### 1. Introduction

### 2. Hydraulic Model Experiment

### 2.1 Overview of the Hydraulic Model Experiment

*ρ*) are 0.123 and 0.032, respectively. where

_{V}*S*(

*f*) is the wave spectral density,

*H*is the significant wave height,

_{S}*T*is the significant wave period, and

_{S}*f*is the frequency.

*H*= 5.33 cm,

_{S}*T*= 0.95 s) and wave case 16 (

_{S}*H*= 11.0 cm,

_{S}*T*= 1.88 s), respectively.

_{S}*K*), transmission coefficient (

_{R}*K*), and energy dissipation coefficient (

_{T}*K*) were calculated to analyze hydraulic characteristics according to the cross-section changes in submerged rigid vegetation. The reflection coefficient (

_{D}*K*) was calculated using the incident/reflected wave separation method proposed by Goda and Suzuki (1976). The transmission coefficient (

_{R}*K*) and energy dissipation coefficient (

_{T}*K*) were calculated using Eqs. (2) and (3), respectively. where

_{D}*H*is the incident wave height and

_{O}*H*is the transmitted wave height.

_{T}### 2.2 Results of the Hydraulic Model Experiment

*K*,

_{R}*K*, and

_{T}*K*were calculated according to the density, width, and multi-row arrangement conditions of submerged rigid vegetation. Fig. 5 shows the distribution of the coefficients according to the wave steepness. Figs. 5(a), 5(b), and 5(c) represent the cases of

_{D}*K*,

_{R}*K*, and

_{T}*K*, respectively.

_{D}#### 2.2.1 Reflection coefficient

*B*= 133 cm,

*ρ*= 0.123), Case R3 (

_{V}*B*= 266 cm,

*ρ*= 0.123), and Case R4 (

_{V}*B*= 133, 133 cm,

*ρ*= 0.123), which had the same density. It decreased slightly for Case R2 (

_{V}*B*= 133 cm,

*ρ*= 0.032), which had lower density. This is because the reflected wave was generated similarly when the density of the vegetation zone was the same and the transmission area of waves increased as the vegetation density decreased.

_{V}#### 2.2.2 Transmission coefficient

*B*= 266 cm,

*ρ*= 0.123) and Case R4 (

_{V}*B*= 133, 133 cm,

*ρ*= 0.123), where the density was the same and the width of the vegetation zone increased compared to that in Case R1 (

_{V}*B*= 133 cm,

*ρ*= 0.123), and in Case R2 (

_{V}*B*= 133 cm,

*ρ*= 0.032), where the width was identical and the density decreased. This is because an increase in the density and width of the vegetation zone induces more breaking waves at the crest of the vegetation zone.

_{V}#### 2.2.3 Energy dissipation coefficient

*B*= 133 cm,

*ρ*= 0.032), where the width and density of the vegetation zone were low, and showed a tendency to increase as the width and density of the vegetation zone increased. Additionally, there was no significant difference in hydraulic characteristics between Case R3 (

_{V}*B*= 266 cm,

*ρ*= 0.032) and Case R4 (

_{V}*B*= 133, 133 cm,

*ρ*= 0.032), which had the same density and width of the vegetation zone. In the case of submerged rigid vegetation, it was assumed that the multi-row arrangement condition does not affect the wave energy attenuation of the incident wave significantly.

_{V}### 3. Overview of the Numerical Model Experiment

### 3.1 Governing Equations

##### (6)

*u*and

*w*are the flow velocities in the

*x*and

*z*directions, respectively.

*q*

^{*}is the flow density of the wave source.

*M*and

_{x}*M*are inertial resistance.

_{z}*E*and

_{x}*E*are laminar flow resistance.

_{z}*D*and

_{x}*D*are turbulence resistance.

_{z}*γ*is the volumetric porosity.

_{v}*γ*and

_{x}*γ*are the area permeabilities along the

_{z}*x*and

*z*directions, respectively.

*t*is the time,

*g*is the gravitational acceleration,

*ρ*is the fluid density,

*p*is the pressure,

*β*is the wave attenuation coefficient,

*F*is the volume ratio occupied by the fluid in each grid, and

*ν*is the sum of the kinematic viscosity coefficient and kinematic eddy viscosity.

_{t}### 3.2 Vegetation Drag

*RV*) is given by Eq. (8). where

_{i}*Φ*is the vegetation density,

*C*is the drag coefficient, and

_{D}*C*is the inertial resistance.

_{M}*C*, Eq. (9), which was presented by Lee et al. (2017a) by analyzing the experimental results of Wu and Cox (2015), was applied.

_{D}*C*= 1.5, as proposed by Sakakiyama and Kajima (1992), was applied. where

_{M}*KC*is the Keulegan–Carpenter number.

### 3.3 Verification of the Numerical Wave Tank

*H*/

_{S}*L*= 0.02). Overall, it was assumed that the calculated values reproduced the experimental values properly. In Fig. 8, the dotted lines show the ±10% range of the values of the hydraulic and numerical model experiments. Overall, it can be seen that the results of the numerical model experiment reproduced the reflection coefficient, transmission coefficient, and energy dissipation coefficient of the hydraulic model experiment appropriately.

_{S}### 4. Results of the Numerical Model Experiment

*R*), width (

*B*), density (

*ρ*), and arrangement distance (

_{V}*V*) were considered for the cross-section conditions of the vegetation zone.

_{d}*ν*is the kinematic viscosity coefficient,

*ν*is the kinematic eddy viscosity of SGS, ∆ is the filter length scale, and

_{l}*S*is the deformation tensor at the grid size. As for the

_{ij}*C*value, Schumann (1987) proposed a value ranging from 0.07 to 0.21, but 0.1 was used in this study because it was applied in a similar study (Christensen and Deigaard, 2001; Okayasu et al., 2005).

_{S}### 4.1 Flow Field, Vorticity Distribution, Turbulence Distribution, and Wave Height Distribution According to the Crest Depth

### 4.2 Flow Field, Vorticity Distribution, Turbulence Distribution, and Wave Height Distribution According to the Width

### 4.3 Flow Field, Vorticity Distribution, Turbulence Distribution, and Wave Height Distribution According to the Density

### 4.4 Flow Field, Vorticity Distribution, Turbulence Distribution, and Wave Height Distribution According to the Arrangement Distance

### 5. Conclusions

From the hydraulic model experiment, the reflection coefficient, transmission coefficient, and energy dissipation coefficient according to the cross-section changes in vegetation were calculated. Additionally, the flow field, vorticity distribution, turbulence distribution, and wave height distribution according to the cross-section changes in vegetation were analyzed using the numerical wave tank, which was validated by comparing the results obtained from the hydraulic model experiment.

The results of the hydraulic model experiment showed that the reflection coefficient decreased as the density of the vegetation zone decreased. This is because the transmission area of waves increased in front of the vegetation zone. Additionally, we found that the transmission coefficient increased as the density and width of the vegetation zone increased. This is because more breaking waves were induced at the crest of the vegetation zone as the density and width of the vegetation zone increased.

The results of the numerical model experiment showed that the distribution and intensity of vorticity and turbulence increased at the crest and back of the vegetation zone as the crest depth of the vegetation zone decreased and width and density increased, thereby increasing the wave height attenuation performance.

When the total width of the vegetation zone was identical, the vorticity distribution, turbulence distribution, and wave height distribution according to the multi-row arrangement and arrangement distance were found to be similar. However, we believe that more examinations are required for wave energy attenuation characteristics according to the multi-row arrangement and arrangement distance of the submerged rigid vegetation.