## Abstract

Experiments are the traditional techniques used in coastal engineering to study complex wave structure interactions. However, with the advent of high-performance computing, even performing 1:1 scale numerical simulations has become a reality. The progress aids in extending the parametric investigation or repeating the procedure for comparable structures. In this study, a numerical model in OpenFOAM^{®} with waves2Foam wave boundary conditions is used to simulate wave structure interactions at seawalls with varied geometrical configurations of recurved parapets. The numerical model is validated by employing ForschungsZentrum Küste (FZK)'s large-scale (1:1) experiments. The validated model is then applied to the plain parapet and vertical wall to understand better overtopping behaviour, pressure distribution, and structural loads. Numerical modelling is used in this study to visualise and assess intrinsic parameters such as the velocity profile, vorticity, air entrapment, and entrainment better to understand the dissipation characteristics of seawalls with recurved parapets.

## HIGHLIGHTS

For the first time, on a large scale, an OpenFOAM

^{®}-based numerical model is applied considering breaking wave loads.A Reynolds-averaged Navier–Stokes equations solver (interFoam) in OpenFOAM

^{®}with waves2Foam is employed.A detailed analysis of impact loads at seawalls retrofitted with different types of parapets is made.

Simulated velocity profile, vorticity, and impact pressure distribution are compared.

## INTRODUCTION

*et al.*2020, 2021; Nazarnia

*et al.*2020; Safari Ghaleh

*et al.*2021; Truong

*et al.*2021). With the increased threat of climate change and associated sea-level rise, extreme climatic events such as wave-induced overtopping are expected to be more acute in nearshore areas due to the reduction of crest freeboard of existing sea defences (IPCC 2018; O'Sullivan

*et al.*2020; Salauddin & Pearson 2020). A parapet is an attachment placed on top of the vertical seawall as a retrofitting element in order to mitigate wave overtopping characteristics over the structure. Different types of parapets, such as plain (Ravindar & Sriram 2021) and recurved (Ravindar

*et al.*2018, 2019, 2021; Stagonas

*et al.*2020), as shown in Figure 1, are employed and categorised based on different combinations of overhang length, exit angle, and curvature. When attached to a vertical seawall, a parapet reduces wave overtopping by deflecting the uprushing water in the seaward direction with their exit angle.

Much research effort has been made to investigate the efficiency of such retrofitting interventions in reducing wave overtopping at seawalls (Kortenhaus *et al.* 2003; Pearson *et al.* 2004; Dong *et al.* 2018, 2020a, 2020b, 2021a; Martinelli *et al.* 2018), producing empirical prediction guidance for the estimation of overtopping at seawalls with parapets. In addition to overtopping characteristics, previous research also focused on wave impact pressures at seawalls with recurve retrofitting (Kisacik *et al.* 2012; Castellino *et al.* 2018; Ravindar *et al.* 2018, 2019, 2021; Stagonas *et al.* 2020; Dong *et al.* 2021b; Ravindar & Sriram 2021; Salauddin *et al.* 2021).

However, the scaled model experiments to estimate wave impact pressure are always associated with inseparable scale effects created due to limitations in Froude scaling (Ravindar *et al.* 2021). For this reason, the large-scale experiments which are performed in 1:1 are popular due to their reliable results. However, the large-scale facilities are limited and expensive; and carrying out a detailed parametric study is not always feasible. Particularly with the advancement in computational modelling with high-power machines, the numerical model could be trained and validated with experiments (Imanian & Mohammadian 2019; Salih *et al.* 2019; Yang *et al.* 2019; Shamshirband *et al.* 2020; Safari Ghaleh *et al.* 2021). Therein, the numerical model would be capacitated to run the parametric study or test various similar structures.

Nevertheless, comprehensive numerical studies devoted to understanding wave impact characteristics on parapets in breaking wave conditions are still very limited. The existing works of literature are reviewed to gain from lessons learnt.

Numerical modelling of breaking wave interaction on vertical structures is a challenging task due to the air–water mixture's involvement associated with air entrapment and air entrainment. The hydrodynamic flow induces the change in the ambient pressure, which leads to the compression and expansion of the air pocket and its closure of it. Earlier works on modelling wave impact pressures are carried out based on the pressure–impulse theory by Cooker & Peregrine (1991, 1995) and an incompressible potential flow theory by Zhang *et al.* (1996). The numerical simulation on the arc crown wall using the Volume of Fluid (VOF) based on the Body-Fitted Co-ordinate (BFC) was used to simulate the wave hydrodynamic characteristics and compared with the experiments (Li *et al.* 2011). Then, a major breakthrough in estimating the aeration effects involved in breaking wave impact on the vertical wall was performed by Bredmose *et al.* (2009) by the combination of the water phase using the potential flow theory and the air phase using a weakly compressible flow model. The flow details, however, could not be captured accurately due to the assumptions about the potential flow. Because turbulence effects were not taken into account, the broken wave could not be replicated using their numerical model. This phenomenon was verified by Plumerault *et al.* (2012), as well as Ma *et al.* (2014), who implemented Navier–Stokes-based two-phase compressible models, which demonstrated good accuracy for shock wave propagation and breaking (Plumerault *et al.* 2012). Wemmenhove *et al.* (2015) extended the idea by simulating single-phase (water) and two-phase (incompressible water and compressible air) models and compared them with experiments. The comparison revealed that the single-phase model considerably overstated the pressures, and two-phase modelling and compressibility effects could improve in wave loading simulation. However, no numerical work has comprehensively reproduced all of the breaking situations of wave–wall structure interactions (Liu *et al.* 2019), to the best of the author's knowledge.

Liu *et al.* (2019) recently used a two-dimensional (2D) computational fluid dynamics (CFD) model to predict various breaking wave impacts on the vertical seawall. To account for the abrupt shift in fluid characteristics and the surge in compressibility over the free surface, the Ghost Fluid Method was used. The vertical wall was subjected to four distinct types of breaking wave impacts. The wave elevation, pressure, and force distribution were compared between numerical and experimental data. For all impact types, good agreement was recorded between models and experiments (see Liu *et al.* 2019). Because wave impacts on a vertical wall are very sensitive to breaking wave height, wave profiles, and breaking point, Liu *et al.* (2019) advised prioritising the precise prediction of waveshapes in computations of seawall's breaking wave impacts.

As evident from past studies, numerical investigations of wave impacts on plain vertical seawalls are complicated, which get exacerbated by the addition of a parapet onto the structure due to further interactions between the deflected water from the parapet and incoming waves. To the best of the authors’ knowledge, only a few numerical investigations studied the wave impact pressure and forces on recurved parapets for breaking wave conditions.

Mamak and Guzel (2013) used the boundary element method for the governing (Reynolds-averaged Navier–Stokes (RANS)) equation's numerical solution. The application of the pressure–impulse theory to curved seawalls was a unique component of this work. The results indicated the possibility of a pressure–impulse model to predict wave impact pressure distribution on curved seawall models accurately. Furthermore, Castellino *et al.* (2018) performed numerical modelling on recurved parapets under non-breaking waves using IHFOAM with the *k*–*ε* turbulence model. IHFOAM is an OpenFOAM^{®}-based three-dimensional (3D) numerical solver (two-phase) developed by IHCantabria for CFD applications.

Castellino *et al.* (2018) showed that a 90° recurved wall might create a considerably high-impact pressure peak compared to other angles. They also stated that the recurved curvature of the wall eliminates the possibility of trapped air in the phenomena; hence, water may be considered to be incompressible. They also claimed that in real-sea conditions, not eliminating entrained air in the fluid generates water compressibility, which the current numerical technique does not address. When considering non-breaking wave circumstances, however, water compressibility is unlikely to have a substantial influence.

Recently, Molines *et al.* (2020) performed comprehensive numerical modelling on wave hydrodynamics on rubble mound breakwaters with different geometries of parapets using OpenFOAM^{®} and validated with model scale experiments. A non-dimensionalised estimator is proposed for calculating the mean overtopping discharge for parapets. Molines *et al.* (2020) compared their results without using a turbulence model, presuming that an appropriate characterisation of the hydrodynamic variables can be obtained without using a turbulence model (Jacobsen *et al.* 2018).

Although plenty of research works have been performed and published on numerical modelling of wave structure interaction processes at plain vertical seawalls, clearly, there is still a lack of literature about numerical modelling of breaking wave impacts on vertical seawalls when retrofitted with parapets.

The unique selling point of the study is that it is one of the foremost large-scale numerical studies which is performed on a 1:1 scale to model the breaking wave impact on recurved parapets. The objective of the study is to develop, train, and validate the large-scale numerical model and further use of the validated model to test for other similar structures.

The paper is organised as follows: The details of experiments like setup, instrumentation, and wave test matrix are provided in Section 2.1, followed by the numerical model and methodology adopted in Section 2.2. Then, in the results and discussion (Section 3), validation of the numerical model is provided by comparing it with the experimental measurements for three predominant breaking types (Ravindar *et al.* 2019) (Section 3.1). The validation is further extended to compare the similarities and differences between three types of recurved parapets (Section 3.2) to study the intrinsic numerical parameters like velocity, vorticity, and pressure distribution. Later, the model is extended to study the dissipation characteristics of recurved, plain parapets and vertical walls. Finally, the salient features, limitations and future recommendations are provided in the conclusion.

## METHODS

### Large-scale laboratory experiments

*et al.*2018, 2019, 2021; Stagonas

*et al.*2020). The large-scale model experiment was conducted in Large Wave Flume (Große Wellenkanal, GWK), in the Coastal Research Centre (ForschungsZentrum Küste, FZK), Hannover, Germany. The flume is 330 m long, 7 m deep, and 5 m high, as shown in Figure 2. Since the material used in the lab construction differs from that used in the field, the word quasi is used.

Three types of recurved parapets were investigated for varying curvature: large (BrL), medium (BrM), and small (BrS) (see Figure 3). The wave return wall is defined by three parameters according to the EurOtop manual (Van der Meer *et al.*): the horizontal distance (*B _{r}*), vertical distance (

*H*) of the upper edge of the seaward to the end of recurved and exit angle. Since the vertical distance (

_{r}*H*) varies for various parapet models, the total height of the model (

_{r}*H*) varies as well. The following were the measurements of the three recurved parapets studied in the large-scale analysis:

_{m}Small (BrS):

*B*= 0.20 m,_{r}*H*= 0.45 m,_{r}*H*= 5.14 m and_{m}*α*= 48°_{e}Medium (BrM):

*B*= 0.40 m,_{r}*H*= 0.57 m,_{r}*H*= 5.26 m and_{m}*α*= 70°_{e}Large (BrL):

*B*= 0.61 m,_{r}*H*= 0.61 m,_{r}*H*= 5.30 m and_{m}*α*= 90°_{e}

The experimental setup was composed of a structure part that includes a recurved parapet mounted on a vertical wall and slope, as shown in Figure 2. The structure was considered a gravity-based rigid system. The parapet was made of steel, and the vertical wall on the seaward side was made of steel covered with Perspex plates for a smooth surface. As shown in Figure 2, the recurved parapet setup is a modular design in which several panels were added to increase the model size from small to large. The flume has a flat bottom and a 1:10 approaching slope that leads to the sea wall's toe.

Sl. no. . | Wave gauges . | x (m)
. | y (m)
. |
---|---|---|---|

1 | WG1 | 50 | 0.25 |

2 | WG2 | 51.9 | 0.25 |

3 | WG3 | 55.2 | 0.25 |

4 | WG4 | 60 | 0.25 |

5 | WG5 | 160 | 0.25 |

6 | WG6 | 161.9 | 0.25 |

7 | WG7 | 165.02 | 0.25 |

8 | WG8 | 170 | 0.25 |

9 | WG9 | 200 | 0.25 |

10 | WG10 | 210 | 0.25 |

11 | WG11 | 220 | 0.25 |

12 | WG12 | 235 | 0.25 |

13 | WG_{bridge} | 3.65 | 0 |

14 | WG_{wavemaker} | 0 | 0 |

Sl. no. . | Wave gauges . | x (m)
. | y (m)
. |
---|---|---|---|

1 | WG1 | 50 | 0.25 |

2 | WG2 | 51.9 | 0.25 |

3 | WG3 | 55.2 | 0.25 |

4 | WG4 | 60 | 0.25 |

5 | WG5 | 160 | 0.25 |

6 | WG6 | 161.9 | 0.25 |

7 | WG7 | 165.02 | 0.25 |

8 | WG8 | 170 | 0.25 |

9 | WG9 | 200 | 0.25 |

10 | WG10 | 210 | 0.25 |

11 | WG11 | 220 | 0.25 |

12 | WG12 | 235 | 0.25 |

13 | WG_{bridge} | 3.65 | 0 |

14 | WG_{wavemaker} | 0 | 0 |

Sl. No. . | Pressure transducer . | x (m)
. | y (m)
. | z (m)
. |
---|---|---|---|---|

1 | PT1 | 243 | 0 | 3.63 |

2 | PT2 | 243 | 0 | 3.96 |

3 | PT3 | 243 | 0 | 4.21 |

4 | PT4 | 243 | 0 | 4.36 |

5 | PT5 | 243 | 0 | 4.51 |

6 | PT6 | 243 | 0 | 4.66 |

7 | PT7 | 243 | 0 | 4.81 |

8 | PT8 | 243 | 0 | 4.96 |

9 | PT9 | 243 | 0 | 5.22 |

10 | PT10 | 243 | 0 | 5.32 |

11 | PT11 | 242.9 | 0 | 5.41 |

12 | PT12 | 242.8 | 0 | 5.48 |

13 | PT13 | 242.7 | 0 | 5.56 |

14 | PT14 | 242.6 | 0 | 5.61 |

15 | PT15 | 242.5 | 0 | 5.65 |

16 | PT16 | 242.4 | 0 | 5.67 |

Sl. No. . | Pressure transducer . | x (m)
. | y (m)
. | z (m)
. |
---|---|---|---|---|

1 | PT1 | 243 | 0 | 3.63 |

2 | PT2 | 243 | 0 | 3.96 |

3 | PT3 | 243 | 0 | 4.21 |

4 | PT4 | 243 | 0 | 4.36 |

5 | PT5 | 243 | 0 | 4.51 |

6 | PT6 | 243 | 0 | 4.66 |

7 | PT7 | 243 | 0 | 4.81 |

8 | PT8 | 243 | 0 | 4.96 |

9 | PT9 | 243 | 0 | 5.22 |

10 | PT10 | 243 | 0 | 5.32 |

11 | PT11 | 242.9 | 0 | 5.41 |

12 | PT12 | 242.8 | 0 | 5.48 |

13 | PT13 | 242.7 | 0 | 5.56 |

14 | PT14 | 242.6 | 0 | 5.61 |

15 | PT15 | 242.5 | 0 | 5.65 |

16 | PT16 | 242.4 | 0 | 5.67 |

In addition to that, incoming and breaking waves were recorded using two high-speed video cameras. In Figure 2, the locations of cameras 1 and 2 are shown. Cameras 1 and 2 were recorded at 300 and 30 fps, respectively. These cameras were extremely helpful in classifying the breaking wave described and for analysing the deflected form of the wave.

The tests were conducted using monochromatic waves for a water depth of 4.1 m for various wave periods and wave heights. The water depth was set so that the highest impact pressures on the structure were observed. Since waves less than 0.5 m do not have enough effect and waves greater than 0.8 m trigger severe loads on the structure, the wave height was investigated between 0.5 m and 0.8 m. Similarly, a wave period less than 4 s caused standing waves to form, whereas a period of more than 8 s caused severe load on the wall. As a result, the test matrix is formed as shown in Table 3, which includes the test definition, wave height, wave period, and duration of each test. Data logging started moments before the wavemaker's initiation for every test, and it was automatically stopped after the duration of the test had passed. Detailed experimental set up and test procedure were reported by Ravindar *et al.* (2018, 2019) and Stagonas *et al.* (2020).

Description . | Wave height (m) . | Wave period (s) . | Duration (s) . |
---|---|---|---|

H05T8 | 0.5 | 8 | 800 |

H06T6 | 0.6 | 6 | 1250 |

H06T8 | 0.6 | 8 | 1250 |

H07T4 | 0.7 | 4 | 1250 |

H07T6 | 0.7 | 6 | 1250 |

H07T8 | 0.7 | 8 | 1250 |

Description . | Wave height (m) . | Wave period (s) . | Duration (s) . |
---|---|---|---|

H05T8 | 0.5 | 8 | 800 |

H06T6 | 0.6 | 6 | 1250 |

H06T8 | 0.6 | 8 | 1250 |

H07T4 | 0.7 | 4 | 1250 |

H07T6 | 0.7 | 6 | 1250 |

H07T8 | 0.7 | 8 | 1250 |

### Numerical modelling

#### Governing equations

^{®}to solve the fluid (air, water) flow, which is written in tensor notation as:where and are the instantaneous velocities in the

*x*and

_{i}*x*directions, respectively,

_{j}*p*is the pressure, and

*ρ*are the fluid dynamic viscosity and density, respectively. The external force applied to the fluid is denoted by

*f*, the gravitational acceleration

_{i}*g*, and the momentum source term is denoted by

_{i}*S*. To absorb incoming waves,

_{i}*S*creates an artificial damping zone. In the interFoam solver, the finite volume approximation to all gradients appearing in Equation (2) is obtained using the ‘Gauss linear’ scheme which corresponds to second-order central-differencing (CD2). The momentum advection is considered in conservative-form with discretised using the ‘Gauss limitedLinearV’ scheme which is a bounded second-order (flux-limited) approximation. The diffusion term in Equation (2) is also approximated using CD2. The solution is advanced in time using a first-order bounded-implicit scheme. In the interFoam solver, OpenFOAM

_{i}^{®}uses the VOF method which is solved by the Multidimensional Universal Limiter for Explicit Solution (MULES) scheme. To track the fluid-free surface, a phase function ‘

*α*’ which is zero in air and unity in fluid is utilised. This function has a range of zero to unity at the interface:where is the dynamic fluid viscosity. Equation (5) can be used to determine ‘

*α*’ where is the velocity field:

^{®}s default solver interFoam and a Pressure Implicit Split Operations (PISO) algorithm are used to solve the incompressible N–S equations. The PISO algorithm included a time derivation term as well as a coupled pressure–velocity equation. The Courant number in PISO algorithms should not be greater than 1. If it is greater than this, the information will enter neighbour cells within the one-time stage, sacrificing accuracy. The time step is dictated by the Courant number, which is dependent on the mesh size and velocity, as given in Equation (6):where is the smallest cell size in the velocity direction,

*u*is the velocity and denotes the maximum permissible time step subjected to the limiting Courant number of 0.25.

#### Waves2Foam

*et al.*(2018) created the waves2Foam toolbox to create and absorb free surface water waves in the numerical water tank. For wave generation and absorption, the waves2Foam employs the relaxation method, which employs an active sponge layer which is governed by the function (Jacobsen

*et al.*2018):where is either or , is obtained from a suitable wave theory and is the relaxation function given by Jacobsen

*et al.*(2018):

The selection of the exponential variation ensures that rapidly transitions from the targeted to the computed value near the boundary while the transition is more gradual near the edge of the relaxation zone bordering the solution domain to ensure numerical stability. Since the present simulations deal with breaking waves interacting a reflecting ‘seaward’ from a recurved wall, only one relaxation zone is implemented adjacent to the inlet boundary of the domain.

#### Numerical wave tank

To get the right combination for a 2D wave generation scenario, the following procedure is established using the Waves2Foam library, as described in this section. The computational domain for wave generation is divided into three sections: wave generation, simulation, and wave absorption. The size of each area is calculated by a number of factors. The fluid does not have enough volume to build the flow properly if the regions are too small, which has a big impact on the results. When the computational domain is big, it adds to the computational cost and difficulty of managing the mesh and the performance.

*et al.*(2020) show that a sufficiently adequate estimation of bulk hydrodynamic variables may be obtained without the use of a turbulence model, so a similar technique is used in this investigation.

#### Convergence study

Grid resolution

*T*= 6 s and a wave height of

*H*= 0.7 m are generated. For the grid resolution analysis, the relative water depth

*d*/

*L*of 0.12 is considered. The wave elevations from the experimental wave gauges are compared with the numerical results from different grid resolutions, as shown in Figure 6. The initial coarse mesh is selected based on the thumb rule, Δ

*x*=

*L*/100; where

*L*is the wavelength and Δ

*x*is the distance of the smallest cell and Δ

*t*=

*T*/6000; where

*T*denotes the wave period, and Δ

*t*denotes the minimum time step. Then, for the test case H07T6BrL:

*L*

*=*35.132 m; Δ

*x*

*=*0.351 m;

*T*= 6 s; Δ

*t*

*=*0.001 s is chosen. Later, the number of grid cells was doubled based on reducing half the element size, such as 0.35 m into 0.17, 0.085, and 0.0425 m, which led to the number of cells as 1 × 10

^{4}cells, 4 × 10

^{4}cells, 1.5 × 10

^{5}cells and 6 × 10

^{5}cells, respectively.

When looking at the results in Figure 6, it is clear that the results begin to converge after the case with 4 × 10^{4} cells. The case with 1.5 × 10^{5} cells and 6 × 10^{5} cells, on the other hand, tends to replicate findings that are similar to those obtained in experiments. The grid of 1.5 × 10^{5} cells was chosen for the current study due to the computational time. As a consequence, a uniform linear grid with element sizes of 0.085 m is used. In the vicinity of the seawall structure, the mesh is refined three times. Near the seawall structure, the most refined grid is Δ*x* = 0.028 m and Δ*y* = 0.028 m, while near the boundary, the most coarse grid is Δ*x* = 0.11 m and Δ*y* = 0.11 m. Table 4 contains information about the grid resolution.

Time resolution

Grid description . | Element size, Δx
. | Number of nodes . | Number of elements . | Refinement level . |
---|---|---|---|---|

Coarse | L/100 = 0.35 | 10,041 | 9540 | 3 |

Medium | L/200 = 0.17 | 40,255 | 39,231 | 3 |

Fine | L/400 = 0.085 | 157,267 | 155,253 | 3 |

Very fine | L/800 = 0.0425 | 617,681 | 613,720 | 3 |

Grid description . | Element size, Δx
. | Number of nodes . | Number of elements . | Refinement level . |
---|---|---|---|---|

Coarse | L/100 = 0.35 | 10,041 | 9540 | 3 |

Medium | L/200 = 0.17 | 40,255 | 39,231 | 3 |

Fine | L/400 = 0.085 | 157,267 | 155,253 | 3 |

Very fine | L/800 = 0.0425 | 617,681 | 613,720 | 3 |

*x*=

*L*/400 for different time step sizes. The wave profile obtained with spacing Δ

*t*=

*T*/6000 is very similar to the wave profile with spacing Δ

*t*=

*T*⁄12,000. Hence, Δ

*t*=

*T⁄*6000 is considered for further simulations. Thus, wave profiles at Δ

*x*=

*L*⁄400 obtained with Δ

*t*=

*T⁄*6000 are found to be good enough and were used in the numerical investigation.

## RESULTS AND DISCUSSION

In this section, the calibrated numerical model is validated using large-scale experiments. The validation is performed for three breaking scenarios to verify the model similar to Liu *et al.* (2019). After the validation, the model is extended to investigate the dissipation characteristics of the recurved parapet, plain parapet, and vertical wall.

### Wave-breaking characteristics

*et al.*(2007). As illustrated in Figure 8, in order to find the shape of the wave that causes the largest impact, the test cases are classified into three breaker kinds in which shock impact pressures would occur. According to Oumeraci

*et al*. (1993) and Kisacik

*et al.*(2012), the three types of breakers discovered in this study are slightly breaking waves (SBW), breaking waves with small air trap (BWSAT), and breaking waves with large air trap (BWLAT). In this section, the results of the large recurved parapet (BrL) are used as a typical reference for the present classification of breakers, and the classification applies to small and medium as well.

The experimental analysis (Ravindar *et al.* 2018; Stagonas *et al.* 2020; Ravindar & Sriram 2021) identified two critical areas in the structure and compared them to the numerical results instead of all the points. Those two crucial points are (i) near to still water level and (ii) edge of the recurved parapet. In the first zone near the still water level, the incoming wave hits the structure, and this is termed as ‘Impact zone’. In the second zone, near the edge of the recurved parapet, the natural flow of uprushing is restricted by the curvature of the recurved parapet. This zone is termed an ‘uprushing zone’.

#### Slightly breaking waves

This type of breaker tends to break by upward sloshing movement and may occur at or near the wall boundary. The water level gently rises and reaches the wall before the wave crest, avoiding interaction with the structure. As a result, no air entrainment is observed, and water simply deflects in the shape of the recurve. The deflected water jet from the vertical section collides with the recurved parapet, causing a significant impact in the zone where the water jet impacts the recurve. In this situation, the vertical wall is only subjected to a steady pulsing load and a curved portion with minimal impact pressure. This type was observed in case 4 with all the parapets, i.e. only till it impacts the vertical wall; after that, due to the difference in curvature, the deformation will be different, which will be discussed in a later section.

*η*

_{i}) and 235 m from wavemaker (

*η*

_{toe}) (i.e. 83 and 8 m from the structure) are selected for the comparison because the numerical simulation is performed in a domain truncated up to a distance of 100 m from the structure. However, the entire physical phenomenon is captured in the domain, which is considered for the simulation. Figure 10 shows the wave elevation before the slope (

*η*

_{i}) and near the structure (

*η*

_{toe}).

The numerical and experimental results of incident (*η*_{i}) wave elevation agree well. The wave elevation near the structure (*η*_{toe}) has a slight variation in the initial duration, and after stabilisation the profile compares well. The discrepancy in the initial duration is due to the different methods of wave generation from the laboratory and the time taken to stabilise the interaction between reflected and incoming waves, which is predominant in the shorter wave period case. The only difference in wave generation between numerical and physical modelling is the type of waves used. In the experiment, a modified trochoidal wave theory is used, whereas in the numerical model Stokes's second-order theory is used as input. The differences are limited only to the initial transient waves, and the full developed waves have minimal effect.

Furthermore, in the dynamic impact part of the vertical wall, the quasi-static pressure due to the rising water can be observed, which is absent in the impact on recurve. After the wave reaches the curved geometry, due to the high momentum from the wave than the gravity, the dynamic pressure would be predominant, and the contact duration during the rising time will be higher than during impact. This was exactly noticed in pressure sensors 11–16, as shown in Figure 12.

#### Breaking waves with small air trap

*η*

_{i}) and near the structure (

*η*

_{toe}) for the H07T6 test case. For wave elevation before the slope (

*η*

_{i}) in Figure 14, the overall profile matches well, but in the initial three wave peaks, slight reflection is observed in troughs of experiment results. This is later self-corrected by the active absorption with the increase in wave height during upcoming wave peaks. Considering wave elevation near the structure (

*η*

_{toe}) in Figure 14, the experimental and numerical results compare well, capturing the reflected and incident part. This is further verified by the amplitude spectrum at the incident and near the structure location.

*et al.*(2012) indicated with respect to the turbulent bore by Goda & Haranaka (1967) and by Oumeraci

*et al.*(1993). The double peak observed may be due to the split-up of entrapped air into bubbles in the free surface that deflect through the boundary of the recurve with splashes, as shown in Figure 16.

#### Breaking waves with large air trap

*η*

_{i}) and near the structure (

*η*

_{toe}) for the H07T8 test case. The influence of the reflected wave is not observed in numerical results, and this can also be observed in wave elevation at incident location (

*η*

_{i}) as shown in Figure 11, where the reflection part in troughs of experimental results is not observed in numerical results. The elevation disparities between computational and experimental results in troughs might be attributed to complex 3D effects in wave breaking wherein turbulence is important, which is obviously a shortcoming of the 2D model utilised in this work. This disparity may be due to the complex 3D effect of lateral sloshing, i.e. the displaced water curled from the recurved parapet interaction with the side walls of the flume. This effect is found in waves with higher wave periods. However, after stabilisation, the numerical wave tank reciprocates the wave profile, which is clearly seen in wave elevation near the structure (

*η*

_{toe}) in Figure 18. The small variations are unavoidable due to the breaking and reflections in the experiments considering these complex phenomena.

*et al.*2007). Furthermore, when the wave rises up, secondary quasi-static pressure is more dominant than the dynamic pressure, indicating that the wave collapses on the structure, thus losing the energy in it and the water mass rises and curls over the recurve. The pressure sensor on the curved parapet is not having any impact pressure that was recorded in SBW and BWSAT, confirming that the energy is lost during the initial impact itself, as shown in Figure 20.

### Comparison of wave elevation and impact pressure

The wave elevation is compared between numerical and experimental results in Figures 10, 14, and 18. The comparison is shown for three prominent breaking cases, SBW corresponding to the test case H07T4, BWSAT corresponding to the test case H07T6 and BWLAT corresponding to the test case H07T8. These three cases are chosen to show the variation in pressure profile for the wave height *H* = 0.7 m with different wave periods *T* = 4, 6, and 8 s.

Based on Figure 21, at the incident location, the numerical simulations are efficient in recreating similar incident wave conditions and the error percentage ranges from 1 to 7%. These error percentages are independent of test parameters and recurved parapet type. The reason which contributes to this is the difference in troughs, mostly due to the different types of wave generations, modified trochoidal theory in experiments, whereas in numerical model, Stokes second-order theory is used as an input.

At the toe location, the error percentage varies from 1 to 19%. The maximum error percentages are found in the H07T4 test case for all recurved parapet types. The reason is that in the case of wave period *T* = 4 s, the transition of reflection characteristics requires approximately three to four wave cycles to stabilise and form repetitive patterns compared to longer periods such as *T* = 6 and 8 s. These transitional wave cycles are slightly varied between numerical and experiments. In other words, the fully reflecting structures similar to the recurved parapet used in the study needs few wave cycles (approximately three to four for *T* = 4 s and reduced with an increase in wave period) to stabilise and form the repetitive pattern between incoming and deflected wave from the parapet. Hence, shorter waves like *T* = 4 s naturally have more transient waves and error percentage contributes due to this difference.

Secondly, higher error ratios occur in large recurved parapet types (3–19%) compared to medium (3–19%) and small (1–16%). The trend of error ratios is in the order of large (90°) > medium (70°) > small (48°) recurved parapets and the reason may be due to an increase in complex interactions between the incident and deflected wave with the increase in recurved exit angle. With the increase in exit angle, the uprushed water falls back closer and flatter; thereby, higher complex interactions are formed. These results clearly indicate that wave parameters and recurved parapet types contribute to the differences in numerical and experiment results. These are similar to findings reported in other literatures (Ravindar *et al.* 2018, 2019; Stagonas *et al.* 2020; Ravindar & Sriram 2021).

where *p _{n}*(

*t*) denotes the experimentally determined instantaneous impact pressure at the location of the

*n*th pressure transducer, Δ

*z*denotes the distance between two sensors, and

_{n}*m*denotes the number of the pressure transducer.

*F*) and experiment (

_{num}*F*) are presented in quartile representation, indicating the median (the halfway point), the 25th and 75th percentiles of force (bottom and top edges of the box) are shown in Figure 23. The dotted lines extend to the maximum and minimum points. The Thumb rule to understand the representation is when the boxes move above the mean line (=1), numerical results under-predict experimental results and vice-versa.

_{expt}From Figure 23, the overall error percentage based on the mean is in the order of large (3–27%) > medium (2–18%) > small (3–12%) recurve parapet. As mentioned in the wave conditions comparison, with an increase in recurve exit angle from 45 to 90°, the complex interactions are created as observed in breaking sequence figures and these interactions reduce the efficiency of the numerical model as it involves air–water mixture and is difficult to recreate numerically as the air is modelled as incompressible in the present simulation results. Regarding test cases, the largest error percentage occurs in the H07T4 case due to differences in stabilisation and reflection characteristics.

*et al.*(2020). The hypothesis is that the recurved parapets have the capacity to alter incoming wave conditions, thereby reducing the overall force on the structure. Secondly, this phenomenon increases with an increase in recurved exit angle. However, it is difficult to understand the hypothesis using physical model experiments; hence the phenomenon is studied using numerical modelling. For example, the uprushing patterns for three recurved parapets are shown in Figure 24, which is helpful in visually observing the angle of uprushing but does not provide details about the velocity or air entrainment. This discrepancy is rectified by using numerical modelling.

^{26}–27 show the wave-breaking process for small (BrS), medium (BrM), and large (BrL) recurved parapets. The abovementioned figures show water surface elevation, velocity profile, and vorticity distribution for respective scenarios. For uniformity, the same ranges are selected.

The commonality among all three recurve parapets is that as the wave approaches the structure, shoaling occurs due to the reduction in water depth, and when critical conditions are reached, the wave begins to break; however, due to the presence of the structure, the wave breaks on the structure with air entrapment. After the wave breaks onto the structure, the jet is formed and uprushes in a trajectory based on the exit angle of the recurved type both in the vertical and horizontal direction. The maximum vertical movement is observed in small recurved parapet types followed by medium and large recurved parapets. This trend is due to the sharp exit angle of 45°, and with an increase in angle, the vertical movement reduces and shifts to increased horizontal movement. The maximum horizontal movement is found in large recurved parapets, followed by the medium and small recurved parapets. The reason is in the large recurved parapet, and the uprushed jet is diverted smoothly from vertical to horizontal direction due to the 90° curvature angle.

*z*/

*d*= 0.965 (ref. Figure 13), and for the BWLAT case is lowered to

*z*/

*d*= 0.885 (ref. Figure 17), which is quantitatively shown in Figure 28 discussing the vertical distribution of impact pressure for different parapets. Furthermore, the amount of air entrained due to breaking waves is also different.

The pattern of wave breaking is heavily influenced by the interaction between the reflected and incoming waves. When the reflected wave and incoming wave interact, there are two possible interferences, either positive or negative interference. Based on the interference, the breaking wave height either increases or decreases. Depending on the breaking wave height, the amount of water drawn back to feed the wave is affected, which determines the location and form of air entrapment and entrainment.

Air entrainment, or free surface aeration, is defined as the entrainment/entrapment of un-dissolved air bubbles and air pockets that are carried away within the flowing fluid. Due to negative interference happening in the large recurved parapet, air mixing happens before the wave reaches the structure, which can be observed in Figure 27.

In small and medium recurved parapets, air entrainment is found near to the structures; contrastingly, in the large recurved parapet, entrained air is mixed in incoming waves. These may be due to the turbulent mixing during interference between incoming and deflected waves, which will be investigated in future studies.

The change in velocity fields due to different uprushing patterns of recurved parapets are shown in the second column of Figures 25,^{26}–27. Initially, the velocity field moves with the wavefront, and when the wavefront overturns and hits the structures, large velocities are formed due to the escape of entrapped air. This phenomenon is common for all recurved parapet types and is verified by the experimental results from Chang & Liu (1998) and Khalifehei *et al.* (2021) and numerical investigation by Xie (2010).

The third column of Figures 25,^{26}–27 shows the vorticity evolution during wave breaking for the three recurved parapet combinations. As the breaking wave propagates from the left to right direction towards the structure, counter-clockwise vortices (i.e. positive vortices) are formed, which move along with the crest. Also, after wave impacts on the structure, a vortex pair containing both clockwise and counter-clockwise vortices are ejected as the jet follows the trajectories of recurved parapets. In summary, three vortices are formed during the uprushing process: (i) incident plunging jet, (ii) reverse flow under the jet, and (iii) due to reflected jet (Xie 2010). The vortices formation is similar for all three parapets.

The variations of impact pressure along the *z*-direction for different parapets corresponding to different breaking conditions are shown in Figure 28. In Figure 28, pressure corresponds to the mean of the local maximum, which is non-dimensionalised using density (*ρ*), acceleration due to gravity (*g*) and incident wave height (*H _{i}*). The vertical locations are non-dimensionalised by water depth (

*d*). A total of 12 pressure locations are plotted for all recurved parapets, and in addition, four extra pressure locations are shown for large recurved parapets.

The variability is displayed for various wave heights and wave periods, which correspond to three breaking scenarios. In each breaking case, all three parapet types are plotted by circle, square, and pentagon markers corresponding to small, medium, and large recurved parapets, respectively. For SBW breaking cases, there is no sufficient impact on the structure in the H05T8 case compared to other cases. The highest impact pressures are found in impulsive loading conditions like BWSAT and BWLAT. Among breaking cases, maximum is observed in H07T6 > H06T6 > H07T8 > H06T8 > H07T4 > H05T8. Among recurved parapets, the maximum impact pressure is large > medium > small recurved parapets. A similar observation of maximum pressure is found in a 90° recurved parapet in other numerical studies (Castellino *et al.* 2018; Molines *et al.* 2020).

### Dissipation characteristics of recurved, plain parapet, and vertical wall

Figure 30 shows the pre-breaking, breaking, and post-breaking scenario of waves on the recurved, plain parapet, and vertical wall for the H07T8 case. The H07T8 case is chosen as a typical example, and the patterns are the same for all the test cases. In Figure 30, the time evolution of the free surface (visualised in the *x*–*z* plane) in the vicinity of the structure is shown in the *x*–*z* plane to compare the free surface elevation generated in the plain, recurved parapets and vertical wall. Based on the one-to-one comparison in Figure 30, it is evident till wave breaking that there is no difference between the parapets, but after impact, the uprushed water changes its course of movement based on the parapet. In the vertical wall, a portion of uprushed water overtops but in the plain and recurved parapet diverted uprushed water moves along the exit angle of the parapet. Exit angle determines the direction of water movement after uprushing.

^{32}–33.

Figures 31,^{32}–33 show mean pressure (average of all the pressure peaks pertaining to a location) comparison between recurved parapet, plain parapet and vertical wall for large (BrL), medium (BrM) and small (BrS) cases. Two key aspects which should be focused on the pressure distribution are the impact location (the location where the maximum peak occurs) in the vertical part (<*z*/*d* = 1.1) and the recurved part (>*z*/*d* = 1.1). In the vertical part, the maximum occurs near the still water level, either slightly above or below, which is common in all three model types (recurved, plain, and vertical wall) and three sizes (large, medium, and small). However, in the parapet, there is a difference in the impact location. In the plain parapet, the maximum pressure for the plain parapet is found to be localised at the *z*/*d* = 1.2 for all three cases. The significance of the location is the first point of contact in the triangular portion of the plain parapet, which restricts the natural uprushing motion of water, whereas in the recurved parapet, the maximum pressure location varies from *z*/*d* = 1.2 to 1.45. In a vertical wall, there is no significant impact pressure at the parapet part as the uprushed water is not restricted, and in a few cases, slight pressure is observed due to the impact of the downfall of uprushed water due to gravity. It is to be emphasised from Figures 31,^{32}–33 that irrespective of the overhang length and exit angle, the maximum pressure at the parapet occurs at the point of contact where the natural flow of uprushed water is altered by the parapet, which can also be termed as a hindrance point.

In correlating the pressure distribution with the dissipation pattern of the plain, recurved parapet, and a vertical wall, the following patterns are inferred. In the vertical wall, the uprushed water rises and falls vertically near the structure, so there is very minimal interaction with the incoming waves. In the plain parapet, after the wave hits the triangular portion, the uprushed water moves in an inclined direction, dissipating the energy by splitting into wave particles, whereas in the recurved parapet, the uprushed water is curled back towards the incoming wave, thereby having maximum interaction with incoming waves among the three model types.

Overall, considering the dissipation and pressure distribution, in the cases of complete prevention of overtopping, including the water particles, the recurved parapet is suitable; however, a plain parapet could be used in all other cases because it attracts lower loads on the structure and provides substantial overtopping.

## SUMMARY AND CONCLUSIONS

In this study, 1:1 quasi-prototype recurved seawall experiments are numerically simulated using OpenFOAM^{®}, an Eulerian mesh-based open-source code. The main objective of the study to develop, train, and validate the numerical model to study the breaking wave impacts on the coastal structure on a large scale is achieved by numerically simulating 1:1 quasi-prototype recurved seawall experiments from FZK using OpenFOAM^{®}, an Eulerian mesh-based open-source code. The discrepancy of testing the limited number of model and test cases in large-scale testing facilities due to expensive nature and time restrictions is surpassed by extending the numerical model to test plain parapets and vertical walls.

The numerical model is validated by comparing the wave elevation and impact pressure between numerical simulation and experimental measurements for three breaking scenarios, including SBW, BWSAT, and BWLAT. The major deviation is reported in estimating the case with a relatively lower wave period, where the deviation increases with the increase in recurved exit angle.

The validated numerical model is used to simulate the breaking wave impacts on the plain parapet and vertical wall, which helped in investigating the intrinsic dissipation characteristics. Overall, among the recurved, plain parapets and a vertical wall, the recurved parapet is suggested for the greater reduction of overtopping and a plain parapet could be used for conditions where dissipation through water droplets is acceptable.

The major advantage of numerical modelling is to visualise the complex wave structure interactions such as air entrapment and entrainment and to understand the variation of water surface, velocity, vorticity, and pressure, enhancing the complete understanding of uprushing characteristics of recurved parapets which were lacking in experiments.

Nevertheless, it is to note that in this study, the numerical simulation is performed using the 2D model, and wave breaking involves complex 3D effects. Future detailed research investigations using a 3D turbulence model would be albeit desirable to study the interaction between different wave-breaking types and different shapes of recurve parapets.

## ACKNOWLEDGEMENT

The authors thank DST-SERB for funding the project through the Extramural Research grant. The authors would also like to thank Dr. Stefan Schimmels and Dr. Dimitris Stagonas for the many discussions on the large scale experiments carried out at GWK by them.

## FUNDING

This research was part of the IDWavImp (Improved Design Perspective for Violent Wave Impacts during Extreme Natural Disaster Events) project, funded by the Science and Engineering Research Board, Department of Science and Technology, grant number 182016003183.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*Tech. Note of the Port and Harbour Res*. Inst, (32), pp. 1–18

*Numerical Modelling of Breaking Waves Under the Influence of Wind*