Numerical study on free-surface flow

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, ISIS-

CFD and is commercialized by NUMECA International

as a part of the FINE/Marine computing suite. The flow

solver solves the Unsteady Reynolds-averaged Navier-

Stokes 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

2 Miguel Piteira Gomes

build the spatial discretisation of the transport equations,

and uses a face based reconstruction method for three-

dimensional 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.

( 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)

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 sub-

layer, 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 wall-

function boundary condition, which replaces the near-

wall 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 two-

dimensional submerged hydrofoil as illustrated in figure

1.

Fig. 1 – Duncan test case

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) 0.8 Depth, H (m) 0.23 Chord, c (m) 0.203 Angle, a ( ) 5 Density water (kg/m) 1000 Density air (kg/m) 1.2 Dynamic viscosity water 0.00114125 Dynamic viscosity air 1.85 10-5 Gravity (m/s2) 9.8 Reynolds number 1.423 105

Froude number 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.

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

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

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


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 finite-

difference 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.

Fig. 4 – Norm between numerical values and measured value of the

wave elevation at x=0.412744 m.

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

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.

.

Fig. 5 – Comparison of wave elevation between flow models

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.

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Título do Eixo

h

(%)


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.

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. 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. 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.


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

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.

Table 2 – Measure of the wave elevation with respect to the depth of

the hydrofoil

immersion (cm) wave elevation

(mm)

Percentage

difference

21.0 13.638 39.84

23.0 9.753 0.00

26.1 5.877 -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


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

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

(%)


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.

References

[1] WILCOX, D.C. (1998) Turbulence Modelling for CFD, 2nd ed. California: DCW Industries.

[2] QUEUTEY, P. and VISONNEAU, M. (2007) An interface captur-ing method for free-surface hydrodynamic flows. Computers & fluids, 36, pp. 1481-1510.

[3] WACKERS, J. et all (2011) Free-Surface Viscous Flow Solution Methods for Ship Hydrodynamics. Arch Comput Methods Eng, 18, pp. 1-41.

[4] DUNCAN, J.H. (1983) The breaking and non-breaking wave resistance of a two-dimensional hydrofoil. J. Fluid Mech, 126, pp. 507-520.

[5] MUSCARI, R. And DI MASCIO, A. (2002) A numerical study of breaking waves. RTO-MP-089.

[6] DI MASCIO, A. BROGLIA, R. and MUSCARI, R. (2007) On the application of the single-phase level set method to naval hydrodynamic flows. Computers & fluids, 36, pp. 868-886.

Technology

Numerical study on free-surface flow