Estimation of the transmission coefficients of wave height and period after smooth submerged breakwater using a non-hydrostatic wave model

Ocean Engineering 122 (2016) 202–214

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Estimation of the transmission coefficients of wave height and period after smooth submerged breakwater using a non-hydrostatic wave model Na Zhang a,b, Qinghe Zhang b,n, Guoliang Zou c, Xuelian Jiang a a

Tianjin Key Laboratory of Soft Soil Characteristics & Engineering Environment, Tianjin Chengjian University, Tianjin 300384, China State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China c Nanjing Hydraulic Research Institute, Nanjing 210024, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 27 January 2016 Received in revised form 23 June 2016 Accepted 24 June 2016

In the present study, new empirical formulas of wave transmission coefficients, including Kp for the mean wave period and Kt for the significant wave height, defined as the ratios between the transmitted and incident values of a smooth submerged breakwater, are proposed based on results simulated using the SWASH non-hydrostatic wave model. According to the analysis of the numerical results, both Kt and Kp are less than 1.0, and the transmitted peak period is close to the incident peak period in most cases. The reduction of wave height is mainly caused by energy loss due to wave breaking, and the reduction of the mean period is mainly caused by the generation of high-frequency harmonics after the submerged breakwater. Moreover, both Kt and Kp have good linear correlations with the relative submergences, and Kp has a good linear relationship with wave steepness. The relative crest width and slope have no significant effect on the transmission coefficient of the submerged breakwater compared with the effects of the relative submergences and wave steepness. Based on the factors mentioned above, new empirical 2 ) of Kt and Kp between measurements and formulas are formed. The coefficients of determination ( RCor values calculated using the new empirical formulas are 0.90 and 0.95, respectively, representing an improvement over the models of Van der Meer et al. (2003) and Carevic et al. (2013). The sensitivities of Kt and Kp to the variables in the empirical formulas are also investigated. According to the sensitivity analysis, a simplified formula of Kt is suggested based on the relative submergence and breaker parameter, and a simplified formula of Kp is proposed based on the relative submergence and wave steepness. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Smooth submerged breakwater Non-hydrostatic wave model Wave transmission coefficients Mean wave period

1. Introduction Submerged breakwaters are structures commonly used to reduce waves and protect shorelines in coastal engineering, and the transmitted wave parameters are often the focus of the practical design of submerged breakwaters (Van der Meer et al., 2000). Therefore, the estimation of the transmitted wave parameters after a smooth submerged breakwater in constructed is an important issue, and several approaches have been presented to estimate the wave parameters of a constructed submerged breakwater. One commonly used method employs physical model experiments. For example, Van der Meer and Daemen (1994) and Van der Meer et al. (2000, 2003, 2005) conducted a series of experiments on the wave transformation of low-crested structures (LCS). Van der Meer et al. (2000) concluded that both wave height n

Corresponding author. E-mail address: [email protected] (Q. Zhang).

http://dx.doi.org/10.1016/j.oceaneng.2016.06.037 0029-8018/& 2016 Elsevier Ltd. All rights reserved.

and period should be used to determine the required breakwater height for acceptable run-up due to its large dependence on the wave period. Van der Meer et al. (2005) investigated the wave transmission and reflection of low-crested structures and found that the transmitted peak period is close to the incident peak period, but the mean period may decrease considerably. Carevic et al. (2013) showed that the reduction of the wave period has a strong relationship with the wave steepness and relative submersion of a smooth submerged breakwater according to their experimental results in a wave flume. A second option for estimating the transmitted wave parameters is the use of numerical wave models, which mainly include the Navier–Stokes equation model, a nonlinear potential flow model based on the Laplace equation, the Boussinesq equation model and the mild slope equation model. The 2D depth-integrated models (e.g., Boussinesq-type model and mild-slope equation) can reasonably simulate wave transformation in shallow water (Avgeris et al., 2004; Tsai et al., 2006); however, the models insufficiently describe rapid changes in the dynamic pressure

N. Zhang et al. / Ocean Engineering 122 (2016) 202–214

∞

Nomenclature

Rc ξop α B h Hm0

Kp Kt

m0 , m2

203

breaker parameter, ξop = tan α / sop breakwater slope angle, [deg] breakwater crest width, [m] total depth, [m] significant wave height, Hm0 = 4 m0 . Hm0 − i is the incident significant wave height Hm0 − t is the transmitted significant wave height, [m] transmission coefficient of the mean wave period, Kp = T0,2 − t /T0,2 − i transmission coefficient of the significant wave height, Kt = Hm0 − t /Hm0 − i ∞ zero and second spectral moments, m0 = ∫ S ( f ) df , 0

process, e.g., for steep topography or short period waves. The 3D numerical models based on solving the Reynolds-Averaged Navier–Stokes (RANS) equations with a free surface captured by the Volume-of-Fluid (VOF) method have advantages for simulating wave transmission over the submerged breakwater (Zou and Peng, 2011; Zhang et al., 2012; Peng et al., 2013). The models can describe turbulent energy loss; however, the computational time cost is large and the efficiency is low, restricting their application in practical engineering (Ma et al., 2014b). An alternative approach to the VOF model mentioned above is the non-hydrostatic wave model, which assumes that the free surface is a single value function with horizontal coordinates. This assumption simplifies the Navier–Stokes equations and improves the calculation efficiency of the free surface elevation (Ma et al., 2014b). Previous studies suggested that the non-hydrostatic wave model is capable of simulating wave interactions with porous structures, wave breaking, nonlinear wave dynamics and infragravity wave motions in the surf zone (Ma et al., 2014a, 2014b, 2012; Smit et al., 2013; Zijlema and Stelling, 2008; Smit et al., 2014; Rijnsdorp et al., 2014). A third option is the use of an empirical formula that is derived from the main controlling mechanisms of the transmitted wave parameters. Most research has concentrated on establishing the wave transmission coefficient, i.e., the ratio between the transmitted and incident significant wave height based on a large amount of experimental data from the physical model. In addition to wave height, a good estimation of the transmitted wave period is also required for design. Van der Meer et al. (2000) proposed a prediction method for the spectral shape after wave overtopping or transmission from smooth structures. It was assumed that the part of the spectrum with frequencies smaller than 1.5 fp is similar than the incident spectrum, and the total energy is 60% of the total transmitted energy. The part of the spectrum with frequencies larger than 1.5 fp can be described by a uniform energy of up to 3.5 fp (Van der Meer et al., 2000). However, Van der Meer et al. (2005) showed that the main weakness of this model is that energy transfer to higher harmonics is independent of incident wave parameters or construction geometry. Carevic et al. (2013) further improved the method by assuming that the transfer of energy from lower to higher frequencies vanishes linearly with a decrease in the relative submergence. According to the improved model of the transmitted spectrum, a satisfactory description of the mean period reduction over a smooth impermeable breakwater is obtained (Carevic et al., 2013). However, some uncertainties may exist in the improved Van der Meer et al. (2000) model, such as the functional definition of the fmax/fp ratio, which is very sensitive to the model results (Carevic et al., 2013). Therefore, it is important

2 RCor

m2 = ∫ f 2 S ( f ) df 0 distance from the crest to the water level (negative if submerged), [m] 2 the coefficient of determination, RCor =

(∑

N ¯ i = 1 yi − y

(

2

) ( u^i − u¯ ) )

2 2 ∑iN= 1( yi − y¯ ) ⋅∑iN= 1 u^i − u¯

(

)

Sop

wave steepness, Sop = 2πHm0 − i/gTp2

T0,2

mean wave period, T0,2 = m0 /m2 . T0,2 − i is the incident mean wave period, T0,2 − t is the transmitted mean wave period, [s] peak wave period, Tp − i is the incident peak wave period, Tp − t is the transmitted peak wave period, [s] N the fitted value, u¯ = N−1 ∑i = 1 u^i N the measured value, y¯ = N−1 ∑i = 1 yi

Tp u¯ y¯

to further improve the estimation accuracy of the mean period reduction. In this paper, we present an empirical formula of the wave transmission coefficient, including both the wave height and the mean period based on the simulated results of wave interactions with a smooth submerged breakwater using a non-hydrostatic wave-resolving model, the SWASH (Simulating WAves till SHore) model, which was originally developed by Delft University. The open source program can be obtained from http://swash.source forge.net (Zijlema et al., 2011). SWASH has been applied to resolve wave dynamics in surf zones in both laboratory- and field-scale problems (Smit et al., 2013). The arrangement of this paper is as follows. Section 2 describes the non-hydrostatic model. Section 3 presents the model validation. The results are analyzed in Section 4. Section 5 presents a method for the estimation of the transmission coefficients Kt and Kp . A sensitivity analysis of the transmitted coefficients is discussed in Section 6. According to the sensitivity analysis, the simplified expressions of Kt and Kp are proposed in Section 7. A summary of this study is given in Section 8.

2. Non-hydrostatic wave model 2.1. Governing equations This study is based on the SWASH model, which is governed by the nonlinear shallow water equations with non-hydrostatic pressure. The governing equations of the two-dimensional vertical model in the Cartesian coordinate system are as follows:

∂u ∂w + =0 ∂x ∂z

(1)

∂u ∂u2 ∂wu ∂τ ∂τ g ∂ζ 1 ∂q + + + + = xx + xz ∂t ρ ∂x ρ ∂x ∂x ∂z ∂x ∂z

(2)

∂w ∂uw ∂w 2 ∂τ ∂τ 1 ∂q + + + = zz + zx ρ ∂z ∂t ∂x ∂z ∂z ∂x

(3)

where u and w are the flow velocities in the x- and z-directions, respectively, ζ is the free-surface elevation, ρ indicates a constant density, t is time, g is the acceleration of gravity, q is the nonhydrostatic pressure and τij represents the horizontal turbulent stresses.

204

N. Zhang et al. / Ocean Engineering 122 (2016) 202–214

The turbulent stresses are as follows:

∂u ∂w ∂u ∂w , τxz = νt , τzx = νt , τzz = 2νt ∂x ∂x ∂z ∂z

τxx = 2νt

(4)

where vt is the horizontal eddy viscosity due to wave breaking. The free surface elevation can be obtained by integrating Eq. (1) over the water depth and using the kinematic condition at the free surface. The final free surface equation is given as follows: ζ

∂ζ ∂ + ∂t ∂x

∫−d udz = 0

(5)

where d is the still water depth. The bottom friction is obtained through the Manning formula:

cf =

n2g (6)

h1/3

where cf is the dimensionless bottom friction coefficient, n is the Manning's roughness coefficient (in m−1/3s ) and the water depth is h = ζ + d. 2.2. Boundary conditions 2.2.1. Free surface When the surface tension is ignored at the free surface, the pressure boundary condition is written as follows.

q|z = ζ = 0

(7)

2.2.2. Bottom The normal velocity, wb , is imposed through the kinematic condition (Eq. (8)) of the bottom condition.

wb = − u

∂d ∂x

(8)

2.2.3. Open boundaries At the inflow, the internal wave generation method is applied to generate waves (Larsen and Dancy, 1983; Lin and Liu, 1999; Zou and Zhang, 2012). The free surface equation with a mass source function is prescribed as follows:

∂ζ ∂hu + = fs ( t ) ∂t ∂x

Fig. 1. Sketch of the wave-generating position and wave damping absorber.

waves are generated at x = xs . Furthermore, sponge layers are set up along the left boundary ( x = 0) and outflow boundary ( x = xe ) to absorb wave energy propagation out of the model domain. The sponge layers employed for numerical wave dissipation are improved in SWASH 2.0 (Zou and Zhang, 2013). Compared with the original numerical wave dissipation method, an exponential decay function is used for the momentum equations in the sponge layer region, and the proposed damping coefficients more efficiently reduce the wave energy of reflective waves in the sponge layer (Zou and Zhang, 2013). 2.2.4. Closed boundaries A free-slip condition with both the normal velocity and tangential stresses set to zero is imposed. For more details regarding the numerical framework, see Zijlema et al. (2011) and Smit et al. (2013).

3. Model validation In this section, the non-hydrostatic wave model SWASH is validated against laboratory measurements. First, the simulated transmitted wave spectrum over a smooth low-crested structure is compared with the physical model results of Van der Meer et al. (2000). Second, the model is tested against laboratory measurements of transmitted wave parameters based on a smooth submerged breakwater (Carevic et al., 2013). 3.1. Model validation of a transmitted wave spectrum over a smooth low-crested structure

(9)

(11)

3.1.1. Experimental conditions A transmitted wave spectrum over a smooth low-crested structure was experimentally investigated in the wave flume of Delft Hydraulics (Van der Meer et al., 2000). In the physical model experiment, the JONSWAP spectrum is used as the incident wave spectrum, and two tests are used for numerical validation. In test 1, the incident wave height ( Hm0 − i ) is 1.83 m, the mean wave period (T0,2 − i ) is 5.13 s, the peak wave period (Tp − i ) is 6.37 s and the water depth is 5.2 m. In test 2, Hm0 − i is 2.02 m, T0,2 − i is 5.38 s, Tp − i is 6.71 s and the water depth is 5.6 m. The submerged breakwater is made of plywood in the model and asphalt in nature. The crest width is 2 m, the height is 5.7 m and the slope is 1:4 (Fig. 2). Various wave gauges measured the incident and wave transmission. The model scale is 1:15, and all results are given in prototype values (Van der Meer et al., 2000).

where εi is the phase of the ith wave mode. In the wave-generating region, Eq. (9) is valid when the free surface is calculated using Eq. (5) at each time step. Eq. (5) is also employed to solve the free surface equation in the non-wavegenerating region. According to our tests, the wave surface is relatively stable when the source width is 2 Δx (two times the grid space) in the wave-generating region. As illustrated in Fig. 1, the

3.1.2. Parameter setting of the numerical simulation The length of the numerical wave flume is 70 m. The computational grid size is 0.02 m, and the vertical direction is divided into 2 layers. The incident wave parameters in the numerical simulation are the same as those of the physical model experiment. The initial time step is 0.005 s, and the simulation time is 20 min. In addition, the transmitted significant wave height ( Hm0 − t ), mean

where fs is the mass source function. For a regular wave, this can be written as follows:

fs ( t ) =

2ζ0 Cg sin ( ω 0 t ) Δx

(10)

where ζ0 is the wave amplitude of the target wave, Cg is the wave group velocity of the target wave, ω0 is the wave frequency of the target wave and Δx is the grid space in the x-direction. An irregular wave is composed of a series of linear waves with different frequencies and wave amplitudes. Therefore, an irregular wave train can be obtained by superposing different wave modes from i ¼ 1 to n: n

fs ( t ) =

∑ i=1

2ζi Cg sin ( ωi t + ϵi ) Δx

N. Zhang et al. / Ocean Engineering 122 (2016) 202–214

205

Fig. 2. Experimental setup of wave transmission over a low-crested structure (Van der Meer et al., 2000).

Fig. 3. Comparison of the measured and simulated transmitted wave spectra (a: test 1 and b: test 2).

period ( T0,2 − t ), peak period ( Tp − t ) and transmitted wave spectrum are calculated based on 300 wave periods. The bottom friction coefficient is 0.01. The width of the sponge layer is 10 m, which is set at two ends of the numerical wave flume to eliminate the influence of wave reflection. Gauges of incident and transmitted waves are positioned six and three wavelengths offshore from the breakwater to avoid nonlinearities and produce reliable wave analysis. 3.1.3. Comparison between the simulated and measured results As shown in Fig. 3(a) and (b), the simulated wave spectral shape agrees well with the measured shape. Similar to the measured spectrum, the simulated spectrum also has a clear peak. The measured and simulated spectral peak frequencies ( fp ) are 0.15 Hz and 0.15 Hz in test 1 and 0.15 Hz and 0.15 Hz in test 2, respectively. Compared with the measured spectrum, there is also energy present in larger frequencies of the simulated spectrum, which mainly range from 1.5 fp to 3.5 fp . The simulated Hm0 − t , T0,2 − t and Tp − t values calculated based on the spectrum are 0.63 m, 3.48 s and 6.59 s in test 1 and 0.83 m, 3.82 s and 6.81 s in test 2, respectively. The maximum relative error (absolute value) between the measured and simulated wave parameters is not more than 3.28% (Table 1). A comparison between the results of simulations and experiments shows that the numerical model can accurately calculate the transmitted wave spectrum over a smooth low-crested structure. 3.2. Model validation of the smooth submerged breakwater 3.2.1. Experimental conditions The experiment involving the smooth submerged breakwater was conducted in the wave flume of Carevic et al. (2013), and two sets of experiments are used in the numerical validation. In the physical experiments, the JONSWAP spectrum ( γ ¼3.3, σa ¼0.07, σ b ¼0.09) is used as the incident wave spectrum, and the irregular wave is generated by a piston-type wave generator with an active wave absorption control system. Hm0 − i is 0.058 m, T0,2 − i is 0.72 s, Tp − i is 0.81 s and the water depth is 0.4 m in test 1. In test 2, Hm0 − i Table 1 The error between the calculated and measured transmitted wave parameters. Test

Test 1

Test 2

Wave parameters

Hm0 − t (m)

T0,2 − t (s)

Tp − t (s)

Hm0 − t (m)

T0,2 − t (s)

Tp − t (s)

Measured values Simulated values Relative error (%)

0.61 0.63 3.28

3.41 3.48 2.05

6.62 6.59 -0.45

0.81 0.83 2.47

3.75 3.82 1.87

6.80 6.81 0.15

is 0.105 m, T0,2 − i is 0.92 s, Tp − i is 1.10 s and the water depth is 0.45 m. The wave channel width is 1.0 m, and the height is 1.1 m. The submerged breakwater is made of wood and is built on a flat bottom. The crest width is 0.2 m, the height is 0.35 m and the slope is 1:2 (Fig. 4). 3.2.2. Numerical parameter setting The length of the numerical wave flume is 60 m. The computational grid size is 0.02 m, and the vertical direction is divided into 2 layers. The initial time step is 0.005 s, and the simulation time is 20 min. In addition, the wave height and spectrum are calculated based on 300 waves. The bottom friction coefficient is 0.01. The width of the sponge layer is 5 m and is set at the two ends of the numerical wave flume to eliminate wave reflection and re-reflection. 3.2.3. Result validation As shown in Figs. 5 and 6(a), the simulated wave spectral shape of the incident wave agrees well with the theoretical wave spectrum. Meanwhile, the simulated wave spectral shape of the transmitted wave agrees well with the measured shape, as shown in Figs. 5 and 6(b). According to the above results, the non-hydrostatic wave model can reasonably describe the transmitted wave spectrum over a smooth low-crested structure and a smooth submerged breakwater. The numerical model does not require a complex parameter setting for a smooth structure, only a Manning coefficient of 0.01. Therefore, SWASH can be used to simulate wave transmission over a smooth submerged breakwater instead of a physical model.

4. Analysis of results The numerical flume of the smooth submerged breakwater based on SWASH is used to analyze the different factors that influence the transmission coefficients of the submerged breakwater (Fig. 7). As shown in Fig. 7, gauges P1–P5 are established to analyze the wave parameter evolution through the smooth submerged breakwater. G1 and G4 are used to calculate the incident and transmitted wave parameters, which is consistent with Fig. 4. The incident wave parameters (e.g., Hm0 − i and T0,2 − i ) and the transmitted parameters (e.g., Hm0 − t and T0,2 − t ) are defined in the list of symbols according to the principle of spectrum. The incident wave parameters are calibrated without a submerged breakwater using a numerical wave tank. The calibrated incident wave spectrum exhibits good agreement with the theoretical spectrum, and the maximum relative error is only 2%. The transmitted wave

N. Zhang et al. / Ocean Engineering 122 (2016) 202–214

G1

G2G3

G4 3.6m

0.6m

4.4m

0.1m

dissipation chamber

Z

1.1m

206

X SWL

0.2m

1:2

1:2

0.35m

d1=0.40m d2=0.45m

Fig. 4. Experimental setup of the wave flume with a smooth submerged breakwater (Carevic et al., 2013).

parameters are measured with the submerged breakwater using a numerical wave tank. In this paper, two transmission coefficients, including the ratio between the transmitted and incident significant wave height ( Kt ) and the ratio between the transmitted and incident mean wave period ( Kp ), are discussed. According to previous studies (Van der Meer et al., 2000, 2003; Carevic et al., 2013), the main factors influencing the transmission coefficients of the submerged breakwater are the relative submergence depth ( Rc /Hm0 − i ), relative crest width ( B /Hm0 − i ), wave steepness ( Sop ) and slope. In the following section, we investigate how the variables affect the transmission coefficients of the smooth submerged breakwater based on an analysis of the simulated results. The incident wave parameters generated by the numerical tests are shown in Table 2. 4.1. Effects of Rc /Hm0 − i This section discusses the influences of different Rc /Hm0 − i (0.0, 0.5,  1.0 and  1.5) values on Kt , Kp and Tp − t /Tp − i (the ratio between the transmitted peak period and the incident peak period) of the smooth submerged breakwater. 4.1.1. Transmission coefficient of the significant wave height According to Fig. 8(a), the transmission coefficient of the significant wave height is less than 1.0 when the Rc /Hm0 − i varies from 0.0 to  1.5, suggesting that the transmitted significant wave height after the smooth submerged breakwater is smaller than that of the incident wave. The reduction in wave height is caused by the energy loss due to wave breaking. Additionally, if the other parameters are held constant, Kt increases as the absolute value of Rc /Hm0 − i ( Rc /Hm0 − i ) increases due to the decrease in wave energy loss with increasing Rc /Hm0 − i . However, the transmission coefficient increases slowly and is close to 1.0 in the range of  1.0– 1.5 of Rc /Hm0 − i , as shown in Fig. 8(a). Furthermore, Kt is equal to 1.0 when Rc /Hm0 − i = − 2.5 in test 29. This is mainly because wave breaking does not occur when Rc /Hm0 − i is sufficiently large, and the submerged breakwater is no longer the main reason for the loss of wave energy. When Rc /Hm0 − i varies from 0.0 to  1.5, Kt increases by 78% at Sop = 0.04 , 65% at Sop = 0.03, 64% at Sop = 0.02 and 43% at Sop = 0.01. Therefore, Rc /Hm0 − i is an important factor that affects the transmission coefficient. The correlation coefficient of Kt with Rc /Hm0 − i is  0.90, which indicates a strong negative linear correlation between Rc /Hm0 − i and the transmission coefficient.

4.1.2. Transmission coefficient of the mean wave period According to Fig. 8(b), the transmission coefficient of the mean period is less than one when Rc /Hm0 − i varies from 0.0 to  1.5; thus, the transmitted mean wave period after encountering a smooth submerged breakwater is smaller than that of the incident wave. The reduction in the mean period occurs because it covers the waves at all frequencies, including high-frequency harmonics formed by waves over the submerged breakwater. The generation process of the high-frequency harmonic wave can be explained as follows. When the wave propagates over the submerged breakwater, the nonlinear effect of the wave is suddenly enhanced with the rapid change in the water depth, leading to the generation of a higher frequency harmonic wave and more energy being present at higher frequencies than that of the incident spectrum. With the increase in water depth after the submerged breakwater, the nonlinear effect of the wave becomes weak, and part of the wave energy transfers from a higher frequency to a lower frequency. The changes in the wave spectrum mentioned above are shown in Fig. 9. The mean period of the incident wave is 1.57 s, Hm0 − i is 0.1 m and the water depth is 0.5 m (P1). When the waves propagate to the middle of the slope after the submerged breakwater (P4), the mean wave period decreases to 0.82 s, which is almost half of that of the incident wave. Additionally, the significant wave height is 0.1 m, and the water depth is 0.35 m. When the waves propagate to G4, the mean wave period is 1.26 s, the wave height is 0.1 m and the water depth is 0.5 m. Further clarification is shown in Fig. 10, in which the spatial variations of the normalized potential energy of the total, primary and high frequency components are plotted. According to Beji and Battjes (1993), the total energy can be obtained by adding the primary energy and the higher frequency energy. The primary energy is calculated using an integration range between 0.00 Hz and 0.76 Hz (1.5 fp ), and the higher frequency energy is calculated for frequencies larger than 1.5 fp . The energies are normalized with respect to the total measured at G1. According to Fig. 10, the energy distribution parameter (e.d.p.) defined by Briganti et al. (2003) is calculated to describe the variation in the energy associated with f > 1.5 fp between the incident and transmitted spectra. The e.d.p. is 0.44, suggesting that the energy is transferred toward a higher frequency ( f > 1.5 fp ). In addition, the generation of a superharmonic wave transfers the wave energy from a lower frequency to a higher frequency and leads to a reduction in the mean period. The conclusion confirms those of Beji and Battjes (1993) and Carevic et al. (2013). If the other parameters are held constant, Kp increases with

Fig. 5. Comparison of the theoretical and numerical incident wave spectra G1 (a) and the measured and simulated transmitted wave spectra G4 (b) in test 1.

N. Zhang et al. / Ocean Engineering 122 (2016) 202–214

0.003

0.003

simulated theoretical

0.001

simulated meas

0.002 E(m s)

0.002

E(m s)

207

0.001

0.000

0.000 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

f(Hz)

f(Hz)

Fig. 6. Comparison of the theoretical and simulated incident wave spectra G1 (a) and the measured and simulated transmitted wave spectra G4 (b) in test 2.

increasing Rc /Hm0 − i because less energy is transferred from the lower frequency to higher frequency, as the nonlinear influence becomes weaker with increasing Rc /Hm0 − i . This can be further explained by the parameter Et1.5/Et , which shows how much energy is located in the higher frequency ( f > 1.5 fp ). When Rc /Hm0 − i varies from 0.0 to  1.5, Et1.5/Et ranges from 0.44 to 0.18 at Sop = 0.04 . The results agree with the predicted results of Van der Meer et al. (2000) and Carevic et al. (2013). However, Kp increases slowly and is close to 1.0 in the range of 1.0 to  1.5 of Rc /Hm0 − i , as shown in Fig. 8(b). Furthermore, Kp is equal to 1.0 when Rc /Hm0 − i = − 1.67 (test 25) and Rc /Hm0 − i = − 2.5 (test 29). A smaller or even no reduction of the mean wave period suggests that the transmitted waves are not changed by high frequency harmonics. Finally, when Rc /Hm0 − i varies from 0.0 to 1.5, Kp increases by 20% at Sop = 0.04 , 22% at Sop = 0.03, 30% at Sop = 0.02 and 31% at Sop = 0.01. Therefore, Rc /Hm0 − i is an important factor that affects the transmission coefficient. The correlation coefficient between Kp and Rc /Hm0 − i is  0.77, which indicates a strong linear correlation between Rc /Hm0 − i and the transmission coefficient. 4.1.3. Peak period As observed in Fig. 8(b), the ratio of Tp − t /Tp − i computed by SWASH ranges from 1.0 to 1.02 when Rc /Hm0 − i varies from 1.5 to 0.0 and Sop = 0.04 ; thus, the transmitted peak period is affected less by Rc /Hm0 − i . According to the results of Carevic et al. (2013), Tp − t /Tp − i ranges from 0.97 to 1.02 when Rc /Hm0 − i changes from 0.53 to  1.56 and Sop = 0.02~0.04 . These results are in good agreement with the conclusions of Van der Meer et al. (2000), and the ratio of Tp − t /Tp − i is close to 1.0, especially for transmission coefficients larger than 0.2. 4.2. Effects of Sop This section focuses on the influence of different Sop values (0.01-0.06) on Kt , Kp and Tp − t /Tp − i of the smooth submerged breakwater. 4.2.1. Transmission coefficient of the significant wave height Fig. 11(a) shows that Kt decreases with increasing Sop . The wave steepness is an important parameter affecting wave breaking. Wave breaking occurs more easily at larger values of wave steepness (steep waves) than at values of smaller wave steepness (gentle waves). Therefore, the loss of wave energy of a gentle wave

is smaller than that of a steep wave, which is why Kt decreases as Sop increases. When Sop varies from 0.01 to 0.04, Kt is reduced by 29% at Rc /Hm0 − i ¼0.0, 26% at Rc /Hm0 − i ¼  0.5, 14% at Rc /Hm0 − i ¼ 1.0 and 11% t Rc /Hm0 − i ¼  1.5, suggesting that Sop is another important factor that affects the transmission coefficient of the significant wave height, in addition to Rc /Hm0 − i . The correlation coefficient of Kt is  0.38, indicating that Sop and Kt have a weak linear correlation. 4.2.2. Transmission coefficient of the mean wave period According to Fig. 11(b), Kp increases with increasing Sop because larger values of Sop presuppose smaller values of L according to Table 2, which causes weak generation of high-frequency harmonics. In the case of large values of L in relation to Rc , nonlinear effects are more pronounced. Fig. 8(b) illustrates that Kp depends on Rc /Hm0 − i in the same way it depends on Rc /L . Therefore, Kp is less affected by the high-frequency harmonics at larger values of wave steepness (Beji and Battjes, 1993). When Sop varies from 0.01 to 0.04, Kp increases by 34% at Rc /Hm0 − i ¼ 0.0, 28% at Rc /Hm0 − i ¼ 0.5, 24% at Rc /Hm0 − i ¼ 1.0 and 23% at Rc /Hm0 − i ¼  1.5, indicating that Sop is also an important factor that affects Kp , in addition to Rc /Hm0 − i . The correlation coefficient of Kp is 0.77, indicating that the linear correlation between Sop and Kp is good. 4.2.3. Peak period As shown in Fig. 11(b), the ratio of Tp − t /Tp − i computed by the numerical model ranges from 0.97 to 1.02 when Sop changes from 0.01 to 0.06 and Rc /Hm0 − i = − 0.5; thus, the transmitted peak period is less affected by Sop . Based on the results of Carevic et al. (2013), Tp − t /Tp − i ranges from 0.94 to 1.02 when Sop changes from 0.02 to 0.06 and Rc /Hm0 − i = − 0.49~ − 0.57. These results are in good agreement with the conclusions of Van der Meer et al. (2000), and the peak of the transmitted spectrum is more or less the same as that of the incident spectrum. 4.3. Effects of B /Hm0 − i This section focuses on the influence of different B /Hm0 − i (1.08.0) on Kt and Kp of the smooth submerged breakwater. 4.3.1. Transmission coefficient of the significant wave height Fig. 12(a) shows that if the other parameters are held constant, Kt decreases with increasing B /Hm0 − i because more time is required for waves to cross over the crest of a wider-top submerged

Fig. 7. Numerical flume layout of the smooth submerged breakwater.

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Table 2 The incident wave parameters of the numerical analysis. The incident wave was generated by the JONSWAP spectrum ( γ = 3.3, σa = 0.07, σ b = 0.09 ). Rc is the submersion of the wave crown, and h is the total water depth.

Rc ¼ 0.00 m, h ¼ 0.35 m

Rc ¼  0.05 m, h ¼ 0.40 m

Rc ¼  0.10 m, h ¼ 0.45 m

Rc ¼  0.15 m, h ¼ 0.50 m

Test

Hm0 − i (m)

T0,2 (s)

Tp (s)

Test

Hm0 − i (m)

T0,2 (s)

Tp (s)

Test

Hm0 − i (m)

T0,2 (s)

Tp (s)

Test

Hm0 − i (m)

T0,2 (s)

Tp (s)

1 2 3 4 17 18 19 20

0.10 0.10 0.10 0.10 0.06 0.08 0.09 0.10

1.04 1.27 1.51 1.76 0.66 0.77 0.88 0.93

1.30 1.60 1.88 2.19 0.69 0.85 0.98 1.12

5 6 7 8 21 22 23 24

0.10 0.10 0.10 0.10 0.06 0.08 0.09 0.11

1.02 1.23 1.45 1.68 0.66 0.77 0.88 0.98

1.26 1.54 1.80 2.11 0.69 0.85 0.98 1.19

9 10 11 12 25 26 27 28

0.10 0.10 0.10 0.10 0.06 0.08 0.11 0.12

1.01 1.21 1.41 1.62 0.66 0.77 0.91 1.01

1.25 1.50 1.75 2.01 0.69 0.85 1.08 1.24

13 14 15 16 29 30 31 32

0.10 0.10 0.10 0.10 0.06 0.08 0.11 0.12

0.98 1.18 1.37 1.57 0.66 0.77 0.91 1.01

1.25 1.46 1.71 1.96 0.69 0.85 1.08 1.24

along a steep slope. Therefore, the milder the slope, the smaller the value of Kt . When the waves cross over the submerged breakwater, Kt is reduced by 12% as the slope varies from 1.0 to 3.0. Therefore, the slope has no significant effect on the transmission coefficient of the submerged breakwater.

breakwater, making complete wave breaking more likely. This process will lead to greater loss of wave energy and further reduction of Kt . When B /Hm0 − i varies from 1 to 4, Kt is reduced by 12.5% at Sop = 0.04 and 6.9% at Sop = 0.02, which shows that B /Hm0 − i has no significant effect on the transmission coefficient of the submerged breakwater compared with the effects of Rc /Hm0 − i and Sop . However, when B /Hm0 − i varies from 1 to 8, Kt is reduced by 23.4% at Sop = 0.04 and 15.3% at Sop = 0.02; thus, the effect of B /Hm0 − i on Kt is not negligible for structures with very wide submerged crests. This conclusion is in accordance with that of Van der Meer et al. (2003).

4.4.2. Transmission coefficient of the mean wave period Fig. 13(a) shows that Kp decreases with increasing slope because more wave energy is transferred from lower frequencies to higher frequencies in the process of wave transmission over the breakwater along a mild slope compared to over a steep slope. This is demonstrated by the spatial evolution of T0,2 in Fig. 13(c). When the slope varies from one to three, Kp is reduced by 5%, which indicates that the slope has no significant effect on Kp compared with those of Rc /Hm0 − i and Sop . In general, Rc /Hm0 − i is the main factor that affects the Kt value of the smooth submerged breakwater, and Sop is the secondary factor. Rc /Hm0 − i and Sop are the main factors that affect Kp . B /Hm0 − i and slope have no significant effects on the transmission coefficient of the submerged breakwater. Moreover, the transmitted peak period is similar to the incident peak period, which shows that Tp − t /Tp − i is not sensitive to the different influence factors.

4.3.2. Transmission coefficient of the mean wave period Fig. 12(b) shows that when B /Hm0 − i varies from one to eight as the other parameters remain unchanged, Kp is reduced by 6.0% at Sop = 0.04 and 10.9% at Sop = 0.02, indicating that B /Hm0 − i has no significant effect on Kp compared with those of Rc /Hm0 − i and Sop . 4.4. Effects of slope This section focuses on the influence of different front slopes (1.0, 1.5, 2.0, 2.5 and 3.0) on Kt and Kp of the smooth submerged breakwater.

5. Estimation of the transmission coefficients

4.4.1. Transmission coefficient of the significant wave height Fig. 13(a) shows that Kt decreases with increasing slope because wave breaking occurs more easily along the mild slope in shallow water, which is demonstrated by the spatial evolution of the wave height ( Hm0 ) through the flume in Fig. 13(b). When the slope changes from 1.0 to 3.0, the critical breaking wave height of the mild slope is less than that of the steep slope. This result is consistent with the conclusion of Collins and Weir (1969). Then, waves break on the smooth slope and wave height decreases as water depth decreases. When the breaking waves cross over the crest into the water behind the structure, the breaking distance of the mild slope is much longer than that of the steep slope. This process leads to more energy dissipation along a mild slope than

According to the analysis in Section 4, the factors that influence the transmission coefficients of a submerged breakwater are Rc /Hm0 − i , Sop , B /Hm0 − i and slope. In this section, using these variables, the empirical formulas of the transmission coefficients of the submerged breakwater are obtained with functions of nonlinear expression using the least square method (Bilau et al., 2004; Ghasemi et al., 2014). 5.1. Estimation of Kt The form of the nonlinear expression is derived from the 1.2

1.2 1.0

a

1.0 0.8

0.8 0.6

0.2

Kp

Sop=0.01 Sop=0.02 Sop=0.03 Sop=0.04 Sop=0.06

0.4

0.6

Sop=0.01 Sop=0.02 Sop=0.03 Sop=0.04 Sop=0.06 Sop=0.04 for Tp-t/Tp-i

0.4 0.2 0.0

0.0 -1.5

-1.0

-0.5 Rc/Hm0-i

0.0

Tp-t/Tp-i

Kt

b

-1.5

-1.0

-0.5 Rc/Hm0-i

Fig. 8. Numerical simulation results of Kt , Kp and Tp − t /Tp − i in relation to Rc /Hm0 − i ( B /Hm0 − i is 2.0, slope is 1:2).

0.0

N. Zhang et al. / Ocean Engineering 122 (2016) 202–214

0.006

209

0.006

P1

P2 0.004

E(m2 s)

E(m2 s)

0.004

0.002

0.002

0.000

0.000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.006

0.006

P4

P3 0.004

E(m2 s)

E(m2 s)

0.004

0.002

0.002

0.000

0.000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.006

0.006

P5

G4 0.004

E(m2 s)

E(m2 s)

0.004

0.002

0.000

0.002

0.000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

f (Hz)

f (Hz)

Fig. 9. Spectral evolution of a simulated wave ( Rc /Hm0 − i = − 1.5, Sop = 0.01, fp = 0.5Hz ).

in (Eqs. (12) and 13):

normalized energy

2.0 total energy primary energy high frequency energy

1.5

Kt = − 0.3R c /H i +0.75 ⎡⎣ 1 − exp ( −0.5ξop ) ⎤⎦ for ξop < 3

1.0

Kt = − 0.3R c /H i + ⎡⎣ B/Hi ⎤⎦

* ⎡⎣ 1 − exp −0.5ξop ⎤⎦ *0.75 for ξop ≥ 3

−0.31

0.5

-3

-2

-1

0 1 Distance(m)

2

3

4

5

Fig. 10. Spatial variations in the normalized potential energy of a simulated wave ( Rc /Hm0 − i = − 1.5, Sop = 0.01, fp = 0.5Hz ).

)

Kt = a1R c /H m0 − i +a2 B/H m0 − i +a3 exp ( a 4 ξop ) + a5

formula of the transmission coefficient of the smooth submerged breakwater proposed by Van der Meer et al. (2003), which is given

(14)

2.0

2.0

1.8

1.4

1.4

1.0

1.2 1.0

1.8 1.6 1.4 1.2 1.0

0.8

0.8

0.8

0.6

0.6

0.6

0.4 0.01

0.4 0.01

0.02

0.03

0.04 Sop

0.05

0.06

0.4 0.02

0.03

0.04

0.05

Sop

Fig. 11. Numerical simulation results of the transmission coefficients in relation to Sop ( B /Hm0 − i is 2.0, slope is 1:2).

0.06

Tp-t/Tp-i

1.2

1.6

Kp

1.6

Rc/Hm0-i = 0.0 Rc/Hm0-i = -0.5 Rc/Hm0-i = -1.0 Rc/Hm0-i = -1.5 Rc/Hm0-i = -0.83 Rc/Hm0-i = -1.25 Rc/Hm0-i = -0.5 for Tp-t/Tp-i

1.8

Rc/Hm0-i = 0.0 Rc/Hm0-i = -0.5 Rc/Hm0-i = -1.0 Rc/Hm0-i = -1.5 Rc/Hm0-i = -0.83 Rc/Hm0-i = -1.25

(13)

where Kt is the transmission coefficient of the significant wave height of the smooth submerged breakwater. Kt ranges from 0.4 to

2.0

Kt

(

where ξop is the breaker parameter and ξop = tan α / sop . The above formula was divided into two parts. The first is for breaking waves ( ξop < 3), where the influence of the crest width is not present. The other is for non-breaking waves ( ξop ≥ 3). Kt varies from 0.075 to 0.8. Here, we propose a new form of the formula based on Table 2 and the analysis in Section 4, as follows:

0.0 -4

(12)

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1.0

1.0 Sop=0.04 Sop=0.02

0.9 0.8

0.8

0.7

Kp

Kt

Sop=0.04 Sop=0.02

0.9

0.6

0.7 0.6

0.5 0.4

0.5 1

2

3

4

5

6

7

8

1

2

3

B/Hm0-i

4

5

6

7

8

B/Hm0-i

Fig. 12. Numerical simulation results of the transmission coefficients in relation to B /Hm0 − i ( Rc /Hm0 − i is -0.5, slope is 1:2).

1.0, and ξop < 4.5. a1~a5 are coefficients determined using the nonlinear least square method, as follows:

a1 = − 0.2720, a2 = − 0.0130, a3 = 0.1003, a 4 = 0.3343 and a5 = 0.2654. To indicate how well the data fit a statistical model of the transmission coefficients between the measured and calculated values (Cameron and Windmeijer, 1996; Kurz-Kim and Loretan, 2 , is used as 2014), the coefficient of determination, denoted as RCor follows: 2 RCor =

(∑

N i=1

2

( yi − y¯ ) ( u^i − u¯ ) )

(

N N ∑i = 1 ( yi − y¯ )2⋅ ∑i = 1 u^i − u¯

2

)

(15)

N N where y¯ = N−1 ∑i = 1 yi , u¯ = N−1 ∑i = 1 u^i , yi is the measured value and ^ u is the fitted value.

i

2 of the transmission coefficients As shown in Fig. 14(a), the RCor between the numerical results produced by SWASH and the results calculated by Eq. (14) is 0.98, indicating an excellent model fit. To examine the predictability of Eq. (14), we apply the formula to the measured data from Carevic et al. (2013), which is not used

2 to fit Eq. (14) (Fig. 14(b)). The value of RCor is 0.90, suggesting that the measured data fit the formula well. In addition to the comparison of the values calculated using Eq. (14) with the measured values of Carevic et al. (2013), the calculated results of Van der Meer et al. (2003) are also compared with 2 the same measured data, as shown in Fig. 14(c). The RCor of the transmission coefficients between the measured and the calculated values by Van der Meer et al. (2003) is 0.86. Therefore, the formula proposed in this paper can effectively estimate the transmission coefficient of the smooth submerged breakwater, and the estimation accuracy is further improved.

5.2. Estimation of Kp According to the analysis in Section 4, Kp has a better linear relationship with wave steepness, which is different from Kt . Therefore, the linear term of the wave steepness is added to the nonlinear expression of Kt , which is the formula for Kp , as shown in Eq. (16):

Kp = a1R c /H m0 − i +a2 B/H m0 − i +a3 sop + a 4 exp ( a5 ξop )

(16)

Fig. 13. Numerical simulation results of the transmission coefficients in relation to slope (a), the spatial evolution of Hm0 (b) and T0,2 (c) ( Rc /Hm0 − i is -0.5, B /Hm0 − i is 2.0 and Sop is 0.04).

Kt -calculated

N. Zhang et al. / Ocean Engineering 122 (2016) 202–214

211

1.0

1.0

1.0

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

calculated by Eq. (14) y=x

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Kt -simulated by SWASH

0.2 0.0 0.0

calculated by Eq. (14) y=x 0.2

0.4

0.6

0.8

1.0

Kt -measured by Carevic et al. (2013)

calculated by Van der Meer et al. (2003) y=x

0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Kt -measured by Carevic et al. (2013)

Fig. 14. The comparison of Kt based on the numerical results of SWASH with calculated values from Eq. (14) (a), measurements from Carevic et al. (2013) with calculated values according to Eq. (14) (b) and the calculated values of Van der Meer et al. (2003) (c).

where Kp is the transmission coefficient of the mean wave period for a smooth submerged breakwater, Kp varies from 0.50 to 1.05 and ξop < 4.5. a1~a5 are coefficients obtained by the nonlinear least square method, as follows:

a1 = −0.1137, a2 = −0.0151, a3 = 3.1843, a 4 = 0.7108, a5 = − 0.0405. 2 The RCor of the transmission coefficients of the mean wave period between the numerical results and the calculated values is 0.96, which indicates an excellent model fit. Then, the measured data from Carevic et al. (2013) are applied to examine the pre2 value of 0.95 dictability of Eq. (16), as shown in Fig. 15(b). The RCor suggests that the measured data agree well with the data calculated using Eq. (16). The Kp calculated based on the formula of Carevic et al. (2013) is also compared with the same measured values, as shown in Fig. 15(c). The empirical formula of Carevic et al. (2013) is an improved analytical model for the calculation of transmitted spectra based on the Van der Meer et al. (2000) model. It was assumed that the transfer of energy from lower to higher frequencies vanishes linearly with decreasing relative submergence −Rc /Hm0 − i . The energy transferred to higher frequencies is assumed to be uniformly distributed between 1.5 fp and 3.2 fp (Carevic et al., 2013). 2 of the transmission coefficients of the mean wave period The RCor between the measurements and the calculated values by Carevic et al. (2013) is 0.93. Therefore, the formula of Kp proposed in this paper can effectively estimate the transmission coefficient of the smooth submerged breakwater, and the estimation accuracy is further improved.

6. Sensitivity analysis According to Section 5, Rc /Hm0 − i , B /Hm0 − i and ξop are the

Kp-calculated

1.1 1.0

1.1

a

1.0

variables in the formula of Kt in Eq. (14), and Rc /Hm0 − i , B /Hm0 − i , Sop and ξop are the variables in the formula of Kp in Eq. (16) for a smooth submerged breakwater. In this section, we examine the contributions of these variables. To do so, the contribution of one variable is analyzed by ignoring the other variables in (Eqs. (14) 2 values of the transmission and 16) in the sensitivity study. The RCor coefficients between the simulated results and the sensitivity study are shown in Table 3. 6.1. Contribution of each variable to Kt According to the results shown in Fig. 16 and Table 3, most of the data are distributed below the line of y¼x for Rc /Hm0 − i , B /Hm0 − i or ξop . Therefore, Kt based on only one parameter is less than that 2 based on all parameters. For Rc /Hm0 − i , the RCor of Kt is 0.85, suggesting that Rc /Hm0 − i makes a significant contribution to Kt . For 2 values of Kt are 0.16 and 0.25, respectively. B /Hm0 − i or ξop , the RCor 2 of Kt is 0.97, and the calculated Kt is For Rc /Hm0 − i and ξop , the RCor close to the measured value. Therefore, the contribution of B /Hm0 − i to Kt is small enough to be ignored. According to the above analysis, the contribution of Rc /Hm0 − i is the largest, followed by that of ξop , and the contribution of B /Hm0 − i is the smallest in Eq. (14). 6.2. Contribution of each variable to Kp According to the results in Fig. 17, similar to those for Kt , most of the Kp values are distributed below the line of y¼x based on Rc /Hm0 − i , B /Hm0 − i , Sop or ξop . Therefore, Kp based on only one parameter is less than that based on all the parameters. For Rc /Hm0 − i and Sop , the values of Kp are 0.60 and 0.59, respectively. Therefore, Rc /Hm0 − i and Sop have almost the same significant 2 contributions to Kp . For B /Hm0 − i and ξop , the RCor values of Kp are 2 of 0.93 0.10 and 0.15, respectively. For Rc /Hm0 − i and Sop , the RCor between the numerical Kp and the calculated Kp based on the case 1.1

b

1.0

0.9

0.9

0.9

0.8

0.8

0.8

0.7

0.7

0.7

0.6

0.6

0.5 0.4

calculated by Eq. (16) y=x

0.5

0.6 calculated by Eq. (16) y=x

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

0.4 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Kp -simulated by SWASH

Kp -measured by Carevic et al. (2013)

0.5

c

calculated by Carevic et al. (2013) y=x

0.4 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Kp -measured by Carevic et al. (2013)

Fig. 15. The comparison of Kp based on SWASH simulations using values calculated with Eq. (16) (a), measurements from Carevic et al. (2013) with calculated values according to Eq. (16) (b) and measurements from Carevic et al. (2013) with calculated values from Carevic et al. (2013) (c).

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N. Zhang et al. / Ocean Engineering 122 (2016) 202–214

Table 3 2 of K between the numerical results and the calculated values based on the terms in Eq. (14) of the cases of the contributions of R /H RCor t c m0 − i (case 1), B / Hm0 − i (case 2), ξop 2 of K between the numerical results and the calculated values based on the terms in Eq. (16) of the cases of (case 3), Rc /Hm0 − i þ ξop (case 4) and all variables (case 5). RCor p

contributions of Rc /Hm0 − i (case 1), B /Hm0 − i (case 2), Sop (case 3), ξop (case 4), Rc /Hm0 − i þ Sop (case 5), Rc /Hm0 − i þ Sop þ ξop (case 6) and all variables (case 7). Case name

2 RCor

Kp

Kt 1

2

3

4

5

1

2

3

4

5

6

7

0.85

0.16

0.25

0.97

0.98

0.60

0.10

0.59

0.15

0.93

0.94

0.96

1.2

calculated by Rc/Hm0-i

Kt -calculated

1.0

calculated by B/Hm0-i

0.8

calculated by

0.6

calculated by Rc/Hm0-i+

0.4

y=x

op op

0.2 0.0 -0.2 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Kt -simulated by SWASH Fig. 16. The comparison of Kt based on the numerical results with calculated values (one variable is analyzed by ignoring the other variables) according to Eq. (14).

1.2

calculated by Rc/Hm0-i calculated by B/Hm0-i

1.0

calculated by Sop

Kp -calculated

0.8

calculated by

op

calculated by Rc/Hm0-i+Sop

0.6

calculated by Rc/Hm0-i+Sop+

0.4

op

y=x

0.2 0.0 -0.2 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Kp -simulated by SWASH Fig. 17. The comparison of Kp based on the numerical results with calculated values (one variable is analyzed by ignoring the other variables) according to Eq. (16).

5 indicates that there is a strong correlation between them. However, the calculated Kp is smaller than the numerical Kp . This is mainly due to ignoring the contributions of B /Hm0 − i and ξop , which are similar to the constant terms in Eq. (16), as shown in Fig. 17. 2 of Kp is 0.94, which is almost the For Rc /Hm0 − i , Sop and ξop , the RCor same as that based on Rc /Hm0 − i and Sop . Therefore, B /Hm0 − i and ξop have similar minor contributions to Kp . According to the above analysis, the contribution of Rc /Hm0 − i is the largest, followed by that of Sop , and the contributions of ξop and B /Hm0 − i are the smallest in Eq. (16).

7. Simplification of formulas

and B /Hm0 − i is the smallest in the formula of Kp . Accordingly, the formulas of Kt and Kp are simplified in this section. 7.1. Simplification of the formula of Kt 2 According to the sensitivity analysis in 6.1, the RCor of Kt between the numerical results and calculated values based on the terms in Eq. (14) and ignoring B /Hm0 − i is 0.97, which indicates that the contribution of B /Hm0 − i to Kt is very small. Therefore, a simplified formula is obtained based on Rc /Hm0 − i and ξop , as shown in Eq. (17).

Kt = −0.2736R c /H m0 − i +0.4348 exp ( 0.1438ξop ) − 0.1552 2 RCor

According to Section 6, the contribution of Rc /Hm0 − i is the largest, followed by that of ξop , and the contribution of B /Hm0 − i to Kt is small enough to be ignored. Moreover, the contribution of Rc /Hm0 − i is the largest, followed by that of Sop , and the contribution of ξop

(17)

The of Kt between the numerical results and calculated 2 of Kt values based on Eq. (17) is 0.97 for data fitting. The RCor between the measurements and calculated values based on Eq. (17) is 0.87 for data prediction. According to the analysis in 5.1, the 2 of Kt between the measurements and calculated values based RCor

N. Zhang et al. / Ocean Engineering 122 (2016) 202–214

on Eq. (14) is 0.90, which improves Eq. (17) by using all of the variables. Therefore, both (Eqs. (14) and 17) are recommended.

213

submerged breakwater with different armor blocks should be developed.

7.2. Simplification of the formula of Kp Acknowledgments 2 According to the sensitivity analysis in 6.2, the RCor of Kp between the numerical results and calculated values based on the terms in Eq. (16) and ignoring B /Hm0 − i and ξop is 0.93, which indicates that the contributions of B /Hm0 − i and ξop to Kp are very small. Therefore, a simplified formula can be obtained based on Rc /Hm0 − i and Sop , as shown in Eq. (18).

Kp = − 0.1079R c /H m0 − i +4.0830sop + 0.5691

(18)

2 The RCor of Kp between the numerical results and calculated 2 values based on Eq. (18) is 0.94 for data fitting. The RCor of Kp between the measurements and calculated values based on Eq. (18) is 0.93 for data prediction. According to the analysis in 5.2, the 2 of Kp between the measurements and calculated values based RCor on Eq. (16) is 0.95, which improves Eq. (18) by using all of the variables. Therefore, both (Eqs. (16) and 18) are recommended.

8. Conclusions In this paper, numerical simulations are performed to investigate the transmission coefficients of the mean wave period Kp and significant wave height Kt after passing over a smooth submerged breakwater based on the non-hydrostatic wave model SWASH. According to the analysis of the numerical results, both Kt and Kp are less than 1.0 in most cases. The reduction of wave height is mainly caused by energy loss due to wave breaking, and the reduction of the mean period is mainly caused by the generation of high-frequency harmonics after the submerged breakwater. Moreover, both Kt and Kp have good linear correlations with Rc /Hm0 − i , and Kp has a good linear relationship with Sop , which is different from Kt . B /Hm0 − i and slope have no significant effect on the transmission coefficients of the smooth submerged breakwater. New empirical formulas of the wave transmission coefficients, Kp for the mean wave period and Kt for significant wave height are 2 proposed based on the simulated results. The RCor values of Kt and Kp between the numerical results and the calculated values are 2 0.98 and 0.96, respectively, for data fitting. The RCor values of Kt and Kp between the measurements and the calculated results are 0.90 and 0.95 respectively, which represents an improvement comparing with the models of Van der Meer et al. (2003) 2 2 ( RCor = 0.93 for Kp ) over = 0.86 for Kt ) and Carevic et al. (2013) ( RCor a smooth submerged breakwater. The sensitivities of Kt and Kp to the variables in the empirical formulas are also investigated. According to the sensitivity analy2 of Kt between the numerical results and calculated sis, the RCor values based on the terms in the formula of Kt proposed in this paper and ignoring B /Hm0 − i is 0.97, which indicates that the contribution of B /Hm0 − i to Kt is very small. Therefore, a simplified formula of Kt is obtained based on Rc /Hm0 − i and ξop . Similarly, the 2 of Kp between the numerical results and calculated values RCor based on the terms in the formula of Kp proposed in this paper and ignoring B /Hm0 − i and ξop is 0.93, which indicates that the contributions of B /Hm0 − i and ξop to Kp are very small. Therefore, a simplified formula of Kp is obtained based on Rc /Hm0 − i and Sop . The empirical formula proposed in this paper can be used to estimate the transmission coefficients of the smooth submerged breakwater. In the future, the wave dissipation effect with different armor blocks of submerged breakwater should be discussed. An empirical formula of the transmission coefficients of a

This research was supported by the National Natural Science Foundation of China under Grant (51509177); the Science and Technology Program of Tianjin, China under Grant (2013-8); the Natural Science Foundation of Tianjin, China under Grant (14JCYBJC22100); and the National Natural Science Foundation of China under Grant (51409185).

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