1   Introduction

In the last half-century, floating structures of various configurations are being studied due to their significant importance in the field of ocean and coastal engineering. Different types of rigid floating structures are constructed near the coastal region, which is aimed at ocean space utilization for various human activities, marine activities, and coastal zone management such as sea walls, jetties, piers, breakwaters, floating platforms for oil exploration and ship navigation, wave energy devices, etc. One of the most interesting floating breakwater types is the box type which is effective in moderate conditions (McCartney, 1985; Bhattacharjee and Guedes Soares, 2011; Gadelho et al., 2018; Islam et al., 2019). When the floating structures are constructed in a coastal region/nearshore, it is important to develop a three-dimensional body for application in engineering practice. Therefore, the three-dimensional mathematical modeling of hydrodynamic analysis based on the analytical study of floating structures and comparative analysis is of recent interest to coastal researchers and engineers to design an effective floating breakwater.

The study of the radiation problem on floating rectangular structures can provide fundamental information on the hydrodynamic characteristics of wave forces, added mass, and radiation damping of the floating structure. Under the excitation of waves, a rigid floating body will exhibit six degrees of freedom (DOF), namely three translations and three rotational motions. Translation motions are referred to as heave, sway, and surge, and rotational motions are referred to as pitch, yaw, and roll. Physically, vertical (heave), horizontal (sway), rotational (roll) motions, reflection coefficients, and transmission coefficients are of primary importance to analyze the hydrodynamic interaction of ocean waves with floating rectangular structures in three-dimension wave incidence.

In this class of problems, the mathematical difficulty is the non-homogeneous boundary condition at the structure surface which can be handled with the help of the partial differential solution method. Therefore, the analytical solutions shown in the literature have generally been obtained using boundary matching techniques to solve the problem under linearized water wave theory for predicting the hydrodynamic characteristics of floating structures to set the benchmark. The MEFEM has been used in the study of the hydrodynamic interaction of floating rectangular structures due to its considerable accuracy and less use of computer memory and as well as processing time in three dimensions (Mohapatra and Guedes Soares, 2021; 2022). Further, the series solutions using the eigenfunction expansion attend the convergence for the wave forces and hydrodynamic coefficients of a floating structure in two-dimensions and oblique wave cases (Bhattacharjee and Guedes Soares, 2011; Guo et al., 2018; Islam et al., 2019).

Previously, many papers have been involved in the hydrodynamic analysis of floating rectangular structures to obtain hydrodynamic coefficients, wave forces, reflection coefficients, and transmission coefficients based on analytical/semi-analytical methods. Drimer et al. (1992) developed a simplified model associated with a fully linearized hydrodynamic problem to study the performance of a two-dimensional box-type floating breakwater in finite water depth. Lee (1995) presented an analytical solution for the heave radiation of a floating rectangular structure to analyze the hydrodynamic effects of the submergence and width of the structure. Zheng et al. (2004) used the MEFEM to analyze the radiation and diffraction of linear water waves by a rectangular buoy in the water of finite depth in two dimensions. Zheng et al. (2006) used an eigenfunction expansion method to investigate the oblique wave radiation by a floating rectangular structure in finite water depth over a flat seabed. Cho (2016) analyzed the effects of porosity and protruding depth of the vertical side plates on the transmission coefficients and motion response of a floating rectangular breakwater based on matched eigenfunction expansion method. Elchahal et al. (2009) developed a diffraction-radiation boundary value problem (BVP) to study the hydrodynamic performance of the moored floating rectangular breakwater by considering the effect of the side wall, structural parameters, mooring stiffness, angle of incidence on the transmission coefficients and dynamic motion of the floating breakwater. Recently, Guo et al. (2018) analyzed the oblique wave diffraction by a long rectangular rigid floating structure over flat bottom based on linearized water wave theory. Gadelho et al. (2018) studied the CFD analysis of fixed floating box-type structures and compared it with linearized Boussinesq analytical solution and experimental data by analyzing the wave elevations before and after the floating box-type structure and wave forces on the box over the flat bottom. Islam et al. (2019) studied the comparison of CFD, linearized analytical of the oblique wave, and experimental data results of a heaving box-type floating structure. Bhattacharjee and Guedes Soares (2011) studied the effect of the oblique angle and bottom on wave diffraction by a floating rectangular structure over stepped bottom with a rigid wall. Islam et al. (2019) studied the wave radiation by a floating box-type structure over the flat bottom in two dimensions. All the investigations discussed above are assumed to be in two-dimensional and oblique wave incident wave analysis.

From the above literature, it is confirmed that there is no analysis of the wave-induced forces based on the three-dimensional mathematical model solution and comparison with BEM code NEMOH to date to the authors' knowledge.

Therefore, in the present study, the main and new contributions compared with Guo et al. (2018) and Islam et al. (2019) are the 1) mathematical formulation associated with wave radiation by a three-dimensional floating rectangular box and the analytical expression for the radiational potentials (by adding finite structural width), 2) analytical expressions for the hydrodynamic coefficients, added mass, damping coefficients, and wave forces for the heave, sway and roll motions, 3) comparison between the present and the independent numerical BEM code based NEMOH simulations, 4) analysis of the wave forces and hydrodynamic coefficients on heave, sway, and roll motions for modes of oscillations, structural width, and comparison among them from the analytical solutions. Further, the results for the reflection and transmission coefficients for different values of structural width and drafts are presented and also analyzed. Finally, some significant concluding remarks from the present analysis are discussed.

2   Mathematical formulation of floating rectangular box model in 3D

The mathematical model of the BVP is considered in a three-dimensional Cartesian coordinate system (x, y, z) with x - z being in the horizontal plane that coincides with undisturbed water surface and the y-axis being in the vertical downward positive direction. The wave radiation is due to the heave, sway, and roll motions of a rectangular box-type structure floating on the free surface of length 2l, width b, and draft d. The origin o is assumed to be the middle point of the box (as in Figure 1). Hence, the whole fluid domain is divided into three regions as defined by: (l < x < ∞, 0 < y < h, 0≤z≤b), (- l < x < l, d < y < h, 0≤z≤b) and (-l < x < -∞, 0 < y < h, 0≤z≤b) are referred to as R₁, R₂, and R₃, respectively.

Figure  1  Nomenclature of the problem

Download: Full-Size Img

The water is assumed to be inviscid, and incompressible and the flow is irrotational and simple harmonic in time with angular frequency ω. Thus, there exists a velocity potential Φ (x, y, z, t) = Re {ϕ (x, y, z) exp (- iωt)}, where Re denotes the real part of the complex expression and ϕ (x, y, z) denotes the spatial velocity potential which satisfies the three-dimensional Laplace equation as given by

where $\nabla_{x y z}^2=\partial^2 / \partial x^2+\partial^2 / \partial y^2+\partial^2 / \partial z^2$.

The velocity potential ϕ (x, y, z) is decomposed into the incident wave potential, the diffracted potential (due to the stationarity of the structure), and the radiation potential (due to the heave, sway, and roll motions of the floating structure) are denoted by ϕ_I, ϕ_D, and ϕ_R respectively. So the total potential ϕ (x, y, z) can be expressed as

where S = 1, 2 and 3 stand for heave, sway, and roll motions, respectively.

The incident wave potential of linear waves propagating to the negative x- and z-direction is given by

where γ_m = mπ/b for m = 1, 2, 3, ..., are referred to as primary, secondary, tertiary, and other higher modes, respectively with p_m0² = (k₀² - (γ_m)²) > 0 to ensure that progressive wave solution exists (as in Mohapatra and Guedes Soares, 2019), g signifies the gravitational constant, I_m0 is the incident wave amplitude with the dimension of unit length, and wavenumber k₀ associated with the incident wave which satisfies the gravity wave dispersion relation as ω² = gk₀ tanh (k₀h).

Small motions of the floating three-dimensional rectangular box shown in Figure 1 are assumed. If the amplitude of the motion of the rectangular box is assumed as I_R^(S), the radiation potential ϕ_R^(S) (x, y, z) in three-dimensions

where φ_R^(S)(x, y, z) is the spatial velocity potential satisfying the three-dimensional Laplace equation as

Combining the kinematic and dynamic boundary conditions, the linearized boundary condition at the mean free surface is obtained as

where K = ω²/g. The no-flow conditions at the rigid bottom boundary are given by

The non-homogeneous boundary condition due to the heave, sway, and roll motions of the floating structure is given by

Further, the conditions on the rigid structural boundary are described as

where (x₀, y₀) is the centre of rotation for roll motion and δ is Kronecker delta function as defined by

Since the structure is not moving, the no-flow condition at z = 0, b yield

Finally, the radiation condition is assumed to take the form

The next Section will obtain the analytical expressions for radiated potentials in each region and their solutions by satisfying the governing equations along with the relevant boundary conditions.

3   Analytical solution based on the MEFEM

The analytical solutions for the radiation potentials are obtained using the MEFEM of the discussed BVP. The radiation potentials in R₁, R₂ and R₃ are denoted by $\varphi_{R 1}^{(S)}$, $\varphi_{R 2}^{(S)}$ and $\varphi_{R 3}^{(S)}$, respectively. The expressions for the radiation potentials in each region are obtained by applying the method of separation of variables and the unknown coefficients associated with the radiation potentials are then determined by invoking the continuity condition of velocity and pressure at the structural boundary x = ±l and 0 ≤ z ≤ b.

3.1   Analytical expressions for the radiation

Using the Fourier expansion formulae in three regions R₁, R₂ and R₃, the radiation potentials $\varphi_{R 1}^{(S)}(x, y, z)$, $\varphi_{R 2}^{(S)}(x, y, z)$ and $\varphi_{R 3}^{(S)}(x, y, z)$ respectively satisfying the three-dimensional Laplace Equation (5) along with the relevant boundary conditions (7)‒(13) are expanded as

where, the first part in Equation (15) is the particular solution of the radiation potential $\varphi_{R 2}^{(S)}, p_{m n}^2=k_n^2-\left(\gamma_m\right)^2$ and $\alpha_{m n}^2=\beta_n^2-\left(\gamma_m\right)^2$ with $\alpha_{m 0}^2=\beta_0^2-\left(\gamma_m\right)^2>0$ and the eigenvalues k₀ is same as defined in Equation (3) and k_n = ik_n, β_ns are satisfy the following dispersion relations

Further, the eigenfunctions u_mn (y, z) and v_mn (y, z) associated with Equations (14)‒(16) are given by

are orthogonal concerning their intervals as defined by

where

It may be noted that the gravity wave dispersion relations as in Equation (17) have one positive real root and infinitely many roots contribute to the velocity potentials as in Equations (14) and (16), where, the positive real root corresponds to the progressive wave mode and infinitely many roots corresponds to evanescent modes.

The unknown coefficients $A_{1 m n}^{(S)}, A_{2 m n}^{(S)}, A_{3 m n}^{(S)}$ and $B_{2 m n}^{(S)} {'}\mathrm{s}$ for n = 0, 1, 2, ... and m = 1, 2, ... associated with the radiation velocity potentials in Equations (14)‒(16) are determined by using the condition of continuity of velocity and pressure at x = ±l along 0 ≤ z ≤ b.

Continuity of velocity at the vertical interface x = l as

Continuity of velocity at the vertical interface x =- l as

Further, the continuity of pressure at the vertical interfaces x = ±l is given by

Substituting the velocity potentials from Equations (14) and (15) into the continuity of velocity condition (23a), multiplying both sides by the eigenfunction u_mj (y, z) and then integrating with their respective intervals and keeping in mind the orthogonal relation u_mj for j = 0, 1, 2, ..., m = 1, 2, ... in Equation (23), yields

where, the infinite series is truncated after N-terms and as the variable is a dummy j is replaced by n. Proceeding as in Equations (25a) and (25b) and using Equations (15) ‒ (16) into the velocity continuity condition (23b), and applying the orthogonal relation (21), result in

Further, substituting Equations (14)‒(15) into the pressure continuity condition (24a), and applying the orthogonal relation (22), obtained as

Similarly, by substituting Equations (15)‒(16) into the pressure continuity condition (24b), and using the orthogonal relation (21), one can be obtained as

To solve the system of linear equations through numerical methods, the infinite series sums are truncated after the finite number of N-terms present in Equations (26)‒(28). Thus, a linear system of equations 4 (N + 1) is obtained to solve for 4 (N + 1) number of unknown coefficients A_1mn^(S)'s, A_2mn^(S)'s, A_3mn^(S)'s and B_2mn^(S)'s for motion mode S that are associated with the eigenvalues k_n and β_n present in the expressions (14)‒(16). Once the unknown coefficients are determined, then the full solution will be obtained in terms of radiation potentials.

3.2   Analytical expressions for wave forces and hydrodynamic coefficients

The wave forces in the frequency domain can be obtained by use of the known incident and the radiation potentials via the application of the Haskind theorem. The radiation force due to the heave, sway, and roll motions of the floating rectangular structure can be calculated from the radiation potentials in which the hydrodynamic coefficients such as added mass and damping coefficients are involved.

The wave forces in terms of incident and radiation potentials can be expressed as

where j=1, 2, S_b denotes the wetted surface in xyz-plane and n_j is the inward normal to the floating structure. Now, substituting Equations (3), (14), and (16) into Equation (29), which yields

where F_I^j and F_D^j denote the forces due to the incident wave and because of the diffracted wave, respectively, with F_e1, F_e2 and F_e3 are refer to the vertical, horizontal, and torque acting on the floating structure. F_I and F_D denote the forces due to the incident wave and caused by the diffracted wave, respectively, and these expressions are

where

3.3   Hydrodynamic coefficients such as added mass and radiation damping coefficients

Here, the radiation force is the force due to the motion of the floating structure which can be calculated from the radiation potentials. Now, the expression for the radiation force acting on and along the width of the structure is

where M_S,j and N_S,j are the added mass and damping coefficients of the floating structure due to the motion mode S in jth direction, respectively and which expressions are given by

with Re and Im denote to the real and imaginary parts of a complex expression, respectively with f_S,j are given by

The non-dimensional added mass, damping coefficients, wave forces, and phase angle of wave forces of heave, sway, and roll motions are defined by

where I_m0 is the same as in Equation (3). The next subsequent sections will check the computational accuracy of the solutions and several numerical results on hydrodynamic coefficients and wave forces of the heave, sway, and roll motions of the floating structure will be analyzed in different cases

4   Numerical model formulation and description

The description of the detailed numerical model principle is deferred here as it is the open-source BEM code NEMOH software (see for detail Babarit and Delhommeau, 2015).

The mathematical model adopted in this numerical implementation is similar to the one presented by Bispo et al. (2022a, b), bearing in mind that the numerical model presented here is designed as a rigid single module structure with no articulations. Based on the linear theory of waves, the total velocity potential with time-dependent can be written as Φ = ϕe^−iωt, where ϕ is the total time-independent velocity potential, ω is the wave frequency rad/sec and t is the time. Then, one can describe the fluid with the velocity potential that can be decomposed into three parts ϕ_I, ϕ_D, and ϕ_R, respectively, based on the assumption of an ideal fluid and linearity and the velocity potential can be expressed as in Equation (2), satisfying the following boundary conditions:

where g is the gravitational constant and r is the radial distance from the origin.

In this numerical BVP, the fluid domain is represented by Ω, S_f is the free surface, S_B is the bottom surface, S_∞ represents the boundary at infinity, and the wetted surface of the body is given by S_b, n is the unit vector normal to the wetted surface, pointing outwards, and V_Sb is the fluid velocity on the wet surface of the floating structure, as can be seen in Figure 2.

Figure  2  Schematic diagram of the floating body and the computational domain

Download: Full-Size Img

Once the velocity potential, ϕ, is obtained, the added mass and the radiation damping can be computed. On the other hand, the excitation forces, F_Wj, can be related to the incident and diffraction potentials for the jth degree of freedom by:

in which ρ is the fluid density, ω is the wave frequency, ${\overline{S_b}}$ is the mean wet surface of the body, and i is the complex unit.

The generalized added mass and radiation damping are given by:

where a = A_jk and b = B_jk are the added mass and radiationdamping coefficients of the jth mode induced by kth mode. For six degrees of freedom, the equation of motion of the free-floating body is presented in the frequency domain, and for unitary wave amplitude as (see Newman, 1977)

where M is a matrix of mass and inertias, matrix K is the hydrostatic stiffness matrix and it is defined similarly to the mass matrix, for 6 DOF. The added mass and radiation damping matrices, respectively represented by A(ω) and B (ω), are matrices of 6×6 size. The vector X represents the displacement, from which follows that Ẋ and Ẍ are its first- and second-time derivatives, respectively. Lastly, F_W (ω) is the vector of wave excitation forces.

5   Results and discussions

To make sure the correctness of the present analytical solution, the analytical results of wave forces are compared with numerical BEM code NEMOH simulations. Hereafter, all computations are executed throughout the paper by considering gravitational constant g=9.8 m/s² and water density ρ=1 025 kg/m³ unless mentioned otherwise.

5.1   Comparison of analytical and numerical BEM code NEMOH simulations

Figures 3(a)‒(c) show the comparison of present and numerical BEM code NEMOH of vertical wave force (heave motion) result with m=1, and versus non-dimensional wavenumber k₀h. It is seen that the results of the vertical wave forces from the analytical solution are a similar trend and have a good agreement with the results from numerical BEM code NEMOH.

Figure  3  Comparisons of vertical wave force F₁ among analytical, numerical BEM code NEMOH and experimental datasets

Download: Full-Size Img

On the other hand, Figure 3(d) compares the result of vertical wave force acting on the floating structure between present (analytical) and experimental data available in Rodriguez and Spinneken (2016) versus non-dimensional wavenumber k₀b. It is observed that the non-dimensional vertical force is well agreed for intermediate values of k₀b and the trend is similar between the present analytical and existing published experimental data. To quantify the comparison, the analytical result in Figure 3(d) represents 90% of the predicted value of the vertical force and this simple approximation can be considered a good fit for 0.5≤k₀b≤0.9, whereas the discrepancy is less than 10% in the ranges k₀b < 0.4 and k₀b > 1.0.

5.2   Wave forces, hydrodynamic coefficients, and their comparison analysis for different design parameters

To understand the effect of design parameters associated with the BVP, this subsection will present several numerical results on the vertical wave force, horizontal wave force, and torque on the floating rectangular box-type structure by analyzing the effect of mode of oscillation, non-dimensional length, width, and draft due to the heave, sway, and roll motions in different cases.

Figure 4 represents the effect of modes of oscillation m on the non-dimensional vertical wave force F₁, horizontal wave force F₂, torque F₃, comparison between F₁, F₂, and F₃ of heave, sway, and roll motions with b/h=0.6, l/h = 0.5 and d/h=0.6 versus non-dimensional wavenumber k₀h. It is observed that the non-dimensional horizontal force, vertical wave force, and torques increase with an increase in modes of oscillation m. This is attributed to the fact that a group of small amplitude waves is generated at the edges of the floating box which reach a certain amount of incident wave energy that makes higher loads on the box. Further, it is also found that the vertical force values are the least among the horizontal wave force and torque that is F₁ < F₃ < F₂. In addition, in Figure 4(d), the vertical wave force is 0.3 m, while the torque is 0.1 m which is 20% larger.

Figure  4  Effect of modes of oscillation on the non-dimensional wave forces F₁, F₂, F₃ versus k₀h and comparison among them

Download: Full-Size Img

In Figure 5, the effects of non-dimensional structural length l/h on the non-dimensional vertical wave force F₁, horizontal wave force F₂, torque F₃, and comparison of F₁, F₂, F₃ for d/h = 0.5, b/h = 0.8, and m = 1 versus k₀h are plotted. It is observed that the vertical and horizontal wave forces become higher for longer structures whilst, this effect is the opposite trend in the case of torque to that of vertical and horizontal wave forces. In Figure 5(d), the comparison of wave forces indicated that the horizontal wave force is 4.2 m, while the vertical load is 3.6 m which is 60% larger. Further, in Figures 5(b)‒(d), it is seen that the wave forces attained minimum values for a certain wavenumber k₀h, which is due to phase change of incident and reflected waves leading to constructive/destructive interference of the waves at the edges of the floating box.

Figure  5  Effect of structural length on F₁, F₂, F₃, and comparison among them for m=1

Download: Full-Size Img

Figure 6 plots the effect of non-dimensional structural width b/h on the non-dimensional vertical wave force F₁, horizontal wave force F₂, torque F₃, and their comparison with m = 1, d/h = 0.5 and l/h = 0.8 versus k₀h. In general, it is observed that for larger structural widths, the vertical and horizontal wave forces and torques decrease with occurring significant effect for smaller values k₀h. Also, the vertical wave force values are the least among the horizontal wave force and torque that is F₁ < F₃ < F₂. From Figures 5 ‒ 6, it is concluded that the size of the floating box is important, especially when considering the wave forces as the horizontal force to the vertical force may change significantly depending on the length and width of the structure.

Figure  6  Effect of the structural width on the F₁, F₂, F₃, and their comparison for m=1

Download: Full-Size Img

In Figure 7, the effects of the draft d/h on the non-dimensional vertical wave force F₁, horizontal wave force F₂, torque F₃, and comparison of F₁, F₂, and F₃ with m = 1, l/h = 0.8, and b/h = 0.8 versus k₀h are plotted. It is observed that the vertical wave forces become higher with a smaller draft whilst, the horizontal force increases with an increase in the values of d/h. However, the pattern of vertical force is opposite in trend to that of horizontal force. This is due to the deeper drafts leads smaller force in vertical and more horizontal wave force on the floating box. Further, the observations in Figure 7(d) are similar to Figure 6(d).

Figure  7  Effect of the structural drafts on the F₁, F₂, F₃, and comparison among them with m = 1

Download: Full-Size Img

Figure 8 presents the effect of non-dimensional width b/h on the non-dimensional added mass C_aj for the heave, sway, and roll motions of the structure with m = 1, d/h = 0.5 and l/h = 0.8 versus k₀h. From Figures 8(a) ‒ (c), it is seen that the added mass of heave, sway, and roll becomes higher for wider structure whilst, in the case of sway motion, the variations for different values of width become negligible for higher values of non-dimensional wavenumber. This is because the mass of the structure increases which leads to the more added mass.

Figure  8  Effect of width on the added masses C_aj and comparison among them with m = 1, l/h = 0.8 and d/h = 0.5

Download: Full-Size Img

Figure 9 depicts the effect of non-dimensional width b/h on the non-dimensional damping coefficients C_dj for the heave, sway, and roll motions of the structure with m = 1, d/h = 0.5 and l/h = 0.8 versus k₀h. It is observed that the damping coefficients of heave, sway, and roll decrease with an increase in non-dimensional structural width b/h whilst, the damping coefficients of roll increase with an increase in structural width k₀h ≤ 2.5. This is attributed to the fact that as the structural width is increased while the draft and mode of oscillation (primary mode m=1) are kept fixed as a result the mass of the structure increases too which leads to less damping. The observations of the damping coefficients heave and sway is similar to those of Zheng et al. (2006).

Figure  9  Effect of width on the damping coefficients C_dj and comparison among them for m = 1, l/h = 0.8 and d/h = 0.5

Download: Full-Size Img

5.3   Effect of design parameters on the reflection and transmission coefficients

Figure 10 plots the effect of non-dimensional width b/h on the reflection and transmission coefficients for m = 1, l/h = 0.8 and d/h = 0.5 versus k₀h. It is observed that the influences of the width on the reflection and transmission coefficients are appreciable. The maximum reflection coefficient and the minimum transmission coefficients greatly increase and decrease respectively with increases in the values of structural width.

Figure  10  Effect of structural width on the reflection and transmission coefficients for m = 1, l/h = 0.8 and d/h = 0.5

Download: Full-Size Img

In Figure 11, the influences of the d/h on the reflection coefficient K_r and the transmission coefficient K_t with m = 1, l/h = 0.8, and b/h = 0.5 versus k₀h are plotted. From Figure 11, it is seen that the effects of the width on the reflection and transmission coefficients are significant. The maximum reflection coefficient and the minimum transmission coefficients greatly increase and decrease, respectively with increases in the values of the structural draft. This is because less wave energy passes below the floating box as it expected wave reflection becomes more in the upstream region.

Figure  11  Effect of the structural draft on the reflection and transmission coefficients for m = 1, l/h = 0.8 and b/h = 0.5

Download: Full-Size Img

In Figure 12, the effects of the structural length l/h on the reflection coefficient K_r and transmission coefficient K_t with m = 1, l/h = 0.8, and b/h = 0.5 versus k₀h are plotted. For a smaller value of k₀h, the variations and effects of the structural length on the K_r and K_t are significant whilst, for k₀h ≥ 5, the values of the K_r becomes zero and K_t attends one which leads physically true. Further, k₀h ≤ 5, the reflection coefficients increase whilst, the pattern of the transmission coefficients becomes reveres to that of K_r. This is because the structure with a longer length has a larger projected area, which results in a larger hydrodynamic force that leads to higher reflection and lower transmission.

Figure  12  Effect of l/h on the K_r and K_t for m = 1, l/h = 0.8 and b/h = 0.5

Download: Full-Size Img

6   Conclusions

The new contributions to the present compared with the previous research by Guo et al. (2018) (associated with diffraction) and Islam et al. (2019) (related with radiation) in oblique waves are the mathematical formulation on wave radiation by 3D floating box, analytical expressions for radiation potentials in three-dimensions and comparison with numerical BEM code based NEMOH simulations. Further, the results of wave forces and hydrodynamic coefficients such as added mass, damping coefficients, and wave quantities (reflection and transmission coefficients) for different design parameters are analyzed. From the present analysis, the following important conclusions are drawn.

1) In the verification of the analytical results, it is observed that the analytical results of vertical wave forces for different design parameters are well agreed with the numerical NEMOH and supported by the existing published experimental data sets.

2) It is observed that the vertical and horizontal forces and torques become higher for higher modes of oscillation m which because of the group of small amplitude waves generated at the edges of the floating box.

3) The comparisons for the different structural widths, lengths, drafts, and modes of oscillations on the wave forces for the heave, sway, and roll motions revealed that the wave forces among them hold the relation F₁ < F₃ < F₂.

4) The comparison between the non-dimensional added masses of heave, sway, and roll motions indicated that C_a2 < C_a3 < C_a1, whilst the damping coefficients hold the relationship C_d2 < C_d3 < C_d1 irrespective of k₀h.

5) It is observed that the reflection coefficients become higher for wider structures and this effect attends lower when the draft of the structure increases, whilst the trend of the transmission coefficient becomes reveres to that of reflection coefficients.

6) The above analysis indicated that the size of the floating box is important when considering the wave forces as the horizontal force to the vertical force may change significantly depending on the modes of oscillations, structural length, and width.

7) The present analysis will be helpful to model a large floating breakwater by articulating several floating box modules over analytical and numerical approaches to analyze the performance for the floating breakwater.