Study on Submerged Upper Arc-Shaped Plate Type Breakwater

Based on the wave radiation and diffraction theory, this paper investigates a new type breakwater with upper arc-shaped plate by using the boundary element method (BEM). By comparing with other three designs of plate type breakwater (lower arc-shaped plate, single horizontal plate and double horizontal plate), this new type breakwater has been proved more effective. The wave exiting force, transmission and reflection coefficients are analyzed and discussed. In order to reveal the wave elimination mechanism of this type of breakwater, the velocity field around the breakwater is obtained. It is shown that: (1) The sway exciting force is minimal. (2) When the ratio of the submergence and wave amplitude is 0.05, the wave elimination effect will increase by 50% compared with other three types of breakwater. (3) The obvious backflow is found above the plate in the velocity field analysis.


Introduction
With the ocean engineering projects expanding towards deep sea, the design and construction of ocean platforms are becoming more complicated, and the environmental conditions that the offshore structures face are becoming more challenging. Higher standard breakwaters to protect the offshore structures are therefore required. The traditional coastal breakwater could not meet the needs in deep water. The plate type breakwater, however, as a new type of floating breakwater, wins much attention in the field of marine engineering for its low cost, convenient construction, insusceptibility to depth and geological conditions. In the analysis of plate type breakwater, transmission and reflection coefficients are very important parameters to the evaluation of the effectiveness of wave elimination, while the distribution of the velocity flow field reveals the wave elimination mechanism.
Many researchers have carried out numerical and experimental studies on the single horizontal plate (Parsons and Martin, 1992;Wang, 2001;Roy and Ghosh, 2006;Hou et al., 2010). Stoker (1957) proposed an analytical method to study the transmission and reflection coefficients of a fixed plate under the influence of long-wave effect. Ursell (1947) investigated the wave transmission and reflection coefficients for a single plate. Qiu and Wang (1986) proposed analytical formulation of the transmission and reflection coefficients of a single plate at any depth based on the wave energy theory. Wang K. et al. (2010Wang K. et al. ( , 2001 studied the transmission and reflection coefficients, added mass and damping of floating and submerged single horizontal plate and double horizontal plate using the boundary element method, and discussed the wave elimination mechanism of submerged plate using the numerical wave tank technique. Burke (1964) analyzed and solved problems of wave scattering of an underwater plate in deep water using the Wiener-Hopf method. Siew and Hurley (1977) solved the wave motion of an underwater plate in shallow water.
For multi-plate type and plate-pontoon type breakwaters, Wang et al. (2006) carried out a physical experiment to study the wave elimination of multi-layer breakwater. Usha and Gayathri (2005) used the matching eigenvalue function method to study the two-dimensional transmission and reflection of single horizontal plate in shallow water based on the potential theory. Wang and Shen (1999) applied the eigenvalue approximation to study transmission and reflection of multiple horizontal plates. Wang Y.X. et al. (2010) studied the performance of wave elimination and motion response of fixed plate-pontoon type breakwater by the model test. Gayen and Mondal (2016) consider the reflection and transmission of the surface water waves with two inclined identical permeable plates by two hyper-singular integral equations of the second kind. Koh and Cho (2016) studied the heave motion response of a circular cylinder with the dual damping plates by analytical method. Wang and Zhang (2018) obtained the hydrodynamic coefficients of double submerged inclined plates by the boundary element method.
The double plate breakwater has better wave dissipation effects than the single plate breakwater, although the latter is of lower cost. The current plate type breakwaters, such as the single plate, double plate or plate-pontoon type, could not combine the advantages of lower cost and higher effectiveness when considering the change of water depth caused by, for example, tide. It is known that the submergence of the plate is a critical factor affecting the wave elimination. A new type breakwater with upper arc-shaped plate is proposed in this study to adapt to the water depth variation which is more effective and more economical. Section 2 elaborates the mathematical formulation and numerical methods. The results and discussion are presented in Section 3, and the conclusions and remarks are given in Section 4. Fig. 1 is the sketch of different two-dimensional breakwaters with upper arc-shaped plate, single horizontal plate, double horizontal plates, and lower arc-shaped plate in finite water depth. The origin of the two-dimensional global Cartesian coordinate system OXY is located on the free sur-

Calculation model
face, and the local coordinate system oxy is located on the geometrical center of the plate. The incident wave is travelling along the X axis towards the positive direction. The positive normal vector is defined as pointing from the body surface into the flow field. In the truncated wave tank model, the depth of water is H, AB and CD represent the open boundary for the wave outlet and inlet, respectively, and CB ( ) and S 0 are the free surface and body surface. The submergence of the horizontal plates or the distance between the free surface and the highest point on the arc-shaped plate is . The thickness and length of the plate are denoted by TT and , where is the half length of the plate, and denotes the gap between the double plates in Fig. 1c. In Fig. 1a and Fig. 1d, denotes the angle of the upper arcshaped plate and lower arc-shaped plate relative to the x axis. In order to improve the wave elimination effect, an upper arc-shaped plate is investigated in the study as seen in Fig. 1a and a series of numerical simulation results of the wave transmission and reflection coefficients, wave exciting force and velocity distribution around the plate are obtained.

Governing equation and boundary conditions
Suppose that the fluid is inviscid, incompressible and the motion is irrotational, the floating body will move harmonically under the effect of wave, and the velocity potential of the fluid can be expressed as: where denotes the real part of complex variables, is time, , and is the circular frequency of the wave: where is the acceleration of gravity, is the wave number and is the wave number of deep water. The velocity of fluid can be defined as: The velocity potential can be decomposed into various components based on the linear superposition: where is the incident potential, is the diffraction potential, (j=1, 2, 3) are the radiation potentials for the sway (in x), heave (in y) and roll (rotation about the axis perpendicular to the x-y plane) respectively, and is the magnitudes of motions. The incident potential can be expressed as: where A is the amplitude of incident wave. The governing equation and boundary conditions can be expressed as: for the radiation potential while denotes the components of normal vector and for .

Boundary integral equation ϕ
D ϕ j and in Eq. (4) can be obtained by establishing a boundary integral equation on the body surface by applying the Green theorem which leads to: In Eq. (7), is the field point, is the source point, is the Green function, and C is the solid angle. The Green function can be expressed in the following form: Substituting the boundary conditions to Eq. (7), the following equation can be obtained: 2.4 Wave exciting force The wave diffraction force and wave scattering force can be obtained by: 2.5 Transmission coefficient and reflection coefficient The transmission and reflection coefficients can be obtained from the wave scattering force on the plate, e.g. the vertical wave force (in the heave) is:  .
x Similarly, the horizontal or sway wave force (in direction) on the plate is: where and are the real and imaginary parts of , and the phase of is: .
R T R F The transmission coefficient and reflection coefficient can then be defined as: 2.6 Analysis of the velocity field The velocity of the fluid can be calculated from the potential by: where denotes the total velocity potential , including the incident potential , radiation potential , and diffraction potential . The velocity potential at any point P in the fluid domain can be solved from the following equation: 2.7 Boundary element discretization Supposing that the physical quantities are of linear distribution along the elements, the physical variables can be expressed as follows: N 1 (ς) and N 2 (ς) where are the shape functions and can be expressed as: In Eq. (18), are the potentials on element node , is the element length, is the local coordinate, and and are the global coordinates of the element node.
Substituting Eqs. (18) and (19) into Eq. (9) yields: When the radiation and diffraction potentials are solved, the hydrodynamic coefficients and flow field can be obtained.

Formulation of the velocity field
Supposing that the fluid domain is discretized by 4-node quadrilateral elements, the physical variables at any point can be expressed as follows: where, Finally, Eq. (24) can be written as: [J] where is the Jacobian transformation.

Numerical models and results verification
H S F A truncated numerical wave tank with finite water depth was constructed as shown in Fig. 1, the length of the wave tank is 6 times of the water depth , and the origin is located on the free surface. Through the numerical testing, the meshing along different tank boundaries are determined as follows: (1) on the wave inlet CD and outlet AB: 60 elements on each boundary, 360 elements on the free surface ; (2) upper and lower arc plate: 80 elements on the top side and the same number on the bottom side, 40 elements on each plate end; (3) single horizontal plate: 30 elements on each plate ends and 60 elements on the top side and the same number on the bottom side; (4) double horizontal plate: 180 elements on each plate with the same meshing as the single horizontal plate. Fig. 2. Comparison of the transmission coefficient with that of Hsu and Wu (1998) (H=1.0 m, HS=0.2 m, TT=0.04 m, B=2.0 m). In order to improve the numerical precision, double nodes are placed on the interaction of different fluid boundaries and each end of the plate. This kind of double nodes have the same position and velocity potential, but belong to different boundaries with different normal vectors. For the single horizontal thin and rigid plate, Fig. 2 shows the transmission coefficient in comparison with that of Hsu and Wu (1998) as defined in Eq. (15). In Fig. 2, denotes nondimensional wave number and water depth , submergence of plate , plate thickness and length of plate . The result agrees very well and the efficiency of current method has been confirmed.

Wave exciting forcē
The wave exciting force for different breakwaters in three directions is given in Fig. 3, where non-dimensional wave exciting force is defined as . The figure shows that the trend of the wave exciting force for different breakwaters in the same direction is basically similar, and the wave exciting force for the heave is significantly larger than that for the sway and roll. For the heave, the wave exciting forces for four types of breakwater change with the increase of , and in the range of , the upper arc-shaped plate has larger force than others. In other wave frequency ranges, the difference between different types of breakwater is not significant. The wave heave force for the four different types of breakwater all reach their maximum at . For the sway, when , the wave force increases with the KB/2 KB/2 > 0.6 KB/2 = 1.8 0 < KB/2 < 0.9 0.9 < KB/2 < 2.0 KB/2 = 1.1 increase of , and the double horizontal plate has the largest force almost in the entire frequency range, which is mainly due to the cross-sectional area of this type of breakwater being twice as large as others. It is worth noting that, due to its curved design, the upper arc-shaped plate has smaller wave force, especially for and even close to zero at . For the roll, the double horizontal plate has larger wave force for , while the lower arc-shaped plate has larger wave force for and reaches the peak value 0.15 at . Fig. 4 shows the transmission and reflection coefficients of different breakwaters. As can be seen from the figure, the transmission coefficient of upper arc-shaped plate type is smaller than that of the other three types of breakwater in the whole wave frequency range, demonstrating that this type breakwater is more effective in the wave elimination, especially in the range of with 12% increase in the wave elimination for . HS = 0.05 m KB/2 = 0.5 Fig. 5 shows the transmission and reflection coefficient of different breakwaters when the submergence , in which the wave elimination of the upper arc-shaped plate structure is the most effective. For , the transmission coefficient can reach 0.15 with 50% increase in the wave elimination.

Analysis of the velocity field
To further explore the mechanism of the wave elimination of upper arc-shaped plate type breakwater, the effect- KB/2 = 0.6 ive transmission is chosen as an example to analyze the flow field and investigate the velocity change of water particles in the process of the wave elimination. The calculation covers the whole flow field but the results given are only focused in the area of 1.2 m×0.6 m around the plate as the velocity variation in this area is the most significant. The velocity distributions caused by the motions in the sway, heave, roll and diffraction are shown in Fig. 6.

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WANG Ke et al. China Ocean Eng., 2019, Vol. 33, No. 2, P. 219-225 In the x or sway direction as shown in Fig. 6a, the velocity at the lower part of the arc-shaped plate is larger than that of the upper part and the velocity around the two ends of the plate is the largest. Because of the symmetry of the plate, the sway velocity field is anti-symmetric. In the vertical heave direction (Fig. 6b), the water particles around the plate move in a symmetric manner about the y-axis, with smaller velocity at the two ends and larger velocity in the middle with similar values on the upper and lower sides of the plate. As for the roll-induced velocity field (Fig. 6c), back-flow occurs on the upper side of the plate and because of its geometry, the interaction between the back-flow and incident flow is more drastic than that of the horizontal plate. For the velocity due to diffraction (Fig. 6d), the back flow mainly exist around the tail of the plate and for this type of submerged plate, the diffraction is more important and the total velocity in Fig. 6d is mainly determined by diffraction and therefore the velocity fields in Fig. 6d and Fig.  6e are similar. From Fig. 6d and Fig. 6e, it is found that with the changing water depth above the upper arc plate, an enforced shallow water effect prevents the water particle from flowing horizontally and a rotational flow appears in the middle of the plate and then moves upward to the free surface. This is the main mechanism and advantage of upper arc-type breakwater. The transmitting coefficients in Fig. 5a also demonstrate this phenomenon.

Concluding remarks
KB/2 = 0.6, HS = 0.05 m An effective upper arc-shaped plate type breakwater is proposed in this paper as a new concept floating breakwater which combines the advantages of single and double plate breakwaters. By using the boundary element method, the transmission and reflection coefficients, wave exciting force and velocity field were obtained. It is found that when , the transmission coefficient is only 0.15, which is 50% smaller than that of the horizontal single plate. This is because the specially designed upper arc shape enhances the back flow above the plate and at the same time reduces the sway force on the plate.