Modeling of Solitary Wave Interaction with Curved Face Seawalls Using Numerical Method
This paper considers the solitary wave interaction with seawalls of different curved geometries and sloped faces using numerical modeling. This interaction was simulated using the Finite Volume Method-Volume of Fraction (FVM-VOF) approach. To model the turbulent free-surface flow, coupled VOF and k-ε-RNG methods were used. To validate the model, the numerical results for a conventional sloping seawall were compared with the available experimental data. Then the interaction of solitary waves and seawalls with different sinusoidal, logarithmic, and parabolic functions front faces and linear slope was modeled. The results showed that for these seawalls in general by increasing the solitary wavelength, the wave reflection coefficient (Cr) increases. However, the wave run-up on seawalls demonstrates an oscillatory decrease. Furthermore, for parabolic walls in comparison to conventional linear sloping seawalls, the wave run-up and wave reflection increased by 4.1% and 4.7%. For sinusoidal walls, the wave run-up and wave reflection increased by 5% and 1.8%. For logarithmic walls, the wave run-up and wave reflection increased by 6.3% and decreased by 1.1%, respectively. This means that wave run-up on logarithmic walls is more than that of the sinusoidal, parabolic, and sloped walls. The simulation results revealed that normalized maximum run-up increases with an increase in normalized incident wave height for all types of curved walls.
Seawalls are structures constructed almost parallel to the coastline to separate the land area from the sea region. They are used to protect the shoreline from erosion and reduce the risk of long-term effects of waves. The real effects of these structures on beaches can be estimated through the erosion process. After the waves break, a portion of the remaining energy will energize the bore that will run-up the face of the beach or sloped shore structure. Figure 1 depicts this process where run-up R is the maximum height above the still water level to which water rises and h0 is the water depth in front of the slope. These structures are only protecting their own vicinity and do not have any protective action on adjacent areas exposed to littoral drift. The front walls of these structures can be designed and constructed vertically or sloped in different shapes such as linear, stepped wise, or curved, although for performance purposes a combination of the above sections may be applied. Usually, the crest elevation is selected for these structures allowing a limited overtopping. It is possible to select this elevation so that no overtopping occurs even in storm conditions. However, such a high structure not only blocks the sea view but is also not economical.
Weber  proposed a conceptual design of a curved seawall with the combination of a parabolic and a circular arc which brings a smooth change in the direction of propagation from horizontal to vertical and vice versa to reduce the wave-induced pressures. Freeman and Le Mehaute  and Iwasaki and Togashi  improved the computational efficiency of this model. However, their predicted run-up values were not in good agreement with measured data. One of the leading studies in this area is the work of Synolakis , who provided analytical solutions and lab work in the field of wave run-up. Kobayashi et al. [5, 6] further estimated the run-up of broken periodic waves on rough slopes using friction coefficients of bedding. Kobayashi and Wurjanto  developed a numerical model to predict the decrease of wave reflection due to the increase of wave overtopping on sloped coastal structures. Yeh et al.  performed laboratory work on wave run-up on the sloping bed and concluded that broken waves would move faster on the dry surface. Zelt and Raichlen  found that bed friction effects during the inundation phase were important for numerical simulations and experiments of solitary wave run-up on a dry horizontal shore. Subramanya and Grilli  proposed a very robust solution for the calculation of run-up of solitary waves with an initial height close to the limiting wave height. Murakami et al.  proposed a new type of circular arc nonovertopping seawall and measured the pressures and forces on the seawall due to regular waves. Kâno Lu and Synolakis  based on asymptotic results concluded that solitary waves can interact with piecewise linear topographies in a counterintuitive way. It means that for a composite beach with the vertex below the equilibrium surface, the run-up of solitary waves would depend only on the slope closest to the shoreline. Kamikubo et al.  investigated the characteristics of the curved seawall and reproduced the fluid flow near the seawall through numerical simulation using the finite volume method (FVM). The run-up and run down of nonbreaking and breaking solitary waves on a plane beach were investigated by solving the nonlinear shallow-water equations. Numerical results of the free surface evolution, the shoreline movement, the maximum run-up height, and the particle velocity were compared with experimental data. Furthermore, variations of energy transformation and dissipation with respect to the incident wave heights and slopes have been discussed; see, e.g., Lin et al.  and Li and Raichlen [15, 16]. Based on energy conservation, Li and Raichlen  proposed an empirical formula to predict the maximum run-up height of breaking solitary waves on a plane slope. The reliability of the formula was confirmed by comparing predictions with experimental data. Kobayashi and Karjadie  developed numerical models for finite-amplitude, shallow-water waves with arbitrary incident angles to predict detailed wave motions on a slop in the vicinity of the still waterline. Christakis et al.  developed and used a new numerical model of wave dynamics based on volume of fluid (VOF) techniques for wave impacts at coastal structures during the critical period of impact and breaking. Hubbard and Dodd  presented a 2D numerical model using an upwind finite volume technique to simulate wave run-up and overtopping. Allsop et al.  studied and validated prediction methods for wave overtopping discharges and velocities for steep, battered, composite, and vertical seawalls. Park et al.  developed a large eddy simulation (LES) model by using a finite differential algorithm to investigate solitary wave run-up on a vertical wall, flow over a broad-crested weir, and regular waves overtopping on a sloping seawall. Nwogu and Demiirbilek  presented the results of a combined laboratory and numerical investigation into the role of infragravity motions in the wave run-up process over fringing coral reefs. They developed a numerical model based on the Boussinesq equations applied to laboratory data and described the complex changes to the wave spectrum over the reef flat due to nonlinear wave-wave interactions and wave breaking as well as run-up at the shoreline. Anada et al.  performed an experimental study to measure the run-up and overtopping of three different types of curved-front face seawall models. They also measured the wave dynamic pressures exerted on these seawalls. Sælevik et al.  used particle image velocimetry (PIV) to investigate the run-up of solitary waves at straight and composite beaches with different inclinations. They compared the results with numerical simulations using a Navier-Stokes’ solver with zero viscosity. Didenkulova et al.  studied the run-up of long solitary waves of different polarities on a beach in the case of composite bottom topography. They showed that nonlinear effects are more strongly preferred for the run-up of a wave with negative polarity (wave trough). Stockdon et al.  investigated the run-up and its components using a parameterized model developed by comparing run-up observations with offshore wave height, wave period, and local beach slope. The results indicated that the parameterized predictions of the setup may need modification for extreme conditions, and numerical simulations can be used to extend the validity of the parameterized predictions of the infragravity swash. Rehman et al.  investigated the reduction in run-up heights of solitary waves on steep beaches by installing parallel rows of submerged breakwaters. They observed that three rows of breakwaters six seawall heights apart resulted in a significant decrease in wave run-ups as compared to two rows. Yao et al.  studied the effects of reef morphologies (fore-reef slope, back-reef slope, reef-flat width, and reef crest width) on tsunami-like solitary wave transformation and run-up and proposed an empirical formula based on experimental data. Subramaniam et al.  studied the effects of dike curvature on wave run-ups on regular wave attacks by employing a numerical model. They studied various dike-opening angles and compared them with physical model test results. In addition to the mesh-based methods that need to readjust or rezone the grid after each time step, smoothed particle hydrodynamics (SPH) is a meshless particle method with strong self-adaptability  which was initially proposed for astrophysical problems by Gingold and Monaghan . Lots of work was conducted to increase the accuracy and stability of the standard SPH scheme. Mahmoudi et al. [33, 34] modeled the periodic wave breaking process on a plane slope using the weakly-compressible SPH (WCSPH) method. Rostami and Ketabdari  proposed the WCSPH method to solve the continuity and momentum equations with laminar viscosity and the subparticle scale (SPS) turbulence model. Fathi and Ketabdari  used the SPH method to examine the run-up and overtopping of solitary waves on semicircular breakwaters (SBW). He et al.  improved a meshless method and presented a numerical investigation of solitary wave breaking over a slope by using a finite particle method (FPM) and demonstrated that FPM performs better than SPH qualitatively and quantitatively.
Such broad literature shows that most studies were carried out on sloping beaches and seawalls with vertical faces restricted to the action of regular or random waves. Posttsunami conditions has added a new dimension to the problem of the response of such structures to shallow-water waves. Solitary waves represent closely the characteristics of a tsunami. This promoted the authors to undertake the present study. Therefore, this study uses the FVM-VOF method by FLOW-3D® software  to introduce a numerical model of solitary wave-seawall interaction and explain the effects of sinusoidal, logarithmic, and parabolic functions of seawall front faces on wave run-up and reflection.
2. Governing Equations and Numerical Solution
In this paper, a numerical simulation was undertaken using the Reynolds-averaged Navier-Stokes equations (RANS) with the k-ε-RNG renormalization group turbulent model. To model the complex geometric boundary by the fractional area/volumes obstacle representation FAVOR technique Hirt and Sicilian , the general continuity and momentum equations for incompressible turbulent flows are formulated with the area and volume fraction functions:
In (2), and is calculated from the following:
denotes the ensemble-averaged or so-called time-averaged properties, is the velocity component in the subscript direction, the subscripts = 1, 2 represent x and z directions, respectively, p is the pressure intensity, is the fluid density, is the gravitational acceleration, and is the absolute viscosity. is the fractional volume open to the flow and A is the fractional area open to flow in the subscript direction. The above governing equations are reduced to standard RANS equations as both and A are set to unity. The Reynolds stress term in (2) is expressed bywhere is the eddy viscosity, k is the turbulent kinetic energy, and is the Kronecker delta function such that when i = j; , when. .
In (4), the eddy viscosity is related to the effect of the space and time distribution of the turbulent motion, which is solved here using the renormalization group method k-ε-RNG model. The k-ε-RNG turbulent model was proposed by Yakhot and Orszag  improving on the k-ε model. The transport equations of the k-ε-RNG model are expressed in a Cartesian coordinate system aswhere k and ε represent the turbulent kinetic energy and turbulent energy dissipation, respectively. One of the major advantages of the RNG theory is that the important turbulent coefficients are theoretically determined rather than being adjusted empirically. According to Orszag et al. , the turbulence transport coefficients shown in the above equations are summarized in Table 1.
The numerical solutions are implemented by FLOW-3D® , which utilizes the true VOF method of Hirt and Nichols  to accurately track the free water surface and models efficiently the solid geometries using the FAVOR technique. Several studies applied Flow-3D to successfully model the issues of the interaction of waves and structures, such as Choi et al. , Jin and Meng , and Dentale et al. . Also, several studies on solitary wave characteristics, such as in Yang and Chen , Zhang et al. [47, 48], and Guo et al. , was investigated and in this paper, the solution of solitary wave derived from Boussinesq equations was used as the incident wave, which was expressed aswhere η is the free surface elevation, h0 is the still water depth, H is the wave height, and c is the wave celerity expressed by
3. Case Study
In this research, four seawalls with sinusoidal, logarithmic, parabolic, and sloping faces were studied. The values of wave height and depth of water used in modeling are presented in Table 2.
Based on Table 2, a solitary wave was modeled with five different heights: 15, 17.5, 20, 22.5, and 25 cm. Also, a constant water depth in front of the seawalls was modeled with five different heights: 35, 40, 45, 50, and 55 cm. The curvatures of sinusoidal, logarithmic, and parabolic walls were modeled with the powers of 2, 3, 5, and 8 and for the sloped seawall, the gradient was 1 : 1. In Figure 2, a 2D view of sloped and curved walls is presented. In FLOW-3D® software, in Cartesian coordinates, there are six distinct levels to define boundary conditions. Considering the positive direction of the axes, these levels include z-up, z-down, y-back, y-front, x-right, and x-left. These boundary conditions are defined for each computational block. The boundary conditions used in this research are specified in Table 3. A curved seawall with meshes and boundary conditions is shown in Figure 3.
For 2D modeling in this paper, only x-right, x-left, z-up, and z-down are considered. The initial water body was set in a static state. As for the boundary conditions, the tangent shearing stress of the free surface was set as zero; the normal stress was in equilibrium with the atmospheric pressure, and all of the solid surfaces were treated using the no-slip boundary condition. The variation of the turbulent energy and the turbulent dissipation on the free surface boundary was set to zero in the normal direction. In FLOW-3D® software, the solitary wave is modeled using proper input for its generation. Figure 4 shows these inputs.
To define a solitary wave at the mesh boundary, in the wave attributes area, wave height was characterized. The solution of the solitary wave is derived for an infinite reservoir with a flat bottom, which is assumed to exist outside the computational domain and immediately adjacent to the mesh boundary. The mean fluid depth defines the undisturbed fluid depth in the reservoir. It defaults to the difference between the fluid elevation (i.e., the z-coordinate of the undisturbed surface) and the minimum z-coordinate of the computational domain. Users need to define the mean fluid depth if it is different from its default value. The solitary wave is initially located outside the computational domain. By default, the initial distance from the crest to the wave boundary is one half of the wavelength. The wavelength (L) of a solitary wave is the extent of the wave in its propagation direction. It is measured between two points at opposite sides of the crest where the surface displacement is 1% of the wave height. If that distance is different from its default value, we can specify it in its input box. Based on Figure 4 and Table 2, the values of wave height and mean fluid depth were determined as input to the software. The distance from solitary wave crests to the boundary and current velocities parameters has no numeric value.
4. Model Validation
Synolakis  generated the solitary waves in a laboratory flume. He measured wave run-up R at a plane beach of slope 1 in 19.85 for relative wave height H/h0 ranging 0.005–0.633, in which H is the offshore wave height and h0 is the constant offshore depth beyond the plane beach. Based on his work, the recorded run-up values tended towards two distinct asymptotic forms depending upon whether the solitary wave had broken or not, given bywhere α and β are empirical coefficients. The values of these coefficients are α = (11.0, 1.12) and β = (1.22, 0.59) for the lower and upper asymptotes, respectively. In this paper, these experimental data were used to verify the numerical simulation of solitary waves. The experimental results from Synolakis’s study have been widely used in simulation verification, e.g., Lynett et al. . The referenced experimental water depth (ho) for the simulation was 0.21 m; the ratio of the wave height to water depth H/ho was 0.28, and the beach slope was 1/20. To verify the numerical model, in Figure 5 normalized values of 1 : 1 sloping seawall with 0.015 × 0.015 mesh sizes were compared with the laboratory work of Hall and Watts . Lower and upper asymptotes modified by Borthwick et al. [52, 53] based on the outputs of Synolakis  were also plotted in relevant diagrams. The existence of a good agreement between numerical and experimental results is evident. The error in the same ratio of H/d is in the range of 2% to 4%.
Before setting up the model runs, it is required to perform a sensitivity analysis to determine the optimum number of meshes to minimize the computational costs and time. Therefore, a concave seawall with a wave height of 0.30 m and a water depth of 0.35 m was investigated for different mesh sizes. The results are presented in Figure 6. It is evident from this figure that for mesh sizes less than 0.015 × 0.015, the wave’s run-up on this seawall, remains nearly constant. Therefore, all the models in this study were analyzed with 0.015 × 0.015 mesh sizes.
Figures 7show the simulation results of the wave run-up on Figure 8 slope. The circular points and solid lines in this figure represent the experimental data from Synolakis  and numerical simulation results respectively. A comparison among the waveform variations demonstrates that the numerical accuracy could be obtained by using the computational mesh of 0.015 m (x) × 0.015 m (z). The existence of a good agreement between experimental results of Sylonakis data and numerical modeling by using the FVM-VOF method is evident. The amount of error is in a range of 3% to 6%.
To verify the accuracy of the proposed numerical scheme, the computer code FLOW-3D® was applied to reproduce the proposed formula provided by Munk . The propagation of a solitary wave with an incident wave height of H = 0.25 m on a plane beach model of concave seawall with a radius of 1 meter was investigated. The still water depth of the seawall was d = 0.35 m. The solitary wave profile generated by the FVM-VOF method agrees well with the theoretical solution. The effective wavelength of the solitary wave is about 6 m when evaluated by (11). Figure 9 shows a comparison of the numerical results by using the computational mesh of 0.015 m × 0.015 m and the theoretical data of the free surface profile at 1.5 seconds after solitary wave generation. Solid lines are numerical results (FVM method), and symbols are theoretical data . The existence of a good agreement between results is evident as the error for the peak of the solitary wave is in a range of 5% to 7%.
The number of models analyzed in this study for each type of wall was 16. So, for four seawalls a total number of 208 models were investigated based on Table 2. In theory, a solitary wavelength (L) is infinite. However, in practice, a limited L is needed. For measurement of solitary wave L in a range of 3 to 8 m is about 85% wavelength. It is calculated as follows in which H and h0 are the wave height and water depth respectively:.
Figures 10 shows the comparison between the wave reflection Figure 11 coefficient, which is the ratio of the height of the reflected wave to the incident wave (Cr), and the wave run-up (R) on walls with sinusoidal and sloped front face functions, respectively. It can be seen that for this type of wall, with increasing (H/h0), Cr decreases. The average Cr on parabolic walls is about 18.4% higher than that of the sloping walls and can be expressed as a power function by Y = AXB. The coefficients range of this function for walls with all powers are A = 0.47 ∼ 0.61 and B = −0.37 ∼ −0.46 and the R2 coefficients for these functions are 0.94. However, R increases by increasing (H/h0). The average R on parabolic walls is about 4.5% higher than that of the sloping walls and can be expressed as a power function by Y = AXB. The coefficients range of this function for walls with all powers are A = 3.54 ∼ 3.84 and B = 0.60 ∼ 0.69 and the R2 coefficients for these functions are 0.98. For parabolic function front faces wall, the average amount of Cr and R to the power of 2 and 3 is the lowest and highest respectively and by increasing the power, these amounts are increased.
The incident and corresponding reflected wave profiles from parabolic and sloped walls for (H/h0) in the range of 0.27 to 0.71 were investigated. The results show that this kind of wall with the powers of 2, 3, 5, and 8 reflected the wave as much as 34%, 18%, 26%, and 32% less than the incident wave, respectively. Also, the comparison of this type of wall to the powers of 2, 3, 5, and 8 with sloping walls was reduced by 7%, 12%, 15%, and 9%, respectively. In Figure 12 for H/h0 = 0.71, the incident wave and the corresponding reflected waves are shown.
Figures 13 shows these parameters on the sinusoidal Figure 14function front faces and sloped walls. It can be seen that for this type of wall, with increasing (H/h0), Cr decreases. The average Cr on sinusoidal walls is about 7.7% higher than that of the sloping walls and it can be expressed as a power function by Y = AXB. The coefficients range of this function for walls with all powers are A = 0.39 ∼ 0.47 and B = −0.46 ∼ −0.54 and the R2 coefficients for these functions are 0.94. However, R increases by increasing (H/h0). The average R on sinusoidal walls is about 5.1% higher than that of the sloping walls and can be expressed as a power function by Y = AXB. The coefficients range of this function for walls with all powers is A = 3.52 ∼ 3.78 and B = 0.58 ∼ 0.65 and the R2 coefficients for these functions are 0.98. For sinusoidal function front faces wall, the average amount of Cr and R to the power of 2 and 3 is the lowest and highest respectively and by increasing the power, these amounts are increased.
The incident and corresponding reflected wave profiles from sinusoidal and sloped walls for (H/h0) in the range from 0.27 to 0.71 were investigated. The results show that this kind of wall to the powers of 2, 3, 5, and 8 reflected the wave as much as 40%, 28%, 36%, and 40% less than the incident wave, respectively. Also, the comparison of this type of wall to the powers of 2, 3, 5, and 8 with sloping walls was reduced by 1%, 13%, 5%, and 1%, respectively. In Figure 15 for H/h0 = 0.71, the incident wave and the corresponding reflected waves are shown.
Figures 16 show these parameters on the logarithmic Figure 17function front faces and sloped walls. It can be seen that for this type of wall, with increasing (H/h0), Cr decreases. The average Cr on logarithmic walls is about 5.1% lower than that of the sloping walls and can be expressed as a power function by Y = AXB. The coefficients range of this function for walls with all powers are A = 0.25 ∼ 0.44 and B = −0.51 ∼ −0.84 and the R2 coefficients for these functions are 0.94. However, R increases by increasing (H/h0). The average R on logarithmic walls is about 6.4% higher than that of the sloping walls and can be expressed as a power function by Y = AXB. The coefficients range of this function for walls with all powers is A = 3.51 ∼ 3.87 and B = 0.56 ∼ 0.65 and the R2 coefficients for these functions are 0.99. For the logarithmic function front faces wall, the average amount of Cr and R to the power of 2 and 3 is the lowest and highest respectively and by increasing the power, these amounts are increased.
The incident and corresponding reflected wave profiles from logarithmic and sloped walls for (H/h0) in the range from 0.27 to 0.71 were investigated. The results show that this kind of wall with the powers of 2, 3, 5, and 8 reflected the wave as much as 51%, 34%, 42%, and 48% less than the incident wave, respectively. Also, the comparison of this type of wall to the powers of 2, 5, and 8 with sloping walls was increased by 11%, 1%, and 7% and for the wall with the power of 3 was reduced by 7%, respectively. In Figure 18 for H/h0 = 0.71, the incident wave and the corresponding reflected waves are shown.
The normalized wave run-up (R/h0) versus normalized wave height (H/h0) for parabolic, sinusoidal, and logarithmic walls is shown in Figure 19. It was found that with increasing the ratio Figure 18 of H/h0, R/h0 also increases. These values can be represented by a power function of Y = AXB. The values of the coefficients for parabolic, sinusoidal, and logarithmic walls Figure 20 for this function Figure 21 are (A = 3.54 ∼ 3.84, B = 0.60 ∼ 0.69), (A = 3.52 ∼ 3.79, B = 0.58 ∼ 0.65), and (A = 3.51 ∼ 3.88, B = 0.57 ∼ 0.65), respectively. The average value R/h0 against H/h0 for the parabolic walls with the powers of 2, 3, 5, and 8 was about 3%, 4%, 5%, and 5% higher than the sloped wall, respectively, for sinusoidal wall with the powers of 2, 3, 5, and 8 respectively was about 4%, 4%, 5%, and 7%, and for logarithmic wall with the powers of 2, 3, 5, and 8 respectively was about 5%, 6%, 7%, and 9% higher than that of the sloped wall.
In the present study, a numerical FVM-VOF model was developed to study the interaction of solitary waves with curved and sloped seawalls. The RANS model was used to consider turbulent flow. 208 models of seawalls with parabolic, sinusoidal, logarithmic, and sloped faces were analyzed. These analyses were examined for the scaled model solitary wavelength in the range from 3 to 8 m.
In light of the most important results of this research, the following conclusions can be drawn:(i)As the wavelength increases, the wave reflection is increased with oscillation nature for curved faces and sloped walls as a power function.(ii)Among all models, parabolic walls with the power of 3 had the highest amount of wave reflection and logarithmic walls with the power of 2 had the lowest one.(iii)As the wavelength increases, the wave run-up is decreased as a power function.(iv)Among all models, logarithmic walls with the power of 8 had the highest wave run-up and parabolic walls with the power of 2 had the lowest one.(v)As normalized wave height (H/h0) increases, normalized run-up (R/h0) also increases as a power function for curved face walls and sloped ones.(vi)Numerical results show that the parabolic seawalls have the highest amount of wave reflection and turbulence on the seaside.
List of Notations
|:||Fractional area open to the flow|
|:||Turbulence constant (dimensionless)|
|:||Turbulence constant (dimensionless)|
|:||Turbulence constant (dimensionless)|
|:||Turbulence constant (dimensionless)|
|:||Turbulence constant (dimensionless)|
|:||Turbulence constant (dimensionless)|
|:||Turbulence constant (dimensionless)|
|:||Turbulent kinetic energy|
|:||Coefficient of determination|
|:||Reynolds stress term|
|:||Instantaneous velocity in direction x|
|:||Instantaneous velocity in direction z|
|:||Fractional volume open to the flow|
|:||Effective viscous vorticity|
|:||Kinematic (laminar) viscosity|
|:||Free surface elevation|
|:||Kronecker delta function|
|:||Turbulence Reynold’s stress|
|:||Turbulent energy dissipation|
|:||Ensemble-averaged or time-averaged properties.|
The data used and/or analyzed are not available.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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