A study of spar buoy floating breakwater - Angelfire

Ocean Engineering 31 (2004) 43–60 www.elsevier.com/locate/oceaneng

A study of spar buoy floating breakwater Nai-Kuang Liang , 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

Corresponding author. Tel.: +886-2-236-92-034; fax: +886-2-239-25-294. E-mail address: [email protected] (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.


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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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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.


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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 Kr2 Kt2 : For a single-layer structure or fence, Mei (1983) proposed the theoretical result as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 þ 2ð4=3Þðf =khÞðHi =LÞ ð1Þ Kt ¼ ð4=3Þðf =khÞðHi =LÞ 1 Kr ¼ Kt

ð2Þ 2

where f is the dissipation coefficient and is equal to ðð1=CPÞ 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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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 RðH0 Þ

H1t cosðkx rtÞ H1t ¼ H0 TðH0 Þ 2 Hr gr2 ¼ 2 cosðkð2S xÞ þ rtÞ H2r ¼ H1t RðH1t Þ 2 gt1 ¼

H2t cosðkx rtÞ H2t ¼ H1t TðH1t Þ 2 Hr gr3 ¼ 3 cosðkðx þ 2SÞ rtÞ H3r ¼ H2r RðH2r Þ 2 gt2 ¼

H3t cosðkðx þ 2SÞ þ rtÞ 2 Hr gr4 ¼ 4 cosðkð4S xÞ þ rtÞ 2

gt3 ¼

gt4 ¼

H4t cosðkðx þ 2SÞ rtÞ 2

ð3Þ ð4Þ ð5Þ ð6Þ ð7Þ

H3t ¼ H2r TðH2r Þ

ð8Þ

H4r ¼ H3r RðH3r Þ

ð9Þ

H4t ¼ H3r TðH3r Þ

ð10Þ

... where TðHi Þ ¼

1 þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2ð4=3Þðf =khÞðHi =LÞ ð4=3Þðf =khÞðHi =LÞ

RðHi Þ ¼ 1 TðHi Þ

ð11Þ ð12Þ

The total number of reflected and transmitted waves are determined as follows: grTotal ¼ gr1 þ

1 X gt2iþ1 ;

x 0

ð13Þ

x 2S

ð14Þ

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:


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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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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 W

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 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.


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 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 TE cosh

ð34Þ

FZ5 ¼ TE sinh

ð35Þ

M5 ¼ TEL0 cosh

ð36Þ

Buoy bottom wire tension (i ¼ 6): this tension is divided into x and z components: FX6 ¼ T2X

ð37Þ

FZ6 ¼ T2Z

ð38Þ

M6 ¼ 0

ð39Þ

After rearrangement, we have the following equations: the force balance equation in the x direction: TE cosh T2X ¼ WFX

ð40Þ

The force balance equation in the z direction: TE sinh T2Z ¼ WFZ þ Wg qVg

ð41Þ

The moment balance equation: TEL0 cosh ¼ WFM

ð42Þ

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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 r1 Þdz þ qCMX AU_ ðz 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.


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


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.


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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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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 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

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.


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

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

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58


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.


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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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