Many low-lying tropical and subtropical reef-fringed coasts are vulnerable
to inundation during tsunami events. Hence accurate prediction of tsunami
wave transformation and run-up over such reefs is a primary concern in the
coastal management of hazard mitigation. To overcome the deficiencies of
using depth-integrated models in modeling tsunami-like solitary waves
interacting with fringing reefs, a three-dimensional (3-D) numerical wave
tank based on the computational fluid dynamics (CFD) tool
OpenFOAM^{®} is developed in this study. The Navier–Stokes
equations for two-phase incompressible flow are solved, using the large eddy simulation (LES) method for turbulence closure and the volume-of-fluid (VOF)
method for tracking the free surface. The adopted model is firstly validated
by two existing laboratory experiments with various wave conditions and reef
configurations. The model is then applied to examine the impacts of varying
reef morphologies (fore-reef slope, back-reef slope, lagoon width,
reef-crest width) on the solitary wave run-up. The current and vortex
evolutions associated with the breaking solitary wave around both the reef
crest and the lagoon are also addressed via the numerical simulations.

A tsunami is an extremely destructive natural disaster, which can be generated by earthquakes, landslides, volcanic eruptions and meteorite impacts. Tsunami damage occurs mostly in the coastal areas where tsunami waves run up or run down the beach, overtop or ruin the coastal structures, and inundate the coastal towns and villages (Yao et al., 2015). Some tropic and subtropical coastal areas vulnerable to tsunami hazards are surrounded by coral reefs, especially those in the Pacific and Indian oceans. Among various coral reefs, fringing reefs are the most common type. A typical cross-shore fringing reef profile can be characterized by a steep offshore fore-reef slope and an inshore shallow reef flat (Gourlay, 1996). There is also possibly a reef crest lying at the reef edge (e.g., Hench et al., 2008) and/or a narrow shallow lagoon existing behind the reef flat (e.g., Lowe et al., 2009a). Over decades, fringing reefs have been well known to be able to shelter low-lying coastal areas from flood hazards associated with storms and high surf events (e.g., Cheriton et al. 2016; Lowe et al., 2005; Lugo-Fernandez et al., 1998; Péquignet et al., 2011; Young, 1989). However, after the 2004 Indian Ocean tsunami, the positive role of coral reefs in mitigating the tsunami waves has attracted the attention of scholars who conduct postdisaster surveys (e.g., Chatenoux and Peduzzi, 2007; Ford et al., 2014; Mcadoo et al., 2011). There is consensus among the researchers that, in addition to the establishment of the global tsunami warning system, the cultivation of coastal vegetation (mangrove forest, coral reef, etc.) is also one of the coastal defensive measures against the tsunami waves (e.g., Dahdouh-Guebas et al., 2006; Danielsen et al., 2005; Mcadoo et al., 2011). Numerical models have been proven to be powerful tools to investigate tsunami wave interaction with the mangrove forests (e.g., Huang et al., 2011; Maza et al., 2015; Tang et al., 2013, and many others). Comparatively speaking, their applications in modeling coral reefs subjected to tsunami waves are still very few.

Over decades, modeling wave processes over reef profiles face several challenges such as steep fore-reef slope, complex reef morphology and spatially varied surface roughness. Local but strong turbulence due to wave breaking in the vicinity of the reef edge needs to be resolved. Among various approaches for modeling wave dynamics over reefs, two groups of models are the most pervasive. The first group focuses on using the phase-averaged wave models and the nonlinear shallow water equations to model the waves and the flows, respectively, in field reef environments, and typically the concept of radiation stress (Longuet-Higgins and Stewart, 1964) or vortex force (Craik and Leibovich, 1976) is used to couple the waves and the flows (e.g., Douillet et al., 2001; Kraines et al., 1998; Lowe et al., 2009b, 2010; Van Dongeren et al., 2013; Quataert et al., 2015). As for modeling tsunami waves at a field scale, we are only aware of Kunkel et al.'s (2006) implementation of a nonlinear shallow water model to study the effects of wave forcing and reef morphology variations on the wave run-up. However, their numerical model was not verified by any field observations. The second group aims at using the computationally efficient and phase-resolving model based on the Boussinesq equations. This depth-integrated modeling approach employs a polynomial approximation to the vertical profile of velocity field, thereby reducing the dimensions of a three-dimensional problem by one. It is able to account for both nonlinear and dispersive effects at intermediate water level. At a laboratory scale, Boussinesq models combined with some semiempirical breaking wave and bottom friction models have been proven to be able to simulate the motions of regular waves (Skotner and Apelt, 1999; Yao et al., 2012), irregular waves (Nwogu and Demirbilek, 2010; Yao et al., 2016, 2019) and infragravity waves (Su et al., 2015; Su and Ma, 2018) over fringing reef profiles.

The solitary wave has been employed in many laboratory/numerical studies to model the leading wave of a tsunami. Compared to the aforementioned regular/irregular waves, the numerical investigations of solitary wave interaction with the laboratory reef profile are much fewer. Roeber and Cheung (2012) was the pioneer study to simulate the solitary wave transformation over a fringing reef using a Boussinesq model. Laboratory measurements of the cross-shore wave height and current across the reef as conducted by Roeber (2010) were reproduced by their model. More recently, Yao et al. (2018) also validated a Boussinesq model based on their laboratory experiments to assess the impacts of reef morphologic variations (fore-reef slope, back-reef slope, reef-flat width, reef-crest width) on the solitary wave run-up over the back-reef beach. Despite the above applications, several disadvantages still exist in using the Boussinesq-type models: (1) Boussinesq equations are subjected to the mild-slope assumption – thus using them for reefs with a steep fore-reef slope is questionable, particularly when there is a sharp reef crest located at the reef edge; (2) wave breaking could not be inherently captured by Boussinesq-type models – thus empirical breaking model or special numerical treatment is usually needed; (3) Boussinesq models could not resolve the vertical flow structure associated with the breaking waves due to the polynomial approximation to the vertical velocity profile.

To remedy the above deficiencies of using Boussinesq-type models to
simulate the solitary processes (wave breaking, bore propagation and run-up)
over the fringing reefs, we develop a 3-D numerical wave tank based on the
computational fluid dynamics (CFD) tool OpenFOAM^{®} (Open Field Operation and Manipulation) in
this study. OpenFOAM^{®} is a widely used open-source CFD code in
the modern industry supporting two-phase incompressible flow (via its solver
interFoam). With appropriate treatment of wave generation and absorption, it
has been proven to be a powerful and efficient tool for exploring
complicated nearshore wave dynamics (e.g., Higuera et al., 2013b). In this
study, the Navier–Stokes equations for an incompressible fluid are solved.
For the turbulence closure model, although large eddy simulation (LES) demands more computational
resources than Reynolds-averaged Navier–Stokes (RANS) equations, it computes the large-scale unsteady motions
explicitly. Importantly, it could provide more statistical information for
the turbulence flows in which large-scale unsteadiness is significant (Pope,
2000). Thus the LES model is adopted by considering that the breaking-wave-driven flow around the reef edge/crest is fast and highly unsteady. The free-surface motions are tracked by the widely used volume-of-fluid (VOF) method.

In this study, we first validate the adopted model by the laboratory experiments of Roeber (2010) as well as our previous experiments (Yao et al., 2018). The robustness of the present model in reproducing such solitary wave processes as wave breaking near the reef edge/crest, turbulence bore propagating on the reef flat and wave run-up on the back-reef beach is demonstrated. The model is then applied to investigate the impacts of varying reef morphologies (fore-reef slope, back-reef slope, lagoon width, reef-crest width) on the solitary wave run-up. The flow and vorticity fields associated with the breaking solitary wave around the reef crest and the lagoon are also analyzed by the model results. The rest of this paper is organized as follows. The numerical model is firstly described in Sect. 2. It is then validated by the laboratory data from the literature as well as our data in Sect. 3. What follows in Sect. 4 are the model applications for which laboratory data are unavailable. The main conclusions drawn from this study are given in Sect. 5.

To simulate breaking-wave processes across the reef, the LES approach is
employed to balance the need of resolving a large portion of the turbulent
flow energy in the domain while parameterizing the unresolved field with a
subgrid closure in order to maintain a reasonable computational cost. The
filtered Navier–Stokes equations are essential to separate the velocity field
that contains the large-scale components, which is performed by filtering
the velocity field (Leonard, 1975). The filtered velocity in the

For incompressible flow, the filtered continuity and momentum equations are
as follows:

The residual stress is usually calculated by a linear relationship with the
rate-of-strain tensor based on the Boussinesq hypothesis. The one-equation
eddy viscosity mode, which is supposed to be better than the well-known
Smagorinsky model for solving the highly complex flow and shear flow (Menon
et al., 1996), is employed in the present study. Based on the one-equation
model (Yoshizawa and Horiuti, 1985), the subgrid stresses are defined as
^{®} user guide (OpenFOAM Foundation, 2013).

The presence of the free-surface interface between the air and water is
treated through the commonly used VOF method (Hirt and Nichols, 1981), which
introduces a volume fraction and solves an additional modeled transport
equation for this quantity. The general representation of fluid density

In the present solver interFoam, the algorithm PIMPLE, which is a mixture of
the PISO (Pressure Implicit with Splitting of Operators) and SIMPLE
(Semi-Implicit Method for Pressure-Linked Equations) algorithms, is employed
to solve the coupling of velocity and pressure fields. The MULES
(multidimensional universal limiter for explicit solution) method is used
to maintain boundedness of the volume fraction independent of the underlying
numerical scheme, mesh structure, etc. The Euler scheme is utilized for the time
derivatives, the Gauss linear scheme is used for the gradient term and the Gauss linear corrected scheme is selected for the Laplacian term. The detailed implementation
can be founded in the OpenFOAM^{®} user guide (OpenFOAM Foundation, 2013).

Wave generation and absorption are essentials for a numerical wave tank, but
they are not included in the official version of OpenFOAM^{®}.
Therefore, supplementary modules were developed by the other users, e.g.,
waves2Foam (Jacobsen et al., 2012) and IH-FOAM (Higuera et al., 2013a). In
this study, the IH-FOAM is selected because it employs an active wave
absorbing boundary and does not require an additional relaxation zone as
used by waves2Foam. Meanwhile, it supports many wave theories including the
solitary wave theory. The free surface and velocity for a solitary wave
generation in IH-FOAM are (Lee et al., 1982)

The first set of laboratory experiments serving as validation is from Roeber (2010), who reported two series of experiments conducted at Oregon State University, USA, in separate wave flumes. In this study, we only reproduce their experiments in the large wave flume, which is 104 m long, 3.66 m wide and 4.57 m high. As illustrated in Fig. 1a, the two-dimensional (2-D) reef model, starting at 25.9 m from the wavemaker, was built by a plane fore-reef slope attached to a horizontal reef flat 2.36 m high followed by a back-reef vertical wall. Both the waves and flows across the reef profile were measured by 14 wave gauges (wg1–wg14) and 5 ADVs (acoustic Doppler velocimeters), respectively. Only two scenarios for the reef with and without a trapezoidal reef crest subjected to two incident waves are reported in this study (see also Table 1). The large wave flume experiments enable us to test our model's ability to handle relatively large-scale nonlinear dispersive waves together with wave breaking, bore propagation and associated wave-driven flows. For a more detailed experimental setup, see Roeber (2010).

Experiment settings for

Reef configuration and wave condition for the tested scenarios.

The second set of 2-D reef experiments for model validation comes from our
previous work (Yao et al., 2018). These experiments were conducted in a
small wave flume 40 m long, 0.5 m wide and 0.8 m high at the Changsha University
of Science and Technology, P. R. China. As shown in Fig. 1b, a plane slope
was built 27.3 m from the wavemaker and it was truncated by a horizontal
reef flat 0.35 m high. A back-reef beach of

By considering a balance between the computational accuracy and efficiency,
the computational domain (Fig. 2a) is designed to reproduce the main aspects
of the laboratory settings. We calibrate the model with the principle that the computed leading solitary wave height at the most offshore gauge should
exactly reproduce its measurement. For a solitary wave, wave length (

Numerical grids and boundary conditions of the numerical domain.

Structured mesh is used to discretize the computational domain. The
discretization is kept constant in the spanwise (

For LES modeling solitary wave breaking over reefs, it is crucial to
examine the Reynolds number (

Variation of the maximum dimensionless free-surface elevation (

To evaluate the performance of the model, the model skill value is adopted
and calculated by Wilmott (1981):

Figure 4 compares the computed and the measured cross-shore distribution of
the free-surface elevations (

Dimensionless free-surface elevations (

Time series of dimensionless free-surface elevations (

Figure 5 illustrates the computed and measured time series of dimensionless
free-surface elevations (

Time series of dimensionless streamwise velocity (

Dimensionless free-surface elevations (

Figure 6 depicts the time series of streamwise velocity (

Time series of dimensionless free-surface elevations (

As previously introduced, the reef profile of Scenario 2 is identical to
that of Scenario 1 except for a reef crest located at the reef edge. The
cross-shore distribution of dimensionless free-surface elevations (

Figure 8 compares the measured and simulated times-series of dimensionless
free-surface elevations (

Time series of dimensionless streamwise velocity (

As for Scenario 2, Roeber (2010) only reported one location of flow
measurement on the seaside face of the reef crest. Figure 9 presents the
time series of dimensionless streamwise velocity (

Time series of dimensionless free-surface elevations (

Time series of dimensionless free-surface elevations (

The experiments of Yao et al. (2018) only measured the time series of wave records at limited locations (G1–G8) across the reef as well as the maximum wave run-up on the final beach. Figure 10 compares the computed and measured time series of free-surface elevations for Scenario 3. The overall agreement between the simulations and experiments for G1–G8 is very good, with the skill values at all locations larger than 0.9. When the solitary wave travels from the toe (G2) to the middle of fore-reef slope (G3), it gets steeper due to the shoaling effect. Wave breaking starts at a location right before the reef edge (G4), and the surf zone processes extend over the reef flat in the form of a moving bore. Thus, from G5 to G8, the wave time series show saw-shaped profiles and there is a cross-shore decrease in the leading solitary wave height. Such features of the breaking waves are also well captured by the model. Note that the second peak in the time series of G7 is due to wave reflection from the back-reef beach, and the incident and reflected waves are not fully separated from each other at G8 because this location is too close to the beach. The predicted and measured wave run-ups are 0.122 and 0.109 m, respectively, for this scenario. Compared to the Boussinesq model employed by Yao et al. (2018), no significant difference in the predicted time series could be found for the present Navier–Stokes-equation-based model.

Figure 11 depicts the same comparison of wave time series but for the reef profile with a lagoon (Scenario 4). Again, the model performance for this scenario is fairly good (all skill values larger than 0.9). The predicted and measured wave run-ups are 0.123 and 0.116 m, respectively, for this scenario. Notable mismatch only appears for those small wave oscillations generated by the reflected wave propagating out of the lagoon to the reef flat (i.e., from G8 to G6). But our model seems to be superior to the model of Yao et al. (2018) in reproducing those oscillations at G7 and G8. We finally remark that the tail of the leading solitary wave, particularly from G1 to G4, is below the initial water level in the laboratory data, which is due to the water lost to form the generated wave crest around the paddle of the wavemaker. However, such a phenomenon is not observed in the numerical results because we generate a theoretical solitary wave in the numerical domain as indicated by Eq. (11).

In this section, we apply the well-validated LES model to examine the
variations of reef morphological parameters (fore-reef slope, back-reef
slope, lagoon width, reef-crest width) that may affect the wave run-up (

The predicted wave run-up on the back-reef beach (

Comparison of wave-driven current and vorticity on the

Generally, Fig. 12a shows that

One advantage of the current Navier–Stokes-equation-based model over the
depth-integrated models is its ability to resolve the vertical flow
structure under breaking waves, particularly around the complex reef
geometry. Based on the reef profile of Yao et al. (2018), Fig. 13 shows the
simulated wave-driven current and vorticity on the

Comparison of wave-driven current and vorticity on the

Figure 14 compares the computed wave-driven current and vorticity on the

To remedy the inadequacies of using the depth-integrated models to simulate
the interaction between tsunami-like solitary waves and fringing reefs, a 3-D
numerical wave tank, solving the Navier–Stokes equations with the LES for
turbulence closure, has been developed based on the open-source CFD tool
OpenFOAM^{®}. The free surface is tracked by the VOF method. Two
existing laboratory experiments with the wave, flow and wave run-up
measurements based on different fringing reef profiles are employed to
validate the numerical model. Simulations show that the current
Navier–Stokes-equation-based model outperforms the commonly used
Boussinesq-type models in view of its capability to better reproduce the
breaking waves and wave-driven current on the reef flat. The model is then
applied to investigate the impacts of varying morphologic features on the
back-reef wave run-up. The flow and vorticity fields associated with the
breaking solitary wave around the reef crest and the lagoon are also
analyzed via the numerical simulations.

Model results shows that wave run-up on the back-reef slope is most sensitive to the variation of the back-reef slope, less sensitive to the lagoon width, and almost insensitive to the variations of both the fore-reef slope and the reef-crest width within our tested ranges. The existence of a reef crest or a lagoon can notably alter the wave-driven current and vortex evolutions on the reef flat. These findings demonstrate that low-lying coastal areas fringed by coral reefs with steep back-reef slopes and larger lagoons are expected to experience larger wave run-up near the shoreline; thus they are more susceptible to coastal inundation during a tsunami event.

Most of the simulations are made available in the figures of this paper. The raw simulated data used in this study are available upon request (zzdeng@zju.edu.cn).

YY and LC performed the data analyses and wrote the paper. TH prepared the experimental data. ZD and HG performed the numerical simulations.

The authors declare that they have no conflict of interest.

This research has been supported by the National Natural Science Foundation of China (grant nos. 51679014 and 11702244), the Hunan Science and Technology Plan Program (grant no. 2017RS3035), and the Open Foundation of the Key Laboratory of Coastal Disasters and Defense of Ministry of Education (grant no. 201602).

This paper was edited by Maria Ana Baptista and reviewed by two anonymous referees.

^{®}, Coast. Eng., 71, 102–118, 2013a.

^{®}, Coast. Eng., 71, 119–134, 2013b.

^{®}, Int. J. Numer. Meth. Fluids, 70, 1073–1088, 2012.

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