Numerical Investigation on Hydrodynamic Processes of Extreme Wave Groups on Fringing Reef

Abstract

The low-lying reef islands distributed in the tropical and subtropical coastal regions are highly vulnerable to the devastating damages of surges and waves during the severe weather events. Over the past two decades, extreme waves have caused tremendous loss and damages to the tropical and subtropical coastal regions. Previous research has focused on the wave hydrodynamics of tsunami waves, as well as regular and irregular waves on the fringing reefs. The complex wave hydrodynamics of extreme waves on the fringing reefs are rarely studied. By applying the nonhydrostatic numerical flow solver (NHWAVE), transformation and breaking process of the crest- and trough-focused wave groups on the fringing reef are analyzed in this study. Influences of the major factors, i.e., water depth, significant wave height, peak wave period, forereef slope and backreef slope, and ridge width, are discussed in detail. The results show that there are complex interactions between the fringing reef and the focused wave group. Breaking waves of high intensity can form at the reef crest. Meanwhile, due to the wave breakings at the reef crest and bottom friction of the reef flat, the local wave height can be effectively reduced. Within the complex wave hydrodynamics of focused waves on the fringing reef, most of the wave energy can be dissipated. In addition, hydrodynamic difference between the crest- and trough-focused waves on the fringing reef is very limited. The research results of this study will further help researchers to better understand the wave hydrodynamics of extreme waves over the fringing reefs.

1. Introduction

The low-lying reef islands distributed in the tropical and subtropical coastal regions are highly vulnerable to the devastating damages of surges and waves during the severe weather events. In the past few decades, extreme waves have caused tremendous loss and damages to the tropical and subtropical coastal regions [1,2,3]. Due to the special topography of coral reef [4,5], most of the wave energy can be dissipated within the complex wave hydrodynamics on the fringing reef [6,7,8,9]. Fringing reefs are widely recognized as natural buffer zones that can effectively protect the coastal communities behind the fringing reef [10,11,12,13]. Therefore, it is necessarily important to study the wave hydrodynamics of extreme surges and waves over the fringing reefs.

Previous research has focused on the wave hydrodynamics of tsunami-like waves [14,15], regular waves [16,17] and irregular waves [18,19,20] on the fringing reefs. Through field investigation [21,22,23,24,25], theoretical analysis [26,27], physical laboratory [28,29], numerical simulation [30,31], the relevant wave-induced setup [32], sea and swell waves [33], infragravity waves [34,35,36,37], and wave runup height [38] have been systematically analyzed. Besides, the effects of human activities, such as the excavation pits and seawalls, on the hydrodynamics of fringing reefs were evaluated through field surveys [39], experiment work [40], and numerical modellings [41]. Zhu et al. [42] and Qu et al. [43] discussed the effects of reef flat permeability on the wave transformation and setup. Moreover, Guo et al. [44] recently conducted a numerical study to analyze the influences of blowing wind on the wave hydrodynamics of tsunami-like wave on the fringing reef. However, few studies were conducted to investigate the wave hydrodynamics of extreme waves on the fringing reefs. Known as freak or focused waves, extreme waves are often unpredictable. Their huge wave heights can pose a great threat to the safety of the fringing reef and its inhabitants. According to the observation data from 2006 to 2010 [3], extreme waves have appeared in both deep and shallow waters. Although the wave runup process [45], wave–vegetation attenuation characteristics [46], and wave impacting loads [47] of focused wave groups have been extensively studied, hydrodynamics of the focused waves on the fringing reef are still insufficient. Recently, some experiment works were conducted by Zheng [48] and Fan et al. [49] to investigate the hydrodynamics of focused waves on a fringing reef model. However, due to the insufficient experimental settings, the collected data are not enough to conduct a comprehensive analysis. Therefore, it is necessary to strengthen the research work on the hydrodynamic characteristics of focused waves on the fringing reefs. To make up for the shortcomings of previous research, transformation, breaking and runup processes of the focused waves on the fringing reef were fully investigated in this study based on the nonhydrostatic flow solver (NHWAVE) [50,51]. Influences of the major factors, i.e., water depth, significant wave height, peak wave period, forereef slope and backreef slope, and ridge width, are analyzed in detail.

The rest of this paper can be organized as follows. Section 2 describes the information regarding the nonhydrostatic flow solver. Parameters regarding the wave energy evaluation are defined in Section 3. Model verifications are performed in Section 4. Section 5 gives a comprehensive analysis of complex hydrodynamic characteristics of focused waves on the fringing reef. Section 6 summarizes the main research results of this study.

2. Numerical Wave Model

The nonhydrostatic flow solver (NHWAVE) [50] was applied to simulate the wave transforming, breaking, and runup processes of focused waves on the fringing reef, whose governing equations can be read as

$$\frac{\partial D}{\partial t}+\frac{\partial Du}{\partial x}+\frac{\partial Dv}{\partial y}+\frac{\partial \omega }{\partial \sigma }=0$$

(1)

$$\frac{\partial \mathit{U}}{\partial t}+\frac{\partial \mathit{F}}{\partial x}+\frac{\partial \mathit{G}}{\partial y}+\frac{\partial \mathit{H}}{\partial \sigma }={\mathit{S}}_h+{\mathit{S}}_p+{\mathit{S}}_{\tau }+{\mathit{S}}_c$$

(2)

where $\mathit{U}=\left(Du,Dv,Dw\right)$, and the well-balanced fluxes are defined as:

$$\mathit{F}=\left(\begin{array}{c}Duu+\frac{1}{2}g{\zeta }^2+gh\zeta \\ Duv\\ Duw\end{array}\right),\mathit{G}=\left(\begin{array}{c}Duv\\ Dvv+\frac{1}{2}g{\zeta }^2+gh\eta \\ Dvw\end{array}\right),\mathit{H}=\left(\begin{array}{c}u\omega \\ v\omega \\ w\omega \end{array}\right)$$

The flux terms at the right side of the Equation (2) can be defined as:

$${\mathit{S}}_h=\left(\begin{array}{c}gD\frac{\partial h}{\partial x}\\ gD\frac{\partial h}{\partial y}\\ 0\end{array}\right),{\mathit{S}}_p=\left(\begin{array}{c}-\frac{D}{\rho }\left(\frac{\partial p}{\partial x}+\frac{\partial p}{\partial \sigma }\frac{\partial \sigma }{\partial x}\right)\\ -\frac{D}{\rho }\left(\frac{\partial p}{\partial y}+\frac{\partial p}{\partial \sigma }\frac{\partial \sigma }{\partial y}\right)\\ -\frac{1}{\rho }\frac{\partial p}{\partial \sigma }\end{array}\right),{\mathit{S}}_{\tau }=\left(\begin{array}{c}D{S}_{{\tau }_x}\\ D{S}_{{\tau }_y}\\ D{S}_{{\tau }_z}\end{array}\right)$$

In the above formulations, $t$ is time. $x,y,z$ represent the Cartesian coordinates. $u,v,w$ are the corresponding velocities in the $x$, $y$, and $z$ directions, respectively. $\omega$ represents the velocity in the $\sigma$ coordinate, and $\sigma =\left(z+h\right)/D$. The net water depth ($D$) is calculated as $D=h+\zeta$, and ζ is the water elevation. $h$ is the water depth. $g$ is the acceleration of gravity. $\left|\mathit{u}\right|$ represents the magnitude of the velocity, and $\left|\mathit{u}\right|=\sqrt{u^2+v^2+w^2}$. $p$ is the pressure. $\rho$ is the density. In addition, $D{S}_{{\tau }_x}$, $D{S}_{{\tau }_y}$ and $D{S}_{{\tau }_z}$ are the diffusion source terms. The drag forces exerted by the rough reef surfaces can be modelled as ${\mathit{S}}_c=D{\mathit{F}}_c$. ${\mathit{F}}_c=\frac{1}{2}{C}_f\left|\mathit{u}\right|\mathit{u}$ represents the drag force. $C_f$ represents the surface drag coefficient.

The nonlinear $k-\epsilon$ turbulence closure [52] is applied to determine the eddy viscosity. The governing equations for turbulent kinetic energy ($k$) and dissipate rate ($\epsilon$) are read as:

$$\frac{\partial Dk}{\partial t}+\mathsf{\nabla }·\left(Duk\right)=\mathsf{\nabla }·\left[D\left(\nu +\frac{v_t}{{\sigma }_k}\right)\mathsf{\nabla }k\right]+D\left(P_s-\epsilon \right)$$

(3)

$$\frac{\partial D\epsilon }{\partial t}+\mathsf{\nabla }·\left(Du\epsilon \right)=\mathsf{\nabla }·\left[D\left(\nu +\frac{v_t}{{\sigma }_{\epsilon }}\right)\mathsf{\nabla }\epsilon \right]+\frac{\epsilon }{k}D\left(C_{1\epsilon }{P}_s-C_{2\epsilon }\epsilon \right)$$

(4)

The turbulence viscosity $v_t$ can be calculate as $v_t=C_{\mu }\frac{k^2}{\epsilon }$. ${\sigma }_k$,${\sigma }_{\epsilon }$, $C_{1\epsilon }$, $C_{2\epsilon }$, and $C_{\mu }$ are empirical parameters [53]. As suggested by [53], ${\sigma }_{\epsilon }$ = 1.3,${\sigma }_k$ = 1.0,$C_{\mu }$= 0.09, $C_{1\epsilon }$ = 1.44, and $C_{2\epsilon }$= 1.92. Besides, $P_s$ is shear production.

The combined finite volume and finite difference methods [50] are applied to solve the governing equations of NHWAVE. The flux terms of the momentum equations are calculated by the classical shock-capturing HLL-TVD scheme. Additionally, the convective flux terms of the turbulence model are approximated by the second-order hybrid linear and parabolic approximation (HLPA) method. In order to obtain second-order time marching, the two-step 2nd-order nonlinear Runge-Kutta method is used. To satisfy the stability requirement, the time step size is determined adaptively based on the Courant-Friedrichs-Lewy (CFL) condition. For all simulations in this study, CFL number is set as 0.1. Readers can refer to [50,51] for more details of the present nonhydrostatic flow solver.

In this study, both crest- and trough-focused waves were numerically generated using the Dirichlet inlet boundary conditions through the superposition of the linear wave components. The first-order free surface, ${\zeta }_1$, is defined as:

$${\zeta }_1={\displaystyle \sum }_{i=1}^N{a}_{i}cos{\theta }_i$$

(5)

where $a_i$ is the amplitude of each wave component. ${\theta }_i$ is the phase of each wave component, which can be defined as:

$${\theta }_i=k_{i}x-{\omega }_{i}t-{\epsilon }_i$$

(6)

where $k_i$ is the wave number and ${\omega }_i$ is the angular frequency of each wave component. ${\epsilon }_i$ is the phase angle of each wave component.

When the focused waves are generated, ${\epsilon }_i$ is chosen in such a way that each of the wave components will focus at the desired time $t_f$ and location $x_f$ as:

$${\epsilon }_i=k_i{x}_f-{\omega }_i{t}_f$$

(7)

By applying the linear superposition method, wave amplitude ($a_i$) of each wave component can be expressed in terms of the wave amplitude $a_F$ at the actual focal location and the wave spectrum $S_i\left(\omega \right)$:

$$a_i=a_F\frac{S_i\left(\omega \right)\Delta \omega }{{{\displaystyle \sum }}_{i=1}^N{S}_i\left(\omega \right)\Delta \omega }$$

(8)

The JONSWAP spectrum [54] is applied to determine the wave spectrum $S_i\left(\omega \right)$ for the focused wave group. The significant wave height ($H_s$), the peak angular frequency (${\omega }_p$), and the number of component ($N$) are the input parameters of the JONSWAP spectrum. $S_i\left(\omega \right)$ is a function of angular frequency (${\omega }_i$), which determines the distribution of the input wave energy as:

$$S_i\left(\omega \right)=\frac{5}{16}{H}_s^2 {\omega }_p^4{\omega }_i^{-5}\mathit{exp}\left(\frac{-5}{4}{\left(\frac{{\omega }_i}{{\omega }_p}\right)}^{-4}\right){{\rm Y}}^{\mathit{exp}\left(\frac{-{\left({\omega }_i-{\omega }_p\right)}^2}{2{\sigma }^2{\omega }_p^2}\right)}{A}_{{\rm Y}}$$

(9)

where $\gamma$ is the dimensionless peak shape parameter and $\gamma =3.3$.$A_{\gamma }$ is the normalizing factor and $A_{\gamma }=1-0.287\ln \left(\gamma \right)$. For the crest-focused waves, positive significant wave height ($H_s$) is used. When the trough-focused waves are considered, negative wave height (−$H_s$) is applied.

Additionally,

$$\sigma =0.07,when {\omega }_i<{\omega }_p$$

(10)

$$\sigma =0.09,when {\omega }_i>{\omega }_p$$

(11)

The frequency interval ($\Delta \omega$) is given as:

$$\Delta \omega =\frac{{\omega }_s-{\omega }_e}{N}$$

(12)

where ${\omega }_s$ and ${\omega }_e$ are the upper and lower limits of the angular frequency range.

The first-order horizontal velocity $u_1$ and vertical velocity $w_1$ are defined as the sum of individual wave components as:

$$u_1={\displaystyle \sum }_{i=1}^N{a}_i{\omega }_i\frac{cosh\left(k_i\left(z+h\right)\right)}{sinh\left(k_{i}d\right)}cos{\theta }_i$$

(13)

$$w_1={\displaystyle \sum }_{i=1}^N{a}_i{\omega }_i\frac{sinh\left(k_i\left(z+h\right)\right)}{sinh\left(k_{i}h\right)}sin{\theta }_i$$

(14)

Once the wave steepness of a focused wave group is relatively large, wave–wave interactions become more apparent [47]. Then, second-order focused wave must be used by considering the wave–wave interactions. The second-order wave components must be added to the first-order components of the water surface elevation and the velocities as:

$$\zeta ={\zeta }_1+{\zeta }_2$$

(15)

$$u=u_1+u_2$$

(16)

$$w=w_1+w_2$$

(17)

The second-order wave components are implemented using the second-order irregular wave theory [55] as formulated by Ning et al. [56]. Additionally, the total number of the wave components are set as $N$ = 60 for all the following computations based on carefully numerical experiments.

3. Wave Energy Evaluation

During the wave propagating, transforming, and breaking of focused wave group over the fringing reef, different kinds of wave energy are to be lost and transferred. Same as [46], the potential energy ($PE$), kinetic energy ($KE$), and total energy ($TE$) are calculated as:

$$KE=\iint \rho \frac{u^2+w^2}{2}dxdz$$

(18)

$$PE=\iint \rho gzdxdz-{\left[\iint \rho gzdxdz\right]}^{t=0}$$

(19)

$$TE=KE+PE$$

(20)

To evaluate the wave energy dissipating rate, wave energy dissipation rate is defined as:

$$D_E=\frac{E_{T,I}-E_{T,O}}{E_{T,I}}$$

(21)

where $E_{T,I}$ represents the wave energy entering the region of fringing reef and $E_{T,O}$ represents the wave energy leaving the study region.

Additionally, wave energy flux ($E_f$) is also assessed along the simulation domain to evaluate the spatial variation of wave energy, which can be defined as:

$$E_f={{\displaystyle \int }}_{-h}^{\zeta }p·udz$$

(22)

4. Model Verification

4.1. Propagation of Focused Wave

In this section, to calibrate the reliability of NHWAVE, the generation and propagation of focused waves are simulated. The measured data [56] were used to calibrate the computed water elevations and velocities at different wave gauges and velocimeters. The experiment was performed in a wave flume at Dalian University of Technology. To determine the actual focusing position of the focused waves, 35 wave height meters were laterally arranged along the wave flume. By gradually adjusting the input focus position, the actual focus position ($x_f$) always happens at $x$ = 11.4 m away from the paddle. The horizontal velocity at 0.15 m below the water surface and at the actual focusing position ($x$ = 11.4 m) was measured by an acoustic doppler velocimeter (ADV). In the experiment, the water depth ($h$) was set as 0.5 m. The length of simulation domain is 15 m. Grid resolution ($dx$) of 0.02 m was used in the horizontal direction. In addition, 40 grid layers were used. Similar to the physical laboratory, the actual focusing position ($x_f$) is always fixed at 7.2 m in the computation.

Table 1. Wave parameters.

Run	${\mathit{T}}_{\mathit{p}}$(s)	${\mathit{H}}_{\mathit{s} }$(m)
1	1.20	0.0626
2	1.20	0.1246
3	1.25	0.175
4	1.25	0.2062

lists the wave parameters for the four groups of experiment run. Figure 1 plots the time series of water elevation recorded at the actual focusing position under different peak wave periods and significant wave heights. As depicted in Figure 1, both the peak values and time history of predicted water elevations agree well with the measured data. In addition, Figure 2 depicts the time series of horizontal velocities recorded by the velocimeter. The computed horizontal velocities are in good agreement with the measured data, but there are some differences at some time instances when the wave steepness is high. As shown above, the capability of nonhydrostatic flow solver in simulating the propagations of crest- and trough-focused waves has been well verified.

4.2. Transformation of Focused Wave Group on Fringing Reef

Wave hydrodynamics of the transforming and breaking processes of focused waves over the fringing reef are simulated in this section. The measured data collected by Zheng [48] were used to validate the computed water elevations and velocities at different wave gauges and velocimeters. Figure 3 depicts the experimental layout. The smooth plexiglass is applied to make the surfaces of reef slopes and reef-flat, as well as the reef crest, which can effectively reduce the friction forces at the bottom. A 0.6 m high vertical wall is fixed at the right side of the reef flat. The slope is fixed at 35.6 m from the paddle. The reef crest is 1.41 m wide and 0.085 m high. The offshore and onshore side slopes of reef crest are both 1:4. 32 wave height meters are laterally distributed in the wave flume to record the water elevations. Four acoustic doppler velocimeters (ADV) are installed before and after the reef crest to record the wave-breaking induced velocities. In the simulation, the water depth is 1 m, which produces a 0.15 m submergence water depth ($hr$) at the reef flat. The length of simulation domain is 45 m. To verify the requirements of grid resolution, three groups of grid were generated, i.e., dx = 0.08, 0.04, and 0.02 m. Besides, 40 grid layers are used. At the same time, the focusing position is $x_f$ = 37.45 m, and the focusing time is $t_{f }$= 70 s. Figure 4 shows the time series of water elevation at different wave height meters. As depicted in Figure 4, the predicted water surface elevations are in good agreement with the measurements. After the wave group entering the simulation domain, propagation of the focused waves in the flat bottom region can be well predicted (WG1 and WG6). As focused waves approach the forereef slope, wave transformation happens due to the decreasing in the water depth. The numerical wave model can correctly capture this process (WG7 and WG13). After wave crest breaking at the reef–crest, the wave profile of focused wave group can be greatly distorted (WG18 and WG23). Later, the surge bore induced by the wave-breaking processes is to continually propagate along the reef flat (WG27 and WG32). Figure 5 compares the horizontal velocities at four different velocimeters. It clearly demonstrates that the computed velocities agree well with the measurements during the wave propagating, transforming, and breaking processes. As demonstrated above, the reliability of nonhydrostatic flow solver in simulating the complicated wave hydrodynamics of focused waves on the fringing reef were well calibrated. In addition, computational results of three groups of grids are very close to each other, which verifies the grid resolution (dx) of 0.02 m can meet the mesh requirements and can be applied in the following computations.

5. Results and Discussion

The complex transforming, breaking, and runup of both crest- and tough-focused wave groups on the fringing reef were simulated and discussed in this study. Effects of the major factors, i.e., water depth ($h$), peak wave period ($T_p$), significant wave height ($H_s$), forereef and backreef slopes ($\cot \alpha$ and $\cot \beta$), and ridge width ($W_{rc}$) were well analyzed. Figure 6 depicts the computational layout. The reef is 0.85 m high. The length of reef flat ($L$) is 6 m. The side slopes of offshore and onshore side of the reef crest are both 1:4. For the basic run, forereef and backreef slopes are $\cot \alpha$= 4 and $\cot \beta$= 5, respectively. The ridge base width ($W_{rc}$) is 1.41 m. The length of simulation zone is 40 m. The grid resolution ($dx$) of 0.05 m was used in the horizontal direction. In addition, 40 grid layers were used. In the rest of simulations, the focusing position ($x_f$) and time ($t_f$) always were fixed at 15 m and 20 s by adjusting the input focusing location ($x_p$) and the focusing time ($t_p$), respectively.

5.1. Complex Flow Phenomena

Complex flow phenomena within the propagating and transforming of focused waves over the fringing reef were numerically analyzed based on the setup of basic run. The peak wave period ($T_p$) is 2 s. The significant wave height ($H_s$) is 0.16 m. The offshore water depth is 1 m. Figure 7 shows the water velocity contours at different time instances. When the focused wave groups focus is at the toe of forereef slope, water body with the high-speed can be seen around the wave crests (Figure 7a,e). Due to presence of the reef crest, strong wave breakings can be formed (Figure 7b,f). Later, high-speed breaking surge bore rushes into the reef flat region after wave breakings (Figure 7c,g). Meanwhile, the local wave heights can be effectively decreased by the wave breakings around the reef crest. Once the strongest breaking surge bore runs up at the backreef slope, the water velocity magnitude in the reef flat region will decrease significantly (Figure 7d,h). Figure 7b–d,f–h show the velocity contours at the occurrences of the first-three largest wave runup heights. Overall, the maximum wave runup height of crest-focused waves occurs about 0.9 s earlier than the trough-focused wave group. Figure 8 plots the time series of the water elevations at different wave height meters. After the incident, different wave components enter the simulation domain, and the focused wave profile begins to gradually evolve (Figure 8a). When t = 20 s, focused waves are to focus at the toe of forereef slope (Figure 8b). Because of the shallowing effects of the forereef slope, the water elevations increase a little bit (Figure 8c). After the wave breakings at the reef crest, the water elevation can be sharply decreased (Figure 8d). Attributed to the frictional force at the reef flat, the water elevations will undergo complex evolutions (Figure 8e,f). Figure 9 shows the spatial–temporal evolutions of water elevation along the simulation domain. As depicted in Figure 9, focused wave groups focus at $x_f$= 15 m and $t_f$= 20 s. Meanwhile, strong wave reflections can be observed at the forereef and backreef slopes and the reef crest. Moreover, the transmitted waves will be reflected forth and back among the backreef slope and reef crest, indicating the complicated hydrodynamic characteristics of focused waves on the fringing reef. However, differences in the spatial–temporal evolution of water elevation of the crest- and trough-focused waves are negligible. As seen in Figure 10, the local maximum water elevation along the simulation zone has no significant change, except for a significant decrease at the reef crest. The local maximum water elevation approaches its minimum value at the reef crest. Figure 11 depicts the space distribution of the local maximum depth-averaged velocity, which is nearly constant before the wave focusing at the toe of the forereef slope. When the waves transform at the forereef slope, the magnitude of maximum depth-averaged velocity gradually increases and will approach its peak value once the largest wave breaking occurs. At this moment, the local maximum water elevation approaches its minimum value (Figure 10). Later, the magnitude of maximum depth-averaged velocity tends to decrease slightly due to the friction force at the bottom. Besides, differences in the maximum velocity between the crest- and trough-focused waves can be ignored. Figure 12 depicts the space distributions of the local maximum wave height, which is calculated as the local maximum crest-minus-trough. As demonstrated in Figure 12, the local maximum wave height increases gradually in a slight mode before the occurrence of wave focusing. When the focused wave group transforms at the forereef slope, the local maximum wave height noticeably increases due to the shallowing water effect. Once the wave breakings at the reef crest happen, the local maximum wave height decreases sharply at the reef crest, which can be further decreased by the bottom friction. Differences in the maximum local wave heights of crest- and trough-focused waves can be ignored. Figure 13 depicts the energy fluxes at different locations. It is observed that the time series of the energy flux exhibit a similar temporal variation as that of the focused waves (Figure 13a). When the incident wave components focus at the toe of forereef slope, the energy flux of crest-focused wave at the toe of forereef slope approaches a large value at the focusing time. The energy flux of trough-focused wave also approaches a very large value at the focusing time. The magnitude of energy flux of trough-focused wave is much lower than that of crest-focused wave (Figure 13b). After the occurrences of wave breakings around the reef crest, the magnitude of energy flux tends to decrease sharply (Figure 13c). The magnitude of the energy flux can be further decreased by the bottom friction and the turbulent violence within the agitated water body (Figure 13d). Compared to the energy flux at the backreef slope toe, the peak values of energy flux at the toe of forereef slope can be reduced by 85.5% and 84.3% for the crest-focused wave and trough-focused wave, respectively. Time series of the different types of water wave energies and their conversions are plotted in Figure 14. After the wave components entering the simulation domain, both the kinetic and potential wave energies gradually increase, as well as the total water wave energy. These three types of water wave energies approach their peak values at about $t$= 17.9 s. Attributed to the wave breakings at the reef crest, these three types of wave energies can be decreased at a high rate after $t$= 20.6 s. Later, the wave energies of water body can be further dissipated by the bottom friction and the turbulent violence in the water body. Obviously, temporal evolutions of different types of wave energies of crest-focused wave group are nearly same to that of trough-focused wave group. Nearly 87.8% of total water wave energy can be reduced. Figure 15 shows the runup height of breaking bores at the backreef slope. As different wave components continuously break at the reef crest, the runup height of breaking surge bore shows a similar trend as the wave profile of focused wave group. There also exist π-phase differences in the runup height of breaking surge bores of the crest- and trough-focused waves. Hence, the runup height of breaking surge bore of crest-focused wave group approaches its peak vale earlier than that of trough-focused wave. Nevertheless, peak value of the runup height of breaking bore of the crest-focused wave is close to that of the trough-focused wave. There is only a 1.39% difference. As demonstrated above, there exist complex interactions between the focused wave group and fringing reef. It can be also observed that there are limited differences between the wave hydrodynamics of crest-focused wave and trough-focused wave on the fringing reef. In the following sections, the computational results of crest-focused wave group will be mainly analyzed. The computational results of trough-focused wave group will serve as necessary supplements.

5.2. Influences of Significant Wave Height

Influences of the significant wave height were numerically investigated based on the basic setup in this section. The peak wave period ($T_p$) is 2 s. The offshore water depth is 1 m. Five different significant wave heights were chosen, such as $H_{s }$= 0.08, 0.12, 0.16, 0.2, and 0.24 m, which produce five different relative wave heights, such as $\frac{H_s}{h}$= 0.08, 0.12, 0.16, 0.2, and 0.24. The spatial–temporal evolutions of water elevation along the simulation zone are plotted in Figure 16. When the significant wave height increases gradually, the wave reflecting intensity of focused wave groups at the forereef slope and between the backreef slope and the reef crest can be continuously intensified. As shown in Figure 17, the local maximum water elevation approaches to its peak at the toe of the forereef slope. After the wave breakings at the reef crest, the maximum water elevations change dramatically. It is also seen that the higher incident significant wave height is, the lower relative maximum water elevation (${\zeta }_{max}/H_s$) after wave breakings can be. Apparently, the relative maximum water elevation (${\zeta }_{max}/H_s$) at the reef crest monotonically decreases with the relative significant wave height ($H_s/h$), indicating that the wave breaking strength can be intensified with the significant wave height (Figure 18). As depicted in Figure 19, when the significant wave height gradually increases, the relative local maximum wave height ($H_{i,max}/H_s$) decreases more after the wave breakings at the reef crest. Figure 20 depicts the space distribution of maximum depth-averaged velocity. It clearly demonstrates that the depth-averaged flow velocity gradually increases with the significant wave height, especially at the reef crest. As depicted in Figure 21, the magnitude of maximum depth-averaged velocity at the reef crest gradually increases with the significant wave height ($H_s/h$). When the significant wave height ($H_s/h$) gradually increases from 0.08 to 0.24, magnitude of the maximum depth-averaged velocity at the reef crest increased by 67.4%. Obviously, the total wave energy increases sharply with the significant wave height (Figure 22). The maximum value of total water wave energy can be increased by 88.9%, as the relative significant wave height ($H_s/h$) gradually increases from 0.08 to 0.24. The wave energy dissipation rate is found to gradually increases with the relative significant wave height (Figure 23). Meanwhile, there are no differences between crest-focused wave and trough-focused wave in term of wave energy. The wave energy dissipation rate can be increased by 5.3%, as the relative significant wave height ($H_s/h$) gradually increases from 0.08 to 0.24. Besides, as depicted in Figure 24, the maximum wave runup heights of the crest-focused wave and trough-focused wave can be increased by 64.7% and 62.5%, respectively, as the relative significant wave height ($H_s/h$) gradually increases from 0.08 to 0.24. Moreover, if the relative significant wave height ($H_s/h$) is less than 0.16, the maximum wave runup height of crest-focused wave is identical to that of trough-focused wave. If the relative significant wave height ($H_s/h$) becomes greater than 0.16, maximum wave runup height of the crest-focused wave can become slightly larger than that of the trough-focused wave.

5.3. Influences of Peak Wave Period

In this section, effects of the peak wave period are analyzed. In the simulation, the offshore water depth is 1 m. The significant wave height is 0.16 m. Four different peak wave periods ($T_p$) are chosen, i.e., $T_p$= 1.5, 2, 2.5, and 3 s. Normally, with the increase in peak wave period, the spectrum distribution gradually shifts to the low frequency range. At the same time, the maximum value of wave spectrum increases significantly. Therefore, the increase in peak wave period will generally increase the wave period and wave height of incident wave group. Overall, the incident total water wave energy of the incident waves tends to increase. As depicted in Figure 25, when the peak wave period continuously increases, wave reflecting intensity of the focused wave group at the forereef slope, as well as that between the backreef slope and the reef crest can be significantly intensified. Figure 26 plots the space distribution of local maximum water elevations. If the peak wave period is small ($T_p$= 1.5 s), the incident wave group generally breaks at the reef crest. The maximum water elevation behind the reef crest can be decreased significantly. When the peak wave period increases to a certain extent ($T_p$= 2 s), the incident wave group still breaks at the reef crest. However, the local maximum water elevation behind the reef crest becomes close to that before the reef crest. When the peak wave period increases further ($T_p$= 2.5 and 3 s), the incident wave group breaks first on the forereef slope and then at the reef crest. Besides, the local maximum water elevation behind the reef crest become higher than that before the reef crest. Figure 27 depicts the variations of local maximum water elevation with the peak wave period at the reef crest. Apparently, the relative maximum water elevation (${\zeta }_{max}/H_s$) monotonically increases with the peak wave period. The relative maximum water elevation (${\zeta }_{max}/H_s$) increases about 37.9%, as the peak wave period ($T_p$) increases from 1.5 s to 3 s. As the peak wave period gradually increases, the relative local maximum wave height ($H_{i,max}/H_s$) gradually increases, especially behind the reef crest (Figure 28). This finding indicates that the larger peak wave period of focused wave group is, the greater the damage potential of focused wave group to the coastal region behind the reef can be. Space distribution of the local maximum depth-averaged velocity under different peak wave periods is shown in Figure 29. The local maximum depth-averaged velocity gradually increases with the peak wave period. As the peak wave period increases, location of the maximum depth-averaged velocity gradually moves upstream. Figure 30 shows that the magnitude of maximum depth-averaged velocity at the reef crest increases linearly with the peak wave period, which can be increased by 39.9%, as the peak wave period ($T_p$) increases from 1.5 s to 3 s. Figure 31 depicts the total water wave energies. Obviously, the total water wave energy increases sharply with the peak wave period. The total water wave energy can increase by 72.2%, as the peak wave period ($T_p$) increases from 1.5 s to 3 s. Figure 32 depicts the variation of wave energy dissipation rate with the peak wave period. If the peak wave period ($T_p$) is no greater than 2.5 s, the wave energy dissipation rate does not vary noticeably with the peak wave period. However, the energy dissipation rate can be decreased by 3.9%, as the peak wave period increases from 2.5 s to 3 s. The intensity of wave reflection can be continuously intensified with the peak wave period. The intensity of wave breakings is weakened, which leads to the reduction of wave energy dissipation rate. As shown in Figure 31, total wave energy increases gradually as the peak wave period increases. Therefore, maximum wave runup height increases with the peak wave period. Maximum wave runup heights of the crest-focused wave and trough-focused wave can be increased by 55.6% and 61.1%, respectively, as the peak wave period ($T_p$) gradually increases from 1.5 s to 3 s (Figure 33). Meanwhile, if the peak wave period ($T_p$) is no greater than 2 s, maximum wave runup height of the crest-focused wave is slightly larger than that of the trough-focused wave.

5.4. Influences of Water Depth

Influences of the still water depth are discussed in this section. The peak wave period ($T_p$) is 2 s, and the significant wave height ($H_s$) is 0.16 m. Four different water depths are selected, such as, h = 0.85, 0.925, 1, and 1.075 m, which generate four different submergence water depth at the reef flat, such as $h_r$= 0, 0.075, 0.15, 0.225 m. Figure 34 depicts the space distribution of local maximum water elevation. If the reef crest is dry ($h_r\le 0.075 m$), the relative maximum water elevation (${\zeta }_{max}/H_s$) exhibits a strong oscillation behavior. The relative maximum water elevation (${\zeta }_{max}/H_s$) wave height does not decrease significantly, and it even increased ($h_r=0.000m$). As depicted in Figure 35, when the reef crest is dry, the relative maximum water elevation at the reef crest gradually decreases with the submergence water depth. Once the reef crest is wet, the relative maximum water elevation at the reef crest begins to increase with submergence water depth. As plotted in Figure 36, when the reef crest is dry, the maximum local wave height ($H_{i,max}/H_s$) exhibits an oscillation behavior before the wave breakings at the reef crest ($h_r=0$ and 0.075 m). At the same time, when the submergence water depth is relatively low, the propagating of breaking bore on the reef flat can be significantly distorted and dissipated by the friction force at the bottom, and the local maximum wave height can be rapidly decreased further. When the submergence water depth is relatively large, maximum local wave height does not decrease significantly after wave breakings at the reef crest. Space distribution of local maximum depth-averaged velocity is shown in Figure 37. The peak value of local maximum depth-averaged flow velocity gradually decreases with the water depth. As the submergence water depth gradually decreases, position of the maximum depth-averaged velocity gradually shifts upstream. With the increase in water depth, effective water depth at the reef crest gradually increases. Therefore, local maximum depth-averaged velocity at the reef crest gradually decreases with water depth (Figure 38). Local maximum depth-averaged velocity can be reduced by 50.6% as the submergence water depth ($h_r$) gradually increases from 0 m to 0.225 m, As shown in Figure 39, if the top of reef crest is dry ($h_r$< 0.075 m), water wave energy dissipation rate only slightly increases with the water depth. Once the submergence depth ($h_r$) is greater than 0.075 m, the water wave energy dissipation rate noticeably increases with water depth. The water wave energy dissipation rate can be increased about by 8.3% when the submergence water depth gradually increases from $h_r$ = 0.075 m to $h_r$= 0.225 m. Figure 40 shows the variation of maximum wave runup height with submergence water depth. As depicted in Figure 40, maximum wave runup height increases monotonically with submergence water depth. As the submergence water depth gradually increases from $h_r=$0 m to $h_r=$0.225 m, the maximum wave runup heights of crest-focused wave and trough-focused wave increase by 84.0% and 73.9%, respectively. At the same time, if the submergence water depth becomes greater than 0.15 m, the variation of water depth has no noticeable influences on maximum wave runup height. Besides, if the submergence water depth becomes greater than 0.15 m, maximum wave runup height of the crest-focused wave is slightly greater than that of the trough-focused wave.

5.5. Influences of Reef Slopes

Effects of the forereef and backreef slopes are analyzed in this section. The water depth ($h$) is 1 m. The peak wave periods ($T_p$) is 2 s. The significant wave height ($H_s$) is 0.16 m. When the influences of forereef slope on wave hydrodynamics of focused wave over the fringing reef are considered, the backreef slope is $\cot \beta$ = 5. The ridge base width ($W_{rc}$) is 1.41 m. Six different forereef slopes are selected, such as $\cot \alpha$= 6, 5, 4, 3, 2, and 1. As shown in Figure 41, when the forereef slope is relatively large ($\cot \alpha$ = 1), the wave components tend to break firstly at the forereef slope and break secondly at the reef crest. When the forereef slope is steep, the intensity of wave reflections at the forereef slope is relatively strong, which results in a significant reduction in maximum water elevation before largest wave crest of incident waves approaching the reef crest. When the forereef slope gradually decreases ($\cot \alpha$> 4), the incident waves are to break at the reef crest. However, maximum water elevation at the reef crest only slightly increases with the forereef slope (Figure 42). Figure 43 depicts the space distribution of the local maximum wave height. The local maximum wave height gradually increases with the forereef slope before wave breakings at the reef crest. When the wave breakings occur at the reef crest, the local maximum wave height can be significantly reduced. In addition, when the forereef slope ($\cot \alpha$) is 3, the local maximum wave height at the reef flat is much greater than that of other forereef slope conditions. Since the intensity of the wave reflections increases with the forereef slope, the breaking-induced velocity over the forereef slope and the reef crest tends to decrease with the forereef slope (Figure 44). Figure 45 demonstrates that the local maximum depth-averaged velocity at the reef crest can be decreased by 26.2%, as the forereef slope gradually increases from $\cot \alpha$= 6 to $\cot \alpha$= 1. Influences of the forereef slope on the variation of the water wave energy dissipation rate and wave runup height can be ignored (Figure 46 and Figure 47).

When the influence of backreef slope was considered, the forereef slope was $\cot \alpha$ = 4. Five backreef slopes were chosen, such as $\cot \beta$ = 1, 5, 10, 15, and 20. The space distribution of local maximum water elevation along the simulation zone under different backreef slopes is depicted in Figure 48. As shown in Figure 48, when the backreef slope is large ($\cot \beta$ = 1), the maximum water elevation exhibits very large oscillations. This is caused by the strong wave reflections that exist between the steep backreef slope and the reef crest. When the backreef slope gradually increases ($\cot \beta$ > 5), the oscillation intensity of the maximum water elevation at the reef flat can be significantly weaken. As depicted in Figure 49, the maximum water elevation at the reef crest tends to slightly increase with backreef slope. Besides, influences of the variations of the backreef slope on the maximum depth-averaged velocity can be ignored (Figure 50, Figure 51 and Figure 52). Figure 53 shows the variation of the dissipation rate of water wave energy with the backreef slope. When the backreef slope ($\cot \beta$) is less than 15, the water wave energy dissipation rate increases slightly with the backreef slope. However, when the backreef slope ($\cot \beta$) is greater than 15, the wave energy dissipation rate can be sharply decreased. When the backreef slope ($\cot \beta$) gradually increases from 15 to 1, water wave energy dissipation rate can be reduced by 9.2% due to the enhanced wave reflections. As the backreef slope gradually increases, maximum wave runup height of the trough-focused wave group increases as well (Figure 54). When the backreef slope gradually increases from $\cot \beta$= 20 to $\cot \beta$= 1, maximum wave runup height can be increased by 53.3%. When $\cot \beta$ < 5, maximum wave runup height of crest-focused wave decreases with the backreef slope. While, as the backreef slope gradually increases from $\cot \beta$ = 20 to $\cot \beta$ = 5, maximum wave runup height of the crest-focused wave increases gradually with the backreef slope. It is possibly attributed to the fact that when the backreef slope is $\cot \beta$= 1, there exist strong wave reflections. The superposition of reflected waves and incident waves inhibits the wave runup height to a certain extent.

5.6. Influences of Ridge Width

Effects of the ridge width are discussed in this section. The offshore water depth ($h$) is 1 m. The significant wave height ($H_s$) is 0.16 m. The peak wave periods ($T_p$) is 2 s. Four different ridge widths are chosen, i.e., $W_{rc}$= 1.045, 1.41, 1.775, and 2.14 m, which produce four different relative ridge width, i.e., $\frac{W_{rc}}{L}$= 0.174, 0.235, 0.296, and 0.357. As shown in Figure 55, the wave breaking can be gradually intensified with the ridge width, which results in the decreasing of local maximum water elevation at the reef crest. It is also seen that the local maximum water elevation behind reef crest exhibits significant fluctuations. The maximum water elevation behind reef crest decreases significantly with the ridge width. As depicted in Figure 56, the local maximum water elevation at the reef crest slightly decreases with the ridge width. The maximum water elevation at the reef crest can be decreased by 21.5% when the relative ridge width ($W_{rc}/L$) increases from 0.174 to 0.357. As depicted in Figure 57, the variation of ridge width only has limited influences on the wave breaking position. However, local maximum wave height decreases gradually with the ridge width after the wave breaking position. It is possibly attributed to the fact that the acting period and intensity of the bottom friction at the high-speed water column above the reef crest tend to increase with the ridge width, which finally causes the maximum local wave height to decrease. Meanwhile, when the ridge width continues to increase, the wave breaking process can appear several times at the reef crest. The variation of ridge width has a certain effect on the spatial distributions of local maximum depth-averaged velocity after wave breaking, as depicted in Figure 58. Local maximum depth-averaged velocity at the reef crest is relatively low when the ridge width is 0.174 m. After the breaking surge bore leaving the reef crest, the local maximum depth-averaged velocity can be continuously decreased due to the bottom friction and the turbulent violence within the fluctuating water body. The variation of the ridge width does not have any noticeable influences on the local maximum depth-averaged velocity at the reef crest once the relative ridge width ($W_{rc}/L$) becomes greater than 0.235, as shown in Figure 59. As shown in Figure 60, the variation of ridge width has slight influences on the wave energy dissipation rate. When the relative ridge width ($W_{rc}/L$) increases from 0.174 to 0.357, wave energy dissipation rates of the crest-focused wave and trough-focused wave can be increased by 2.3% and 0.8%, respectively. Figure 61 depicts the variation of maximum wave runup height with the ridge width. It is seen that there exist no differences in the maximum wave runup heights of crest-focused wave and trough-focused wave. If the relative ridge width ($W_{rc}/L$) is less than 0.235, maximum wave runup height increases slightly with the ridge width. Once the relative ridge width ($W_{rc}/L$) is greater than 0.235, maximum wave runup height decreases monotonically with the ridge width. Hence, maximum wave runup height approaches its peak value when $W_{rc}/L$= 0.235.

6. Conclusions

In this study, complex hydrodynamics of the transforming and breaking processes of crest- and trough-focused wave groups were investigated. Influences of the major factors, i.e., water depth, peak wave period, significant wave height, forereef and backreef slopes, and ridge width, were discussed in detail. The research findings are organized as follows:

(1)

By comparing the computational results with the experimental data, it can be seen that the nonhydrostatic numerical wave model (NHWAVE) can not only accurately simulate the generation and propagation of focused waves, but also accurately resolve the transformation and breaking processes of focused waves on the fringing reef. Since the NHWAVE solves the governing equations of incompressible flow without the hydrostatic assumption, it can accurately predict the temporal evolutions of water elevation and the velocity distributions with high accuracy.

(2)

The focused wave group undergoes complicated transformations on the fringing reef. Wave breakings of high intensity can be seen at the reef crest. The local wave heights can be effectively decreased through the wave breakings and bottom frictions. Meanwhile, strong wave reflections can be observed at the forereef and backreef slopes and the reef crest. Most of the wave energy can be lost through the complex wave hydrodynamics of focused waves on the fringing reef (about 87.8%). Meanwhile, there exist very limited differences between the wave hydrodynamics of crest-focused wave and trough-focused wave on the fringing reef.

(3)

As the significant wave height gradually increases, relative local maximum wave height decreases more due to the wave breaking process at the reef crest. Nevertheless, wave energy dissipation rate increases slightly with the significant wave height. The wave energy dissipation rate can be increased by 5.3%. Maximum wave runup heights of the crest-focused wave and the trough-focused wave can be increased by 64.7% and 62.5%, respectively, as the relative significant wave height gradually increases from 0.08 to 0.24.

(4)

The intensity of wave reflections at the forereef slope, as well as that at the backreef slope and the reef crest, can be greatly enhanced with the peak wave period. Although wave energy dissipation rate is slightly affected by the variation of peak wave period, wave breaking point can be noticeably affected. The maximum wave runup heights of the crest- and trough-focused waves can be increased by 55.6% and 61.1%, respectively, as the peak wave period increases from 1.5 s to 3 s.

(5)

When the reef crest is dry, the maximum local wave height exhibits an oscillation behavior before the occurrence of the wave breakings at the reef crest. When the submergence water depth is large enough at the reef flat, the maximum local wave height does not decrease significantly after wave breakings at the reef crest. The maximum wave runup height monotonically increases with water depth.

(6)

When the forereef slope is steep, intensity of the wave reflections at the forereef slope is relatively strong. Since the intensity of wave reflection increases with the forereef slope, breaking-induced velocity over the forereef slope and reef crest tends to decrease. Nevertheless, influences of the variation of forereef slope on wave energy dissipation rate and maximum wave runup height are negligible.

(7)

When the backreef slope gradually increases from $\cot \beta$ = 20 to $\cot \beta$ = 1, maximum wave runup height of the trough-focused wave group increases by 53.3%. Although the maximum wave runup height of crest-focused wave group also increases from $\cot \beta$ = 20 to $\cot \beta$ = 5, and maximum wave runup height of crest-focused wave group tends to decrease with the backreef slope if $\cot \beta$ < 5. It is possibly attributed to the fact that when the backreef slope is $\cot \beta$ = 1, there exist strong wave reflections, and the superposition of reflected waves and incident waves inhibits the wave runup height to a certain extent.

(8)

Wave breakings can be gradually intensified with the ridge width. The maximum local wave height tends to gradually decrease with ridge width after wave breaking point. It is possibly attributed to the fact that the acting period and intensity of the bottom friction at the high-speed water column above the reef crest tend to increase with the ridge width, which finally causes maximum local wave height to decrease. The maximum wave runup height approaches its peak value at when the $W_{rc}/L$ = 0.235.

It is hoped that the above research results will further help the academic community to better understand the complex wave hydrodynamics of extreme waves on the fringing reef.