Ocean Engineering 31 (2004) 43–60

www.elsevier.com/locate/oceaneng

A study of spar buoy ﬂoating breakwater

Nai-Kuang Liang Ã, Jen-Sheng Huang, Chih-Fei Li

Institute of Oceanography, National Taiwan University, Taipei, P.O. Box No. 23-13, Taipei 106,

Taiwan, ROC

Received 26 February 2002; accepted 23 May 2003

Abstract

A ﬂoating breakwater produces less environmental impact, but is easily destroyed by large

waves. In this paper, the spar buoy ﬂoating breakwater is introduced with a study on the

wave reﬂection and transmission characteristics and mooring line tension induced by the

waves. Mei (The Applied Dynamics of Ocean Surface Waves, Wiley, New York (1983) 740

p) proposed a theoretical solution for the reﬂection and transmission coeﬃcients as the wave

propagates through a one-layer slotted barrier. For a multiple-layer fence system, the analytical solution is proposed linearly. The results show that the theoretical computations agree

well with the experimental trends. For a multiple-layer fence system, the transmission coeﬃcients become maximal as the layer spacing to wavelength ratio moves to 1/2. Conversely,

the coeﬃcients become minimal, as the ratio moves to 0.3. To estimate the maximum tension of the mooring line, both numerical calculations and laboratory experiments were executed. The numerical calculation results were similar to the experimental results.

# 2003 Elsevier Ltd. All rights reserved.

Keywords: Floating breakwater; Spar buoy; Semi-closed pipe; Vena-contracta; Wave transmission; Slant

wire tension

1. Introduction

Breakwaters are used in near shore sea areas to produce wave amplitude

reduction in areas such as harbors, ﬁshing ports, marinas, power plant in and outtakes and oﬀshore cage aquaculture support bases. The traditional breakwater is

composed of caissons, rubble mounts or a hybrid. This breakwater type could

change the original near shore current system and destroy littoral sand balance and

Ã

Corresponding author. Tel.: +886-2-236-92-034; fax: +886-2-239-25-294.

E-mail address: liangnk@ccms.ntu.edu.tw (N.-K. Liang).

0029-8018/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0029-8018(03)00107-0

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N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 1. Schematic diagram of single spar buoy.

ecological system. The breakwater construction is expensive and time-consuming.

Breakwaters are also diﬃcult to remove. The traditional breakwater is required for

highly stable harbor. A ﬂoating breakwater can be employed for shore facilities

that require a lower level of stability. Many studies have been produced on ﬂoating

breakwaters (Twu and Lee, 1983; Guo et al., 1996; Murali and Mani, 1997; etc.).

The ﬂoating breakwater has low sheltering eﬃciency and maintenance diﬃculties.

The ﬂoating breakwater has therefore been seldom used.

The ﬁrst author proposed a spar buoy ﬂoating breakwater design, i.e. the Semiclosed Pipe Floating Breakwater (SPFB), registered as a new type patent in

Taiwan, ROC (Liang, 2000). A pipe made of polyethylene is closed at one end.

Holes are drilled for anchoring at the other end. The semi-closed pipe is aerated

from the open end. This pipe becomes a tautly moored spar buoy if the water is

deep enough. To suppress spar buoy pitching, two slant wires are anchored at the

top of the buoy (Fig. 1). There is pretension in the slant wire. Successive spar

buoys are installed on a line like a slotted vertical column fence (Fig. 2). More fences can be added to increase the sheltering eﬀect. A rod is used to pierce the lower

end of the pipe with used tires piled on it to enlarge the cross section and protect

the pipe (Fig. 3). Several application possibilities are suggested in Section 2 of this

work.

There are two questions that should be answered, i.e. the wave sheltering eﬀect

(or wave transmission) and the maximum tension of the slant wire during huge

waves. Theoretical and experimental studies are presented in Sections 3 and 4

(Huang, 2002; Li, 2002).

Fig. 2. Schematic diagram of spar buoy fences.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

45

Fig. 3. Schematic diagram of practical spar buoy.

2. Practical design concept and possible applications

For a small island with tourism value, such as the Tung-Sa corral reef island in

the northern South China Sea, a multiple-layered semi-closed pipe fence system

could be used to build a breakwater and established a simple harbor (Fig. 4). The

environmental impact of such a breakwater is minimal, the cost is the lowest and

the breakwater fence can be easily removed. There are many islands in the South

Paciﬁc where the sea is rather calm year round. A ﬂoating breakwater is to provide

eﬀective shelter in these areas.

A beach for swimming is an important recreation area across the world. However, many beaches are open only part of the year due to high waves. An oﬀshore

ﬂoating breakwater could increase the beach utilization rate.

Traditional breakwaters are commonly old and dangerous in large waves. Often

the harbor basin or entrance is not stable enough due to poor breakwater design.

A spar buoy ﬂoating breakwater can be installed outside of the weak part of the

old breakwater in the former case. In the latter case, such a breakwater could be

installed at a proper location that the entrance becomes calm and ships can easily

Fig. 4. Schematic diagram of a simple harbor with ﬂoating breakwater.

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N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

come into and out of the harbor. Ships will be unharmed even if they collide with

the ﬂoating breakwater.

3. Theoretical approach

As regards to the wave sheltering of the spar buoy ﬂoating breakwater, an

assumption is made for simplicity that ﬁxed vertical pipes are assumed to simulate

the aerated semi-closed pipes in studying the wave reﬂection and transmission

characteristics. There is much published literatures on vertical slotted barrier wave

shelters. Wiegel (1960, 1961) proposed the power transmission theory which states

that if the energy dissipation and reﬂection of waves transmitted through the

porous portion of the barrier is neglected, the wave transmission coeﬃcient is

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ

Ht =Hi ¼ b=B ¼ P. P is the porosity and is equal to b/B, where B is equal to

D þ b (Fig. 5). Hi is the incident wave height, and Ht is the transmitted wave

height. Hayashi et al. (Hayashi et al., 1966; Hayashi et al., 1968) proposed a transmission coeﬃcient Kt and a reﬂection coeﬃcient Kr for a closely spaced pile breakwater. The long wave assumption considers that only the horizontal water particle

current exists. A jet ﬂow in the slot and a vena-contracta could take place (Fig. 5).

Mei (Mei et al., 1974, Mei, 1983) proposed a solution for the transmission coefﬁcient under the long wave assumption (shallow water wave). Their study pointed

out that the velocity variation in the jet ﬂow could result in energy losses and the

wave steepness, porosity and relative depth are the main factors. Referring to Mei’s

theory (1983), Kriebel (1992) integrated the momentum equation in the water

depth direction and obtained a transmission coeﬃcient solution for any water

depth. The solution can approach Mei’s result for a shallow water wave. Several

researchers (Williams et al., 2000; Suh et al., 2001; Zhu and Chwang, 2001) executed serial studies on the reﬂection of an absorbing-type caisson breakwater. This

type of breakwater is a caisson with permeable thin structures that are installed

at equal spacing. As the S=L ¼ ð2n þ 1Þ=4, in which n ¼ 0,1,2,3, . . .and L is the

Fig. 5. Schematic diagram of vena-contracta through slotted pile barriers.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

47

wavelength, the reﬂection wave height is minimal. Conversely, as S=L ¼ n=2, the

reﬂection becomes maximal. There is little literature on the slant wire tension.

3.1. Wave sheltering

The reﬂection coeﬃcient is K r ¼ H r =H i and the transmission coeﬃcient is K t ¼

H t =H i where Hr is the wave height of the reﬂected wave. The energy loss coefﬁcient is

ELOSS ¼ 1 À Kr2 À Kt2 :

For a single-layer structure or fence, Mei (1983) proposed the theoretical result as:

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

À1 þ 1 þ 2ð4=3Þðf =khÞðHi =LÞ

ð1Þ

Kt ¼

ð4=3Þðf =khÞðHi =LÞ

1 À Kr ¼ Kt

ð2Þ

2

where f is the dissipation coeﬃcient and is equal to ðð1=CPÞ À 1Þ and C is the

vena-contracta coeﬃcient.

For multiple-layer fences, it is assumed that the successive incident, transmitted

and reﬂected waves are linearly superimposed (Huang, 2002). A two-layer fence

case is used as an example (Fig. 6). As the incident wave g0 passes the 1st fence,

the 1st reﬂected wave gr1 and the 1st transmitted wave gt1 are generated. As the 1st

transmitted wave passes the 2nd fence, the 2nd reﬂected wave gr2 and the 2nd transmitted wave gt2 take place. As the 2nd reﬂected wave propagates to the 1st fence,

Fig. 6. Schematic diagram of the linear superimposition of wave components in two-layer fence system.

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N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

the 3rd reﬂected wave gr3 and the 3rd transmitted wave gt3 come out, and so on.

There will be theoretically inﬁnite number of reﬂected and transmitted waves. They

are:

gr1 ¼

H1r

cosðkx þ rtÞ

2

H1r ¼ H0 Á RðH0 Þ

H1t

cosðkx À rtÞ H1t ¼ H0 Á TðH0 Þ

2

Hr

gr2 ¼ 2 cosðkð2S À xÞ þ rtÞ H2r ¼ H1t Á RðH1t Þ

2

gt1 ¼

H2t

cosðkx À rtÞ H2t ¼ H1t Á TðH1t Þ

2

Hr

gr3 ¼ 3 cosðkðx þ 2SÞ À rtÞ H3r ¼ H2r Á RðH2r Þ

2

gt2 ¼

H3t

cosðkðx þ 2SÞ þ rtÞ

2

Hr

gr4 ¼ 4 cosðkð4S À xÞ þ rtÞ

2

gt3 ¼

gt4 ¼

H4t

cosðkðx þ 2SÞ À rtÞ

2

ð3Þ

ð4Þ

ð5Þ

ð6Þ

ð7Þ

H3t ¼ H2r Á TðH2r Þ

ð8Þ

H4r ¼ H3r Á RðH3r Þ

ð9Þ

H4t ¼ H3r Á TðH3r Þ

ð10Þ

...

where

TðHi Þ ¼

À1 þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þ 2ð4=3Þðf =khÞðHi =LÞ

ð4=3Þðf =khÞðHi =LÞ

RðHi Þ ¼ 1 À TðHi Þ

ð11Þ

ð12Þ

The total number of reﬂected and transmitted waves are determined as follows:

grTotal ¼ gr1 þ

1

X

gt2iþ1 ;

0

ð13Þ

x ! 2S

ð14Þ

x

i¼1

gtTotal ¼ gt2 þ

1

X

gt2i ;

i¼2

This principle can be applied to any layered fence system.

3.2. Tension of slant wire

A two-dimensional rectangular coordinate system is assumed (Li, 2002). As

shown in Fig. 7, x is the horizontal axis and z the vertical axis. The origin is at

point a, which is the anchor point of the slant wire.

For simplicity, the assumptions are as follows:

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

49

Fig. 7. Sketch deﬁnition for wave propagation on an anchored spar buoy.

1. The wire elongation and buoy deformation are very small and can be neglected.

2. The diameter of the wire is small. The drag, inertial, buoyancy and gravity

forces are all neglected.

3. Only waves are considered and there is no current.

4. The entire system is in a static state.

5. The entire buoy is submersed in the water.

The environmental forces acting at the buoy or pipe are as shown in Fig. 8. They

are gravity, buoyancy, tension and wave forces. Because the wire cannot sustain

compressive force, the right slant wire is idle, as the wave force directs to the right,

and vice versa. The force balance equations for the positive wave force are as

Fig. 8. Sketch deﬁnition for environmental forces on a spar buoy.

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N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

follows:

For the x direction

6

X

Fxi ¼ 0

ð15Þ

i¼1

For the z direction

6

X

Fzi ¼ 0

ð16Þ

i¼1

For the moment

6

X

Mi ¼ 0

ð17Þ

i¼1

where Fxi is the force in the x direction, Fzi the force in the z direction and Mi the

moment referring to the lower end of the buoy. The sub-index i indicates the various environmental forces, introduced as follows:

Gravity (i ¼ 1):

Fx1 ¼ 0

ð18Þ

~g

Fz1 ¼ ÀW

ð19Þ

M1 ¼ 0

ð20Þ

˜ is the mass of the buoy and g the gravitational acceleration.

in which W

Buoyancy force (i ¼ 2):

Fx2 ¼ 0

ð21Þ

Fz2 ¼ qVg

ð22Þ

M2 ¼ 0

ð23Þ

where q is the water density and V the volume of the buoy.

Drag force (i ¼ 3): according to the Morison equation, we have

ð r2

1

qCDX DðUÞjUjdz

Fx3 ¼

r1 2

1

qCDZ AðW ÞjW j

2

ð

1 r2

W¼

W dz

L0 r1

ð r2

1

qCDX DðUÞjUjðz À r1 Þdz

M3 ¼

r1 2

Fz3 ¼

ð24Þ

ð25Þ

ð26Þ

ð27Þ

where r1 is the z-coordinate of the buoy lower end, r2 the z-coordinate of the buoy

upper end, D the spar buoy diameter, A is the cross-sectional area, CDX is the horizontal drag coeﬃcient, CDZ is the vertical drag coeﬃcient, U is the horizontal velo the average vertical velocity of the water particles

city of the water particles, W

and L0 the spar buoy length.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

51

Inertial force (i ¼ 4): according to the Morison equation, the inertial forces are

as follows:

ð r2

Fx4 ¼ qCMX AU_ dz

ð28Þ

r1

_

Fz4 ¼ qCMZ V W

ð r2

_ ¼ 1

_ dz

W

W

L0 r1

ð r2

M4 ¼ qCMX AU_ ðz À r1 Þdz

ð29Þ

ð30Þ

ð31Þ

r1

CMX ¼ 1 þ kMX

ð32Þ

CMZ ¼ 1 þ kMZ

ð33Þ

_ is the average vertical acceleration of the water particles, K

where W

MX the horizontal added mass coeﬃcient and KMZ the vertical added mass coeﬃcient.

Left slant wire tension (i ¼ 5): the slant tension TE is decomposed into x and z

components:

FX5 ¼ ÀTE cosh

ð34Þ

FZ5 ¼ ÀTE sinh

ð35Þ

M5 ¼ ÀTEL0 cosh

ð36Þ

Buoy bottom wire tension (i ¼ 6): this tension is divided into x and z components:

FX6 ¼ ÀT2X

ð37Þ

FZ6 ¼ ÀT2Z

ð38Þ

M6 ¼ 0

ð39Þ

After rearrangement, we have the following equations: the force balance equation in the x direction:

ÀTE cosh À T2X ¼ ÀWFX

ð40Þ

The force balance equation in the z direction:

ÀTE sinh À T2Z ¼ ÀWFZ þ Wg À qVg

ð41Þ

The moment balance equation:

ÀTEL0 cosh ¼ ÀWFM

ð42Þ

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N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

where

WFX ¼

ð r2

r1

1

qCDX DðUÞjUjdz þ

2

ð r2

qCMX AU_ dz

ð43Þ

r1

1

_

qCDZ AðW ÞjW j þ qCMZ V W

2

ð r2

ð r2

1

qCDX DðUÞjUjðz À r1 Þdz þ qCMX AU_ ðz À r1 Þdz

¼

r1 2

r1

WFZ ¼

ð44Þ

WFM

ð45Þ

Eqs. (40), (41) and (42) are the governing equations for numerically calculating

the slant wire tension TE.

4. Laboratory experiments and comparison with theories

These experiments were carried out at the wave ﬂume at the Institute of Oceanography, National Taiwan University. This ﬂume has the following dimensions: 17

m in length, 0.8 m in height and 0.6 m in width. The wave maker is piston type

with a 1:6 slope at the end of the ﬂume to eliminate the reﬂection waves. Capacitance wave meters and tension meters were used to measure the wave and tension.

The data acquisition was accomplished using a personal computer.

4.1. Wave sheltering

The layout of the wave sheltering experiment is shown in Fig. 9. The ﬁxed vertical cylinders used to simulate the spar buoy ﬂoating breakwater were made of PVC

pipe, 3.5 cm in diameter. The pipes were ﬁxed in a steel framework mounted on the

ﬂume. The pipe spacing was 0.5 cm. The porosity P was equal to 0.125 (0.5/4). In

this experiment, the water depth h was a constant, i.e. 45 cm. The model wave

Fig. 9. Schematic diagram of wave sheltering experiment setup.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

53

period was between 0.8 and 1.2 s, of which the corresponding wavelength was

between 0.99 and 2 m. The wave height ranged from 5 to 15 cm. The Goda and

Suzuki (1976) method was employed to separate the incident and reﬂected wave

components in front of the wave barrier (Huang, 2002). As mentioned in Section

3.1, the vena-contracta coeﬃcient C was an empirical constant. From the literature,

the C constant is a function of the slot shape and varied between 0.5 and 1.0. Mei

(1983) suggested that for a sharp-edge oriﬁce C ¼ 0:6 þ 0:4P2 . Hayashi et al.

(1966) compared the experimental result with the theoretical calculation by substituting C ¼ 0:9 or 1:0. According to Fig. 10, C ¼ 1:0 is a better choice. From Fig.

10, as the wave steepness Hi/L increases, Kr increases, Kt decreases and ELOSS

increases. However, Kr, Kt and ELOSS gradually approach constant, as the wave

steepness Hi/L increases. As shown in Fig. 11, the comparisons for the two-layer

fence reveal that Kr, Kt and ELOSS oscillate with the relative spacing S/L in a

sinusoidal wave. As S=L ¼ 1=4, the Kr and Kt values are minimal but ELOSS

becomes maximal. Conversely, as S=L ¼ 1=2, the Kr and Kt values become maximal but ELOSS becomes minimal. However, for the experimental Kt value, the minimum is at S=L ¼ 0:3 instead of 0.25. The results are shown in Fig. 12 for the threelayer fence system. Both for theory and experiment Kr, Kt and ELOSS also oscillate

with the relative spacing. As S=L ¼ 1=2, the Kr and Kt values become maximal but

ELOSS becomes minimal. This is the same as the two-layer fence system. However,

as S=L ¼ 1=4, the Kr, Kt and ELOSS become a little diﬀerent from that in the twolayer fence system. The Kr and Kt minimums appear at the two sides of the point

S=L ¼ 1=4 for the theoretical calculations. This phenomenon is not clear for the

Fig. 10. Comparisons of theory (solid curve for C ¼ 1:0 and dotted curve for C ¼ 0:6) and experiment

(symbols) in the one-layer fence system.

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N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 11. Comparisons of theory (solid curve) and experiment (symbols) in the two-layer fence system.

Fig. 12. Comparisons of theory (solid curve) and experiment (symbols) in the three-layer fence system.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

55

Fig. 13. Schematic diagram of slant wire tension experiment setup.

experiment data. The minimum of the Kt experimental value which is smaller than

that for the calculated value also appears at about S=L ¼ 0:3.

4.2. Maximum tension of slant wire

This experiment was carried out in the same wave ﬂume (Fig. 13). There are

three kinds of models. The 1st model is composed of a spar buoy (40 cm long,

3.5 cm in diameter and 110 g weight) with three nylon wires, of which two are

54 cm length and the other 4.3 cm (Fig. 14). The 2nd model adds a soft pipe to the

spar buoy in the 1st model to simulate used car tires in Fig. 3 (Fig. 15). The dimensions of the soft pipe are 30 cm in length, 6.3 cm in outer diameter and 5 cm in

inner diameter. The 3rd model adds a ﬁxed pipe fence used in the previous wave

sheltering experiment, of which one pipe is substituted by the 1st model buoy

(Fig. 16). The water depth in the experiment was 47.6 cm. There are four wave periods, i.e. 0.8, 1.0, 1.2, and 1.5 s, and ﬁve wave heights, i.e. 3.0, 4.0, 5.0, 6.0, and

7.0 cm, in the experiment (Li, 2002).

The slant wire tension variation for the 1st model is shown in Fig. 17. The corresponding theoretical result is shown in Fig. 18. Because only the positive half

cycle of the particle velocity is considered for the left slant wire, only the half cycle

wire tension is calculated. We were interested in the maximum tension TEmax .

Fig. 14. The 1st slant wire tension model.

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N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 15. The 2nd slant wire tension model with a soft pipe.

A non-dimensional comparison between the experimental and numerical data is

shown in Fig. 19, where B0 ¼ qVg À Wg. They coincide with one another well. The

experimental data for the 2nd model are shown in Fig. 20. The maximum tension

is larger than that in the 1st model. This is obvious due to the enlarged diameter.

In the 3rd model, the maximum tension is a little larger than that in the 1st model

(Fig. 21). The gap between adjacent pipes is 0.5 cm.

5. Discussions and conclusions

The reﬂected waves in the two-layer fence system are calculated as follows:

grTotal ¼ gr1 þ gt3 þ gt5 þ gt7 þ Á Á Á

for x

0

As S=L ¼ 1=4, the phase lag between gr1 and gt3 is p and the super-position reduces the wave. Although gt5 has a phase lag of 2p with gr1 and strengthens the superposition, it does not have an inﬂuence because gt5 is much smaller than gt3 due to its

two more reﬂections. As regards to the total transmission wave, the superposed

wave is mainly composed of gt2 and gt4 . As the phase lag is p, i.e. S=L ¼ 1=4, the

superposed wave is the minimum. However, gt4 is much smaller than gt2 . Hence, the

oscillation amplitude of Kt is smaller than that for Kr (Fig. 11). Another reason to

explain that Kr and Kt are minimal as S=L ¼ 1=4 is that two adjacent fences are

both the reﬂection wall and node point for one another. At the node point, the

horizontal velocity of the water particles in a partial standing wave is the greatest.

This results in larger energy loss at the slotted barrier.

Fig. 16. The 3rd slant wire tension model.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 17. Experimental results of 1st slant wire tension model.

Fig. 18. Theoretical results of 1st slant wire tension model.

57

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N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 19. Comparison between non-dimensional experimental and numerical data for the 1st model.

The theoretical calculation for the maximal slant wire tension was veriﬁed

by the laboratory experiment. Using the numerical calculation, the maximal

slant wire tension is inﬂuenced mainly by the pipe diameter and is almost not

Fig. 20. Theoretical results of the 2nd slant wire tension model.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

59

Fig. 21. Experimental results of the 3rd model slant wire tension.

aﬀected by the net buoyancy for the same wave condition. A prototype estimation

is as follows: water depth ¼ 10 m, wave height ¼ 7:8 m, wave period ¼ 12 s,

wave length ¼ 113 m, pipe diameter ¼ 0:5 m, pipe length ¼ 9 m, middle anchor

wire length ¼ 1 m, slant wire length ¼ 12:5 m, distance between the slant wire

anchor and the middle anchor ¼ 7:5 m, pipe and tire weight ¼ 200 kg, tire diameter

¼ 0:6 m, tire column length ¼ 8 m. The maximum slant wire tension is estimated

to be 3.3 tons. In practical use, the slant wires should be pre-tensioned so that the

buoy will be more stable and the wire connection will grind less. To lower the

demand of derricks, geotubes or geobags made of geotextile and sand can be used

for the anchorage. The following conclusions were made:

1. The proposed ‘Semi-closed Pipe Floating Breakwater’ is feasible for simple harbors for ﬁshing, cage farming, yachts, or as a supplementary breakwater for a

traditional breakwater or a beach for swimming. This breakwater is economical

and environmentally benign.

2. The transmission coeﬃcient Kt is a function of the porosity P, the relative spacing S/L and the number of layers. For a three-layer breakwater Kt can be kept

under 0.3, as P is equal to 0.125 and S=L ¼ 0:3.

3. The maximum slant wire tension is inﬂuenced mainly by the pipe diameter and

the wave, not the net buoyancy of the spar buoy. For an 8 m height wave with a

12 s period and 0.6 m pipe diameter and 10 m water depth, the maximum tension is about 4 tons. In the practical use, the wire should be pre-tensioned so

that the wire connection parts grind less. To lower the demand of derricks, geotubes or geobags made of geotextile and sand can be used for the anchorage.

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Mei, C.C., 1983. The Applied Dynamics of Ocean Surface Waves. Wiley, New York, pp. 740.

Mei, C.C., Liu, P.L.-F., Ippen, A.T., 1974. Quadratic loss and scattering of long waves. Journal Waterway, Port, Coastal and Ocean Division, ASCE 100, 217–239.

Murali, K., Mani, J.S., 1997. Performance of cage ﬂoating breakwater. Journal of Waterway, Port,

Coastal and Ocean Engineering 123 (4), 172–179.

Suh, K.D., Choi, J.C., Kim, B.H., Park, W.S., Lee, K.S., 2001. Reﬂection of irregular waves from perforated-wall caisson breakwaters. Coastal Engineering 44, 141–151.

Twu, S.W., Lee, J.S., 1983. Wave transmission in shallow water through the arrangements of net-tubes

and buoyant balls. Proceedings, The Seventh Conference on Ocean Engineering, Taipei, Taiwan,

Rep. of China, Vol. II, pp. 26-1–26-21 (in Chinese).

Wiegel, R.L., 1960. Transmission of waves past a rigid vertical thin barrier. Journal of the Waterways

and Harbors Division, ASCE WW1, 1–12.

Wiegel, R.L., 1961. Closely spaced piles as a breakwater. Dock and Harbor Authority 42 (491), 150.

Williams, A.N., Mansour, A.M., Lee, H.S., 2000. Simpliﬁed analytical solution for wave interaction

with absorbing-type caisson breakwaters. Ocean Engineering 27, 1231–1248.

Zhu, S., Chwang, A.T., 2001. Investigations on the reﬂection behaviour of a slotted seawall. Coastal

Engineering 43, 93–104.

www.elsevier.com/locate/oceaneng

A study of spar buoy ﬂoating breakwater

Nai-Kuang Liang Ã, Jen-Sheng Huang, Chih-Fei Li

Institute of Oceanography, National Taiwan University, Taipei, P.O. Box No. 23-13, Taipei 106,

Taiwan, ROC

Received 26 February 2002; accepted 23 May 2003

Abstract

A ﬂoating breakwater produces less environmental impact, but is easily destroyed by large

waves. In this paper, the spar buoy ﬂoating breakwater is introduced with a study on the

wave reﬂection and transmission characteristics and mooring line tension induced by the

waves. Mei (The Applied Dynamics of Ocean Surface Waves, Wiley, New York (1983) 740

p) proposed a theoretical solution for the reﬂection and transmission coeﬃcients as the wave

propagates through a one-layer slotted barrier. For a multiple-layer fence system, the analytical solution is proposed linearly. The results show that the theoretical computations agree

well with the experimental trends. For a multiple-layer fence system, the transmission coeﬃcients become maximal as the layer spacing to wavelength ratio moves to 1/2. Conversely,

the coeﬃcients become minimal, as the ratio moves to 0.3. To estimate the maximum tension of the mooring line, both numerical calculations and laboratory experiments were executed. The numerical calculation results were similar to the experimental results.

# 2003 Elsevier Ltd. All rights reserved.

Keywords: Floating breakwater; Spar buoy; Semi-closed pipe; Vena-contracta; Wave transmission; Slant

wire tension

1. Introduction

Breakwaters are used in near shore sea areas to produce wave amplitude

reduction in areas such as harbors, ﬁshing ports, marinas, power plant in and outtakes and oﬀshore cage aquaculture support bases. The traditional breakwater is

composed of caissons, rubble mounts or a hybrid. This breakwater type could

change the original near shore current system and destroy littoral sand balance and

Ã

Corresponding author. Tel.: +886-2-236-92-034; fax: +886-2-239-25-294.

E-mail address: liangnk@ccms.ntu.edu.tw (N.-K. Liang).

0029-8018/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0029-8018(03)00107-0

44

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 1. Schematic diagram of single spar buoy.

ecological system. The breakwater construction is expensive and time-consuming.

Breakwaters are also diﬃcult to remove. The traditional breakwater is required for

highly stable harbor. A ﬂoating breakwater can be employed for shore facilities

that require a lower level of stability. Many studies have been produced on ﬂoating

breakwaters (Twu and Lee, 1983; Guo et al., 1996; Murali and Mani, 1997; etc.).

The ﬂoating breakwater has low sheltering eﬃciency and maintenance diﬃculties.

The ﬂoating breakwater has therefore been seldom used.

The ﬁrst author proposed a spar buoy ﬂoating breakwater design, i.e. the Semiclosed Pipe Floating Breakwater (SPFB), registered as a new type patent in

Taiwan, ROC (Liang, 2000). A pipe made of polyethylene is closed at one end.

Holes are drilled for anchoring at the other end. The semi-closed pipe is aerated

from the open end. This pipe becomes a tautly moored spar buoy if the water is

deep enough. To suppress spar buoy pitching, two slant wires are anchored at the

top of the buoy (Fig. 1). There is pretension in the slant wire. Successive spar

buoys are installed on a line like a slotted vertical column fence (Fig. 2). More fences can be added to increase the sheltering eﬀect. A rod is used to pierce the lower

end of the pipe with used tires piled on it to enlarge the cross section and protect

the pipe (Fig. 3). Several application possibilities are suggested in Section 2 of this

work.

There are two questions that should be answered, i.e. the wave sheltering eﬀect

(or wave transmission) and the maximum tension of the slant wire during huge

waves. Theoretical and experimental studies are presented in Sections 3 and 4

(Huang, 2002; Li, 2002).

Fig. 2. Schematic diagram of spar buoy fences.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

45

Fig. 3. Schematic diagram of practical spar buoy.

2. Practical design concept and possible applications

For a small island with tourism value, such as the Tung-Sa corral reef island in

the northern South China Sea, a multiple-layered semi-closed pipe fence system

could be used to build a breakwater and established a simple harbor (Fig. 4). The

environmental impact of such a breakwater is minimal, the cost is the lowest and

the breakwater fence can be easily removed. There are many islands in the South

Paciﬁc where the sea is rather calm year round. A ﬂoating breakwater is to provide

eﬀective shelter in these areas.

A beach for swimming is an important recreation area across the world. However, many beaches are open only part of the year due to high waves. An oﬀshore

ﬂoating breakwater could increase the beach utilization rate.

Traditional breakwaters are commonly old and dangerous in large waves. Often

the harbor basin or entrance is not stable enough due to poor breakwater design.

A spar buoy ﬂoating breakwater can be installed outside of the weak part of the

old breakwater in the former case. In the latter case, such a breakwater could be

installed at a proper location that the entrance becomes calm and ships can easily

Fig. 4. Schematic diagram of a simple harbor with ﬂoating breakwater.

46

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

come into and out of the harbor. Ships will be unharmed even if they collide with

the ﬂoating breakwater.

3. Theoretical approach

As regards to the wave sheltering of the spar buoy ﬂoating breakwater, an

assumption is made for simplicity that ﬁxed vertical pipes are assumed to simulate

the aerated semi-closed pipes in studying the wave reﬂection and transmission

characteristics. There is much published literatures on vertical slotted barrier wave

shelters. Wiegel (1960, 1961) proposed the power transmission theory which states

that if the energy dissipation and reﬂection of waves transmitted through the

porous portion of the barrier is neglected, the wave transmission coeﬃcient is

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ

Ht =Hi ¼ b=B ¼ P. P is the porosity and is equal to b/B, where B is equal to

D þ b (Fig. 5). Hi is the incident wave height, and Ht is the transmitted wave

height. Hayashi et al. (Hayashi et al., 1966; Hayashi et al., 1968) proposed a transmission coeﬃcient Kt and a reﬂection coeﬃcient Kr for a closely spaced pile breakwater. The long wave assumption considers that only the horizontal water particle

current exists. A jet ﬂow in the slot and a vena-contracta could take place (Fig. 5).

Mei (Mei et al., 1974, Mei, 1983) proposed a solution for the transmission coefﬁcient under the long wave assumption (shallow water wave). Their study pointed

out that the velocity variation in the jet ﬂow could result in energy losses and the

wave steepness, porosity and relative depth are the main factors. Referring to Mei’s

theory (1983), Kriebel (1992) integrated the momentum equation in the water

depth direction and obtained a transmission coeﬃcient solution for any water

depth. The solution can approach Mei’s result for a shallow water wave. Several

researchers (Williams et al., 2000; Suh et al., 2001; Zhu and Chwang, 2001) executed serial studies on the reﬂection of an absorbing-type caisson breakwater. This

type of breakwater is a caisson with permeable thin structures that are installed

at equal spacing. As the S=L ¼ ð2n þ 1Þ=4, in which n ¼ 0,1,2,3, . . .and L is the

Fig. 5. Schematic diagram of vena-contracta through slotted pile barriers.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

47

wavelength, the reﬂection wave height is minimal. Conversely, as S=L ¼ n=2, the

reﬂection becomes maximal. There is little literature on the slant wire tension.

3.1. Wave sheltering

The reﬂection coeﬃcient is K r ¼ H r =H i and the transmission coeﬃcient is K t ¼

H t =H i where Hr is the wave height of the reﬂected wave. The energy loss coefﬁcient is

ELOSS ¼ 1 À Kr2 À Kt2 :

For a single-layer structure or fence, Mei (1983) proposed the theoretical result as:

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

À1 þ 1 þ 2ð4=3Þðf =khÞðHi =LÞ

ð1Þ

Kt ¼

ð4=3Þðf =khÞðHi =LÞ

1 À Kr ¼ Kt

ð2Þ

2

where f is the dissipation coeﬃcient and is equal to ðð1=CPÞ À 1Þ and C is the

vena-contracta coeﬃcient.

For multiple-layer fences, it is assumed that the successive incident, transmitted

and reﬂected waves are linearly superimposed (Huang, 2002). A two-layer fence

case is used as an example (Fig. 6). As the incident wave g0 passes the 1st fence,

the 1st reﬂected wave gr1 and the 1st transmitted wave gt1 are generated. As the 1st

transmitted wave passes the 2nd fence, the 2nd reﬂected wave gr2 and the 2nd transmitted wave gt2 take place. As the 2nd reﬂected wave propagates to the 1st fence,

Fig. 6. Schematic diagram of the linear superimposition of wave components in two-layer fence system.

48

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

the 3rd reﬂected wave gr3 and the 3rd transmitted wave gt3 come out, and so on.

There will be theoretically inﬁnite number of reﬂected and transmitted waves. They

are:

gr1 ¼

H1r

cosðkx þ rtÞ

2

H1r ¼ H0 Á RðH0 Þ

H1t

cosðkx À rtÞ H1t ¼ H0 Á TðH0 Þ

2

Hr

gr2 ¼ 2 cosðkð2S À xÞ þ rtÞ H2r ¼ H1t Á RðH1t Þ

2

gt1 ¼

H2t

cosðkx À rtÞ H2t ¼ H1t Á TðH1t Þ

2

Hr

gr3 ¼ 3 cosðkðx þ 2SÞ À rtÞ H3r ¼ H2r Á RðH2r Þ

2

gt2 ¼

H3t

cosðkðx þ 2SÞ þ rtÞ

2

Hr

gr4 ¼ 4 cosðkð4S À xÞ þ rtÞ

2

gt3 ¼

gt4 ¼

H4t

cosðkðx þ 2SÞ À rtÞ

2

ð3Þ

ð4Þ

ð5Þ

ð6Þ

ð7Þ

H3t ¼ H2r Á TðH2r Þ

ð8Þ

H4r ¼ H3r Á RðH3r Þ

ð9Þ

H4t ¼ H3r Á TðH3r Þ

ð10Þ

...

where

TðHi Þ ¼

À1 þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þ 2ð4=3Þðf =khÞðHi =LÞ

ð4=3Þðf =khÞðHi =LÞ

RðHi Þ ¼ 1 À TðHi Þ

ð11Þ

ð12Þ

The total number of reﬂected and transmitted waves are determined as follows:

grTotal ¼ gr1 þ

1

X

gt2iþ1 ;

0

ð13Þ

x ! 2S

ð14Þ

x

i¼1

gtTotal ¼ gt2 þ

1

X

gt2i ;

i¼2

This principle can be applied to any layered fence system.

3.2. Tension of slant wire

A two-dimensional rectangular coordinate system is assumed (Li, 2002). As

shown in Fig. 7, x is the horizontal axis and z the vertical axis. The origin is at

point a, which is the anchor point of the slant wire.

For simplicity, the assumptions are as follows:

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

49

Fig. 7. Sketch deﬁnition for wave propagation on an anchored spar buoy.

1. The wire elongation and buoy deformation are very small and can be neglected.

2. The diameter of the wire is small. The drag, inertial, buoyancy and gravity

forces are all neglected.

3. Only waves are considered and there is no current.

4. The entire system is in a static state.

5. The entire buoy is submersed in the water.

The environmental forces acting at the buoy or pipe are as shown in Fig. 8. They

are gravity, buoyancy, tension and wave forces. Because the wire cannot sustain

compressive force, the right slant wire is idle, as the wave force directs to the right,

and vice versa. The force balance equations for the positive wave force are as

Fig. 8. Sketch deﬁnition for environmental forces on a spar buoy.

50

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

follows:

For the x direction

6

X

Fxi ¼ 0

ð15Þ

i¼1

For the z direction

6

X

Fzi ¼ 0

ð16Þ

i¼1

For the moment

6

X

Mi ¼ 0

ð17Þ

i¼1

where Fxi is the force in the x direction, Fzi the force in the z direction and Mi the

moment referring to the lower end of the buoy. The sub-index i indicates the various environmental forces, introduced as follows:

Gravity (i ¼ 1):

Fx1 ¼ 0

ð18Þ

~g

Fz1 ¼ ÀW

ð19Þ

M1 ¼ 0

ð20Þ

˜ is the mass of the buoy and g the gravitational acceleration.

in which W

Buoyancy force (i ¼ 2):

Fx2 ¼ 0

ð21Þ

Fz2 ¼ qVg

ð22Þ

M2 ¼ 0

ð23Þ

where q is the water density and V the volume of the buoy.

Drag force (i ¼ 3): according to the Morison equation, we have

ð r2

1

qCDX DðUÞjUjdz

Fx3 ¼

r1 2

1

qCDZ AðW ÞjW j

2

ð

1 r2

W¼

W dz

L0 r1

ð r2

1

qCDX DðUÞjUjðz À r1 Þdz

M3 ¼

r1 2

Fz3 ¼

ð24Þ

ð25Þ

ð26Þ

ð27Þ

where r1 is the z-coordinate of the buoy lower end, r2 the z-coordinate of the buoy

upper end, D the spar buoy diameter, A is the cross-sectional area, CDX is the horizontal drag coeﬃcient, CDZ is the vertical drag coeﬃcient, U is the horizontal velo the average vertical velocity of the water particles

city of the water particles, W

and L0 the spar buoy length.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

51

Inertial force (i ¼ 4): according to the Morison equation, the inertial forces are

as follows:

ð r2

Fx4 ¼ qCMX AU_ dz

ð28Þ

r1

_

Fz4 ¼ qCMZ V W

ð r2

_ ¼ 1

_ dz

W

W

L0 r1

ð r2

M4 ¼ qCMX AU_ ðz À r1 Þdz

ð29Þ

ð30Þ

ð31Þ

r1

CMX ¼ 1 þ kMX

ð32Þ

CMZ ¼ 1 þ kMZ

ð33Þ

_ is the average vertical acceleration of the water particles, K

where W

MX the horizontal added mass coeﬃcient and KMZ the vertical added mass coeﬃcient.

Left slant wire tension (i ¼ 5): the slant tension TE is decomposed into x and z

components:

FX5 ¼ ÀTE cosh

ð34Þ

FZ5 ¼ ÀTE sinh

ð35Þ

M5 ¼ ÀTEL0 cosh

ð36Þ

Buoy bottom wire tension (i ¼ 6): this tension is divided into x and z components:

FX6 ¼ ÀT2X

ð37Þ

FZ6 ¼ ÀT2Z

ð38Þ

M6 ¼ 0

ð39Þ

After rearrangement, we have the following equations: the force balance equation in the x direction:

ÀTE cosh À T2X ¼ ÀWFX

ð40Þ

The force balance equation in the z direction:

ÀTE sinh À T2Z ¼ ÀWFZ þ Wg À qVg

ð41Þ

The moment balance equation:

ÀTEL0 cosh ¼ ÀWFM

ð42Þ

52

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

where

WFX ¼

ð r2

r1

1

qCDX DðUÞjUjdz þ

2

ð r2

qCMX AU_ dz

ð43Þ

r1

1

_

qCDZ AðW ÞjW j þ qCMZ V W

2

ð r2

ð r2

1

qCDX DðUÞjUjðz À r1 Þdz þ qCMX AU_ ðz À r1 Þdz

¼

r1 2

r1

WFZ ¼

ð44Þ

WFM

ð45Þ

Eqs. (40), (41) and (42) are the governing equations for numerically calculating

the slant wire tension TE.

4. Laboratory experiments and comparison with theories

These experiments were carried out at the wave ﬂume at the Institute of Oceanography, National Taiwan University. This ﬂume has the following dimensions: 17

m in length, 0.8 m in height and 0.6 m in width. The wave maker is piston type

with a 1:6 slope at the end of the ﬂume to eliminate the reﬂection waves. Capacitance wave meters and tension meters were used to measure the wave and tension.

The data acquisition was accomplished using a personal computer.

4.1. Wave sheltering

The layout of the wave sheltering experiment is shown in Fig. 9. The ﬁxed vertical cylinders used to simulate the spar buoy ﬂoating breakwater were made of PVC

pipe, 3.5 cm in diameter. The pipes were ﬁxed in a steel framework mounted on the

ﬂume. The pipe spacing was 0.5 cm. The porosity P was equal to 0.125 (0.5/4). In

this experiment, the water depth h was a constant, i.e. 45 cm. The model wave

Fig. 9. Schematic diagram of wave sheltering experiment setup.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

53

period was between 0.8 and 1.2 s, of which the corresponding wavelength was

between 0.99 and 2 m. The wave height ranged from 5 to 15 cm. The Goda and

Suzuki (1976) method was employed to separate the incident and reﬂected wave

components in front of the wave barrier (Huang, 2002). As mentioned in Section

3.1, the vena-contracta coeﬃcient C was an empirical constant. From the literature,

the C constant is a function of the slot shape and varied between 0.5 and 1.0. Mei

(1983) suggested that for a sharp-edge oriﬁce C ¼ 0:6 þ 0:4P2 . Hayashi et al.

(1966) compared the experimental result with the theoretical calculation by substituting C ¼ 0:9 or 1:0. According to Fig. 10, C ¼ 1:0 is a better choice. From Fig.

10, as the wave steepness Hi/L increases, Kr increases, Kt decreases and ELOSS

increases. However, Kr, Kt and ELOSS gradually approach constant, as the wave

steepness Hi/L increases. As shown in Fig. 11, the comparisons for the two-layer

fence reveal that Kr, Kt and ELOSS oscillate with the relative spacing S/L in a

sinusoidal wave. As S=L ¼ 1=4, the Kr and Kt values are minimal but ELOSS

becomes maximal. Conversely, as S=L ¼ 1=2, the Kr and Kt values become maximal but ELOSS becomes minimal. However, for the experimental Kt value, the minimum is at S=L ¼ 0:3 instead of 0.25. The results are shown in Fig. 12 for the threelayer fence system. Both for theory and experiment Kr, Kt and ELOSS also oscillate

with the relative spacing. As S=L ¼ 1=2, the Kr and Kt values become maximal but

ELOSS becomes minimal. This is the same as the two-layer fence system. However,

as S=L ¼ 1=4, the Kr, Kt and ELOSS become a little diﬀerent from that in the twolayer fence system. The Kr and Kt minimums appear at the two sides of the point

S=L ¼ 1=4 for the theoretical calculations. This phenomenon is not clear for the

Fig. 10. Comparisons of theory (solid curve for C ¼ 1:0 and dotted curve for C ¼ 0:6) and experiment

(symbols) in the one-layer fence system.

54

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 11. Comparisons of theory (solid curve) and experiment (symbols) in the two-layer fence system.

Fig. 12. Comparisons of theory (solid curve) and experiment (symbols) in the three-layer fence system.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

55

Fig. 13. Schematic diagram of slant wire tension experiment setup.

experiment data. The minimum of the Kt experimental value which is smaller than

that for the calculated value also appears at about S=L ¼ 0:3.

4.2. Maximum tension of slant wire

This experiment was carried out in the same wave ﬂume (Fig. 13). There are

three kinds of models. The 1st model is composed of a spar buoy (40 cm long,

3.5 cm in diameter and 110 g weight) with three nylon wires, of which two are

54 cm length and the other 4.3 cm (Fig. 14). The 2nd model adds a soft pipe to the

spar buoy in the 1st model to simulate used car tires in Fig. 3 (Fig. 15). The dimensions of the soft pipe are 30 cm in length, 6.3 cm in outer diameter and 5 cm in

inner diameter. The 3rd model adds a ﬁxed pipe fence used in the previous wave

sheltering experiment, of which one pipe is substituted by the 1st model buoy

(Fig. 16). The water depth in the experiment was 47.6 cm. There are four wave periods, i.e. 0.8, 1.0, 1.2, and 1.5 s, and ﬁve wave heights, i.e. 3.0, 4.0, 5.0, 6.0, and

7.0 cm, in the experiment (Li, 2002).

The slant wire tension variation for the 1st model is shown in Fig. 17. The corresponding theoretical result is shown in Fig. 18. Because only the positive half

cycle of the particle velocity is considered for the left slant wire, only the half cycle

wire tension is calculated. We were interested in the maximum tension TEmax .

Fig. 14. The 1st slant wire tension model.

56

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 15. The 2nd slant wire tension model with a soft pipe.

A non-dimensional comparison between the experimental and numerical data is

shown in Fig. 19, where B0 ¼ qVg À Wg. They coincide with one another well. The

experimental data for the 2nd model are shown in Fig. 20. The maximum tension

is larger than that in the 1st model. This is obvious due to the enlarged diameter.

In the 3rd model, the maximum tension is a little larger than that in the 1st model

(Fig. 21). The gap between adjacent pipes is 0.5 cm.

5. Discussions and conclusions

The reﬂected waves in the two-layer fence system are calculated as follows:

grTotal ¼ gr1 þ gt3 þ gt5 þ gt7 þ Á Á Á

for x

0

As S=L ¼ 1=4, the phase lag between gr1 and gt3 is p and the super-position reduces the wave. Although gt5 has a phase lag of 2p with gr1 and strengthens the superposition, it does not have an inﬂuence because gt5 is much smaller than gt3 due to its

two more reﬂections. As regards to the total transmission wave, the superposed

wave is mainly composed of gt2 and gt4 . As the phase lag is p, i.e. S=L ¼ 1=4, the

superposed wave is the minimum. However, gt4 is much smaller than gt2 . Hence, the

oscillation amplitude of Kt is smaller than that for Kr (Fig. 11). Another reason to

explain that Kr and Kt are minimal as S=L ¼ 1=4 is that two adjacent fences are

both the reﬂection wall and node point for one another. At the node point, the

horizontal velocity of the water particles in a partial standing wave is the greatest.

This results in larger energy loss at the slotted barrier.

Fig. 16. The 3rd slant wire tension model.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 17. Experimental results of 1st slant wire tension model.

Fig. 18. Theoretical results of 1st slant wire tension model.

57

58

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

Fig. 19. Comparison between non-dimensional experimental and numerical data for the 1st model.

The theoretical calculation for the maximal slant wire tension was veriﬁed

by the laboratory experiment. Using the numerical calculation, the maximal

slant wire tension is inﬂuenced mainly by the pipe diameter and is almost not

Fig. 20. Theoretical results of the 2nd slant wire tension model.

N.-K. Liang et al. / Ocean Engineering 31 (2004) 43–60

59

Fig. 21. Experimental results of the 3rd model slant wire tension.

aﬀected by the net buoyancy for the same wave condition. A prototype estimation

is as follows: water depth ¼ 10 m, wave height ¼ 7:8 m, wave period ¼ 12 s,

wave length ¼ 113 m, pipe diameter ¼ 0:5 m, pipe length ¼ 9 m, middle anchor

wire length ¼ 1 m, slant wire length ¼ 12:5 m, distance between the slant wire

anchor and the middle anchor ¼ 7:5 m, pipe and tire weight ¼ 200 kg, tire diameter

¼ 0:6 m, tire column length ¼ 8 m. The maximum slant wire tension is estimated

to be 3.3 tons. In practical use, the slant wires should be pre-tensioned so that the

buoy will be more stable and the wire connection will grind less. To lower the

demand of derricks, geotubes or geobags made of geotextile and sand can be used

for the anchorage. The following conclusions were made:

1. The proposed ‘Semi-closed Pipe Floating Breakwater’ is feasible for simple harbors for ﬁshing, cage farming, yachts, or as a supplementary breakwater for a

traditional breakwater or a beach for swimming. This breakwater is economical

and environmentally benign.

2. The transmission coeﬃcient Kt is a function of the porosity P, the relative spacing S/L and the number of layers. For a three-layer breakwater Kt can be kept

under 0.3, as P is equal to 0.125 and S=L ¼ 0:3.

3. The maximum slant wire tension is inﬂuenced mainly by the pipe diameter and

the wave, not the net buoyancy of the spar buoy. For an 8 m height wave with a

12 s period and 0.6 m pipe diameter and 10 m water depth, the maximum tension is about 4 tons. In the practical use, the wire should be pre-tensioned so

that the wire connection parts grind less. To lower the demand of derricks, geotubes or geobags made of geotextile and sand can be used for the anchorage.

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