1 Introduction

In Da-yao Bay of Dalian, China, a comb-type caisson breakwater has been built to meet local engineering requirements (Niu et al., 2003). A comb-type caisson consists of a rectangular caisson and two vertical side plates,as idealized in Fig.1. In practice, the side plate may be directly connected to the seabed (non-open type) or partially immersed (open-type). In comparison with a traditional vertical wall breakwater, comb-type caisson breakwater features better stability against sliding and requires lower foundation bearing capacity (Li et al., 2002). Knowledge on the hydrodynamic performance of comb-type caisson breakwaters bears significance for engineering design and applications.

Some researchers have studied oblique wave interaction with comb-type caisson breakwaters based on experimental tests and numerical simulations. Zhu et al.(2001) summarized the wave action mechanism and sheltering function of comb-type caisson breakwaters. Li et al.(2002) and Dong et al. (2003) carried out a series of experimental tests to examine the hydrodynamic performance of comb-type caisson breakwaters. They developed simple empirical formulas to estimate wave forces acting on comb-type caisson breakwaters. Zhang et al. (2002)used finite element method to analyze wave-induced stress distributions on side plates of comb-type caissons. Fang et al. (2010) numerically examined reflection and transmission coefficients and wave forces on open-type combtype caisson breakwaters. In addition to experimental tests and numerical simulations, analytical studies on hydrodynamic performance of comb-type caisson break

waters have been rarely reported. Evans and Linton (1993)developed an analytical solution for edge waves along a periodic rectangular coastline, which is actually similar to comb-type caisson breakwater. However, Evans and Linton (1993) only focused on edge waves propagating along the coastline and decaying exponentially away from the coastline and did not examine scattering waves and wave forces acting on the structure.

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This study developed an analytical solution for oblique wave scattering by comb-type caisson breakwaters. Different from previous numerical solutions of comb-type caisson breakwaters, the present analytical method directly develops series expressions of velocity potentials to better understand the hydrodynamic performance of comb-type caisson breakwaters. The present problem can also be solved by other semi-analytical methods, such as the composite model developed by Zhao and Teng (2004).Initially, only the non-open comb-type caisson breakwater(idealized in Fig.1) is examined. The boundary value problem for the present problem is formulated by introducing a periodic boundary condition (e.g., Evans and Fernyhough, 1995; Linton and McIver, 2001; Porter and Evans, 2005; Cal et al., 2014). Then, boundary value problem is solved by matched eigenfunction expansions,and reflection coefficient and wave forces acting on breakwater are estimated. In Section 3, convergence of analytical solution is examined. The solution is validated by a multi-domain boundary element method (BEM) solution (e.g., Ang, 2007; Liu et al., 2012; Chuang et al.,2015; Brebbia and Walker, 2016). Section 4 provides numerical examples with discussions. Some valuable results are recommended for engineering design. Finally, the main conclusions of this study are drawn.

2 Analytical Solution

2.1 Boundary Value Problem

For the present solution based on potential theory, no wave energy dissipation occurs near the breakwater. All incident wave energy is reflected by the breakwater along different directions. Thus, total reflection coefficient of comb-type caisson breakwater is unity in theory. Our calculation results also show that total reflection coefficients estimated by Eq. (29) at different conditions equal to unity.

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Fig.1 Idealized sketch for oblique waves interaction with a comb-type caisson breakwater (only six entire caissons are plotted in the y-direction).

The present problem is solved by linear potential theory. Wave motion in the whole fluid domain is described by velocity potential Φ(x, y, z, t). By further considering linear time-harmonic incident waves, time factor and depth-dependent variable can be separated from velocity potential:

where Re denotes real parts of variables, φ(x, y) is a complex spatial velocity potential for plane waves, ω represents angular frequency, and k is the incident wave number and satisfies the following equation:

in which g is gravitational acceleration. Linear free surface condition of water wave motions and non-penetration condition on the seabed have been used for developing Eqs. (1) and (2).

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Spatial velocity potential φ(x, y) satisfies Helmholtz equation:

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Magnitude of total horizontal wave force Fy in the y-direction and acting on each comb-type caisson is given by the following calculations:

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with m=0, ±1, ±2, ··, velocity potential must also satisfy far-field radiation conditions.

As shown in Fig.1, geometry of comb-type caisson breakwater varies periodically (with periodicity l) along the y-direction. Thus, velocity potential satisfies the following periodic boundary condition (Evans and Fernyhough, 1995; Linton and McIver, 2001; Porter and Evans,2005; Liu, 2016):

where ky=ksinθ0 is the component of wave number k in the y-direction. Using this periodic boundary condition, we can solve the present problem only in the strip of y∈[0, l]and then extend the solution to other fluid domains. For the convenience of solution, we divide the fluid domain in the strip of y∈[0, l] into two regions: region 0, the fluid domain in front of the breakwater (x ≤ −B, 0 ≤ y ≤l); and region 1, the fluid domain between two adjacent caissons(−B ≤ x ≤ 0, 0 ≤ y ≤ 2a).

2.2 Matched Eigenfunction Expansions

In the outer region 0, the series solution of velocity potential satisfies Eqs. (3), (7), and the far-field radiation condition and can be written as follows:

where kx=kcosθ0 is the component of wave number k in the x-direction; Sm (m = 0, ±1, ±2,··) are unknown expansion coefficients; Em(y) are eigenfunctions in the y- direction and given by the following equations:

On the right-hand side of Eq. (8), the first term denotes incident waves from open sea, and the second term includes two different modes. The modes corresponding to γm ＞ 0 are evanescent modes, which decay exponentially with increasing distance from the breakwater. The modes corresponding to γm ＜ 0 are reflected waves propagating along different directions. Following Fernyhough and Evans (1995) and Liu et al. (2016), the total number of reflected waves equals M1 + M2 + 1 with

where An (n = 0, 1, 2,··) are unknown expansion coefficients, and Cn (y) are eigenfunctions given by the following:

Thus, the total number of reflected waves is only determined by the wave number k, wave incident angle θ0,and comb-type caisson length l.

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In the inner region 1, velocity potential that satisfies boundary conditions in Eqs. (3), (5) and (6) features the following form:

where Int denotes the integer part of variables. In addition,the included angle θn (−M1 ≤ n ≤ M2) between propagation direction of nth reflected wave and x-axis is determined by

Eigenfunctions Cn(y) and Em(y) satisfy the following orthogonal relations:

where the asterisk denotes the complex conjugate. These relations will be used in determining unknown expansion coefficients in velocity potentials.

Preceding expansions of velocity potentials agree with symmetric velocity potentials in the work of Fernyhough and Evans (1995) on wave reflection and transmission by equally spaced rectangular blocks. Fernyhough and Evans(1995) used Galerkin approximation method to determine unknown expansion coefficients in velocity potentials and considered cube-root singularity of fluid velocity near block corners. For the present engineering analysis of comb-type caissons, we use a simple and direct matched eigenfunction expansion method to determine unknown expansion coefficients in velocity potentials and to estimate hydrodynamic quantities (reflection coefficients and wave forces) of comb-type caisson breakwater.

On the interface between two regions (x = −B), velocity potentials satisfy continuous pressure and velocity conditions:

Inserting velocity potentials in Eqs. (8) and (11) into Eqs. (16) and (17), respectively, yields the following:

Multiplying both sides of Eq. (18) by Cn(y) and integrating with respect to y from 0 to 2a, then using Eq. (15)and truncating m and n to ±M and N, respectively, we obtain the following:

Multiplying both sides of Eq. (19) by ） and integrating with respect to y from 0 to l, then using Eq. (14)and truncating m and n to ±M and N, respectively, yields:

We simultaneously solve Eqs. (20) and (23) using Gaussian elimination method to determine expansion coefficients Sm and An. Once velocity potentials are determined,all hydrodynamic quantities can be estimated.

2.3 Hydrodynamic Quantities

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where ρ represents fluid density, and Cg stands for wave group speed (identical for all propagation waves). As previously explained in Eq. (8), nth reflected wave leaves the comb-type caisson breakwater at an included angle θn to the x-axis. However, nth reflected waves cannot be directly observed physically because all reflected waves are superposed (See Fig.5). Reflection coefficient Rn of nth reflected wave is defined as follows:

Total reflection coefficient of all reflected waves is defined according to the following:

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Fig.1 shows the interaction between oblique incident waves and a comb-type caisson breakwater. The breakwater is constructed by an array of uniform comb-type caissons. For each comb-type element, length of rectangular caisson is b, and length of side plate equals a. Thus,total length of a comb-type caisson is l = 2a + b. Width and length of a wave chamber formed by adjacent caissons are B and 2a, respectively. Water depth is a constant d.Cartesian coordinate system is adopted with x-y plane in the still wave level and z-axis measuring vertically upward. Origin is located at the intersection of a side plate, a rectangular caisson vertical face, and still wave level. A wave train with wave height H and wavelength L approaches the breakwater at an angle θ0 (0 ≤ θ0 ＜ π/2) to the positive x-axis.

Wave pressure on comb-type caissons can be calculated by the Bernoulli equation:

Then, wave forces are determined by integrating wave pressures along the surface of breakwater.

Energy flux is defined as the rate of working hydrodynamic force on a vertical plane of unit width perpendicular to wave propagation direction, whereas average energy flux is determined by averaging energy flux over a wave period (Dean and Dalrymple, 1991). Average energy fluxes of the incident wave and nth reflected wave in the x-direction are respectively given by the following:

Magnitude of total horizontal wave force Fx in the xdirection and acting on each comb-type caisson is calculated by the following equation:

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where

Fx1, Fx2, and Fx3 correspond to wave forces acting on the up-wave side plate, rectangular caisson, and lee-wave side plate, respectively.

On surfaces of rectangular caissons and side plates,velocity potential satisfies non-penetration boundary conditions:

Dimensionless total wave forces CFx and CFy in the xand y-directions, respectively, are defined as follows:

When wave chamber width B equals zero, the present comb-type caisson breakwater becomes a traditional vertical wall breakwater. Under this condition, total dimensionless horizontal wave force acting on a caisson can be explicitly calculated by the following (Li and Teng, 2015,Chapter 2.2):

3 Validations

3.1 Convergence Examination

To ensure convergence of the present analytical solution, truncated numbers M and N used in Eqs. (20) and(23), must be carefully determined. In our calculations,we set 2M = N. Table 1 summarizes typical calculation results for the dimensionless wave force CFx at different values of M. Calculation conditions are as follows: a = 3 m,b = 12 m, B = 7 m, d = 10 m, and θ0 = π/4. As shown in Table 1, M = 50 ensures solution convergence. Thus, truncated number M = 50 is used in the following calculations.

Table 1 Convergence of CFx with increasing values of M

Truncated number M kd=CFx 0.5 kd=1.0 kd=2.0 kd=4.0 kd=6.0 5 0.9098 0.6929 0.4049 0.1513 0.0751 10 0.9098 0.6928 0.4065 0.1511 0.0750 20 0.9098 0.6928 0.4070 0.1511 0.0750 40 0.9098 0.6928 0.4072 0.1511 0.0750 50 0.9098 0.6928 0.4073 0.1511 0.0750 60 0.9098 0.6928 0.4703 0.1511 0.0750

3.2 Limiting Cases

For normal incident waves (θ0 = 0), calculation results for dimensionless wave force CFy in the y-direction are constantly zero. Reflection coefficients of positive and negative nth (n ≠ 0) reflected waves are the same, and zeroth reflected wave propagates along the normal direction of breakwater. All these results should be valid in physics.

Free surface elevation at any point in the fluid region is estimated by the following equation:

The present calculation results of CFx at B = 0 are the same as that calculated by Eq. (40).

3.3 Comparisons with Multi-Domain BEM Solution

Apart from the analytical solution, we also use a multidomain BEM method to solve wave interaction with breakwater. The present multi-domain BEM solution is slightly modified by our previous solution for a perforated caisson breakwater with perforated partition walls(Liu et al., 2016). Multi-domain BEM solution is obtained in the strip fluid domain of y∈[0, l], which is the same as that of the analytical solution. However, a fictional boundary is set at the position distant from the breakwater. Hankel function of the first kind of zero order is adopted as fundamental solution of Helmholtz Eq. (3).Boundary curves of inner fluid regions are discretized by constant elements. Hydrodynamic quantities of CFx, CFy,and Rn calculated by the present analytical solution and multi-domain BEM solution are compared in Fig.2 at: a =3 m, b = 10 m, B = 7 m, d = 10 m, and θ0= π/3. As shown in the figure, lines and dots denote analytical and multi- domain BEM solutions, respectively. The two solutions agree excellently. This agreement indicates accuracy of procedure of the present analytical solution. In comparison with multi-domain BEM solution, the solving procedure of the present analytical solution is much simpler.The present analytical solution directly provides the series solution of velocity potentials and thus can yield better insights into wave scattering process.

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4 Results and Discussion

In this section, we present numerical examples to examine the hydrodynamic performance of comb-type caisson breakwater. Reflection coefficients, surface elevations near the breakwater, and wave forces acting on the breakwater are all examined. In the following calculations, wave depth d = 15 m and rectangular caisson length b = 10 m are used unless otherwise noted.

4.1 Reflection Coefficients and Wave Surface Elevations

Fig.3 illustrates variations in reflection coefficient Rn for n-th order reflected wave versus the dimensionless wave number kd. As shown in this figure, only one reflected wave (zero-order reflected wave) exists for low-frequency incident waves (small kd), and reflection coefficient R0 equals to unity. When incident wave number increases, diffuse reflection with multiple reflected waves occurs, and variations in reflected coefficients versus kd become complex. As shown in Fig.3a, the nega-tive first-order reflected wave occurs at kd = 4.05. As depicted in Fig.3b, the negative first-order and second-order reflected waves occur at kd = 3.15 and 6.15, respectively.By further comparing Figs.3a (l = 12 m) and 3b (l = 14 m),diffuse reflection occurs at smaller kd with increasing caisson length l.

Fig.2 Comparisons between the present analytical solution and multi-domain BEM solution.

Fig.3 Reflection coefficient Rn of comb-type caisson breakwater: θ0 = π/4.

For the comb-type caisson breakwater examined in Fig.3a, dimensionless surface elevation(A is incident wave amplitude) near the breakwater at kd = 1.5 and kd =4.5 are plotted in Figs.4 and 5, respectively. Given the periodicity of breakwater, only three entire comb-type caissons are plotted. As presented in Fig.4, only one reflected wave exists in front of the breakwater, i.e., the zeroth-order reflected wave. Then, free surface elevation leaving a gap from the comb-type caisson breakwater is similar to specular reflection at a traditional vertical wall breakwater. As presented in Fig.5, diffusion reflection occurs with two reflected waves along different propagation directions. Reflection coefficients R0 and R−1 in Fig.5 reach 0.524 and 0.852, respectively. As a result of diffusion reflection, surface elevation in Fig.5 is more complicated than that in Fig.4. Diffusion reflection may significantly change the wave forces acting on comb-type caisson breakwater.

Fig.4 Contours of dimensionless free surface elevation and B = 12 m.

Fig.5 Contours of dimensionless free surface amplitudeand B = 12 m.

4.2 Total Wave Forces on Comb-Type Caissons

Fig.6 shows the dimensionless total wave force CFx in the x-direction at different wave chamber width B. As displayed in the figure, curve of B = 0 m denotes a traditional vertical wall breakwater, and the corresponding wave force CFx decreases monotonously with increasing wave number kd. However, wave force CFx on comb-type caisson breakwater can attain a valley value with increasing wave number kd. In addition, the wave number kd corresponding to the valley value of CFx decreases significantly with increasing chamber width B. For long period waves at small values of kd, wave forces acting on comb-type caisson breakwater and traditional vertical wall breakwater are almost the same. At moderate values of kd, the wave force acting on comb-type caisson breakwater is significantly reduced. This result bears significance in enhancing breakwater stability. However, the wave force acting on comb-type breakwater may reach a peak value for high-frequency waves and may be larger than the wave force acting on traditional vertical wall breakwater. This condition mainly results from diffusion reflection.

Fig.6 Dimensionless total wave force CFx at different wave chamber widths B: θ0 = π/4.

Fig.7 Dimensionless total wave force CFy at different wave chamber widths B: θ0 = π/4.

Fig.7 presents the dimensionless total wave force CFy in the y-direction at different wave chamber width B.When wave chamber width equals zero, wave force CFy is also zero. Thus, this result is not plotted in Fig.7. The figure shows that with increasing wave number kd, wave force CFy can reach a large peak. When wave chamber width B increases, peak value of CFy also increases significantly, and the wave number corresponding to the peak decreases. Notably, a large peak forms at kd = 1.5–3.0, which corresponds to usual engineering conditions.Thus, one should pay more attention to the peak value of CFy in practical engineering design.

As presented in Figs.6 and 7, only two side plate lengths of a = 2 m and a = 4 m are considered. Wave forces CFx and CFy at different side plate lengths are further examined in Fig.8. The figure shows that when side plate length increases, total wave force CFx generally decreases slightly, but the decrease in peak value of CFy is notable.In comparison with side plate length a, wave chamber width B exhibits more significant effect on wave force CFx by comparing Fig.8 with Figs.6 and 7.

Fig.9 shows wave forces CFx and CFy at different wave incident angles θ0. For normal incident waves (θ0 = 0),wave force CFy in the y-direction is zero and is not plotted in Fig.9b. Fig.9a displays that when wave number kd is smaller than 2.0, waves with larger incident angle can result in smaller wave force CFx. However, this result is the opposite for high-frequency incident waves. Fig.9b indicates that when wave incident angle increases, peak value of CFy first increases, attains its maximum value,and then decreases. Overall, wave forces are not necessarily reduced when wave incident angle increases based on the analyses above.

Fig.8 Dimensionless total wave forces CFx and CFy at different side plate lengths a: θ0 = π/4 and B = 12 m.

Fig.9 Dimensionless total wave forces CFx and CFy at different wave incident angles θ0: a = 4 m and B = 12 m.

4.3 Wave Forces on Side Plates

The two side plates are weak components in a combtype caisson under wave actions. Thus, we further examine wave forces acting on the side plates. In the present study, we define dimensionless wave forces on the side plates as follows: CFx1=|Fx1|/ρgHld (acting on the side plate located on the up-wave side) and CFx3=|Fx3|/ρgHld(acting on the side plate located on the lee-wave side).Fig.10 plots calculation results of CFx1 and CFx3. As shown in the figure, wave forces acting on the two side plates are close, and they are far smaller than the total wave force CFx on the whole comb-type caisson. For lowfrequency waves, wave frequency (wave number) exerts minimal effects on CFx1 and CFx3. When wave number kd exceeds a certain value, values of CFx1 and CFx3 rapidly decrease with increasing kd. Finally, the trends of wave forces on two sides move toward consistency.

Fig.10 Dimensionless total wave forces CFx1 and CFx3 acting on the side plates: θ0 = π/4 and B =12 m.

5 Conclusions

We have used matched eigenfunction expansion method to develop an analytical solution for oblique wave scattering by a comb-type caisson breakwater. Periodic boundary condition has been directly incorporated into the solution. Hydrodynamic quantities of reflection coefficients,surface elevations, and wave forces acting on the combtype caisson breakwater have been estimated. Convergence of the present analytical solution is satisfactory.Hydrodynamic quantities calculated by the present analytical solution are in excellent agreement with numerical results by a multi-domain BEM solution.

Given the periodicity of comb-type caisson breakwater,diffusion reflections with multiple reflected waves along different propagation directions have been observed. Reflection coefficient of each reflected wave and free surface elevations around the breakwater have been shown.Numerical examples have also been presented to examine wave forces acting on com-type caissons. Among wave chamber widths, side plate length, and wave incident angle, wave chamber width may pose the most significant effect on total wave forces acting on comb-type caissons.The total normal wave force acting on a suitably designed comb-type caisson can be much smaller than that on a traditional caisson. This result bears significance in enhancing breakwater stability. Total wave forces acting on the two side plates are close and much smaller than the total normal wave force on whole comb-type caissons.The results given in this study should be valuable for engineering design and application of comb-type caissons.This study have only considered linear waves. In future work, numerical simulations for highly nonlinear wave interaction with comb-type caisson breakwaters may be conducted.

Acknowledgements

This study was supported by the National Natural Science Foundation of China (Nos. 51490675, 51322903 and 51279224.

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