In this section we will review the literature on numerical simulation of porous materials with a special focus on coastal structures. Due to space limitations, only the most relevant references will be covered. For a more extensive review the reader is referred to Higuera (2015) and Losada et al. (2016).
There are two main approaches to simulate porous structures numerically. First, in the direct approach, all the individual elements that form the porous materials are represented in the numerical domain, either meshed around (Dentale et al., 2014, 2018), or placed as solids in meshless methods (Altomare et al., 2014). Although this approach can offer valuable insights and detailed flow patterns around individual elements, it is limited to the largest elements only (e.g., pre-cast concrete units). The smaller materials in the inner layers will often present random sizes and arrangements, making it impossible to represent such geometries numerically.
The second method is the averaged approach. In it, some simplifications are made (e.g., homogeneous and isotropic materials) and the Navier-Stokes equations are modified accordingly, via averaging. Two main averaging procedures can be found in literature, namely, time-averaged volume-averaged Navier-Stokes equations (de Lemos and Pedras, 2001) and volume-averaged RANS (VARANS) equations (Liu et al., 1999). In this section will be focussing on the second approach, as it is more widely used in our field.
Flow through porous media has been studied in multiple science and engineering branches. The initial advances in volume-averaged RANS equations, which will be introduced in the next section, were developed in the late 60s for the petroleum industry Whitaker (1967); Slattery (1967). This link is natural, as petroleum extraction involves extremely complex multiphase flow through porous media. Other fields, as the chemical and manufacturing industries, have also made significant advances during the years, as porous media is widely used in, e.g., catalysts. Therefore, significant developments can also be found in heat transfer (Kaviani, 1995).
The application of porous media flow in coastal engineering problems started as early as Madsen (1974), based on eigenfunction expansions. This approach has been applied for a long time (Dalrymple et ah, 1991; Losada et ah, 1993). Other solutions based on the mild- slope equation and depth-integrated models were also developed. The reader is referred to Losada et ah (2016) for a complete discussion on these methods.
Although the first truly numerical modelling application to coastal structures was implemented by Sakakiyama and Kajima (1992), it was not until Lin and Liu (1998); Liu et ah (1999), applied a set of Navier-Stokes equations, similar to the ones used today, and the Volume Of Fluid (VOF) technique (Hirt and Nichols, 1981), with the introduction of COBRAS (Cornell breaking waves). Prior to this breakthrough work, similar models as VOF- break (Troch and De Rouck, 1998) did not include the porosity inside the equations, merely porous media-induced drag forces. However, Liu et ah (1999) presented a main limitation: porosity had not been accounted for in the differential operators during the development, thus, spatial variations in it were not taken into account. Improvements such as including a volume-averaged к — e turbulence model were introduced in Hsu et ah (2002), based on the closure developed by Nakavama and Kuwahara (1999) for heat transfer applications. Nevertheless, the differential operators were kept independent of porosity. The model IH2VOF, derived from COBRAS, was later applied to simulate all types of coastal structures (Lara et ah, 2006, 2008), also with porosity outside of the differential operators. Finally, Hur et ah (2008) presented a 3D model in which the RANS equations were not volume-averaged, but included drag terms and face and volume fractions in the porous cells to account for the effects of the porous materials.
A boom in numerical modelling started from 2010, boosted by more accessible and efficient computations. The new generation of models solved the previous issues with the porosity gradients, extended the range of applicability to three-dimensions and shifted in most cases to a finite volume discretizat ion. Examples of the newly developed models include COMFLOW (Wellens et ah, 2010), IH3VOF (Lara et ah, 2012) and OpenFOAM® (Higuera et ah, 2014a; Jensen et ah, 2014). Other models with different discretizations are Flow3D (finite differences) (Vanneste, 2012; Vanneste and Troch, 2012) and the tool presented in Larese et ah (2015) (finite elements).
The development of these models was accompanied by a thorough study of the volume- averaged equations. In that sense del Jesus (2011) performed an extensive work and derived a set of VARANS equations, including a new volume-averaged к — lo SST turbulence closure model. His work was applied in Lara et ah (2012) and del Jesus et ah (2012) and later implemented in OpenFOAM® (Higuera et ah, 2014a,b). Jensen et ah (2014) re-developed the equations in a different way, comparing the results with Lara et ah (2012) and del Jesus et ah (2012), concluding that they are almost identical. The set of VARANS equations presented in Higuera (2015), currently implemented in the olaFlow model (Higuera, 2017), follow the same approach in Jensen et ah (2014), with the difference that the derivation includes the time variation of porosity, an important feature in some cases that will be applied in Section 6.
Although this review has so far been centred in Eulerian Navier-Stokes type of modelling, it must be noted that other methods exist in literature too. For example, porous media flow has been modelled following the averaged approach with the Smooth Particle Hydrodynamics (SPH) method (Shao, 2010; Ren et ah, 2016) and with the explicit Moving Particle Semi- implicit (MPS) method (Sun et ah, 2018). The Lattice Boltzmann method has also been applied (Espinoza-Andaluz et ah, 2017), but in this case for only direct approach simulations.