Study on Hydrodynamic Coefficients of Double Submerged Inclined Plates

Added mass and damping coefficients are very important in hydrodynamic analysis of naval structures. In this paper, a double submerged inclined plates with ‘/ \’ configuration is firstly considered. By use of the boundary element method (BEM) based on Green function with the wave term, the radiation problem of this special type structure is investigated. The added mass and damping coefficients due to different plate lengths and inclined angles are obtained. The results show that: the added mass and damping coefficients for sway are the largest. Heave is the most sensitive mode to inclined angles. The wave frequencies of the maximal added mass and damping coefficients for sway and roll are the same.


Introduction
In the past decades, many studies concerning the hydrodynamic characteristics of the thin rigid plate have been conducted (Lopes and Sarmento, 2002;Koh and Cho, 2016;Farina et al., 2017). Also for the potential flow theory, some viscous damping predictions are incorporated in the roll motion of a floating body (Salvesen et al., 1970;Downie et al., 1988;Kristiansen and Faltinsen, 2010). Although most of the above work were focused on horizontal or vertical plates (Li et al., 2010;Shen et al., 2012;Teng et al., 2010;Wu et al., 2009;Kang and Wang, 2002;Zhang, 1990;An and Faltinsen, 2013;Sinha et al., 2003), few investigation is reported on inclined plates. However, the inclined plate is commonly used in ocean engineering, for example, inclinedplate type breakwaters, ship fin stabilizers, wings of underwater vehicles, and paddler type wave-maker. All these could be simplified as inclined plate models.
In Shaw's (1985) pioneer hydrodynamic work about inclined plate, he considered the thin plate being vertically inserted into water with small inclination angle, the transmission and reflection coefficients were thus obtained by a perturbation expansion method. Based on Shaw's method, Mandal and Chakrabarti (1989) improved the first-order correction on the transmission and reflection coefficients by Green theorem. Mandal and Kundu (1990) extended the above algorithm to a submerged slightly inclined plate. By constructing hyper-singular integral equation and approximating discontinuous velocity potential on both sides of the thin plate by Chebyshev polynomials. Parsons and Martin (1994) studied the transmission and reflection coefficients of inclined thin plate that partly extended the above free surface in deep water. Midya et al. (2001) extended the methods of Parsons and Martin (1994) to a submerged inclined plate in shallow water. The transmission and reflection coefficients as well as wave forces were obtained. Kharaghani and Lee (1986) studied inclined plate partly extending above the free surface in heading wave. The wave field was divided into three sub-domains and the finite element method was utilized to obtain the near-field velocity potential while the far-field velocity potential was matched through the eigenfunction expansion method. The transmission and reflection coefficients, wave pressure, and motion responses of plate were obtained. Similarly, Sobhani et al. (1988) investigated wave transmission of two different types of inclined plate and the result was verified by the model test. Cho and Kim (2008), combining the boundary element method with eigenfunction expansion method, studied the transmission coefficient, reflection coefficient, and wave forces of the floating inclined plate fixed before the vertical wall. This method solved the problem by multi-domain boundary element method for the inner field and eigenfunction expansion method for the far field respectively, and the velocity and pressure were matched on the interblock boundaries.
The studies mentioned above were mostly focused on wave transmission and reflection coefficients. As for the thickness of the inclined plate, Parsons and Martin (1994) and Midya et al. (2001) established a hyper-singular integral equation to take the different values of the velocity potential on either side of the thin plate as the unknown, thus the thickness of the thin plate was zero. The mathematical derivation of this method was sophisticated and difficult to apply to real engineering problems. Gayen and Mondal (2016) also used the method of Parsons and Martin (1994) to water wave interaction with two symmetric inclined permeable plates. Some literatures mentioned above did not give detailed explanation on the treatment of the plate thickness. According to numerical experience (Kharaghani and Lee, 1986;Sobhani et al., 1988), if the plate thickness is assumed very small, the CPU cost will be very high and the eigenfunction expansion method or other analytical methods used to solve the far field problem can be fairly complicated.
To completely overcome the above difficulties, we extends the algorithm to inclined plates with '/ \' configuration based on successful work on radiation and diffraction problem of horizontal and vertical plate with the thickness of 0.005 m (Wang et al., 2011a(Wang et al., , 2011b in the present study. Similar to the methods of Saad and Schultz (1986), Saber-iNajafi and Zareamoghaddam (2008), a fast iterative solver, generalized minimum residual (GMRES) method is used to solve the problem of large number elements in BEM. Section 2 elaborates the mathematical formulae and numerical methods. Results are discussed in Section 3, and conclusion and remarks are drawn in Section 4.

Governing equation and boundary conditions
Double inclined plates with '/ \' configuration are located below free surface and each plate is assumed thin and rigid (see Fig. 1). The Cartesian coordinate system is stationary relative to the undisturbed free surface. The origin is located on the static water surface with the y axis positively upward. The inclined angle between the y axis and the plate is α. The length of each plate is B; the thickness is T T ; the submergence between the top ends of either plate is H s ; the horizontal distance between the top ends of two plates is T s .
Suppose that the fluid is inviscid and incompressible, its motion is irrotational and the object can move harmonically.

ϕ(x, y)
The fluid velocity can be expressed as the gradient of space complex velocity potential , which satisfies the following governing equation and boundary conditions: Based on linear motion assumption, the complex velocity potential can be decomposed as follows: where is the object motion amplitude, is wave frequency, denotes the radiation potential with unit amplitude and . ϕ j The radiation potential satisfies the following boundary condition: ∂ϕ n 1 , n 2 , and n where are components of a normal vector on the object surface, and is the rotation center of the object. The added mass and damping coefficients are given by:

Boundary integral equation ϕ j
The radiation potential in Eq. (4) can be solved by the following boundary integral equation according to Green's Theorem.
where , is the field point, is the source point, is Green function and C is solid angle. The Green function can be expressed in the following form where is the wave number in deep water.

Discretization of the boundary integral equation
Divided integral surface of the object into a series of linear elements, the coordinates and physical quantities on each element can be expressed as follows: are shape functions and can be given as: (10) (x 1 , y 1 ) and (x 2 , y 2 ) In Eq. (9), are node coordinates of element j, l N is the element length and ζ is local coordinate.
Substituting Eq. (9) and Eq. (10) into Eq. (5), and Eq. (5) can be written in following form: where log r 1 log r 2 I C In Eq. (12), the coefficients containing term can be obtained through analytic methods. and the wave term are regular with no singularity and can be obtained directly through numerical methods (Wang et al., 2011a).

Numerical verification
To verify the efficiency of present method, a reflection coefficient comparison of single submerged inclined plate is carried out with that of Midya et al. (2001), where d is submergence, h is water depth, a is half of the plate length and the inclined angle α is redefined as in Fig. 2. The plate thickness is 0.005 m, and the total element number on the plate is 200. From Fig. 2, it is found that the two results agree very well.
cients of double submerged inclined plates with different inclined angles for sway, heave and roll. The axis x presents the dimensionless plate length . As shown in Fig. 3, within the range of , the added mass for sway increases to the peak of 0.25 at and then decreases to 0.13. For , the added mass remains nearly constant with the increase of . For large inclined angle of the plate, the added mass for sway will decrease. It is shown that the added mass coefficients of and are much closer than that of and   WANG Ke, ZHANG Zhi-qiang China Ocean Eng., 2018, Vol. 32, No. 1, P. 85-89 . From Fig. 4 for , the damping for sway is zero, which means there is no outgoing waves. With the increase of , the damping increases to of 0.09 at KB/2 = 0.5 and then decreases. Within the range of , the damping with different inclined angles almost coincides, which indicates that the change of inclined angles has less influence on the damping. For , the increasing inclined angle will decrease the damping.
Figs. 5 and 6 show the added mass and damping coefficients for heave. In the figures, the coefficients for heave are both small. However, the added mass and damping coefficients for heave are most sensitive to the change of the inclined angle. As shown in Fig. 5, when the inclined angle , the added mass for heave is nearly zero. When the inclined angle , the maximum value of the added mass for heave is 0.06. In Fig. 6, within the range KB/2 < 0.7, the damping for heave is 0.0015 when . The damping coefficient increases with inclined angles for . When , the damping remains below 0.0005, while the maximum damping reaches about 0.0015 when , nearly three times that of . When , the damping reaches the peak of 0.0065, nearly 13 times that of . Another phenomenon worth to be noted is that when , the damping for heave is zero, which means there is no outgoing waves for all inclined plates.
Figs. 7 and 8 show the added mass and damping coefficients for roll. It is shown that the change of added mass and damping coefficients for roll is similar to that for sway. The wave frequencies at the extreme values of the added mass and damping coefficients for sway and roll are the same. The added mass for roll reaches the peak of 0.049 when while the damping reached the maximum of 0.017 when . Within the whole frequency range, the added mass for roll will increase inclined angles. However, the inclined angles only influence the damping within the range of 0.2<KB/2<1.4. Thus, the added mass and damping coefficients for sway are critical.

Concluding remarks
By use of the boundary element method, the added mass and damping coefficients of the double submerged inclined plates are investigated .The findings are as follows: (1) The added mass and damping coefficients for sway are the largest.
(2) The added mass and damping coefficients for heave are most sensitive to the change of the inclined angle.
(3) The wave frequencies of maximal values of the ad-    ded mass and damping coefficients for sway and roll are the same.