along the diffuser wall whenever the SST turbulence model is used.

This mesh has proved to be fine enough to lead to mesh independ-

ent results for velocity profiles as well as turbulence profiles.

ments in the radial and axial directions, respectively. The slice o +is 2 wide and the wall resolution assures 0.52 < y < 1.38 every-

where along the diffuser wall whenever the SST turbulence model

is used. This mesh has proved to be fine enough to lead to mesh in-

dependent results for velocity profiles as well as turbulence pro-

files.

INLET RADIAL VELOCITY AND TURBULENCE

The first simulations of the conical diffuser were made using a

zero radial velocity on the inlet boundary condition which was

thought to be a reasonable choice since the inlet plane is located in

a cylindrical duct, 25 mm upstream from the diverging section.

However, the unsatisfying results obtained quickly showed the im-

portance of the radial component of velocity in the development of

the whole flow field. A rotating cylinder was then added upstream

of the domain to simulate the swirl generator used in the experi-

mental study. This addition revealed the disregarded presence of a

small, but essential, radial velocity component at the inlet plane.

Adding this numerically computed velocity component dramati-

cally improves the agreement with the experimental data even

though Ur peaks at less that 3% of the average inlet axial velocity.

Next, the common approximation

U = U tan θr z

where θ is a function of the diffuser half-angle defined by θ = θ

wall (r/R) is compared with the other two cases. Figure 2 illustrates

the velocity profiles near the exit of the diffuser for the three radial

velocity profiles specified at the inlet. Clearly, the effect of Ur must

not be neglected.

The results presented in Figure 2 can be examined using the

continuity equation expressed in cylindrical coordinates. Under the

hypothesis of axisymmetry, the equation simplifies to ∂/θ∂ term

drops and the equation simplifies to

We deduce from this expression that the presence of a non-

zero radial velocity thus imposes the spatial rate of change of the

one is decelerating. It can be noted on Figure 2a that the near-wall

ABSTRACT

This numerical study aims to assess the influence of boundary conditions and mesh refinement on the flow topology in draft tubes, us-

ing the commercial code ANSYS CFX and two equations turbulence models, especially Menter's SST. The final objective of this investiga-

tion is to improve the RANS predictions of draft tubes and more specifically the flow characteristics associated with a significant turbine effi-

ciency drop occasionally observed near the best efficiency point in rehabilitation projects. The first step of the study is to reproduce a

well-documented test case, the swirling flow inside a conical diffuser available on the ERCOFTAC's database.

Among the most important parameters, it is found that the inlet radial velocity must be specified with great care for this test case since

it is directly related to wall separation or core flow recirculations. Further, we observe that the outlet treatment used to simulate a dis-

charge to ambient air mainly impacts the outlet pressure distribution. Actual draft tube geometry from the Chute-à-la- Savane project is

then used to confirm and extend the results validity. One of the most important results obtained in this geometry comes from the inlet tur-

bulence. It is found that this parameter alone is sufficient to modify the flow topology and radically change the draft tube's efficiency. The

apparent sensitivity of the physics associated with the efficiency loss near the design point of some rehabilitated turbines thus requires for

a very careful and advised, complete specification of the boundary conditions.

Key words: Draft tube, RANS modelling, boundary conditions sensitivity.

radial gradient of Ur is much greater for the approximated curve

than for the computed one, which leads to an increased velocity

peak on the corresponding Us profile as shown in Figure 2b. Also

note that the grey zone in Figure 2a defines a region where the flow

should be accelerating according to the computed profile but is in-

stead decelerating, causing the axial velocity peak to be closer to

the diffuser wall. Since the mass flow must be the same in all three

cases, a higher near-wall velocity leads to a lower level of kinetic en-

ergy around the axis. Accordingly, a recirculation bubble is shown

by the solid curve.

Opposite from this behaviour, imposing no radial velocity in the

inlet plane implies that no kinetic energy is transferred toward the

boundary layer, and its weakness eventually leads to the separa-

tion of the flow from the wall, as visible at location S7 presented.

The fluid is thus forced to flow in the middle of the diffuser yielding

higher speeds in this area and very poor comparison with experi-

mental data. From these evidences, we conclude that the inlet ra-

dial velocity component is of high importance in correctly repro-

ducing this swirling flow.

Interestingly, additional verifications proved that the inlet ra-

dial velocity affects the flow predicted by the k-ε turbulence model

in a different manner, most probably due to the use of wall func-

tions. Using equation (2) in a k-ε model yields the same conclusion

than with the SST, but imposing Ur = 0 leads to a k-ε solution that is

much better than for the SST case. This is thought to explain the ac-

ceptable agreement between the k-ε solution with no inlet radial ve-

locity and the experimental data reported in the past by some au-

thors (e.g. Mauri [3] and Page et al. [10]). However, other authors

using the SST turbulence model have faced difficulties in matching

the experimental results with this test case [11,12]. It thus ap-

pears that the inlet radial velocity is needed to re-energize the

boundary layer in a SST simulation, and that it is the wall function

approximation in k-ε model with no inlet radial velocity that com-

pensates with its intrinsic, increased boundary layer robustness.

However, relying on log-layer wall functions to compensate for the

lack of precision of the inlet boundary should not be viewed as reli-

able in our opinion.

Since in most cases no information is available on the inlet tur-

bulence parameters, qualified guesses also have to be made to esti-

mate them as accurately as possible. To assess the role played by

these assumptions, five different approaches, divided in two cate-

gories are compared and summarized in Table 1. The first category

uses the available measured profile of turbulent kinetic energy k

while it estimates the turbulence dissipation rate ε using two differ-

ent equations. The first of them is proposed by Armfield et al [13] in

his numerical study of this test case, and the second is taken from a

one-equation model reported in Cousteix & Aupoix [14]. The other

approach considered is to make approximations on both k and ε us-

ing constant turbulence intensity and a length scale being a frac-

tion of the inlet diameter, Di.

Results presented in Figure3 confirm that the imposed turbu-

lent parameters do have an influence on the solution's accuracy.

For the present test case, best results are obtained using the mea-

INTRODUCTION

In assessing turbine global efficiency, the importance of the

draft tube simulation is widely recognized among the scientific and

industrial communities, especially when dealing with low-head

power plants. In the past years, many research projects focussed

on draft tubes simulations and proposed some guidelines on the cal-

culations parameters to use. Among them, the FLINDT [1] and Tur-

bine-99 [2] projects as well as related doctoral thesis such as

Mauri's [3] or Cervantes' [4] are a good source of information on

this subject. The main objective of Turbine-99 was to assess the po-

tential of CFD to accurately predict draft tubes flows while FLINDT

focused specifically on the particular efficiency drop phenomenon

described below. There is also an ongoing measurement campaign

taking place at Laval University's Laboratory for Hydraulic Ma-

chines aiming to characterize the flow within the turbine and to

help fine-tuning computer simulations. These references, how-

ever, do not provide sufficient details on the specific impact of some

key simulation parameters and modelling approaches involved.

The present paper addresses these issues and summarizes the

work done as part of a Master Degree [5].

Since draft tube calculation confronts the CFD analyst to an in-

let plane located inside the region of interest and to an outlet

boundary also very close to it, a good understanding of their indi-

vidual effect is essential. Such knowledge helps understand the

sources of the errors induced when the entire geometry cannot be

modelled or, in other cases, it may also allow minimising the use of

unnecessary buffer zones or computing domain extensions while

being conscious of the resulting effect on the investigated result.

In some rehabilitation projects, the behaviour of the existing

draft tube is difficult to anticipate in relation with the newly de-

signed runner. The interaction between those components has

sometimes been observed to cause a sudden drop in the efficiency

curve near the best efficiency point [1,6]. This phenomenon unfor-

tunately seems to be highly sensitive and approximate numeri-

cal simulations have most often failed to predict it properly. In the

present paper, the impact of various calculation parameters recon-

sidered keeping mind the volatility of this particular phenomenon.

METHODOLOGY

The first test case chosen to evaluate those parameters is the

ERCOFTAC's [7] swirling flow in a conical diffuser. It is particularly

well suited for this task due to the availability of the detailed exper-

imental data of Clausen et al. [8] and to its similarities with actual

draft tube flows. The paper then addresses the Chute-à-la-Savane

draft tube case since this particular geometry is known to cause the

efficiency drop mentioned previously. The results help to confirm

the conclusions of the first test case and to extend the study to new

parameters as well.

The numerical investigation was conducted using ANSYS CFX

11.0 which solves three- dimensional Reynolds-averaged Navier-

Stokes (RANS) equations. Turbulence is modelled using the imple-

mented SST model of Menter [9] and results are in some cases com-

pared with a standard k-є turbulence model with wall functions.

PART 1 : CONICAL DIFFUSER TEST CASE

The ERCOFTAC's test case features a swirling flow of air enter-oing a 10 half-angle diffuser that discharges to ambient atmo-

sphere. The area ratio of the geometry is 2.84 and the level of rota-

tion imposed to the inlet flow was carefully adjusted during the ex-

periment to avoid wall separation as well as the appearance of

recirculation bubble in the core flow. The inlet swirl number,

often used to quantify the level of rotation in the flow in relation

to the axial component of velocity is in this case considered repre-

sentative of many draft tube flows. The geometry is schematically

reproduced in Figure 1 along with the two sets of coordinate axes

used. The velocity components U , U and U are associated with the r θ z

standard cylindrical coordinates (r, θ, z) whereas Us is the wall par-

allel velocity component in the xs direction.

The numerical results presented in the following sections were

all obtained on a one-cell thick pseudo-2D mesh with 115 × 165 ele-oments in the radial and axial directions, respectively. The slice is 2

+wide and the wall resolution assures 0.52 < y < 1.38 everywhere

1F.A. Payette2V. De Henau

3G. Dumas4M. Sabourin

SENSITIVITY OF DRAFT TUBE FLOWPREDICTIONS TO BOUNDARY CONDITIONS

ARTIGOS TÉCNICOS TECHNICAL ARTICLES

1 ALSTOM Hydro - felix-antoine.payette@power.alstom.com2 ALSTOM Hydro - vincent.deheneau@power.alstom.com

3 Université Laval - gdumas@gmc.ulaval.ca4 ALSTOM Hydro - michel.sabourin@power.alstom.com

26 27

drrU

drUUrSw

ZR

ZR

20

20

∫∫

= θ

Fig 1: Sketch of the diffuser with experimental measurement stations andreference systems used. Dimensions are in mm.

0

1 =∂

∂+∂

∂rr

rUr

Z

U Z

Fig 2: Radial velocities imposed at the diffuser inlet and resultingvelocity profiles at station S7.

Table 1: Turbulence parameters imposed at the inlet boundary.

ARTIGOS TÉCNICOS TECHNICAL ARTICLES

sured k profile and Armfield's equation for ε or, alternatively, using

the combination I=0.05 with Le=10%Di. However, a fundamental

difference exists between these two modelling approaches. The ef-

fective viscosity is the sum of the fluid viscosity and the eddy vis-

cosity, the latter being related to both k and ε in the following way:

The effective viscosity is therefore modified by the turbulent

quantities imposed at the diffuser inlet. As can be seen in Figure

3b, it is considerably larger in the case using Le = 10%Di, despite

the similarity in the predicted Us profiles. As expected, a higher vis-

cosity tends to damp the fluctuations in the flow as visible in region

A of Figure 3a. The opposite behaviour is also seen in region B

where the fluidity of the flow in the case Le = 0.1%Di allows the

boundary layer to briefly separate at station S4, as showed by the

local negative velocity, but to reattach before station S7.

Despite the good results noted here, using the approximation

Le = 10%Di appears to be a risky operation due to the resulting

damping that tends to makes the flow artificially robust, and be-

cause of the difficulty to determine the correct turbulence length

scale to privilege in other applications. Moreover, the problematic

approximation does not seem to lie in the eddy length scale itself

but rather in the constant turbulence intensity imposed. Looking

back at Armfield's equations presented in Table 1, it can be noted

that this model also needs a fraction of the inlet diameter to char-

acterize the turbulence. It thus seems that the error may mostly be

attributed to the use of a uniform intensity I, which gives a turbu-

lence kinetic energy, profile very different from the measurements.

In this case, imposing Le = 10%Di only hides the induced error and

should not in any case be associated with a reliable modelling pro-

cedure.

TREATMENT OF THE OUTLET CONDITION

The impact of the outlet geometry is determined by comparing

five extensions downstream of the diffuser. The size and geometry

of the discharge tanks added (with solid and/or permeable walls)

are shown on Figure 4, the simplest being no extension at all.

As a first step, boundaries 1 & 2 are specified to be standard no-

slip walls as to represent a physical tank. The effect of varying the

geometry is in most cases negligible for the axial and tangential ve-

locity profiles, but the pressure distribution in the outlet plane is in-

deed changed. On the left hand side of Figure 5, it is seen that the

wall pressure normalized as,

begins to be affected by the extension geometry at 70% of the

diffuser wall length, L. In the outlet plane, the pressure distribution

along r varies considerably between all simulations as shown by

the right hand side of the figure.

The geometry of the extension thus affects the pressure results

near the diffuser's exit. However, the shape of the extension do-

Fig 4: Outlet extensions geometry.

i

atmwallp

U

PPC

ρ2/1

−=

main is not the only parameter be considered to model the test

case properly. The boundary conditions imposed on the extension

also have to be as accurate as possible and this is why two addi-

tional simulations using the medium tank are presented on Figure

5. In both cases, the standard no-slip walls are removed on bound-

ary 1 and 2. In the first case, a zero total pressure is imposed on the

extension's back and top (boundary 1 & 2) to make the best repre-

sentation possible of a large volume of fluid at rest. In the other

case, only the back of the tank (boundary 1) uses this condition

while boundary 2 is a free slip-wall. Both pressure curves are

superposed but differ from the one using standard no-slip walls.

This leads to the conclusion that the treatment of the extension's

boundaries is significant, but in this particular case, the results re-

main unaffected as long as fluid entrainment by the free jet is free

to occur. It is to be noted that the flow topology within the exten-

sion box is considerably altered between the two cases, but this

has no effect on the results inside the diffuser itself since the veloci-

ties are small (see flow fields presented in [5]).

The radial variation of the pressure in the outlet plane is not to

be neglected since it is directly related to the machine's efficiency.

One way to quantify the quality of a diffuser is to look at the amount

of kinetic energy converted into pressure via the recovery coeffi-

cient, defined as

The static pressures P1 and P2 can be determined from an inte-

gration of the values in the entire plane or from an average of the

wall pressure values. These two options were evaluated for the dif-

fuser with extensions using no slip walls and results are presented

in Table 2. The variables χwall(1) and χwall(2) use the parietal pres-

sure at positions defined in Figure 4. The first of these two coeffi-

cients includes the last portion of the geometry, thus being more af-

fected by the extension than the same measure taken farther in-

side the diffuser. Although the velocity profiles inside the diffuser

are not significantly affected by the extension's shape, it is impor-

tant to keep in mind that it is likely to influence the pressure recov-

ery coefficient, especially when calculated from wall pressure and

close to the outlet.

Fig 5: Pressure evolution along the diffuser wall andprofile in the outlet plane.

2

12

2

1

−=

refA

Q

PPX

ρ

PART 2 : CHUTE-A-LA-SAVANE DRAFT TUBE STUDY

The Chute-à-la-Savane draft tube has been selected for a com-

parative study aiming to confirm and refine the investigation on the

sensitivity of numerical results to calculation parameters. This par-

ticular draft tube was chosen despite the absence of experimental

results since it is known to clearly present the efficiency loss de-

scribed in Loiseau et al. [6].

MESH SIZE

The first step conducted was to evaluate mesh independency to

confirm the validity of the results and to give an idea of the refine-

ment needed to properly evaluate draft tube performances. Three

meshes containing 1.15, 2.02 and 3.60 millions of structures hexa-

hedral elements were used. Quite surprisingly, all three meshes led

to similar results in terms of performances as well as flow topology.

Obviously, the most refined case gave a little more details in the

flow field but the main flow characteristics were present in the

three cases, confirming that an acceptable mesh independency

had been reached. The relative error made on the IEC losses, eval-

uated as

is of the order of 2.5%, the absolute values ranging from

1.93% to 1.89% and 1.88% as the mesh is refined. We thus infer

that refining the mesh to a very high number of elements is not the

parameter that has the most influence on the flow topology. In

cases where only a good approximation is sought and computing

time is a critical factor, it seems reasonable to use a moderate-size

mesh. For the following investigations, the intermediate 2.02M ele-

ments mesh is used.

AXISYMMETRY OF THE INLET BOUNDARY CONDITION

When experimental data is available at the inlet boundary, it is

often measured on a single axis. Although this is well suited for

axisymmetric geometries such as that of the ERCOFTAC's conical

diffuser, it is found that making a 1D approximation only has a mod-

erate impact on the computed CEI losses. The error induced by

such an approximation in the case of the Chute-à-la-Savane draft

tube is just a little more that 0.1%.

INLET TURBULENCE LEVEL

The conical diffuser test case showed that two different meth-

ods of specifying the inlet turbulence led to similar results. It was

then said that using a turbulence intensity of 5% and an eddy

length scale equal to 10% of the runner diameter is risky since it

tends to artificially smooth the flow characteristics. Results ob-

tained in the Chute-à-la-Savane geometry are very convincing on

this point. Figure 6 shows contours of the axial velocity in the draft

tube channels. In the first case, the turbulence was taken from a

simulation including the distributor and runner. In the second case,

imposing the combination I & Le leads to a much more uniform dis-

tribution of the flow where the recirculation bubble blocking the

right channel disappears. As a consequence, the calculated losses

considerably decrease from 1.89% to 1.42%.

Table2:Diffuser's recovery coefficients for various extension boxes(no slip walls).

n

out

outintot

gH

A

QPP

lossesIECρ

ρ

+−

=

2

,

2

1

_

It is clearly seen from this example that modifying the inlet tur-

bulence level can, by itself, modify the whole flow field even if the

imposed velocity profiles are unchanged. In the present case, us-

ing 10% of the inlet diameter as the eddy length scale leads to an

obviously too viscous flow and the calculated losses are underesti-

mated of 0.5% (absolute), which could lead to costly penalties in a

contractual context.

FINAL CONSIDERATIONS

Simulating diffusers using computational fluid dynamics has al-

ways been a difficult task due to the unstable nature of the flows un-

der adverse pressure gradients. It has been showed by studying

two test cases — the ERCOFTAC's swirling flow in a conical diffuser

and the Chute-à-la-Savane draft tube — that the imposed bound-

ary conditions have an important impact on the calculated flow to-

pology. Among these parameters, the inlet radial velocity was

shown to have the greatest impact on the conical diffuser's flow

since it is directly related to the appearance of wall separation or

core flow recirculation. The second most important parameter is

probably the inlet turbulence and its role is best seen in the draft

tube geometry. Overestimating the incoming turbulent mixing arti-

ficially rises the viscosity and leads to a much more uniform flow

which tends to underestimate the draft tube losses. The third as-

pect to recall is the outlet treatment. The addition of an outlet dis-

charge extension did not influence much the velocity profiles inside

the conical diffuser, but it did modify the outlet static pressure dis-

tribution and consequently altered the calculated recovery coeffi-

cient. It is not excluded that, in some cases, it might even have an

impact on velocity profiles within the diffuser, especially in the last

part of the geometry.

On the other hand, refining the mesh and imposing an

axisymmetric boundary condition did not appear to have a first or-

der influence on the computed results. These two parameters

should however be taken into account as precisely as possible

when suspecting the presence of very sensitive phenomena such

as the efficiency drop near the best efficiency point.

ACKNOWLEDGEMENT

The first author would like to thank the Natural Sciences and En-

gineering Research Council of Canada for its financial support to

this research project.

BIBLIOGRAPHICAL REFERENCES

[1] AVELLAN, F., 2000, Flow Investigation in a Francis Draft

Tube : The FLINDT Project, Proceedings of the Hydraulic Machinery

and Systems, 20th IAHR Symposium, Charlotte, USA.

[2] Turbine-99 website : www.turbine99.org

[3] MAURI, S., 2002, Numerical Simulation and Flow Analysis

of an Elbow Diffuser, Doctoral Thesis, École Polytechnique de

Lausanne, Switzerland.

Fig. 6 : Axial velocity contours in the draft tube channels withturbulence calculated from a) upstream components and

b) the approximation I=5% and Le=10%Di.

ευ2

Kt ∝

Fig 3: Effect of the inlet turbulence on axial velocity profilesand eddy viscosity.

28 29