Floating forest: A novel breakwater-windbreak structure against wind and wave hazards

Abstract

A novel floating breakwater-windbreak structure (floating forest) has been designed for the protection of vulnerable coastal areas from extreme wind and wave loadings during storm conditions. The modular arch-shaped concrete structure is positioned perpendicularly to the direction of the prevailing wave and wind. The structure below the water surface acts as a porous breakwater with wave scattering capability. An array of tubular columns on the sloping deck of the breakwater act as an artificial forest-type windbreak. A feasibility study involving hydrodynamic and aerodynamic analyses has been performed, focusing on its capability in reducing wave heights and wind speeds in the lee side. The study shows that the proposed 1 km long floating forest is able to shelter a lee area that stretches up to 600 m, with 40%–60% wave energy reduction and 10%–80% peak wind speed reduction.

Graphical abstract

Keywords

floating structure / breakwater / windbreak / hydrodynamic / CFD

1 Introduction

Global climate change in recent years has led to warmer oceans which in turn produce intense cyclones or hurricanes by extracting energy from the warm surface waters [1–3]. These natural hazards have caused immense damage to coastlines and coastal cities [4–6]. Breakwaters are generally constructed for reducing wave elevation on their leeward side, thus protecting the shoreline from damage caused by extreme waves, and creating an area of calm waters for safe berthing of vessels, fish farming, and floating infrastructure. Breakwaters use two main mechanisms for wave reduction: one is by reflecting the waves at their sea-facing walls, and the other is by utilization of the viscosity of the water and so to dissipate the wave energy by inducing turbulence and wave breaking.

When compared with fixed breakwaters that usually have heavy gravitational bases and are installed permanently at a preselected site, floating breakwaters are attracting more attention recently because of their mobility, cost effectiveness in deep coastal waters, relatively smaller benthic footprints and their reduced impacts on the water quality and current flow [7,8]. Further, to increase the efficiency of breakwaters and to reduce their costs, new breakwater concepts have been proposed and tested [9–12]. However, most of the floating breakwater designs only target wave energy reduction; extreme winds will apply large wind forces to any structure behind the breakwater, and may impose undesirable overturning moments on structures which increases the risk of collapse. Further, case reports of cyclone destructions on coastal cities emphasize wind damage (e.g., [13]).

Historically, on-land windbreaks have been constructed to lessen damages by strong winds. They usually act as barriers to provide a sheltered lee side area for humans and their assets against strong wind events [14]. Forests, hills and lagoons can act as natural windbreaks in storms. On-site observations have shown significant contribution of natural windbreaks to mitigation of wind speeds in their lee [15,16]. Alternatively, artificial man-made windbreaks have been developed where natural windbreaks are not available. Fences comprising screens, supported by structural members are one of the most common types of artificial windbreaks globally [17]. On-land windbreaks have seen wide applications for the protection of agricultural farms, agroforestry and urban infrastructure. However, they also have the shortcoming of large space occupation, especially when constructed near cities where land resources are more economically valuable.

To combine the advantages of current wind and wave reduction technologies while minimizing their shortcomings, a novel mega floating breakwater-windbreak structure (kept in position by caissons) is proposed for the first time. The installation of arrays of tubes on the deck area of the breakwater together with the breakwater itself act as a windbreak. The tube layers act as an artificial forest which gives the structure its name. Besides, the structure has other novel features: modularized segments of floating concrete breakwater hulls in arch shapes in plan view; a tilted deck to create beach run-up for better wave energy dissipation; tubes connected to the internal channels inside the breakwater hull which has openings on the sea-facing wall, thereby forming oscillating water columns for wave energy dissipation. The hull of the breakwater structure and caissons are made from marine concrete for its high strength, large inertia against motion and low cost in construction and maintenance. The tubes are made from HDPE or plastic composite materials which are durable in a marine environment. These tubes are connected to the internal channels inside the hull that end with openings on the vertical sea-facing wall of the hull, allowing the dissipation of wave energy through oscillating columns.

This paper will demonstrate the feasibility of such a floating forest concept and its benefits through a case study involving the environmental conditions of an Australian coastal site. The environmental conditions at the site as well as the wind and wave reduction targets are first introduced. Separate hydrodynamic and aerodynamic analyses are then performed to study the responses of the breakwater and windbreak of the floating forest under wave and wind actions. The hydrodynamic analysis involves solving a wave diffraction-radiation problem with linearized 3D boundary element method (BEM). The wave field, wave load, rigid body motion and hydrodynamic pressure distribution on the breakwater hull are calculated and analyzed. An aerodynamic analysis of the flow field around the windbreak is carried out separately by using computational fluid dynamics (CFD), with the view to assess the wind reduction in the lee of the floating forest. Both mean and turbulent flow fields are examined, and peak wind speeds are calculated.

2 Geometry of floating forest and site conditions

2.1 Site conditions

The site chosen for the case study is the Gold Coast in Queensland, Australia. As a prominent coastal city relying mainly on tourism, it has a long coastline with valuable beach resources that are fragile to strong wave and wind events but are key to the local economy [18]. Historically, it has suffered from several cyclone attacks that resulted in widespread structural damages, landscape destructions and floods, causing casualties, economic losses and disruption to local industries [19]. The gust wind speed of these cyclones can reach as high as 60 m/s. Based on existing published Metocean data [20], the design wave condition is fitted with a JONSWAP spectrum, with characteristic parameters as shown in Table 1. The directional distribution of the wave spectrum is not yet known, and the variation of the wave height with respect to heading is consequently not considered in the study. The water depth is assumed to be 40 m.

2.2 Detailed geometry of floating forest

Figure 1 presents a global plan view of the proposed floating forest. It comprises several concrete arch shape segments that are kept in place by caissons at their ends, facing the incoming prevailing wave and wind direction. Wind turbines may be installed on the caissons to harvest wind energy. The dimensions of the 3-segment floating forest are summarized in Table 2. The draft as well as the width of the floating forest are later determined by parametric hydrodynamic studies in Section 3. HDPE tubes are installed on the deck of the arch segments. Figure 2 presents plan view and section view of the arch shape floating segment. Note that only the outer geometry is given in Fig. 2, without structural details. From Fig. 2, it can be seen that the cross-section of the side hull of the module can be separated into a lower rectangular part and an upper triangular part providing a tilted deck area on which the tubes can be installed.

The interior of the hull is divided into a number of sections along the direction of the length by internal walls as is displayed in Fig. 3, and also divided into several sections along the direction of the width. The deck area is also subdivided according to the hull divisions. Each section of the deck area is in the shape of a smaller arch. Based on the layout of the tubes and internal channels, these sections are categorized into two modules: 3-4-3 and 4-3-4 as is shown in Fig. 4. This staggered arrangement of the tubes is designed to maximize wind speed reductions. As it is shown in Fig. 2(a), both of the modules are composed of alternately arranged sections with 3 layers and 4 layers of tubes, and details of the 3-tube and 4-tube sections are illustrated in Figs. 5 and 6, respectively. Taking the module 3-4-3 in Fig. 5 as an example, three layers of tubes are arranged in the direction of the width on the deck. The hollow tubes on the tilted deck are connected to the internal channels in the lower hull. The channels go through the outer wall and allow water to enter and rise in the curved channels before finally being expelled from the top of the hollow tubes or flowing back out from the entrance.

3 Hydrodynamic analysis of breakwater-windbreak structure subject to ocean waves

To study the performance and the dynamic responses of the floating forest under wave action, a linearized 3D hydrodynamic study was carried out using the BEM [21–23]. This section focuses on determining the wave field and wave forces on the floating forest. So, the hydrodynamic study only considers the structure below the still water line. However, the influence of porosity due to the tubular water channels is neglected in the hydrodynamic study for simplicity. The mechanism and resultant influence of porosity have been studied separately in detail [24].

3.1 Mooring system and motion equation

Considering the water depth of 40 m and the large dimensions of the floating forest, using caissons as the mooring system is a cost-effective option. Figure 7 shows the proposed mooring system layout for the breakwater modules. The breakwater sections are held in place by the caissons at two sides, and marine rubber fenders are installed in the gaps along the caisson walls to absorb somewhat the large horizontal forces exerted by the breakwater on the caissons. The rubber fenders are denoted as L1, L2, L3 and R1, R2, R3, to receive the left end and right end portions of the breakwater sections respectively, as shown in Fig. 7. Figure 7 also depicts that the caissons are arranged with a 45° heading in the present coordinate system, and the fender arrays are installed in the radial and tangential direction with respect to the breakwater arch. The mooring system allows free motion of the floating breakwater in the vertical direction so as to accommodate tidal variation, while the horizontal and rotational motions are largely restrained.

In the motion analysis of the floating breakwater, the marine fenders are simplified to be linear elastic and smooth, so that they impose reaction forces to the breakwater in normal directions of the fenders while the amplitudes of the forces obey Hooke’s Law. No friction forces are applied to the breakwater during relative motion. The simplifications guarantee a linear stiffness matrix, so that the motion equation may be solved in the frequency domain as:

(1)[\mathit{M}+\mathit{A}(\omega )]\ddot{\mathit{x}}+\mathit{B}(\omega )\dot{\mathit{x}}+({\mathit{K}}_{\text{h}}+{\mathit{K}}_{\text{f}})\mathit{x}=\mathit{F}(\omega ),

where M is the mass matrix of the breakwater, A(ω) and B(ω) are the frequency dependent added mass and radiation damping induced by wave diffraction and radiation; K_h and K_f are the hydrostatic stiffness and fender stiffness, respectively. F(ω) is the wave excitation force. The three breakwater sections can be analyzed separately since there is no mechanical coupling between the sections. As a result, each matrix M, A(ω), B(ω), K_h and K_f has a dimension of 18 by 18, while the force vector F(ω) has 18 components. The rigid body motions are arranged in the sequence of x, y, z, r_x, r_y, r_z, to represent surge, sway, heave, pitch, roll and yaw.

As it can be seen in Fig. 7, a heavy coupling effect between the degree of freedoms (DOFs) can be expected, as the normal direction of the caissons and fenders is at 45° with respect to the longitudinal line of the breakwater. The stiffness of the fender support can be derived by analyzing the contact forces during breakwater motion. Here we suppose that the breakwater is seamlessly in contact with each of the fenders at its equilibrium position, and no pre-compressive force is applied to the system. Under the assumption of small motion amplitudes and ignoring small coupling terms, the components of the stiffness of the marine fenders can be written as:

(2)\begin{array}{rl}{K}_{11}=& K_{\text{rad}}{\sin }^2{\theta }_{\text{f}}+K_{\tan }{\sin }^2{\theta }_{\text{f}}+K_{\text{rad}}\sin {\theta }_{\text{f}}\cos {\theta }_{\text{f}},\\ K_{12}=& -K_{\text{rad}}\sin {\theta }_{\text{f}}\cos {\theta }_{\text{f}}-K_{\tan }\sin {\theta }_{\text{f}}\cos {\theta }_{\text{f}}+K_{\text{rad}}{\cos }^2{\theta }_{\text{f}},\\ K_{14}=& K_{41}=-K_{\text{rad}}\sin {\theta }_{\text{f}}\cos {\theta }_{\text{f}}(z_{\text{l3}}-z_{\text{c}})\\ & -K_{\tan }\sin {\theta }_{\text{f}}\cos {\theta }_{\text{f}}(z_{\text{r2}}-z_{\text{c}})+K_{\text{rad}}{\cos }^2{\theta }_{\text{f}}(z_{\text{r1}}-z_{\text{c}}),\\ K_{16}=& K_{61}\approx K_{\text{rad}}\sin {\theta }_{\text{f}}\cos {\theta }_{\text{f}}(x_{\text{l3}}-x_{\text{c}})\\ & +K_{\text{rad}}\sin {\theta }_{\text{f}}\cos {\theta }_{\text{f}}(x_{\text{r2}}-x_{\mathrm{c}})+K_{\text{rad}}{\cos }^2{\theta }_{\text{f}}(x_{\text{r1}}-x_{\text{c}}),\\ K_{22}=& 2K_{\text{rad}}{\cos }^2{\theta }_{\text{f}},\\ K_{24}=& K_{42}=-K_{\text{rad}}{\cos }^2{\theta }_{\text{f}}(z_{\text{l3}}-z_{\text{c}}),\\ K_{66}=& K_{\text{rad}}{\cos }^2{\theta }_{\text{f}}[(x_{\text{l3}}-x_{\text{c}}{)}^2+(x_{\text{r1}}-x_{\text{c}}{)}^2],\end{array}

where θ_f = π/4 is the orientation angle of the marine fenders, as illustrated in Fig. 7. K_rad and K_tan are the stiffness of marine fender arrays arranged in radial and tangential directions, respectively. The contact points of the 6 marine fender arrays are denoted by (x_li, y_li, z_li) and (x_ri, y_ri, z_ri). The rotational center has the coordinates of (x_c, y_c, z_c), which is the same as the center of gravity. The other terms in the stiffness matrix are either zero or considered to be negligible compared with the hydrostatic stiffness. As it can be seen from Eq. (2), the resultant stiffness matrix of the marine fenders is asymmetrical, as the sway motion leads to symmetric fender reaction forces while surge motion induces asymmetrical ones. The gravity-based caissons sit on the seabed and are assumed to be stationary, and thus the fender stiffness matrix of each breakwater section is consequently independent from each other. The global stiffness matrix of the fenders comprises three identical 6 by 6 matrices, which are combined diagonally.

The added mass, damping and wave excitation forces from wave diffraction and radiation are calculated by using the BEM-based program HydroSTAR. The frequency-dependent 6-DOF motions are determined from Eq. (1). The theory of linear wave diffraction and radiation is briefly described in Section 3.2, and results are given in Section 3.3.

3.2 Linear wave diffraction-radiation model

The linear BEM is selected for determining wave diffraction and radiation for its high efficiency. It is acknowledged that the wave response can be nonlinear in an extreme storm, but the complexity and large scale of the global analysis results in a high computational cost for alternative approaches.

Under the assumption of the linear potential flow theory, that the water flow is non-viscous, incompressible and irrotational, the wave diffraction and radiation problem is solved by obtaining the solution of the Laplace equation:

(3)\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{z}^2}=0,

where ϕ is the total velocity potential. For a non-porous structure and flat non-porous seabed, the boundary conditions are given by:

(4)\begin{array}{rl}& \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }z}=0,\text{on the seabed},\\ & \frac{\mathrm{\partial }{\varphi }_{\text{D}}}{\mathrm{\partial }n}=-\frac{\mathrm{\partial }{\varphi }_{\text{I}}}{\mathrm{\partial }n},\text{on the breakwater surface},\\ & \frac{\mathrm{\partial }{\varphi }_{\text{R}i}}{\mathrm{\partial }n}=v_{i\text{n}},\text{on the breakwater surface},\\ & \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }z}=\frac{{\omega }^2}{g}\varphi ,\text{on the free surface}.\end{array}

In Eq. (4), n is the surface normal of the breakwater pointing seawards, g the gravitational acceleration, ω is the angular frequency of the wave, v_in the normal velocity on the breakwater surface of ith DOF and ϕ_I the incident wave potential given by

(5){\varphi }_{\text{I}}=-\frac{\mathrm{i}gA}{\omega }\frac{\cosh [k(z+h)]}{\cosh (kh)}{\text{e}}^{\mathrm{i}k(x\cos \beta +y\sin \beta )},

where A is the wave amplitude, k the wave number determined by dispersion relation, h the water depth, β the incident wave direction. The total wave potential, ϕ, can be considered as having three wave components:

(6)\varphi ={\varphi }_{\text{I}}+{\varphi }_{\text{D}}+\sum _{j=1}^6{\varphi }_{\text{R}j},

where ϕ_I and ϕ_D denote the incident and diffracted wave potential, ϕ_Rj with j = 1–6 are the radiation wave potentials induced by the 6-DOF rigid body motion. For a stationary breakwater, only incident and diffracted wave components are considered.

The velocity potential on each panel of the discretised breakwater is calculated by solving the boundary integral equation in Ref. [25]. and the hydrodynamic parameters can be derived afterwards. Figure 8 shows the mesh used in the hydrodynamic study, with the three breakwater sections marked as B1, B2, and B3. Note that the caissons are modeled for illustrative purpose only and will not be considered in the numerical simulation, because the narrow gaps between the caissons and the breakwater modules may lead to numerical instabilities. Considering that the sizes of the caissons are much smaller than that of the breakwater, it is reasonable to assume that the breakwater hulls contribute most of the blockage effect on the incident waves. The breakwaters are modeled in full scale, with 1100 panels on each breakwater. The added mass and damping terms are given by Ref. [26]:

(7){\omega }^2{A}_{ij}+\mathrm{i}\omega B_{ij}=\rho {\omega }^2\underset{S_{\text{B}}}{\iint }{\varphi }_{\text{R}i}{n}_j\text{d}S.

The excitation forces are expressed as the integration of incident and diffracted wave potential on the breakwater surface S_B:

(8)F_j=\mathrm{i}\omega \rho \underset{S_{\text{B}}}{\iint }({\varphi }_{\text{I}}+{\varphi }_{\text{D}})n_j\text{d}S,

where n_j is the surface normal of jth DOF. The added mass, damping and wave force terms are separately calculated for each breakwater section. The wave elevation η at random points around the breakwater can be calculated by:

(9)\eta =\frac{\mathrm{i}\omega \varphi }{g}{}_{z=0}.

The calculated added mass, potential damping and wave excitation forces are integrated into the motion Eq. (1), and the breakwater motion calculated at each frequency. Table 3 lists all the parameters required for the hydrodynamic analyses, in which the mass terms are expressed with radii of gyration. The breakwater and fender layout are symmetric with respect to the Y axis, so the following relationships hold: x_{\text{l}i}=-x_{\text{r}i},y_{\text{l}i}=y_{\text{r}i},z_{\text{l}i}=z_{\text{r}i}. The two wave directions represent oblique sea state and beam sea state, respectively.

3.3 Numerical results

The frequency dependent hydrodynamic parameters are presented in the form of response amplitude operator (RAO), which is the response under a unit incident wave elevation. The performance of the breakwater is first evaluated by investigating the wave field behind the breakwater. To emphasize the influence of breakwater motion on the wave field, the results for the floating breakwater with present mooring design are compared with those for a presumed pile-supported stationary breakwater with the same geometry. Figure 9 shows the contours of wave elevation RAOs near the breakwater with and without considering motion for a beam sea condition. The corresponding wave elevation RAOs in oblique wave condition are shown in Fig. 10. It can be observed that the stationary breakwater is able to create a larger area with small wave amplitudes, while the floating breakwater creates intermittently altering high and low center of wave elevations.

Figure 11 plots the influence of the floating breakwater in the far-field. The wave elevation RAOs in the range y ~ [−200 m, 100 m], [−500 m, −200 m] and [−800 m, −500 m] are plotted inFigs. 11(a)–11(c), respectively. It can be noticed that the variations of the wave elevation amplitude are gradually smaller when measured farther away from the structure, with the variation range changing from 0.8 to 1.6 in Fig. 11(a), to 0.6–1.2 in Fig. 11(b) and finally to 0.85–1.15 in Fig. 11(c). The smaller variations of wave RAOs from 1.0 indicate a declining influence of the structure, and the wave response is therefore gradually returning to the original incident wave.

The results in Figs. 9–11 show that the breakwater sheltering effect is spatially varied, and the structures to be protected by the breakwater should be placed where the wave elevation is smallest. As an example, an efficiently sheltered area can be found at around the site bounded by −300 m < x < −100 and −200 m < y < 0 m as well as the site bounded by 100 m < x < 300 m and −200 m < y < 0 m. Each site has an area of 40000 m ² and is suitable for most offshore structures to be located. For the wave period T = 9.4 s considered, which is the lower bound of the 1-year return wave period, these areas can achieve a wave elevation RAO of 0.4−0.6, which means a reduction of wave height by 40% to 60%.

The breakwater performance is also sensitive to the incident wave period. It has been proven in previous studies [27,28] that shorter period waves tend to be blocked more by the breakwater. To quantify the breakwater performance under an irregular sea state, an average wave transmission coefficient is calculated by taking the average of wave elevation RAO at all sampling points in the most sheltered area as mentioned above. The average transmission coefficients under beam sea and oblique sea states are hence plotted with respect to the wavelength normalized with breakwater width L/B. Furthermore, a wave response spectrum can be calculated by applying the incident JONSWAP spectrum as specified in Table 1:

(10)S_{\text{T}}(\omega )=k_{\text{t}}(\omega {)}^2{S}_{\text{I}}(\omega ),

where S_I and S_T are the spectra of incident and transmitted wave. The transmission coefficients under regular sea states and response spectra under irregular sea states are presented in Fig. 12. It can be seen that the wave transmission coefficients of a floating breakwater range from 0.4 to 0.8 in beam sea conditions while those of a stationary breakwater are around 0.3–0.6. The spectral calculation shows that up to 40% of the wave energy will be blocked by a floating breakwater while the value is 53% in the case of a stationary breakwater. The results indicate that the stationary breakwater has a better performance in terms of wave blockage. Considering the large wave force on the breakwater, however, a fully restricted breakwater will increase the cost of the mooring system.

The motion of the breakwater and the wave excitation forces are studied next. Figures 13 and 14 demonstrate the 6-DOF wave excitation forces calculated for each breakwater section. The results are compared with the results of a single breakwater section. The comparison highlights the multi-body interaction among the breakwater sections. First, a sharp increase in both surge and sway excitation forces F_x and F_y can be observed for the multi-body breakwater, which is possibly induced by the narrow gap resonance between the neighboring breakwater sections. In addition, it can be also seen that the symmetry in the y direction is no longer satisfied for the sections B1 and B3 (as shown in Fig. 8) located at the corners. Consequently, nonzero excitation force F_x and moment M_y exist in a beam sea condition, which is different from the single breakwater case.

The rigid body motion of the breakwater is jointly influenced by the hydrodynamic coupling effect seen in the wave excitation forces and added masses, and the mechanical coupling effect induced by the marine fenders. The motion RAOs are plotted in Figs. 15 and 16 for oblique sea and beam sea conditions, respectively. It can be seen that for both sea conditions, the rotation motion roll and pitch are below 0.5°/m. The yaw motion is negligible, which means the adverse effect induced by the geometric eccentricity is minimized for the present design parameters. The translational motions are more pronounced. Resonant periods can be identified as 3.0 s for surge, 2.0 s for sway and 11.5 s for heave motion. The surge and sway motion are far away from the peak wave period and small motion amplitudes can be expected under the JONSWAP wave spectrum. The heave resonant frequency is in the range of the 1-year return wave period and deviates from the 100-year peak wave period. However, the heave motion is not restricted and will consequently not influence the mooring force.

It can be seen that the surge motions of B1 and B3 are not exactly identical in beam sea conditions, even though the geometrical symmetry ensures equal wave excitation force as suggested in Fig. 14(a). This is due to the large coupling term K₁₂, K₁₄ and K₁₆ in the stiffness matrix and the different phase angles between motions. With the large coupling stiffness, the surge, sway, roll and yaw motions are no longer independent from each other, as they would be in symmetric floating body cases, leading to more complex motion responses. It can be seen that the pitch motions of B1 and B3, which are less affected by the marine fenders, have almost the same amplitude for all wave periods considered. The slight differences are due to the nonzero coupling terms in the added mass matrix and the hydrodynamic coupling between the breakwater modules. To better quantify the different responses in oblique sea and beam sea conditions, the maximum RAO amplitudes in the calculated wave period range are summarized in Table 4. Note that the maximum values may appear in different breakwater modules. The results show that maximum pitch and roll motion appear in oblique sea condition while maximum surge, sway, heave and yaw appear in beam sea condition. Considering that the beam sea condition is the dominant wave direction where most of the wave energy concentrates, the corresponding surge, sway, heave and roll responses are more important factors to consider in the design of the breakwater.

Figure 17 demonstrates the added mass components normalized with respect to mass. The added mass terms indicate the influence of radiation waves on inertial forces. It should be noted that the global added mass matrix has the size of 18 × 18, and only selected terms are plotted. Figures 17(a)–(c) show the diagonal terms for sway, roll and yaw motion, while Fig. 17(d) shows the off-diagonal terms between sway and roll motion for each body. It can be seen that the added mass of B1 and B3 are exactly the same and are only slightly different from those of B2. This indicates that added mass is predominently determined by the breakwater geometry, while the coupling between breakwaters has negligible effect. Figures 17(e) and 17(f) show the influence of the sway and roll motion of B1 on the motion of B2 respectively, which indicate the coupling effect between different breakwater modules. These coupling terms can be either positive or negative, reflecting the in-phase or out-of-phase inertial force imposed by B1 on B2.

The RAOs plotted in Figs. 9–17 are amplitudes of each parameter, which may not appear at the same time due to the phase lag with respect to the incident waves. By using the sway excitation force F_y as an example, Fig. 18 plots the distribution of hydrodynamic pressure and correlated wave surface at the time when F_y reaches the maximum. At this instant, the sea-facing wall is subjected to positive hydrodynamic pressure while the lee side wall is under a negative hydrodynamic pressure. The pressure difference between the two vertical walls thus reaches the maximum. Figure 19 plots the instant wave surface corresponding to the maximum wave force. It shows that the maximum wave force is reached while the wave is distributed in an asymmetrical pattern along the y direction, with wave crest at the lee side and wave trough at the sea facing side. It can be seen from Fig. 18 that at this time instant, most area of the sea facing wall is subjected to negative hydrodynamic pressure, whereas the pressure on the shore-facing wall is mostly positive. In the frequency domain, this phenomenon is indicated by similar or same RAO phase response. Similarly, F_y reaches zero when the wave is symmetric along the y direction, with wave crest or trough at the middle. The distribution of hydrodynamic pressure ultimately leads to wave excitation forces as well as wave shear force and bending moment, which will serve as inputs for detailed structural analysis.

It should be noted that the analyses under the assumption of non-porous wall tend to be conservative for the purpose of mooring system analysis. It has been discovered experimentally [24] that the porosity induced by the tubes as in Figs. 5 and 6 is sufficient in reducing the horizontal wave forces imposed on the breakwater under all wave periods. Meanwhile, the wave elevations at the upstream side are reduced while those at the downstream side are not visibly affected [24]. As a result, a calmer sea on the lee side can be created by adopting the porous breakwater design.

4 Aerodynamic analysis of windbreak through CFD

CFD is being used more frequently in recent times to analyze the wind flow field around structures due to its advantages of exhibiting the full picture of the flow field without requiring dedicated experimental facilities and considerable labor [29,30]. The three main categories of CFD methods are Reynolds-averaged Navier–Stokes Simulation (RANS), Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS). DNS computes both the mean flow and fluctuations by directly solving the governing equations while LES solves a part of the turbulent fluctuations that is related to relatively large eddies, leaving the rest of the turbulent fluctuations to be modeled by sub-scale models. RANS only focuses on the mean flow and has the turbulence modeled by turbulence models. Although DNS provides the most precise solution, RANS still remains the mainstream method when it comes to industrial applications because of the limitation of computing resources and the exponential growth of the number of grids needed to resolve the smallest eddy at a high Reynolds number, which is often the case for engineering projects. Therefore, a 3D steady-state RANS analysis will be conducted to study the flow field around the floating forest with a focus on the mean and peak wind speed reductions in the lee of the floating forest.

Only the part of the floating forest that is above the mean sea level is considered for the aerodynamic analysis. The approach wind flow is perpendicular to the longitudinal axis of the floating forest. A length scale of 1/200 is chosen to scale down the prototype of the floating forest to be consistent with the scale model used by the present authors in a wind tunnel experiment for validation purpose. The computational domain, the coordinate system as well as the floating forest structure for wind analysis are shown in Fig. 20. The floating forest model has a height above the bottom of the computational domain of H* = 0.1925 m. This corresponds to a Reynolds number of 159395 when taking the freestream velocity as 13 m/s, which is large enough to ensure that the scaling will not impact the representation of the situation in reality. As the entire floating forest spans over 1000 m, only a typical section of the structure that contains both the row with three tubes and the row with four tubes was considered in order to save computational time. The width of the model was therefore chosen to be 0.46H*. In this way, the side effect at the corner of the floating forest is neglected but will only have minimal impacts in the flow behind the central part of the forest as the floating forest is sufficiently long.

The selected domain size is 12H* in height, 47.26H* in length and the width equated to the width of the floating forest model, which leaves 11H* above the floating forest model, 12H* to the inlet and 34.8H* to the outlet, satisfying the requirements stated in best practice guidelines [29,31,32]. A symmetric boundary condition was chosen for the top, the left and right side of the domain. The surface of the floating forest model and the bottom of the domain are treated as wall boundary conditions. The scale of roughness of the floating forest surface is relatively smaller than that of the bottom of the domain, which is the sea surface in reality. Therefore, the floating forest surface was simplified to be a smooth wall whereas the bottom of the domain has an aerodynamic roughness length {\mathit{\text{z}}}_0\text{=}5\times {10}^{-6}\text{m} in the model scale. After trying several sets of values of sand grain roughness height k_{\text{s}} and roughness constant C_{\text{s}}, k_{\text{s}} = 0.038 m and C_{\text{s}} = 0.1 were finally chosen since they produce roughness effects equivalent to the desired {\mathit{\text{z}}}_0. The inlet of the domain was characterized by a mean wind speed profile, a turbulence kinetic energy profile and a specific turbulence dissipation rate profile. After iteratively running simulations in an empty domain as large as the domain shown in Fig. 20, a final set of profiles that are fully developed and aligned with the roughness of the bottom of the domain is achieved as shown in Fig. 21. The longitudinal mean wind speeds at different heights {\overline{U}}_{\text{0,}\mathit{\text{z}}} are normalized by the mean wind speed at the height of H*, {\overline{U}}_{\text{0,}{\text{H}}^{\text{*}}}, while the local standard deviation of the longitudinal fluctuating wind speed {\sigma }_{\text{u0,}\mathit{\text{z}}}, representing the turbulence, is normalized by the mean wind speed at the same height {\overline{U}}_{\text{0,}\mathit{\text{z}}}. The theoretical equation of the normalized mean wind speed following the log-law is given by

(11)\frac{{\overline{U}}_{0,\mathit{\text{z}}}}{{\overline{U}}_{\text{0,}{\text{H}}^{\text{*}}}}=\frac{\ln \left({\displaystyle \frac{\mathit{\text{z}}}{{\mathit{\text{z}}}_0}}\right)}{\ln \left({\displaystyle \frac{H^{\ast }}{{\mathit{\text{z}}}_0}}\right)}.

By assuming the shear velocity u^{\text{*}}\text{=}0.4{\sigma }_{\text{u0,}\mathit{\text{z}}} [33], the normalized standard deviation, which is also called the turbulence intensity, is given by

(12)\frac{{\sigma }_{\mathrm{u}0,\mathit{\text{z}}}}{{\overline{U}}_{0,\mathit{\text{z}}}}=\frac{{\sigma }_{\mathrm{u}0,\mathit{\text{z}}}}{{\displaystyle \frac{u^{\text{*}}}{\kappa }}\ln \left({\displaystyle \frac{\mathit{\text{z}}}{{\mathit{\text{z}}}_0}}\right)}\text{=}\frac{1.05}{\ln \left({\displaystyle \frac{\mathit{\text{z}}}{{\mathit{\text{z}}}_0}}\right)}.

As it can be seen from Fig. 21, both the normalized mean wind speed and the turbulence intensity profiles in simulations finally applied at the domain inlet match well with the targeted theoretical equations.

A cut-cell mesh was built for the simulation, which has structural grids at the majority part of the domain and avoids unnecessary grid refinement that would occur far away from the structure in a purely structural mesh [34,35] as is shown in Fig. 22. The refined area around the floating forest model extends to 0.52H* in its front, 1.04H* in its lee and 1H* above it as shown in Fig. 22, providing sufficient resolutions in places where gradients of the flow radically change. A set of mesh sensitivity studies have been conducted with the smallest grid sizes of 0.003, 0.002, and 0.001 m. There was no discernible difference between the mesh with the smallest grid size of 0.002 and 0.001 m. The mesh with the smallest grid size of 0.002 m was then taken for the final simulation. The total number of cells was around 6.63 million.

The classical SIMPLE algorithm was chosen for the pressure and velocity coupling. Discretization of the convection and viscous terms in the governing equations were taken through second-order discretization schemes. A hybrid-initialization was applied before the iterative solving process. The residuals of the calculation were 10⁻¹⁰ for the x, y, z velocity, 10⁻⁵ for the continuity, 10⁻⁸ for the turbulence kinetic energy and 10⁻⁶ for the specific dissipation rate, which were way below the recommended values by Ref. [36]. The generalized k−ω (GEKO) model was chosen to be the turbulence model as it fuses the advantages of other existing k−ω and k−ε models with tuneable coefficients associated with typical physical phenomena [37]. Based on the results measured in the validation experiment in the wind tunnel, the CMIX and CBF_TUR coefficients, which are related to the flow mixing and separation were tuned to be 1 and 0.2, respectively. The coefficient of determination, R², calculated between the simulation and the validation experiment for the mean wind speeds at 187 points with x/H* ranging from 1 to 18 and y/H* ranging from 0 to 3 is 0.7, showing that the simulation was capable of predicting the flow in the lee side of the floating forest.

Three plane sections were considered for the CFD analysis; i.e., plane section containing a row of four tubes, plane section containing a row of three tubes and a plane section containing the gap between two rows of tubes as shown in Fig. 23. The mean flow field in the plane section containing a row of four tubes is presented in Fig. 24. The local mean wind speed {\overline{U}}_{x,\mathit{\text{z}}} was normalized by using the mean wind speed at the same height in the approach flow, i.e., {\overline{U}}_{0,\mathit{\text{z}}}, in the reduction ratio contour of the mean wind speed in the lee of the floating forest. The height z and the horizontal distance x have been converted back to full scale for better interpretation. As the mean flow fields for the other plane sections are similar with minor difference only appearing before x = 50 m, they will not be presented. A small recirculation bubble, identified by the 0-contour line, exists below the height of the floating forest. The 20% mean wind speed reduction area (i.e., 0.8 contour line) covers the region bounded by 0 < x < 550 m and 0 < z < 40 m, while the 10% mean wind reduction area (i.e., 0.9 contour line) extends beyond x = 800 m. The contour line of 1 goes beyond x = 800 m and z = 100 m, meaning a small reduction of mean wind speeds still exist beyond x = 800 m.

As the reduced energy of the mean flow could possibly flow to turbulence according to the governing equation, it is also important to examine the turbulence characteristics. Figure 25 shows the normalized standard deviation of the longitudinal fluctuation {\sigma }_{\mathrm{u}x,\mathit{\text{z}}}/ {\overline{U}}_{0,\mathit{\text{z}}}. The height z and the horizontal distance x are also at full scale. It can be seen that the area in the lee of the floating forest has higher turbulence than the area at the same height before the structure, suggesting that the reduction of mean wind speeds does result in increments in turbulence. Similar to the normalized mean wind speed results, there is no significant difference in {\sigma }_{\mathrm{u}x,\mathit{\text{z}}} for the different plane sections considered. The largest {\sigma }_{\mathrm{u}x,\mathit{\text{z}}}/ {\overline{U}}_{0,\mathit{\text{z}}} = 0.25 circles the region of 40 m < x < 400 m and 0 < z < 40 m, covering the area where most mean wind speed reductions occur. The 0.2 contour line starts from the tip of the floating forest platform and goes up to the height of the floating forest and ends at x = 800 m while the 0.15 contour line goes up from z = 20 m at x = −200 m to z = 70 m and maintains this height until x = 800 m. This means that increments in longitudinal turbulent fluctuations still exist until x = 800 m, which echoes the reduction of mean wind speeds at this farthest point of analysis.

To see the combined effect of the reduction in mean wind speeds and the increment in turbulence, we further examine the peak wind speed ratio in Fig. 26 since it is usually the peak wind speed that will be used when designing structures for peak loads. Here, we follow an established definition of the peak wind speed ratio [38], which is given by {\displaystyle \frac{{\overline{U}}_{x,\mathit{\text{z}}}\text{+}{\sigma }_{\mathrm{u}x,\mathit{\text{z}}}}{{\overline{U}}_{0,\mathit{\text{z}}}\text{+}{\sigma }_{\mathrm{u}0,\mathit{\text{z}}}}}. Similar to the normalized mean wind speed, this ratio shows how much peak wind speed is reduced as compared to that without the floating forest. Generally, the contours of the peak wind speed ratio follow the contours of the normalized mean wind speed while the percentage of reduction is less because of the influence of the turbulence increment. Below the height of the floating forest, the 0.8 contour extends to x = 400 m, while the 0.95 contour goes up to x = 800 m, and the 1.0 contour extends beyond x = 800 m. Considering that the wind pressure acting on structures is proportional to the square of the wind speed, the structures within 400 m distance behind the floating forest can benefit from 36% reduction of wind pressures up to around 40 m in height, which covers the height range of normal mid-rise structures. The structures within x < 600 m will experience a 20% reduction in wind pressure. Even structures located as far as 770 m behind the floating forest can still benefit from a 10% wind pressure reduction. Multiply this sheltered length by the length of the floating forest, one obtains an 831600 m ² sheltered area that is relatively calm during wind hazard events.

When designing a structure, engineers usually use the peak wind speed at 10 m height from the ground (or the mean sea level) for calculating wind pressures on structures according to structure design codes worldwide. Therefore, in Table 5, we extract the 10 m height peak wind speeds at several locations behind the floating forest from Fig. 26. It can be seen that the floating forest is able to reduce the incoming high peak wind speed by at least 20% for a distance of 400 m behind the floating forest and by at least 10% for a distance of 600 m.

5 Conclusions and future work

A novel breakwater-windbreak structure, ‘floating forest’, to reduce both wave and wind impacts at its lee side, is proposed. A feasibility study is performed in order to evaluate the performance and dynamic responses of the floating forest under waves and winds. Separate hydrodynamic and aerodynamic analyses have been carried out for the breakwater and windbreak.

A linear BEM based on potential flow theory has been applied to study the wave diffraction and radiation of the breakwater. With linear mooring fender stiffness matrix, the problem is solved in the frequency domain, and hydrodynamic parameters such as wave excitation forces, motion and wave field around the body have been calculated. The results show that a 1 km long floating breakwater is able to provide a sheltered area of around 60000 m² at its lee side, where the wave energy is reduced by 40% to 60%. A stationary breakwater is of course more efficient in wave energy reduction but at the cost of restricting large wave excitation forces. The rigid body motion of the floating breakwater is jointly influenced by fender stiffness, wave excitation forces and its geometry. Under present fender design, the breakwater has very short natural frequencies in surge and sway motion which are safely away from the peak wave period. The rotational motions (roll, pitch, and yaw) are found to be small in all calculated wave periods. Heave is found to be the most significant motion, but it does not greatly influence the fender reaction forces and consequently the structural safety.

Wind flow analysis of the floating forest leeward region through CFD suggests the efficacy of the floating forest for reducing wind speeds and subsequently lessening wind damages of structures in its sheltered region. The mean wind speed is significantly reduced in the lee while the turbulence is increased, resulting in peak wind speed reduction over a large area that extends to more than 770 m behind the floating forest. At around 200 m behind the floating forest and below its height, 60% peak wind speed reduction is achieved. At about 400 m behind the floating forest, 20% peak wind speed reduction is observed. As the distance from the floating forest increases, this wind sheltering effect gradually decays but is still present at a distance of 600 m with 10% peak wind speed reduction. This percentage of wind speed reduction is significant for designing structures in the sheltered area given that the wind pressure is proportional to the square of the wind speed.

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