Interface capture of a multi-fluid flow

1. June, 2012
École Centrale de Nantes
Numerical study on free-surface flow
Miguel Piteira Gomes
Abstract
The following report aims to study some of the parameters involved in free-surface flow. The study was conducted for a
submerged hydrofoil which generates a steady wave-train behind it. The wave profile was computed numerically and
compared to experimental results. A study on grid convergence was conducted for the
SST model was performed
based on the convergence of the Drag. The influence of the turbulence model was analysed by comparison with a laminar
model. A study on the relation between the depth of the hydrofoil and the wave height was performed. The study also
broaches on the influence of the discretisation schemes.
1. Introduction
In 1982, James Duncan presented some results on several
tests performed on a two-dimensional hydrofoil. In this
project we shall try to replicate some of his numerical
procedures and make comparisons with the experimental
values obtained in his study.
The study of a submerged hydrofoil is interesting on the
perspective of the simulation of free-surface flow. The
solution of such problem is to find the location of the
material interface between water and the atmosphere.
This is now a matured topic and it is used in industrial
marine applications namely for the design and
optimisation of ship hulls.
In this study we shall use the ISIS-CFD solver to
simulate this flow, and to understand some of the
particularities of such a problem, and how parameters
related to the set-up or to the numerical methods, can
affect the results.
This report was divided in four parts, considering this
first part to be a short introduction to the subject.
The second part presents a short reference to the flow
solver. The objective of this section is to present the
governing equations and the discretisation schemes used
by the solver, as well as some theoretical aspects that
might make it easier to understand the results, but would
be too ponderous on the presentation of the results.
The third section is the main section of this report, where
the five cases that constitute this study shall be presented
and discussed.
The first case presents the first results for the wave
pattern on a coarse mesh, comparing it with experimental
values. It also explains what boundary conditions and
numerical schemes were used in the subsequent studies.
The second case study has to do with the accuracy of our
results, based on a grid convergence study. The objective
is to understand whether we can trust the results we
obtain numerically.
The third study as to do with the uncertainty of the
turbulence model. Here we shall investigate whether we
can obtain better results using a laminar model for a flow
which is clearly turbulent.
We then proceed with a study to investigate a possible
relation between the depth of the submerged hydrofoil
and the height of the waves generated by it.
Finally, in the fifth case we shall look at the influence of
the discretisation schemes for the transport equations on
the same turbulence model.
The fourth section reports to the conclusions made from
all the computations conducted throughout the case
studies, with the objective to also summarise the main
points observed in the results section in a concise
manner.
2. General background
2.1. Governing equations
The flow solver used in this study was created by EMN
(Equipe Modélisation Numérique) under the name, ISISCFD and is commercialized by NUMECA International
as a part of the FINE/Marine computing suite. The flow
solver solves the Unsteady Reynolds-averaged NavierStokes equations which in the multi-phase continuum, for
an impressible viscous fluid under isothermal conditions,
can be written in the form of equations 1, for mass
conservations, and 2, for momentum conservation.
( 1)
( 2)
In equations 1 and 2, is the control volume, bounded
by the surface , moving with velocity
. The mean
quantities and represent respectively the velocity and
pressure fields. is the identity matrix,
represents the
viscous stress tensor components and
is the gravity
vector.
The flow solver is based on the finite volume method to

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Miguel Piteira Gomes
build the spatial discretisation of the transport equations,
and uses a face based reconstruction method for threedimensional unstructured meshes with an arbitrary
number of constitutive faces.
When the grid is moving the space conservation law in
equation 3 must also be satisfied.
( 3)
2.2. Interface capturing techniques
Thirty years have passed since Duncan presented his
results for the two-dimensional submerged airfoil in [4].
Nowadays, several methods exist to simulate the viscous
free-surface flow, which can be qualified based on the
discretisation methods used for the water surface. The
two main types of methods for water discretisation are:
fitting methods, where the computational mesh is
deformed, making the water surface as a boundary; and
capturing methods, where the water surface is located in
the interior of the mesh.
Capturing methods can then be divided in two types,
depending on the definition of the surface that intersects
the grid dividing the two fluids. The original type, used
the reconstruction process to find this surface, and the
level set technique has played a large role on this method,
as a tracking device to locate the actual position of the
surface.
The second type of capturing methods does not use
reconstruction, instead after calculating the volume
fraction of each fluid in the cells, a numerical
discontinuity is imposed at the interface, and this latter
shall be the technique used by our flow solver.
The idea of this capturing method is to use a scalar
indicator function, which assumes values between zero
and one for the volume fraction. The values in between
the two integers represent a mixture between the two
fluids, with a direct indication of the relative proportion
of fluid occupying the cell, and thus the value at the
interface in equal to 0.5. As explained in [3] the great
advantage of this technique is that only one transport
equation has to be solved to determine the proportion of
fluid in each cell, and this is equation 4.
2.3. Boundary conditions
The turbulence model used throughout this project was
the near-wall low-Reynolds SST
model. Near-wall
means that the flow is computed up to the viscous sublayer, and thus the mesh needs to be fine enough to
capture the flow. However, for free surface flow
simulation, the flow solver provides an innovative wallfunction boundary condition, which replaces the nearwall low Reynolds number formulation. This method is
employed to avoid any difficulties related with the
behaviour of the interface in the vicinity of the wall for
very fine grids, and it was used as a boundary condition
for the hydrofoil. It is based on the law of the wall as the
constitutive relation between velocity and surface shear
stress. The wall functions are then determined from the
absolute value of the surface shear stress yielding new
equations for and at the grid points close to the body.
At the top and outflow boundaries of our control volume
we shall require the model to impose hydrostatic pressure
with the fluid at rest. The alternative to this boundary
condition would be to update the hydrostatic pressure
given the water height at the boundaries. At the top this
frozen pressure boundary condition shall not represent a
numerical problem, but at the outflow boundary, as the
mesh is refined, the model shall capture the interface to a
larger extent, which will lead to some reflection from the
value at the boundary to the values for the pressure near
the boundary. The only way to correct this would be to
increase our domain of interest, as we want to consider
that the flow is at rest in the far field. To update the value
of the hydrostatic pressure is not an option either as
instead of trying to dissipate the value of the pressure at
the boundary we would be assuming that it is different
than the value when the fluid is at rest.
3. Case studies
As mentioned in the introduction five test cases will be
presented in this section, based on different aspects that
involve free-surface flow computations. This study was
based in the Duncan tests performed on a twodimensional submerged hydrofoil as illustrated in figure
1.
( 4)
The drawback is on the accuracy of this approach, which
will rely on the discretisation schemes used, as they may
be too diffusive in the vicinity of the interface. This shall
motivate our final study on the influence of the
discretisation scheme.
The effective flow physical properties, that is, the
dynamic viscosity and density, shall be obtained with
respect to the volume fraction, as translated by equation
5.
(5)
Fig. 1 – Duncan test case

3. Numerical study on free-surface flow
3
Unless specifically mentioned the properties used are
summarised in table 1. For the fourth case, the relation
between the depth of the hydrofoil and the wave height is
related to a direct change in the parameter H in figure 1.
Table 1 – General parameters
Incoming velocity, U (m/s)
Depth, H (m)
Chord, c (m)
Angle, a ( )
Density water (kg/m)
Density air (kg/m)
Dynamic viscosity water
Dynamic viscosity air
Gravity (m/s2)
Reynolds number
Froude number
0.8
0.23
0.203
5
1000
1.2
0.00114125
1.85 10-5
9.8
1.423 105
0.567
3.1. Interface capturing
The primordial objective of this study was to identify the
main parameters the may play a role in the interface
capture of a multi-fluid flow. In this sub-section we are
interested in computing the interface between the two
fluids, using the method explained in section two.
Hence, we shall model our two-dimensional domain
using a rectangular box which shall account for the
presence of the two fluids, and compute the interface by
calculating the volume fraction of each fluid. Hence
special care needs to be taken on the grid, as the region
where the free surface may lie must account with a finer
mesh, just as the region close to the hydrofoil where a
wall-function will be applied.
schemes highly depend on this factor, a very short time
step of 0.003 seconds was used for 2000 iterations.
The hydrofoil was considered to go from rest to -0.8m/s
in one second on a 1/4 sinusoidal ramp.
Using the parameters above the pattern of the flow at the
interface was found to be the one illustrated by figure 2.
In figure 2, the results are scaled by the reference length
of the hydrofoil. The pattern was very well captured by
the model, but the amplitude of the waves seems to be
quite underestimated. However, this result was obtained
using a coarse mesh.
The following study will analyse whether it is possible to
obtain better results with finer meshes.
3.2. On error estimation
In order to provide estimates for the accuracy of our
measurements, two key aspects should be addressed: the
iteration convergence, and the grid convergence. As
explained well in [1], the iteration convergence error is
defined as the difference between the current iterate and
the exact solution to the difference equations. The
difference between successive iterates is often used as a
measure of the error in the converged solution, but this is
in itself inadequate. A small relaxation factor can always
give a false indication of the error, and in our
computations a relaxation factor of 0.5 is used for the
volume fraction, and 0.3 for the computation of the
pressure.
Fig. 3 – Convergence of the forces applied on the hydrofoil
Fig. 2 – Comparison between the wave patterns
Regarding boundary conditions in the far field, an inflow
velocity of zero m/s was defined; prescribed frozen
pressures at the top and outflow boundaries of the
domain was set; and slip condition at the bottom.
As the convergence of the numerical discretisation
In this study we shall not calculate the difference
between the final value of the drag force and its
analytical value, because we do not know the analytical
value. As the difference between successive iterates is
said to be inadequate we shall not do that either.
However we shall look at the evolution of forces with the
time iteration as a measurement of the iteration
convergence, as these forces should stabilize. The reason
we will not look at the residuals is due to the fact that our

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Miguel Piteira Gomes
results will never converge, as it can be seen in [5].
As figure 3 shows, the forces converge in time at about
5000 time iterations. As it was stated in section 3.1 we
are using a time step of 0.003 seconds, thus at
we have performed
iterations.
The
oscillations observed from here onwards shall continue
indefinitely due to the several aspects, such as the way
we compute the free surface and the reflection from the
outflow boundary condition.
As we impose an abrupt change from zero to one within
two cells, for capturing the interface, some oscillations
will rise on the computations of the error, and as our
discretisation increases the effect of the outflow
boundary condition, registering the hydrostatic pressure
with the fluid at rest at the outflow boundary, will cause a
reflection from the result at the boundary on the results
near the boundary. This is not ideal as we want to capture
the wave from the front and we want to consider that the
far field is at rest.
The study of grid convergence relies on the existence of
discretisation errors related to the finite size of the finitedifference cells. These errors represent the difference
between the solution to the differential equation and the
exact continuum solution to the differential equations.
As we do not have an analytical solution for this case we
shall base our study on the computation of the norm
between the measure value for the value of the highest
wave elevation and the numerical value computed at the
same location, for different meshes. All the parameters
and flow model will remain constant and the number of
cells doubled in each direction for each case. Figure 4
shows the wave elevation with respect to the cell size, h.
The value taken for the cell size was based on dividing
the number of cells used in the x-direction by 100.
amplitude higher than the second finest mesh. As the
value of the cell size approaches zero the results should
converge to a constant difference to the experimental
values and the variation observed cannot allow us to
declare that we have grid convergence. It is thus not
possible to prove the numerical accuracy in any of the
grids.
The large percentage difference to the experiments
indicates that we either do not have very accurate
experimental values, or that our model is not capable to
fully estimate the real flow.
From this study we could also observe an increase on the
waves’ amplitude with the grid refinement. As the
turbulent model always yields wave amplitudes smaller
than those observed in experiments, a smaller difference
to the experimental value means a higher value for the
wave elevation.
3.3. Turbulence Model
In the previous sub-section we have said that there might
be a problem with our turbulent model, as it does not
seem to fully capture the wave elevations when
comparing very fine grids to numerical experiment
results.
As this model has proven to behave satisfactorily for
separated flows over airfoils, as commented in [2], we
shall not compare our results with another turbulence
model. Instead, in this section we shall compare the
results with a laminar model, as according to the research
community the Laminar model seems to have a more
accurate behaviour for this particular type of flow.
.
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Título do 30
Eixo
25
20
(%)
15
10
5
0
0
1
2
3
4
h
Fig. 4 – Norm between numerical values and measured value of the
wave elevation at x=0.412744 m.
Fig. 5 – Comparison of wave elevation between flow models
The study of grid convergence is done to insure that a
fine enough grid has been used in order to reduce the
error to an acceptable level. In this case we can see that
our results differ from the experiments in about 23%, for
very fine meshes. However, they do not seem to converge
as the finest mesh presents a value for the wave
From a physical point of view this is extremely hard to
understand, just by looking at the Reynolds number we
can see that we are in the presence of a turbulent flow, so
it would make no sense to use a laminar model to capture
a turbulent flow. Hence, this study may help us to
understand slightly better the particularities of this flow.

5. Numerical study on free-surface flow
In the laminar study all the parameters remained the
same, and in order to attempt a convergence of the forces
10 000 iterations in time were used. As a laminar model,
it would not be possible to impose a wall function at the
hydrofoil, and instead the no slip condition was used. The
comparison between the wave elevations for the two
models is illustrated by figure 5. There, we can see that
the laminar model overestimates the amplitude of the
wave but still gives a very accurate prediction of the
flow, which is quite surprising. From the first case, we
saw that the
SST model underestimates the
amplitude of the wave meaning a lower value for the
forces applied on the hydrofoil. The fact that the laminar
model can obtain such prediction for the wave elevation
for a turbulent flow indicates that the viscous forces are
dominant over the inertial forces. As we are in the
presence of a turbulent flow, characterised by a
dispersive behaviour it would be interesting to look at the
forces applied on the hydrofoil for the laminar case.
Figures 6 and 7 show the convergence of the forces in the
x and y directions with respect to time for the two cases.
5
Fig. 7 – Convergence of the lift in time
The difference in amplitude of the waves is directly
connected to a difference in forces. Higher waves
correspond to higher forces. In the laminar model there is
no convergence of these forces in time, and this is due to
the fact that we are using a laminar model to capture an
erratic flow. As the value of the Reynolds number is of
the order of five, this yields the vortex shedding shown in
figure 8, for the velocity field.
Fig. 6 – Convergence of the drag in time
In figure 6 it comes to reason why we cannot use a
laminar model for turbulent flow. Turbulence flows are
dispersive by nature, due to the nonlinearity of the
convection terms, as the velocity field will generally
depend on the transported variable. However, by
superposition we can see that the turbulent model
captures the mean of the laminar behaviour, with lower
values justified by the difference in amplitude of the
waves.
The drag force is connected to the viscous stress, and it is
a source of heat for the flow, the power of this force is
equal to the power of energy that is dissipated into heat, a
result that comes from the conservation of energy.
In laminar flows the viscous forces are predominant, thus
for the laminar model to work it is expected that the two
curves exhibit the same pattern.
Fig. 8 – Velocity field in the x-direction for the laminar model
On the other hand the turbulent model captures the real
flow and provides mean values for the forces that
relatively converge in time. The use of the word relative
lies in the fact that due to several reasons, mainly
discussed in the previous section, we shall never observe
an absolute convergence. However, the difference
between results for the wave elevation is smaller than it
would be expected.

6. 6
Miguel Piteira Gomes
to study the influence of these parameters, in order to
understand where experimental errors may play a role.
In this section we shall study whether the depth of the
hydrofoil, measured by H in fig.1, will have a relation
with the wave elevation.
In this study we have used a fine mesh, with 10 000
iterations in time in order to make sure our results would
be as accurate as possible.
Three cases were studied, with h assuming the values of
21.0 cm, 23.0 cm (standard case), and 26.1 cm. The wave
patterns for the three cases are illustrated in figure 10.
From figure 10 we can be sure that the immersion of the
hydrofoil does have a relation with the wave height. We
shall not attempt to find a trend-line as too many
parameters, but we shall quantify this relative difference
in terms of percentage.
Fig. 9 – Mean velocity field in the x-direction for the turbulent model
Even though, the laminar model presents accurate results
we cannot expect that it behaves accurately with an
increase in the velocity of the hydrofoil. In fact, as it has
not been designed to capture turbulent flows, it is still
difficult to accept it as an option to be considered.
Also, from an engineering point of view, it is also
interesting to compare the value of the mean velocity
given in figure 9 with the instantaneous velocity captured
by the laminar flow in figure 8. However, the results have
shown that our turbulent model lacks expertise and we
cannot trust the results as depicting the real flow.
3.4. Relation immersion vs. wave height
The study of a submerged hydrofoil involves several
parameters which may influence the generation type of
the steady wave-train behind it.
Fig. 10 – Comparison between wave heights for changing depth
As we are comparing our numerical results it is important
Table 2 – Measure of the wave elevation with respect to the depth of
the hydrofoil
immersion (cm)
wave elevation
(mm)
Percentage
difference
21.0
23.0
26.1
13.638
9.753
5.877
39.84
0.00
-39.74
From table 2 we see that for a difference of 2 cm going
upwards and 3.1 cm going downwards we observe a
relative difference of almost 40%. This is quite a large
difference which implies that the depth of the hydrofoil is
a key parameter to take into account in this study. It also
means that to compare results from cases where the
hydrofoil is not exactly at the same location will prove
senseless.
3.5. Influence of discretisation schemes
In this study we have been using the
SST model
proposed by Menter in 1993. The model itself is very
popular, but the discretisation schemes selection can
condition the efficiency of the model thus the need for
this type of analysis. As stated in section 2, we have been
using in this project the Blended Reconstructed Interface
Capturing Scheme (BRICS), used to compute the value
of the volume fraction at the interface, and the
AVLSMART for the convective fluxes.
As special care needs to be paid to the diffusivity on the
vicinity of the interface, this new method was thought to
be a blend from several methods, each with its own
advantages. Details of this discretisation scheme are fully
explained in [2], but intrinsically we can say it lies in the
following specific requirements: to assume face bounded
reconstructions for face based topologies; to avoid
unrealistic oscillations that arise from the sharp
discontinuity from zero two one within two cells; to find
an acceptable compromise between accuracy, obtained
with the CDS scheme, and boundedness, obtained with
the UDS scheme; to introduce downwind information

7. Numerical study on free-surface flow
and change any smooth gradient into a step function by
means of compressive differencing scheme such as DDS;
and to, hopefully, eliminate the Courant number
limitations.
However, can we prove the efficiency of this
discretisation method? The answer to this question lies in
figure 11, where we compare the results obtained for the
wave elevation with the AVLSMART method for the
convective fluxes and the BRICS method for the volume
fraction transport equation, with the results obtained with
an upwind method for the convective fluxes and the GDS
method for the volume fraction.
(%)
Fig. 11 – Norm between numerical values and measured value of the
wave elevation at x=0.412744 m for two numerical scheme approaches.
Figure 11 shows that with the approach that we have used
in all the studies of this report we had better results than
if we had used the upwind scheme for the convective
fluxes and the Gama scheme for the volume fraction. We
see in figure 11 that we obtain the same result with a
coarse mesh for the first approach as using a mesh four
times finer for this approach. In terms of industrial
practice this is an important remark as a coarser mesh
will require less computational effort.
Fig. 12 – Wave elevation for two different discretisation schemes on the
same mesh
7
Moreover, when analysing the results computed using the
upwind discretisation for the convective fluxes, we
observed some irregularities that can be associated to the
problem of convergence for this discretisation scheme,
when 2000 time iterations were applied, figure 12, and
could also be influenced by the outflow boundary
condition.
As discussed in section 2.2, the interface capturing using
the transport equation for the volume fraction brings
simplicity to the model. However, as it was mentioned,
the accuracy of this approach will rely on the
discretisation schemes used, as they may be too diffusive
in the vicinity of the interface. Figure 12 illustrates such
statement proving that the discretisation scheme will also
affect the accuracy of the results as the upwind scheme
underestimates even more the amplitude of the waves.
4. Conclusions
In this report we have started by looking at the wave
pattern obtained numerically using a coarse mesh. From
observing the results, which were scaled by chord of the
hydrofoil, it was concluded that the pattern was very well
captured by the model, although the amplitude of the
waves was underestimated when compared to
experimental values.
Then it was found that we cannot fully trust our
numerical results as several conditions affect our
solution, such as the strict way the interface is imposed
by demanding an abrupt change from zero to one within
two cells, for the volume fraction transport equation,
yielding some oscillations on the convergence of the
forces; or even as the outflow boundary condition
defined as a frozen pressure will cause some reflection
that will affect the results as the discretisation increases.
As the value of the cell size approached zero the results
did not converge to a constant value, thus it was not
possible to prove the numerical accuracy in any of the
grids.
In the
SST model it was also observed an increase
on the waves’ amplitude with the grid refinement.
As it has been discussed by the research community,
some modelling errors exist for this particular case. The
laminar model registered higher wave amplitudes than
the experiments, and the full turbulent models registered
lower amplitudes than the experiments.
From the study of the models it was possible to recognise
that the viscous forces are dominant and the inertial
forces do not play a large role, as the laminar model
provides surprisingly good results for the wave pattern.
Results for the velocity fields and convergence of forces
for the two models was also interesting on a point of
view of what turbulence modelling adds to numerical
simulation.
The study of the depth of the submerged hydrofoil as a
parameter that influences the height of the generated
waves, proved positive, experiencing differences in
between wave heights of 40% when moving it on about

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Miguel Piteira Gomes
8% relatively to the free surface. This large difference
implied that the depth of the hydrofoil is a key parameter
to take into account in this study, and that to compare
results from cases where the hydrofoil is not exactly at
the same location will prove to be inadequate.
The importance of the numerical schemes for
discretisation of the transport equations plays a large role
in industrial applications where accurate results are
needed in short time. It was also seen that three main
ways to capture the interface in the free-surface flow
exist. The approached used by the numerical solver used,
of implementing a transport equation for the volume
fraction adds simplicity to the problem, as it is dependent
only on the type of discretisation scheme. The last point
here presented was to verify that the BRICS scheme
eliminates this potential problem on using this approach.
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[2]
[3]
[4]
[5]
[6]
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& fluids, 36, pp. 1481-1510.
WACKERS, J. et all (2011) Free-Surface Viscous Flow Solution
Methods for Ship Hydrodynamics. Arch Comput Methods Eng,
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DUNCAN, J.H. (1983) The breaking and non-breaking wave
resistance of a two-dimensional hydrofoil. J. Fluid Mech, 126,
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