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# Numerical model

The computational fluid dynamics (CFD) model olaFlow (Higuera, 2017) is used in this work. olaFlow is an open source project conceived as a continuation of the work in Higuera (2015). The model is developed within the OpenFOAM® framework (.Tasak, 1996; Weller et al., 1998) to enable the simulation of wave hydrodynamics and wave-structure interaction, including porous structures, in the coastal and offshore fields.

Wave generation is linked with active wave absorption at the boundaries (Higuera et al., 2013a), thus minimizing the computational cost with respect to relaxation zones and passive wave absorption approaches. The VARANS modelling was developed and validated in Higuera et al. (2014a), although the present implementation follows Higuera (2015). Further validation and applications of the model can be found in Higuera et al. (2013b, 2014b). Additional features, such as wave generation with moving boundaries to reproduce experimental wave tanks are also included and can be found in Higuera et al. (2015).

olaFlow solves the VARANS equations introduced in Section 3.3 for two incompressible phases (e.g., water and air) using a finite volume discretization. The implementation is as follows:

The newly introduced variables are the dynamic pressure, defined as p* = ppgjTj, where p is the total pressure; gj is the acceleration due to gravity and ry is the position vector.

Reynolds stresses in Equation 13 are modelled using Boussinesq hypothesis, linking them with the velocity gradient through a turbulent eddy viscosity (щ) which follows the isotropy assumption and is given by the turbulent model selected. The viscous term in Equation 18 is computed with the effective dynamic viscosity (//eff), which comprises the molecular dynamic viscosity of the fluid (p) and a turbulent dynamic viscosity (pt = put) components.

The last term in the second line of Equation 18 was introduced in Brackbill et al. (1992) to transform the surface tension force into a body force acting at the cells belonging to the interface between both fluids. Here a is the surface tension coefficient, к is the curvature of the free surface and a is the VOF indicator function introduced in Section 3.2.

The free surface between water and air is captured with the VOF technique (Hilt and Nichols, 1981). The implementation of VOF in OpenFOAM® is algebraic and only first order accurate. This method does not involve surface tracking or geometrical reconstruction, therefore, it is computationally efficient. The VOF advection equation is:

In order to obtain physical results, the scalar field a needs to be conservative, strictly bounded between 0 and 1 while maintaining a sharp interface. The first two conditions are achieved with a special solver called MULES (Multidimensional Universal Limiter with Explicit Solution). The reader is referred to Marquez (2013) for a complete description of the solver. The last term in the VOF equation does not appear in Equation 12. This additional term is a numerical artifact that produces compression forces at the interface (0 < a < 1) in order to prevent the diffusive behaviour of Equation 19 (Rusche, 2002). The new variable (it?) is a compression velocity normal to the interface, dependent on a user-defined compression enhancement constant and especially designed to avoid creating high artificial velocities.

Further details about olaFlow and the numerical implementation involved in OpenFOAM® can be found in Higuera (2015).

# Applications: Solitary wave impacting into a rubble mound breakwater

In the first application case the numerical model will be applied as a design tool to evaluate different structural alternatives to protect a rubble mound breakwater against the action of a large solitary wave. Initially the original structure will be tested to establish the reference solicitations, including the instantaneous force on the caisson, overtopping and the reflection coefficient. The model will then be applied to simulate an array of cases, in which different parametrized alternatives are built on top of the original geometry to reduce the solicitations of the solitary wave. The goal is to determine the suitable alternatives in terms of reducing the forces, enhancing stability and limiting overtopping. The two selected alternatives will be compared with the original case in detail.

## Numerical setup

The structure tested is similar to the one built for in the experiments by Guanche et al. (2009). The design of the caisson has been modified, because the original structure was designed to avoid any displacements during testing, thus it was heavily overdesigned. The new breakwater is shown in Figure 3.

The breakwater comprises a primary and secondary armour layers (1:2 slope, 12 cm and 10 cm thick, respectively) and a core. The physical properties of the porous media materials, measured in the laboratory, are gathered in Table 1. The new geometry of the

Figure 3: Sketch of the breakwater geometry including alternatives A and B.

Table 1: Physical properties and friction parameters of porous media materials.

 Material D50 (cm) Porosity a 0 Primary armour layer 12.0 0.50 50 0.6 Secondary armour layer 3.5 0.49 50 2 Core 1.0 0.49 50 1.2

concrete crown wall is L-shaped, which will reduce the high original safety factors (SF). The crown wall (1.04 m x 0.3 m) is located 25 m from the wave generation boundary and founded on top of the core at z = 0.7 m. The front vertical wall is 10 cm thick, and the bottom slab is 20 cm thick. The still water level is set at h = 0.8 m, therefore the breakwater has a positive freeboard of 20 cm.

The numerical wave flume is 30 m long (a,’-direction) and 1.3 m high (^-direction) and has a horizontal bottom. The mesh cell size in the vertical direction is constant and equal to 1 cm. The horizontal cell size varies in a geometric progression from the wave generation boundary (2 cm at x = 0 m) to x = 20 m, remaining constant and equal to 1 cm until the flume end (x = 30 m). These parameters yield a two-dimensional mesh of 350,000 cells.

A solitary wave of steepness II/h = 0.5 has been generated with the third order theory developed by Grimshaw (1971). The opposite end of the flume (x = 30 m) carries an active wave absorption boundary condition (Higuera et ah, 2013a) to prevent the transmitted wave from reflecting back. The effective wavelength, given к = J('MIis L = 8.21 m, therefore, the solitary wave propagates a distance of approximately 3 wavelengths before reaching the breakwater.

The friction parameters a and /3 used in the numerical simulations are included in Table 1. These values have been obtained by best-fit (Higuera, 2015, Chapter 7.2). The reader is referred to this reference for a complete validation study of the original case, including free surface elevation at different locations on the flume and pressure sensors mounted on the caisson, inside and outside the porous media.

Two different alternatives, shown in Figure 3, have been proposed and parametrized to enhance the initial design. The first alternative (A) consists of an additional armour layer on top of the primary layer. This alternative depends on 2 parameters: the freeboard (/) and the width of the berm (b). The second design (B) consists of a detached breakwater. This alternative also depends on / and b. plus an additional parameter s to control the separation between the toes of both structures. In either case the additional material is the same as in the primary armour layer, to avoid introducing further uncertainties in the case, and the new slopes are also 1:2.

The /, b and s parameters will be made dimensionless from here on in, dividing them by the water depth (h). The values considered are as follows: / range is [-0.8, -0.6, -0.4, -0.2, 0]; b range is [0.2, 0.4, 0.6, 0.8, 1]; and s range is [0, 1, 2, 3].

All the numerical experiments have been simulated in parallel using 6 cores of a Xeon (2.50 GHz) workstation. Turbulence modelling was connected and set to the volume-averaged кш SST model modified with the features presented in Larsen and Fuhrman (2018) and Devolder et al. (2017). The simulations are 15 seconds long, and each of the 125 cases computed takes less than 2 hours to compute.

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