Experimental and Numerical Study on the Hydrodynamic Characteristics of Solitary Waves Passing Over A Submerged Breakwater

In this study, solitary waves passing over a submerged breakwater are investigated both experimentally and numerically. A total of 9 experimental conditions are carried out, including different incident wave heights and water depths. Numerical simulations are performed using a high-order finite-difference model solving Navier-Stokes (N-S) equations. The predicted water wave elevation, velocity and pressure show good agreement with experimental data, verifying the accuracy and capacity of the numerical model. Furthermore, parametric studies are conducted by numerical modelling to examine the effects of the geometrical features of submerged dike on hydrodynamic characteristics around the breakwater.


Introduction
In coastal and ocean engineering, submerged coastal structures have been extensively constructed instead of the emerged coastal structures as the society has a growing concern with the coastal environment. Meanwhile, it brings more challenges to the structure design due to the complicated interaction between the submerged structure and the wave. To understand the complicated mechanism of wave and submerged structure interaction and optimize the structure design, various studies have been carried out with the approaches of theoretical analysis, laboratory and field experiments, and numerical simulations (Madsen and Mei, 1969;Seabra-Santos et al., 1987;Lin, 2004;Mansouri and Aminnejad, 2014).
Solitary wave is widely used to describe certain physical phenomena and hydrodynamics of sea state, such as tsunamis, storm surges, and other long free surface waves. Great efforts have been made to study the interaction between solitary wave and submerged breakwaters for deep physical insights. Mei (1985) investigated the energy loss due to a solitary wave passing an abrupt junction. A solitary wave interacting with a bottom mounted rectangular dike was studied by Chang et al. (2001) experimentally. Ex-perimental investigations on wave transmission at submerged breakwater with smooth and stepped slopes were carried out by Lokesha et al. (2015). The complicated interaction between the solitary wave and the submerged structure still need further understanding albeit some studies have been done and some achievements have been made. In most of the studies, the free surface variation is the only object of interest. The distribution of velocity and pressure, which is essential in the design of submerged breakwater, is still unclear. The investigation of hydrodynamic characteristics around submerged dikes under different structures is also vital to the optimum structural design. Among all the study approaches, the theoretical approach only works for the simple case, while the experimental approach is very expensive. With the advance of computational techniques and the development of numerical solution to partial differential equations, numerical simulation of the wave transformation over a submerged breakwater has become practical. The accuracy and efficiency of the numerical model in predicting the wave properties as well as the hydrodynamic behavior of submerged breakwaters are challenging because of the complexity of the physical processes, especially considering wave shoaling, reflection, flow separation and breaking (Ning et al., 2016;Shen and Wan, 2016).
The main purpose of this study is to provide reliable predictions of solitary wave transformation and run-up over a submerged breakwater using both the high-order accuracy numerical method and physical experiments. Wave surface elevations, velocity and pressure around the structure were carefully measured. A high precision numerical model based on N-S equations was employed to reproduce the experiments to cross-check both physical and numerical methods. A CIP-based method is proved to be a powerful tool and can be applied to the study of the wave-structure interaction in coastal engineering Wang et al., 2018). In this paper, two different problems were mainly studied: (1) the interaction between solitary wave and a submerged breakwater; (2) the hydrodynamic characteristics of breakwaters with different structures. The effects of the existence of submerged structures on the wave characteristics and high-order harmonic waves are discussed in detail.
The paper is structured as follows. In the next two sections, the experimental setup and the numerical model are introduced. In Section 4, numerical results and experimental data are presented and compared with each other. In Section 5, new series of simulations are performed considering various seaward/leeward slopes and crest lengths to study the effect of geometry of the submerged breakwater on hydrodynamic characteristics around the breakwater. Finally, some general conclusions are drawn.

Experimental setup
Physical experiments were performed in a large wavecurrent flume in the Laboratory of Coastal and Offshore Engineering, Zhoushan Campus, Zhejiang University, China. The glass-wall wave flume is 75.0 m in length, 1.8 m in width, 2.0 m in depth. At one end of the wave flume, an electric motor-driven piston-type wave maker is equipped and the absorbing devices are set at the other end. The trapezoidal submerged breakwater was placed at x=32.0 m away from the wave maker with the height of 0.36 m, the crest length of 1.5 m and the bottom length of 3.08 m. The width of the trapezoidal submerged breakwater was the same as the width of the wave flume, i.e. 1.8 m. The initial water depths were fixed at 0.5/0.55/0.6 m in the experiment. The wave conditions are shown in Table 1.
The sketch of experiments setup is shown in Fig. 1 Fig. 2 shows the arrangement of the pressure sensors on the surface of the breakwater. Eight pressure sensors (diaphragm size of 10 mm and full scale of 40 kPa), referred to as P1-8, are placed along the central line of the flume to measure wave pressure on the submerged breakwater. The first pressure sensor is placed on the seaward slope at x=32.75 m away from the wave generator, and the last pressure sensor is installed on the leeside at x=34.33 m with the same vertical levels as the first one. Six pressure gauges are installed on the top of the submerged dike. The sampling frequency of the pressure sensor is 1 kHz. Three vectrino velocimeters are installed to observe the variation of velocity along the flume (labeled V.1, V.2, V.3 in Fig. 1 and marked by blue diamond in Fig. 2). The vectrino velocimeter uses the Doppler effect to measure current velocity. A frequency of 60 Hz is used for data acquisition in V.1 velocity gauge. The V.2 and V.3 velocimeters work with the frequency of 25 Hz.

Experimental repeatability
The quality of repeatability and the effect of three-dimension are verified to ensure accuracy and reliability of the experimental measurements. Fig. 3 depicts the experimental data of wave elevations of Case 9 from two separate experiments with the identical experimental conditions. From Figs. 3a-3c, it is clear to observe that all experimental data appear as one curve, which indicates that the experiments are done with an excellent repeatability and the threedimensional effect can be ignored. Fig. 4 shows the comparison of horizontal velocity in two different experiments of Case 9. Excellent repeatability of the velocity is also obtained. All the above results indicate that the experimental data are applicable to the following study.

Numerical method
A CIP-based model (Hu and Kashiwagi, 2004;Zhao and Hu, 2012) is employed in this study to predict strongly nonlinear wave-structure interactions. The model is built to solve Navier-Stokes equations in a Cartesian grid.

Governing equations
With a viscous incompressible flow, the governing equations are mass conservation equation and the Navier-Stokes momentum equations, expressed as: where t, u i , p and x i represent time, velocities, hydrodynamic pressure and spatial coordinates, respectively. f i is the body-force components and S ij is the viscous term given by ρ μ where and are the density and viscosity, respectively. The wave-structure interaction is treated as a multiphase problem that includes water, air and solid body. A volume fraction field (m =1, 2, and 3 indicate water, air and solid, respectively) is adopted to represent and capture the phase interface. The evolution of the volume function is governed by ∂ϕ where is set in a range between 0 and 1. The cell is full of air if equals 0 and the cell is full of water if equals 1. In addition, anything between 0 and 1 is considered to be the interface. Overall, the sum of the volume functions of gas, liquid and solid is united everywhere. The volume function for solid is determined by a Lagrangian method in which a fixed rigid body is assumed, then the volume function for air is determined by . After the volume function is calculated, the physical property , such as the density and viscosity in every grid can be calculated by the following formula

Flow solver
In this study, a staggered grid system is used for the spatial discretization and a fractional method is applied to solve Navier-Stokes equations. The first step is to calculate the advection term, as shown in Eq. (6) A CIP approach is employed to solve the above equations (Yabe et al., 2001). The basic principle of CIP method is an interpolation in a grid by a cubic polynomial with the value and differential on the grid node.
where X is the unknown coefficient function, x i is the cubic polynomial to approximate the spatial profile of u i in an upwind cell. The superscript * denotes the intermediate time level after the advection step. The second step is to solve the diffusion term, momentum source term and damping term by a central difference scheme.
where u i ** is the solution of this diffusion term.
In the end, the pressure term is solved by the projection method: where the superscript n+1 denotes the new time step. The red-black successive over-relaxation approach (SOR) is employed to solve the above Poisson equation.
After the total flow field updated, the free surface is captured by a volume of fluid (VOF) type method, the tangent of hyperbola for interface capturing with slope weighting (THINC/SW) scheme. Compared with the original VOF method, the THINC/SW scheme first proposed by Xiao et al. (2011) uses a smoothed Heaviside function with a variable steepness parameter, which can keep the sharpness of the interface and significantly improve the geometrical accuracy. The solid boundary is treated by an immersed boundary method (IBM) proposed by Peskin (1972). More details about the flow solver can be found in Hu and Kashiwagi (2004) and Zhao and Hu (2012).

Solitary wave generation
In this study, a numerical paddle wave maker is set at the left side of wave tank to generate the solitary wave. The approximate solution of solitary wave profile (Boussinesq, 1872) near the wave paddle can be described as follows ξ where H, h, c, and are the amplitude of the solitary wave, still water depth, wave celerity and wave-paddle trajectory, respectively. The wave-paddle velocity can be calculated as: (14)

Results and discussion
In this section, computational predictions of interactions between solitary waves and a submerged breakwater are carried out. Solitary wave generation, free surface variation, flow velocities and wave pressure are mainly analyzed. Numerical results are compared with experimental measurements. The effect of geometrical characteristics of submerged breakwater is further studied numerically.
In the numerical simulation, the entire computational domain is discretized using a non-uniform grid system. To show the convergence of the adaptive spatial discretization, three non-uniform grids are used to perform the refinement test. The grids information is listed in Table 2. The validation is first carried out in Case 1 for G1. Fig. 5 depicts the free surface time history at G1 in different grids, respectively. We can see that the results obtained using the fine and intermediate meshes are almost identical. However, for the coarse grid, there is a little underestimation and deviation. The mesh convergence is achieved using the intermediate mesh. Time step dt is 0.0025 s which is dynamically determined to satisfy the criterion of CFL number. The total simulation time is up to 30 s to obtain fully developed wave field.

Free surface elevation
Figs. 6a-6c show the comparison of the transformation of solitary waves when propagating over the submerged   of St.5 (x=35.08 m), which is on the leeside toe of the submerged breakwater, strong deformation in the wave shape can be found by the appearance of the secondary crests at the trailing side of the primary wave. As the wave propagates along the wave flume, the wave height decreases and a train of well developed dispersive waves can be clearly seen at St.6 (x=37.18 m), which is similar to Seabra-Santos et al. (1987). Although the tendency of deformation is similar in Figs. 6a-6c, some differences can be found with the increases of the wave nonlinearity. The wave crests become more steep and the secondary crests become more noteworthy at St.5 (x=35.08) in Fig. 6a. Besides, compared with Figs. 6b and 6c, more dispersive waves can be found at St.6 (x=37.18 m) in Fig. 6a, which means that with the decrease of initial water depth, more dispersive tailed wave will be released when the solitary waves travel over a submerged breakwater. Despite the slight difference at St. 5 and St. 6 in Fig. 6, the overall numerical results agree well with experimental data.

Flow velocities
The temporal variations of horizontal velocities are measured at three points around the submerged breakwater. Point V.1 is located at x=33.54 and 0.44 m above the flume bottom, and Point V.2 is 0.75 m downstream of Point V.1 with the same vertical position. Point V.3 is located at x=35.08 m and 0.36 m above the bottom of flume. Comparison between the numerical results and the experimental data are shown in Fig. 7. Great agreements are obtained. However, it needs to notice that a rapid change of velocity can be found at Point V.2 in Fig. 7, which induces the loss of some data (the maximum sample rate of the ADV is 25 Hz). The rapid change of velocity is possibly caused by the flow separation and vortex generation, which has been studied by many researchers through physical experiments (Chang et al., 2001;Li and Ting, 2012) and numerical simulations (Peng et al., 2012;Lin et al., 2014).

Wave pressure
In this section, the distribution of dynamic pressure around the submerged breakwater is shown in Fig. 8. Case 9 is considered here. Pressure sensors with a frequency of 1000 Hz is used for data acquisition. The pressure sensors are making measurements to a precision of five parts in a thousand of the full scale (40 kPa). It needs to be noted that, due to the large measurement scale of the pressure sensor, almost all of the experimental data have a fluctuation range of around 400 pa. The comparison in pressure field distribution between the numerical results obtained by the present model and the experimental data is shown in Fig. 8. Despite some minor differences between the experimental data and the numerical results in wave profiles, the numerical results of the present model show a good match to the experimental data. Generally speaking, the present CIP-based model has proved to be a robust tool to well represent the hydrodynamic characteristics in terms of the wave transformation, velocity field and wave pressure.

Effect of the geometrical characteristics of a submerged breakwater
Following Ting and Kim (1994), in this study, the front side and back side of the obstacle are defined as the seaward and leeward slope, respectively. In order to study the effect of seaward/leeward face slopes and crest length of the submerged trapezoid breakwater on hydrodynamic performance around the dike, a series of simulations based on the present model are carried out in this section. The solitary wave with wave height of 0.1 m under the water depth of 0.6 m is considered here. The two typical phenomena: dispersive waves and flow separation are also discussed below. Computations are performed for the seaward/leeward slope of the breakwater (h 0 /L 0 ) ranging from 1:0 to 1:22 and the crest length of the breakwater (L) ranging from 1 to 10 folds of h 0 , where h 0 is the height of the breakwater and L 0 /L 1 is the projection of the seaward/leeward slope onto a horizontal plane. Numerical layout of the submerged breakwater with various seaward/leeward face slopes and crest lengths is shown in Fig. 9. It is worth pointing out that the right corner of the crest is constant (at x=34.29 m) throughout the simulations. The relative positions of the right corner of the crest, the wave gauges (St.1-St.6), velocity gauges (V.1-V.3) and pressure gauges (P1-P8) are invariable.

Effect of seaward slope (h 0 /L 0 )
In this part, twelve different seaward slopes changing from gentle (h 0 /L 0 =1: 22) to steep (h 0 /L 0 =1: 0) are considered to study the effect of the seaward slope on the hydrodynamic performance around the submerged breakwater. Meanwhile, the leeward slope and crest length of the submerged breakwater keep constant. Fig. 10 shows the time series of the free surface variation around the submerged breakwater for different seaward slopes. From Fig. 10, some interesting findings can be obtained: (1) With the decrease of h 0 /L 0 , the effect of seaward slope to the reflected wave is diminishing. Meanwhile, the time interval of the reflected wave propagating to the current position is longer, which means that the time delay of the reflected wave is more serious. What is more, the reflected wave height is small at St.1, as shown in Fig. 10a. (2) It can be clearly observed from Figs. 10a-10f that the lines which are determined by the steep seaward slope (h 0 /L 0 >1:10) coincide with each other with small discrepancy, which indicates that the seaward slope actually had little effect on the free surface elevation when h 0 /L 0 >1:10.
(3) As shown in Figs. 10b-10f, with the decrease of h 0 /L 0 , especially when h 0 /L 0 ≤1:10 (gentle slope), the time delay and the nonlinear effects of the solitary wave become more and more serious due to the significant shoaling effect. (4) Higher wave height will appear at St. ) and the extreme wave height will occur closer to the right corner of the dike, with the decrease of h 0 /L 0 . (5) From Fig. 10f, the wave height shows little change at St.6 with the decrease of h 0 /L 0 . while a train of more serious dispersive waves can be observed.
Figs. 12 and 13 present the variation of the velocities at V.1/2/3 and the wave pressure on the crest of the sub-merged breakwater for different kinds of seaward slopes. As shown in Figs. 12a-12c and Figs. 13a-13c, some typical phenomena can be found which are similar to the free surface elevation. With the decrease of the seaward slope gradient, the asymmetry of the velocity/pressure lines becomes obvious and the time delay of the solitary wave becomes serious. We can also find that the lines which are determined by the steep seaward slope (h 0 /L 0 >1:10) coincide   with each other with small discrepancy, which indicates that the seaward slope actually has very little effect on the wave velocity and wave pressure when h 0 /L 0 >1:10.
The vorticity fields of three typical seaward slopes (h 0 /L 0 =1:0, vertical slope; h 0 /L 0 =1:4, steep slope; h 0 /L 0 =1:18, gentle slope) at indicated time are shown in Fig. 14. Similar to Hsu et al. (2004), the vorticity is calculated by the following equation . From Fig. 14, it can be observed that flow separation modifies the fluid kinematics around the edges of the submerged obstacle, which also can be found in previous studies (Ting and Kim, 1994;Hsu et al., 2004;Lin et al., 2014). To illustrate this phenomenon, four indicated instants are selected, i.e., t=6.00 s, 6.72 s, 7.44 s and 8.80 s. Starting from the time t=6.00 s when the wave crest approaches the breakwater, it can be observed that flow is separated from the edge of the vertical slope and reattaches on the surface of the breakwater in Fig. 14a-1. The shear layers at the steep (h 0 /L 0 =1:4) and gentle slopes (h 0 /L 0 =1:18) of the breakwater are separated from the wall in Figs. 14(b/c-1). When the wave crest is leaving (t=6.72 s), a separation bubble and a secondary vortices are forming at the top of the vertical slope (as shown in Fig. 14a-2). The shear layers at the leeward slopes of the breakwater are separated from the wall obviously and then generate vortices into the flow (as shown in Figs. 14(a/b/c-2)). In Figs. 14(a/b/c-3), as the solitary wave moves away from the breakwater (t=7.44 s), the vortices at the leeward corner of the breakwater form recirculating regions and then induce secondary vortices on the wall. The vortices, induced by the seaward vertical slope, move to the downstream surface of the wave from the seaward corner of the breakwater in Fig. 14a-3. When the solitary wave moves far away from the breakwater (t=8.80 s), the vortices dissipate gradually due to viscous effect as the driven flow disappears. Moreover, it is observed that the decreased ratio of h 0 /L 0 makes the vortices at the seaward slope of the breakwater become weak and have few impacts on the vortices at the leeward slope of the breakwater.

Effect of crest length (L)
To investigate the influence of the crest length (L) on hydrodynamic performance around the submerged dike, various crest lengths are considered. Six crest lengths (L) ranging from one-fold (L= h 0 ) to ten-fold of h 0 (L= 10h 0 ) are chosen. Meanwhile, the seaward/leeward slope of the submerged dike keeps constant. Also, the right corner of the crest is constant (at x=34.29 m). Fig. 15 shows the computed time series of the free surface elevation around the submerged dike for different crest

262
ZHAO Xi-zeng et al. China Ocean Eng., 2019, Vol. 33, No. 3, P. 253-267 Besides, a higher wave height will appear at St. ) and the extreme wave height will occur closer to the right end of the breakwater with the increase of the crest length. Similar to Fig. 10f, Fig. 15f shows that more dispersive wave will appear with the increase of crest length (L). Figs. 16 and 17 illustrate the time series of the flow velocity at locations V.1/2/3 and wave pressures at locations P.6/7/8 for different crest lengths of the submerged dike. Similar to Fig. 12 and Fig. 13, the increase of crest length also makes the asymmetry of the velocity/pressure lines become more obvious and the time delay of the solitary wave become more serious, as shown in Fig. 16 and Fig. 17. Fig. 18 dipicts the velocity distribution and vorticity fields of different crest lengths (L) of the breakwater at indicated time, as the solitary wave passes over the submerged breakwater. When the solitary wave approaches the breakwater at t=6.00 s, the shear layers at the seaward slopes and the top of the dike are separated from the wall of the breakwater in Figs. 18(a/b-1). As the wave crest arrives at the top of the dike (t=6.24 s), the vortices at the seaward and leeward slopes become more distinct in Fig. 18a-2, due to the shorter length of the crest (one-fold).  show that when the wave crest moves forward and arrives at the right corner of the breakwater (t=6.48 s), the vortices at the seaward slope are becoming weaker and the vortices at the leeward slope are becoming stronger. When the solitary wave moves away (t=6.72 s), the vortices at the left and right corner of the one-fold breakwater and the right corner of the ten-fold breakwater form recirculating regions and then induce secondary vortices on the wall (as shown in ). With the solitary wave moves far away (t=7.44 s), the vortices dissipate gradually in Figs. 18(a/b-5). As shown in Fig. 18, a shorter crest length will have a more significant effect on the vortices at the leeward slope at the early stage as wave passing over the submerged breakwater. However, as the solitary wave moves away, the shorter crest has limited effect on the size and strength of the vortices at the leeward slope.

Effect of leeward slope (h 0 /L 1 )
In this section, the effect of the leeward slope on hydrodynamic performance around the submerged dike is also considered. Twelve leeward slopes are chosen ranging from gentle (h 0 /L 0 =1:22) to steep (h 0 /L 0 =1:0). Besides, the seaward face slope and crest length of the submerged breakwater are constant. Fig. 19 shows time series of free surface elevation around the submerged breakwater for different leeward slopes (h 0 /L 0 changes from 1:0 to 1:22). The results show similar consistency at wave gauges St.1-St.6 with different leeward slopes. Little difference can be found when wave arrives at the last wave gauge St.6 located at the downstream of the breakwater. With the decrease of h 0 /L 1 , a train of more serious dispersive waves can be observed. Fig. 20 presents the time series of the velocities at V.1-V.3 for different leeward slopes (h 0 /L 1 ). We can see that all numerical data appear as one curve in Fig. 20a, which indicates that the leeward slope has limited influence on the velocity field ahead of it. Owing to the seriously separated flows around the corner of steep leeward slopes (h 0 /L 1 =1:0/1:2), the velocity profiles remain strongly distor-  ted in the region. Two typical irregular curves in Fig. 20b can be found. V.3 is closer to the top of the leeward slope of the breakwater and the velocity at V.3 becomes stronger with the decrease of h 0 /L 1 .
The numerical time series of the wave pressure on the crest of the submerged breakwater for different leeward slopes (h 0 /L 1 ) are shown in Fig. 21. The results show similar consistently with different leeward slopes, and only minor difference is found at P7. The decrease of the h 0 /L 1 will induce a limited pressure increase near the right top corner of the submerged breakwater.
Numerical vorticity fields of different leeward slopes (h 0 /L 1 ) at indicated times (t=6.00 s, 6.72 s, 7.44 s and 8.80 s) are shown in Fig. 22. From Fig. 22, we can find that the change of the leeward slope almost has no influence on the vortices at the seaward slope. However, with the increase of leeward slope (h 0 /L 1 ), the vortices at the corner of the leeward slope will grow in both size and strength. Fig. 23 in-dicates the simulated velocity distribution of vertical leeward slope when the solitary wave progates near the submerged breakwater. The development and growth of the vortex structure behind the dike can be seen clearly. As solitary wave passes by, the vortex grows in both size and strength. The core of the vortex tends to move downstream in the direction of wave propagation and rise upward towards the free surface. This phenomena also can be seen in the reference of Ma et al. (2016).

Conclusions
In this study, laboratory experiments and numerical simulations on wave propagation over a trapezoidal submerged breakwater are performed to investigate hydrodynamic characteristics around the submerged breakwater. The experiments are conducted in a glass-walled wave flume of different water depths. An in-house code is applied to reproduce the experiments to cross-check both physical and numerical Furthermore, the influence of the geometry of the submerged trapezoidal breakwater on the hydrodynamic characteristics around the breakwater is studied numerically. Series of numerical simulations are performed to calculate the wave transmission, flow velocity, wave pressure and vorticity field for different h 0 /L 0 (seaward slope), h 0 /L 1 (leeward slope), L (crest length). It is found that: (1) With the decrease of h 0 /L 0 (seaward slope), the nonlinear effects of solitary wave become stronger. The extreme wave height occurs closer to the right corner of the breakwater and a train of stronger dispersive waves can be observed. Also, the decrease of h 0 /L 0 makes the vortices at the seaward   slope of the breakwater weaker. However, it has less influence on the vortices at the leeward slope of the breakwater.
(2) As the crest length L increases, stronger nonlinearity is observed at St.2-St.6. The extreme wave height appears closer to the right end of the breakwater and more dispersive waves appear with a large scale. Besides, a short crest length has an significant effect on the vortices of the leeward slope at the early stage as wave passing over the submerge breakwater. However, when the solitary wave moves away, the short crest has limited effect on the size and strength of the vortices at the leeward slope. (3) With the decrease of the leeward slope ratio h 0 /L 1 , the results show similar phenomenon at wave gauges St.1-St.5 and a train of more serious dispersive waves can be seen at St.6. The leeward slope almost has no influence on the vortices at the seaward slope. However, with the increase of h 0 /L 1 , the vortices at the leeward slope grow in both size and strength.