A coupled wave–vegetation simulation is presented for the moving effect of the coastal vegetation on tsunami wave height damping. The problem is idealized by solitary wave propagation on a group of emergent cylinders. The numerical model is based on general Reynolds-averaged Navier–Stokes equations with renormalization group turbulent closure model by using volume of fluid technique. The general moving object (GMO) model developed in computational fluid dynamics (CFD) code Flow-3D is applied to simulate the coupled motion of vegetation with wave dynamically. The damping of wave height and the turbulent kinetic energy along moving and stationary cylinders are discussed. The simulated results show that the damping of wave height and the turbulent kinetic energy by the moving cylinders are clearly less than by the stationary cylinders. The result implies that the wave decay by the coastal vegetation may be overestimated if the vegetation was represented as stationary state.

A huge tsunami in Southeast Asia caused catastrophic damage and claimed more than 200 000 people in December 2004. Cochard et al. (2008) pointed out that this event has stimulated a debate about the role coastal ecosystems, such as mangrove forests and coral reefs, played in protecting low-lying coastal area. For example, Baird (2006) questioned the effectiveness of the coastal forests or reefs on the reduction of the damage caused by the tsunami. However, Danielsen et al. (2005) reported that areas with coastal tree vegetation were markedly less damaged than areas without. Iverson and Prasad (2007) also indicated that developed areas were far more likely to be damaged than forested zones. Several studies (Hiraishi and Harada, 2003; Harada and Kawata, 2004; Teh et al., 2009) have shown that tsunami wave heights, velocities, and energies were significantly reduced as the wave propagates through mangrove forests. Nevertheless, Wolanski (2006) has noted that mangroves probably cannot protect the coast against a tsunami wave greater than a threshold level based on some evidence from observations of the Indian Ocean tsunami. Based on the field observations, Shuto (1987) and Yanagisawa et al. (2009) found that single trees or even entire forests could be destroyed through tilting, uprooting, bending, or trunk breaking by tsunami. Because tsunamis remain a threat to lives and property along the most coasts of the world, it remains important for estimating the effectiveness of the coastal vegetation on the tsunami impact.

Cylinder cell arrangement (left), field length (right), and locations of wave probes for the computations.

FAVOR technique geometry of cylinders and constructed computational
rectangular grid with 0.001 m

Comparison of free-surface evolution between numerical and
experimental using field length of 1.635 m and

Comparison between the numerical results using the field
length of 0.545 m and

Wave surface evolutions for stationary and moving cylinders,

Comparison of wave height evolutions for moving cylinders with different spring constants.

Comparison of the maximum deflection angle of moving cylinders with different spring constants.

Comparison of wave height damping ratio between stationary and moving cylinders.

The snapshots of the velocity distribution for stationary cylinders
(left) and moving cylinders (right),

Comparison of the horizontal velocity profile for moving and
stationary cylinders as wave crest passes each section,

Snapshots of TKE for stationary cylinders (left) and moving
cylinders (right),

Snapshots of DTKE for stationary cylinders (left) and moving
cylinders (right),

The time evolution of surface elevation and TKE at each section for
stationary cylinders,

The time evolution of surface elevation and TKE at each section for
moving cylinders,

Comparison of vertical profiles of TKE as wave crest passes
through each section,

Many numerical and experimental approaches have been developed in recent years to help understand the tsunami wave interactions with coastal vegetation. Coastal tree vegetation was idealized by a group of rigid cylinders in most investigations. Huang et al. (2011) performed both experiments and a numerical model by considering solitary wave propagation on emergent rigid cylinders and found that dense cylinders may reduce the wave transmission because of the increased wave energy dissipation into turbulence in cylinders. By using both direct numerical simulation and a macroscopic approach, Maza et al. (2015) simulated the interaction of solitary waves with emergent rigid cylinders based on the arrangement of laboratory experiments of Huang et al. (2011). Previous approaches (e.g. Anderson et al., 2011; Huang et al., 2011; Maza et al., 2015; Wu et al., 2016) assumed that the idealized mangrove vegetation is stationary and neglected the plant motion with the wave.

Several works have investigated the hydraulic resistance of coastal
vegetation involving the flexible effect of plants. Zhang et al. (2015)
pointed out that the prop roots under tidal hydrodynamic loadings in a
mangrove environment can be regarded as fairly rigid on account of a large
Young modulus. However, Augustin et al. (2009) indicated that motion of the
flexible elements is an important factor on wave attenuation based on flume
tests considering both stiff and flexible parameterized tree models under
wave action. Husrin (2013) found that the trunk of a mangrove, with its
strength properties, may behave as a stiff or flexible structure which also
governs its relative contribution to the total energy dissipation under
tsunami and storm wave action. Coastal pines, which are typical of coastal forest
vegetation, have longer trunk than mangroves; Husrin and
Oumeraci (2013) indicated that they are more deflected when subject to
similar flow velocity compared to mangroves. Husrin et al. (2012) and
Strusińska et al. (2013, 2014) examined the tsunami attenuation by
coastal vegetation under laboratory conditions for mature mangroves using
parameterized trees, including flexible tree models. Maza et al. (2013)
presented a new numerical model for the interaction of waves and flexible
moving vegetation which couples the flow and the plant motion by considering the
plant deformation using RANS equation with

Some mangrove roots and branches at the growing stage are hanging from the canopy to the flow; this causes the prop roots to oscillate in the water. This study presents a numerical simulation that consider vegetation motion coupled with tsunami waves to investigate the wave-damping performance. We model the motion of the vegetation by attaching rigid cylinders to torsional connectors under wave action, which is similar to the experimental work of Kazemi et al. (2015). This is also a simplified way to represent some movements of mangroves induced by sediment scour, tilting, or uprooting states. A direct numerical model based on computational fluid dynamics (CFD) is presented in this paper for simulating the wave-damping characteristics by both stationary and moving vegetation.

Among a number of open-source CFD codes available, IHFOAM (Higuera et al., 2013, 2014) is specially designed for coastal engineering applications. IHFOAM was used by Maza et al. (2015) for direct numerical simulation of a solitary wave interacting with stationary vegetation. Alternatively, the model Flow-3D (Flow Science, Inc., 2012) is applied in this paper to conduct numerical simulations that include vegetation motion under wave action. Flow-3D provides exclusively the FAVOR (fractional area/volumes obstacle representation) technique (Hirt, 1993) and a general moving object (GMO) model that is capable of simulating the rigid body motion dynamically coupled with fluid flow. The FAVOR technique retains rectangular elements with a simple Cartesian grid system and has been shown to be one of the most efficient methods to treat immersed solid bodies (Xiao, 1999). The free water surface tracking in the model is accomplished by using the volume of fluid (VOF) method (Hirt and Nichols, 1981).

Referring to previous literature, the problem is idealized by a solitary wave
passing on a group of emergent rigid cylinders. Considering the fluid to be
incompressible, the continuity and momentum equations for a moving object
formulated with area and volume fraction functions are given as

The eddy viscosity

Referring to Yakhot et al. (1992), the turbulent transport equations of the
RNG

For coupling the rigid body motion dynamically with fluid flow, the GMO model is adopted here. Compared with the continuity
equation for stationary obstacle problems,

In computing the coupling of fluid and rigid body interaction, the velocity and pressure of fluid flow are first solved. The hydrodynamics forces on the rigid body are then obtained and used to calculate the velocity of the rigid body. Then the volume and area fractions are updated according to the new position of the rigid body, and the source term can be calculated using Eq. (8). The flow field is computed repeatedly until the convergence is achieved. A similar GMO model has been applied for the numerical simulation of coupled motion of solid body and waves, e.g. in Bhinder et al. (2009), Dentale et al. (2014), and Zhao et al. (2014).

As for the boundary conditions for solving the governing equations of flow, the normal stress is in equilibrium with the atmospheric pressure while shearing stress is zero on the free surface. All of the solid surfaces were treated using the no-slip boundary condition. The variation of the turbulent energy and the turbulent energy dissipation on the free-surface boundary was set as zero in the normal direction. The solution of solitary wave derived from Boussinesq equations was employed as the incident wave.

Huang et al. (2011) conducted laboratory experiments in a wave flume for the
solitary waves interacting with emergent rigid vegetation. The vegetation
was represented by a group of cylinders which were made of Perspex tubes with a
uniform outer diameter of 0.01 m. The present computations used the same
geometric configuration of Huang's laboratory works. The water depth was
uniform and equal to

Two different uniform computational meshes around the cylinder field, 0.002 and 0.001 m, respectively, were used to test the numerical accuracy and the sensitivity to grid size. Figure 2 shows that the FAVOR technique resolved successfully the geometry of cylinders using these two computational grids constructed. It indicates that FAVOR efficiently uses 29 and 17 points to define each cylinder for the mesh of 0.001 and 0.002 m, respectively.

Figure 3 shows the comparison of free-surface evolution between the present
numerical results and experimental measurements for an incident wave height

The above comparisons demonstrated that the present numerical model is capable of simulating accurately the wave evolution by the group cylinders. The following simulations are performed for a solitary wave passing through both the stationary and moving cylinders. The characteristics of the surface elevation evolution, the flow field variation, and the turbulent kinetic energy (TKE) are analysed and compared between stationary and moving cylinders. The numerical domain and the arrangement of cylinders used in the following simulations are the same as in previous section. The uniform fine mesh with 0.001 m is used for the following computations.

The moving cylinders induced by waves are set up by the GMO model for coupling the cylinder's motion and fluid flow dynamically.
Similar to Kazemi et al. (2015), each cylinder end was simplified by
attaching a torsion spring connector on the bottom in the model. The use of
torsion spring could not completely reproduce the natural bending behaviour
of the mangrove tree, but it allows the cylinders to move with the passing
wave. Peltola et al. (2000) and Husrin (2013) indicated that the deflection
angles for a broken trunk may range from 23 to 42

The numerical free-surface evolutions along the stationary and moving
cylinders, respectively, are shown in Fig. 5. The spring constant is set by

Figure 6 shows the comparison of wave height evolution for moving cylinders
with different spring constants, which can be seen that the results of moving
and stationary cylinders are almost identical as

Figure 8 shows the variation of the wave height damping ratio,

Figure 9 shows the snapshots of velocity distribution at the centre line of
the tank for moving and stationary cylinders as the solitary wave crest passes through gauges G3 to G6 for an incident wave height

The TKE will be generated and will dissipate during the wave when
interacting with the group of cylinders. The turbulent kinetic energy (

Comparison of total TKE evolution along cylinder array,

Figures 11 and 12 display the snapshots of the spatial distribution of the
TKE and the DTKE for stationary and moving cylinders, respectively, when the wave
crest passes through gauges G3 to G6. It shows that the turbulent kinetic
energy starts generating and dissipating after the wave crest impinges on the
front row of cylinders. It can be seen that the characteristics of spatial
distribution of TKE and DTKE for moving cylinders or stationary cylinders are
very similar. Figures 13 and 14 display the time variations of TKE at each
section (

Figure 13 also shows that there is a time lag between the occurrence of maximum TKE and the maximum wave elevation for stationary cylinders, but there is almost no lag for moving cylinders. That is, the maximum TKE is produced after the wave crest passes each section for the case of stationary cylinders. However, for moving cylinders, the maximum TKE occurs at the wave crest when passing each section. It can also be seen in Fig. 14 that multiple peaks of the TKE evolution exist in the case of moving cylinders during the return to its original position, and the peak values decrease with time to zero.

Figure 15 shows the comparisons of vertical profiles of TKE between moving and
stationary cylinders as the wave crest passes through gauges G3 to G6. It
confirms that almost no TKE is produced at

A numerical simulation based on the three-dimensional RANS equations and RNG

The data used for the validation were cited from Huang et al. (2011) and Maza et al. (2015). To access the numerical dataset, please contact the corresponding author.

The authors declare that they have no conflict of interest.

The authors would like to express their appreciation to Maria Maza, University of Cantabria, Spain, and the anonymous reviewer for their valuable comments and suggestions. Edited by: M. Gonzalez Reviewed by: M. Maza and one anonymous referee