Floating Breakwater Response to Waves Action Using

Floating Breakwater Response to Waves Action Using a Boussinesq Model Coupled with a 2DV Elliptic Solver E. V. Koutandos1; Th. V. Karambas2; and C. G. Koutitas3 Abstract: The hydrodynamic behavior of fixed and heave motion floating breakwaters is studied in the present paper, using a finitedifference, mathematical model based on the Boussinesq type equations. The flow under the floating breakwater is treated separately as confined flow. The pressure field beneath the floating structure is determined by solving implicitly the Laplace equation for the potential ⌽ of the confined flow using the appropriate boundary conditions. The dynamic equation of heave motion is solved with the consequent adjustments of the continuity equation in the case of a heave motion floating breakwater. Numerical results, concerning the efficiency of fixed and heave motion floating breakwaters, are compared to experimental results satisfactorily. The ability of the numerical model to predict the pressure field beneath the floating structure and the vertical force acting on it is thoroughly examined by making comparisons of the numerical results with large-scale experimental data. The experiments were conducted in the CIEM flume of the Catalonia Univ. of Technology, Barcelona, Spain. The final goal is to study floating breakwaters efficiency in shallow and intermediate waters. DOI: 10.1061/共ASCE兲0733-950X共2004兲130:5共243兲 CE Database subject headings: Floating structures; Breakwaters; Boussinesq equations; Mathematical models; Wave action.

Introduction The main function of a floating breakwater is to attenuate the wave action. Such a structure cannot stop all the wave action. The incident wave is partially transmitted, partially reflected, and partially dissipated. Energy is dissipated due to damping and friction and through the generation of eddies at the edges of the breakwater. Due to wave energy the breakwater can be put in motion and a radiated wave is produced, which is propagated in both directions: offshore and onshore. The movement of the breakwater is specified in terms of the anchoring which defines the degrees of freedom of the breakwater. There are a number of performance studies dealing with the hydrodynamic problem of floating breakwaters in deep and intermediate water depth. Linear models and analytical solutions have been developed, which describe the full hydrodynamic problem by Hwang et al. 共1986兲, Williams et al. 共1991兲, Drimer et al. 共1992兲, Bhatta and Rahman 共1993兲, Isaacson and Bhat 共1998兲, Williams et al. 共2000兲, and Kriezi et al. 共2001兲. A coupled solution for diffraction and body movement is proposed in order to eliminate the error introduced by the linear approach of the problem 共Isaacson 1982a,b; Fugazza and Natale 1988; and Gottlieb 1

Dept. of Civil Engineering, Division of Hydraulics and Environmental Engineering, Aristotle Univ. of Thessaloniki, Thessaloniki 54006, Greece. 2 Dept. of Marine Sciences, Univ. of the Aegean, Mytilene, 81100, Greece 共corresponding author兲. E-mail: [email protected] 3 Dept. of Civil Engineering, Division of Hydraulics and Environmental Engineering, Aristotle Univ. of Thessaloniki, Thessaloniki 54006, Greece. Note. Discussion open until February 1, 2005. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on December 19, 2001; approved on July 11, 2003. This paper is part of the Journal of Waterway, Port, Coastal, and Ocean Engineering, Vol. 130, No. 5, September 1, 2004. ©ASCE, ISSN 0733950X/2004/5-243–255/$18.00.

and Yim 1995兲. A limited number of studies have dealt with the interaction of the floating body with oblique waves 共Isaacson and Bhat 1998; Sannasiraj et al. 1998兲. The current behind the floating structure has been also studied by Isaacson and Cheung 共1993兲, while overtopping has been studied by Isaacson 共1982a,b兲. Different models have been studied which calculate the forces and the mooring system of a floating breakwater such as Niwinski and Isaacson 共1983兲, Yamamoto et al. 共1982兲, Yamamoto 共1982兲, Nossen et al. 共1991兲, Isaacson and Bhat 共1994兲, and Yoon et al. 共1994兲. However a limited number of experimental studies exist: Sutko and Haden 共1974兲, Tolba 共1998兲, Isaacson and Bhat 共1998兲, and Christian 共2000兲. In the present work a finite difference depth-averaged wave propagation model is developed coupled with a 2DV model for the determination of the pressure field beneath the floating structure in order to investigate the efficiency and the hydrodynamic behavior of fixed-heave motion floating breakwaters.

Basic Hydrodynamic Assumptions The scope of the present paper is to study the dynamic behavior of floating breakwaters, to reveal the magnitude of the forces acting on the structure vertically, and to estimate their efficiency 共i.e., transmission coefficients兲 for waves in shallow and intermediate water depth mobilizing water masses to depths on the order of some meters. The basic hydrodynamic assumptions for the model formulation are: 1. The wave propagation model is based on the Boussinesq type equations. The model is applied in shallow and intermediate waters for which h/L⬍0.5. The floating breakwater by its movement produces radiated waves that have frequencies in the same order as the resonant frequencies of the floating structure 共period should be around 1–3 s兲. Thus a dispersive wave model is required in order to describe both low and high frequency waves and their interaction.

JOURNAL OF WATERWAY, PORT, COASTAL AND OCEAN ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2004 / 243

2.

The water depth assumed in the region of the floating breakwater is the equivalent water depth producing the same specific discharge as the investigated wave in a shallow or intermediate water depth environment due to the differentiation in the horizontal velocity profile, in order to determine the mobilization of the water mass beneath the breakwater. By equating the depth average horizontal velocity from linear wave theory with that from long wave theory, it is concluded that

冉冊

2␲ L tanh h (1) 2␲ L where h eq⫽equivalent depth in intermediate water depth and L⫽wavelength. A similar expression was also adopted by Losada et al. 共1997兲 and Mendez et al. 共2001兲 for the determination of an effective water depth for wave propagation over porous structures. 3. The semi-immersed body is a rectangle that allows periodic mobilization of the water masses between its keel and the sea bed, regulated by the pressure differences on its upstream and downstream sides. Due to the above considerations the proposed numerical model is valid in the intermediate waters (h/L⭐0.5). heq⫽

Governing Equations Free Surface Wave Motion Away from Floating Structure According to the notations of Fig. 1 the mathematical model outside the area occupied by the floating breakwater is synthesized by the following equations 共Madsen et al. 1991 and Karambas 1999兲: mass continuity ⳵␨ ⳵q ⫹ ⫽0 ⳵t ⳵x

(2)

x momentum

冉

⳵u ⳵ 3u ⳵ 2u ⳵u ⳵␨ h 2 ⳵ 3 u 2 h h ⫹d ⫹u ⫹g ⫽ ⫹B x b ⳵t ⳵x ⳵x 3 ⳵x 2 ⳵t ⳵x⳵t ⳵x 2 ⳵t ⫹g

⳵ 3␨ ⳵x 3

冊

⫹2B b hd x

冉

⳵ 2u ⳵ 2␨ ⫹g 2 ⳵x⳵t ⳵x

冊

(3)

where ␨⫽surface elevation; q⫽volume flux q⫽u(h⫹␨); u⫽depth mean horizontal velocity; h⫽water depth; and d x ⫽bottom slope. The dispersion terms proportional to B b extend the applicability of the model into intermediate waters. According to Madsen and Sørensen 共1992兲, B b is set equal to 1/15, a value that gives the closest match to linear theory dispersion relation for values of h/L 0 covering the range of intermediate waters (h/L 0 ⭐0.5).

Confined Flow in Breakwater Area On the breakwater sides the continuity equation is modified by the breakwater motion and the specific discharge on the two ends of the structure is composed as follows: q⫽q u ⫹q s

(4)

Fig. 1. Basic notations and boundary conditions

where q u ⫽underflowing discharge component referring to the closed conduit flow; and q s ⫽rate of water masses dislocation due to the heave motion of the structure in the case of a heave motion floating breakwater q s ⫽⫾

dx 2 B dt 2

(5)

where x 2 ⫽vertical displacement of the floating body; and B⫽breakwater width. The underflowing discharge component q u is calculated by considering the following equation for the closed conduit between the breakwater keel and the sea bed: ⳵q u ␦P ⫽⫺d ⳵t ␳BC m

244 / JOURNAL OF WATERWAY, PORT, COASTAL AND OCEAN ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2004

(6)

where ␦ P⫽pressure difference on the breakwater sides; and C m ⫽dynamic mass coefficient 关 O(C m )⫽1.5– 2 兴 according to experimental evidence of Fugazza and Natale 共1988兲 and d the difference between ‘‘efficient’’ water depth and the breakwater draught (d⫽h eq⫺dr). C m remains the basic calibration parameter influencing the dynamics of the wave masses oscillating under the breakwater and consequently effecting the radiated and transmitted part of the wave energy. It is deduced from experimental studies by Sutko and Haden 共1974兲 that the assumption of oscillating flow, Eq. 共6兲, is valid when the ratio B/L, where L the wavelength, is between 0.1 and 0.4. For the calculation of the pressure on the breakwater sides, the following pressure distribution is adopted 共Peregrine 1972; Dingemans 1997兲: p bouss⫽␳g 共 ␨⫺z 兲 ⫹␳

冉

共 h⫹z 兲 2 ⫺ 共 h⫹␨ 兲 2 2

冊冉冊 ⳵ 2u ⳵x ⳵t

(7)

Eq. 共7兲 is a well known pressure distribution by the Boussinesq theory which has been derived directly from the depth integrated vertical momentum equation without the assumption of progressive waves, and consequently is valid both for progressive and standing waves. The specific pressure distribution in the edges of the floating structure is partially violated mainly due to vortices in the upstream submerged horizontal part of the breakwater. Reflection and partially standing waves in the region of the floating structure are taken into account in the numerical model through the final synthesis of the free surface elevation in the edge of the breakwater. The degree of violation of the specific hypothesis is revealed in the next sections where a thorough investigation of the pressure acting on the structure is presented. Numerical results are compared against large scale experimental results conducted in the CIEM flume of the Catalonia Univ. of Technology, Barcelona, Spain. The validation of the numerical model will present its ability to predict the pressure field beneath the structure and the vertical force acting on it. For the heave motion model the ability of the numerical model to predict the dynamics of the structure is tested. The prediction of the dynamics of the structure is linked to the precise prediction of the phase angle between the standing wave in the front part of the breakwater and the motion of the structure.

Pressure Field Determination in Floating Breakwaters Region The pressure field beneath the floating structure is determined solving the Laplace equation for the potential of the flow ⌽, in a 2DV, space variable computational field with time variable boundary conditions ⵜ 2 ⌽⫽0

(8)

The following boundary conditions are applied: 1. At the sea bed 关Newmann type boundary condition 共Fugazza and Natale 1988兲兴

2.

Fig. 2. Variation of C t and C r against B/L. Restrained model (dr/h⫽1/5, B/h⫽1/2).

d⌽ ⫽0 (9) dz At the right and the left boundary condition of the 2DV computational region the potential is a function of time and depth as the potential of the flow is distributed in the specific boundaries in every time step (10) ⌽⫽f 共z,t兲 where the distribution of the potential of the flow in depth follows the following expression:

3.

pbouss ⳵⌽ ⫹gz⫹ ⫽0 (11) ⳵t ␳ in which p bouss⫽pressure distribution as predicted by the Boussinesq theory 关Eq. 共7兲兴. At the top boundary condition which coincides with the floating breakwaters keel for the case of fixed structure the derivative of ⌽ normal to the boundary is zero. For the heave motion floating breakwater case the derivative of ⌽


Fig. 3. Variation of C t and C r against B/L. Restrained model (dr/h⫽1/4, B/h⫽1/2).

normal to the boundary equals the vertical velocity of the structure 关Newmann type b.c. 共Fugazza and Natale 1988兲兴. d⌽ ⫽0 dz d⌽ dx 2 ⫽ dz dt

共 fixed structure兲共 heave motion兲

(12) (13)

Fig. 4. Variation of C t and C r against B/L. Heave motion model (dr/h⫽1/5, B/h⫽1/2).

The computational 2DV field and the boundary conditions are shown in Fig. 1. After the solution in the 2DV model has converged in each time step, we can dissociate the pressure term and finally calculate the pressure in every computational node using the linearized Bernoulli’s equation ⳵⌽ p ⫹gz⫹ ⫽0 ⳵t ␳


(14)

The integration of the pressure beneath the floating structure provides the buoyancy acting on the floating breakwater.

Dynamic Equation for Vertical Motion In the case of a heave motion floating breakwater the dynamic equation for the vertical motion of the breakwater is solved, in order to calculate the vertical displacement x 2 . The vertical motion is described by Newton’s law in the z direction m

d 2x 2 dt 2

⫽B u ⫺R st⫺K s ⫺W

(15)

where B u ⫽buoyancy acting on the floating structure which is known from the 2DV model; K s ⫽equivalent vertical spring coefficient in case of vertically restrained motion; W⫽weight of the structure; and R st⫽resistance force due to the vertical motion of the structure in the water R st⫽C d ␳S

冏冏冒

dx 2 dx 2 dt dt

2

(16)

where C d ⫽2.

Numerical Solution The set of differential equations 共2兲 and 共3兲 describe the wave field around the floating breakwater and they have to be combined with Eq. 共6兲共modified momentum equation in the floating breakwater region兲 in order to provide Eqs. 共8兲–共14兲共2DV model兲 with the necessary information. The numerical solution of the wave model is based on the fourth-order time predictor–corrector scheme proposed by Wei and Kirby 共1995兲共which uses a fourthorder predictor–corrector finite differences method to advance in time, and discretizes first-order spatial derivatives to fourth-order accuracy兲. Wave generation is implemented inside the computational domain using the source function method as described by Wei et al. 共1999兲. This method employs a mass source term in the continuity equation that acts on a limited ‘‘source region’’ while it is combined with wave damping sponge layers at the boundaries. In the case of regular waves with angular frequency ␻, the expression for the source function term yields f s 共 x,t 兲 ⫽D s exp关 ⫺␤ s 共 x⫺x s 兲兴 2 sin共 ␻t 兲

(17)

where x s ⫽center of source function; ␤ s ⫽source shape coefficient; and D s ⫽magnitude of the source function. In the next step the Laplace equation is solved. The domain beneath the floating structure is discretized into grids of size dx and dz. The discretization step dz in the case of a heave motion breakwater is time and space variable in order to adjust the computational 2DV field to the physical domain which is restructured in time due to the motion of the floating structure. If ⌽ i, j is used to denote the grid-point value of the potential, standard discretization of Eq. 共8兲 using second-order finite differences yields ⌽ i⫹1,j ⫺2⌽ i, j ⫹⌽ i⫺1,j dx 2

⫹

⌽ i, j⫹1 ⫺2⌽ i, j ⫹⌽ i, j⫺1 dz 2

⫽0 (18)

The resulting system of equations is solved using an iterative method. Finally, after the estimation of the pressure field beneath the structure 关using Eq. 共14兲兴, the equation of motion 共15兲 is solved in the heave motion model case, which in return modifies the con-

Fig. 5. Variation of C t and C r against B/L. Heave motion model (dr/h⫽1/4, B/h⫽1/2).

tinuity equation in the floating breakwater region 关Eq. 共4兲兴, and adjusts the 2DV computational field.

Model Verification The verification of the numerical model is divided into two separate parts. In the first part the basics assumptions of linear 关i.e., excluding nonlinear terms in Eq. 共3兲兴 dispersive waves propaga-


Fig. 7. Pressure sensors position on floating breakwater during experiments

In the second part the degree of violation of the hypothesis made in the section ‘‘Confined Flow in Breakwater Area’’ is revealed concerning the b.c. in the 2DV module for the vertical distribution of pressure in the edges of the breakwater. Two experiments are presented for the determination of the pressure acting vertically on the breakwater, one for a fixed and one for a heave motion model. Experimental results concerning pressure time series, pressure distributions, buoyancy time series and heave motion time series are collectively tested against the corresponding numerical results.

Comparison Against Experimental Results of Tolba (1998)—Analysis The transmission and the reflection coefficients (C t ⫽H t /H i , C r ⫽H r /H i , where H t ⫽transmitted wave height; H r ⫽reflected wave height; and H i ⫽incident wave height兲, of the specific ex-

Fig. 6. Variation of heave motion height against B/L. Comparison of experimental and numerical results 关共A兲 dr/h⫽1/5, 共B兲 dr/h⫽1/4].

tion and of the closed conduit flow under the floating breakwater are collectively tested against experimental results for shallow and intermediate water depth 共Tolba 1998兲 for restraint and heave motion models. The influence of certain geometric characteristics of the structure is determined. The influence of wave characteristics, the dynamics of heave motion models which lead to resonance phenomena, is also investigated.

Fig. 8. Fixed breakwater under wave forcing during experiments (H i ⫽0.2 m, dr/h⫽1/5, T⫽2.67 s)


Fig. 9. Heave motion breakwater under wave forcing during experiments (H i ⫽0.2 m, dr/h⫽1/5, T⫽2.67 s)

perimental results which cover the area of shallow and intermediate water depth for the same B/dr ratio and the same range of B/L are very well comparable to the ones given by the present model for a range of values for C m 1.5–2.0 as it is shown in Figs. 2, 3, 4, and 5. Tolba in his experimental analysis examined the behavior of restrained models 共Figs. 2 and 3兲 and models in heave motion with no vertical restraint 共Figs. 4 and 5兲. In his work he gave an extensive report on the energy dissipation in the floating breakwater’s area. The experiments were performed in a wave flume 24 m long, 0.3 m wide, and 0.5 m high. In Figs. 2 and 3 the variation of C t and C r over B/L is presented. In these graphs the fact that the floating breakwater operates more efficiently under shorter period wave action, with greater values of the B/L ratio, is clearly shown. In the figures the analytical solution by Drimer et al. 共1992兲 is also shown. The analytical solution is not very close to the model predictions and the experimental results, since in the latter cases the energy relation is not satisfied due to viscous effects. The viscous effects depend on the geometric charac-

Fig. 10. Time series of pressure on pressure sensors 3, 4, 5, and 6 on submerged keel of breakwater. Comparison of experimental and numerical results 共restrained model, H i ⫽0.2 m, dr/h⫽1/5, T⫽2.67 s). JOURNAL OF WATERWAY, PORT, COASTAL AND OCEAN ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2004 / 249

Fig. 11. Pressure distribution on submerged keel of breakwater during wave period. Comparison of experimental and numerical results 共restrained model, H i ⫽0.2 m, dr/h⫽1/5, T⫽2.67 s).

teristics of the structure, the velocity field, and the wave period. All the above effects parametrized through C m . Comparing Figs. 2 and 3 the influence of the draught of the structure is also revealed. The minimum C t in Fig. 2 (dr/h⫽1/5) is 0.3, while in Fig. 3 (dr/h⫽1/4), the corresponding value is 0.18. By comparing Figs. 4 and 5 with Figs. 2 and 3 we can see the effect of the heave motion in the performance of the floating breakwater. In both cases the breakwater presents the same trends

in the specific graphs with a slightly better performance in the heave motion model case. The performance of the heave motion floating breakwater is directly linked to the phase difference between the partially standing wave formed in the front part of the structure and the motion of the heave motion breakwater. When the radiated waves cancel the transmitted waves the heave motion breakwater performs better than the restrained one. It is remarkable in all cases that reflection is very high and reaches values


Fig. 12. Time series of pressure on pressure sensors 3, 4, 5, and 6 on submerged keel of breakwater. Comparison of experimental and numerical results 共heave motion model, H i ⫽0.2 m, dr/h⫽1/5, T⫽2.67 s).

greater than 90% in some cases. This reveals the fact that the floating breakwater operates in a reflective manner. The evaluation of reflection and transmission coefficients is done numerically using the following methodology. First, the computational wave field without the floating breakwater is generated and the surface elevation at a point upstream of the section on which the floating breakwater is going to be put, is recorded. The computational field is generated in the next step with the breakwater put into the appropriate section. The record of the total surface elevation at the same point upstream of the section of the breakwater provides the synthesized incident-reflected wave. The subtraction of these two time series yields the reflected wave time series from which the reflection coefficient is evaluated. As in the experiments of Tolba 共1998兲, the transmission coefficient is evaluated using the time series of the transmitted wave one wavelength downstream of the structure. Almost identical results are obtained using the time series two or three wavelengths downstream of the structure. In Fig. 6 the dynamic behavior of the heave motion model is presented. In these graphs the fact that the floating breakwater

moves in phase with longer waves maximizing its motion is shown. The phase angle difference between the partially standing wave formed in the front part of the structure and the motion of the heave motion breakwater has been studied by Tolba 共1998兲 using video analysis. It was found that the floating structure moves in phase with longer waves but out of phase with shorter period waves. The model describes satisfactorily the dynamic behavior of the vertically moving structure and slight resonance phenomena that occur in the region of the eigen period of vertical oscillation of the floating structure. The maximum vertical motion of the breakwater occurs when B/L⬍0.1 when the wave period is less than 1.5 s close to the self period of heave motion of the floating breakwater. Generally it is shown that the numerical model can describe satisfactorily wave reflection and transmission in the vicinity of a fixed-heave motion floating breakwater and the dynamics of a heave motion model. This confirms the fact that the numerical model predicts the phase angle difference between the partially standing wave formed in the front part of the structure and the motion of the heave motion structure.


Fig. 13. Pressure distribution on submerged keel of breakwater during wave period. Comparison of experimental and numerical results 共heave motion model, H i ⫽0.2 m, dr/h⫽1/5, T⫽2.67 s).

Experiments in LIM-UPC The experiments were conducted in the CIEM flume of the Catalonia Univ. of Technology, Barcelona in order to investigate the pressure field beneath the floating structure and validate the numerical model results. The dimensions of the flume are 100 m length, 5 m depth, and 3 m width. The floating breakwater was placed in the horizontal part of the flume in 2 m depth. The length of the breakwater was 2 m, the height 1.5 m, and the transverse

length 2.8 m. For the heave motion model iron rails were attached to the walls of the flume in order to restrain horizontal and rotational motion of the structure while greased pneumatic wheels were attached on the structure in order to allow unrestrained vertical motion with the minimum friction 共Fig. 9兲. A position sensor was placed on the structure in order to record the vertical motion of the floating breakwater. An HR Wallingford wedge type, wavemaker was used while the experimental equipment consisted of a


number of HR Wallingford wave gages, Huba Control pressure transducers, and two-component Delft Hydraulics current meters. Four pressure sensors 共pressure Sensors 3, 4, 5, and 6兲 were attached on the horizontal submerged part of the breakwater 共Fig. 7兲. The sampling frequency during the experiments was 20 Hz. The fixed-heave motion model experiments covered the range of shallow and intermediate waters (0.04⬍h/L⬍0.30). The draught of the floating breakwater was 0.4 m (d r /h⫽1/5). The wave height was 0.2 m. The shortest wave period was 2.04 s (B/L⫽0.32) while the longest wave period was 6.23 s (B/L ⫽0.0672). In Figs. 8 and 9, respectively, the fixed-heave motion floating breakwater under wave forcing is shown. Comparison of Numerical and Experimental Analysis Two characteristic experiments are chosen for the validation of the numerical model concerning the pressure distribution acting on the breakwater and the degree of violation of the hypothesis made in the section ‘‘Pressure Field Determination in Floating Breakwater Region’’ in the 2DV module b.c. for the vertical distribution of pressure in the edges of the breakwater. One experiment was for a fixed and one was for a heave motion floating breakwater. The wave characteristics are as follows: T⫽2.67 s and H i ⫽0.2 m. The width of the breakwater B is 2 m while the draught dr is 0.4 m. In Fig. 10 the time series of the pressure on pressure Sensors 3, 4, 5, and 6 is shown for the restrained model. Comparison of experimental and numerical results is presented. The numerical results are satisfactorily close to the corresponding experiment with a slight overprediction of the pressure on the downstream part of the breakwater. In Fig. 11 the pressure distribution acting on the submerged keel of the breakwater is shown during one wave period. Comparison of experimental and numerical results is presented. We can see that the numerical model presents the same range of values for the pressure as in the experimental results. Morphologically the numerical distributions are similar to the corresponding experimental ones. The corresponding figures for the heave motion model are presented in Figs. 12 and 13. It is shown that the numerical model tends to overpredict slightly the pressure acting on the heave motion floating structure. The comparisons of the experimental and numerical distributions of the pressure 共Fig. 13兲, yields the conclusions that morphologically the numerical distributions are similar to the corresponding experiment in the same range of values. In Fig. 14 the heave motion time series is shown. A comparison of experimental and numerical results is presented. The numerical model presents a slight overestimation of the heave motion height as a result of the overprediction of the pressure acting on the floating structure. In Fig. 15 the time series of the vertical force acting on the floating breakwater is presented for the restrained model 关Fig. 15共A兲兴 and for the heave motion model 关Fig. 15共B兲兴. A comparison of experimental and numerical results is presented. Numerical results are slightly higher again as a result of the overestimation of the pressure acting on the floating structure. Comparing the two cases examined, the fixed and the heave motion model cases, it is concluded that the mean vertical force acting on the breakwater is of the same range. In the fixed breakwaters case though the hydrodynamic components of the pressure are much higher. It is shown from the distribution of the pressure on the submerged keel that the maximum pressures appear in the front submerged part of the floating breakwater in both cases examined.

Fig. 14. Time series of heave motion of floating breakwater. Comparison of experimental and numerical results 共heave motion model, H i ⫽0.2 m, dr/h⫽1/5, T⫽2.67 s).

The satisfactory performance of the numerical model reveals the fact that the degree of violation in the hypothesis made in the section ‘‘Pressure Field Determination in Floating Breakwater Region’’ in the 2DV module b.c. for the vertical distribution of pressure in the edges of the breakwater is not crucial and therefore the model is able to describe the pressure field beneath a fixed-heave motion floating breakwater.

Conclusions A finite difference numerical model is developed and tested for the investigation of the hydrodynamic behavior, the efficiency and the magnitude of the vertical forces acting on fixed-heave motion floating breakwaters. The influence of certain geometric characteristics of the structure is determined. The influence of wave characteristics, the dynamics of heave motion floating breakwaters which lead to resonance phenomena, is also investigated by comparing numerical results with experimental data. Two large scale experiments, conducted at LIM-UPC, are presented for the determination of the pressure acting vertically on the breakwater: one for a fixed and one for a heave motion model. Experimental results concerning pressure time series, pressure distributions, buoyancy time series, and heave motion time series are collectively tested against the corresponding numerical results. The numerical model performs satisfactorily concerning the performance, the dynamic behavior, and the prediction of the pressure field beneath a fixed-heave motion floating breakwater. The following conclusions can be derived: 1. The ratios B/L and dr/h are the most important parameters in the performance of the floating breakwater. The structure operates more efficiently in intermediate waters under the action of shorter period waves.


3.

4.

5.

6.

oscillation of the structure with the incident wave, in longer waves, allows the transmission of the major part of the wave energy. Resonance phenomena occur when the period of the incoming wave is close to the self period of vertical oscillation of the heave motion floating breakwater. The floating breakwater operates in a reflective manner. Reflection coefficients reach values higher than 0.9 in intermediate waters. Comparing the two cases examined, the fixed and the restrained model cases, it is concluded that the mean vertical force acting on the breakwater is of the same range. In the fixed breakwaters case though the hydrodynamic components of the pressure are much higher. It is shown from the distribution of the pressure on the submerged keel that the maximum pressures appear in the front submerged part of the floating breakwater.

Acknowledgments The writers acknowledge the financial support of EU through the program ‘‘Improving the Human Research Potential-Large Scale Infrastructures,’’ provided to the first writer for conducting experiments at LIM/UPC. The writers would also like to thank the technical personnel of LIM/UPC for their assistance and hospitality during the period of experiments.

References

Fig. 15. Time series of vertical force acting on floating breakwater. Comparison of experimental and numerical results 关共A兲 restrained model and 共B兲 heave motion model, H i ⫽0.2 m, dr/h⫽1/5, T ⫽2.67 s].

2.

The performance and the dynamic response of the heave motion floating breakwater is directly linked to the phase angle difference between the partially standing wave formed in the front part of the structure and the motion of the heave motion breakwater. The heave motion structure moves in phase with longer waves maximizing its vertical oscillation, while the opposite occurs under forcing of the shorter period waves. This mechanism is also directly linked to the performance of the heave motion breakwater since the in phase

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