# Efficiency and accuracy of the domain and functional decomposition strategies for the wave-structure interaction problem

Sithik Aliyar, Guillaume Ducrozet, Benjamin Bouscasse, V. Sriram, Pierre Ferrant

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To cite this version:

Sithik Aliyar, Guillaume Ducrozet, Benjamin Bouscasse, V. Sriram, Pierre Ferrant. Efficiency and accuracy of the domain and functional decomposition strategies for the wave-structure interaction problem. Ocean Engineering, 2022, 266, pp.112568. 10.1016/j.oceaneng.2022.112568. hal-04491023

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# Efficiency and accuracy of the domain and functional decomposition strategies for the wave-structure interaction problem

Sithik Aliyar<sup>a,b</sup>, Guillaume Ducrozet<sup>b</sup>, Benjamin Bouscasse<sup>b</sup>, Sriram Venkatachalam<sup>a</sup> and Pierre Ferrant<sup>b</sup>

## ARTICLE INFO

**Keywords:**

Numerical modelling

CFD

OpenFOAM

nonlinear wave structure interactions

focusing wave

SWENSE

## ABSTRACT

Extreme waves can be modelled in a numerical wave tank (NWT) as a focusing wave in order to generate events that are crucial for designing any offshore structures. The focused wave generation implemented in the completely nonlinear potential solver Higher Order Spectral method (HOS-NWT) with the time-reversal technique was used in the present work. The OpenFOAM based solvers *foamStar* and *foamStarSWENSE* which use the domain decomposition and the functional decomposition approach, respectively, were used to couple with a HOS-NWT to perform the focusing wave interaction with the structure (in this case, a cylinder). The incident waves from HOS-NWT are blended into a *foamStar* and *foamStarSWENSE* subdomain, where the waves will propagate and interact with the structure within their domain. In the *SWENSE* methodology, the total field in the computational domain is divided into the incident field and a complementary field. *foamStar* solves for the total flow fields, and *foamStarSWENSE* solves for the complementary (total minus incident) flow fields. The Volume of Fluid (VoF) method was selected to capture the interface. 2D parametric studies for focused wave generation were carried out with both solvers, and 3D wave interactions for different mesh types (coarse, medium, and fine) were investigated to understand the convergence for such transient type problems. For verification, the case's uncertainty was quantified using the Richardson extrapolation approach, and validation was then done by comparing the results to the experiment. The computational efficiency of the two coupling techniques was also compared, and recommendations for solver improvements are made. In general, when solving the problem of wave generation and propagation in 2D and 3D conditions, *foamStarSWENSE* surpassed *foamStar* in accuracy. On the other hand, when the simulation included a structure in the domain, the accuracy of the solution in *foamStar* and *foamStarSWENSE* was nearly the same, but the *foamStar* performed better in terms of computational cost. Overall, both coupling approaches generate a realistic depiction of focused wave interaction and force over the cylinder at a minimum computational cost.

## 1. Introduction

Coastal and offshore engineering problems are generally solved using empirical formulations, physical model tests, and numerical methods. Numerical approaches are preferred because they can address issues that have no empirical formulations and are often less expensive than model tests. The governing equations that are used to categorize numerical models are FNPF (Fully Nonlinear Potential Flow models), CFD (Navier Stokes (NS) Computational Fluid Dynamics model), NLSW (Nonlinear Shallow Water equations models) and BT (Boussinesq-type models). CFD and FNPF models are the most robust among the models described for the wide range of applications. The NS-based CFD model can be solved by the conventional finite volume method (Jacobsen et al. (2012)), finite difference method (Hur and Mizutani (2003)) or meshless particle-based methods (Lind et al. (2012); Rijas et al. (2019)). In the context of wave-structure Interaction (WSI), air entrapment, viscous and turbulent effects mostly matter only in the close vicinity of the body. CFD models (Devolder et al. (2017); Kim et al. (2005a)) can accurately capture these effects, but the computational cost is significant due to the high spatial and temporal resolution requirement throughout the computational grid and the need for longer domains for certain applications (Vandebeek et al. (2018); Yan et al. (2019); Ransley et al. (2020)). When wave propagation over long distances or viscous effects around the structure can be

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<footnote><sup>a</sup>Dept of Ocean engineering, IIT Madras, Chennai, Tamil Nadu, India-600036</footnote>

<footnote><sup>b</sup>École Centrale de Nantes, LHEEA Lab. (ECN and CNRS), Nantes, France-44321</footnote>

<footnote>* This document is the results of the research project funded by the Joint doctoral fellowship by IITM-ECN</footnote>

<footnote>Sithik.Aliyar@ec-nantes.fr (S. Aliyar); guillaume.ducrozet@ec-nantes.fr (G. Ducrozet); benjamin.bouscasse@ec-nantes.fr (B. Bouscasse); vsriram@iitm.ac.in (S. Venkatachalam); pierre.ferrant@ec-nantes.fr (P. Ferrant)</footnote>

<footnote>ORCID(s): 0000-0003-0188-4613 (S. Aliyar)</footnote>
ignored, FNPF models are routinely used to solve the problems. FNPF models can be solved using the boundary element method (Grilli et al. (1989)), finite element technique (Wu and Hu (2004); Yan and Ma (2007); Sriram et al. (2006)), finite difference method (Engsig-Karup et al. (2009, 2012); Bihs et al. (2020)), finite volume method (Lin et al. (2021)), spectral element method (Engsig-Karup et al. (2016)), and high order spectral method (Ducrozet et al. (2012b)). In comparison to CFD models, these models have proved to require less computational effort at a given level of accuracy (Nematbakhsh et al. (2015)). Düz et al. (2016) analyzed the capability of different potential models to couple with viscous solvers and concluded that none of them was superior to others. In recent years, considerable effort has gone into developing near- and far-field coupled hybrid models that effectively with overcome the shortcomings of both the CFD and the FNPF models. The goal is to use CFD wave models near the body and FNPF wave models far away from the body where viscous effects are minimal (indeed, the main objective of the FNPF model is wave generation). The scientific community has developed both meshless (Sriram et al. (2014); Fourtakas et al. (2018); Yan and Ma (2017)) and mesh-based NS models (Campana and Lalli (1992); Colicchio et al. (2006); Jacobsen et al. (2012)) coupled with FNPF over the years. See Sriram and Ma (2021) for a detailed discussion of multiple hybrid coupled models developed for WSI problems.

These FNPF-CFD coupling models are classified based on domain decomposition (DD) and functional decomposition (FD) strategies. The DD strategy divides the computational domain into parts and applies different mathematical models in each part. In most DD-based WSI problems, the computational domain is decomposed into a viscous inner sub-domain and a potential outer sub-domain. The information (velocity, pressure and surface elevation) will be transferred through either relaxation zones or a sharp interface. Jacobsen et al. (2012) introduced relaxation zones on coupling boundaries to gradually blend the information between the two models. In these zones, the weight function is applied for the smooth transition of flow quantities. Higuera et al. (2013) proposed a sharp interface instead of relaxation zones, which is computationally advantageous, hence advantageous of the reduction in the computational domain, but it has been reported that there is a high chance of reflection. Also, based on how the information between the solvers is transferred, either one-way coupling (weak coupling) or two-way coupling (strong coupling) can be used. In one way coupling (Guignard et al. (1999); Lachaume et al. (2003); Grilli et al. (2004); Christensen et al. (2009); Jacobsen et al. (2012); Hildebrandt et al. (2013); Higuera et al. (2013); Paulsen et al. (2014); Monroy et al. (2016); Choi et al. (2020)), information is transferred only from the potential solver into the viscous solver, but in two way coupling (Tahara et al. (1992); Campana and Lalli (1992); Chen and Lee (1999); Guillerm (2001); Iafrati and Campana (2003); Colicchio et al. (2006); Hamilton and Yeung (2011); Sriram et al. (2014); Fredriksen (2015); Lu et al. (2016); Siddiqui (2017); Verbrugghe et al. (2018); Agarwal et al. (2021b)), the information is transferred both ways, from potential to viscous and from viscous to potential solvers. Two-way coupling is beneficial since the size of the computational domain of the viscous solver can be drastically reduced. However, it requires iterations between the two models on the common interface. The advantage of one-way coupling is that no such iterations are needed, but it needs a longer viscous domain to avoid the reflection from outer boundaries. *foamStar* (Choi et al. (2020)), a DD-based one-way coupled solver developed by Bureau Veritas and Ecole Centrale Nantes with the background of interDymFoam from the OpenFOAM 5.0 package, is one of the solvers used for the present investigation. Choi et al. (2018b); Kim (2021a) can be referred to developments and validations of the solver.

The fundamental concept of FD is Dommermuth (1993) proposal to use the Helmholtz decomposition to separate the velocity field into the rotational and irrotational parts to investigate the free surface flow. There are two categories under this decomposition, based on whether the structure is considered in the potential solver or not. In the first category, the WSI problem is divided into a potential part and a viscous part: the entire problem is first solved by a potential solver and then corrected by adding the viscous correction (Kim et al. (2005a,b); Edmund et al. (2013); Rosemugury et al. (2016)). One drawback of this strategy is that the potential solver must first solve the entire problem before applying viscosity correction. As a result, challenges such as higher-order waves and complex interactions are still constraints in this classification. In the second category, the total unknown ($\chi$) is decomposed into the incident part ($\chi_I$) and the complementary part ($\chi_C$). Only the incident flow is modelled in the incident part (Wave only), leaving all the interaction with structure calculated by the viscous solver as the complementary part. The common name among researchers for this classification is SWENSE (Spectral Wave Explicit Navier Stokes Equations), proposed by Ferrant et al. (2002) and actively developed by Gentaz (2004); Luquet et al. (2003); Li et al. (2018); Kim (2021a). The NS equation modified into the SWENS equation is solved to yield the complementary fields ($\chi_C$). The advantage of this method is that the wave models directly provide incident wave solutions, minimising the problem's complexity and cost. For a detailed derivation of single-phase and two-phase SWENSE, see Luquet et al. (2007) and Li et al. (2018) respectively. The applications in single-phase SWENSE over the years can be found in Luquet et al. (2007); Monroy
et al. (2010). The proposed two-phase SWENSE method (Li et al. (2021c)) is implemented on top of foamStar and is called foamStarSWENSE. The only difference is that in this solver, the NS equations in foamStar are replaced by SWENSE. Recent developments of foamStarSWENSE can be read as the introduction of irregular waves from the Higher-Order Spectral method (HOS) (Choi et al. (2018a)), and higher-order forces estimation on a vertical cylinder in the circular domain (Li et al. (2018)).

<figure><img src="image_4.png" /><figcaption>**Figure 1:** Schematic representation of DD (top figure) and FD (bottom figure) coupling methods in the numerical wave tank (NWT). *u*, *p*, *η* represent velocity, pressure and surface elevation, respectively and the subscript 'c' represents the complementary field variables</figcaption></figure>

To outline, Figure 1 represents the numerical wave tank (NWT) with the potential and the CFD model along with the relaxation zones. The top figure represents DD one way coupling, and the bottom represents the FD approach without the structure in the potential domain. The potential solution is blended into the CFD domain inside the inlet relaxation zone in the DD. The blended properties are carried forward inside the CFD domain by solving the NS equation. The CFD solution is blended to zero in the outlet relaxation zone to account for one-way coupling. In the FD, potential solutions are available across the domain; therefore, incident potential solutions are taken across the grid and processed at each instant. Inside the computation domain, only complementary fields are solved using SWENSE, and waves diffracted/radiated from the body are suppressed in the relaxation zones.

The cylinder is a commonly adopted foundation form for the oil and gas industries and recently for offshore wind turbines (such as Jacket legs, Jack up rigs, monopiles, SPAR etc.). To evaluate the reliability and durability of offshore structures, one must consider the load acting on the structures during extreme waves, such as focusing waves or freak/rogue waves, which are more frequent than rare (Liu and Pinho (2004)). Although simple regular and irregular waves over a cylinder have been studied for a long time, the kinematics of steep waves over a cylinder is still an active field of research. The focusing wave is widely recognized as a realistic depiction of an extreme wave in both experimental and numerical studies. The linear and second-order wave theories often used in routine design may not be sufficient for a reliable assessment since focusing waves sometimes entail considerable nonlinearity. The impact of extreme waves on structures shows a nonlinearity of higher than 5th order (Zang et al. (2010)), such that wave loads may be amplified by a factor of almost three when a slamming force caused by a focusing wave impact is involved. This interaction between the cylinder and the focusing wave, which has received much attention recently (Sriram et al. (2021c); Hildebrandt and Sriram (2014)), was chosen to validate the hybrid coupled models (DD and FD) in the present study. The OpenFOAM-based solvers foamStar and foamStarSWENSE were applied for the DD and FD approach respectively. Both the solvers were coupled with the potential solver HOS-NWT (Ducrozet et al. (2012b)) to generate focusing waves.
This paper aims to validate the coupling models by comparing the pressure and force over the cylinder with the experimental data. A further aim is to shed light on the accuracy and efficiency of the two approaches along with a convergence study for such transient problems using the Richardson extrapolation technique. This is a numerical analysis technique for estimating the error in the solution by solving the problem with two different grid sizes, provided the functional form of the solution is known (Rao, 2001). The rest of the paper is organized as follows. Section 2 addresses the mathematical formulation implemented in the present work, and Section 3 presents the focusing wave generation and its validation in the two solvers. The following section details the problem formulation for fixed cylinder interaction with focusing waves. In the first part, the convergence study between different mesh types is carried out based on the Richardson extrapolation technique, and then the results are validated with the experiment. Finally, conclusions summarise and compare the efficiency of the two coupling strategies.

## 2. Numerical methods

For the present study, the foamStar solver (based on the DD strategy) and foamStarSWENSE solver (based on the FD strategy) were chosen to compare the DD and FD strategies. The mathematical formulations of each solver are discussed in the following subsections.

### 2.1. Governing equations - foamStar

foamStar (Choi et al. (2020)) is an in-house code co-developed by Bureau Veritas and Ecole Centrale de Nantes for solving WSI problems. It was developed in the framework of the standard multi-phase solver interDymFoam (Rusche (2002); Márquez Damián (2013)) from the CFD package OpenFOAM Version 5.0. The improvements of foamStar over interDymFOAM are its wave generation module and the mechanical solver to solve the dynamics of rigid and flexible bodies in waves. The solver is based on the RANS-CFD model and the governing equations based on incompressible fluid are written as,

$$ \nabla \cdot \mathbf{u} = 0 \qquad (1) $$

$$ \frac{\partial(\rho\mathbf{u})}{\partial t} + \nabla \cdot (\rho\mathbf{u}\mathbf{u}^T) = -\nabla p_d + \nabla \cdot [\mu(\nabla\mathbf{u} + \nabla\mathbf{u}^T)] - (\mathbf{g} \cdot \mathbf{x})\Delta\rho \qquad (2) $$

where $\mathbf{u}, \mathbf{x}$ and $\mathbf{g} = [0, 0, -g]^T$ are the fluid velocity, position and gravitational acceleration vector, respectively. The dynamic pressure $p_d = p - \rho\mathbf{g} \cdot \mathbf{x}$ is as introduced in Rusche (2002). Based on the recommendation of Li et al. (2021a) from a similar problem, the cylinder surface boundary condition and turbulence modelling techniques do not appear to change wave elevations or impact pressure, implying that turbulence effects are minor. Hence the turbulent model was not applied in this study.

### 2.2. Governing equations- foamStarSWENSE

The SWENSE method proposed by Ferrant et al. (2002) defines three primitive fields in a WSI problem, introducing the decomposition as,

$$ \chi = \chi_I + \chi_C \qquad (3) $$

where the total field $\chi$ (pressure, velocity and free surface elevation) is decomposed into the incident field ($\chi_I$) and the complementary field ($\chi_C$). The total field ($\chi$) represents the real flow governed by the incompressible Navier Stokes equation (Eq (1),Eq (2)). The incident field ($\chi_I$) concerns the propagation of incident waves in the computational domain without structures. The incompressible Euler equation governs the incident flow. The complementary field ($\chi_C$) is governed by SWENSE and represents the field generated by the structure and the viscosity of the fluid. Li et al. (2021c) developed a two-phase SWENSE solver named foamStarSWENSE in which the governing equations are derived by subtracting the Euler equations from the NS equations. The final form of SWENSE is given as follows,

$$ \nabla \cdot \mathbf{u}_C = 0 \qquad (4) $$
$$ \frac{\partial \rho \mathbf{u}_C}{\partial t} + \nabla \cdot (\rho \mathbf{u} \mathbf{u}_C) + \rho \mathbf{u}_C \nabla \cdot \mathbf{u}_I = -\nabla p_C + \nabla \cdot (\mu(\nabla \mathbf{u}_C + \nabla \mathbf{u}_C^T)) - \frac{p_I}{\rho_I} \nabla \rho \quad (5) $$

where $\rho = \rho_I$ is the density, $\mu$ represents the dynamic viscosity; $\mathbf{u}_C = \mathbf{u} - \mathbf{u}_I$ and $p_C = p - p_I^*$; subscript C represents complementary field variables and I represents incident field variables and $p_I^*$ as density weighted pressure given as,

$$ p_I^* = \frac{\rho}{\rho_I} p_I \quad (6) $$

Li et al. (2021c) showed that the density weighted pressure is a crucial element in the development of a robust numerical solution and their work can be referred to for further details.

## 2.3. Shared Formulations

The foamStar and foamStarSWENSE solvers share certain similar techniques such as free surface tracking, relaxation zones and incident wave modeling. The common properties and parameters are summarized below.

### 2.3.1. Free surface tracking

The interface between the two immiscible fluids is tracked using the VoF (Volume of Fluid) approach developed by Hirt and Nichols (2021). It is based on a scalar function alpha (in OpenFOAM, it is $\alpha(t, \mathbf{x})$) that represents the volume of the fraction of fluid in the computational cell. In general, the two fluids will be water and air. Hence, for $\alpha$ is one, the control volume (CV) is filled with water, and for $\alpha$ is zero, it is air-filled. A CV is considered on the free surface when $\alpha$ lie between 0 and 1. The transport equation for $\alpha$ is written as

$$ \frac{\partial \alpha}{\partial t} + \nabla \cdot (\alpha \mathbf{u}) + c_{\alpha} \nabla \cdot (\mathbf{u}_n \alpha (1 - \alpha)) = 0 \quad (7) $$

where the third term on the left-hand side is the artificial compression to keep the interface sharp. $\mathbf{u}_n$ is normal fluid velocity to the interface, and $c_{\alpha}$ is the compression coefficient.

$$ \rho = \alpha \rho_w + (1 - \alpha) \rho_a \quad (8) $$

$$ \mu = \alpha \mu_w + (1 - \alpha) \mu_a \quad (9) $$

The mixture density in the governing equation is modified for two-phase based on the VoF scheme as $\rho$ and viscosity $\mu$ with the subscripts a and w representing air and water, respectively.

The VoF field is not decomposed in SWENSE because the complementary field's bounds are difficult to establish, hence the total VoF remains bounded between 0 and 1. The transport equation of VoF in SWENSE is the same as the one mentioned in Eq. (7).

### 2.3.2. Relaxation zones

The simplest method of wave generation in one way DD (foamStar) is to impose wave velocity and free surface elevation from potential wave models at the inlet of the CFD domain. In foamStar, the wave generation and absorption methods are based on an explicit scheme that relaxes the computed solution towards a given target flow field (Jacobsen et al. (2012); Seng (2012); Mayer et al. (1998); Engsig-Karup (2006); Choi et al. (2018a)), and the scheme requires a weight function that varies between 0 and 1 in the relaxation zone. It is achieved as follows:

$$ \zeta = w \zeta_{\text{target}} + (1 - w) \zeta_{\text{CFD}} \quad (10) $$

where $\zeta$ is either the velocity field ($\mathbf{u}$) or the VoF field ($\alpha$), $w$ is the weight function, and $\zeta_{\text{target}}$ will be the given target flow field usually set to be either the incident wavefield or calm-water. Two common choices of weight function are the
polynomial weight function (Mayer et al. (1998); Engsig-Karup (2006); Choi et al. (2018a)) and the exponential weight function (Jacobsen et al. (2012); Seng (2012); Choi et al. (2018a)). The present study adopted the explicit relaxation scheme using the polynomial weight function given as,

$$w(x) = -2x^3 + 3x^2 \qquad (11)$$

here $x \in [0, 1]$ is the normalised coordinate inside the relaxation zone. The above technique can be applied to the wave absorption zone. If the target values ($\zeta_{target}$) are set to no waves, the entire wavefield is damped, and if the target values are set to incident waves, then the scattered waves are damped. Choi et al. (2018a) can be referred to for some examples of 3D wave generation of a realistic sea spectrum.

Similar relaxation zones are also presented in *foamStarSWENSE* with $\zeta$ as a complementary variable. The generation of waves in the SWENSE method does not define specific zones for wave generation. Instead, it imposes the incident wave solution explicitly in the entire computational domain and solves the disturbances of the incident waves as a complementary correction. The Spectral wave models used for wave generation are based either on the stream function theory (Fenton (2014); Ducrozet et al. (2019)) for regular wave propagation or High-Order Spectral methods - HOS (Ducrozet et al. (2012b, 2016a)) for complex sea states. They have been demonstrated to be accurate and efficient in several studies. The advantage of using spectral wave models (HOS-NWT) is that they also help the SWENSE method adjust the zone of the incident wave above the free surface, such that the same functional decomposition can be used in the entire CFD computational domain (Li et al. (2018)).

### 2.3.3. Wave model: HOS-NWT

The wave field is described with a dedicated numerical model based on FNPF. This ensures an accurate description of the wave physics and a reduced computational effort compared to a full RANS description. In the present study, the wave solver is based on the High-Order Spectral (HOS) method (Domermuth and Yue (1987); West et al. (1987)). Thanks to its pseudo-spectral formalism, this approach exhibits interesting convergence properties. Associated with the use of Fast Fourier Transforms (FFTs), HOS models have been shown to be the most efficient wave solvers (Ducrozet et al. (2012a)).

Validation was performed based on comparison with the experiments (see Sec 4.2). The wave field is generated with the wave maker and absorber close to the opposite wall to control incoming waves. Their reproduction is achieved with a dedicated Numerical Wave Tank (NWT) that includes all the features of the physical wave tank. The open-source solver HOS-NWT, Version 1.2, that was developed to this end and has been widely validated on different wave conditions (Ducrozet et al. (2012b); Hasan et al. (2018)) was adopted in the present study.

### 2.3.4. Grid2Grid

For both the DD and FD strategies, incident waves generated from HOS are solved independently, not alongside with *foamStar* and *foamStarSWENSE*. The results of the incident waves from HOS computations are stored for further use. The pseudo-spectral formalism permits the storage of modal coefficients of the quantities of interest that can be reconstructed with a linear combination of basis functions. The *foamStar* and *foamStarSWENSE* solvers then use these modal coefficients to reconstruct the incident wave information on their mesh. The modal amplitudes are firstly transformed into uniformly spaced discretized points by IFFT and then interpolated into the solvers mesh. This interpolation is needed as HOS simulation uses fewer points per peak wavelength than required for CFD solvers. One of the consequences of the interpolated velocity field is that it has a large divergence, which breaks the boundness of the VoF field and causes instability problems, especially in *foamStarSWENSE* (Li et al. (2021c)). This instability can be avoided by using interpolation with the zero-padding technique, i.e. introducing additional zero amplitude modes at the end of the spectrum, adding extra spatial points, making the HOS grid finer at the post-processing stage. With this method, velocity divergence is drastically reduced. The interpolation mentioned above is made available to the public in an open-source Grid2Grid (Choi et al. (2017)). The ability to simulate regular and irregular waves by both *foamStar* and *foamStarSWENSE* methods is tested and reported in (Choi (2019); Li et al. (2021c)).

## 2.4. Solution algorithm

The key steps of the implementation of the above procedure are summarized for both the DD and FD solvers in this subsection. For the graphical description of the algorithm in *foamStar* and *foamStarSWENSE* solver, Figure 4.5 in Kim (2021b) and Figure 13 in Li et al. (2021b) can be referred respectively.
### 2.4.1. foamStar

The foamStar algorithm between each time step is as follows.

* Wave generation and absorption in the relaxation zone are updated based on the weight function Eq. (10)

* In the PIMPLE outerloop, if the body exists, body motion (floating body) is solved and the computational mesh is updated accordingly

* The VoF convection equation (Eq. (7)) is solved with the bounded discretization schemes. It updates the transportation and interface properties based on the updated VoF field and interface

* In the inner PISO loop, the fluid equations Eq.(1), (2)) are solved. Particularly, after the momentum equation (Eq. (2)) is solved, it solves the pressure equation derived from the continuity equation (Eq.(1)) and gives the pressure and the velocity.

* After solving for fluid parameters and VoF ($\alpha$), it is relaxed in the relaxation zone to impose the target values. Thus, the fluid velocity is also relaxed in the relaxation zone at the end of PISO.

* If there is a turbulence model, it will be solved at every PIMPLE iteration or only at the last PIMPLE iteration. This option is controlled using a user selectable option.

* The PIMPLE loop repeats the PIMPLE iteration up to the point when the solver meets the tolerance requirements or until the solver completes a specified number of iterations.

### 2.4.2. foamStarSWENSE

The structure of the foamStarSWENSE solver supports the subsequent algorithm in its conservative form (Li et al. (2021c)).

* At the beginning of each time step, the incident wave properties $u_I$ and $p_I$ are updated from potential wave solvers and mapped on the CFD mesh. The total velocity is reconstructed with $u = u_I + u_C$

* The VoF field (Eq.(7)) is solved and the modified incident wave pressure is updated with new density field Eq.(8) & Eq.(9).

* In the PISO loop, the complementary velocity field is solved from Eq.(5) first (the momentum prediction step). Secondly, the complementary pressure is solved. The pressure field corrects the flux of the complementary velocity. After the correction, the flux is used to reconstruct the complementary velocity field at the cell centre.

* The solution is then blended to the target values in the relaxation zones to attenuate the complementary waves in the far-field Eq.(10).

* In the PIMPLE loop, outer nonlinear iterations are made to achieve convergence of the VoF, the velocity, and the pressure before moving to the next time-step.

## 3. Uncertainty Analysis

The accuracy of numerical results is sensitive to the discretization of the governing equations. Grid generation and the simulation setup, therefore, need to follow established strategies that ensure sufficiently accurate results. The first part of this section introduces and discusses concepts of error estimation based on systematic grid refinement. The second part of the section introduces the quantitative determination of model coefficients by the cross-correlation analysis of time series.
## 3.1. Convergence study

The numerical results of CFD methods are sensitive to both spatial and temporal discretization. In transient 3D problems (focusing wave over the cylinder), the number of unknowns is a function of the grid spacing to the power of four. At the same time, the error is typically a function of the grid spacing to the power of one or two, depending on the order of the approximation schemes (Oberhagemann (2017)). Insufficient grid resolution or poor quality grids may lead to inaccurate results or give wrong tendencies. Even with very fine discretizations, one may still not obtain sufficiently converged, i. e. grid-independent, results, mainly due to the asymptotic accuracy. Beyond a certain level of refinement, further reduction of the discretization error will not be worth the additional computational effort. In the present work, the Richardson extrapolation, a widely accepted approach for uncertainty and error estimation of steady or quasi-steady problems, was applied for this transient problem.

In numerical analysis, the literature shows that there are different procedures to verify and validate the accuracy of the simulations. A laymen approach is to compare the results of simulations with experimental results, and consider that the simulations are valid if they resemble the experiments. A second approach is to use numerical convergence. The procedure is typically performed by refining the mesh until the solution no longer varies and normally a peak magnitude or error over an entire duration will be used to achieve convergence. Finally, some authors combine numerical convergence and validation with experimental results. In this paper, grid convergence was evaluated based on the Richardson extrapolation (Richardson and Gaunt (1927)) method along with verification of the experiment and discussion of its uncertainty.

The Richardson extrapolation method, also known as 'h² extrapolation', is a method for obtaining a higher-order estimate of the continuum value (value at zero grid spacing) from a series of lower-order discrete values. Based on Roache (1997), any quantity from a numerical simulation is expressed as:

$$ \phi = \phi_{exact} + g_1 h + g_2 h^2 + g_3 h^3 + \dots \qquad (12) $$

where h is the grid spacing and gᵢ are the functions defined in the continuum and thus are independent of the grid spacing. φ<sub>exact</sub> is the continuum value at zero grid spacing. Let the mesh types coarse, medium and fine be represented as 1, 2 and 3 respectively.

Let the grid refinement ratio r be written as,

$$ r_{1-2} = \frac{h_1}{h_2}, r_{2-3} = \frac{h_2}{h_3} \qquad (13) $$

Without considering the absence of an odd power in Eq. 12 may be rewritten and generalized to p<sup>th</sup> order methods as:

$$ \phi_{exact} \approx \phi_3 + \frac{\phi_2 - \phi_3}{r^p - 1} \qquad (14) $$

The apparent order of the method and absolute error can be calculated as

$$ p = \frac{1}{\ln(r)} \left| \ln \left| \frac{\epsilon_{32}}{\epsilon_{21}} \right| + q(p) \right| \qquad (15) $$

where ε<sub>32</sub> = φ<sub>3</sub> - φ<sub>2</sub>, ε<sub>21</sub> = φ<sub>2</sub> - φ<sub>1</sub> are the absolute errors of the variable of interest φ obtained with the three different meshes and the function q(p) is defined as

$$ q(p) = \ln \left( \frac{(r_{2-3})^p - s}{(r_{1-2})^p - s} \right) \qquad (16) $$

where s is the parameter related to monotonic or oscillatory behaviour of the solution (Roache (1997)) as the grid is refined and it is calculated as,

$$ s = \frac{\epsilon_{32}/\epsilon_{21}}{|\epsilon_{32}/\epsilon_{21}|} \qquad (17) $$
As can be seen, Eq.(15) and Eq.(16) must be solved iteratively. Once the preceding values are obtained, the extrapolated value of the solution can be obtained as

$$ \phi_{ext}^{2-3} = \frac{(r_{2-3}^p)\phi_3 - \phi_2}{(r_{2-3}^p) - 1} \quad (18) $$

This extrapolated solution ($\phi_{ext}$) will be shown in the later stage of the paper that demonstrates the use of this technique to verify the grid refinement's convergence and establish the procedure for the evaluation of convergence in the Spatio-temporal problem.

## 3.2. Cross Correlation

Because the event being studied is a focusing wave, both the magnitude and phase shift play a vital part in understanding the convergence in this study. In order to account for this, the following Cross Correlation procedure was adopted. The cross-correlation between time series corresponding to the predicted and measured time series, which offers a correlation factor $C$ as a function of a time-lag $\tau$, was used for deterministic comparison. The maximum correlation factor value and its related time-lag can be regarded as the shape and amplitude correspondence of the two time series, as well as an estimate of the time shift between them, respectively. A variant of the correlation function was calculated by sliding the child time window over a parent time window as

$$ C_{xy}(\tau) = \int_{-\infty}^{+\infty} \gamma_x(t)\gamma_y(t+\tau)dt \quad (19) $$

where $\gamma_x(t)$ and $\gamma_y(t)$ represent the discrete-time series for the parent and child time windows, respectively and $\tau$ denotes the delay time. The time range for the summations in the equation is maintained the same, and if the child sampling frequency is of a different length, then cubic interpolation is carried out to match the parent sampling frequency. The advantage of the proposed technique is that besides measuring the similarity between the signals, it can also measure the time shift between them. In the present case, the cross-correlated time series is normalized with an autocorrelation coefficient at zero lag, such as

$$ \overline{C_{xy}(\tau)} = \frac{C_{xy}(\tau)}{\sqrt{C_{xx}(0)C_{xx}(0)}} \quad (20) $$

where $C_{xy}(\tau)$ represents the covariance of two real continuous time series of, say, experiment and any of the numerical time series. and $C_{xx}$ is the sample variance of $\gamma_x(t)$. In other words covariance is normalised by autocorrelation values with respect to $\gamma_x$ at zero time lag. The time series are perfectly matched when the maximum cross-correlation value is one. The difference in value from one, on the other hand, represents the quantification of inaccuracy.

# 4. Focusing wave modeling

The good quality of wavefield reproduction is the prime requirement for any WSI simulations. The foamStar solver, which simulates the wave system, needs fine meshes to propagate the wave accurately. Instead, incident wave propagation in the foamStarSWENSE solver is much easier as wave models explicitly give incident waves (Li et al. (2021c)). This incident focusing wave modelling is intentionally designed to test the stability and accuracy of the solvers. Since both solvers are reliant on HOS for wave generation, the HOS wave is validated first, followed by a discussion of other parameters that are required for both the solvers.

## 4.1. Test case setup

The validation of wave generation and propagation simulations was carried out in 2D NWT to determine the most appropriate parameters for the solvers. In the present work, the experimental results from Sriram et al. (2021a) were used for comparison. The wave considered for validation is a unidirectional focusing wave group consisting of 32
<figure><img src="image_5.png" /><figcaption>**Figure 2:** 2D NWT computational domain for parametric studies with representation of HOS and CFD (NS and SWENSE) domain</figcaption></figure>

wave components. The frequencies of the wave components range from $f_l = 0.34$ Hz to $f_u = 1.02$ Hz. The ratio of bandwidth to the central frequency was 1. Consequently, the computational domain for HOS was made similar to the experimental tank, measuring 2.2m in width and 0.7m water depth. Only the length of the domain was reduced to 50m (in the experiment, it is 110m) as it is already far from the focusing point (25m) and longer than that is less interesting to be examined. The wave absorption in the HOS is provided by a beach that prevents reflection from the opposite wall. The two dimensional CFD domain was $2.75m \times 1.4m$ (length x height) to maintain the corresponding domain for the next wave structure interaction study. Here the CFD represents both the *foamStar* and *foamStarSWENSE* solvers. Other domain parameters are shown in Table 1 and the standard computational domain is shown in Figure 2. The figure shows that in the coupling section, the focusing wave is generated in HOS-NWT and is coupled with CFD NWT using *Grid2Grid*. In the experiment, one wave gauge (WP5) is located 0.57 m in front of the focusing location, the second wave gauge (WP6) is in line with the focusing location, and the third wave gauge (WP7) is located 0.71 m behind the focusing location. A similar setup is maintained in the HOS and both the solvers, as shown in the Figure 2. These wave gauge readings were used to compare and validate the input focusing wave. As in the work of Hildebrandt and Sriram (2014), throughout the study, the time was normalized by the transient zero up crossing wave period ($T = 2.08$ s in this case) measured at the focusing point, while the wave elevations were normalized by the maximum wave amplitude recorded ($\eta_{max}$) at WP6.

## 4.2. Wave reproduction using Time Reversal

The accurate reproduction of the incident wavefield is achieved thanks to the Time-Reversal (TR) methodology applied to the reproduction of waves in water tanks (Chabchoub and Fink (2014); Ducrozet et al. (2016b, 2020)). The method's main idea is to make use of the time-symmetry of the physical laws of conservative systems, such as wave propagation. The TR method results in a simple way to reproduce any temporal free-surface profile at an arbitrary location in a wave tank.
<table><caption>**Table 1**

General properties for numerical computations</caption><thead><tr><th>Items</th><th>Value</th></tr></thead><tbody><tr><td>Depth, d (m)</td><td>0.700</td></tr><tr><td>η<sub>max</sub> (m)</td><td>0.21</td></tr><tr><td>HOS NWT dimensions (m)</td><td>50 X 2.2 X 0.7</td></tr><tr><td>Diameter of Cylinder (D)(m)</td><td>0.22 m</td></tr><tr><td>2D CFD NWT (m) Length X Depth</td><td>2.75 X 1.4</td></tr><tr><td>Freesurface Refinement Zone depth (m)</td><td>0.44</td></tr><tr><td><strong>Wave characteristics</strong></td><td><strong>Value</strong></td></tr><tr><td>Frequency range, Δf (Hz)</td><td>0.34 - 1.02</td></tr><tr><td>Central frequency, f<sub>c</sub> (Hz)</td><td>0.68</td></tr><tr><td>Bandwidth ratio, Δf/f<sub>c</sub> 1</td><td></td></tr><tr><td>Number of components</td><td>32</td></tr><tr><td>Wave steepness</td><td>0.002</td></tr></tbody></table>

This two-step procedure uses a target free surface elevation $\eta_t(t)$ that one wants to reproduce. In the present configuration, the wave-only experimental measurement of the wave gauge at the focusing location was used. The first step uses this temporal elevation to generate the wavefield in the tank, which was measured at the target location $x_t$. The measured signal $\eta_m(t)$ is time-reversed and generated in the same wave tank during a second step. Due to the time-invariance of the governing equations, it is expected that the target free-surface elevation $\eta_t(t)$ was reproduced at the target location $x_t$.

Figure 3 shows the results of the TR validation technique using wave probe WP6 (the focusing point) in comparison to the experiment (Figure 3a), and Figure 3b zooms in on the focusing duration. The accuracy of the principle is fulfilled, but some minor discrepancies were observed which can be attributed to the small breaking occurring in this configuration. However, it is noticeable that there is an excellent agreement between experiments and HOS-NWT simulations. This potential wave generated will be fed into the *foamStar* and *foamStarSWENSE* solvers for their wave requirements.

<figure><img src="image_6.png" /><figcaption>**Figure 3:** Validation of Time Reversal (TR) principle of focusing wave generation with Experimental results</figcaption></figure>

## 4.3. Parametric studies

The parameters identified as important in the present study are: initiation time of simulation, cell size, time step, size of relaxation zones and mesh size outside FS. The computational domain for the parametric study is a two dimensional NWT as illustrated in Figure 2 for the two solvers *foamStar* and *foamStarSWENSE*.

### 4.3.1. Influence of HOS initiation time

The duration of the wave recorded in the reference experiment is 85s, with the focusing time at 39 s (see Figure 3b). However, because the WSI's residual effects beyond the focusing time are of limited interest in this investigation, the HOS duration was set to 50s. Simulating this long duration in CFD would be computationally very costly as the time
<figure><img src="image_7.png" /><figcaption>**Figure 4:** Comparison between focusing wave generated in the *foamStar* and *foamStarSWENSE* domain for different HOS initiation times at WP 6</figcaption></figure>

step requirement is also very fine to capture the focusing wave. It should be noted that focused waves are localized in time, and energetic frequency components are only in the zone of interest after a specific time. Since the CFD domain is small, it is interesting to optimize the initiation time only for the focusing time. Because of the coupling advantage, the simulation can be adjusted to start at a random time in the HOS time scale rather than at 0 because that will be the first timestep in CFD to initialise the domain. To understand its influence, four test cases were considered

* Case 1 - 3s simulation starting at 37.5s to 40.5s

* Case 2 - 3s simulation starting at 39.5 to 42.5s

* Case 3 - 5s simulation starting at 37.5s to 42.5s

* Case 4 - 50s simulation starting at 0s

The wave probe at the focusing location WP6 is compared for these four test cases, and the results are illustrated in Figure 4a for the *foamStar* solver and in Figure 4b for the *foamStarSWENSE* solver. To understand the time influence at the scale of numerical duration itself, the vertical and horizontal axes are preserved as dimensionalized ones. Similarly, vertical scale non-dimensioning is distinct in each case since peaks also vary along with the duration. The results show that the simulation duration has no significant effect, whether it be 50s, 5s, or 3s. But since it is a localised wave, it is important to generate the wave before the focusing point; hence we cannot start the simulation too late (Case 2). Also, no phase difference was observed in any case tested in the two solvers. Based on the findings, a 5s simulation with a start time of 37.5s was chosen for the 3D interaction study, saving a significant amount of computational time.

### 4.3.2. Influence of mesh size and time step

The computational meshes and time steps were examined to understand their influence on the focusing wave generation. Mesh parameters are identified in terms of lateral spacing $\Delta x$, vertical spacing in the free surface zone $\Delta z$ and time step $\Delta t$. With the analytic wave fluid velocities, representative cell-based Courant numbers (Co) and
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 5:** Comparison of focusing wave generated in *foamStar* and *foamStarSWENSE* with HOS wave for different cell based Reynolds numbers and cell based Courant numbers (Re and Co)</figcaption></figure>

<table><thead><tr><th>Test Case Name</th><th>Cell Reynolds number (Re)</th><th>Cell Courant number (Co)</th></tr></thead><tbody><tr><td>fsRe1e5,SwRe1e5</td><td>1.13 10<sup>5</sup></td><td>1 0.75 0.5 0.25 0.1</td></tr><tr><td>fsRe8e4,SwRe8e4</td><td>8.52 10<sup>4</sup></td><td>1 0.75 0.5 0.25 0.1</td></tr><tr><td>fsRe5e4, SwRe5e4</td><td>5.68 10<sup>4</sup></td><td>1 0.75 0.5 0.25 0.1</td></tr><tr><td>fsRe3e4, SwRe3e4</td><td>2.84 10<sup>4</sup></td><td>1 0.75 0.5 0.25 0.1</td></tr><tr><td>fsRe1e4, SwRe1e4</td><td>1.13 10<sup>4</sup></td><td>1 0.75 0.5 0.25 0.1</td></tr></tbody></table>

**Table 2**

Mesh and time discretization based on cell based Re and Co for the convergence tests. fs stands for *foamStar* and Sw stands for *foamStarSWENSE*

cell-based Reynolds numbers ($Re_Δ$) (Choi (2019)) can be defined as

$$Co = \sqrt{Co_x^2 + Co_z^2}, \quad Re_Δ = \sqrt{Re_{Δx}^2 + Re_{Δz}^2} \qquad (21)$$

where

$$Co_x = \frac{u_{wave} \Delta t}{\Delta x}, \quad Co_z = \frac{u_{wave} \Delta t}{\Delta z}, \quad Re_{Δx} = \frac{u_{wave} \Delta x}{\nu}, \quad Re_{Δz} = \frac{u_{wave} \Delta z}{\nu} \qquad (22)$$

$u_{wave}$ and $w_{wave}$ are the maximum horizontal and vertical velocity, in this case at the focusing point it is 1.4 m/s and 0.5 m/s respectively. The test matrix for this study is given in Table 2. The aspect ratio of one was maintained for all the cases and varied the cell size from Re 1.13 10⁴ to 1.13 10⁵. For each Re, different Co are tested for both the *foamStar* and *foamStarSWENSE* solver. The solver's wave probe at the focusing location was compared with the HOS probe at the same location using cross-correlation (Eq.(19)), and the outcome is shown in Figure 5a and Figure 5b for *foamStar* and *foamStarSWENSE* respectively. The first figure illustrates the *foamStar* cross-correlation with HOS, and in almost all the cases, the efficiency of *foamStar* increases when refining the mesh and downsizing the Co. The greatest accuracy is observed for mesh fsRe3e4 and fsRe1e4 in the order of 0.99–0.995. The second figure highlights the *foamStarSWENSE* correlation with HOS for different Re and Co. A coarser mesh in the *foamStarSWENSE* (SwRe8e4, SwRe1e5) has a good accuracy (0.995 – 0.998), matching with the finest mesh in *foamStar*. For a finer mesh, *foamStarSWENSE* converges towards the same value, even for a small timestep. These cases prove the essential advantage of the SWENSE method, i.e., allowing the use of coarse mesh to simulate incident waves: *foamStarSWENSE* solver can use a grid 2–4 times coarser than that of *foamStar* (in each dimension) to achieve the same accuracy for the application of incident waves. The less accurate results seen for fsRe1e5 at a Co of 1 appear to have a 40% amplitude loss at the focusing point, but it appears to fit neatly with the wave in the entire inspection time (total simulation time 5s), therefore taking the value of 0.97. Given the facts, mesh Re5e4 to Re1e4 for cell-based Co less than 0.25 achieves an accuracy better than 0.99 in both solvers and will be studied further in the WSI investigation in the presence of structure.
<figure><img src="image_10.png" /><figcaption>**Figure 6:** Computational domain for inlet and outlet relaxation zones for different lengths based on $\lambda$ (Transient wave length calculated based on transient wave period T)</figcaption></figure>

#### 4.3.3. Influence of Relaxation zone

The inlet relaxation zone blends the velocity and VoF field into the CFD domain from the HOS domain and also absorbs the reflected/scattered waves coming from the body. As described earlier, the philosophy of generating waves is different in *foamStarSWENSE*; subsequently, only the *foamStar* solver was studied for this parameter. Choi et al. (2018b) suggested the length of the inlet ($L_{Inlet}$) zone should be 1-1.5 times the wavelength in the case of regular/irregular waves. This yields 30-40% of the cells in the relaxation zone, which increases the computational cost. If $L_{Inlet}$ can be reduced, it lessens the number of cells and reduces the computational cost. However in cylinder interaction with a focusing wave, only wave group of relatively short time duration will pass through. It is expected that the reflections from the structure will not affect the focusing wave. Hence, the only objective of the inlet zone is to blend the focusing wave properly. A study on different $L_{Inlet}$ keeping the length of the CFD domain and outlet zone constant was carried out (see Figure 7). The $L_{Inlet}$ was measured in terms of transient wave length ($\lambda$) evaluated based on the zero-crossing wave period ($T \approx 2.08$s) using dispersion relation. The zone length varies from $0.04\lambda$ to $0.4\lambda$. The wave probe WP6 comparison for these different cases are shown in Figure 7a. It can be observed that there is only a marginal influence of less than 0.5% variation in amplitude when the inlet length is $0.04\lambda$, and no other discrepancy was observed.

<figure><img src="image_11.png" /><figcaption>**Figure 7:** Comparison between focusing wave generated in *foamStar* for different relaxation zone lengths</figcaption></figure>

The purpose of the outlet relaxation zone is to absorb the input or scatter waves from the body. Similar to the previous study, $L_{Inlet}$ and length of the CFD domain were retained while $L_{Outlet}$ was changed from $0.04\lambda$ to $0.4\lambda$ (Figure 7). The results shown in Figure 7b prove that there is no significant deviation in amplitude and phase and also
<figure><img src="image_12.png" /><figcaption>Figure 8: Comparison between focusing wave generated in foamStar and foamStarSWENSE for different cell size ratios between the zones</figcaption></figure>

that there is no reflection. Hence, short $L_{Inlet}$ and $L_{Outlet}$ of 0.04λ will be used for the subsequent interaction studies, reducing the computational time further.

### 4.3.4. Influence of cells outside the free surface

By altering the cell size at outer zones distant from the free surface zone, the final parametric study aims to investigate the influence of cell size above and below the free surface on wave generation (Figure 2). For this the Cell size ratio was set as,

$$ \text{Cell size ratio (CR)} = \frac{\Delta x_{Out}}{\Delta x_{FS}} $$

where FS represents the free surface zone and Out represents the outer zone. Since the aspect ratio in this study is maintained as 1, cell size in the two directions are the same. For the requirement of a 3D interaction study at a later stage, four test cases with CR ratio's of 1, 2, 5 and 10 were examined. Based on the size of cells in the free surface zone, the width of outer cells was modified as per CR. Like the earlier studies, the wave probe WP6 at the focusing location was chosen to investigate the wave variation for both solvers. Figure 8a and Figure 8b show that there is almost no influence on cell variation above and below the free surface. The results are also correlated with HOS and the outcome is depicted in Figure 9. The solution is more accurate at CR1 than in other situations, but the variation is too small to be a concern. As a result, it is important to highlight that the CR variation is kept between 2 to 10, depending on the mesh type used for the future interaction study. In any case, it has been proved that this will have no effect on the outcome.

<figure><img src="image_13.png" /><figcaption>Figure 9: Cross Correlation between numerical wave based different CR setups and HOS for foamStar and foamStar-SWENSE solvers</figcaption></figure>
<caption>**Table 3**</caption>

Test Matrix and its nomenclature for fixed cylinder interaction with non breaking focusing wave

<table><thead><tr><th>Mesh</th><th>TimeStep (s)</th><th>Co</th><th>TestCase (Nomenclature)</th></tr></thead><tbody><tr><td>Mesh 1 (M1)</td><td>0.0001,0.001, 0.01</td><td>0.003,0.03,0.3</td><td>foamStar- fsCo0.3M1..</td></tr><tr><td>Mesh 2 (M2)</td><td>0.0001,0.001, 0.01</td><td>0.007, 0.07, 0.7</td><td>foamStarSWENSE -SwCo0.3M1.. foamStar- fsCo0.7M2..</td></tr><tr><td>Mesh 3 (M3)</td><td>0.0001,0.001, 0.01</td><td>0.01, 0.1, 1</td><td>foamStarSWENSE -SwCo0.7M2.. foamStar- fsCo0.1M3..</td></tr><tr><td>Mesh 4 (M4)</td><td>0.001, 0.01</td><td>0.1,1</td><td>foamStarSWENSE -SwCo0.1M3 foamStar- fsCo0.1M4.. foamStarSWENSE -SwCo0.1M4</td></tr></tbody></table>

# 5. Focusing wave interaction with fixed cylinder

This section validates the fixed cylinder subjected to the previously validated unidirectional focusing wave. In addition to comparing the wave probe readings, pressure and force across the cylinder are also validated for both solvers. Convergence studies on various mesh setups are also discussed in this section, as well as the efficiency of the solvers.

## 5.1. Test case setup

The NS/SWENSE domain is placed inside the HOS domain (same domain length as in section 4), where viscous effects play a significant role near the structures. The diameter of the cylinder (D) is 0.22m. The length of the CFD domain was set at 2.75m (12D -including 1D for the inlet zone and 1D for the outlet zone, see section 4). The width and water depth of the CFD domain are the same as those of the HOS domain, with the only difference being the air zone above FS of 0.7m. Note that here, CFD represents both the NS and SWENSE domain. The overall computational domain is shown in Figure 10.

<figure><img src="image_14.png" /><figcaption>**Figure 10:** Typical 3D computational domain representation with wave Wave Probes (WP) and Pressure sensors Probes (PP) locations around the cylinder</figcaption></figure>

Recalling the setup, there are seven wave gauges WP1 to WP7, with the first wave gauge placed 4.98m from the wave paddle and the following three wave gauges (WP2-WP4) in the middle from the focusing location. WP5 is located 0.57m in front of the cylinder centre (focusing point), WP6 is located at a point of the tank cross section in line with the centre of the cylinder and WP7 is located 0.71m behind the cylinder. Only three-wave gauges WP5 to WP7 fall in
<figure><img src="image_15.png" /><figcaption>Mesh 1</figcaption></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>Mesh 2</figcaption></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /><figcaption>Mesh 3</figcaption></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /><figcaption>Mesh 4</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Figure 11:** Mesh configuration of three levels Coarse, medium and fine mesh with refinement ratio of 2 for each case and Mesh 4 without free surface refinement to investigate the performance of SWENSE</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>(a) Before interaction</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>(b) During interaction at focusing time</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>(c) After interaction</figcaption></figure>

<caption>**Figure 12:** Typical interaction of focusing wave on the fixed cylinder and its velocity description over the flow field</caption>

the CFD zone. Eight pressure transducers are used, five are located at 0° with equal spacing in the front face, and the remaining three are located at 20°, 90° and 180° near the still water level as shown in Figure 10.

Finally, to perform a convergence study, three different mesh configurations with different discretization levels were generated using the OpenFOAM utility SnappyHexMesh. The grids were generated as an unstructured mesh but control was established over the mesh growth with a refinement ratio of 2. The free surface zone was similar to the results from the parametric studies with three configurations chosen (Re5e4, Re3e4 and Re1e4). The presence of cylinder alters these configurations slightly, while maintaining them at the same level of refinement. These configurations will be represented by Mesh 1 (M1), Mesh 2 (M2) and Mesh 3 (M3) hereafter. An illustration of all these configurations is given in Figure 11. In summary, Mesh 1 (coarse mesh) has 31,248 elements, Mesh 2 (medium mesh) has 226,072 elements, and Mesh 3 (coarse mesh) has 1,751,736 elements. An additional mesh Mesh 4 (based on Re1e5) was made similar to Mesh 3 but without any free surface refinement to investigate the advantage of the SWENSE methodology. The grid size parameter *h* for each mesh case is 40mm, 20mm, and 10mm, and their corresponding grid refinement ratio is 2. The CR ratio is maintained at 2,5,10 for each mesh type respectively (From section 4).
Validation of Potential-Viscous coupling strategies

<figure><img src="image_26.png" /><figcaption>(a) Waveprobe WP5</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>(b) Waveprobe WP6</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>(c) Waveprobe WP7</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>(d) Wave Probe mean error (with respect to the wave probes WP5 - WP7) cross correlation in relation with fsCo0.01M3</figcaption></figure>

<caption>**Figure 13:** Wave probe comparison between different Mesh types M1, M2, M3 and M4 with the *foamStar* solver</caption>

Unlike the previous parametric studies (See 4.3.1) that used cell-based Co, three unique time steps of 0.01s, 0.001s, and 0.0001s were investigated for each mesh to investigate the influence of temporal variation in each solver based on physical Co. This time, however, the timesteps chosen are based on the outcome of cell-based Co with values below 0.25. As mentioned, 'Co' in this study is the physical Courant number assessed based on the maximum velocity of the wave and minimum cell size in the mesh. More details about the test matrix are presented in Table 3. The *foamStar* and *foamStarSWENSE* solvers were tested for each combination of mesh type and Courant number, and the results are reviewed in the following sections.

**5.2. Results and discussion**

This section is split into three parts. The first part compares different mesh combinations with the finest mesh and its verification in order to propose a temporal problem convergence technique. The second part compares experimental results with different mesh type combinations. The final subsection compares the computational efficiency of the two solvers.

5.2.1. *Validation with finest mesh combination - fsCo0.01M3*

A 3D simulation of the interaction between a focused wave and a vertical circular cylinder was performed in the NWT. The interaction modifies the kinematics and the flow field around the cylinder as shown in Figure 12. The figure shows the free surface with velocity magnitude when the focused wave interacts with the cylinder for different time instants for a typical case. First, the focused wave crest interacts with the cylinder causing a rise of water level in front of the cylinder. Thus wave crest velocity increases in the vicinity of the cylinder. Second, the focused wavefront impacts the cylinder, resulting in increased water levels on both sides of the cylinder. On the downstream sides, separate wavefronts formed behind the cylinder rush and collide with each other forming small water jets along the centerline of the cylinder along the wave direction.
Validation of Potential-Viscous coupling strategies

<figure><img src="image_30.png" /><figcaption>(a) PP2</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>(b) PP3</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>(c) PP4</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>(d) PP5</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>(e) PP6</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>(f) PP7</figcaption></figure>

**Figure 14:** Pressure time history of the fixed cylinder in non-breaking focusing waves for different mesh types simulated by the *foamStar* solver

In order to investigate the changes in the free surface, pressure and force during the interaction, the numerical results for different cases are examined. This subsection is divided into two parts. The first part focuses on *foamStar*, and the following part is for *foamStarSWENSE* solver. In each part, along with wave elevation, pressure and force and its correlation along with grid convergence are also discussed. Assuming that the finest mesh M3 with the smallest Co (fsCo0.01M3) tends to be the most accurate, it is considered a reference case for the cross-correlation study with other mesh types, and the accuracy of the assumption is validated in the following subsection.

### *foamStar*

The numerically captured free surface elevation probe readings are shown in Figure 13. Note that only three wave gauges WP5, WP6 and WP7, fall within the computational domain and hence are presented. The wavefront before the focusing point, at the focusing point and after the focusing point is measured by the WP5, WP6 and WP7 probes respectively. For reference purposes, the HOS wave generated in HOS-NWT and experimental results are also shown. Most of the mesh types replicate the focused wave profile precisely, while slight numerical damping is observed only
<figure><img src="image_36.png" /><figcaption>**Figure 15:** Pressure time history of the fixed cylinder in non-breaking focusing waves for different mesh types simulated by the *foamStar* solver for PP8 (left) and mean correlation coefficient for overall PP1-PP8 (right)</figcaption></figure>

<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /><figcaption>**Figure 16:** Force recorded for a fixed cylinder in non-breaking focusing waves by the *foamStar* solver (left) and its correlation coefficient with fsCo0.01M3 (right)</figcaption></figure>

<figure><img src="image_39.png" /></figure>

in M1 and M4. The reason may be that coarser free surface meshes are unable to capture the smaller wavelengths in the wave and leading to slight phase shift across the mesh types. The broken wave recorded in WP7 shows the inefficiency of some test cases to capture the leeward wave well, but only the finer mesh M3 with the finest time step captures all flow field characteristics very well. This is proven in the cross-correlation relation presented in Figure 13d. When the cross-correlation of each wave probe reading is taken, it appears that the differences between the wave probe readings are almost identical. Consequently, the mean of all the coefficients was selected for the sake of concision. It should be noted that the *foamStarSWENSE* evaluation in the following section follows a similar procedure. The plot's X-axis is the computational cost, and it can be seen that the finer mesh has a higher computational cost than the coarser ones. The meshes M1, M2 and M3 with coarser time steps have a similar accuracy of approximately 0.991 but reducing the Co improves their respective accuracies. The increase in accuracy from M1 to M2 is higher than from M2 to M3, showing that the solution is nearly converged. The mesh type M4 corresponds poorly to the focused wave because fewer cells are in the free surface refinement region. However, with a medium time step, this mesh is able to reach 0.99, demonstrating the ability of this one-way DD coupling to construct a decent estimate even with the coarsest mesh. As already mentioned, the only purpose of this mesh is to enable comparison with the performance of the *foamStarSWENSE* solver which will be discussed in later sections.

The ability of the different meshes to reproduce the focused wave pressure on the cylinder is evaluated here by comparing the time variation of the dynamic pressure recorded on the submerged as well as exposed surfaces of the cylinder (Figure 14). The time is normalized by the transient wave period (T), and the dynamic pressures are normalized by $\rho c^2$, where $\rho$ is the water density and $c$ is the maximum fluid velocity at the focusing point (i.e. 1.4 m/s obtained from HOS). The simulated time histories of the submerged pressure probes (PP2-PP3) in stagnation point are expected
<figure><img src="image_40.png" /><figcaption>(a) *foamStar* - Wave probe grid convergence study</figcaption></figure>

<figure><img src="image_41.png" /><figcaption>(b) *foamStar* - Pressure grid convergence study</figcaption></figure>

<figure><img src="image_42.png" /><figcaption>(c) *foamStar* - Force grid convergence study</figcaption></figure>

<caption>**Figure 17:** Extrapolation of coefficient with the Richardson method and its corresponding convergence for *foamStar*</caption>

to follow the wave elevation. Accordingly, all the test cases exhibit similar trends but with similar discrepancies to those observed in the wave elevation study. A pressure drop is observed for the probes around the cylinder (PP6, PP7 and PP8) due to flow acceleration and wake effects. Although the free surface refinement is poor in some cases (M1 and M4), but with very fine refinement around the cylinder, the majority of the test cases correctly reproduce the pressure loading, as shown in Figure 14 and Figure 15. There is a small difference exists in simulated peak pressure in PP4 and PP5 (air probes), which depends only on good wave run-up over the cylinder during the focusing event. Hence the probe PP5 probe was chosen to study the cross correlation estimation in comparison with the fsCo0.01M3 for the convergence behavior. From the figure, it can be observed that decreasing the Courant number increases accuracy for all the mesh types. However, refining from a medium to a fine timestep has only a negligible impact on the pressure estimation. Overall, the accuracy of the results increases when refining the mesh from M1 (0.99) to M3 (0.999) and moves towards convergence.

The force on the cylinder was also investigated. Figure 16 depicts the time history of the force recorded for each mesh type during the focused wave interaction. In the experiment, the cylinder experienced the "ringing" phenomenon after impact, which cannot be simulated numerically as the body is assumed to be rigid. So the time trace is compared only up to impact and the details after the interaction are ignored. The simulated forces are normalized for the convenience of comparison with the factor $ρgr^3$, where r represents the radius of the cylinder. Similar to the wave and pressure results, the test case with less refinement in the free surface exhibits a discrepancy (the force is over predicted) but further refinement in both space and time provided the converged result. The convergence is proved in the cross-correlation (Figure 16b) as accuracy improved for each refinement from 0.997 to 0.999. The mesh type M4 is the least accurate among the mesh types, but its coefficient is 0.992, as shown in the time series. This demonstrates a crucial point: the coarsest mesh with the coarsest time step (with $10^3$ computational time) can predict the force acting on the cylinder with a deviation of less than 10% (based on peak magnitude comparison). This highlights the benefit of DD coupling in overcoming the high computational cost required, if the simulation is solely a CFD problem.

The Richardson extrapolation technique was used to verify the grid convergence and verify the M3 mesh type as a converged configuration. The convergence was estimated based on the cross-correlation coefficient values obtained for each mesh type (Mesh 1 (M1), Mesh 2 (M2) and Mesh 3 (M3)) under the time step of 0.001s (fixing a time step). The convergence study was carried out for all the parameters (surface elevation, pressure and force) and the outcome is presented in Figures 17a, 17b and 17c. The extrapolated coefficient shows that the coefficients values tend to be zero ($h \to 0$), which would correspond to infinite mesh resolution. The extrapolated coefficient from Mesh 2 and Mesh 3
<figure><img src="image_43.png" /><figcaption>(a) Waveprobe WP5</figcaption></figure>

<figure><img src="image_44.png" /><figcaption>(b) Waveprobe WP6</figcaption></figure>

<figure><img src="image_45.png" /><figcaption>(c) Waveprobe WP7</figcaption></figure>

<figure><img src="image_46.png" /><figcaption>(d) Wave Probe mean error cross correlation with respect to fsCo0.01M3</figcaption></figure>

<caption>**Figure 18:** Wave probe comparison between different Mesh types M1, M2, M3 and M4 in the *foamStarSWENSE* solver</caption>

reaches a value of 0.999 for wave elevation. The slope of convergence from Mesh 1 to Mesh 2 looks to be 0.17%, and from Mesh 2 to Mesh 3 it appears to be 0.03% (L1 norm). Similarly, between Mesh 3 and the extrapolated coefficient, the slope is 0.01%. This demonstrates that the solution has already reached convergence. A similar trend is observed for the pressure, and the only exception is that accuracy is slightly better.

Lastly, the extrapolated coefficient for force is estimated at 0.9999. The solution for the force appears to have already converged, since the grid convergence from Mesh 1-2, Mesh 2-3 and Mesh 3-extrapolated coefficient is found to be 0.04%, 0.02% and 0.01% respectively. This observation highlights two interesting findings. One is that even the coarsest mesh has a decent estimation of force and the second is that all the results have already converged and have a smaller slope than other parameters. The results confirm that an accurate enough solution had been already found with the finest mesh, justifying the decision to keep fsCo0.01M3 as a base result for cross-correlation. Because the simulation duration considered for the comparison is not the peak time but the overall duration of 5s, the cross-correlation number may appear much closer to one. Overall, all of the results in this study performed very well, with differences only at the peak crest.

## *foamStarSWENSE*

This section aims to demonstrate the two-phase *foamStarSWENSE* solver's ability for focussing wave interaction with the cylinder. Both the mesh geometry and the wave conditions are analogous to those followed in the *foamStar* simulation. Similarly, surface elevation, pressure, and force are compared among different mesh types to validate the results. The mesh types' results were cross-correlated with fsCo0.01M3 to guarantee a comparison between the solvers. A grid convergence study was also carried out in *foamStarSWENSE* to confirm the convergence of the mesh types. The non-dimensional surface elevation, pressure and force are also identical to those used with *foamStar* solver.

Figure 18a to Figure 18c outline the surface elevation for the wave probes WP5 to WP7, respectively. The mesh types M1 and M4 are numerically damped a little even before to tap the cylinder. This difference is observed everywhere, both during and even after interaction with the cylinder, as shown by WP6 and WP7. To recap, one of the main advantages of *foamStarSWENSE* is the explicit treatment of incident waves; hence, even a coarse mesh in
Validation of Potential-Viscous coupling strategies

<figure><img src="image_47.png" /><figcaption>**Figure 19:** Pressure time history of fixed cylinder in non breaking focusing waves recorded by *foamStarSWENSE* for different mesh types</figcaption></figure>

the far-field can produce the incident wave information accurately. This is visible in the cross-correlation study since compared with the finest mesh in *foamStar*, even the coarsest mesh types M1 and M4 accuracy range is 0.9995 and 0.9985 respectively. This holds good even with the coarser time step (it was 0.984 in *foamStar*). This confirms that with *SWENSE*, the size of the computational domain and number of cells can be drastically reduced for such problems, reducing the computational cost.

Figure 19 represents the dynamic pressure variation on submerged and surface probes for different mesh types in *foamStarSWENSE*. The submerged pressure probes show valid agreement among different mesh types with specific discrepancies coming from waves. Other than PP8, the probes around the cylinder in the wake region show good agreement with each other. The difference arises in PP8, where the formation of water jet occurs as the coarser mesh types under predict the pressure. A similar disagreement arises in the air probes PP4 and PP5, where the coarser
<figure><img src="image_48.png" /></figure>

<figure><img src="image_49.png" /><figcaption>**Figure 20:** Pressure time history comparison (PP8) for fixed cylinder in non breaking focusing waves for different mesh types in *foamStarSWENSE* and mean correlation coefficient for overall PP1-PP8</figcaption></figure>

<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /><figcaption>**Figure 21:** Force comparison for fixed cylinder in non breaking focusing waves between different mesh types under *foamStarSWENSE*</figcaption></figure>

mesh analysis of the pressure were also under predicts pressure slightly near the focusing point. The cross-correlation coefficient of the pressure probe PP5 is plotted against computational cost as shown in Figure 20b. It can be perceived that refining the mesh and time improves accuracy and progress towards convergence. The coarser mesh types have an accuracy range of 0.995. The only exception is mesh M4, which had an accuracy of only 0.985 in *foamStar*, against 0.99 in *foamStarSWENSE*. Another relation for force measured among all mesh types is given in Figure 21a. The simulated results agree well among different test cases, and as mentioned earlier, there are some discrepancies exist for coarser mesh types. Each time series is compared with the force obtained from fsCo0.01M3, and the resulting correlation coefficient is shown in Figure 21b. The *foamStarSWENSE* solver seems to predict the force closely enough even for the coarser mesh types (M1 and M4). The accuracy of the results is in the range of 0.994 -0.998, and the results clearly show that reducing the timestep has very little impact on the accuracy, whereas refining the mesh leads to a more comprehensive improvement in the result. In the Mesh M4, the result shows a decrease in accuracy when the timestep is refined because of the phase shift observed in the result.

Finally, to verify the convergence for this study, the corresponding Richardson extrapolation technique was used. Figure 22 illustrates the grid convergence and the extrapolated solution for wave elevation, pressure and force for the *foamStarSWENSE* cases. In all cases, the extrapolated solution derived using Mesh 2 and Mesh 3 parameters is almost 1. Because of SWENSE's technique, the slope of the convergence is modest in the case of wave elevation. Pressure and force behaviour are shown to have a similar pattern, but pressure is more precise than force. Mesh 1-2 and Mesh2-3 have slopes of 0.14% and 0.013% for pressure and 0.135% and 0.04% for force, respectively. Despite the fact that
the slope in the force parameter is rather high, it is converged with the slope between Mesh 3 and the extrapolated coefficient of 0.015%.

Mesh 1's refinement around the cylinder is coarser, hence its accuracy coefficient with *foamStarSWENSE* is often much lower than with *foamStar* (in *foamStar* 0.999 and *foamStarSWENSE* 0.9968). However, refining a cylinder for Mesh 2 indicates a significant improvement whereas refinement to Mesh 2 in *foamStar* showed only a minor improvement, indicating that it is already close to the converged solution. With Mesh 3, the convergence of *foamStarSWENSE* has been nearly achieved but *foamStar* has already achieved convergence. Only with higher refinement near the cylinder can *foamStarSWENSE* make a reliable force prediction. However, even with the coarser refinement near the cylinder, *foamStar*'s force estimation is fairly accurate. This is illustrated in the next section with a comparison based on computational efficiency.

<figure><img src="image_52.png" /><figcaption>**Figure 22:** Extrapolation of coefficient with the Richardson method and its corresponding convergence for *foamStarSWENSE*</figcaption></figure>

### 5.2.2. Validation with experiments

In this subsection, both the *foamStar* and *foamStarSWENSE* methods are validated by comparing the test cases with the experiment conducted by Sriram et al. (2021a). The cylinder with all the pressure sensors and wave gauges is the same as in Figure 10. The model scale of the experiments is 1:30, corresponding to the Alpha Ventus offshore wind farm in the North Sea. The duration of the recording interval is 60 s, the sampling frequency is 100 samples per second for wave gauges and 9600 samples for pressure. There is a constant phase shift of 0.14s with respect to the experiment (as reported in (Sriram et al. (2021b))). In the previous results and discussion, the experimental time series for surface elevation, pressure and force obtained on the cylinder were already shown for reference. The numerical simulation results agree favourably with the experiment but with some differences. It should also be remembered that above [Section 5.2.1], the coefficient for all the mesh types was given in correlation with the finest mesh (fsCo0.01M3) in the *foamStar*. Now all the simulated results are correlated with the experimental time series. Figure 23 compares the correlation for wave elevation, pressure and force with *foamStar* and *foamStarSWENSE*. None of the results converge towards one showing the uncertainty in comparison with the experiment. Wave and pressure converge towards 0.995 and 0.99, respectively and force towards 0.998. The uncertainty between the experiment and numerical simulation can be attributed to factors such as experimental errors (Sriram et al. (2021a)), HOS generated
<figure><img src="image_53.png" /><figcaption>(a) Wave elevation correlation - *foamStar*</figcaption></figure>

<figure><img src="image_54.png" /><figcaption>(c) Pressure correlation - *foamStar*</figcaption></figure>

<figure><img src="image_55.png" /><figcaption>(e) Force correlation - *foamStar*</figcaption></figure>

<figure><img src="image_56.png" /><figcaption>(b) Wave elevation correlation - *foamStarSWENSE*</figcaption></figure>

<figure><img src="image_57.png" /><figcaption>(d) Pressure correlation - *foamStarSWENSE*</figcaption></figure>

<figure><img src="image_58.png" /><figcaption>(f) Force correlation - *foamStarSWENSE*</figcaption></figure>

<caption>**Figure 23:** Comparison of correlations between different Mesh types M1, M2, M3 and M4 of *foamStar* (left) and *foamStarSWENSE* (right) with experimental data. (From top to bottom) Wave elevation, pressure and force</caption>

focusing wave, CFD assumptions (discretization, time scheme, schemes) and so on. Furthermore, the behaviour of *foamStar* and *foamStarSWENSE* is similar to the observations made earlier. The accuracy of *foamStar* increases on refining the mesh and time, but *foamStarSWENSE* shows no drastic improvement in wave and pressure accuracy when refining the mesh. An improvement is observed in the force due to an increase in refinement around the cylinder. Another advantage of the *foamStarSWENSE* solver is that on the coarsest grid (M4), the solution accuracy in wave and force prediction is significantly better than that of *foamStar* solver on the same mesh. This of the two solvers shows that the *foamStarSWENSE* model is better in all mesh configurations for accurate wave propagation inside the computational domain, as it was proven that refining did not improve the results (Figure 23b), which were already converged. As a result, any configuration from M1 to M4 can be chosen with the minimal computational cost for the wave propagation problem. However, by refining the free surface region in *foamStar*, the waves improved a lot (Figure 23a), eventually reaching the same convergence point as that of *foamStarSWENSE*. Hence to achieve a wave profile similar to *foamStarSWENSE*, either M2 or M3 with less Co must be chosen in the *foamStar* solver. For the pressure observed over the cylinder, the solvers showed only a marginal accuracy improvement, and no drastic difference was observed (Figure 23c and 23d). Hence both models with a finer refinement near the cylinder are good enough to
measure the pressure accurately. Lastly, the appropriate model for force prediction over the cylinder is discussed in the
following section along with its computational cost.

**5.2.3. Computational efficiency**

<figure><img src="image_59.png" /><figcaption>**Figure 24:** Correlation between different Mesh types M1, M2, M3, and M4 as well as the computational cost by comparing force recorded by the solvers *foamStar* and *foamStarSWENSE* with the experiment</figcaption></figure>

Current advances in computational power and software architecture are inadequate, as even preliminary CFD simulation is still exorbitant. The primary advantage of this hybrid coupling is faster simulation than the full/conventional NS solvers while still achieving a comparable level of accuracy. In this problem, the discretization for parallel simulations is maintained at 1 CPU for every 30000 cells. All the simulations are run with the same residual tolerances and schemes. Simulations with fewer cells were carried out using the workstation and very refined test cases using the cluster (supercomputer LIGER, at Centrale Nantes, France). In particular, most of the in-house codes are OpenMP or MPI paralleled. Considering the variation in the number of CPUs (Np) for different test cases, in an attempt to compare the solvers and mesh types, a CPU effort defined as (Agarwal et al. 2021a) $\frac{T_{CPU} Np}{T_{SIM}}$ was used. $T_{CPU}$ is the overall execution time for the simulation and $T_{SIM}$ represents the duration of the simulation (in the present case, 5s). The Np for mesh types M1, M2, M3 and M4 are 1, 8, 58, 1 respectively. In this way, the computational cost is comparable between the test cases and solvers in order to understand the solver efficiency. The comparison for each mesh type is plotted in Figure 24 with the computational cost found earlier on one vertical axis and the cross-correlation coefficient on the other vertical axis. The plotline indicates the accuracy in terms of coefficient in each case, while the bar chart represents the recorded computational cost. Both solvers yield reasonably accurate answers with minimal errors having reasonable accuracy. The accuracy of *foamStar* is slightly better than that of the *foamStarSWENSE* solver for mesh types M1 and M2 (because of the coarser grid around the cylinder). The computational cost of *foamStarSWENSE* is always 1.1-1.25 times higher than of *foamStar* for a similar setup. The computational cost of the *foamStarSWENSE* solution is slightly higher for the finer mesh M3, but accuracy is comparable to that of the *foamStar* solver thanks to a higher refinement around the cylinder. The coarser time step (Co 0.7 and Co 1) *foamStarSWENSE* prediction outperforms the *foamStar* prediction in both M2 and M3, demonstrating its advantage. Finally, the *foamStarSWENSE* mesh M4 shows that *foamStarSWENSE* is more accurate than the *foamStar* solver for a similar setup. Despite the fact that the computational cost is still higher than that of *foamStar*, it can be clearly seen that *foamStarSWENSE* with this
configuration takes 100 times less time to estimate results to obtain the same accuracy (check M3 - all Co). Hence *foamStarSWENSE* can be used to make an initial guess in an industrial estimation of any offshore wave structure problem at a minimal cost and with high precision. However, *foamStarSWENSE* is more expensive than *foamStar* for similar meshes and timesteps, a drawback that needs to be investigated in the future.

## 6. Conclusion

In the marine and offshore industries, CFD tools are frequently used to simulate wave structure interactions. However, traditional CFD tools need a large computational domain to ensure adequate development of nonlinear wave propagation and to avoid boundary effects caused by waves interacting with any fixed or floating structures in the domain. Additionally, very fine mesh elements are needed to prevent excessive numerical dissipation during wave propagation. Due to the considerable increases in computational costs caused by these factors, standard CFD methodologies are not practical for regular simulations of wave-structure interactions. The development of coupled hybrid models (most notably FNPF and CFD) that successfully address CFD's drawbacks has recently received much attention. These FNPF-CFD coupling models are classified based on domain and functional decomposition methods. These domain and functional decomposition approach based solvers (*foamStar* and *foamStarSWENSE*) have been validated in the OpenFOAM framework based on the focused wave kinematics generated from the HOS-NWT using the time-reversal technique. Both coupling strategies are currently one-way schemes, implying that information is exchanged from HOS to CFD but not vice versa. Here CFD represents both the NS/SWENSE solvers. The CFD domain is restricted to the zone where vorticity and viscous effects are predominant, and a potential flow solver is considered for input waves only. Initially, both approaches were applied to a 2D NWT to generate a focusing wave from HOS to CFD. The study has addressed HOS initiation time, relaxation zone, spatial and temporal requirements, and different cell ratios. It was found that, due to the coupling, HOS initiation can start immediately before the focusing time and that the simulation does not need to be started from the beginning. This saves a substantial amount of computational time. Focusing wave's spatial and temporal requirement in the CFD domain was also explored based on cell-based Reynolds and Courant numbers. In addition, the problem's relaxation zone has been significantly reduced, saving computational cost.

The coupling approach was then applied in a 3D wave structure interaction study representing a fixed surface-piercing circular cylinder interacting with the focusing wave. Three different mesh types (coarse (M1), medium (M2) and fine (M3)) were developed with a refinement ratio of 2. The cross-correlation coefficient was adopted to identify the relative error difference between the time series, thus taking care of the difference in the wave profile, phase shift and peak magnitude. The parameters compared were the wave elevation, pressure and force over the cylinder. The mesh types results were initially compared with the finest mesh combination (fsCo0.01M3), and a grid convergence study was carried out based on the Richardson extrapolation. The simulated results were also validated with experimental data. In comparison to addressing a similar issue in the complete CFD problem, both coupling algorithms yield good agreement with experimental results and achieve a significant reduction in CPU time. Even the coarsest mesh (M1 and M4) of the *foamStar* and *foamStarSWENSE* solvers can predict the forces over the cylinder with a difference of only 10%. The *foamStarSWENSE* solver estimates the wave and pressure over the cylinder more accurately on multiple occasions (M3 and M4) than *foamStar*, but it takes 1.25-1.5 times longer computational time. Another advantage is that the accuracy of the mesh type without any free surface refinement (M4) in *foamStarSWENSE* is similar to medium mesh refinement in *foamStar*. Thanks to an explicit treatment of incident waves in SWENSE, even a coarse mesh in the far-field can accurately produce the incident wave information and finer refinement near the cylinder. In no instances, however can the performance of *foamStar* claimed to be lower than that of *foamStarSWENSE*. It consistently produced good results in all types of meshes with a modest computational cost, despite the fact that refining increased the accuracy of the results.

In general, *foamStarSWENSE* (FD based solver) outperformed *foamStar* (DD based solver) in both the 2D and 3D conditions when solving the problem of wave generation and propagation. However, the *foamStar* and *foamStarSWENSE* (DD and FD-based solvers) performed similarly to each other when the problem had a structure in the domain, while *foamStar* outperformed the *foamStarSWENSE* in terms of computational cost. *foamStarSWENSE* solver needs to be improved in terms of processing speed to fully benefit from its explicit handling of incident waves to surpass the DD-based solvers in computational efficiency. Overall, the *foamStar* and *foamStarSWENSE* solvers performed well for focused wave generation and interaction with a vertical cylinder. Future research will focus on
the interaction of a forward-moving cylinder (similar to current) with a focused wave and the interaction of a moored
floating body with waves to understand the solvers capabilities.

7. ACKNOWLEDGEMENT

This work was performed in the framework of the Chaire Hydrodynamique et Structure Marines CENTRALE NANTES - BUREAU VERITAS. The first author acknowledges IITM and LHEEA, ECN for financial support for this Ph.D. study. This work was partly supported by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" Program (ANR-16-IDEX-0007).

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CRediT authorship contribution statement

**Sithik Aliyar**: Conceptualization of the study, Methodology, Software, Writing - Original draft preparation.
**Guillaume Ducrozet**: Methodology, Data curation, Writing - review draft preparation.
**Benjamin Bouscasse**: Methodology, Conceptualization of the study, Writing - review draft preparation.
**Sriram Venkatachalam**: Conceptualization of the study, Writing - review draft preparation.
**Pierre Ferrant**: Conceptualization of the study, Writing - review draft preparation.