Interaction of oblique waves and a rectangular structure with an opening near a vertical wall

With the method of separation of variables and the eigenfunction expansion employed, an analytical solution is presented for the radiation and diffraction of a rectangular structure with an opening near a vertical wall in oblique seas, in which the unknown coefficients are determined by the boundary conditions and matching requirement on the interface. The effects of the width of the opening and the angle of incidence on the hydrodynamic characteristics of a rectangular structure with an opening near a vertical wall are mainly studied. The comparisons of the calculation results with wall-present and with wall-absent are also made. The results indicate that the variation trends of the heave added mass and excitation force with wall-present are almost the same as those with wall-absent, and that the peak values in the former case are obviously larger than those in the latter due to the reflection of the vertical wall.


Introduction
The rectangular structure with a moon pool is widely used in the engineering ships due to its simple structure. Thus it is very necessary to investigate the wave interaction with rectangular structures. Up to now, many researchers have studied this problem. The studies include the wave radiation due to oscillations of the horizontal rectangular structures (Bai, 1977;Lee, 1995), the wave radiation and diffraction by an infinitely long rectangular structure floating on the free surface (Zheng et al., 2004). Compared with the intact rectangular structure, the moon pool structure has attracted more and more researchers to investigate its complicated inside fluid movement, such as the water motions in a moon pool (Aalbers, 1984), the piston and sloshing modes in moon pools (Molin, 2001), the wave radiation and diffraction problem of a two-dimensional rectangular body with an opening at its bottom . By considering most of incident waves are oblique in the actual seas, some researchers have studied the wave interaction with the structure in oblique seas with analytical method. The studies include that the wave radiation by an infinitely long rectangular structure floating on the free surface in ob-lique seas (Zheng et al., 2006), the radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas of finite depth (Zheng et al., 2007), the wave radiation and diffraction by a floating rectangular structure with an opening at its bottom in oblique seas (Yang et al., 2017), the wave force on an infinitely long circular cylinder in an oblique sea (Bolton and Ursell, 1973), the interaction of oblique waves with an infinite cylinder (Garrison, 1984), the wave diffraction by thin vertical barriers in waters of finite depth in oblique seas (Mandal and Dolai, 1994), and the effects of a fixed vertical barrier on obliquely incident surface waves in deep water (Evans and Morris, 1972).
The focuses of the above works are on the floating bodies which are in unbound domains. However, there are many cases that the structures are in front of a vertical wall. Thus we should study the effect of a vertical wall on the structures. The related studies include the hydrodynamic coeffcients for an oscillating rectangular structure on a free surface with sidewall (Hsu and Wu, 1997), the interactions between regular obliquely incident waves and vertical wall (Li et al., 2001), the interactions between irregular ob-liquely incident waves and vertical wall (Li et al., 2002), and the wave radiation and diffraction by a two-dimensional floating body with an opening near a side wall .
In this paper, we investigate the linear wave radiation and diffraction of a rectangular structure with a moon pool in front of a vertical wall in oblique seas. Firstly, the whole fluid domain is divided into seven subdomains, and the expressions for the velocity potentials in each subdomain are obtained by the method of separation of variables and the eigenfunction expansion. The wave excitation forces are calculated from the incident and scattered potentials. Secondly, the effects of the width of opening and the angle of incidence for the hydrodynamic characteristics on the rectangular structure with a moon pool are studied mainly. In addition, the calculation results in this paper are compared with those in the absence of a vertical wall, which are obtained from Yang et al. (2017). Finally, the conclusions are given.

Governing equation and boundary conditions
An infinitely long rectangular structure with the width 2b and draught h 1 is floating on the waters with constant depth h. The structure has a hole with the width 2a at its bottom. A Cartesian coordinate system shown in Fig. 1 is employed with the origin at the undisturbed water surface. The z-axis points upwards and the x-axis directs to the right.
Here it is assumed that the structure is infinitely long in the y direction and the incident wave direction forms an angle θ (0°<θ<90°) with the x-axis.
Assuming the fluid is inviscid, incompressible, the flow is irrotational, and the incident wave and the motions of the structure are small, the fluid flow can thus be described by the theory of velocity potential. According to Yang et al. (2017), the total velocity Φ can be expressed as: where k is the wave number, t is the time, Re [ ] denotes the real part of a complex function, , ω is the angular frequency, and are the spatial velocity potentials 2,3,4) due to the incident and scattered waves, are the spatial velocity potentials for unit body displacement corresponding to heave (vertical), sway (horizontal) and roll (rotational) motions, and are the corresponding motion amplitudes, and is the amplitude of the incident wave. Each of these potentials satisfies the following Helmholtz equation ) and the boundary conditions , g is the acceleration of gravity, S b is the wetted body surface. The incident wave can be written as: The dispersion relation is expressed as: Similar to Zhang and Zhou (2013), the following boundary conditions on the body surface can be obtained ∂ϕ (l) where δ jl is the Kronecker delta function given by: 3 Solution procedure

Expressions for the potentials
To obtain the radiated and diffracted potentials, the fluid domain is divided into seven subdomains, i.e. I (  ), II  ,  III , IV , V , VI and VII ( ), respectively. The radiated potentials and the diffracted potentials in the seven subdomains  )   7   are denoted by  ,  ,  ,  ,  , , and , respectively. The method of separation of variables is applied in each subdomain to obtain the expressions for the unknown radiated potentials.
Based on the method of separation of variables and taking into account the boundary conditions, the potential in each subdomain can be expressed as: , and are the unknown coefficients, λ n , γ n , β n , μ n , α n and ν n are given by , n = 1, 2, 3, ...
, are the particular solutions, according to Yang et al. (2017), they can be written as: (30) 3.2 Solution for the unknown coefficients Using the continuity conditions of the velocity and pressure at x=±b and x=±a, we have (43) By substituting the expressions for the velocity poten-tial, Eqs. (13)-(19) into Eqs. (32)-(43) and then using the orthogonality of the trigonometric functions, the differential equations can be converted into a series of linear system of equations. For practical computation, we shall take the first N terms in the infinite series. This can yield a linear system of 12N equations, and then the radiated potentials and the diffracted potentials in each subdomain can be computed by the method of Gaussian elimination. The reader can find the relevant details in Zhou et al. (2013).

Wave radiation force
Once the velocity potential is known, the hydrodynamic force on a body can be obtained from the integration of the pressure based on the linear Bernoulli equation over the wetted body surface. The wave radiation force can be written as: in which ρ is the density, τ ij is the force in mode i due to the motion of unit amplitude in mode j.
The added mass μ ij and damping coefficients λ ij can be calculated by the wave radiation force. They are generally called hydrodynamic coefficients, and can be nondimensionalized

Wave excitation force
Wave excitation force is the force due to the incident wave acting on a body which is stationary. It can be computed from the incident wave potential and the diffracted potential. We have the vertical component the horizontal component and the rolling moment about the origin (0, 0) .
(49) The wave excitation force can be non-dimensionalized

Convergence study
The infinite series in subsection 3.2 are truncated to keep the first N terms. A numerical experiment is performed to test the convergence of the hydrodynamic coefficients and wave excitation forces with N terms. The results with N=50 and N=100 terms are in very good agreement, which means that N=50 can provide sufficient accuracy. In the following, all the hydrodynamic coefficients and wave excitation force were calculated with N=50.

Symmetry of hydrodynamic coefficients
It is well known that τ ij =τ ji (e.g. Mei, 1989). These identities can be used for further verification. Comparisons for the coupling between sway and roll motions are presented in Fig. 2, and very good agreement can be found. In the paper, D is taken as 3h 1 for all of the cases with wall-present.

Discussions
In this subsection, the effects of the width of opening, the angle of incidence on hydrodynamic coefficients and wave forces are discussed. In addition, the comparisons between the hydrodynamic coefficients and wave forces in the presence of a vertical wall and those in the absence of a vertical wall are made. The purpose is to understand the hydrodynamic behavior of the structure under different conditions in oblique seas, which is important for ocean engineering.

Hydrodynamic effects of the angle of incidence
Three different cases are considered, namely, the angle of incidence θ is taken as 15°, 45°, and 75°, respectively, to illustrate the hydrodynamic effects of the angle of incidence. The heave added mass and damping coefficient at different angles of incidence are shown in Figs. 3a and 3b, respectively. It can be seen from Fig. 3a that there are peaks in μ 11 when 2kb ≈ 0.575, 0.617, 0.662 for θ=15°, 45°, 75°, respectively. Figs. 3a and 3b show that the variation trends of hydrodynamic coefficients are similar at different angles of incidence. From Figs. 3c to 3f, it is found that the added mass for the sway and roll motion modes at different angles of incidence is quite different when 2kb>2.25. However, the difference of the damping coefficients is small. More specifically, from Figs. 3c and 3e, we can see that the added mass has a large fluctuation when the angle of incidence is 15° or 45°. All these suggest that the difference of the angle of incidence does significantly affect the resonant frequency of the sway and roll modes.
The real part and imaginary part of the vertical force at different angles of incidence are shown in Figs. 4a and 4b, respectively. It can be seen from Fig. 4a that the absolute values of the peaks in Re(f 1 ) are 0.906, 0.660, 0.234 for θ=15°, 45°, 75°, respectively, followed by a decline of 27.2% and 64.5%, respectively; and from Fig. 4b that the absolute values of the peaks in Im(f 1 ) are 0.852, 0.573, 0.155 for θ=15°, 45°, 75°, respectively, followed by a decline of 32.7% and 72.9%, respectively. Figs. 4c-4f indicate that the results of real and imaginary parts of the horizontal force and the roll moment are also obviously different from each other at different angles of incidence. Figs. 4c and 4e show that the absolute values of the peaks of the real parts of horizontal force component and roll moment are 1.891, 1.607, 1.186 and 1.527, 1.298, 0.958 for θ = 15°, 45°, 75°, respectively, when 2kb = 0.535; the declines are 15. 02%,26.20% and 15.00%,26.19%,respectively. Figs. 4d and 4f show that the frequencies of the first peak of the imaginary parts of the horizontal force component and roll moment are similar to those of the real parts, and that the curve of the imaginary part appears obviously different Fig. 2. Symmetry of the added mass and damping coefficients between the coupled sway and roll motions with θ=30°, h/h 1 =3, b/h 1 =0.5, and a/b=0.5. Fig. 3. Effect of the angle of incidence on hydrodynamic coefficients with h/h 1 =3, b/h 1 =0.5, a/b=0.5 when the angle of incidence varies.
The variation of the heave hydrodynamic coefficients and the wave excitation forces with the angle of incidence are shown in Fig. 5. From Figs. 5a and 5b, we can see that the heave added mass has a little variation and that the damping coefficient has no change basically, when 2kb is taken a larger value. It shows that the heave hydrodynamic coefficients vary apparently with the increase of the angle of incidence if 2kb is close to the value at which the peak of the heave added mass appears. Figs. 5c and 5d indicate that both the vertical and horizontal forces decrease with the increase of the angle of incidence although 2kb is taken different values. Specially, the effects of the angle of incidence on the excitation forces are very obvious when the angle of incidence is between 45° and 90°.

Hydrodynamic effects of the width of the opening
The simulations are made when a/b are taken 0.2, 0.4, 0.6, and 0.8, respectively. The results for the heave added mass and damping coefficient at different a/b are shown in Figs. 6a and 6b, respectively. It can be seen from Fig. 6a that there are peaks in μ 11 at 2kb ≈ 0.498, 0.535, 0.617, and 0.617 for a/b = 0.2, 0.4, 0.6, and 0.8, respectively. It is shown in Fig. 6b that  The real part and the imaginary part of the vertical force in the case of different a/b are shown in Figs. 7a and 7b, respectively. Since the force is from the pressure over the horizontal bottom, the opening evidently has a large effect. It can be seen from Fig. 7a that the absolute values of the peaks in Re(f 1 ) are 2.864, 1.111, 0.518, and 0.138 for a/b = 0.2, 0.4, 0.6, and 0.8, followed by a decline of 61.2%, 53.3%, and 73.4%, respectively; and from Fig. 7b the absolute values of the peaks in Im(f 1 ) are 1.684, 1.020, 0.473, and 0.127, followed by a decline of 39.4%, 53.6%, and 73.2%, respectively. Figs. 7c-7f indicate that the results of the real part of the horizontal force and the rolling moment are basically the same at different a/b, and that the results of the imaginary part appear obviously different. Fig. 8 shows two curves of hydrodynamic coefficients of the rectangular structure with a moon pool in oblique seas, one is in the presence of a vertical, and the other is in the absence of a vertical wall. The results in the absence of a vertical wall are obtained from the formulae in Yang et al. (2017). From Figs. 8a and 8b, we can see that the peak val- ues of the heave added mass and damping coefficient in the presence of a vertical wall are approximately twice those in the absence of a vertical wall, respectively, which is due to the reflection of the vertical wall on the right side of waters. Figs. 8d and 8f show that the first peak values of the damping coefficient for the sway and roll modes in the presence of a vertical wall are close to those in the absence of a vertical wall, respectively; Figs. 8c and 8e show that there are several peak values in the curves of the added mass for the sway and roll modes in the presence of a vertical wall.

Comparison of hydrodynamic characteristics with a vertical wall and without a wall
The real parts and the imaginary parts of wave excitation force for different motion modes are shown in Fig. 9.   From Figs. 9a and 9b, it is shown that the absolute values of the peaks of the real part and the imaginary part of the vertical force in the presence of a vertical wall are obviously larger than those in the absence of the vertical wall at 2kb<1, which is also due to the reflection of the vertical wall on the right side of water. From Figs. 9c-9f, it is indicated that the variation trends of the curves of the wallpresent horizontal force and the roll moment are basically the same as those with wall-absent, respectively. We can see that some fluctuations appear in the curves with wall-presence, and that the peak values of the wall-present horizontal forces and roll moments are also obviously larger than those with wall-absence at 2kb<1.

Conclusions
With the method of separation of variables and the eigenfunction expansion, an analytical solution is presented for the radiation and diffraction of a rectangular structure with an opening in front of a vertical wall in oblique seas. The calculated hydrodynamic coefficients and wave excitation forces in the presence of vertical wall are discussed in detail. The calculation results show that the angle of incidence has little effect on the damping coefficients and the variation trends of the heave added mass but a large effect on the added mass for the sway and roll motions. It is shown that the width of the opening has little effect on the damping coefficients but a large effect on the absolute value of the peak of the added mass. Owing to the opening at the structure bottom, both the outside and inside fluid movement can be regarded as a whole along the vertical movement, and the width of the opening thus has a little effect on the horizontal force and sloshing frequency, but a large effect on the vertical force and piston frequency.
The calculation results are also compared with the data of Yang et al. (2017). When 2kb<1, the peak values of the excitation force and the heave added mass obviously increase in the presence of a vertical wall. This is due to the reflection of the vertical wall. It is illustrated that the vertical wall on the right side of water indeed has obvious effects on the motion of rectangular structure with a moon pool in oblique seas. In this study, the calculation results are obtained from the analytical method, which is on the basis of liner wave theory. Thus, the solution should be further examined with the numerical solution or the data of physical experiment, especially when θ>45° and a/b<0.2.