# Numerical Study of Solitary Wave Interaction with a Submerged Semicircular Cylinder

Authors: Qiaoling Ji; Yu Wang; Guowei Zhang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The propagation on submerged structures of solitary wave, as a typical nonlinear wave, has guiding significance for the design and operation of coastal engineering. This paper presents a numerical model based on Navier-Stokes equations to study the interaction of the solitary wave with a submerged semicircular cylinder. A multiphase method is utilized to deal with water and air phase. The model uses the CIP (Constrained Interpolation Profile) method to solve the convection term of the Navier-Stokes equations and the THINC (Tangent of Hyperbola for Interface Capturing) scheme to capture the free surface. Three representative cases different in relative solitary wave height and structure size are simulated and analyzed by this model. By comparing the surface elevations at wave gauges with the experimental data and the documented numerical results, the present model is verified. Then, the wave pressure field around the submerged semicircular cylinder is presented and analyzed. At last, the velocity and vorticity fields are demonstrated to elucidate the characteristics of wave breaking, flow separation, and vortex generation and evolution during the wave-structure interaction. This work presents the fact that this numerical model combining the CIP and THINC methods has the ability to give a comprehensive comprehension of the flow around the structure during the nonlinear interaction of the solitary wave with a submerged structure.


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## Body

## 1. Introduction

The interaction between waves and obstacles has an important impact on the safety and stability of coastal structures. Particularly, submerged breakwaters are widely used in recent years for its ability to obstruct waves and dissipate wave energy. Consequently, many studies have been done on the interaction between waves and submerged structures. Among them, laboratory experiment with expensive measurement equipment like LDV is taken as a main means. Recently numerical model is increasingly gaining favor with the advantages of low operation cost and more detailed flow field information.In the early studies of wave transmission across submerged structures, attention was mainly paid to the periodic waves. For example, Losada et al. [1] used an eigenfunction expansion method to obtain the transmission and reflection coefficients of waves impinging obliquely on a vertical thin barrier. Subsequently, experimental study on the periodic wave interaction with submerged structures was also carried out by Losada et al. [2]. However, most waves in the ocean and coastal environment are nonlinear, and it is known that solitary waves have more nonlinear characteristics than the periodic waves. Thus, the study of the interaction between solitary waves and submerged structures is of guiding significance for the design and operation of coastal engineering. Lin [3] studied the transmission, reflection and energy dissipation coefficients of solitary waves with rectangular obstacles by numerical modeling and introduced the integral of energy flux to calculate these wave coefficients. Christou [4] utilized a Boundary Element Method (BEM) to study the interaction between nonlinear regular waves and rectangular submerged breakwaters and showed good performance in terms of the water surface profile and the harmonic generation. Rectangular structures are the focus of these researches. At the same time, semicircular structure, as a new type breakwater, has gained popularity in engineering because of its good stability with a zero moment at the center of the semicircle and small foundation stress. Yuan and Tao [5] reported that semicircular breakwaters were constructed in many places in China. The characteristics of the interaction of solitary waves and semicircular structures were studied in experimentally, theoretically and numerically, such as Cooker et al. [6], Xie [7], Yuan and Tao [5], Peng and Mizutani [8], and Liu and Li [9]. Cooker et al. [6] studied this problem extensively through experiments and a numerical model based on a boundary-integral equation. Their calculations failed to confirm the flow separation behind the cylinder and could not continue when the surface became very steep. Yuan and Tao [5] combined the BEM and finite difference method (FDM) to study the wave force on the semicircular breakwater and deduced a simplified formula to calculate the total wave force of the breakwater. Klettner and Ian [10] calculated the forces on the cylinder and energy dissipation caused by a submerged cylinder. Great progress had been made in the above researches, but most of them focused on the wave forces, reflection, and transmission coefficients.The wave transmission was accurately calculated in the above studies. However, solitary wave usually breaks during the transmission across submerged structures, which is difficult to calculate by the potential flow theory [6]. Moreover, the vortices around the submerged structures could not be simulated by the potential flow theory. Because the vortices could cause local scour, which might affect the safety and stability of the submerged structures, attention has been turned to viscous wave-structure interaction. Studies have been done to describe the vortex structures. The majority were restricted to the flow across a submerged rectangular obstacle. Zhuang and Lee [11] combined the BEM and a vorticity stream function to simulate the flow field on the lee side of a submerged dike. The model predicted good results for the wave profiles but performed bad in velocities. Huang and Dong [12] used two-dimensional Navier-Stokes equations to simulate the nonbreaking of solitary waves transmitting over submerged dikes and to study the influence of wave height and dike size on the flow fields and forces. Chang et al. [13] studied the nonbreaking and breaking solitary waves over rectangular structures by coupling the Reynolds averaged Navier-Stokes equations and k-ε turbulence equations. Kasem and Sasaki [14, 15] used Weighted Essentially Nonoscillatory (WENO) and level set methods to simulate the interaction of solitary waves with rectangular and semicircular submerged obstacles. They revealed various features including the separation vortices, large free surface deformations, and flow fields. In recent years, Zarruk [16] utilized a numerical model combining the Volume of Fluid and Large Eddy Simulation (VOF-LES) to study the vortex shedding and evolution in the process of a solitary wave passing across a submerged cylindrical structure.The Constrained Interpolation Profile (CIP) method was firstly put forward to solve general hyperbolic equations by Takewaki et al. [17] as a cubic-interpolated pseudo-particle method. It is a semi-Lagrangian method and was further developed by Yabe and Aoki [18], Yabe et al. [19], Nakamura [20], Yabe et al. [21], Xiao [22], Utsumi and Yabe et al. [23], and Toda [24] in terms of convergence speed and mass conservation. The CIP method could simplify the calculation process and greatly improve the calculation efficiency, because a nonuniform, staggered Cartesian grid system is established which does not need to be adaptive and dependent on the profile of free surface and solid boundary. Furthermore, comparing with conservative algorithms such as ENO which needs more cells to improve high accuracy, the interpolation function only needs the value and spatial gradient of one cell, so the compact structure and precision of subcell can be guaranteed. That is why the CIP method could deal with complex free surface when waves break under violent fluid flow. In addition, the CIP method uses only one set of governing equations without restriction of time step to solve interaction among incompressible fluids, solid, and compressible gas, which can greatly shorten the calculation time. To overcome the disadvantage of poor mass conservation in the CIP method, Xiao et al. [25] presented a Tangent of Hyperbola for Interface Capturing (THINC) method to capture the free surface. Hu and Kashiwagi [26, 27] applied the CIP and THINC methods to simulate the strongly nonlinear wave-structure interaction problem. Subsequently, much work had been done on the application of the CIP method to the interaction of waves and structures, such as He et al. [28], Liao and Hu [29], Zhao et al. [30], Ji et al. [31], and Wang et al. [32]. This work is an extension of the numerical model based on CIP and THINC methods to the interaction of a solitary wave interaction with a submerged semicircular cylinder.This paper presents a CIP-based model and applies it to study the transformation and vortices evolution of solitary waves over a semicircular cylinder. The paper is organized as follows. Section2 describes the numerical model based on the Navier-Stokes equations. A fractional step method is employed to separate the momentum equations. The CIP (Constrained Interpolation Profile) method is utilized in the calculation of the advection step. The THINC (Tangent of Hyperbola for Interface Capturing) scheme, as a method to capture the free surface, is described. Section 3 summarizes the numerical calculation results and experimental observations of Cooker et al. [6]. Numerical calculation conditions for this problem are also made clear in this section. In Section 4, numerical calculation results are demonstrated and discussed. Surface elevations at wave gauges are calculated and compared to experimental measurement and other numerical results for the model verification. The pressure distribution, velocity, and vorticity fields along with the free surface profile around the semicircular cylinder are shown. Wave breaking, wave pressure, flow separation, and vortices near the semicircular cylinder are analyzed in detail. The conclusions are given in Section 5.


## 2. Methodology

In order to analyze the effect of multiphase flow including water and air, a viscous and incompressible flow is considered in this numerical model. The Navier-Stokes equations are used as the governing equations:(1)∂ui∂xi=0(2)∂ui∂t+ui∂ui∂xj=-1ρ∂σij∂xj+fiwhereui (i=1, 2) means the component of velocity on the axisxi;xi (i=1,2) means the Cartesian coordinate. σij=-p+2μSij, where Sij=1/2∂ui/∂xj+∂uj/∂xi.fi means external force, such as gravity.The governing equations are solved on a staggered Cartesian grid system. The momentum equation (2) is separated into three steps by applying a fractional step method, including one advection step and two nonadvection steps.Advection Step (3) ∂ u i ∂ t + u j ∂ u i ∂ x j = 0Nonadvection Step (i) (4) ∂ u i ∂ t = - 2 μ ρ ∂ S i j ∂ x j + f iNonadvection Step (ii) (5) ∂ u i ∂ t = - 1 ρ ∂ p ∂ x iInstead of calculatinguin→uin+1 directly from (2), intermediate values of ui∗ and ui∗∗ are provided. uin→ui∗ by the advection step by (3), ui∗→ui∗∗ by the nonadvection step (i) by (4), and ui∗∗→uin+1 by the nonadvection step (ii) by (5) are calculated.In the advection step, the CIP (Constrained Interpolation Profile) scheme is utilized. The basic concept of the CIP scheme is to construct a profile inside the computational grid cell by a polynomial function to preserve it during the advection. Instead of using at least three points to construct the interpolation function like usual high-order schemes, the CIP scheme only needs two points to approximate the interpolation function in one dimension. The construction is accomplished by using the grid value and its spatial derivative in two grid points. To explain the CIP method, we use a universal 1D advection equation as follows:(6)∂f∂t+u∂f∂x=0Many difference schemes can be used to obtain the numerical solution of (6); see Figure 1 and Yabe et al. [21]. The initial profile, solid line of Figure 1(a), moves like the dashed line. The solution at the grid points is denoted by the solid circles. If we remove the solid and dashed lines and keep the circles as in Figure 1(b), it is hard to imagine the original profile and a profile of the solid line in Figure 1(c) would appear when a linearly interpolated function is used to construct the profile, such as the first order upwind scheme. This is because we neglect the profile of the solution inside the grid cell. The CIP method uses the spatial derivatives of the grid points to construct a third order interpolation and a profile as the solid line in Figure 1(d) is retrieved.Figure 1
The principal of the CIP method. (a) The solid line is the initial profile and the dashed line is an exact solution after advection. (b) The discretized points. (c) When (b) is linearly interpolated, numerical diffusion appears. (d) In the CIP method, the spatial derivative also propagates and the profile inside a grid cell is retrieved. (e) CIP scheme as a semi-Lagrangian method [21].
(a) (b) (c) (d) (e)Foru>0, the profile for fn inside the upwind cell [xi-1,xi] can be approximated by(7a)Finx=aix-xi3+bix-xi2+cix-xi+di(7b)ai=gin-gi-1nΔx2-2fin-fi-1nΔx3bi=2gin+gi-1nΔx-3fin-fi-1nΔx2ci=gindi=fin(7c)fin+1=Finxi-uΔtgin+1=dFinxi-uΔtdxEquation (7a) is a cubic-interpolated function for the profile inside the cell. Then the profile at time stepn+1 in Figure 1(e) is obtained by shifting the profile at time stepnby –uΔt, which is called a semi-Lagrangian approach; see Figure 1(e).By differentiating (3) with respect toxi we obtain (8)∂∂t+uj∂∂xi∂uj∂xi=-∂ui∂xj∂uj∂xiThe advection calculation contains the solution of the two equations of (3) and (8). The right hand side term of (8) can be solved in the nonadvection step (ii). The solution of ui∗ in this step can be expressed as follows:(9)ui∗=Fixi-uinΔtwhere Fi is the cubic polynomial to approximate the profile inside an upwind cell. More details about this interpolation function can be found in Yabe et al. [19] and Hu and Kashiwagi [26].In the nonadvection step (ii), the velocity pressure coupling is dealt by the following Poisson equation:(10)∂∂xi1ρ∂pn+1∂xi=1Δt∂ui∗∗∂xiEquation (10) is assumed to be applicable for water and air phase and is solved by the Successive Over-Relaxation (SOR) method. The solution provides the velocity and pressure distribution in the whole computational domain. Thus, this treatment does not need a boundary condition at the interface between water and air.The free surface can be determined by the solution of the following equation ofϕ1:(11)∂ϕ1∂t+ui∂ϕ1∂xi=0where ϕ1 means the density function of water phase. This is an advection equation and many schemes have been proposed to solve it. In this paper, a THINC (Tangent of Hyperbola for Interface Capturing) scheme proposed by Xiao et al. [25] is used. Figure 2 depicts the one-dimensional concept of the scheme.Figure 2
The one-dimensional concept of the THINC scheme: the interface is identified by the step-jump falling in a cell where the density functionϕ¯in has a value between 0 and 1; △up indicates distance between the position of the jump xi¯ and the cell boundary.In THINC scheme, the 1D advection equation ofϕ (11) is revised into a conservation form as follows:(12)∂ϕ∂t+∂uϕ∂x=ϕ∂u∂xThis transformation can make sure the mass conservation as a VOF-like function. It uses a hyperbolic tangent functionFi(x) to approximate the profile inside the grid cell as follows: (13)Fix=α21+γtanh⁡βx-xi-1/2Δxi-xi¯where xi¯ is the intermediate point of leap transformation of hyperbolic tangent function and is determined by ϕ¯in=1/Δxi∫xi-1/2xi+1/2Fixdx; see Figure 2. The parameters of α, β, γ are defined to modify the diffusion, sharpness, and slope of the function; for more details see Xiao et al. [25].After the piecewise approximation functionsFi(x) have been computed for all mesh cells; the density function ϕ is updated by the formulation of flux form as(14)ϕ¯in+1=ϕ¯in-1Δxigi+1/2-gi-1/2+ΔtΔxiϕ¯inui+1/2n-ui-1/2nwhere gi+1/2 denotes the flux across boundary x=xi+1/2 during tn+1-tn, as marked in Figure 2.Numerical studies have showed that the THINC scheme performs better than the CIP in terms of reducing numerical diffusion, avoiding numerical oscillation and retaining a compact thickness of the free surface [25]; Hu et al., [27]; Ji et al., [33].


## 3. Numerical Modeling Solitary Wave over a Submerged Semicircular Cylinder

### 3.1. Experimental Results

This interaction between a solitary wave and a submerged semicircular cylinder has been studied numerically and experimentally by Cooker et al. [6]. A number of phenomena were found in their calculations, such as crest-crest exchange (C-C), forward breaking (F-B) and backward breaking (B-B). They summarized the results in a (a/d,R/d) parameter space, as shown in Figure 3.Figure 3
The(a/d, R/d)parameter space.a/dandR/dare the dimensionless incident wave amplitude and semicircle radius, respectively. The red circles indicate experiments [6].In Figure3, the region ofa/d∈(0.2, 0.4) andR/d∈(0.5, 0.8) is characterized by the crest-crest exchange (C-C) phenomenon. The amplitude of the incident wave increases gradually and forms a transmitted wave after passing the semicircular cylinder without breaking. Then the transmitted wave continues to propagate forward and the crest increases slowly. At the same time, the incident wave is attenuated to a small wave and propagates backward as a reflected wave. The region ofa/d∈(0.3, 0.5) andR/d∈(0.5, 0.7) is controlled by the backward breaking (B-B) phenomenon. After the main wave crest passes across the cylinder, the free surface of the wave tail gets steeper and breaks backward to the cylinder. Their numerical method fails to resolve the free surface profile with a high curvature. The region ofa/d∈(0.2, 0.6) andR/d∈(0.4, 0.9) is dominated by the forward breaking (F-B) phenomenon. The incident wave is also separated into two waves when passing across the obstacle. The front part of the transmitted wave breaks forward. The rear part of it is steep and may break backward after the forward breaking. Their computations cannot continue to solve the breaking motion. In addition to the above three phenomena, the wave train (W-T) and Tanaka instability (T) phenomenon are also marked in the parameter space. The wave train (W-T) phenomenon usually occurs when solitary wave interacts with small and medium-sized obstacle. The influence of obstacles on the solitary wave is little, and only slight disturbance can be found. In the Tanaka instability (T) phenomenon, the solitary wave is easy to break even when the obstacle is small.Cooker et al. [6] chose three typical cases of the solitary wave interaction with a submerged cylinder to check their numerical predictions. Table 1 lists the experimental parameters of each case and the measurement positions in each case. The submerged semicircular cylinder was cemented to the concrete bed of the wave tank, with the fixed radius value of R=20.4cm. Three capacitance wave gauges were placed at nine positions in different runs to obtain the wave surface heights, as shown in Figure 4 [6]. The wave tank was 70m long with a piston-type wave maker at one end and a wave absorbing beach to damp out passing waves.Table 1
Parameters for three experimental cases.
Case Case 1 Case 2 Case 3 R/d 0.6 0.7 0.8 a/d 0.311 0.514 0.191 Measurement positions 1,2,3 4,5,6 1,2,3Figure 4
Schematic diagram of the experimental wave tank [6].The crest-crest exchange was observed in case 1 (a/d=0.311,R/d=0.6) of their experiments. The predicted wave elevations agree well with the experimental data for most gauges except for a small discrepancy near the cylinder. In case 2(a/d=0.514,R/d=0.7), both backward breaking (B-B) and forward breaking (F-B) occurred during the experiments. The transmitted wave had strongly steep surface and broke forwards. Unexpectedly, the backward breaker did not tip over though its surface was steep, which was the initial stage of the backward breaking. Instead it passed back across the cylinder, followed by one further wave. In Cooker’s simulation, the drawn-down of the free surface around the cylinder was overestimated. In case 3 (a/d=0.191,R/d=0.8), backward breaking (B-B) occurred but without forward breaking (F-B). Although the free surface was draw down deeply and had a steep surface before the forward breaker, the wave did not break backwards. Their predictions were bad after the wave crest passed the cylinder. They concluded that the discrepancies between the experimental data and their numerical results may have connection with the flow separation around the cylinder.


### 3.2. Numerical Calculation Setup

A piston-type hydraulic wave generator is used to generate the solitary wave, and the motion of the paddle was derived from Grimshaw (1976). An artificial dissipation zone is placed on the region at the outflow boundary by adding a fictitious damping source term to vertical momentum Navier-Stokes equation [27]. Also a coarse mesh is used near the outflow boundary to damp the wave energy. Since the submerged structure is stationary in this problem, the blocked-off regions method [34] is used in a Cartesian grid system. This method does the same discretization for all grid points. The velocity inside the solid structure is assigned to zero, and a nonslip boundary condition is satisfied on the solid boundary.In our calculations, the simulation wave tank is shortened to 12.0m to save computation time, and the submerged semicircular cylinder is set around 5.2m from the wave maker. The computation domain is 12.0m×1.3m inx and z directions. Figure 5 shows the calculation mesh near the region of the submerged cylinder with two index skipping in each direction. Five grids for case 1 were tested to check the grid independency of the numerical results. Figure 6 gives the comparison of the surface elevation at gauges 1 and 4 for different grids. We found that finer grids would enhance the calculation quality to only small extent. For the sake of computational efficiency, a nonuniform grid is used so that finer grids are distributed in the regions where finer resolution is demanded. The finer grid resolution of 0.5cm inx direction and 0.1cm in y direction is arranged around the wave maker, free surface, and the top of the semicircular cylinder. The computational domain is divided by a grid of 758×740 cells inx andz directions, respectively.Figure 5
Numerical grids in the computation domain near the structure.Figure 6
Surface elevation at gauges 1 and 4 for different grids (case 1).
(a) (b)The total computation is performed fort=7.5 s. A time step △t=0.005s is initially set and the length of each time step is dynamically adjusted during computation by the algorithm so it is not easy to compare the computational efficiency with other methods. The simulations have been done on an Intel-based personal computer with four CPU (with a 3.00 GHz Core(TM) i5-2320 CPU). The total run time including the time for writing the flow information files is 147.69 minutes for case 1, 112.14 minutes for case 2, and 151.98 minutes for case 3, respectively.


## 3.1. Experimental Results

This interaction between a solitary wave and a submerged semicircular cylinder has been studied numerically and experimentally by Cooker et al. [6]. A number of phenomena were found in their calculations, such as crest-crest exchange (C-C), forward breaking (F-B) and backward breaking (B-B). They summarized the results in a (a/d,R/d) parameter space, as shown in Figure 3.Figure 3
The(a/d, R/d)parameter space.a/dandR/dare the dimensionless incident wave amplitude and semicircle radius, respectively. The red circles indicate experiments [6].In Figure3, the region ofa/d∈(0.2, 0.4) andR/d∈(0.5, 0.8) is characterized by the crest-crest exchange (C-C) phenomenon. The amplitude of the incident wave increases gradually and forms a transmitted wave after passing the semicircular cylinder without breaking. Then the transmitted wave continues to propagate forward and the crest increases slowly. At the same time, the incident wave is attenuated to a small wave and propagates backward as a reflected wave. The region ofa/d∈(0.3, 0.5) andR/d∈(0.5, 0.7) is controlled by the backward breaking (B-B) phenomenon. After the main wave crest passes across the cylinder, the free surface of the wave tail gets steeper and breaks backward to the cylinder. Their numerical method fails to resolve the free surface profile with a high curvature. The region ofa/d∈(0.2, 0.6) andR/d∈(0.4, 0.9) is dominated by the forward breaking (F-B) phenomenon. The incident wave is also separated into two waves when passing across the obstacle. The front part of the transmitted wave breaks forward. The rear part of it is steep and may break backward after the forward breaking. Their computations cannot continue to solve the breaking motion. In addition to the above three phenomena, the wave train (W-T) and Tanaka instability (T) phenomenon are also marked in the parameter space. The wave train (W-T) phenomenon usually occurs when solitary wave interacts with small and medium-sized obstacle. The influence of obstacles on the solitary wave is little, and only slight disturbance can be found. In the Tanaka instability (T) phenomenon, the solitary wave is easy to break even when the obstacle is small.Cooker et al. [6] chose three typical cases of the solitary wave interaction with a submerged cylinder to check their numerical predictions. Table 1 lists the experimental parameters of each case and the measurement positions in each case. The submerged semicircular cylinder was cemented to the concrete bed of the wave tank, with the fixed radius value of R=20.4cm. Three capacitance wave gauges were placed at nine positions in different runs to obtain the wave surface heights, as shown in Figure 4 [6]. The wave tank was 70m long with a piston-type wave maker at one end and a wave absorbing beach to damp out passing waves.Table 1
Parameters for three experimental cases.
Case Case 1 Case 2 Case 3 R/d 0.6 0.7 0.8 a/d 0.311 0.514 0.191 Measurement positions 1,2,3 4,5,6 1,2,3Figure 4
Schematic diagram of the experimental wave tank [6].The crest-crest exchange was observed in case 1 (a/d=0.311,R/d=0.6) of their experiments. The predicted wave elevations agree well with the experimental data for most gauges except for a small discrepancy near the cylinder. In case 2(a/d=0.514,R/d=0.7), both backward breaking (B-B) and forward breaking (F-B) occurred during the experiments. The transmitted wave had strongly steep surface and broke forwards. Unexpectedly, the backward breaker did not tip over though its surface was steep, which was the initial stage of the backward breaking. Instead it passed back across the cylinder, followed by one further wave. In Cooker’s simulation, the drawn-down of the free surface around the cylinder was overestimated. In case 3 (a/d=0.191,R/d=0.8), backward breaking (B-B) occurred but without forward breaking (F-B). Although the free surface was draw down deeply and had a steep surface before the forward breaker, the wave did not break backwards. Their predictions were bad after the wave crest passed the cylinder. They concluded that the discrepancies between the experimental data and their numerical results may have connection with the flow separation around the cylinder.


## 3.2. Numerical Calculation Setup

A piston-type hydraulic wave generator is used to generate the solitary wave, and the motion of the paddle was derived from Grimshaw (1976). An artificial dissipation zone is placed on the region at the outflow boundary by adding a fictitious damping source term to vertical momentum Navier-Stokes equation [27]. Also a coarse mesh is used near the outflow boundary to damp the wave energy. Since the submerged structure is stationary in this problem, the blocked-off regions method [34] is used in a Cartesian grid system. This method does the same discretization for all grid points. The velocity inside the solid structure is assigned to zero, and a nonslip boundary condition is satisfied on the solid boundary.In our calculations, the simulation wave tank is shortened to 12.0m to save computation time, and the submerged semicircular cylinder is set around 5.2m from the wave maker. The computation domain is 12.0m×1.3m inx and z directions. Figure 5 shows the calculation mesh near the region of the submerged cylinder with two index skipping in each direction. Five grids for case 1 were tested to check the grid independency of the numerical results. Figure 6 gives the comparison of the surface elevation at gauges 1 and 4 for different grids. We found that finer grids would enhance the calculation quality to only small extent. For the sake of computational efficiency, a nonuniform grid is used so that finer grids are distributed in the regions where finer resolution is demanded. The finer grid resolution of 0.5cm inx direction and 0.1cm in y direction is arranged around the wave maker, free surface, and the top of the semicircular cylinder. The computational domain is divided by a grid of 758×740 cells inx andz directions, respectively.Figure 5
Numerical grids in the computation domain near the structure.Figure 6
Surface elevation at gauges 1 and 4 for different grids (case 1).
(a) (b)The total computation is performed fort=7.5 s. A time step △t=0.005s is initially set and the length of each time step is dynamically adjusted during computation by the algorithm so it is not easy to compare the computational efficiency with other methods. The simulations have been done on an Intel-based personal computer with four CPU (with a 3.00 GHz Core(TM) i5-2320 CPU). The total run time including the time for writing the flow information files is 147.69 minutes for case 1, 112.14 minutes for case 2, and 151.98 minutes for case 3, respectively.


## 4. Results and Discussion

In this section, the present numerical model capabilities in simulating the nonlinear interaction of the solitary wave with a submerged semicircular cylinder are demonstrated. Numerical results, including free surface elevations, wave pressure, and velocity, and vorticity fields are presented. The model is verified by comparing the free surface elevation with the experimental data and the published computational results. Flow characteristics are analyzed in detail.


### 4.1. Verification of the Free Surface Elevation

Figure7 shows graphs of the time histories of the local free surface elevation at the corresponding wave gauges. Nondimensional variables are used as η/d in y axis and tg/h inx axis. For validation, the free surface elevations calculated by this model are presented along with the experimental data by Cooker et al. [6] and the numerical predictions by Kasem & Sasaki [14, 15]. Encouraging agreement is found between our numerical computed results and the experimental data by Cooker et al. [6]. A slight discrepancy is observed at the gauge 3 in case 1 and 3 after the main crest has passed. There are several sources of error for this discrepancy mentioned by Cooker et al. [6]. One is caused by the horizontal movement of the gauges when the free surface is steep at the experimental measurement positions. There may also be hysteresis in the gauge responses due to the water surface motion. Besides, the complexity of the flow separation and vortex evolution could be another error causation of the discrepancy between the numerical calculations and measurement results. In Cooker et al. [6], an irrational flow model was used to calculate the free surface elevations, and the computation could not continue when the wave is breaking, while our model can proceed the computation when the surface motion gets violent and even when the wave is breaking. Kasem and Sasaki [14, 15] calculated the free surface elevations of cases 2 and 3. In their results, good agreement is found at most wave gauges except that there is a slight discrepancy at the gauge 4 in case 2 and gauge 3 in case 3. By comparing our calculation results with results of Kasem and Sasaki [14, 15], we find that our model predicts almost as well as that of Kasem and Sasaki [14, 15]. Moreover, our model predicts better than that of Kasem and Sasaki [14, 15] at gauge 4 in case 2, where Kasem and Sasaki [14, 15] overestimated the wave crest by around 10%-20%. We can conclude that our numerical model has a good performance in prediction the solitary wave transformation over a submerged structure.Figure 7
Comparison of numerical computed wave surface profiles (—) with experimental results by Cooker et al. [6] (▲) and calculated results by Kasem et al. (2010) (- -) at wave gauges.


### 4.2. Wave Pressure Distribution

The pressure fields at six critical time instants for three cases are shown in Figure8. We can see that the pressure around the cylinder reaches the highest as the wave crest is passing upon it and decreases gradually after the wave has passed away for three cases. Among them, the pressure on the top of the cylinder of case 1 with moderate relative wave heighta/dis higher than other cases, indicating that the pressure acting on the structure is affected not only by the nonlinearity of the solitary wavea/dbut also by the relative structure sizeR/d. After the main crest has passed the cylinder, there is a low pressure region near the top lee side of the cylinder as the reflected wave propagates backward the cylinder for three cases. It disappears after the reflected wave breaks or transmits over the cylinder. The size and strength of the low pressure seem to be strongly related to the nonlinearity of the solitary wavea/d because case 2 has a stronger low pressure region. This may have a connection with the high velocity distribution and vortex generation near this area. More details will be discussed in the following section.Figure 8
Pressure fields (units: Pa) at six critical time instants for three cases during the solitary wave across the structure. (a) Case 1:a/d =0.311,R/d=0.6. (b) Case 2:a/d =0.514,R/d=0.7. (c) Case 3:a/d =0.191,R/d=0.8.
(a) (b)  (c)


### 4.3. Evolution of Velocity and Vorticity Fields

In Figure9, the velocity fields along with the free surface profiles around the submerged semicircular cylinder are presented. Distributions of both air velocity and water velocity were calculated by the present multiphase model. The transformation of the solitary wave and the flow separation and vortex generation and evolution near the obstacle can be clearly observed from the figures.Figure 9
Simulated water velocity fields and free surface profile around the semicircular cylinder. (a) Case 1:a/d =0.311,R/d=0.6. (b) Case 2:a/d =0.514,R/d=0.7. (c) Case 3:a/d =0.191,R/d=0.8.
(a) (b)  (c)For case 1, the solitary wave does not break because both the relative solitary wave amplitude and structure size are relative small. In Figure9(a), from the timet=3.8s tot=4.3s, as the solitary wave passes across the cylinder, a second crest grows on the lee side of the cylinder and continues to propagate forward. At the mean time the incident wave crest decays, reverses its direction, and propagates backward as a reflected wave, which can be observed at timet=4.6s tot=5.2s. At timet=4.6s, there is a small drop of the water surface behind the cylinder as the two newly formed waves move apart in opposite directions. This phenomenon is called crest-crest exchange by Cooker et al. [6]. The separation vortex is formed behind the semicircular cylinder at timet=4.3s after the solitary wave has passed over the cylinder. As seen from timet=4.6s tot=5.2s in Figure 9(a), the vortex keeps growing and rising to the free surface, and the vortex is stretched to both forward and backward sides. Meanwhile another small vortex comes into being on the top of the semicircular cylinder att=5.0s as the reflected wave propagates backward.For case 2 in Figure9(b), as the solitary wave passes over the cylinder, the wave surface becomes steep at the timet=3.8s. And the front of the solitary wave tends to break forward after it transmits across the cylinder between the timet=4.0s tot=4.4s. While this happens, the rear surface of the transmitted wave also steepens and tends to break backward between the timet=4.2s tot=4.4s. Subsequently, another small reflected wave appears and rolls over backward aroundt=4.7s-5.0s, which causes a drawn-down of the water surface and air bubbles entrapped. This phenomenon is called forward breaker and backward breaker by Cooker et al. [6]. Similar to case 1, the flow separation and vortex can also be observed after the solitary wave passes across the cylinder at timet=3.8s, and the vortex gets growing and rising, except that the size and strength are a little bigger than that of case 1. The air-water mixing is observed as the reflected wave rolls over backward. This phenomenon is not mentioned in the experiment description of Cooker et al. [6], yet it is mentioned in other experiments for a solitary wave passing over a rectangular structure [32, 35, 36]. The air is trapped into the water surface at the initial stage of the backward breaking at the time instantt=4.7s. Then the air bubbles are carried down into the deep water by the vortex motion at the time instantst=4.8s-5.0s. Along with the clockwise motion of the vortex, the entrapped air bubbles get to spiral to the water surface at the time instantst=5.2s-5.6s and go down into the deep water again at the time instantt=6.1s.For case 3 in Figure9(c), there is also a crest-crest exchange, as seen at time instantt=4.6s. After it passes over the cylinder, the tail of the solitary wave gets steeper and finally breaks backward onto the cylinder, as seen at time instantst=4.6s-5.4s. This breaking process gives rise to a backwash which propagates backward as like a reflected wave, as seen at time instantst=5.4s-5.8s. This phenomenon is called backward breaker by Cooker et al. [6], which may lead to a deep draw-down of the free surface at the rear of the transmitted wave. Similarly, the vortex generates at time instantt=4.6s after the main crest has passed the semicircular cylinder. It keeps growing and existing for a long time till the transmitted wave has left far away from the cylinder, as seen at time instantt=6.8s. The air-water mixing stages are also seen at time instantst=5.4s-6.8s.To further understand the evolution of the vortical structures, the temporal variation of these above vortices was also calculated in terms of vorticity. The vorticity is defined as the curl of the velocity field∂v/∂x-∂u/∂y. According to the definition, the negative vorticity has the clockwise circulation, and the positive has the counterclockwise circulation. The vorticity fields induced by a solitary wave passing over a submerged semicircular cylinder are shown in Figure 10 to illustrate the vortex generation and evolution. At the initial stage of the development of the flow separation, a strong shear layer is observed on the top of the weather and lee side surfaces of the cylinder. As the wave crest approaches the cylinder, the shear layer has partially separated from the surfaces of the cylinder, giving way to the formation of two small vortices of a counterclockwise vortex at the weather side and a clockwise vortex at the lee side. When the wave crest passes over the structure, the vortices are convected downstream, as shown at time instant oft=4.0s-4.3s in Figure 10(a), att=3.8s in Figure 10(b), and att=5.0s in Figure 10(c). As a reflected wave propagates backward, a new separating shear layer generates and rolls up into small counterclockwise vortices, cutting off the supply of the former vortex at the lee side which moves upward, as shown at time instant oft=4.6s in Figure 10(a), att=4.0s-4.2s in Figure 10(b), and att=5.2s-5.4s in Figure 10(c). For case 2 and 3, the reflected wave is breaking upon the semicircular cylinder, some air is entrapped into water, and a counterclockwise vortex is induced around the free surface, as shown at time instant oft=4.0s-4.2s in Figure 10(b), and att=5.0s-5.2s in Figure 10(c). And then the vorticity distributions become complicated at the free surface after the wave breaking for cases 2 and 3. By comparing the vorticity strength of the three cases, we can find that the strength is larger when the relative wave height is larger, which indicates that the solitary wave nonlinearity has important influence on the vortical strength as we expected.Figure 10
Simulated water vorticity fields and free surface profile around the semicircular cylinder. (a) Case 1:a/d =0.311,R/d=0.6. (b) Case 2:a/d =0.514,R/d=0.7. (c) Case 3:a/d =0.191,R/d=0.8.
(a) (b)  (c)


## 4.1. Verification of the Free Surface Elevation

Figure7 shows graphs of the time histories of the local free surface elevation at the corresponding wave gauges. Nondimensional variables are used as η/d in y axis and tg/h inx axis. For validation, the free surface elevations calculated by this model are presented along with the experimental data by Cooker et al. [6] and the numerical predictions by Kasem & Sasaki [14, 15]. Encouraging agreement is found between our numerical computed results and the experimental data by Cooker et al. [6]. A slight discrepancy is observed at the gauge 3 in case 1 and 3 after the main crest has passed. There are several sources of error for this discrepancy mentioned by Cooker et al. [6]. One is caused by the horizontal movement of the gauges when the free surface is steep at the experimental measurement positions. There may also be hysteresis in the gauge responses due to the water surface motion. Besides, the complexity of the flow separation and vortex evolution could be another error causation of the discrepancy between the numerical calculations and measurement results. In Cooker et al. [6], an irrational flow model was used to calculate the free surface elevations, and the computation could not continue when the wave is breaking, while our model can proceed the computation when the surface motion gets violent and even when the wave is breaking. Kasem and Sasaki [14, 15] calculated the free surface elevations of cases 2 and 3. In their results, good agreement is found at most wave gauges except that there is a slight discrepancy at the gauge 4 in case 2 and gauge 3 in case 3. By comparing our calculation results with results of Kasem and Sasaki [14, 15], we find that our model predicts almost as well as that of Kasem and Sasaki [14, 15]. Moreover, our model predicts better than that of Kasem and Sasaki [14, 15] at gauge 4 in case 2, where Kasem and Sasaki [14, 15] overestimated the wave crest by around 10%-20%. We can conclude that our numerical model has a good performance in prediction the solitary wave transformation over a submerged structure.Figure 7
Comparison of numerical computed wave surface profiles (—) with experimental results by Cooker et al. [6] (▲) and calculated results by Kasem et al. (2010) (- -) at wave gauges.


## 4.2. Wave Pressure Distribution

The pressure fields at six critical time instants for three cases are shown in Figure8. We can see that the pressure around the cylinder reaches the highest as the wave crest is passing upon it and decreases gradually after the wave has passed away for three cases. Among them, the pressure on the top of the cylinder of case 1 with moderate relative wave heighta/dis higher than other cases, indicating that the pressure acting on the structure is affected not only by the nonlinearity of the solitary wavea/dbut also by the relative structure sizeR/d. After the main crest has passed the cylinder, there is a low pressure region near the top lee side of the cylinder as the reflected wave propagates backward the cylinder for three cases. It disappears after the reflected wave breaks or transmits over the cylinder. The size and strength of the low pressure seem to be strongly related to the nonlinearity of the solitary wavea/d because case 2 has a stronger low pressure region. This may have a connection with the high velocity distribution and vortex generation near this area. More details will be discussed in the following section.Figure 8
Pressure fields (units: Pa) at six critical time instants for three cases during the solitary wave across the structure. (a) Case 1:a/d =0.311,R/d=0.6. (b) Case 2:a/d =0.514,R/d=0.7. (c) Case 3:a/d =0.191,R/d=0.8.
(a) (b)  (c)


## 4.3. Evolution of Velocity and Vorticity Fields

In Figure9, the velocity fields along with the free surface profiles around the submerged semicircular cylinder are presented. Distributions of both air velocity and water velocity were calculated by the present multiphase model. The transformation of the solitary wave and the flow separation and vortex generation and evolution near the obstacle can be clearly observed from the figures.Figure 9
Simulated water velocity fields and free surface profile around the semicircular cylinder. (a) Case 1:a/d =0.311,R/d=0.6. (b) Case 2:a/d =0.514,R/d=0.7. (c) Case 3:a/d =0.191,R/d=0.8.
(a) (b)  (c)For case 1, the solitary wave does not break because both the relative solitary wave amplitude and structure size are relative small. In Figure9(a), from the timet=3.8s tot=4.3s, as the solitary wave passes across the cylinder, a second crest grows on the lee side of the cylinder and continues to propagate forward. At the mean time the incident wave crest decays, reverses its direction, and propagates backward as a reflected wave, which can be observed at timet=4.6s tot=5.2s. At timet=4.6s, there is a small drop of the water surface behind the cylinder as the two newly formed waves move apart in opposite directions. This phenomenon is called crest-crest exchange by Cooker et al. [6]. The separation vortex is formed behind the semicircular cylinder at timet=4.3s after the solitary wave has passed over the cylinder. As seen from timet=4.6s tot=5.2s in Figure 9(a), the vortex keeps growing and rising to the free surface, and the vortex is stretched to both forward and backward sides. Meanwhile another small vortex comes into being on the top of the semicircular cylinder att=5.0s as the reflected wave propagates backward.For case 2 in Figure9(b), as the solitary wave passes over the cylinder, the wave surface becomes steep at the timet=3.8s. And the front of the solitary wave tends to break forward after it transmits across the cylinder between the timet=4.0s tot=4.4s. While this happens, the rear surface of the transmitted wave also steepens and tends to break backward between the timet=4.2s tot=4.4s. Subsequently, another small reflected wave appears and rolls over backward aroundt=4.7s-5.0s, which causes a drawn-down of the water surface and air bubbles entrapped. This phenomenon is called forward breaker and backward breaker by Cooker et al. [6]. Similar to case 1, the flow separation and vortex can also be observed after the solitary wave passes across the cylinder at timet=3.8s, and the vortex gets growing and rising, except that the size and strength are a little bigger than that of case 1. The air-water mixing is observed as the reflected wave rolls over backward. This phenomenon is not mentioned in the experiment description of Cooker et al. [6], yet it is mentioned in other experiments for a solitary wave passing over a rectangular structure [32, 35, 36]. The air is trapped into the water surface at the initial stage of the backward breaking at the time instantt=4.7s. Then the air bubbles are carried down into the deep water by the vortex motion at the time instantst=4.8s-5.0s. Along with the clockwise motion of the vortex, the entrapped air bubbles get to spiral to the water surface at the time instantst=5.2s-5.6s and go down into the deep water again at the time instantt=6.1s.For case 3 in Figure9(c), there is also a crest-crest exchange, as seen at time instantt=4.6s. After it passes over the cylinder, the tail of the solitary wave gets steeper and finally breaks backward onto the cylinder, as seen at time instantst=4.6s-5.4s. This breaking process gives rise to a backwash which propagates backward as like a reflected wave, as seen at time instantst=5.4s-5.8s. This phenomenon is called backward breaker by Cooker et al. [6], which may lead to a deep draw-down of the free surface at the rear of the transmitted wave. Similarly, the vortex generates at time instantt=4.6s after the main crest has passed the semicircular cylinder. It keeps growing and existing for a long time till the transmitted wave has left far away from the cylinder, as seen at time instantt=6.8s. The air-water mixing stages are also seen at time instantst=5.4s-6.8s.To further understand the evolution of the vortical structures, the temporal variation of these above vortices was also calculated in terms of vorticity. The vorticity is defined as the curl of the velocity field∂v/∂x-∂u/∂y. According to the definition, the negative vorticity has the clockwise circulation, and the positive has the counterclockwise circulation. The vorticity fields induced by a solitary wave passing over a submerged semicircular cylinder are shown in Figure 10 to illustrate the vortex generation and evolution. At the initial stage of the development of the flow separation, a strong shear layer is observed on the top of the weather and lee side surfaces of the cylinder. As the wave crest approaches the cylinder, the shear layer has partially separated from the surfaces of the cylinder, giving way to the formation of two small vortices of a counterclockwise vortex at the weather side and a clockwise vortex at the lee side. When the wave crest passes over the structure, the vortices are convected downstream, as shown at time instant oft=4.0s-4.3s in Figure 10(a), att=3.8s in Figure 10(b), and att=5.0s in Figure 10(c). As a reflected wave propagates backward, a new separating shear layer generates and rolls up into small counterclockwise vortices, cutting off the supply of the former vortex at the lee side which moves upward, as shown at time instant oft=4.6s in Figure 10(a), att=4.0s-4.2s in Figure 10(b), and att=5.2s-5.4s in Figure 10(c). For case 2 and 3, the reflected wave is breaking upon the semicircular cylinder, some air is entrapped into water, and a counterclockwise vortex is induced around the free surface, as shown at time instant oft=4.0s-4.2s in Figure 10(b), and att=5.0s-5.2s in Figure 10(c). And then the vorticity distributions become complicated at the free surface after the wave breaking for cases 2 and 3. By comparing the vorticity strength of the three cases, we can find that the strength is larger when the relative wave height is larger, which indicates that the solitary wave nonlinearity has important influence on the vortical strength as we expected.Figure 10
Simulated water vorticity fields and free surface profile around the semicircular cylinder. (a) Case 1:a/d =0.311,R/d=0.6. (b) Case 2:a/d =0.514,R/d=0.7. (c) Case 3:a/d =0.191,R/d=0.8.
(a) (b)  (c)


## 5. Conclusions

A validated semi-Lagrange numerical scheme CIP combined with a THINC scheme for the free surface capture was adopted to investigate numerically a solitary wave passing over a submerged semicircular cylinder. The three typical cases of case 1 (R/d=0.6,a/d =0.311), case 2 (R/d=0.7,a/d =0.514), and case 3 (R/d=0.8,a/d =0.191) were simulated and demonstrated to study the solitary wave flow characteristics around the submerged structure.The violent free surface motions were simulated well by the present model. Free surface elevations at measurement positions were compared with the recorded experimental data [6] and published numerical results of Kasem and Sasaki [14, 15]. The results agree satisfactorily with experimental data and numerical results and even have more accuracy in some cases. Pressure distributions around the cylinder were depicted. Comparisons show that both the relative solitary wave heighta/d and structure sizeR/d have influence on the pressure on the cylinder. A low pressure region near the top lee side of the cylinder was revealed as the reflected wave propagates backward the cylinder, and the size and strength might have relationship with the relative solitary wave heighta/d.The velocity and vorticity fields along with the free surface profiles around the semicircular cylinder at some special moments were depicted to analyze the phenomena of wave breaking, flow separation, and vortex near the structure. The present model well simulated the phenomena of crest-crest exchange, forward breaking, and backward breaking mentioned in Cooker et al. [6], as well as the air-water mixing. Results indicate that the relative structure size and wave amplitude have big influence on the possibility of solitary wave breaking. The vortex generation and evolution were described by the velocity and vorticity fields. The present simulations reveal that the vortices are preserved for a very long time even after the solitary wave crest has passed the structure. The results also reveal the detailed features of complicated vortical structures. There are several dominant vertical structures shed by the submerged structure, and a reversal vortex is induced by the air-water mixing when the solitary wave breaks. As the relative wave amplitude increases, the size and strength of vortices increase.This work presents a simulation tool that can be used to study the wave-structure interaction to obtain a comprehensive comprehension of the flow around submerged structures. It has a potential to aid and improve the current design practices for submerged structures in coastal engineering, which do not account for the effect of the flow and vortex around submerged structures.



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*Source: 3589052-2019-04-02.xml*