Experimental and numerical study of regular waves past a submerged breakwater ＊

Bi-jin Liu , Du Cheng, Zhao-chen Sun Xi-zeng Zhao, Yong Chen, Wei-dong Lin

1. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024,China 2. Ocean College, Zhejiang University, Zhoushan 316021, China 3. School of Civil Engineering and Architecture, Xiamen University of Technology, Xiamen 361024, China

Abstract: Laboratory experiments are performed to investigate the hydrodynamics around a submerged breakwater due to regular incident waves. With an in-house code, a constrained interpolation profile (CIP)-based model is employed to simulate this process. The model is built on a Cartesian grid system with the Navier-Stokes equations using a CIP method for the flow solver and an immersed boundary method (IBM) is used for the treatment of the solid body boundary. A more accurate interface capturing scheme, the tangent of hyperbola for interface capturing/slope weighting (THINC/SW) scheme, is used to track the free surface. The numerical results are compared with experimental data. Reasonably good agreement is achieved in terms of the wave profiles at six measuring stations, the flow velocities at three different space locations and the pressures of eight points on the surface of the submerged breakwater. Moreover,the water mass transfer over the breakwater is discussed using a two-phase VOF model and the wave spectrum is also presented for analysis. It is indicated that the present model can accurately predict the hydrodynamic characteristics of the wave over a submerged bar. Furthermore, the experimental data in the present work can provide reliable basic data, including the wave transformations, the velocities and the dynamic pressures, for the validation of other CFD models.

Key words: Submerged breakwater, CIP method, wave transform, VOF method, immersed boundary method

Introduction

The waves propagating from the deep water to the coastal areas are affected by the bottom bathymetry, such as the submerged breakwaters, resulting in wave refraction, shoaling and breaking. The accurate prediction of the wave properties as well as the hydrodynamic behavior over a submerged breakwater is a challenging task. To maximize the functionality and secure the stability of the submerged structures, the interaction between the water waves and the submerged structures is an essential issue[1].While previous studies accumulated some useful information such as the generation of higher harmonics, the flow separation and the vortex generation, the distributions of the velocity and the impact force, also of fundamental importance in the design of the submerged breakwater, remain an issue to explore. Hence, it is important to study the wave transformation process and its hydrodynamic characteristics around a submerged breakwater[2].

The wave propagation over submerged breakwaters was studied experimentally and numerically.The laboratory experiment is a preferred method because of its easy feasibility and lower cost as compared with the field observation. Ohyama et al.[3]performed laboratory experiments of the nonlinear wave transformation over steep topographies in a 65 m long wave flume. The experimental data may be used to verify the reliability of the numerical models.Apart from the data of the wave deformation, the data of the velocity and pressure fields are still lacking.Experiments in a wave flume were performed by Brossard and Chagdali[4]to analyze the nonlinear interaction between the regular waves and a submerged horizontal plate used as a breakwater. An experimental study was carried out for waves passing an isolated reef terrain in a physical wave flume[5].Experimental investigations of the wave transmission over a submerged breakwater with smooth and stepped slopes were conducted by Lokesha et al.[6].The interactions between the regular waves and the submerged breakwaters with rectangular or trapezoidal sections were investigated experimentally and numerically by Ning et al.[7], with emphasis on the effect of the breakwater on the hydrodynamic behavior. Reliable data were obtained from laboratory experiments, focusing mostly on the wave deformation, the reflection performance and the harmonic components, but not so much on the hydrodynamic characteristics (such as the flow field and the impact force) around the submerged breakwater. To our knowledge, there is no available experimental data that provide the wave deformation and the hydrodynamic characteristics at the same time, for a more convenient and efficient basic data for the validation of a new numerical model.

It is known that the numerical simulation can provide detailed information of the hydrodynamics around a submerged breakwater, which is not easily obtained by physical experiments. Therefore, a well-developed numerical model is desirable to study the detailed evolution of waves passing over the submerged bar. The mild-slope equations[8]and the Boussinesq equations[3,9]were widely used in most early published papers on the wave propagation over a submerged bar. The wave breaking effects cannot be fully represented in the above models. As a choice, a Navier-stokes equations-based flow solver is preferred.

To model the interaction of the water waves and a structure, several difficult issues have to be solved.One is the distorted free surface related with the water wave deformation. Here, this problem can be solved by adopting an accurate VOF-type free surface/interface capturing method. The VOF method is one of the most popular methods in capturing the watersurface, which was first introduced by Hirt and Nichols[10]. Some improved VOF-type schemes were proposed, such as the THINC[11](tangent of hyperbola for interface capturing), the VOF/WLIC[12](WLIC:weighed line interface calculation) and the THINC/SW scheme[13]. The solid body boundary treatment is another similar method. Alternatively, one can use an immersed boundary method (IBM) or a virtual boundary force method in a Cartesian grid system.The immersed boundary method (IBM) was originally developed to investigate the blood flow patterns of human heart. Then, it was introduced to simulate the interactions between the solid objects and the incompressible fluid flows[14]. Besides, the evolution and the transport of the water mass around the breakwater are still not well studied. A great effort is needed to find the internal mechanism. In some recent studies, the water-water interface was traced by applying particle markers in the Eulerian grids and computing those particles in a Lagrangian way and this method was successfully applied to the studies of the breaking wave and the drop impact process[15-16].But they are usually much more complicated than the original VOF. Ye and Zhao[17]proposed a two-liquid method to trace both the free surface and the water-water interface in the Eulerian description. Only little complexity is added in the two-liquid volume of fluid (VOF) method, as compared with the original VOF method.

There are two main aspects in the present paper.First, new series of laboratory observations are carried out for the hydrodynamic characteristics around a trapezoidal submerged breakwater under the condition of the regular incident waves. The experimental data provide a reliable basis for our numerical model.Secondly, an advanced numerical model is employed to simulate the waves propagating over the submerged bar. The present model is governed by the Navier-Stokes equations under the free surface boundary conditions, and a constrained interpolation profile(CIP) method is used for the flow solver. A more accurate volume of fluid (VOF)-type scheme, the THINC/SW scheme, is adopted for capturing the interface and an IBM is used for the boundary treatment of the submerged breakwater. The twoliquid VOF method is employed to analyze the interaction and the evolution of the internal water particles around the breakwater.

Fig. 1 Experimental arragement and locations of wave gages (Sts.1-9) and velocimeters (V.1#-3#), with lengths (m)

Fig. 2 (Color online) Section view A-A and plane view of the distribution of wave gages (green circle), pressure sensors (red triangle), and velocity gages (blue diamond), with lengths (m)

1. Laboratory experiment

1.1 Experimental setup

Physical experiments for waves propagating over a submerged breakwater are performed in a newlybuilt large wave-current flume in the Laboratory of Coastal and Offshore Engineering, Zhoushan Campus,Zhejiang University, China. The glass-wall wave flume is 75 m in length, 1.80 m in width, and 2 m in depth, equipped with an electric motor-driven, pistontype wave maker at one end of the wave flume and absorbing devices at the other end. The trapezoidal submerged breakwater of 0.36 m in height, 1.50 m in crest length and 3.08 m in bottom length is placed atx=32 m . The width of the trapezoidal submerged breakwater is the same as the width of the wave flume,i.e., 1.80 m, so the flow can be treated as two dimensional. The overall setup of the experiment is shown in Fig. 1.

Six wave gauges (Sts.1-6) are placed along the central axis of the wave flume for recording the wave propagation and transformation over the submerged breakwater. Three wave gages (Sts.7-9) are installed at the same locations as the wave gauges Sts.1, 2 and 3, side by side, to verify that the flow is two dimensional. The wave gage St.3 is set above the center of the breakwater, 33.54 m away from the wave maker. The wave gages St.1 and St.7 are placed atx=28 m away from the wave generator to verify the generated wave characteristics and to insure that the incident wave in the experimental region is the right one. The frequency of the data acquisition for the wave gauges is 50 Hz.

Figure 2 shows the arrangement of the pressure sensors on the surface of the breakwater. Eight pressure sensors (diaphragm size of 10 mm and full sca le of 40 kPa ), r efer red as P 1- P8, are pl aced along the central line of the flume to measure the wave breaking pressure on the submerged breakwater. The first pressure sensor is placed on the front slope atx=32.75 m away from the wave generator, and the last pressure sensor is installed on the lee side atx=34.33m with the same vertical level as the first one. Six pressure gages are installed on the top of the submerged breakwater, 0.25 m, 0.50 m, 0.90 m, 1.05 m,1.20 m and 1.35 m, respectively, away from the front corner of the submerged breakwater. The sampling frequency is 1000Hz. As shown in Fig. 1, three vectrino velocimeterslabeled V.1#, V.2#, V.3#, are located at 0.44 m, 0.44 m, 0.36 m, respectively, above the flume bottom. These three velocity gages are installed at x =33.54 m,x=34.29 m andx=35.08 m along thexdirection of the flume to measure the variation of the velocity along the flume.The vectrino velocimeter uses the Doppler effect to measure the current velocity. The frequency of the data acquisition for the V.1# velocity gauges is 60 Hz.The data acquisition frequency of the V.2# and V.3#velocimeters is 25 Hz due to the equipment limitations.Video camera recordings are also used to monitor the occurrence of breaking and the related morphological features of the breakers.

Table 1 Wave conditions

1.2 Initial condition

The still water depth is fixed at 0.50 m, 6 m in the experiment, the water depth above the submerged breakwater is therefore 0.14 m, 0.24 m. The wave elevations at six wave gauges, the fluid velocities at three different locations and the pressure distributions are recorded to study the hydrodynamic characteristics of the wave propagation over the submerged breakwater.

Nine different regular wave conditions (three different wave periods with three different incident wave heights) are investigated in this experiment.These nine wave cases include broken wave cases and non-broken wave cases, for the convenience of analysis of the difference. The wave conditions are shown in Table 1, in whichTis the wave period,h0is the water depth andH0is the wave height.

Fig. 3 Repeatability and 3-D effects of the measured wave elevations in the experiment

Fig. 4 Repeatability of the measured flow velociy in the experiment

1.3 Experimental repeatability and three dimensional effects

The quality of repeatability and three dimensional effects are tested to ensure the accuracy of the experimental measurements. Figure 3 depicts the experimental data of the wave elevations from two separate experiments under identical experimental conditions. It is clearly observed from Figs. 3(a), 3(b)that all experimental data fall on one line. It can be found that the results from St.1 and St.7, St.3 and St.8,St.4 and St.9 are identical, which indicates that the three dimensional effect can be ignored and the experiment can be regarded as under two dimensional wave conditions. On the other hand, the results of the first test are in agreement with those of the second test,which indicates that the experiments have an excellent repeatability. The experimental data of the velocity from two separate experiments under the same experimental conditions are shown in Fig. 4 and an excellent repeatability of the velocity is also observed.

2. Numerical model

The computations presented in this paper are performed by means of a CIP-based method to estimate the water wave and a submerged breakwater interactions. The method is based on the solution of the Navier-Stokes equations with a CIP adopted as the base numerical scheme to obtain a robust flow solver in a Cartesian grid. A turbulence model is not considered and the surface tension is neglected.

2.1 Governing equations

For a viscous incompressible flow, the governing equations are the mass conservation equation and the Navier-Stokes momentum equations, expressed as:

wheret,ui,pandxirepresent the time, the velocities, the hydrodynamic pressure and the spatial coordinates, respectively.fiare the momentum forcing components andSi jis the viscous term given by

whereρandμare the density and the viscosity,respectively.

The numerical model is built under a fixed Cartesian grid system. The interaction between the wave and the submerged breakwater is considered as a multi-phase problem, including the structure, the water and the air. A volume functionmφis used,wherem=1, 2 and 3 indicates the water, the air and the solid, respectively. The total volume function for the whole computation domain is solved by the following advection equation

The water depths in the following discussions are the superposition of the3φ.

2.2 Numerical methods

Following Fu et al.[18], the governing equations are discretized using a high-order finite difference method on a Cartesian grid system and the total discretizing processes are divided into three steps,which are the advection phase, the non-advection phase (1) and the non-advection phase (2). The key elements of the numerical code are given in the following sections for the sake of clarity:

A fractional step scheme is used to solve the Navier-Stokes equations. In the advection phase the spatial derivatives of the velocities are needed to apply the CIP method[19]in the first place, then the pressure is obtained by solving the Poisson equation derived by enforcing the continuity constraint and the final velocity is updated by simple algebraic operations.

In this study, the THINC/SW scheme[13]is applied for the free-surface capturing. The THINC scheme is a VOF type method. Unlike the original VOF method, the THINC scheme uses a smoothed Heaviside function as the characteristic function, the hyperbolic tangent function. In the THINC/SW method, an interface orientation dependent weighting between the original THINC scheme and the first order donor cell scheme is suggested in Ref. [11],which significantly improves the geometrical accuracy of the original THINC scheme.

Fig. 5 Records of video camera of three typical cases

The fractional area volume obstacle representation (FAVOR) method is applied to deal with the solid body boundary condition, as one of the most efficient methods to treat the immersed solid bodies. In the general IB method, the solid and fluid domains are meshed and calculated independently, which is a huge advantage for the simulation of a moveable structure in the fluid. In the present numerical solution procedure, a forcing term is added to the momentum equation to impose the velocity distribution inside and on the solid body boundary, and an alternative way is to do the following updating after the calculation with Eq. (7).

3. Results and discussions

In the numerical simulation, the computational domain is subdivided into a mesh of fixed rectangular cells in the Cartesian coordinates. A non-uniform grid system is employed inxdirection andzdirection,with the minimum grid size of Δxmin=0.02 m and Δymin=0.04 min the vicinity of the submerged breakwater. The computational time step is 0.25×10-3s and the total simulation time is 40 wave periods to obtain a fully developed wave field. It is critical to select a stationary stage of the waveform for the analysis. We start the timer when the first wave reaches the St1 gauge, after 7 periods, the waveform becomes stable. The moment when the 8th wave reaches the St1 gauge is redefined ast=0 s, and most attentions will be paid on the period oft=0s-10s.

3.1 Wave breaking

When applying thewave breaking criteria by Miche (1951)associated with the shoaling, it can be easily judged whether the wave breaks. It is found that in the following four cases, the wave will not break:T=1.34s,H0=0.05 m, 0.08 m and 0.10 m andT=2 s ,H0=0.05m . And in the following three cases, the waves are very close to break:T=2 s,H0=0.08 m, 0.10 m,T=2.68s ,H0=0.05m. In the case ofT=2.68s,H0=0.08 m, 0.10 m one sees the total breaking waves. Figure 5 shows the records of video camera of three typical cases(T=1.34s ,H0=0.05m ,T=2 s,H0=0.08m andT=2.68s ,H0=0.10 m ), representing nonbreaking, close breaking and breaking cases, respec-tively. It is interesting to find that in the case, with the wave close to the breaking criteria according to Miche(1951)'s equation, the wave does break in the experiment. In this case, the white spray of the wave firstly appears at the top of the wave crest, as the wave propagates over the submerged breakwater, and the spilling wave is generated. In the case ofT=2.68s,H0=0.10 m, when the wave approaches the top of the breakwater, the front and back surfaces of the wave become asymmetric. Then, the wave breaks at the toe of the front surface, after that, the whole front surface turns to a broken state of fragmentation.

Fig. 6 (Color online) Original state of water

3.2 Water mass transport

The water mass transport in the process of the wave propagating over a submerged breakwater is simulated by the present CIP-based model. A twoliquid-phase VOF method is used to mark the water phase on the top of the submerged breakwater. Figure 6(a) shows the original state of the water without breakwater. Figure 6(b) shows the original state of the water with breakwater. The blue color part is the water marked.

Figure 7 shows the results of the water transport in three cases (T=1.24 s, 2 s and 2.73 s,H0=0.05 m). As the wave period increases, the Ursell number increases, and the wave nonlinearity increases,and the water mass transport becomes more significant. As can be seen from Figs. 7(a)-7(c), for a short wave, the marked water moves backward as a whole, and the water-water interface is perfectly preserved. As the wave period becomes 2.0 s, the water-water interface begins breaking and the two phases of water is mixing. The water mass near the free-surface moves fast. As for the case ofT=2.68s ,the water-water interface is completely broken, the water in front of the submerged breakwater enters the inside of the marked water near the top of the submerged breakwater. At the back of the submerged breakwater, the marked water breaks into drops of small water mass and these drops of water mass transport to a further position.

Fig. 7 (Color online) Water transports in three wave cases at t=25T

3.3 Free surface elevation

Fig. 8 Time series of computed and measured wave elevations at six positions

Fig. 9 Amplitude spectrums of all six cases at six gauges

The transformation of the regular wave as it propagates over the bar is recorded in the experiment and the data are compared with the numerical results,as shown in Fig. 8. The process of the interaction between the regular waves and the submerged breakwater can be traced from Figs. 5(a)-5(f). The wave periods are 1.47 s, 2.20 s with three different incident wave heights (H0=0.05 m, 0.08 m and 0.10 m).It is noted thatt=0denotes the moment when the wave is fully developed and reached its stable stage.The measured free surface atx=28 m in Fig. 8 is considered to be the incident wave on the submerged breakwater. The waves, as they travel to the crest of the submerged bar (atx=32.79 m ), gradually become slightly higher because of the shoaling effect as shown in Figs. 5(a)-5(c). Comparing with the wavelengths (3.42 m, 5.35 m), the projected length of the front slope (0.79 m) is relatively small. So, the wave profiles atx=32.79 mare still symmetrical in Figs. 8(a)-8(f). The wave deforms due to the shoaling on a reduced water depth as it reaches the top of the b reakwa ter an d p ropag ates ove r the flat bott om with awaterdepthof0.24m.Thewavegagesatx=33.54 m andx=34.29 m in Figs. 8(a)-8(c) give a waveform of the vertical asymmetry. The wave crests become steep and the wave troughs become fat, which means that the nonlinear effects become significant in this shallow-water region over the top of the breakwater because of the decrease of the water depth.The decomposition of the wave due to the wave-bottom interaction begins atx=33.54 m in Figs. 8(a)-8(c), resulting in the appearance of a decomposed wave. The famous phenomenon of high order harmonics is observed above the submerged bar,with complex forms of the transmitted waves as a result of the nonlinear interaction. In Figs. 8(a)-8(c),the transformed incident wave appears with a clearly developed secondary decomposed wave at the end of the bar crest (x=34.29 m). We can see the same phenomenon in the experiment of Ohyama[3]. From this point onwards, as these incident waves and the following decomposed ones move into the leeward slope of the bar, the amplitudes of the waves begin to reduce as a de-shoaling effect and the waves decompose into several smaller amplitude waves. At the location ofx=35.08 min Figs. 8(a)-8(c), which are on the leeside toe of the submerged breakwater, a large deformation of the wave shapes can be observed with the appearance of the secondary crests at the trailing side of the primary wave. As the wave propagates further along the wave flume, the well-developed secondary wave is clearly seen in Figs.8(a)-8(c) atx=37.18 m. Although the deforming process is similar in Figs. 8(a)-8(c), some differences can be seen with the increases of the wave height. The wave crests become steeper and the wave troughs become flatter with the increases of the wave height atx=33.54 m andx=34.29 m in Figs. 8(a)-8(c).Besides, comparing with the waves in Fig. 8(a), a more significant secondary wave crest can be observed atx=35.08 mandx=37.18 m in Figs.8(b), 8(c).

Fig. 10 Comparison of the computed and measured velocities

The qualitative behavior of the wave transformation in Figs. 8(c) forT=2.20 s and in Figs. 8(a),8(b) forT=1.34s are similar but with a little difference in the waveform. One of the obvious differences is that the appearance of a distinct secondary crest and smaller tertiary crests are seen atx=35.08 mandx=37.18 m in Fig. 8(f), as the wave propagates over the submerged breakwater. Moreover, the asymmetry becomes more significant with the increase of the wave period atx=33.54 mandx=34.29 m ,as shown in Figs. 8(a)-8(c). Generally speaking, the numerical results show a good match to the experimental data and the model can provide a good representation of the wave propagation and transformation.

To explore the energy transformation of the wave past the submerged breakwater and for further verification of the CIP-based model, harmonic analysis is crucial. Figures 9(a)-9(f) show the amplitude spectrums in all six cases at six gauges. For the cases ofT0=1.47s , when the wave reaches the St.1 gauge, the wave energy is concentrated in the basic frequency,with scarcely any energy dissipation. As the wave keeps on going past the submerged breakwater, high order harmonics can be observed. In Figs. 9(c2), 9(e2),the highest harmonics reach up to sixth-order. In the meantime, the amplitude of the basic frequency decreases, which indicates that the wave energy transfers to harmonics of higher order. After the wave passes over the submerged breakwater, the amplitudes of the basic frequency and the first few high order harmonics (in the range of 2nd-6th orders) decrease,which means that the wave energy has dissipated. In the cases ofT0=2.20s , before the wave reaches the St.1, the waveform is changed and the wave energy is transferred from the basic frequency, the amplitude of the basic frequency is less than 1.0 at the St.1.

In view of the drastic wave breaking, when the wave propagates to the rear side of the submerged breakwater, the residual wave energy distributes mainly over the basic frequency, the 1st and 2nd order harmonics, and the amplitudes are very close (as shown in Figs. 9(d3), 9(f3)). The results of the CIP-based model agree with the experimental measurements.

3.4 Flow velocities

The temporal variations of the horizontal velocities are measured at three points around the submerged breakwater. The point 1 is located atx=33.54 m,0.44 m above the bottom of the flume, and the point 2is 0.75 m downstream the point 1 with the same water depth. The point 3 is located atx=35.08 m, 0.36 m above the bottom of the flume. Comparisons between the numerical results and the experimental data are shown in Fig. 10. As shown in Figs. 6(a)-6(f) (regular wave), some typical phenomena can be identified,which is similar as the wave elevation. With the increase of the wave period, the asymmetry of the velocity lines becomes evident in Figs. 10(a)-10(f).We can also find that a secondary crest is formed at the point 2 in Figs. 10(b), 10(d), 10(f). The secondary crest becomes clearer as the wave travels to the deeper water behind the submerged breakwater at the point 3,as shown in Figs. 10(b), 10(d) and 10(f). Another special phenomenon can be observed in Figs.10(a)-10(f) is that as the waves come up with a higher wave height, the velocity lines appear with a steep wave crest and a flat wave trough. Moreover, with the influence of the secondary crest, the velocity lines are expected to be more deformed in Figs.10(a)-10(f) with the increase of the wave height. It can be seen in Fig.10 that the results of the simulations agree well with the measurements at both points.

3.5 Pressure

The comparisons between the experimental data and the numerical results of the distribution of the dynamic pressure around the submerged breakwater are shown in Fig. 11. Two regular waves (T=1.47 s,H0=0.08m andT=2.20 s ,H0=0.10 m ) are con-sidered. The frequency of the data acquisition is 1 000 Hz for the pressure sensors. The pressure sensors enjoy a precision of five parts in a thousand of the full scale(40 kPa). It is to be noted that, due to the large measurement scale of the pressure sensor, almost all experiment data have a range of around 400 Pa. As the waves move along the submerged breakwater, the crests of the pressure data lines become steeper and the troughs of the pressure data lines become flatter,as shown in Figs. 7(a), 7(b). Additionally, the pressure data lines become asymmetric about a vertical axis with a steep front face and a gentle rear face at the P2,in Figs.11(a), 11(b). The asymmetric feature is increasingly evident, as the waves move forward on the top of the submerged breakwater, as shown in Figs.11(a), 11(b). The comparison between the numerical results obtained using the present model and the experiment data in Fig. 7 shows a reasonable agreement in the pressure field distribution, when the waves pass by the submerged breakwater.

4. Conclusions

The physical experiment and the numerical simulation for the wave propagation over a trapezoidal submerged breakwater are performed to investigate the hydrodynamic characteristics around the submerged breakwater, subjected to regular incident waves.The experiments are conducted in a glass-walled wave flume with a constant water depth of 0.5 m, 0.6 m.Nine regular wave conditions are considered in this experiment. In order to study the transmission characteristics of the submerged breakwater and provide detailed information on the hydrodynamics around the submerged breakwater, a CIP-based model is applied to simulate this process. Comparisons between the experimental data and the numerical results indicate that the present numerical scheme can accurately predict the wave deformation and the detailed hydrodynamic characteristics around the submerged breakwater, including the distribution of the velocity and the dynamic pressure, as the waves propagate over a submerged bar.

The results can be summarized as follows:

(1) The wave, close to the breaking limit according to the Miche (1951) criteria, breaks in our experiment. This can be explained by the wave-bottom interaction.

(2) The water mass transport in the process of the wave propagating over a submerged breakwater is also analyzed, as the wave nonlinearity increases, the water transport and the inner water exchange become more intense.

(3) Due to the wave shoaling, the wave deforms when the wave propagates over a submerged breakwater. The wave spectrum analysis indicates the transfer of the energy and verifies the ability of the CIP-based model to describe the wave transformation process well.

(4) With the increase of the wave period, the asymmetry of the velocity becomes evident and a secondary crest is formed at the second velocity measurement point.

(5) As the waves move along the submerged breakwater, the crests of the water pressure become asymmetric on both horizontal and vertical components.

The good agreement between the experiment data and the numerical result demonstrates that the present model is feasible for the study of the wave evolution process from deep to shallow water.

Acknowledgements

This work was supported by the Natural Science Foundation of Zhejiang Province (Grant No.LR16E090002), the Fundamental Research Funds for the Central Universities (Grant No. 2018QNA4041).