A finite-volume solver for two-fluid flow in heterogeneous porous media based on OpenFOAM(R)
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A finite-volume solver for two-fluid flow in heterogeneous porous
media based on OpenFOAM R©
O. F. Oxtoby∗, J. A. Heyns and R. Suliman
Aeronautic Systems, Council for Scientific and Industrial Research, Pretoria, South Africa
Abstract
In this paper we derive a finite-volume algorithm for simulating two-fluid flows in
porous media. The algorithm handles arbitrary heterogeneous porosity fields contain-
ing discontinuities without introducing instabilities or oscillations into the solution. For
the two-fluid (free-surface) modelling the volume of fluid (VoF) method is used with the
HiRAC interface capturing algorithm. The developed scheme is evaluated by comparison
with high-fidelity experimental results and found to agree well. The scheme is implemented
in the OpenFOAM R© code and extends upon the capabilities contained therein.
1 Introduction
Packed beds and multiphase flows are two common features of industrial processes such as
furnaces [1] and reactor vessels [2], and yet the combination of the two is spasely covered by
the literature on numerical modelling. Further applications include wave action on harbour
breakwaters [3] and multi-fluid flows in rock beds, which encompasses the forced extraction of
oil in underground wells as well as sparging of non-aqueous pollutants [4].
In this paper our aim is to derive a numerical method for handling arbitrary heterogeneous
porosity fields, including possible discontinuities, without introducing instabilities or spurious
oscillations. At present, only homogeneous porous zones are catered for by the OpenFOAM R©
software; however, implementation of the methods presented below allows porosity to be spec-
ified as just another field, and set up using the standard tools, for example the setFields
utility.
The outline of this paper is as follows. In Section 2, the governing equations for porous multi-
phase flows are presented, then in Section 3 their discresation is derived, including the consis-
tent handling of discontinuities. Finally, Section 4 describes the evaluation problem to which
the algorithm is applied.
∗E-mail: ooxtoby@csir.co.za
1
2 Model Equations
The physical domain to be modelled contains two immiscible viscous incompressible fluids
present in the pores between a solid matrix. Initially, we focus on the fluid regions only. For
clarity, we shall refer to the two fluids as liquid and gas, although the same arguments also apply
unmodified for fluids of the same phase. For modelling purposes we may make the assumption
that there is a small mixed region where the fluids meet, and assume a continuous pressure and
velocity field. We define an indicator fraction α as
α = 1 in the liquid phase,
α = 0 in the gas phase, and
0 < α < 1 in the notional smooth transition region.
(1)
By considering mass conservation of the two fluids, one obtains, firstly, the standard incom-
pressible divergence-free constraint,
∂ui
∂xi
= 0, (2)
where ui are the components of velocity. Secondly, one obtains the interface-convection equa-
tion
∂α
∂t+
∂
∂xi
(αui) = 0. (3)
Interface-capturing algorithms are necessary to maintain the sharpness of the nearly-discontinuous
volume-fraction field α, i.e. to keep the fictional transition region between the two fluids as small
as the discretisation method will allow. This will be detailed in Section 3.1 below.
Considering now the conservation of momentum in the liquid/gas continuum, we obtain the
standard momentum equation
∂ρuj
∂t+
∂
∂xi
(ρuiuj) +∂p
∂xj
=∂
∂xi
[
µ
(
∂ui
∂xj
+∂uj
∂xi
)]
+ ρgj. (4)
Here, ρ = αρl + (1 − α)ρg and µ = αµl + (1 − α)µg are the effective material properties of
the mixture, where ρl, µl and ρg, µg are the density and viscosity of the liquid and gas phase
respectively, gj denotes the body-force acceleration in direction j and p represents the pressure
field. Due to the sharp density gradients inside the convective terms, the numerical treatment of
this equation is delicate. To avoid this, several authors [5, 6, 7, 8] use an alternative formulation
of momentum conservation in the mixture, i.e.
∂uj
∂t+
∂
∂xi
(uiuj) +1
ρ
∂p
∂xj
=∂
∂xi
[
µ
ρ
(
∂ui
∂xj
+∂uj
∂xi
)]
+ gj. (5)
However, Eq. (5) is not in conservative form and, when discretised, suffers from significant
conservation error due to the rapid change in ρ across the interface. We have found this to be
problematic for the stability of violent flows, and therefore retain the momentum-conserving
form (4) as used by several other authors [9, 10, 11, 12, 13]. We detail the numerical expedients
that need to be taken to handle the sharp density gradients in the next section.
2
Having calculated the effective governing equations for the liquid/gas combination, we use the
method of volume averaging pioneered by Whitaker [14] to obtain the governing equations for
the multiphase flow within the rigid porous medium. Using this procedure one obtains expres-
sions for spatially averaged quantities on scales greater than the pore length. Here it is important
to distinguish between the so-called intrinsic and superficial (or Darcy) values. Intrinsic values
are averages over the void region only, and are represented by 〈〉f while superficial values are
averaged over the entire space (assumed zero inside the solid) and denoted 〈〉. Thus they are
related by 〈w〉 = φ〈w〉f , where φ is the porosity of the material. Only intrinsic values are used
herein.
Following the volume averaging procedure, we obtain
∂φ〈ui〉f
∂xi
= 0 (6)
for the continuity equation,
∂φ〈α〉f
∂t+
∂
∂xi
(
〈α〉fφ〈ui〉f)
= 0 (7)
for the interface-capturing equation, and
∂φ〈ρ〉f 〈uj〉f
∂t= −
∂
∂xi
(
φ〈ρ〉f〈ui〉f 〈uj〉
f)
− φ∂〈p〉f
∂xj
+∂
∂xi
(
φ〈τij〉f)
+φ〈ρ〉f〈gj〉f +Di (8)
for the momentum equation, where τ is the viscous stress tensor expressed as
〈τij〉f = 〈µ〉f
[
∂〈ui〉f
∂xj
+∂〈uj〉
f
∂xi
−2
3δij
∂〈uk〉f
∂xk
]
(9)
and D is the porous drag term which models the interaction between the fluid and the mesh.
It was developed by Ergun [15] and proposed in the following form by Radestock and Jeschar
[16]:
Di = −
(
150(1− φ)
dpφ〈µ〉f + 1.75〈ρ〉f |〈u〉f |
)
(1− φ)
dp〈ui〉
f − cA∂〈ui〉
f
∂t. (10)
The coefficients used above are the same as determined in Ergun’s original experiments [15]:
no calibration has been done using these coefficients to fit the numerical results to experimental
data. cA is an effective added mass coefficient taken as 0.34 (1−φ)φ
[17]. This is due to the flow
recirculation caused by the porous particles which acts to increase the effective mass of fluid to
be accelerated.
3 Discretisation
3.1 Surface capturing scheme
As noted previously the HiRAC VoF method [18] is employed for the purpose of capturing the
free-surface. The HiRAC method uses, firstly, the CICSAM method of Ubbink and Issa [10] to
interpolate the volume fraction to edge centres. CICSAM, in turn, blends the Ultimate-Quickest
3
and Hyper-C interpolation methodologies, where the compressive Hyper-C method is selected
when the interface is aligned with the direction of flow, whereas the high-resolution but more
diffusive Ultimate-Quickest method is dominant when the flow is tangential to the interface.
While the CICSAM method keeps smearing to a minimum while preserving the fidelity of the
interface, it is unable to re-sharpen an interface if it begins to smear due to large velocity gra-
dients normal to the interface. To remedy this, the HiRAC method blends in a small amount of
anti-diffusion, using the expression employed in the Inter-Gamma scheme of Jasak and Weller
[19]. The VoF equation (7) is then replaced by
∂φ〈α〉f
∂t= −
∂
∂xi
[
〈α〉fφ〈ui〉f + 〈α〉f(1− 〈α〉f)φ〈uc〉
f∣
∣
i
]
, (11)
where the second term in square brackets is the additional anti-diffusive term. Here, 〈uc〉f =
cα|〈u〉f · n|n, where n is a unit vector normal to the interface calculated as n = ∇α/|∇α| and
α is a smoothed version of the volume-fraction field, as described by Heyns et al. [18]. Finally,
cα selects the amount of anti-diffusion, here set equal to 0.5.
3.2 Incompressible solution scheme
The incompressible flow equations are solved using a pressure-projection scheme; this is de-
rived by approximating the flux at cell faces (Φf ) at the following time-step using the momen-
tum equation (8):
Φf (t+∆t) ≡
[
[
φui(t)]f
+∆t
(
−φuj
∂ui(t)
∂xj
∣
∣
∣
∣
f
−φave
ρave
∂p(t + 12∆t)
∂xi
∣
∣
∣
∣
f
+Sfi (t)
ρave
)]
Ani. (12)
Here and below we drop the 〈〉f spatial averaging operator for brevity. Si =∂φτij∂xj
+ φρgi +Di,
superscript f denotes a value numerically approximated at a cell face, A is the area of the
cell face and n is the outward-pointing unit normal vector. The quantities φave and ρave denote
interpolations of porosity and density to be determined below.
To determine the unknown pressure p(t + ∆t), Eq. (12) is substituted into the discrete weak
form of Eq. (6),∑
Cell faces
Φf = 0. (13)
The resulting equation contains a discrete Laplacian operator acting on the unknown values
p(t + ∆t). Face values of the derivatives are approximated using the compact stencil as in
the method of Rhie and Chow [20] , with the non-orthogonal component evaluated explicitly
using previous-iteration values. The equation is solved using the OpenFOAM R© preconditioned
conjugate gradient solver with implicit lower-upper (ILU) preconditioning, with iterations to
converge the explicit corrections.
The solution procedure is as follows. First the previous value of the face flux Φf is modified
with the artificial compressive velocity as per Eq. (11) to give
Φ′f (t) = Φf (t) + (1− αf)Φf (t)uc|iAni (14)
4
where αf is calculated using the CICSAM scheme [10]. The VoF equation (11) is then discre-
tised as follows using the Crank-Nicolson scheme and solved using explict corrections:
∆α
∆tV =
∑
faces
αfΦ′f (t), (15)
where V denotes the cell volume and ∆α is defined as α(t+∆t)−α(t) and similarly for other
variables below.
Next, the momentum equation
∆(φρui)
∆tV = −
∑
faces
ufi (t+
12∆t)ρf (t + 1
2∆t)Φf (t)
− φ∑
faces
pf (t+ 12∆t)Ani +
∑
faces
φfµfτ fij(t +12∆t)Anj + φρgiV +DiV (16)
is solved using a simple diagonal solver to treat diagonal (e.g. sourceterms) implicitly and the
others explicitly. Here, ρf = ρlαf + ρg(1 − αf) for necessary consistency between mass and
momentum convection terms. The value of p(t + 12∆t) determined as described above is used
to obtain the interpolated face pressure values using the interpolation described below. Other
quantities evaluated at t+ 12∆t are done so using simple averages. Correction iterations are then
done to the pressure as per the PISO scheme [21] or by re-solving both momentum and pressure
equations as per the PIMPLE scheme of OpenFOAM R©.
3.3 Conditions for consistency
The objective of this paper is to develop a discretisation which is transparent to discontinuities
in the porous field, and requires no special specification of the location of these discontinuities.
Therefore, it is important to carefully consider the consistent interpolation of values to cell faces
in order to avoid oscillatory and/or unbounded behaviour, both in respect of the discontinuity in
porosity (and consequently, of velocity) and the discontinuity in density brought about by the
two-fluid flow.
3.3.1 Face porosity and averaged sourceterm
In order to determine the appropriate interpolation to be applied to the porosity and source
terms used to obtain φave and Sfi respectively in Eq. (12), we consider the simplifying case
of steady single-phase flow in a 1D channel with a step change in porosity. We stipulate that
the corresponding solution consists of a step change in velocity and pressure, at the interface
between cells n and m, namely u = um, p = pm to the left of the discontinuity where φ = φm
and u = un, p = pn to the right where φ = φn. The pressure equation Eq. (13) then reduces to
φavepn − pm
∆x= −ρaveU
un − um
∆x+ Sf (17)
and the momentum equation in the half computational cell to the left of the discontinuity reads
φm pf − pm
(1− r)∆x= −ρaveU
uf − um
(1− r)∆x+ Sm. (18a)
5
Similarly, to the right of the discontinuity it reads
φnpn − pf
r∆x= −ρaveU
un − uf
r∆x+ Sn. (18b)
In the above, r is the fraction of the distance between the centres of cell m and cell n at which
the face is located, and ∆x is the distance between the cell centres.
Eliminating the pressures from these equations, we find that
1
φave=
1
φm
(
uf − um
un − um
)
+1
φn
(
un − uf
un − um
)
(19)
and1
φaveSf =
1
φm(1− r)Sm +
1
φnrSn. (20)
The interpolation of uf between um and un depends on the convective interpolation scheme
chosen; this is flexible and user-defined. As per the above, the face interpolation of porosity is
accordingly adjusted to maintain consistency between the momentum and pressure-projection
equations.
3.3.2 Face pressure
In order to determine the appropriate interpolation of pressure, we again consider a 1D channel
with step change in porosity but do not assume steady flow. We also allow for a general two-
fluid flow. Using the same notation conventions as above, we consider the momentum equations
in the two half-cells either side of the porosity discontinuity. This gives:
∆(φmρmum)
∆t= −U
ρfuf − ρmum
(1− r)∆x− φm pf − pm
(1− r)∆x+ Sm, (21a)
∆(φnρnun)
∆t= −U
ρnun − ρfuf
r∆x− φnp
n − pf
r∆x+ Sn. (21b)
Now, we note that
∆(ρφu) = ρ(t +∆t)∆(φu) + φu(t)∆ρ (22)
where, from the interface tracking equation (15), ∆ρ/∆t can be written for the two neighbour-
ing half-cells as
∆ρm
∆t= −U
ρf − ρm
(1− r)∆xand (23a)
∆ρn
∆t= −U
ρn − ρf
r∆x. (23b)
Hence, Eqs (21) simplify to
ρm∆(φmum)
∆t= −ρfU
uf − um
(1− r)∆x− φm pf − pm
(1− r)∆x+ Sm, (24a)
ρn∆(φnun)
∆t= −ρfU
un − uf
r∆x− φnp
n − pf
r∆x+ Sn. (24b)
6
To determine face pressure from the above, we note that ∆(φmum) and ∆(φnun) are both
unknowns which are equal in the 1D case and are assumed to be approximately equal in general.
Therefore it is possible to solve for pf , resulting in the pressure face interpolation
pf = Wpm + (1−W )pn (25)
where
W =rφmρn
rφmρn + (1− r)φnρm
+ρfU [rρn(uf − um)− (1− r)ρm(un − uf)]− r(1− r)∆x(ρnSm − ρmSn)
(rρnφm + (1− r)ρmφn)(pn − pm). (26)
In the multidimensional case, we generalise this to
W =rφmρn
rφmρn + (1− r)φnρm
+ρfΦ/A[rρn(uf
i − umi )− (1− r)ρm(un
i − ufi )]ni − r(1− r)(ρnSm
i − ρmSni )ℓi
(rρnφm + (1− r)ρmφn)(pn − pm). (27)
where ℓ is the vector pointing from cell m to cell n.
3.3.3 Averaged density
Returning to Eqs (24), we now consider the steady flow case (remove the terms on the left hand
side). These equations must then be compatible with Eq. (13), which in the 1D case is
φf
p2 − p1∆x
= −ρaveUun − um
∆x+ Sf (28)
which implies that ρave = ρf .
4 Application and Evaluation
4.1 Two-fluid benchmark case
The following experimental test-case was first presented by Lin [22] and has been used by del
Jesus et al. [3] in order to calibrate drag coefficients. In our model, no calibration is necessary
and we use this test case to evaluate the accuracy of the developed numerical scheme.
The experimental set-up is as shown in Fig. 1. It consists of a tank with an open top of dimen-
sions 89.2 cm (length) by 44 cm (depth) by 58.0 cm (height) containing a porous baffle of width
29.0 cm filling the entire tank depth (44 cm) and placed with its left edge 30.0 cm from the
left hand side of the tank. The left hand side of the tank was initially filled by a water column
confined behind a trap-door and taking up a width of 28.0 cm and the entire 44 cm depth of
the tank. The height of the water column was varied, with experiments using 25.0 cm and 14.0
cm heights used here. An initial pool of water of 2.5 cm height was initially present across the
entire tank bottom.
7
Figure 1: Set-up of porous benchmark problem.
Different materials were used to fill the porous baffle, namely crushed stone and glass beads,
having porosities of φ = 0.49 and 0.39 respectively and effective radius of dp = 1.59 cm and
3 mm respectively. The fluids involved were water and air, having densities ρl = 998.2 kg m−3
and ρg = 1.205 kg m−3 respectively and viscosities µl = 1.002 × 10−3 kg m−1s−1 and µg =1.821× 10−5 kg m−1s−1 respectively.
Due to the symmetrical nature of the experimental set-up, it was assumed that the flow remained
essentially two-dimensional. Simulations were therefore performed on two-dimensional meshes.
Three meshes were used; a coarse structured mesh with cell size of 1 cm (horizontal) by 0.5 cm
(vertical), a fine structured mesh of 0.5 cm × 0.25 cm spacing, and a triangular unstructured
mesh having approximately the same number of elements as the coarse structured mesh, namely
10 500. No-slip boundaries were used on the tank walls and a prescribed total pressure set at
the top (open) boundary.
Results are shown in Figs 2–4 for various initial water column heights and porous materials. In
all cases, snapshots of the free-surface position are plotted at various different times showing
the water seeping through the porous baffle. Figure 2 shows the result for crushed stone with
the properties mentioned above and and initial water height of 25 cm, while in Figs 3 and 4 the
porous material is glass beads and the initial water heights are 25 cm and 14 cm respectively.
Figure 2 shows the comparison between the coarse structured mesh (red line) and unstructured
mesh (black line). The two are in good agreement, but with the irregularity of the mesh intro-
ducing some noise. In Figs 3 and 4, results for the coarse structured mesh are plotted in red
and for the fine structured mesh in black. The two are barely distinguishable from each other,
indicating that mesh independence of the result has been achieved.
In general the numerical and experimental results (blue crosses) show good correlation. There
is some discrepancy in the beginning where the lower portion of the measured interface position
has moved further to the right than the simulated one. This is, however, consistent with it taking
a finite time to open the trap-door constraining the water; indeed Lin [22] states that the process
took approximately 0.1 s.
5 Conclusion
In this paper we have outlined the development of a general heterogeneous porous solver im-
plemented using OpenFOAM R© technology. Validation against experimental results has been
8
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8
t = 0 s
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8
t = 0.2 s
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8
t = 0.4 s
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8
t = 0.6 s
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8
t = 0.8 s
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8
t = 1 s
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8
t = 1.2 s
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8
t = 1.4 s
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8
t = 1.6 s
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8
t = 1.8 s
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8
t = 2 s
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8
t = 2.2 s
Figure 2: Free-surface position – experimental vs numerical. Porous material: crushed stone,
initial water height: 25 cm. Mesh comparison: coarse structured (red) vs unstructured (black).
9
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8
t = 0 s
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8
t = 0.4 s
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8
t = 0.8 s
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8
t = 1.2 s
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8
t = 1.6 s
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8
t = 4 s
Figure 3: Free-surface position – experimental vs numerical. Porous material: glass beads,
initial water height 25 cm. Mesh comparison: coarse (red) vs fine (black) structured.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8
t = 0 s
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8
t = 0.4 s
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8
t = 0.8 s
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8
t = 1.2 s
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8
t = 1.6 s
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8
t = 4 s
Figure 4: Free-surface position – experimental vs numerical. Porous material: glass beads,
initial water height 14 cm. Mesh comparison: coarse (red) vs fine (black) structured.
10
provided for a test case involving a discontinuous porosity field and both structured and un-
structured meshes. Source code for a version of the solver implemented as an extension to the
interFoam solver will be made available on www.opensim.co.za.
Acknowledgements
This work was funded by the Council for Scientific and Industrial Research (CSIR) on Young
Researcher Establishment Fund Grant nr. YREF 2011 36.
References
[1] Mehdi Kadkhodabeigi. Modeling of Tapping Processes in Submerged Arc Furnaces. PhD
thesis, Norwegian University of Science and Techology, Trondheim, 2011.
[2] H.A. Jacobsen. Chemical Reactor Modeling. Springer, Berlin, 2008.
[3] M. del Jesus, J. L. Lara, and I. J. Losada. Three-dimensional interaction of waves and
porous coastal structures: Part I: Numerical model formulation. Coastal Engineering,
64:57–72, 2012.
[4] P. Bastian. Numerical computation of mulitphase flows in porous media. Habilitationss-
chrft, Christian-Albrechts-Universitat Kiel, 1999.
[5] Y. Andrillon and B. Alessandrini. A 2D+T VOF fully coupled formulation for the calcula-
tion of breaking free-surface flow. Journal of Marine Science and Technology, 8:159–168,
2004.
[6] R. Panahi, E. Jahanbakhsh, and M. S. Seif. Development of a VoF-fractional step solver
for floating body motion simulation. Applied Ocean Research, 28:171–181, 2006.
[7] L. U. Lin, L. I. Yu-cheng, and T. Bin. Numerical simulation of turbulent free surface flow
over obstruction. Journal of Hydrodynamics, 20(4):414–423, 2008.
[8] D. Liu and P. Lin. Three-dimensional liquid sloshing in a tank with baffles. Ocean Engi-
neering, 36:202–212, 2009.
[9] F. Ozkan, Ma. Worner, A. Wenka, and H. S. Soyhan. Critical evaluation of CFD codes for
interfacial simulation of bubble-train flow in a narrow channel. International Journal For
Numerical Methods In Fluids, 55:537–564, 2007.
[10] O. Ubbink and R. I. Issa. A method for capturing sharp fluid interfaces on arbitrary
meshes. Journal of Computational Physics, 153:26–50, 1999.
[11] J. Liu, S. Koshizuka, and Y. Oka. A hybrid particle-mesh method for viscous, incompress-
ible, multiphase flows. Journal of Computational Physics, 202:65–93, 2005.
[12] T. Wacławczyk and T. Koronowiczy. Modeling of the wave breaking with CICSAM and
HRIC high-resolution schemes. In S. Wesseling, E. Onate, and J. Periaux, editors, Euro-
pean Conference on Computational Fluid Dynamics ECCOMAS CFD, 2006.
11
[13] I. R. Park, K. S. Kim, J. Kim, and S. H. Van. A volume-of-fluid method for incompressible
free surface flows. International Journal for Numerical Methods in Fluids, 61:1331–1362,
2009.
[14] S. Whitaker. Diffusion and dispersion in porous media. American Institute of Chemical
Engineers Journal, 13:420–427, 1967.
[15] S. Ergun. Fluid flow through packed columns. Chemical Engineering Progress, 48(2):89–
94, 1952.
[16] J. Radestock and R. Jeschar. Theoretische Untersuchung der gegenseitigen Beeinflus-
sung von Temperatur- und Stromungsfeldern in Schuttungen. Chemie Ingenieur Technik,
43(24):1304–1310, 1971.
[17] M.R.A. Van Gent. Formulae to describe porous flow. In Communications on Hydraulic
and Geotechnical Engineering, No. 1992-02, number 92-2, Delft, 1992. Delft University
of Technology.
[18] J. A. Heyns, A. G. Malan, T. M. Harms, and O. F. Oxtoby. Development of a compressive
surface capturing formulation for modelling free-surface flow using the volume-of-fluid
approach. International Journal for Numerical Methods in Fluids, 71:788–804, 2013.
[19] H. Jasak and H. Weller. Interface tracking capabilities of the Inter–Gamma differencing
scheme. Technical report, CFD research group, Imperial College, London, 1995.
[20] C.M. Rhie and W.L. Chow. Numerical study of the turbulent flow past an airfoil with
trailing edge separation. AIAA Journal, 21(11):1525–1532, 1983.
[21] R. I. Issa. Solution of the implicitly discretised fluid flow equations by operator-splitting.
Journal of Computational Physics, 62:40–65, January 1986.
[22] Pengzhi Lin. Numerical modeling of breaking waves. PhD thesis, Cornell University,
1998.
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