Coastal Engineering 83 (2014) 243–258

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Coastal Engineering

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Three-dimensional interaction of waves and porous coastal structuresusing 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° 15, Parque Cientifico y Tecnologico de Cantabria, 39011 Santander, Spain

⁎ Corresponding author.E-mail address: losadai@unican.es (I.J. Losada).

0378-3839/$ – see front matter © 2013 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.coastaleng.2013.08.010

a b s t r a c t

a r t i c l e i n f o

Article history:Received 16 April 2013Received in revised form 22 August 2013Accepted 26 August 2013Available online 5 October 2013

Keywords:CFDRANSOpenFOAMIHFOAMWave–structure interactionTwo phase flowPorous media

In this paper and its companion (Higuera et al., 2014–this issue), the latest advancements regarding Volume-averaged 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 arigorous implementation of the equations. Turbulence modelling is also addressed for k-� and k-ω SST modelswithin the porous media. The numerical model is validated for a wide range of cases including a dam breakand wave interaction with porous structures both in two and three dimensions. In the second part of thispaper the model is applied to simulate wave interaction with a real structure, using an innovative hybrid (2D–3D) methodology.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

One of the determining factors to generalize the use of numericalmodels for coastal engineering is that themost advanced ones can han-dle flow through porous media, thus being able to simulate any struc-tural typology. Should the models lack porous media flow, they wouldonly be suitable to calculating impervious coastal structures.

Following this reasoning, the importance of porous media flow isclear, as the vast majority of coastal structures have a porous portion.Rubble mound breakwaters have armour layers that are built of con-crete pieces or crushed rocks. Even vertical breakwaters, which maybe seen as an impervious structure, have a porous foundation which af-fects the stability of the caisson due to the uplift pressure. Wewill com-ment on the different approaches to simulate porous media in thefollowing paragraphs.

The first approach worth mentioning is the Smoothed Particle Hy-drodynamics (SPH) method. Works by Dalrymple and Rogers (2006)and Shao (2006) can be remarked as the first real applications of SPHto coastal engineering.

SPH methods are in an early stage of development. Recently (Shao,2010) presented a precursory application of wave interaction with po-rous media. The main drawback is that it can only be applied in 2D,which restricts the range of applicability, as we will comment on thesecond part (Higuera et al., 2014–this issue).More recentworks present

ghts reserved.

significant advancements, as Akbari and Namin (2013). However stillno one has published a 3D porous model for SPH.

The other relevant approach is Reynolds-Averaged Navier–Stokes(RANS) equations. Unlike the previous method, RANS is an Eulerian ap-proach, as these equations represent the continuum properties ratherthan the behaviour of individual particles. RANS equations have beenextensively used for coastal engineering applications. The first remark-able application was presented almost 20 years ago by van Gent et al.(1994), and it should be noted that it already included flow through po-rous media.

The period of time in which RANS has been applied to coastal engi-neering is very long compared to SPH. Therefore, RANS codes have al-ready been able to deal with a great number of applications. A brieflist of such cases includes all kinds of wave generation and absorption(Higuera et al., 2013a; Jacobsen et al., 2012; Lara et al., 2011; Lin andLiu, 1999; Troch and De Rouck, 1999) andwave interactionwith coastalstructures (del Jesus et al., 2012; Guanche et al., 2009; Higuera et al.,2013b; Lara et al., 2006; Lara et al., 2008; Lara et al., 2012; Losadaet al., 2008; Luppes et al., 2010).

One of the strong points of RANS is that they are accessible to thewhole community through commercial codes, but also free and opensource models are available. Some examples of CFD codes applied tocoastal engineering include IH-2VOF Lara et al. (2006), IH-3VOF Laraet al. (2012), COMFLOW Luppes et al. (2010), VOFbreak Troch and DeRouck (1999) or OpenFOAM® Higuera et al. (2013b). However, to theauthors' knowledge and until this work, there is no three-dimensionalopen source model available in which porous media flow is treated fortwo-phase flows.

244 P. Higuera et al. / Coastal Engineering 83 (2014) 243–258

This paper is structured as follows. After this introduction, the differ-ent ways of implementing porous media flow in RANS models arediscussed. Then, further development of existing derivations leads to animplementation procedure in OpenFOAM®. Next, the model IHFOAM isvalidated using a wide range of cases, including a dam break and oscilla-tory flow experiments both in two- and three-dimensions. Finally, theconclusions of this work are highlighted.

2. Porous media equation discussion

Themainmethods to treat porous media flow in numerical modelsare described in the first part of this section. Then, the VARANS equa-tions, as developed in del Jesus et al. (2012), are introduced. Finally,OpenFOAM® is described and the implementation of VARANS inOpenFOAM® is detailed.

2.1. Porous media flow for Navier–Stokes equations

There is not a universal or unique way to simulate flow through po-rous media, therefore, in this section we will introduce the two mainmethods to treat porousmedia flow in NS numericalmodels, themicro-scopic and macroscopic approaches.

The most intuitive way to simulate the flow through a porousmate-rial is the microscopic approach. In it each of the elements of which thematerial is formed (e.g. each of the concrete blocks of a breakwater, eachof the stones…) is represented in the mesh. It is impossible to applysuch procedures in our field for several reasons: there is no way tohave the complete and exact description of the geometry, and it is notpossible to mesh with such a great variation of scales (from blocks tosand grains). Furthermore, it is of greater interest to understand theglobal effects of porousmedia in the flow than obtaining an accurate so-lution of the flow within.

The second procedure is the macroscopic approach, which relies inobtaining a mean behaviour of the flow within the porous media byaveraging its properties along control volumes. Volume averaging NSequations allows considering the porous zone as a continuous medi-um, characterized by its macroscopic properties only, thus eliminatingthe need of a detailed description of its complex geometry. This simpli-fication, however, introduces new terms in the equations that need tobe modelled.

Averaging the NS equations can be done in several ways. This paper isfocused inVolume-averagedReynolds-averagedNavier–Stokes (VARANS)equations Hsu and Liu (2002), but time-averaged volume-averagedmethods also exist, as presented in de Lemos (2006).

The VARANS equations can have different terms, depending on the as-sumptions applied by the author. For example, the work presented in Hsuand Liu (2002) has been a reference for almost 10 years. It is based on theprevious work by Liu et al. (1999), and extends it to include a k-� turbu-lence model closure within the porous media, as presented by Nakayamaand Kuwahara (1999), which made it the most suitable formulation forcoastal engineering at the time. However, porosity is taken out of the dif-ferential operators, which is not applicable if spatial gradients of porosityexist.

The most recent advance is the VARANS formulation presented indel Jesus et al. (2012). Thiswork includes a discussion about the differentequations found in literature, commenting on the underlying assump-tions and ranges of application. A newmodel called IH-3VOF is developedto simulate two-phase flows within porous media, solving a new set ofVARANS equations and the volume of fluid (VOF) technique.

del Jesus et al. (2012) extend the range of applicability of VARANS tothe most general scenario, in which spatial variation of porosity is alsotaken into account. The equations are developed keeping the porosityinside the differential operators. This approach is very important forcoastal engineering, as the structures can present several layers of dif-ferent porous materials. Otherwise, the flux across the interfaces ofsuch porous media would not be accurately solved.

The advantages of VARANS equations are numerous. The solvingprocess yields very detailed solutions, both in time and space. Pressureand 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 complexinteractions among the different processes are also taken into consider-ation. Finally, the effects of turbulence within the porous zones can alsobe easily incorporated with closure models.

There is also another approach to consider porous media flow,presented in Hur et al. (2008). Although the NS equations are notvolume-averaged, the resistance due to the porous material is repre-sented in a similar fashion using drag forces. The momentum equa-tion is also modified to include area and volume fractions torepresent the porosity.

2.2. VARANS equations in IH-3VOF

The VARANS equations proposed by del Jesus et al. (2012) andimplemented in IH-3VOF are presented next. They include conservationof mass (Eq. (1)), conservation of momentum (Eq. (2)) and the VOFfunction advection equation (Eq. (3)).

∂∂xi

ui

n¼ 0 ð1Þ

∂∂t ui þ uj

∂∂xj

ui

n

¼ − nρ

∂∂xi

pþ ngi þ n∂∂xj

ν∂∂xi

ui

n

� �−a ui−b uijuij−c

∂∂t ui

ð2Þ

∂α1

∂t þ ∂∂xi

ui

nα1 ¼ 0: ð3Þ

in which u is the so-called extended averaged or Darcy velocity; n is theporosity, defined as the volume of voids over the total volume; ρ is thedensity; p is the pressure; g is the acceleration of gravity; ν is the kine-matic viscosity; and α1 is the VOF indicator function, and is defined asthe quantity of water per unit of volume at each cell.

The three last elements in Eq. (2) appear as closure terms to accountfor physics that cannot be solved when volume-averaging (i.e. frictionalforces, pressure forces and addedmass due to the individual elements ofthe porous media).

Since Darcy (1856) introduced a study of water flowing throughsand, the study of flow through porous media has been characterizedby drag forces. This first approach included only one linear term (firstterm in Eq. (4)), which was appropriate to model laminar flows. Itwas not until Forcheimer (1901), with the addition of a quadraticterm (second term in Eq. (4)), that the more energetic flows (in termsof larger Reynolds numbers) could be modelled. Polubarinova-Kochina (1962) extended the model proposed by Forcheimer (1901)to account for unsteady flows, adding a third term (third term inEq. (4)), which is transient and represents an inertial acceleration. Thefinal expression of the drag forces, as applied in Eq. (2), is presentednext:

I ¼ a uþ b u juj þ c∂u∂t ð4Þ

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 dependon the physical properties of the material, control the balance betweeneach of the friction terms.

In this work Engelund (1953) formulas, as applied in Burcharth andAndersen (1995), have been used for the friction coefficients. Neverthe-less, and as it is explained later, these parameters need calibration fromphysical tests to obtain results close to reality.

245P. Higuera et al. / Coastal Engineering 83 (2014) 243–258

The reader is referred to del Jesus (2011) for further details regard-ing this section.

2.3. Introduction to OpenFOAM®

OpenFOAM® (interFoam solver) solves the RANS equations out ofthe box for two incompressible phases, and tracks the free surfacemovement using the VOF technique. The pressure and velocity fieldsare obtained solving simultaneously the continuity (Eq. (5)) and mo-mentum conservation (Eq. (6)) equations, while the free surface istracked with Eq. (7)

∇ � U ¼ 0 ð5Þ

∂ρU∂t þ∇ � ρUUð Þ−∇ � μ eff∇Uð Þ¼ −∇p�−g � X ∇ρþ∇U �∇μ eff þ σκ∇α1

ð6Þ

∂α1

∂t þ∇ � U α1 þ∇ � Uc α1 1−α1ð Þ ¼ 0: ð7Þ

In these equations the bold letters indicate a vector field (e.g.U is thevelocity vector); μeff = μ + ρ νturb is the efficient dynamic viscosity; p∗

is the pseudodynamic pressure; and X is the position vector. The lastterm accounts for surface tension effects: σ is the surface tension coeffi-cient; κ ¼ ∇ � ∇α1

∇α1j j is the curvature of the interface. The last term inEq. (7) is a numerical artefact to avoid the excessive diffusion of the in-terface, being Uc a compression velocity.

Since pressure and velocity are coupled, the solution of both fields isobtained with a two step approach. OpenFOAM® has developed aunique methodology, called PIMPLE, which is originated by mergingPISO and SIMPLE algorithms.

OpenFOAM® follows a tracking approach rather than reconstructingthe free surface each time step. The main advantage is that it requiresless computational cost, as only an advection–diffusion equation has tobe solved. On top of this, there is no need to apply a pressure boundarycondition on the free surface. The main drawback is that Eq. (7) diffusesthe 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 0 and 1.OpenFOAM®'s special solver called MULES (Multidimensional Uni-versal Limiter for Explicit Solution) uses a limiter factor on the fluxesof the discretized divergence term to fulfil these restrictions.

For further reference regarding the governing equations and thesolving procedures see Rusche (2002) and Higuera et al. (2013a).

2.4. Numerical implementation of VARANS in IHFOAM

Adapting the equations presented in the previous section toOpenFOAM® formulation is simple, but not straightforward.

As mentioned before, the VARANS equations (Eq. (1)–(3)) are for-mulated in terms of the extended averaged or Darcy velocity (U). Theycan also be formulated in terms of the averaged or intrinsic velocity(U∗), as intrinsic magnitudes refer to the portion of fluid existing withinthe gaps of the solid skeleton of the porousmedia. The transformation isdirect:

U� ¼ Un: ð8Þ

The process to assimilate the VARANS equations in OpenFOAM®

starts by transforming the first equations to bewritten in intrinsic veloc-ities. The purpose of such procedure is clearly seen when comparingEq. (5) and Eq. (1). If OpenFOAM® solved for U∗, both equations wouldbe the same. The same happens with the VOF function advection func-tion. Therefore, this is the assumption that will be applied from now

on: OpenFOAM® will be adapted to solve the equations in intrinsicmagnitudes.

Some adjustments have to bemade in Eq. (2) in order to better iden-tify it with Eq. (6). These include multiplying both sides by ρ, and intro-ducing this variable within the differential operators (which can bedone, as the fluids are considered incompressible by definition). Bothsides are also divided by porosity, and a single term ismultiplied and di-vided by porosity. The result is presented as follows:

1þ cð Þ ∂∂tρui

nþ uj

n∂∂xj

ρui

n¼ − ∂

∂xip

þρgi þ∂∂xj

μ∂∂xi

ui

n

� �−A

ui

n−B

ui

n

�����ui

n

�����ð9Þ

for which replacements of A = ρ a, B = ρ n b have been applied. Notehow the transient friction term has been grouped on the first factor.

It is remarkable that now Eq. (9) and Eq. (6) are identical, with theonly exception of the three drag terms, which have to be added inOpenFOAM®'s momentum equation.

Recalling the previous definition of A and B and applying Engelund(1953) formulas, modified by van Gent (1995) yields:

A ¼ α1−nð Þ3n2

μD250

ð10Þ

B ¼ β 1þ 7:5KC

� �1−nn2

ρD50

ð11Þ

where D50 is the mean nominal diameter of the material. KG is theKeulegan–Carpenter number, which introduces additional friction dueto the oscillatory nature and unsteadiness of the system. It is definedas follows: To

D50

uMn . uM is the maximum oscillatory velocity, and To is the pe-

riod of the oscillation.Regarding the friction parameters,α and β have to be calibrated. The

variation in value for chas beenproven to be of little importance inmostof the cases del Jesus (2011). Therefore, according to recommendationsand previous experience, a value of c = 0.34 has been kept constantthroughout this paper.

Up to this point onlyminor changes are needed to adapt OpenFOAM®

to solve VARANS equations. In fact, a similar approach is carried out inporousInterFoam solver, which just adds some drag forces for additionalfriction.

Such procedure may work fine for single-phase flows or when theporous region is always submerged. If this is not the case and the inter-face of the fluid crosses the porous zone, the total mass of the system isnot conserved. Nevertheless, it can be used for certain applications inwhich porosity is low enough to neglect such effects (e.g. flow throughcertain vegetation fields).

However, and to be rigorous, the effect of porosity has to be intro-duced. The premise is that the amount of fluid that can enter any cellis 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 implementationprocedure has to be compatible with the restrictions that using theVOF function implies, as for example, being bounded between 0 and 1,so that the fluid properties can still be calculated as a weighted average.

From the previous derivations it is clear that themain VOF advectionfunction (Eq. (7)) needs no modifications. Therefore it is the VOF func-tion solver (MULES) element which needs to be adjusted to account forthe porosity.

A new version of MULES called IHMULES has been coded, compiledand linked to IHFOAM solver. The new solver accepts the porosityfield as additional input to modify the cell volumes. It limits the amountof fluid that can enter a cell, depending on its porosity, while ensuringthat the total VOF function value within a cell does not exceed 1.

The aforementioned elements are not the only ones to address inorder to obtain an advanced solver for coastal engineering. Turbulence

246 P. Higuera et al. / Coastal Engineering 83 (2014) 243–258

modelling is also a very important factor to take into account, as theflows interacting with porous structures are most of the times very en-ergetic and turbulence plays a great role as a dissipationmechanism, notonly around, but also inside the porous media. According to del Jesuset al. (2012) the turbulencemodels have to be volume-averaged aswell.

Currently the only turbulencemodel that has a closure developed forporous media is k-�, as presented in Nakayama and Kuwahara (1999).Even though it was initially developed for heat transport through po-rous media, the good results shown in del Jesus et al. (2012) and Laraet al. (2012) indicate that it is reasonable to extrapolate its use to coastalengineering. The k-ω SST model was also volume-averaged in del Jesuset al.'s (2012) work; no closure is provided, though. Both turbulencemodels have been adapted to work in IHFOAM inside and outside theporous media. The details regarding the new formulations are includedin the A.

3. Validation of the model

In this section the new implementation of the VARANS equations forIHFOAM is validated against laboratory measurements. First, a sensitiv-ity analysis of the porous parameters is carried out replicating the well-known Lin (1998) dam break experiments. The model is validated nextin 2D for oscillatory flow in a wave flume, simulating the interaction ofregular waves with a high mound breakwater from Guanche et al.(2009). Finally, the validation is extended for fully 3D wave interactionwith a porous obstacle within a wave tank, as presented in Lara et al.(2012).

Thewave generation and absorption boundary conditions presentedin Higuera et al. (2013a) have been applied in all the cases.

3.1. Two-dimensional porous dam break

These experiments carried out by Lin (1998) have been used asbenchmark cases for numerical models featuring flow through porousmedia. Their simple set-up andwide range of conditionsmake them es-pecially suitable for these purposes.

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4t = 0.00 s t = 0

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.2 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.2 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4t = 1.15 s t = 1

Fig. 1. Validation case: crushed rocks, h = 35 cm. α = 10000, β

3.1.1. Physical experimentsLin (1998) tested a dam break flow through different porous mate-

rials. The experiments were performed inside a glass tank (consideringan idealized 2D behaviour: 89 cm horizontally and 58 cm vertically),which permitted the use of video recording techniques to obtain thefree surface elevation all along the domain. This includes the clearflow region and also the interior of the porous medium.

The physical set-up was always the same, regardless of the porousmedium type or water level tested. The main water body was confinedon the left side of the domain, separated from the porous medium inter-face by amoving gate. This initial region spanned 30 cm in the horizontaldirection. Right next to thewater reservoir the porousmediumextendedfor 29 cm. Finally, therewas another clear flowzone between the porousmedium and the end wall, spanning 30 cm. A base level of water of2.5 cm was set all over the tank bottom, outside the reservoir. A sketchof the initial state can be found in the upper left panel in Fig. 1.

Two different porous materials were tested: crushed rocks and glassbeads, to account for different flow regimes. The flow through the glassbeads (D50 = 0.3 cm and ϕ = 0.39) was found to be laminar, and clos-er to a Darcy flow (del Jesus et al., 2012). However, the flow through thecrushed rocks (D50 = 1.59 cm and ϕ = 0.49) is fully turbulent, as ve-locities and pore size are greater. Three different water levels were test-ed: 15, 25 and 35 cm.

The experiments were started by raising the separation gate be-tween the water and the porous medium.

3.1.2. Numerical experimentsThewhole tank has been reproduced numerically in 2D. The cell size

has been kept constant and equal to 0.5 cm throughout the domain. Themesh is orthogonal and conformal, and is formed by 20,648 cubicelements.

Only the crushed rockmaterial has been considered, as the flowsweare dealing with in coastal engineering are most of the times turbulent.Furthermore, this material is closer to the ones found in rubble moundbreakwaters.

.35 s t = 0.75 s

0.6 0.8

0.6 0.8

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4.55 s t = 1.95 s

= 3. Laboratory data as circles, numerical data as points.

Table 1Sensitivity 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 1.5 3 6

5000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−5

−4

−3

−2

−1

0

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−5

−4

−3

−2

−1

0

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−5

−4

−3

−2

−1

0

1

2

3

4

5

10,000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−5

−4

−3

−2

−1

0

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−5

−4

−3

−2

−1

0

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−5

−4

−3

−2

−1

0

1

2

3

4

5

20,000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−5

−4

−3

−2

−1

0

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−5

−4

−3

−2

−1

0

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−5

−4

−3

−2

−1

0

1

2

3

4

5

247P.H

igueraetal./CoastalEngineering

83(2014)

243–258

Fig. 2. Dam break: turbulent kinetic energy.

248 P. Higuera et al. / Coastal Engineering 83 (2014) 243–258

The system starts from rest state, with the water and porous medi-um set at their initial location. h = 35 cm has been tested, as is themost nonlinear case. Turbulence is modelled using the modified k-�model.

Several porous medium parameters have been considered, as indel Jesus et al. (2012). The starting point has been α = 10000 andβ = 3. All the combinations between α and β for the given originalvalue and half and double these values have been carried out. Eachcase has been simulated for 4 s, and took less than 10 min to run insingle 2.5 GHz processor (regular PC).

3.1.3. ResultsThe results for the best case (h = 35 cm,α = 10000 and β = 3) are

shown next. The free surface elevation at certain times is presented inFig. 1. The porous medium is shaded in grey and contrasts with theclear flow region. Experimental results are shown as black circles,while numerical data is depicted as blue dots. The resolution is veryhigh, which makes them look as a continuous line.

As we can see in Fig. 1 the numerical results agree quite well withthe experimental data. Minor differences arise mainly in the first in-stants, because the lifting process of the gate is not immediate, and nei-ther it is reproduced numerically. Therefore, these discrepancies occurat the initial snapshots, and are greater towards the bottom of thetank, 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. (2012) has beencarried out to figure out the best combination of parameters (whose re-sults have been shown in Fig. 1) and to check their influence on theflow.

The best way to see the influence is to calculate the absolute error ofthe numerical solution with respect to the laboratory results. If this isdone very early in the simulation, the expected errors are high due tothe already mentioned issue with the gate separating the water andthe 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 hasbeen chosen: t = 1.95 s, this way the initial differences are dilutedenough and the system is still evolving.

All the results have been aggregated in Table 1. Each panel shows theabsolute error in water elevation. The process to obtain these values issimple. First the discrete laboratory measurements (black circles inFig. 1) are transformed into a piecewise linear function. Then, for eachnumerical point on the free surface, its difference in height with respectto the laboratory result is calculated and plotted. We will now analysethis table, referring to individual cases using their (α, β) values.

The influence of both friction parameters along with some interest-ing processes can be observed here. Finding the precise balance be-tween the linear and non-linear drag coefficients is crucial to obtainan accurate solution. An excessive drag will lead to a water excess onthe initial reservoir and a deficit on the other side of theporousmedium.This issue can be observed in all cases with β = 6 plus (20,000, 6). Thecontrary effect can also happen, as having less friction increases theflowrate through the porous medium. This case is represented by(5000, 1.5), (5000, 3) and (10,000, 1.5).

The two leftover cases show an evolution close to reality. One cannotdecide which one presents better performance directly, some indicatorshave to be calculated.We have taken into account two of them, calculatedfor the absolute value of the error curve: themaximum absolute error andtheareaunder the curve. For bothmagnitudes the case (10,000, 3) presentssmaller values: 0.97 cm vs 1.08 cm and 0.8611 cm2 vs 0.9420 cm2. Conse-quently, this case is taken as the best performance.

Other interesting facts are that the shape of the errors in the reser-voir are more or less of the same shape for all the cases, only varyingin their relative location with respect to the zero line. It is also remark-able that some of the cases present trapped pockets of air, as (5000, 3)or (5000, 6), appearing detached from the main (top) free surface.

Regarding the turbulence modelling, a snapshot showing the distri-bution of turbulent kinetic energy is presented in Fig. 2. The contour

lines indicate values that are a negative integer power of 10. As it canbe observed, the k is greater closer to the free surface, where the mostenergetic movements take place. The k also increases gradually withinthe porous medium, and especially when it accelerates as it goes out.

In conclusion, the implementation of VARANS equations isOpenFOAM® works as expected, having a close resemblance withreality.

3.2. Regular waves interacting with a high mound breakwater in 2D

The next validation casewas presented in Guanche et al. (2009), andinvolves the interaction of regular waves with a rubble mound break-water (as defined in Kortenhaus and Oumeraci (1998)), although ac-counting for its geometry it is closer to a high mound breakwateraccording to Oumeraci and Kortenhaus (1997).

3.2.1. Physical experimentsThe physical experiments took place in the University of Cantabria's

large flume. This facility is 60.0 m long (from the wavemaker mean po-sition), 2.0 m wide and 2.0 m high.

The flume bottom is horizontal. Originally, two different breakwa-ters were constructed and tested, however, here we are only reproduc-ing the high mound breakwater.

The caisson of the breakwater was made of concrete and spannedthe whole width of the flume. It was 1.04 m long and 0.3 m deep, andits seaward side was located 45 m away from the wave paddle. Thisblock laid on a gravel foundation which was 0.7 m high, and acted asthe core of the breakwater. The core presented a 10 cmbermon the sea-side. A 10 cm thick secondary armour layer made of a different type ofgravel was present on both sides. The principal armour layer was12 cm thick, and included a 14 cm long seaside berm. The slope of theporous materials was 2H/1V. A sketch of the geometry can be seen inFig. 3. The specific physical properties of the porous media are given inTable 2.

A rampwas placed behind the breakwater to dissipate the transmit-tedwaves. This devicewasmade ofmetallicmesh screens, andwill laterbe modelled as another porous medium.

Waves were generated using a piston-type wavemaker, which fea-tured an AWACS system to deal with the waves reflected by the struc-ture. The still water level was kept constant and equal to 0.8 m.Different wave conditions, including regular and irregular sea states,were tested. The reader is referred to Guanche et al. (2009) for furtherdetails.

Fourteen resistive water elevation gauges were placed along thecentreline of the flume. Their location is indicated in Table 3. The firstseven of them were placed in front of the structure. Then, the restwere located on the structure. This set is represented in Fig. 3, as verticaldotted lines. Notice that gauges 8–10 and 14 pierce the porous media,while those placed on top of the caisson lie on top of it. The last free

Fig. 3. High mound breakwater section.

Table 3Free surface gauges location.

FS gauges X (m)

1 17.0

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surface gauge was positioned behind the breakwater, to measure trans-mitted energy.

Pressure was also measured at different locations on the caisson, asshown in Fig. 4, in which the sensors are represented as bold points.The first four pressure cells were placed on the seaside of the concreteprism, two of them within the armour layers and the other two on theclear part. The rest of the gauges are located below the caisson, in con-tact with the porous core.

3.2.2. Numerical experimentsThe flume has been reproduced in its entirety in 2D: 60 m in length

and 1.3 m in height. Three horizontal zoneswith different cell size grad-ing have been defined. In the vertical direction the cell size is constantand equal to 1 cm throughout the flume.

The first zone represents the wave propagation area, and coversfrom the wave paddle (X = 0 m) to X = 40 m, close to the structure.The horizontal cell size in this sector varies from 4 cm close to thewave generation boundary to 1 cmclose to the next area. This gradationsaves computational cost by having less cells where they are not so im-portant, as close to the generation boundary.

The next region is the interest zone, and it is located fromX = 40 mto X = 48 m. The structure lies within, so the cell dimensions are cho-sen to give an adequate resolution. Cell size is constant and equal to1 cm, which provides enough detail to represent the processes takingplace around the structure.

The final zone is where the energy that surpasses the structure is fi-nally dissipated, and it covers the final 12 m. Here, the cell size variesfrom1 cmnear the structure to 3 cm at the endwall. The dissipative per-forated ramp has been replicated by means of another porous medium.

This case has been used to calibrate the porousmedia with oscillato-ry flow. The numerical parameters have already been presented inTable 2. They have been obtained by best fit of the gauges, and willlater be used for the 3D high mound breakwater.

From the initial mesh that has been described, the caisson is re-moved and reproduced as a void in the mesh. The final mesh has over355,000 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 centreslay inside the surface provided, otherwise they are identified as aclear region and keep the value 1). This approach is valid since the po-rous media equations follow a macroscopic approach. The final setup

Table 2Porous media physical properties and best fit parameters.

Material D50 (m) Porosity α β

Primary armour layer 0.12 0.5 5000 2.0Secondary armour layer 0.035 0.493 5000 3.0Core 0.01 0.49 5000 1.0Dissipation ramp 0.035 0.856 5000 3.0

is depicted in Fig. 7. Note that the vertical scale of thefigure has been ex-aggerated. The shape of the outer porous layer is uneven, this is becausea laser scanner was carried out in the laboratory and the original geom-etry has been replicated.

The whole set of laboratory measuring instruments have beenreproduced numerically. Free surface elevation has been probed at 14locations, while pressure measurements have been taken at 10locations.

The free surface elevation is equal to 0.8 m, and 20 cm in height and3 s of period waves have been generated using the cnoidal theory.

Turbulence in this case is modelled using the modified k-�model, sothat the behaviour inside the porous media addresses both macroscale(Forcheimer) and flow patterns.

The simulation has been fulfilled using 2 cores of a standard PC(2.5 GHz), and the 300 s are available after 30 h.

3.2.3. ResultsOne of the challenges of the simulation is tomanage thewave trans-

formation processes that encompass with the high mound breakwatersuch as wave reflected at the structure and the damped energy bymeans of wave breaking and flow percolation through the porousmedia. In order to dealwith the high energy reflected from the structure(90%) reported by Guanche et al. (2009), the boundary conditionspresented by Higuera et al. (2013a) have been used to generate and ab-sorb waves.

Results are presented in Fig. 5 for all the wave gauges reported inTable 3; ten in front of the structure, three over the caisson and an addi-tional one leeward the breakwater to measure the transmitted wave bythe combination of wave overtopping and flow percolation throughoutthe core.

A high degree of accuracy is demonstrated by the model in repro-ducing both wave phase and height. The wave profile, which revealsthe existence of a quasi-steady wave pattern by the combination of an

2 37.23 38.04 38.75 40.06 41.07 42.58 44.09 44.610 44.911 45.0312 45.3113 45.6214 46.84

Fig. 4. Pressure gauge distribution on the caisson.

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incident and reflected wave, is very well caught during the numericalsimulations. Larger discrepancies are observed in gauges 8, 9 and 10,which are located along the seaward breakwater slope. The lack ofbetter information to reproduce the geometry of the porous slopescould induce such slight discrepancies. The use of incompressibletwo-phase flow modelling could also affect the wave evolution in thebreaking zone, as the simulation is performed in a two-dimensionalmesh. This implies that the air cannot escape sideways, as it does in re-ality, thus, the evolution of the wave during breaking can be modified.Such issue is not observed in three-dimensional simulations, as shownlater.

Regarding the overtopping gauges (11, 12 and 13), some discrepan-cies can be observed. Due to the location of sensor 11, very close to theborder of the caisson, it is subjected to the splash of the waves. Thehigher incident waves and the enhanced splash on the physical experi-ments may yield the observed differences. However, the water layer ontop of the caisson due to overtopping (sensors 12 and 13) is reasonablywell captured in shape, especially taking into accounting the limitedmesh resolution, while underestimated. The transmitted wave (gauge14, lower panel) is not well captured by themodel. This may be a directconsequence of the lower wave height reaching the top of the structureand diminished overtopping.

From the results of the first gauges, it can be seen that thewave gen-eration and absorption boundary conditions presented in Higueraet al.'s (2013a) work adequately, as they manage to generate the targetwave while absorbing the reflected energy, as the AWACS does in thelaboratory.

Fig. 6 shows the time series of dynamic pressure. The comparison in-cludes four pressure gauges along the vertical face of the caisson(gauges 1 to 4) and six gauges underneath the caisson (gauges 5 to10). In general, the model is able to predict pressures accurately atevery location. It even captures the momentum damping induced atthe core as waves propagate underneath the caisson towards the lee-ward side of the structure. Only minor discrepancies in height andphase are shown for a few waves along the time series, especially atgauge 4. This might be another side effect of that mentioned under pre-diction of the wave height.

Two snapshots of turbulent kinetic energy around the structure arepresented in Fig. 7. The top panel shows the instant when the firstwave impacts the structure. The k level is greater around the free surfacelocation (shown as a black line) and around the primary rock layer, pre-senting a uniform value along the whole depth. At this point the turbu-lence effects start to penetrate the core.

The bottom panel presents the same situation at a mean stage ofthe simulation. The turbulence distribution both around the free sur-face and the structure has clearly diffused. However, the turbulencelevel is more or less of the same magnitude. A turbulence level build-up is expected for very long simulations, as described in Jacobsenet al. (2012), because the production terms generate turbulence evenfor potential flows. Furthermore, the turbulence models were initiallyformulated for stationary cases, and not for transient ones. It is inter-esting 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

the structure. Nevertheless, as the flow inside is close to laminar, thevast majority of its volume continues to present low values of k.

3.3. Three-dimensional interaction of waves with a porous structure

Now that the capabilities of the model have been proven to work intwo-dimensional cases it is time to extend the simulations to full three-dimensional ones. The interaction of waves with a vertical porous struc-ture from Lara et al. (2012) is now analysed.

3.3.1. Physical experimentsThe physical modellingwas carried out in the University of Cantabria's

wave basin, which is 17.8 m long, 8.6 mwide and 1.0 m high. The waveswere generated by a piston-type wavemaker formed by 10 individualpaddles that can move independently. However, all the cases had normalincidence of waves, so the whole set of paddles behaved as one. Theremaining three walls were fully reflective.

The bottom of the basin was completely flat. A porous structure wasbuilt with a metallic mesh filled with a granular material. The meanstone diameter was 15 mm and the global porosity was 0.51. Theshape of the structure was prismatic: 4 m long, 0.5 m wide and 0.6 mhigh, and resembled a porous vertical breakwater. The porous prismwas attached to one of the lateralwalls, and its seaward facewas located10.5 m away from the wavemaker. A general scheme of the setup ispresented in Fig. 8.

Thewater depthwas kept constant and equal to 0.4 m for all the ex-periments. A set of solitary waves (5, 7 and 9 cm of wave height) andcnoidal regularwaves (5 and 9 cmofwave height; 2 and4 s ofwavepe-riod) were carried out.

Free surface elevation was measured at 15 locations, as depicted inFig. 8. Pressuremeasurementswere also taken at six points on the struc-ture, shown in Fig. 9. These locations had been selected to assess thethree dimensional effects.

3.3.2. Numerical experimentsThe wave tank has been replicated numerically in its whole exten-

sion. The shape of the domain is a box (17.80 × 8.60 × 0.65 m),which makes the mesh orthogonal and conformal. The cell size variesin the X direction to save computational cost: from thewave generationboundary to 1.5 m away from the structure Δx varies from 2 cm to1 cm. Then, it is kept constant and equal to 1 cm around the porouszone to obtain better details. Finally, from 1.5 m leewards the structureto the end of the tank the Δx grows again from 1 cm to 2 cm. In the Yand Z directions the cell size is kept constant: Δy = 2 cm and Δz =1 cm. The total number of cells is slightly greater than 16.5 million.

The porous structure is treated as in the previous cases. The situationresembles the dam break case, as the structure is defined perfectly bythe cells. Therefore, the castellated approach yields to the exact geome-try. The porosity variables are set to the second best valueswhich previ-ously represented crushed rocks: α = 20000 and β = 1.5, as slightlymore accurate results are obtained with these.

Two wave conditions have been selected. First, a solitary wave of9 cm in height is tested. Then, 9 cm in height and 4 s in period cnoidalwaves are generated. In both cases, 20 s are simulated in less than17 h using 128 processors (2.6 GHz).

Free surface elevation and pressure sampling has been carried out atthe locations depicted on the experimental setup.

3.3.3. ResultsIn this section, the model is proven to reproduce the most relevant

hydraulic processes to be considered in wave–structure interaction ina three-dimensional domain, which encompass wave reflection, wavedissipation, and wave transmission resulting from wave penetrationthrough the porous structure, wave diffraction and wave run-up on aporous vertical structure.

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Fig. 10–14 present the comparison of IHFOAM predictions versuslaboratory measurements in a three-dimensional domain. Free surfaceand dynamic pressure time evolution are shown for both cases.

The solitary wave validation is presented in Fig. 10 and 11. Severalwaves can be identified at the figures as a consequence of the reflected

solitarywave at the basinwalls. Themodel shows a high degree of accu-racy in predicting both the free surface and pressure time series. Themodel simulates quite well the energy reflected at the porous structure,as can be seen on gauges 3 and 4 (Fig. 10), located seawards and close tothe porous breakwater. The transmitted and diffractedwave height (see

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Fig. 7. Turbulent kinetic energy level around the structure for the first wave impact and att = 200 s.

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gauges 6, 7, 8 and 12, Fig. 10) is perfectly reproduced numerically. Thepredictions for locations leewards the breakwater (see gauges 9, 10and 11, Fig. 10) show also a perfectmatchwithmeasured data. A lag be-tween the experimental and the numerical data is observed for thereflected wave towards the end of the series. The original solitarywave gets to reflect at several locations on the basin: the end wall, thelateral walls and even on the wavemaker. Different reflections (andre-reflections) are in general terms well reproduced both in amplitudeand shape. The lag was also reported by Lara et al. (2012) and it was at-tributed to discrepancies in severalmeasurements, including the geom-etry of the breakwater, its locationwithin the basin and slight variationsin the location of the wave gauges. Another important factor is the re-flections on the displaced wavemaker, as the numerical domain doesnot vary. The pressure measurements, presented in 11, show a perfectmatch with measurements.

Regarding the turbulence effects, several snapshots which illustratethe evolution of the turbulent kinetic energy around the porous struc-ture are presented in Fig. 12. Two longitudinal transects and a plane5 cm above the bottom are shown. The first snapshot (top left panel)shows the instant in which the solitary wave reaches the structureand starts to penetrate. The turbulent level is still relatively low. Outside

A

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Fig. 8. University of Cantabria's wave basin. Sketch of the geometry and free surface elevation gauges. Auxiliary lines each metre.

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the structure the k values are close to 0 outside the free surface interfacezone and the corners of the structure, in which some vortices arestarting to develop. A uniformbase level of turbulence appears through-out the porousmedium. The largest turbulence values occur near the in-terface of the structure, where thewave 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 medi-um. Themost dissipative zone is locatedwhere the gradients of free sur-face height are greater. It is remarkable that there is also another zoneoutside the porous medium in which k is relatively large, and this isthe area near the corners of the structure, where vortices appear. How-ever, they do not detach as expected, due to using k-� as turbulencemodel, as it can be seen in the two final snapshots (bottom panels).

The regular wave validation is plotted in Fig. 13 and 14. A similar be-haviour to the previous one is found in the comparisons between nu-merical data and laboratory measurements. The agreement is quitehigh, and the model properly reproduces the interactions between theincident and the reflected waves. Wave dissipation at the porousmedia seems to be well simulated because the reflected waves (gauges3 and 4, Fig. 13) and the transmitted waves (see gauges 9, 10 and 11,Fig. 13) appear to be accurately captured in shape and in amplitude.The non-linear interactions between the incident waves with the

C B

Fig. 9. Location of the pressure gauges on the s

multiple reflected waves at the boundaries are also reproduced by themodel, as it can be seen towards the end of the signals presented inFig. 13. The dynamic pressure measurements, plotted in Fig. 14, showalso that the model provides a good representation of the physical pro-cesses at the porous media.

As a summary, the overall agreement in the graphs presented is verygood for both cases, proving that the model is capable of handling athree-dimensional scenario.

4. Conclusions

In this paper a new numerical model called IHFOAM, based onOpenFOAM®, has beendeveloped to dealwith real applications in coast-al engineering. The validation process presented several applications,such as simple flow through a porous medium, but also wave interac-tion with porous structures in 2D and 3D. In the second part (Higueraet al., 2014–this issue), the model will be applied to fulfil the stabilityand 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 treat-ment of two-phase porous media flow in OpenFOAM® (i.e. failure toconserve mass). The special boundary conditions for wave generation

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Fig. 10. Solitary wave: free surface elevation.

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and active wave absorption presented in Higuera et al. (2013a) havealso been included as part of the model.

A review of the current approaches to treat such physics has beenpresented, yielding to the conclusion that the most suitable formulationin this case is VARANS equations, as presented in del Jesus et al. (2012).The equations have been further analysed and implemented successfully

in IHFOAM, including the k-� andk-ω SST turbulencemodels. k-� includesa closure model, therefore the additional turbulence production withinthe porous media is considered. No closure model is available for k-ωSST 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

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Fig. 11. Solitary wave: pressure signals.

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simulated to calibrate the porous media parameters. The sensitivityanalysis yields results similar to those reported in del Jesus et al.(2012). The absolute error in free surface elevation is below 1 cm,which is to be seen as a very good result and as an indicator that theimplementation in IHFOAM has been carried out correctly.

Further validation regarding wave interaction with porous struc-tures has been considered. First, the 2D test case in which regularwaves interact with a high mound breakwater shows very good resultsin the far field. Closer to the structure the comparisons are accurate, al-though discrepancies for wave height and pressure arise mainly due to

Fig. 12. Solitary wave: turbulent

the geometry of the breakwater. The 2D nature of the numerical casecombined with the two-phase flow may also be another of the causes.

IHFOAM is eminently a three-dimensionalmodel, so it is expected tobe used for 3D cases. The three-dimensional comparisons for wave in-teraction with a porous structure in a wave basin show its potential.The results for the solitary wave and the regular wave trains show ahigh degree of accordance with the laboratory data.

A special effort in assessing the effects of turbulence both inside andoutside 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

kinetic energy generation.

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Fig. 13. Regular waves: free surface elevation.

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only one with a closure model. The results were found according to theexpectations: turbulence is greaterwithin the porousmedia, as it is gen-erated when there is flow through it. High turbulent kinetic energylevels are found in the free surface interface region.

In the second part of this paper (Higuera et al., 2014–this issue) a newmethodology which enables the study of real structures in 3D will be

presented. IHFOAM will be applied to study the stability and overtoppingon a high-mound breakwater.

In short, IHFOAMhas proved to be a valuable instrument to assess thethree-dimensional effects in simulations which include porous mediaflow. The accuracy of the results presented in the validation sections indi-cates that IHFOAM is adequate to simulate coastal engineering structures.

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Gauge 2

Time (s)

η (m

)

0 2 4 6 8 10 12 14 16 18 20−500

0

500

1000

Gauge 3

Time (s)

η (m

)

0 2 4 6 8 10 12 14 16 18 20−500

0

500

1000

Gauge 4

Time (s)

η (m

)

0 2 4 6 8 10 12 14 16 18 20−500

0

500

1000

Gauge 5

Time (s)

η (m

)

0 2 4 6 8 10 12 14 16 18 20−500

0

500

1000

Gauge 6

Time (s)

η (m

)

Fig. 14. Regular waves: pressure signals.

257P. Higuera et al. / Coastal Engineering 83 (2014) 243–258

Acknowledgements

Pablo Higuera is indebted to the Spanish Ministry of Education, Cul-ture and Sports for the funding provided in the “Formación deProfesorado Universitario” Grant Program (FPU12-04354).

The work is funded by project BIA2011-26076 of the “Ministerio deCiencia e Innovación” (Spain).

Appendix A. Turbulence models adaptation

Appendix A.1. k-� turbulence model

k-� is a two equation turbulence model that solves one equation forthe turbulent kinetic energy (k, Eq. (A.1)) and another one for the turbu-lent dissipation (�, Eq. (A.2)). The already volume-averaged expressionsprovided in del Jesus (2011) are as follows:

ρ∂∂t kþ uj

∂∂xj

kn

" #

¼ nτ�ijS�ij−ρ�þ n

∂∂xj

μ þ μ t

nσk

� � ∂∂xj

kn

" #þ n CT½ �k

ðA:1Þ

ρ∂∂t �þ uj

∂∂xj

�

n

" #¼ nC�1

�

kτ�ijS

�ij−C�2ρ

�2

k

þn∂∂xj

μ þ μ t

nσ �

� � ∂∂xj

�

n

" #þ n CT½ ��

ðA:2Þ

Where:

τ�ijS�ij ¼ 2

μ t

nS�ij−

23ρknδij

� �S�ij ðA:3Þ

S�ij ¼12

∂∂xj

ui

nþ ∂∂xi

u j

n

!¼ 1

2∂∂xj

u�i þ

∂∂xi

u�j

!ðA:4Þ

μ t ¼ ρCμk2

�: ðA:5Þ

The closure terms are modelled as follows:

CT½ �k ¼ �∞ ðA:6Þ

CT½ �� ¼ C�2�2∞k∞

ðA:7Þ

�∞ and k∞ being:

k∞ ¼ 3:7 1−nð Þn32Xi

u2i ðA:8Þ

�∞ ¼ 39:0 1−nð Þ52n2 Xi

u2i

!1D50

: ðA:9Þ

Following the same procedures explained in Section 2.4:

∂∂t

knþ uj

n∂∂xj

kn¼ 1

ρτ�ijS

�ij−

�

n

þ ∂∂xj

ν þ νt

nσk

� � ∂∂xj

kn

" #þ 1ρ

CT½ �kðA:10Þ

∂∂t

�

nþ uj

n∂∂xj

�

n¼ 1

ρC�1

�

kτ�ijS

�ij−C�2

�2

nk

þ ∂∂xj

ν þ νt

nσ �

� � ∂∂xj

�

n

" #þ 1ρ

CT½ ��:ðA:11Þ

Finally, the new implementation yields:

∂∂t k

� þ u�j∂∂xj

k�¼ 1ρτ�ijS

�ij−�

�þ ∂∂xj

ν þ ν�t

σk

� � ∂∂xj

k�" #

þ 1ρ

CT½ �k ðA:12Þ

∂∂t �

� þ u�j∂∂xj

�� ¼ 1

ρC�1

��

k�τ�ijS

�ij−C�2

��2

k�

þ ∂∂xj

ν þ ν�t

σ �

� � ∂∂xj

��

" #þ 1ρ

CT½ ��ðA:13Þ

258 P. Higuera et al. / Coastal Engineering 83 (2014) 243–258

Where:

ν�t ¼ Cμ

k�2

��ðA:14Þ

k∞ ¼ 3:7 1−nð Þn72Xi

u�2i ðA:15Þ

�∞ ¼ 39:0 1−nð Þ52n5 Xi

u�2i

!1D50

: ðA:16Þ

The implementation is analogous to the momentum equation inVARANS, the only change needed is the addition of the closure terms.

Appendix A.2. k-ω SST turbulence model

The k-ω SST turbulencemodelwas developed byMenter (1994) andits distinctive feature is that it considers a blending of k-� and k-ωmodels. The result is that its performance is as good as k-ωwhenmodel-ling boundary layer flow, and as good as k-� for the free flow region.

As the transformations needed for this turbulence are identical tothe one presented above, only the final expressions are shown:

∂∂t k

� þ u�j∂∂xj

k� ¼ 1ρτ�ijS

�ij−β�ω�k�

þ ∂∂xj

ν þ σkνtð Þ ∂∂xj

k�" #

þ 1ρ

CTSST½ �kðA:17Þ

∂∂tω

� þ u�j∂∂xj

ω� ¼ 1ργνt

τ�ijS�ij−β�ω�2 þ ∂

∂xjν þ σωνtð Þ ∂

∂xjω�

" #

þ2 1−F1ð Þσω21ω�

∂∂xj

k�∂∂xj

ω� þ 1ρ

CTSST½ �ω:ðA:18Þ

Note that β∗ is not an intrinsic magnitude, but a constant of themodel. Other expressions that appear on the previous equations are:

νt ¼a1k

�

max a1ω�;Ω� F2½ � : ðA:19Þ

Two blending functions also appear in the equations F1 = tanh(arg14)and F2 = tanh(arg22). The arguments are calculated as follows:

arg1 ¼ min min max

ffiffiffiffiffik�

p

β�ω�y;500νω�y2

!;4σω2k

�

CDkωy2

" #;10

( )ðA:20Þ

CDkω ¼ max 2nσω21ω�

∂∂xj

k�∂∂xj

ω�;10−10

!ðA:21Þ

arg2 ¼ min max2ffiffiffiffiffik�

p

β�ω�y;500νω�y2

!;100

" #: ðA:22Þ

Notice that the only modification needed is adding the porosity inEq. (A.21).

No closure model for the k-ω SST turbulence model inside porousmedia is found in literature. Therefore the last term in Eq. (A.17) and(A.18) cannot be modelled yet.

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