### 1. Introduction

*Fn*). Mohseni et al. (2018) conducted a CFD simulation using OpenFOAM to investigate the physics of wave scattering and wave run-up around a truncated cylinder, considering various wave steepness. They reported that the formation of lateral edge waves becomes more significant with high-steepness waves.

### 2. Numerical Method

### 2.1 Numerical Method

*u*and

_{i}*x*denote the fluid velocity and coordinate component in the

_{i}*i*

^{th}direction, respectively.

*P*is the fluid pressure, and

*ρ*is the fluid density.

*g*is the gravitational acceleration.

*u*

_{i}^{′}denotes the fluctuating velocity component in the

*i*

^{th}direction and

*μ*is the turbulent viscosity. The

*k*−

*∊*turbulence model was used, and the volume-of-fluid (VOF) method was applied to capture the interface of multiphase flow. Regarding the CFD simulation, the finite volume method (FVM) was used to solve the RANS equations and VOF transport equation. A second-order implicit unsteady scheme was used for temporal discretization. The time step was chosen as 1/1000 of the incident wave period (

*T*) to ensure the Courant number was below 0.1 (Perić and Abdel-Maksoud, 2018).

_{n}### 2.2 Grid System and Analysis Method

*η*

_{1}), second- (

*η*

_{2}), and third- (

*η*

_{3}) order harmonic components of the wave run-up are effectively extracted using DFT.

### 3. Numerical Validation

### 3.1 Verification of Incident Waves

*T*= 7 s, 8 s, 9 s, 10 s, 11 s, and 12 s) with four different wave steepness values (

*T*= 7 s, 8 s, 9 s, 10 s, 11 s, and 12 s) with a constant wave steepness of

*H*/

*λ*= 1/30 were used for the numerical simulation and validation. As noted, a scaled cylinder model was used. Table 1 lists the dimensions of the scaled truncated cylinder and the corresponding wave conditions.

### 3.2 Convergence Study

*T*=7

*s*and

*H/λ*= 1/30. Three different meshes (Meshes A, B, and C) were considered, with a refinement factor (

*r*) chosen as √2. Table 3 provides detailed information for three mesh resolutions.

*∊*=(

_{BA}*η*−

_{B}*η*), and the changes between Meshes B and C are denoted as

_{A}*∊*=(

_{CB}*η*−

_{C}*η*), respectively. Based on the GCI method, the convergence ratio (

_{B}*R*), order of accuracy (

*P*), and GCI index were calculated using the following equations: where

*r*is the refinement factor, and

*p*denotes the theoretical order of the convergence, which is 2.

*FS*is the safety factor of the method; Roche et al. (1994) recommended a constant value of

*FS*as 1.25 when using more than two mesh resolutions.

### 3.3 Validation of Wave Run-Up on a Bare Truncated Cylinder

*η*

_{1}) of wave run-up corresponding to the fundamental wave frequency (

*ω*). In the present study, the first-order harmonic component of wave run-up around the cylinder was normalized to the first-order harmonic amplitude of the incident wave (

*A*

_{1}). Fig. 7 compares normalized first-order harmonic components of wave run-up on the surface of the cylinder with respect to various scattering parameters (

*ka*).

*η*

_{2}) of wave run-up corresponding to the double wave frequency (2

*ω*). Regarding the second-order harmonic components, the nonlinear interaction was more significant, particularly for short waves at the rear and shoulder points, and for long waves at the front and rear points of the truncated cylinder (Mohseni et al., 2018). The mean set up/down denotes the average time history for the free surface elevation. Both the mean set-up/down and the second-order harmonic components of the wave run-up were normalized to

*a*” represents the radius of the cylinder. Fig. 8 presents the validation results of the second-order harmonic components.

### 4. Results and Discussion

*D*denotes the diameter of the truncated cylinder. The cylinder draft and the total damper thickness were assumed to be the same regardless of the damper type. As shown in Fig. 9, type 1 denotes the truncated cylinder with a single damper. From types 2 to 3 cylinders, dual dampers were attached to the truncated cylinders with various separating distances (0.125

*D*and 0.25

*D*). The lower damper was always fixed at the bottom for all cylinders.

### 4.1 Time History of Wave Elevation

*T*=7

*s, H/λ*= 1/30) and Fig. 11 (

*T*=12

*s, H/λ*=1/30) compare the wave run-up time series for four cylinders. Only the time histories of the last five wave periods are plotted in these figures. When the short incident waves (

*T*=7

*s*) arrive, the wave run-up around the cylinders increase because of the presence of the damper (Fig. 10). In the case of the cylinders with dual dampers, the closer the upper damper is to the free surface, the stronger the scattering and amplification effects around the cylinder, leading to a higher run-up. Consequently, the wave elevations around the cylinder increased as the upper damper was raised upward. Regardless of the damper type, the amplification of wave elevation (WP1 and WP2) at the front face of the cylinder was greater than that of the side or rear of the cylinder (WP3, WP4, and WP5), which is also consistent with the trend observed in the bare cylinder. Therefore, the effect of the damper could be considered an enhancement of the original wave run-up characteristics for the bare cylinder. In particular, at the position of WP1, the kinetic energy of the incident wave was converted to a strong run-up because the front cylinder surface and the upper damper work together to prevent the horizontal propagation of the incident wave and change the flow vertically upward. In addition, as the upper damper rises, it can generate a strong vortex flow on the edge of the damper (Fig. 12), significantly altering the wave surface around the truncated cylinder. On the other hand, the effects of dampers on wave elevations around the cylinders were reduced when the wave period was 12 s (Fig. 11). This is because the diffraction effect becomes insignificant under the condition where the wavelength is much longer than the cylinder diameter. Interestingly, the higher position of the upper damper also causes higher wave crests and deeper wave troughs around the truncated cylinder (Figs. 10 and 11). The corresponding physical wave pattern could also be confirmed in Fig.13.

### 4.2 First-Order Components of Wave Run-Up

*η*

_{1}) corresponding to the fundamental wave frequency (

*ω*). Fig. 16 presents the first-order harmonic components of wave run-up around various cylinders with respect to various scattering parameters (

*ka*). The present results obviously showed that a gradual increase in the first-order harmonic components occurred around the cylinder as the position of the upper damper moved closer to the free surface. Such an increment became even more significant under short-wave conditions (

*ka*= 0.657).

*ka*), the amplification of first-order harmonic components became more significant as the upper damper moved upward. The dramatic increase in the crest of total wave run-up could be explained by the first-order harmonic component being the dominant component. Moreover, the first-order harmonic components of wave run-up on the weather side (WP1 & WP2) of the cylinder increased as the scattering parameter increased, following the original characteristics for the bare cylinder. Fig. 17 presents the distributions of first-order harmonic components at different wave probes. In the case of short wave conditions (Fig. 17(a)), there was a consistent decreasing tendency of the first-order harmonic components of wave run-up from the front point (WP1) to the back shoulder point (WP4) and then an increase again at the back point (WP5). In contrast, under long wave conditions, the variations of the first-order harmonic components of wave run-up at various probes became less significant (Fig. 17(b)).

### 4.3 Second-Order Components of Wave Run-Up

*η*

_{2}) at the double wave frequency (2

*ω*), corresponding to the second-order sum-frequency component of the free-surface elevation. Furthermore, the mean set-up/set-down of the free surface elevation could be considered the second-order difference frequency component.