Coastal Engineering 83 (2014) Contents lists available at ScienceDirect. Coastal Engineering

Transcription

1 Coastal Engineering 8 () 8 Contents lists available at ScienceDirect Coastal Engineering journal homepage: Three-dimensional interaction of waves and porous coastal structures using OpenFOAM. Part I: Formulation and validation Pablo Higuera, Javier L. Lara, Inigo J. Losada Environmental Hydraulics Institute IH Cantabria, Universidad de Cantabria, C/Isabel Torres n, Parque Cientifico y Tecnologico de Cantabria, 9 Santander, Spain article info abstract Article history: Received 6 April Received in revised form August Accepted 6 August Available online October Keywords: CFD RANS OpenFOAM IHFOAM Wave structure interaction Two phase flow Porous media In this paper and its companion (Higuera et al., this issue), the latest advancements regarding Volumeaveraged Reynolds-averaged Navier Stokes (VARANS) are developed in OpenFOAM and applied. A new solver, called IHFOAM, is programmed to overcome the limitations and errors in the original OpenFOAM code, having a rigorous implementation of the equations. Turbulence modelling is also addressed for k- and k-ω SST models within the porous media. The numerical model is validated for a wide range of cases including a dam break and wave interaction with porous structures both in two and three dimensions. In the second part of this paper the model is applied to simulate wave interaction with a real structure, using an innovative hybrid (D D) methodology. Elsevier B.V. All rights reserved.. Introduction One of the determining factors to generalize the use of numerical models for coastal engineering is that the most advanced ones can handle flow through porous media, thus being able to simulate any structural typology. Should the models lack porous media flow, they would only be suitable to calculating impervious coastal structures. Following this reasoning, the importance of porous media flow is clear, as the vast majority of coastal structures have a porous portion. Rubble mound breakwaters have armour layers that are built of concrete pieces or crushed rocks. Even vertical breakwaters, which may be seen as an impervious structure, have a porous foundation which affects the stability of the caisson due to the uplift pressure. We will comment on the different approaches to simulate porous media in the following paragraphs. The first approach worth mentioning is the Smoothed Particle Hydrodynamics (SPH) method. Works by Dalrymple and Rogers (6) and Shao (6) can be remarked as the firstrealapplicationsofsph to coastal engineering. SPH methods are in an early stage of development. Recently (Shao, ) presented a precursory application of wave interaction with porous media. The main drawback is that it can only be applied in D, which restricts the range of applicability, as we will comment on the second part (Higuera et al., this issue). More recent works present Corresponding author. address: losadai@unican.es (I.J. Losada). significant advancements, as Akbari and Namin (). However still no one has published a D porous model for SPH. The other relevant approach is Reynolds-Averaged Navier Stokes (RANS) equations. Unlike the previous method, RANS is an Eulerian approach, as these equations represent the continuum properties rather than the behaviour of individual particles. RANS equations have been extensively used for coastal engineering applications. The first remarkable application was presented almost years ago by van Gent et al. (99), and it should be noted that it already included flow through porous media. The period of time in which RANS has been applied to coastal engineering is very long compared to SPH. Therefore, RANS codes have already been able to deal with a great number of applications. A brief list of such cases includes all kinds of wave generation and absorption (Higuera et al., a; Jacobsen et al., ; Lara et al., ; Lin and Liu, 999; Troch and De Rouck, 999) and wave interaction with coastal structures (del Jesus et al., ; Guanche et al., 9; Higuera et al., b; Lara et al., 6; Lara et al., 8; Lara et al., ; Losada et al., 8; Luppes et al., ). One of the strong points of RANS is that they are accessible to the whole community through commercial codes, but also free and open source models are available. Some examples of CFD codes applied to coastal engineering include IH-VOF Lara et al. (6), IH-VOF Lara et al. (), COMFLOW Luppes et al. (), VOFbreak Troch and De Rouck (999) or OpenFOAM Higuera et al. (b). However, to the authors' knowledge and until this work, there is no three-dimensional open source model available in which porous media flow is treated for two-phase flows /$ see front matter Elsevier B.V. All rights reserved.

2 P. Higuera et al. / Coastal Engineering 8 () 8 This paper is structured as follows. After this introduction, the different ways of implementing porous media flow in RANS models are discussed. Then, further development of existing derivations leads to an implementation procedure in OpenFOAM. Next, the model IHFOAM is validated using a wide range of cases, including a dam break and oscillatory flow experiments both in two- and three-dimensions. Finally, the conclusions of this work are highlighted.. Porous media equation discussion The main methods to treat porous media flow in numerical models are described in the first part of this section. Then, the VARANS equations, as developed in del Jesus et al. (), are introduced. Finally, OpenFOAM is described and the implementation of VARANS in OpenFOAM is detailed... Porous media flow for Navier Stokes equations There is not a universal or unique way to simulate flow through porous media, therefore, in this section we will introduce the two main methods to treat porous media flow in NS numerical models, the microscopic and macroscopic approaches. The most intuitive way to simulate the flow through a porous material is the microscopic approach. In it each of the elements of which the material is formed (e.g. each of the concrete blocks of a breakwater, each of the stones ) is represented in the mesh. It is impossible to apply such procedures in our field for several reasons: there is no way to have the complete and exact description of the geometry, and it is not possible to mesh with such a great variation of scales (from blocks to sand grains). Furthermore, it is of greater interest to understand the global effects of porous media in the flow than obtaining an accurate solution of the flow within. The second procedure is the macroscopic approach, which relies in obtaining a mean behaviour of the flow within the porous media by averaging its properties along control volumes. Volume averaging NS equations allows considering the porous zone as a continuous medium, characterized by its macroscopic properties only, thus eliminating the need of a detailed description of its complex geometry. This simplification, however, introduces new terms in the equations that need to be modelled. Averaging the NS equations can be done in several ways. This paper is focused in Volume-averaged Reynolds-averaged Navier Stokes (VARANS) equations Hsu and Liu (), but time-averaged volume-averaged methods also exist, as presented in de Lemos (6). The VARANS equations can have different terms, depending on the assumptions applied by the author. For example, the work presented in Hsu and Liu () has been a reference for almost years. It is based on the previous work by Liu et al. (999), and extends it to include a k- turbulence model closure within the porous media, as presented by Nakayama and Kuwahara (999), which made it the most suitable formulation for coastal engineering at the time. However, porosity is taken out of the differential operators, which is not applicable if spatial gradients of porosity exist. The most recent advance is the VARANS formulation presented in del Jesus et al. (). This work includes a discussion about the different equations found in literature, commenting on the underlying assumptions and ranges of application. A new model called IH-VOF is developed to simulate two-phase flows within porous media, solving a new set of VARANS equations and the volume of fluid (VOF) technique. del Jesus et al. () extend the range of applicability of VARANS to the most general scenario, in which spatial variation of porosity is also taken into account. The equations are developed keeping the porosity inside the differential operators. This approach is very important for coastal engineering, as the structures can present several layers of different porous materials. Otherwise, the flux across the interfaces of such porous media would not be accurately solved. The advantages of VARANS equations are numerous. The solving process yields very detailed solutions, both in time and space. Pressure and velocity fields are obtained cell-wise, even inside the porous zones, so the whole three-dimensional flow structure is solved. Furthermore, non-linearity is inherent to the equations, and therefore all the complex interactions among the different processes are also taken into consideration. Finally, the effects of turbulence within the porous zones can also be easily incorporated with closure models. There is also another approach to consider porous media flow, presented in Hur et al. (8). Although the NS equations are not volume-averaged, the resistance due to the porous material is represented in a similar fashion using drag forces. The momentum equation is also modified to include area and volume fractions to represent the porosity... VARANS equations in IH-VOF The VARANS equations proposed by del Jesus et al. () and implemented in IH-VOF are presented next. They include conservation of mass (Eq. ()), conservation of momentum (Eq. ()) and the VOF function advection equation (Eq. ()). x i u i n ¼ t u u i þ u i j n ¼ n p þ ng ρ x i þ n ν u i a u i x i n i bu i ju i j c t u i α t þ x i u i n α ¼ : in which u is the so-called extended averaged or Darcy velocity; n is the porosity, defined as the volume of voids over the total volume; ρ is the density; p is the pressure; g is the acceleration of gravity; ν is the kinematic viscosity; and α is the VOF indicator function, and is defined as the quantity of water per unit of volume at each cell. The three last elements in Eq. () appear as closure terms to account for physics that cannot be solved when volume-averaging (i.e. frictional forces, pressure forces and added mass due to the individual elements of the porous media). Since Darcy (86) introduced a study of water flowing through sand, the study of flow through porous media has been characterized by drag forces. This first approach included only one linear term (first term in Eq. ()), which was appropriate to model laminar flows. It was not until Forcheimer (9), with the addition of a quadratic term (second term in Eq. ()), that the more energetic flows (in terms of larger Reynolds numbers) could be modelled. Polubarinova- Kochina (96) extended the model proposed by Forcheimer (9) to account for unsteady flows, adding a third term (third term in Eq. ()), which is transient and represents an inertial acceleration. The final expression of the drag forces, as applied in Eq. (), is presented next: I ¼ auþ bujujþc u t where I is the hydraulic gradient (proportional to the drop in pressure), and u is the Darcy velocity. Three coefficients (a, b and c), which depend on the physical properties of the material, control the balance between each of the friction terms. In this work Engelund (9) formulas, as applied in Burcharth and Andersen (99),have beenusedfor thefrictioncoefficients. Nevertheless, and as it is explained later, these parameters need calibration from physical tests to obtain results close to reality. ðþ ðþ ðþ ðþ

3 P. Higuera et al. / Coastal Engineering 8 () 8 The reader is referred to del Jesus () for further details regarding this section... Introduction to OpenFOAM OpenFOAM (interfoam solver) solves the RANS equations out of the box for two incompressible phases, and tracks the free surface movement using the VOF technique. The pressure and velocity fields are obtained solving simultaneously the continuity (Eq. ()) and momentum conservation (Eq. (6)) equations, while the free surface is tracked with Eq. (7) U ¼ ρu þ ðρuuþ ðμ t eff UÞ ¼ p g X ρ þ U μ eff þ σκ α α t þ U α þ U c α ð α Þ ¼ : ð7þ In these equations the bold letters indicate a vector field (e.g. U is the velocity vector); μ eff = μ + ρν turb is the efficient dynamic viscosity; p is the pseudodynamic pressure; and X is the position vector. The last term accounts for surface tension effects: σ is the surface tension coefficient; κ ¼ α j α j is the curvature of the interface. The last term in Eq. (7) is a numerical artefact to avoid the excessive diffusion of the interface, being U c a compression velocity. Since pressure and velocity are coupled, the solution of both fields is obtained with a two step approach. OpenFOAM has developed a unique methodology, called PIMPLE, which is originated by merging PISO and SIMPLE algorithms. OpenFOAM follows a tracking approach rather than reconstructing the free surface each time step. The main advantage is that it requires less computational cost, as only an advection diffusion equation has to be solved. On top of this, there is no need to apply a pressure boundary condition on the free surface. The main drawback is that Eq. (7) diffuses the interface over some cells, which violates a main VOF assumption. However, this process is addressed numerically. The solution of Eq. (7) has to be bounded between and. OpenFOAM 's special solver called MULES (Multidimensional Universal Limiter for Explicit Solution) uses a limiter factor on the fluxes of the discretized divergence term to fulfil these restrictions. For further reference regarding the governing equations and the solving procedures see Rusche () and Higuera et al. (a)... Numerical implementation of VARANS in IHFOAM Adapting the equations presented in the previous section to OpenFOAM formulation is simple, but not straightforward. As mentioned before, the VARANS equations (Eq. () ()) are formulated in terms of the extended averaged or Darcy velocity (U). They can also be formulated in terms of the averaged or intrinsic velocity (U ), as intrinsic magnitudes refer to the portion of fluid existing within the gaps of the solid skeleton of the porous media. The transformation is direct: U ¼ U n : The process to assimilate the VARANS equations in OpenFOAM starts by transforming the first equations to be written in intrinsic velocities. The purpose of such procedure is clearly seen when comparing Eq. () and Eq. (). If OpenFOAM solved for U, both equations would be the same. The same happens with the VOF function advection function. Therefore, this is the assumption that will be applied from now ðþ ð6þ ð8þ on: OpenFOAM will be adapted to solve the equations in intrinsic magnitudes. Some adjustments have to be made in Eq. () in order to better identify it with Eq. (6). These include multiplying both sides by ρ, and introducing this variable within the differential operators (which can be done, as the fluids are considered incompressible by definition). Both sides are also divided by porosity, and a single term is multiplied and divided by porosity. The result is presented as follows: ð þ cþ ρu i t n þ u j ρu i n n ¼ p x i þρg i þ μ u i A u i x i n n B u i u ð9þ i n n for which replacements of A = ρ a, B = ρ n b have been applied. Note how the transient friction term has been grouped on the first factor. It is remarkable that now Eq. (9) and Eq. (6) are identical, with the only exception of the three drag terms, which have to be added in OpenFOAM 's momentum equation. Recalling the previous definition of A and B and applying Engelund (9) formulas, modified by van Gent (99) yields: A ¼ α ð nþ n μ D B ¼ β þ 7: n KC n ρ D ðþ ðþ where D is the mean nominal diameter of the material. KG is the Keulegan Carpenter number, which introduces additional friction due to the oscillatory nature and unsteadiness of the system. It is defined as follows: T o u Mn D. u M is the maximum oscillatory velocity, and T o is the period of the oscillation. Regarding the friction parameters, α and β have to be calibrated. The variation in value for c has been proven to be of little importance in most of the cases del Jesus (). Therefore, according to recommendations and previous experience, a value of c =. has been kept constant throughout this paper. Up to this point only minor changes are needed to adapt OpenFOAM to solve VARANS equations. In fact, a similar approach is carried out in porousinterfoam solver, which just adds some drag forces for additional friction. Such procedure may work fine for single-phase flows or when the porous region is always submerged. If this is not the case and the interface of the fluid crosses the porous zone, the total mass of the system is not conserved. Nevertheless, it can be used for certain applications in which porosity is low enough to neglect such effects (e.g. flow through certain vegetation fields). However, and to be rigorous, the effect of porosity has to be introduced. The premise is that the amount of fluid that can enter any cell is not the same as the cell volume, it depends on the volume of voids, thus on the porosity of the cell. At the same time, the implementation procedure has to be compatible with the restrictions that using the VOF function implies, as for example, being bounded between and, so that the fluid properties can still be calculated as a weighted average. From the previous derivations it is clear that the main VOF advection function (Eq. (7)) needs no modifications. Therefore it is the VOF function solver (MULES) element which needs to be adjusted to account for the porosity. A new version of MULES called IHMULES has been coded, compiled and linked to IHFOAM solver. The new solver accepts the porosity field as additional input to modify the cell volumes. It limits the amount of fluid that can enter a cell, depending on its porosity, while ensuring that the total VOF function value within a cell does not exceed. The aforementioned elements are not the only ones to address in order to obtain an advanced solver for coastal engineering. Turbulence

4 6 P. Higuera et al. / Coastal Engineering 8 () 8 modelling is also a very important factor to take into account, as the flows interacting with porous structures are most of the times very energetic and turbulence plays a great role as a dissipation mechanism, not only around, but also inside the porous media. According to del Jesus et al. () the turbulence models have to be volume-averaged as well. Currently the only turbulence model that has a closure developed for porous media is k-, as presented in Nakayama and Kuwahara (999). Even though it was initially developed for heat transport through porous media, the good results shown in del Jesus et al. () and Lara et al. () indicate that it is reasonable to extrapolate its use to coastal engineering. The k-ω SST model was also volume-averaged in del Jesus et al.'s () work; no closure is provided, though. Both turbulence models have been adapted to work in IHFOAM inside and outside the porous media. The details regarding the new formulations are included in the A.. Validation of the model In this section the new implementation of the VARANS equations for IHFOAM is validated against laboratory measurements. First, a sensitivity analysis of the porous parameters is carried out replicating the wellknown Lin (998) dam break experiments. The model is validated next in D for oscillatory flow in a wave flume, simulating the interaction of regular waves with a high mound breakwater from Guanche et al. (9). Finally, the validation is extended for fully D wave interaction with a porous obstacle within a wave tank, as presented in Lara et al. (). The wave generation and absorption boundary conditions presented in Higuera et al. (a) have been applied in all the cases... Two-dimensional porous dam break These experiments carried out by Lin (998) have been used as benchmark cases for numerical models featuring flow through porous media. Their simple set-up and wide range of conditions make them especially suitable for these purposes.... Physical experiments Lin (998) tested a dam break flow through different porous materials. The experiments were performed inside a glass tank (considering an idealized D behaviour: 89 cm horizontally and 8 cm vertically), which permitted the use of video recording techniques to obtain the free surface elevation all along the domain. This includes the clear flow region and also the interior of the porous medium. The physical set-up was always the same, regardless of the porous medium type or water level tested. The main water body was confined on the left side of the domain, separated from the porous medium interface by a moving gate. This initial region spanned cm in the horizontal direction. Right next to the water reservoir the porous medium extended for 9 cm. Finally, there was another clear flow zone between the porous medium and the end wall, spanning cm. A base level of water of. cm was set all over the tank bottom, outside the reservoir. A sketch of the initial state can be found in the upper left panel in Fig.. Two different porous materials were tested: crushed rocks and glass beads, to account for different flow regimes. The flow through the glass beads (D =.cm and ϕ =.9) was found to be laminar, and closer to a Darcy flow (del Jesus et al., ). However, the flow through the crushed rocks (D =.9cm and ϕ =.9) is fully turbulent, as velocities and pore size are greater. Three different water levels were tested:, and cm. The experiments were started by raising the separation gate between the water and the porous medium.... Numerical experiments The whole tank has been reproduced numerically in D. The cell size has been kept constant and equal to. cm throughout the domain. The mesh is orthogonal and conformal, and is formed by,68 cubic elements. Only the crushed rock material has been considered, as the flows we are dealing with in coastal engineering are most of the times turbulent. Furthermore, this material is closer to the ones found in rubble mound breakwaters t =. s t =. s.. t =.7 s t =. s t =. s t =.9 s Fig.. Validation case: crushed rocks, h =cm. α =, β =. Laboratory data as circles, numerical data as points.

5 Table Sensitivity analysis results. The small figures show absolute error in water elevation (Y axis, in centimetres) along the whole tank (X axis, in metres). α\β. 6, P. Higuera et al. / Coastal Engineering 8 () 8,

6 8 P. Higuera et al. / Coastal Engineering 8 () 8 The system starts from rest state, with the water and porous medium set at their initial location. h =cm has been tested, as is the most nonlinear case. Turbulence is modelled using the modified k- model. Several porous medium parameters have been considered, as in del Jesus et al. (). Thestartingpointhasbeenα = and β =. All the combinations between α and β for the given original value and half and double these values have been carried out. Each case has been simulated for s, and took less than min to run in single. GHz processor (regular PC).... Results The results for the best case (h =cm, α = and β =)are shown next. The free surface elevation at certain times is presented in Fig.. The porous medium is shaded in grey and contrasts with the clear flow region. Experimental results are shown as black circles, while numerical data is depicted as blue dots. The resolution is very high, which makes them look as a continuous line. As we can see in Fig. the numerical results agree quite well with the experimental data. Minor differences arise mainly in the first instants, because the lifting process of the gate is not immediate, and neither it is reproduced numerically. Therefore, these discrepancies occur at the initial snapshots, and are greater towards the bottom of the tank, where the pressure gradient is higher. However, as time advances, these get smaller and virtually unnoticeable. A sensitivity analysis as presented in del Jesus et al. () has been carried out to figure out the best combination of parameters (whose results have been shown in Fig. ) and to check their influence on the flow. The best way to see the influence is to calculate the absolute error of the numerical solution with respect to the laboratory results. If this is done very early in the simulation, the expected errors are high due to the already mentioned issue with the gate separating the water and the porous media. If it is done too late, the errors will be close to zero, as the system tends to equilibrium. A time between both bounds has been chosen: t =.9 s, this way the initial differences are diluted enough and the system is still evolving. All the results have been aggregated in Table. Each panel shows the absolute error in water elevation. The process to obtain these values is simple. First the discrete laboratory measurements (black circles in Fig. ) are transformed into a piecewise linear function. Then, for each numerical point on the free surface, its difference in height with respect to the laboratory result is calculated and plotted. We will now analyse this table, referring to individual cases using their (α, β) values. The influence of both friction parameters along with some interesting processes can be observed here. Finding the precise balance between the linear and non-linear drag coefficients is crucial to obtain an accurate solution. An excessive drag will lead to a water excess on the initial reservoir and a deficit on the other side of the porous medium. This issue can be observed in all cases with β = 6 plus (,, 6). The contrary effect can also happen, as having less friction increases the flowrate through the porous medium. This case is represented by (,.), (, ) and (,,.). The two leftover cases show an evolution close to reality. One cannot decide which one presents better performance directly, some indicators have to be calculated. We have taken into account two of them, calculated for the absolute value of the error curve: the maximum absolute error and the area under the curve. For both magnitudes the case (,, ) presents smaller values:.97 cm vs.8 cm and.86 cm vs.9 cm.consequently, this case is taken as the best performance. Other interesting facts are that the shape of the errors in the reservoir are more or less of the same shape for all the cases, only varying in their relative location with respect to the zero line. It is also remarkable that some of the cases present trapped pockets of air, as (, ) or (, 6), appearing detached from the main (top) free surface. Regarding the turbulence modelling, a snapshot showing the distribution of turbulent kinetic energy is presented in Fig.. The contour Fig.. Dam break: turbulent kinetic energy. lines indicate values that are a negative integer power of. As it can be observed, the k is greater closer to the free surface, where the most energetic movements take place. The k also increases gradually within the porous medium, and especially when it accelerates as it goes out. In conclusion, the implementation of VARANS equations is OpenFOAM works as expected, having a close resemblance with reality... Regular waves interacting with a high mound breakwater in D The next validation case was presented in Guanche et al. (9),and involves the interaction of regular waves with a rubble mound breakwater (as defined in Kortenhaus and Oumeraci (998)), although accounting for its geometry it is closer to a high mound breakwater according to Oumeraci and Kortenhaus (997).... Physical experiments The physical experiments took place in the University of Cantabria's large flume. This facility is 6. m long (from the wavemaker mean position),. m wide and. m high. The flume bottom is horizontal. Originally, two different breakwaters were constructed and tested, however, here we are only reproducing the high mound breakwater. The caisson of the breakwater was made of concrete and spanned the whole width of the flume. It was. m long and. m deep, and its seaward side was located m away from the wave paddle. This block laid on a gravel foundation which was.7 m high, and acted as the core of the breakwater. The core presented a cm berm on the seaside. A cm thick secondary armour layer made of a different type of gravel was present on both sides. The principal armour layer was cm thick, and included a cm long seaside berm. The slope of the porous materials was H/V. A sketch of the geometry can be seen in Fig.. The specific physical properties of the porous media are given in Table. A ramp was placed behind the breakwater to dissipate the transmitted waves. This device was made of metallic mesh screens, and will later be modelled as another porous medium. Waves were generated using a piston-type wavemaker, which featured an AWACS system to deal with the waves reflected by the structure. The still water level was kept constant and equal to.8 m. Different wave conditions, including regular and irregular sea states, were tested. The reader is referred to Guanche et al. (9) for further details. Fourteen resistive water elevation gauges were placed along the centreline of the flume. Their location is indicated in Table. The first seven of them were placed in front of the structure. Then, the rest were located on the structure. This set is represented in Fig., as vertical dotted lines. Notice that gauges 8 and pierce the porous media, while those placed on top of the caisson lie on top of it. The last free

7 P. Higuera et al. / Coastal Engineering 8 () 8 9 Fig.. High mound breakwater section. surface gauge was positioned behind the breakwater, to measure transmitted energy. Pressure was also measured at different locations on the caisson, as shown in Fig., in which the sensors are represented as bold points. The first four pressure cells were placed on the seaside of the concrete prism, two of them within the armour layers and the other two on the clear part. The rest of the gauges are located below the caisson, in contact with the porous core.... Numerical experiments The flume has been reproduced in its entirety in D: 6 m in length and. m in height. Three horizontal zones with different cell size grading have been defined. In the vertical direction the cell size is constant and equal to cm throughout the flume. The first zone represents the wave propagation area, and covers from the wave paddle (X = m) to X = m, close to the structure. The horizontal cell size in this sector varies from cm close to the wave generation boundary to cm close to the next area. This gradation saves computational cost by having less cells where they are not so important, as close to the generation boundary. The next region is the interest zone, and it is located from X = m to X = 8 m. The structure lies within, so the cell dimensions are chosen to give an adequate resolution. Cell size is constant and equal to cm, which provides enough detail to represent the processes taking place around the structure. The final zone is where the energy that surpasses the structure is finally dissipated, and it covers the final m. Here, the cell size varies from cm near the structure to cm at the end wall. The dissipative perforated ramp has been replicated by means of another porous medium. This case has been used to calibrate the porous media with oscillatory flow. The numerical parameters have already been presented in Table. They have been obtained by best fit of the gauges, and will later be used for the D high mound breakwater. From the initial mesh that has been described, the caisson is removed and reproduced as a void in the mesh. The final mesh has over, hexahedral cells. The porous media have been set using a castellated approach (i.e. porosity is set cellwise to the target value for the cells whose centres lay inside the surface provided, otherwise they are identified as a clear region and keep the value ). This approach is valid since the porous media equations follow a macroscopic approach. The final setup Table Porous media physical properties and best fit parameters. Material D (m) Porosity α β Primary armour layer... Secondary armour layer..9. Core..9. Dissipation ramp..86. is depicted in Fig. 7. Note that the vertical scale of the figure has been exaggerated. The shape of the outer porous layer is uneven, this is because a laser scanner was carried out in the laboratory and the original geometry has been replicated. The whole set of laboratory measuring instruments have been reproduced numerically. Free surface elevation has been probed at locations, while pressure measurements have been taken at locations. The free surface elevation is equal to.8 m, and cm in height and s of period waves have been generated using the cnoidal theory. Turbulence in this case is modelled using the modified k- model, so that the behaviour inside the porous media addresses both macroscale (Forcheimer) and flow patterns. The simulation has been fulfilled using cores of a standard PC (. GHz), and the s are available after h.... Results One of the challenges of the simulation is to manage the wave transformation processes that encompass with the high mound breakwater such as wave reflected at the structure and the damped energy by means of wave breaking and flow percolation through the porous media. In order to deal with the high energy reflected from the structure (9%) reported by Guanche et al. (9), the boundary conditions presented by Higuera et al. (a) have been used to generate and absorb waves. Results are presented in Fig. for all the wave gauges reported in Table ; ten in front of the structure, three over the caisson and an additional one leeward the breakwater to measure the transmitted wave by the combination of wave overtopping and flow percolation throughout the core. A high degree of accuracy is demonstrated by the model in reproducing both wave phase and height. The wave profile, which reveals the existence of a quasi-steady wave pattern by the combination of an Table Free surface gauges location. FS gauges X (m)

8 P. Higuera et al. / Coastal Engineering 8 () 8 the structure. Nevertheless, as the flow inside is close to laminar, the vast majority of its volume continues to present low values of k... Three-dimensional interaction of waves with a porous structure Fig.. Pressure gauge distribution on the caisson. incident and reflected wave, is very well caught during the numerical simulations. Larger discrepancies are observed in gauges 8, 9 and, which are located along the seaward breakwater slope. The lack of better information to reproduce the geometry of the porous slopes could induce such slight discrepancies. The use of incompressible two-phase flow modelling could also affect the wave evolution in the breaking zone, as the simulation is performed in a two-dimensional mesh. This implies that the air cannot escape sideways, as it does in reality, thus, the evolution of the wave during breaking can be modified. Such issue is not observed in three-dimensional simulations, as shown later. Regarding the overtopping gauges (, and ), some discrepancies can be observed. Due to the location of sensor, very close to the border of the caisson, it is subjected to the splash of the waves. The higher incident waves and the enhanced splash on the physical experiments may yield the observed differences. However, the water layer on top of the caisson due to overtopping (sensors and ) is reasonably well captured in shape, especially taking into accounting the limited mesh resolution, while underestimated. The transmitted wave (gauge, lower panel) is not well captured by the model. This may be a direct consequence of the lower wave height reaching the top of the structure and diminished overtopping. From the results of the first gauges, it can be seen that the wave generation and absorption boundary conditions presented in Higuera et al.'s (a) work adequately, as they manage to generate the target wave while absorbing the reflected energy, as the AWACS does in the laboratory. Fig. 6 shows the time series of dynamic pressure. The comparison includes four pressure gauges along the vertical face of the caisson (gauges to ) and six gauges underneath the caisson (gauges to ). In general, the model is able to predict pressures accurately at every location. It even captures the momentum damping induced at the core as waves propagate underneath the caisson towards the leeward side of the structure. Only minor discrepancies in height and phase are shown for a few waves along the time series, especially at gauge. This might be another side effect of that mentioned under prediction of the wave height. Two snapshots of turbulent kinetic energy around the structure are presented in Fig. 7. The top panel shows the instant when the first wave impacts the structure. The k level is greater around the free surface location (shown as a black line) and around the primary rock layer, presenting a uniform value along the whole depth. At this point the turbulence effects start to penetrate the core. The bottom panel presents the same situation at a mean stage of the simulation. The turbulence distribution both around the free surface and the structure has clearly diffused. However, the turbulence level is more or less of the same magnitude. A turbulence level buildup is expected for very long simulations, as described in Jacobsen et al. (), because the production terms generate turbulence even for potential flows. Furthermore, the turbulence models were initially formulated for stationary cases, and not for transient ones. It is interesting to remark that the k level is similar on the air and water phases. The turbulent effects have continued to propagate through the core of Now that the capabilities of the model have been proven to work in two-dimensional cases it is time to extend the simulations to full threedimensional ones. The interaction of waves with a vertical porous structure from Lara et al. () is now analysed.... Physical experiments The physical modelling was carried out in the University of Cantabria's wave basin, which is 7.8 m long, 8.6 m wide and. m high. The waves were generated by a piston-type wavemaker formed by individual paddles that can move independently. However, all the cases had normal incidence of waves, so the whole set of paddles behaved as one. The remaining three walls were fully reflective. The bottom of the basin was completely flat. A porous structure was built with a metallic mesh filled with a granular material. The mean stone diameter was mm and the global porosity was.. The shape of the structure was prismatic: m long,. m wide and.6 m high, and resembled a porous vertical breakwater. The porous prism was attached to one of the lateral walls, and its seaward face was located. m away from the wavemaker. A general scheme of the setup is presented in Fig. 8. The water depth was kept constant and equal to. m for all the experiments. A set of solitary waves (, 7 and 9 cm of wave height) and cnoidal regular waves ( and 9 cm of wave height; and s of wave period) were carried out. Free surface elevation was measured at locations, as depicted in Fig. 8. Pressure measurements were also taken at six points on the structure, shown in Fig. 9. These locations had been selected to assess the three dimensional effects.... Numerical experiments The wave tank has been replicated numerically in its whole extension. The shape of the domain is a box ( m), which makes the mesh orthogonal and conformal. The cell size varies in the X direction to save computational cost: from the wave generation boundary to. m away from the structure Δx varies from cm to cm. Then, it is kept constant and equal to cm around the porous zone to obtain better details. Finally, from. m leewards the structure to the end of the tank the Δx grows again from cm to cm. In the Y and Z directions the cell size is kept constant: Δy = cm and Δz = cm. The total number of cells is slightly greater than 6. million. The porous structure is treated as in the previous cases. The situation resembles the dam break case, as the structure is defined perfectly by the cells. Therefore, the castellated approach yields to the exact geometry. The porosity variables are set to the second best values which previously represented crushed rocks: α = and β =.,asslightly more accurate results are obtained with these. Two wave conditions have been selected. First, a solitary wave of 9 cm in height is tested. Then, 9 cm in height and s in period cnoidal waves are generated. In both cases, s are simulated in less than 7 h using 8 processors (.6 GHz). Free surface elevation and pressure sampling has been carried out at the locations depicted on the experimental setup.... Results In this section, the model is proven to reproduce the most relevant hydraulic processes to be considered in wave structure interaction in a three-dimensional domain, which encompass wave reflection, wave dissipation, and wave transmission resulting from wave penetration through the porous structure, wave diffraction and wave run-up on a porous vertical structure.

9 P. Higuera et al. / Coastal Engineering 8 () 8 Sensor. Lab Num Sensor Sensor Sensor Sensor Sensor Sensor Sensor Sensor Sensor Sensor Sensor Sensor Sensor Fig.. Regular waves interacting with a high mound breakwater in D: free surface elevation. Fig. present the comparison of IHFOAM predictions versus laboratory measurements in a three-dimensional domain. Free surface and dynamic pressure time evolution are shown for both cases. The solitary wave validation is presented in Fig. and. Several waves can be identified at the figures as a consequence of the reflected solitary wave at the basin walls. The model shows a high degree of accuracy in predicting both the free surface and pressure time series. The model simulates quite well the energy reflected at the porous structure, as can be seen on gauges and (Fig. ), located seawards and close to the porous breakwater. The transmitted and diffracted wave height (see

10 P. Higuera et al. / Coastal Engineering 8 () 8 Lab Num Sensor Sensor Sensor Sensor Sensor Sensor Sensor Sensor Sensor Sensor Fig. 6. Regular waves interacting with a high mound breakwater in D: pressure series. Fig. 7. Turbulent kinetic energy level around the structure for the first wave impact and at t =s. gauges 6, 7, 8 and, Fig. ) is perfectly reproduced numerically. The predictions for locations leewards the breakwater (see gauges 9, and, Fig. ) show also a perfect match with measured data. A lag between the experimental and the numerical data is observed for the reflected wave towards the end of the series. The original solitary wave gets to reflect at several locations on the basin: the end wall, the lateral walls and even on the wavemaker. Different reflections (and re-reflections) are in general terms well reproduced both in amplitude and shape. The lag was also reported by Lara et al. () and it was attributed to discrepancies in several measurements, including the geometry of the breakwater, its location within the basin and slight variations in the location of the wave gauges. Another important factor is the reflections on the displaced wavemaker, as the numerical domain does not vary. The pressure measurements, presented in, show a perfect match with measurements. Regarding the turbulence effects, several snapshots which illustrate the evolution of the turbulent kinetic energy around the porous structure are presented in Fig.. Two longitudinal transects and a plane cm above the bottom are shown. The first snapshot (top left panel) shows the instant in which the solitary wave reaches the structure and starts to penetrate. The turbulent level is still relatively low. Outside

11 P. Higuera et al. / Coastal Engineering 8 () 8 B A C Fig. 8. University of Cantabria's wave basin. Sketch of the geometry and free surface elevation gauges. Auxiliary lines each metre. the structure the k values are close to outside the free surface interface zone and the corners of the structure, in which some vortices are starting to develop. A uniform base level of turbulence appears throughout the porous medium. The largest turbulence values occur near the interface of the structure, where the wave is impacting. The next time step (top right panel) shows the evolution of the system one second after. The turbulent energy base level has increased inside the porous medium. The most dissipative zone is located where the gradients of free surface height are greater. It is remarkable that there is also another zone outside the porous medium in which k is relatively large, and this is the area near the corners of the structure, where vortices appear. However, they do not detach as expected, due to using k- as turbulence model, as it can be seen in the two final snapshots (bottom panels). The regular wave validation is plotted in Fig. and.asimilarbehaviour to the previous one is found in the comparisons between numerical data and laboratory measurements. The agreement is quite high, and the model properly reproduces the interactions between the incident and the reflected waves. Wave dissipation at the porous media seems to be well simulated because the reflected waves (gauges and,fig. ) and the transmitted waves (see gauges 9, and, Fig. ) appear to be accurately captured in shape and in amplitude. The non-linear interactions between the incident waves with the multiple reflected waves at the boundaries are also reproduced by the model, as it can be seen towards the end of the signals presented in Fig.. The dynamic pressure measurements, plotted in Fig., show also that the model provides a good representation of the physical processes at the porous media. As a summary, the overall agreement in the graphs presented is very good for both cases, proving that the model is capable of handling a three-dimensional scenario.. Conclusions In this paper a new numerical model called IHFOAM, based on OpenFOAM, has been developed to deal with real applications in coastal engineering. The validation process presented several applications, such as simple flow through a porous medium, but also wave interaction with porous structures in D and D. In the second part (Higuera et al., this issue), the model will be applied to fulfil the stability and overtopping analysis of a porous breakwater in three dimensions. The following conclusions can be extracted. IHFOAM has been developed to address the lack of rigorous treatment of two-phase porous media flow in OpenFOAM (i.e. failure to conserve mass). The special boundary conditions for wave generation C B A Fig. 9. Location of the pressure gauges on the structure. Viewpoints according to Fig. (8).

12 P. Higuera et al. / Coastal Engineering 8 () 8 Gauge Gauge.. Lab Num Gauge Gauge Gauge Gauge Gauge Gauge Gauge 9 Gauge Gauge Gauge Gauge Gauge Gauge Fig.. Solitary wave: free surface elevation. and active wave absorption presented in Higuera et al. (a) have also been included as part of the model. A review of the current approaches to treat such physics has been presented, yielding to the conclusion that the most suitable formulation in this case is VARANS equations, as presented in del Jesus et al. (). The equations have been further analysed and implemented successfully in IHFOAM, including the k- and k-ω SST turbulence models. k- includes a closure model, therefore the additional turbulence production within the porous media is considered. No closure model is available for k-ω SST yet. The validation of the model has been carried out for several cases. First, a simple dam break flow through a porous obstacle has been

13 P. Higuera et al. / Coastal Engineering 8 () 8 Gauge Gauge Lab Num Gauge Gauge Gauge Gauge Fig.. Solitary wave: pressure signals. simulated to calibrate the porous media parameters. The sensitivity analysis yields results similar to those reported in del Jesus et al. (). The absolute error in free surface elevation is below cm, which is to be seen as a very good result and as an indicator that the implementation in IHFOAM has been carried out correctly. Further validation regarding wave interaction with porous structures has been considered. First, the D test case in which regular waves interact with a high mound breakwater shows very good results in the far field. Closer to the structure the comparisons are accurate, although discrepancies for wave height and pressure arise mainly due to the geometry of the breakwater. The D nature of the numerical case combined with the two-phase flow may also be another of the causes. IHFOAM is eminently a three-dimensional model, so it is expected to be used for D cases. The three-dimensional comparisons for wave interaction with a porous structure in a wave basin show its potential. The results for the solitary wave and the regular wave trains show a high degree of accordance with the laboratory data. A special effort in assessing the effects of turbulence both inside and outside the porous media has been carried out throughout the paper. Only the k- model has been used in the validation process, as it is the Fig.. Solitary wave: turbulent kinetic energy generation.