Wave Dissipation Characteristics of A Mountain-Type Breakwater

One mountain-type breakwater consisting of two inclined plates and one vertical plate is proposed based on several types of traditional free surface breakwaters, including the horizontal plate, curtain wall, and trapezoidal barriers. The interaction between the regular waves and the fixed free surface mountain-type breakwater is measured in one wave flume (15.0 m×0.6 m×0.7 m). The wave propagation, reflection, and transmission process are simulated using the VOF method and the hybrid SAS/laminar method. The simulated wave profiles are consistent with the experimental observations. For waves with a length smaller than four times width of the mountain-type breakwater, the reflected wave amplitudes are slightly larger than those of the vertical-plate breakwater, while the wave transmission coefficients are all smaller than 0.5, and the wave loss coefficients are larger than 0.7. The wave energy is dissipated by wave breaking on the windward inclined plate, and turbulent flow around the vertical plate and the leeward inclined plate.


Introduction
As one important coastal protection equipment, free surface breakwaters have been extensively studied in the past two decades (Losada et al., 1996;Rajendran, 2002b, 2002a;Li and Lin, 2012;Lara et al., 2012;Teh, 2013). The conventional gravity-type breakwater rests on the sea bottom, and its enormous size and weight provide enough structural stability against waves and storms. Compared with the gravity-type breakwater, the free surface breakwater is connected to the sea bottom by pile/jacket structures (fixed breakwater) or mooring cables (floating breakwater), permitting water circulation beneath the breakwater. Although free surface breakwater has less advantage in storm protection, it is more environmental-friendly and cost-saving, and can serve as a sustainable alternative to the gravity-type breakwater especially in an environmentally sensitive site.
Based on the configuration of the proposed design of free surface breakwaters, Teh (2013) classified them into four categories: solid-type, plate-type, caisson-type, and multipart-type. The solid-type barriers, e.g. box (Koutandos et al., 2005), cylinder (Isaacson et al., 1995) and trapezoidal (Duclos et al., 2004) structures, maintain their stability due to the large volume and effective mass, and restrain incident wave energy mainly by reflection. The plate-type barriers, e.g. plate-type (Ji et al., 2018), T-type (Neelamani and Rajendran, 2002b) and H-type (Neelamani and Vedagiri, 2002) structures, are composed by one or multiple plates, and the waves are reflected by vertical plates and dissipated by wave breaking on horizontal plates. The caisson-type barriers, e.g. U-type (Günaydına and Kebdaşlı, 2004) and Π-type (Koftis and Prinos, 2005) structures, consist of open interference chambers permitting wave interaction taking place in the interior, 'tune' the wave phase and suppress the wave activity. Multipart-type barriers consist of multiple structural elements, e.g. plates (Wang et al., 2006) and bars (Hsiao et al., 2008), and the high porosity limits the wave reflection.
Based on the above ingenious designs of free surface breakwaters, a novel mountain-type breakwater is proposed to combine the advantages of the horizontal plate, caissontype, and trapezoidal barriers. As shown in Fig. 1, the mountain-type breakwater is composed of three uniformsized plates and the connecting structures. The vertical plate 2 is the primary wave reflector, while the two inclined plates can also reflect waves. The two inclined plates (plate 1 and plate 3) not only function as horizontal plates, but also dissipate incident/reflected waves. Additionally, three plates form two trapezoidal chambers, which are similar to the caisson-type barriers.
In this study, the interactions between a fixed mountaintype breakwater (height H = 0.12 m and width W = 0.3 m) and regular waves are experimentally measured and numerically simulated using the VOF method and the hybrid SAS/laminar method. The numerically simulated wave profiles are validated by the measured experimental data. Then the wave dissipation performance of the mountain-type breakwater is tested under regular waves of different wavelengths (L i = 2W, 4W, and 8W) and wave heights (H i = 0.032 m, 0.048 m, and 0.064 m).

Mathematical modeling
The two-dimension incompressible and viscous interfacial flow is governed by the Navier−Stokes equations as follows: where ρ, p, and u denote the fluid density, pressure and fluid velocity, respectively. Effective viscosity μ eff = μ+μ t is the sum of the molecular viscosity and the turbulent eddy viscosity. The turbulence is modeled using the scale-adaptive simulation (SAS) model (Menter and Egorov, 2010), based on the von Karman length scale L vK . The value of turbulent kinetic energy k and turbulent eddy frequency ω are resolved by the following transport equations: The turbulent viscosity is defined as: where S is the magnitude of the shear strain rate, F 2 is the blending function, and the constant a 1 = 0.31. The source term are calculated via The constant β′ equals 0.09. The value of the blending function, F 1 and F 2 , is approximately constant in the nearwall region and tends to be zero away from the wall. , The coefficients α, β, σ k and σ ω are calculated as follows: The source term Q SAS in the ω-equation is specifically defined for the SAS model.
where the constants F SAS = 2.0, ς 2 = 1.755, κ = 0.41 and σ ϕ = 2/3. L is the turbulence length scale, and the ratio of L to the von Karman length scale, L νK , is the key to the switching between the SST and SAS treatments.
The whole computation domain is divided into two regions: one turbulence region near the breakwater, where the SAS model (Menter and Egorov, 2010) is only applied to this region; the other region is relatively stable, where the flow is less disturbed by the breakwater. The turbulence model is turned off in the region away from the breakwater to avoid the unphysical energy loss, which is induced by the small width−length ratio of computation cells near the free surface during the wave propagating process.

Wave simulation
The boundary-wave method was adopted to generate wave based on the volume of fraction (VOF) method (Hirt and Nichols, 1981), which introduces one new continuity equation for the volume fraction of liquid, ψ 1 , ∂ψ Here, the volume fraction of air, ψ 2 , satisfies ψ 1 +ψ 2 ≡ 1. Then any material property (e.g., density and viscosity), ϕ, can be decomposed as: The boundaries can be divided into four types: inlet, outlet, wall, and atmosphere. At the inlet boundary, the values of ψ 1 and ψ 2 are prescribed according to the incident wave profile. Then wave is generated from the inlet and leaves the computation domain through the outlet. The lower boundary and the breakwater boundary are the no-slip walls, and the upper boundary is open to air.
To avoid wave reflection of the outlet boundary, one porous wave absorber is prescribed at the outlet to absorb wave energy before reflection occurs (Zhan et al., 2010(Zhan et al., , 2014. For this aim, one momentum source term is added to Navier−Stokes equations as follows: where α is the permeability. To avoid the abrupt change of the flow resistance in the wave absorption region, the value of 1/α is set up to increase linearly in this region: where x 0 and x e are the x coordinates of the two endpoints of the wave absorption region. Usually, the length of the wave absorption region is set to be twice the wavelengths.

Separation of incident and reflected waves
The incident wave propagated forward through the breakwater in the wave tank and was reflected by the breakwater. Hence, the wave gauges before the breakwater measured the composite waves formed by the incident waves and the reflected waves. To evaluate the performance of the breakwater, the reflected waves need to be separated from the incident waves.
As three very different wavelengths were tested for this study, the distance between the two gauging points is difficult to satisfy the requirement of the widely used two-point method proposed by Goda and Suzuki in 1976. To avoid the singularity problem, one analytical method based on the Hilbert transform function is applied to calculate the reflection coefficient K r (Sun et al., 2002). The transfer function methods directly separate the time series data measured at two wave gauges before the breakwater into the incident and reflected waves (Zhu, 1999). Additionally, compared with the two-point method and the improved three-point method proposed by Mansard and Funke in 1980, the transfer function method does not need to calculate the wave amplitudes at wave gauges, and can be extended to the irregular waves.
For regular waves in this study, the Fourier transformations of the composite surface elevations at two wave gauges ( ) are calculated by using the fast Fourier transform (FFT) method. Let and its relative analytical signal is obtained using the inverse Fourier transformation. The real part of is the same as the real signal . The imagery part of is the Hilbert transform of , i.e.
Then the complex time series, and , are calculated as: η where is the wave number, and . The incident wave and the reflected wave are the real parts of and , respectively. The reflection coefficient is calculated as: ∥ * ∥ where is the modulus operator of the complex signal. The analytical method is realized based on python libraries, including cmath, numpy, and matplotlib. The code is verified by the numerical sample of regular waves provided by Sun et al (2002). The calculated reflected coefficient is K r = 0.319 9, which is very similar to the theoretical value, 0.32.

Numerical implementation
One free surface mountain-type breakwater is positioned in a two-dimensional wave tank, as shown in Fig. 2, where the center of the breakwater is placed at x = 0.0 m. Elevations of the wave surface are monitored numerically at GONG Ye-jun et al. China Ocean Eng., 2020, Vol. 34, No. 6, P. 863-870 865 six gauging points: x = ±1.5 m, ±1.0 m, and ±0.5 m. The computation mesh is symmetrically distributed on two opposite sides of the mountain-type breakwater within the near field region, and the local mesh distribution around the breakwater is shown in Fig. 3. The mesh is refined until the calculated wave amplitudes are similar to the experimental results. For all computation cases, the area-weighted average values of wall y + vary from 1.76 to 4.33. Wave interaction with the mountain-type breakwater is simulated by solving the two dimensional Navier−Stokes equations using the finite volume method, and the zonal hybrid SAS/laminar method is used to simulate the turbulent flow (Chen et al., 2016;Zhan et al., 2017). The CFD package, FLUENT 15, has been utilized to study the wave-structure interaction, and the UDF function for the wave generation method is coupled with the basic solvers of ANSYS Fluent (ANSYS, 2011). The Quadratic Upwind Interpolation of Convective Kinematics (QUICK) algorithm is used to discretize the momentum convection. Pressure Implicit with Splitting of Operators (PISO) algorithm is used for the pressure-velocity coupling with pressure Staggering Option (PRESTO) discretization scheme. Further details of the discretization scheme are given in Zhan et al. (2017).

Experiments
Experiments were carried out in a small wave flume (15 m×0.6 m×0.7 m) at Sun Yat-sen University. A computer-controlled wavemaker generated the regular waves in one end of the flume, and a rubble mound wave energy dissipater at another end absorbed the transmitted wave energy. The still-water level in the wave flume was 37 cm during testing. To photograph the flow around the mountain-type breakwater, the experimental model is made of the transparent PVC plastic. Fig. 2 shows the arrangement of the breakwater model in the wave flume. For the mountain-type breakwater model, the dimensions of the three plates are 0.6 m in length, 0.12 m in width, and 5 mm in depth. The angles between the inclined plates and the horizontal line are both 30°. Three capacitive-type wave probes (Resolution: 0.1% FS; Range: 0−40 cm; Strut diameter: 5 mm; Probe diameter: 0.65 mm) and recorder units were installed before and behind the breakwater. To measure the characteristics of the incident and reflected waves, probes 1 and 2 were positioned in front of the structure at 1.5 m and 1.0 m away from the breakwater model. To measure the transmitted wave characteristics, probe 3 was fixed 1.0 m away from the breakwater model at its lee side. A high-definition camera (Sony pxw-z150) was used to monitor the water surface deformation around the breakwater. The collected data are used to validate the numerical results of incident wave height H i = 0.048 m = 0.4H and wavelength L i = 2W, 4W, 8W. To ensure repeatability of the experiment, each case has been repeated at least three times, to ensure that the measured wave profiles at three wave gauges have good reproducibility.
In addition to the wave height H i = 0.048 m used in experiments, two more wave heights are selected for numerical simulations as shown in Table 1. The higher one is set up as H i = 0.064 m, which is a little larger than the vertical height of the inclined plates, and the lower one is chosen as H i = 0.032 m. Then the initial wave steepness H i /L i varies in the range of 0.013 to 0.11. Usually, the wave breaks if the wave steepness is larger than the extreme value, which is 0.142 for traveling stationery wave according to Stokes and Mitchell's estimation (Kellner and Tilgner, 2014).  Fig. 4 shows the comparison between the simulated wave profiles with the experimental measurements for cases with incident wave height H i = 0.048 m = 0.4H and wavelength L i = 2W, 4W, 8W. The incident wave amplitudes measured at x = −1.5 m and the transmitted wave amplitudes measured at x = 1 m are both in good accordance with the relative experimental data. Though the current numerical method can predict the first wave very well, it fails to capture the secondary wave especially for the cases with L i = 4W and L i = 8W. Better simulation of the turbulent flow around the breakwater may improve the prediction of the secondary wave, but also lead to much higher computation cost. Considering that the secondary wave has less effect on the accuracy of the predicted wave height, which decides the wave absorbing ability of the breakwater, the following computation utilizes the numerical method introduced in Section 2.
Furthermore, Fig. 5 shows that the simulated water sur-face deformation is also similar to the photos taken in the laboratory. Both numerical simulation and wave tank experiments capture the wave run-up and wave breaking on the two inclined plates of the breakwater. However, the water level is not predicted very well in the two open chambers, where wave interaction takes place. The open chamber is formed by one of the inclined plates and the central vertical barrier. Waves climb up along the two inclined plates, splashes onto the vertical barrier, and the induced random turbulent flow influences the flow field prediction within the two chambers.

Comparison against the vertical-plate breakwater
To show the merit of the mountain-type (abbreviated as Mountain in Fig. 6) breakwater, the amplitudes of the reflected waves and the transmitted waves are compared with those of the vertical-plate (abbreviated as VertPlate in Fig. 7) breakwater. The calculation of the reflected wave and the transmitted wave are based on the relatively stable time series data of wave elevation at three wave gauges (x=−1.5 m, −1.0 m, and 1.5 m), to reduce the effects of multi-reflection from the fore-wall. The reflected wave amplitudes during the selected time intervals are shown in Fig. 6. Fig. 6 shows the reflected wave, which is separated from the incident wave using the analytical method described in Section 2.3 (Sun et al., 2002). The separated incident waves of the Mountain cases are similar to the VertPlate cases. Limited by space, the separated incident waves are not included in the context. The reflected wave amplitudes of the Mountain cases are larger than those of the VertPlate cases of L i = 2W and H i = 0.032 m or 0.048 m. As expected, the central vertical plate plays the main role as the wave reflector, while the inclined plate at the windward side only takes a small effect on the wave reflection.
In Fig. 7, the transmitted wave amplitudes of the Vert-Plate cases are much larger compared with the corresponding Mountain cases of L i = 2W and 4W. For long waves of L i = 8W, the performance of the mountain-type breakwater is not superior to the vertical-plate breakwater. Additionally, Fig. 7   The reflection wave coefficient K r = A r /A i is calculated based on the simulated wave amplitudes at two wave gauges (x=−1.5 m and x=−1.0 m), where A i is the incident wave amplitude, A r is the amplitude of the reflected wave. The values of A i and A r are calculated based on the separated incident and reflected waves by analytical method (Sun et al., 2002). The wave transmission coefficient K t = A t /A i is calculated based on the simulated transmitted wave amplitude A t at x = 1.5 m. Then the energy loss coefficient is estimated as , according to the energy conservation law. Fig. 9 shows the effect of the ratio D w /L i on the coefficients K t , K r , and K l for cases with different incident wave heights (H i = 0.032 m, 0.048 m and 0.064 m). A wave is   considered to be a shallow-water wave when the ratio of the water depth and its wavelength is smaller than 0.05. Though the shallow water wave is not considered in this study, it is still obvious that the wave transmission coefficient K t decreases with D w /L i , while the reflection coefficient K r and energy loss coefficient K l do the opposite.
The values of K t are similar for cases of different wave heights, indicating that the wave height has less effect on the wave transmission, compared with the wavelength. The value of the transmission coefficients K t is around 0.2 for a short wave of L i = 2W, around 0.4 for a wave of L i = 4W, and much higher (around 0.9) for the long-wave of L i = 8W due to wave overtopping.
The mountain-type breakwater can effectively dissipate the wave energy of short wave (D w /L i > 0.2, or L i < 8W). It is worth noting that the wave reflection coefficients K r of all cases are smaller than 0.6, while the wave dissipation coefficient K l is larger than 0.7 when D w /L i > 0.2. This indicates that the mountain-type breakwater absorbing wave energy mostly attributes to the wave dissipation instead of the wave reflection by the vertical plate. Especially, values of K r are around 0.2 for waves with L i = 8W. As mentioned in Subsection 4.2, the wave reflection ability of the mountain-type breakwater is similar to that of the vertical-plate breakwater.

Flow field evolution around the breakwater
As discussed above, the strong wave dissipation capability of the mountain-type breakwater is mainly due to the wave breaking on two inclined plates and wave interaction taking place in the two open chambers. To deeply investigate the wave dissipation process, the simulated free surface deformation and the turbulent kinetic energy around the breakwater under long-wave (L i = 8W and H i = 0.048 m) and short wave (L i = 2W and H i = 0.048 m) are compared in Figs. 10 and 11.
For the short wave case of L i = 2W, Fig. 10 shows the contour plot of the turbulent kinetic energy and the vector plot around the breakwater. Wave run-down and run-up are observed on the windward-facing inclined plate in Figs. 10a and 10c. Note that the flow direction is always the same along both the top and the bottom sides of the windward side plate. Fig. 10b also shows the wave breaking on the windward-facing inclined plate. The existence of the vertical plate suppresses the horizontal water flow in the two open chambers. This leads to the flow around the bottom of the vertical plate in Fig. 10b, indicated by higher turbulent kinetic energy near the bottom. Different from the windwardfacing inclined plate, the flow directions are always the opposite along the two sides of the leeward-facing plate, which act as one deflector inducing turbulent flow around its bot-    9. Effect of the ratio D w /L i on K t , K r and K l .
For the long wave case of L i = 8W, the flow evolution process is similar to that of the short wave, as shown in Fig. 11. The greatest difference is that the wave breaking on the windward-facing plate is not obvious in the long-wave case, reducing the energy dissipation due to wave breaking. Secondly, due to the better transmissivity of the long wave, the whole breakwater plays the role as one barrier, and the flow near the free surface is disturbed by the bottoms of the three plates, leading to the energy dissipation by turbulence, which is not well simulated in this study. Additionally, the flow inside the open chamber is less influenced by the wave, such that the water level variation in the open chamber is smaller compared with that in the short wave case.

Conclusion
Validated by the wave tank experiments, compared against the vertical-plate breakwater, the proposed free surface mountain-type breakwater (H i = 0.012 m and W i = 0.3 m) performs well for regular waves with wave length L i ≤ 4W and wave heights H i = 0.032 m, 0.048 m, 0.064 m, where W = 0.3 m is the width of the breakwater. The transmission coefficients Kt are around 0.3 for short waves with L i = 2W, and around 0.4 for waves with L i = 4W. Among the three plates of the mountain-type breakwater, the windwardfacing plate dissipates the wave energy mainly due to the wave breaking, while the vertical plate and the leeward-facing plate play the role as flow deflectors inducing turbulence. However, for long wave with L i = 8W, the breakwater has small effect (K t > 0.9), and the wave energy is dissipated mainly by turbulence.