Flow field analysis of submerged horizontal plate type breakwater

Submerged horizontal plate can be considered as a new concept breakwater. In order to reveal the wave elimination mechanism of this type breakwater, boundary element method is utilized to investigate the velocity field around plate carefully. The flow field analysis shows that the interaction between incident wave and reverse flow caused by submerged plate will lead to the formation of wave elimination area around both sides of the plate. The velocity magnitude of flow field has been reduced and this is the main reason of wave elimination.


Introduction
Owing to the capable of maintaining water exchange, the floating breakwater has important practical engineering interests in the coastal erosion area, aquaculture area, and the peripheral protection area of deep-sea platforms. Therefore, the study of its hydrodynamic characteristics has aroused increasing attention in recent years. The main design concept of this type breakwater is the attenuation of part of the wave energy, which can keep the huge wave forces from acting directly on the marine structures. Besides, the free water exchanges behind the breakwater can keep the sea water clean and marine ecosystem uninfluenced.
Many mathematical derivations and experiments have been done in the study of the hydrodynamic characteristics of floating breakwater. Parson and Martin (1992) established a high order singular integral equation, and studied the diffraction of the submerged horizontal plate by solving the discontinuous velocity potential on its both sides. Neelamani and Reddy (1992), Yu and Chwang (1993) solved the diffraction problem of the horizontal plate in finite water depth using the eigenvalue approximation in finite region. Hu et al. (2002) presented a two-dimensional analytical solution to study the reflection and transmission of linear water waves propagating past a submerged horizontal plate and through a vertical porous wall. Usha and Gayathri (2005) considered the scattering of surface waves by a submerged, horizontal plate or disc, by using eigenfunction expansions within the finite domain. Zheng et al. (2007) studied the hydrodynamic coefficients and wave exciting forces for long horizontal rectangular and circular structures by boundary element method. Dong et al. (2008) studied wave transmission coefficients of floating breakwaters by experiment. Liu et al. (2009) investigated the hydrodynamic performance of a submerged two-layer horizontal plate by the matched eigenfunction expansion method.
However, up to now, no clear and direct elaboration has been made about wave elimination mechanism from the viewpoint of flow field analysis. Therefore, the present study adopts the boundary element method to solve the diffraction problems of the submerged horizontal plate and obtain the distribution of velocity field around the plate. The numerical result confirms that this method is efficient to the analysis of the velocity field around submerged horizontal plate.
In Section 2, mathematical models and numerical scheme concerning the velocity field are elaborated. Section 3 provides the numerical results and discussion. Conclusion is drawn in Section 4. Fig. 1 shows the definition of Cartesian coordinate system oxy and plate location. This coordinate system is stationary and related to the undisturbed position of the free surface. The origin o is located on free surface, positive x -axis is from left to right and y -axis is positive upward. The incident wave is propagating from right to left along x-axis. Besides, plate length is 2 B a  , plate thickness is T T and plate submergence is S H . It is also supposed that the plate is rigid and thin, and normal vector is positively pointing out of the fluid domain. The fluid is assumed to be non-viscous, incompressible and irrotational. Supposing the motion of the object is harmonic oscillation, then, the fluid velocity can be expressed by the gradient of velocity potential Φ and Φ can be defined as

Governing Equation and Boundary Conditions
x y    denotes the real part,  is the frequency of incident wave, t is time, i 1   , and  is the spatial complex velocity potential irrelevant to time. The velocity potential  satisfies Laplace equation and related boundary conditions (Faltinsen, 1991;Wang et al., 2011a).
Based on linear assumption, complex velocity potential  can be decomposed as follows: where i x is the motion amplitude of the object, I  is the incident potential, D  is the diffraction potential, S I D      is the scattering potential and i  is the radiation potential for sway, heave, and roll, respectively. The boundary condition for diffraction and radiation potentials is: where,  is the wave circular frequency, 2 K g   is the wave number, g is the acceleration of gravity, and A is the amplitude of incident wave.

Boundary Integral Equation
The hydrodynamic of submerged horizontal plate can be solved by establishing integral equation on the object surface with Green's theorem. Following is the boundary integral equation about i  and D  :

 
, Q   is source point,   , G P Q is Green's function, and C is solid angle.   , G P Q can be expressed as follows (Wang et al., 2011b):

Flow Field Velocity
If  represents scattering potential S  , radiation potential R 1 2 3 ( , , )     and total velocity potential  respectively, the velocity of flow field ( , ) u v induced by  can be obtained through the following equation: where ( ) p  at any point p can be solved by source point ( ) q  as follows: Supposing the fluid domain is discritized into finite elements, the velocity of any point in the fluid domain can be calculated as: where [ ] J is Jacobian determinant, i N is shape function, and and   are local coordinates.

Flow Field Analysis
For further revealment of the wave elimination mechanism of plate type breakwater, this section explores the velocity of water particles in the whole process of wave elimination. In the calculation, the , where the transmission coefficient is 0.3 (Wang et al., 2011a) and the heave wave exciting force is the largest (Wang et al., 2011b). Since plate is symmetric about y-axis, the generated radiation heave flow field 2 2 ( , ) u v is symmetric and roll flow field 3 3 ( , ) u v is anti-symmetric.
For the roll motion, the flow directions of water below the plate and above the plate are opposite, and the velocity amplitude below the plate is smaller than that above the plate. That is the characteristic of roll motion. When the plate heaves, water particles around the plate also reciprocate vertically at the same time. Since the plate is thin and close to the free surface, the plate length is decisive for the change of velocity field. It can be seen from Fig. 2d that water particle above the plate at crest time is separated into two parts and flow horizontally to the fluid field with opposite direction. This wave-plate interaction repeats reciprocally. The interaction between reverse flow above the plate and the incident wave leads to the formation of wave elimination area in the head-sea of the plate, where fluid velocity changes dramatically. From Fig. 2e we can see that the velocity magnitude in head sea is much larger than that behind the plate. As a result, the reverse flow caused by submerged plate should be the main reason of wave elimination.