Mathematical Physics 5(6), 805-819.
663
Analysis of submerged platform breakwater 2.00
1.50
1.00 z-
050
0.00 O.OlM
0.020
0.040
0.060
0.080
0.100
B/gT2 Fig. 7(c). Variation of Horizontal Force Coefficient /Fxl with B/(gT*) (d/L =0.192, d’/d =0.3, t’/d =0.025)
1.12
-
0.75
-
038
-
-2
0.00
-L% 0.000
0.020
0.040
OMll
B/gT2
0.080
0.100
-
Fig. 7(d). Variation of Vertical Force Coefficient ]Fyl with B/(gT’) (&L =0.192, d’/d =0.3, t’/d =0.025). Drimer, N., Agnon, Y. and Stiassnie, M. 1991. A simplified analytical model for a floating breakwater in water of finite depth. Jourrnal of applied ocean research 14, 33-41. Greene, T.R. and Heins, A.E. 1953. Water Waves over a channel of Infinite Depth. Quarl. of Applied Mathematics 11(2), 201-214. Heins, A.E. 1948. Water Waves over a channel of Finite depth with a Dock.. American .I. of Mathematics 70, 730-748. Heins, A.E. 1950. Water Waves over a channel of Finite depth with a submerged plane barrier.. Canadian J. of Mathematics 2, 210-222. Ijima, T., Ozaki, S., Eguchi, Y. and Kobayashi, A. 1971. Breakwater and Quay wall by Horizontal plates.. Proc. of Coastal Engg. Con& 3, 1537-1556. Leppington, F.C. 1972. On the Radiation and Scattering of Short Surface Waves.. J. of Fluid Mechanics 56(l), 101-l 19.
664
Hin-Fatt Cheong et nl. 0.50 EW QQ44
O.ooO
0.050
0.100
0.150 WgT
FEU
0.200
02.50
2
Fig. 8(a). Reflection Coefficient IRI with B/(gT’) (d/L =0.6, d’/d =0.3, t’/d =0.025). 1.05 EFE 4444
O.OCWl
0.050
0.100
0.150 WgT
FEh.9
0.200
0.250
2
Fig. 8(b). Variation of Transmission Coefficient Iq with B/(gT2) (d/L =0.6, d’/d =0.3, F/d =0.025). Liu, P.L.F and Iskandarani, M. 1989. “Hydodynamic Wave forces on submerged Horizontal plate.“, Proc. of 23rd Cow., IAHR, C51-C64. Nallayarasu, s., Cheong, H.F. and Shankar, N.J. 1992. “Wave induced dynamic pressures and forces on a submerged inclined plate by Finite Element Method.” Proc. of Znr. Conf: on Numerical methods in Engg., Singapore, 1, 113-118. Newman, J.N. 1990. Numerical solutions of the water-wave dispersion relation. Journal of applied ocean research
12(l),
14-18.
Patarapanich, M. 1978. “Wave Reflection from a Fixed Horizontal plate.“, Proc. of the Inr. conj on Water Resources Engg., Asian Institute of Technology, Bangkok, 427-446. Siew, P.F. and Hurley, D.G. 1977. Long Surface Waves Incident on a Submerged Horizontal Plate.. J. of Fluid Mechanics 83, 141-151. Stoker, J.J. 1957. Water waves, Interscience, New York.
665
Analysis of submerged platform breakwater
\
O.OCiI
0.050
0.100
0.150
0.200
0.250
B/gT2 Fig. 8(c). Variation of Horizontal Force Coefficient ]Fxl with B/(gT*) (d/L =0.6, d’/d =0.3, t’/d =0.025).
--I O.ooO
0.050
0.100
0.150
0.200
0.250
WgT2
Fig. 8(d). Variation of Vertical Force Coefficient IFyl with B/(gT*) (d/L =0.6, d’/d =0.3, t’/d =0.025). APPENDIX
A
Matched Asymptotic Expansion Method (LWS) In this method (Siew and Hurley, 1977), the problem domain in consideration is divided into four outer regions as shown in Fig. 2. The solution to the velocity potentials is determined after normalising the governing equations and boundary conditions with an appropriate length scale. Close to the edges of the plate, Schwarz-Christoffel mapping is applied to obtain the solution for the inner region. The transmission (7) and reflection (R) coefficients are obtained by expanding the inner solution to the outer regions and matching them with the outer solutions. The zero order reflection and transmission coefficients and the velocity potentials in various regions are expressed in closed form and are summarised as follows:
666
Hin-Fatt Cheong et al. al = [eik(~+ 0 + Re@+ a2 = [T&M-O
+ O],-iCOl
(22)
le(-iw)
1 a4 =[ueikx +v-ikyiot )I $=
pf+Q
emiot
[
R = $ [kBsin(k’B)-2p( 1 -cos(k’B)]
(23) (24) (25) (26) (27)
where (28)
(29)
(30)
(31) where X is defined as X = 2p(l -cos(k’B))
KBd + (1 + p2) m sin(k’B) + 2i
(32)
and p* is the submergence ratio (d’/d) and 1 is the half width of the plate (B/2). k’ and k are the wave numbers in depth d’and d respectively which can be computed using dispersion relationship given by equation (11). The complex constants R, T, P, Q, U and V can be obtained using the equations (26)-(32) by expressing them in matrix form and the velocity potential in each region can be computed using equations (22)-(25). APPENDIX
B
Finite Element Method (FEM) In this method, the fluid domain is subdivided into 8 noded isoparametric rectangular elements within which the function $ is continuous and bounded. The free surface boundary, and the body boundaries are discretized using 3 noded quadratic line element and the radiation boundary using boundary damper 3 noded quadratic line element. The problem is solved for the complex diffracted potential using variational principle resulting in a set of linear simultaneous equations. A detailed discussion on variational formulation can be found elsewhere Bai (1975). A finite element program based on linear diffraction theory is developed to solve this problem for various wave parameters and plate dimensions.