Analysis of the swirling flow at GAMM Francis runner
outlet for different values of the discharge
Tiberiu CIOCAN*, Romeo F. SUSAN-RESIGA*, Sebastian MUNTEAN**
Corresponding author
Department of Hydraulic Machinery, POLITEHNICA University of Timisoara
Bd. Mihai Viteazu 1, 300222, Timisoara, Romania
tiberiu_ciocan@yahoo.com, resiga@mh.mec.upt.ro
**
Centre for Advanced Research in Engineering Sciences, Romanian Academy –
Timisoara Branch
Bd. Mihai Viteazu 24, 300223, Timisoara, Romania
seby@acad-tim.tm.edu.ro
*
Abstract: Nowadays an important, and still open, issue in Francis runner design is the blade
geometry at the trailing edge, wich directly determines the swirling flow at the runner outlet and
further downstream the pressure recovery and hydraulic losses in the turbine draft tube. In this paper
we present and validate a novel mathematical model for computing the radial profiles of both axial
and circumferential velocity components, respectively, at the runner outlet for Francis hydraulic
turbines within the full operating range. We apply this methodology of computing the swirling flow
downstream the runner, over the GAMM Francis turbine model, where experimental data were
available. We present the computation procedure for the dimensionless discharge and dimensionless
flux of moment of momentum, respectively. We introduce the swirl-free velocity profile in order to
expres the relationship between the axial and circumferential velocity components. This hypothetical
velocity, the swirl-free velocity, is subject of optimization in order to achieve the best performance of
the draft tube over the whole intended operating range.
Key Words: swirling flow, Francis turbine, flux of moment of momentum, swirl-free velocity
1. INTRODUCTION
Nowadays, modern hydraulic turbines meet new challenges associated with variable demand
on the energy market and limited possibilities for energy storage, resulting in the need for
flexibility in their operation. This need for flexibility in the operation of a turbine, means that
there is often a tendency (required) the use of turbines for a wide range of operating regimes,
far from the operating point for which have been designed. Hydraulic turbines, especially
radial-axial turbines (e.g. GAMM Francis turbine [1]) having a rotor with fixed pitch blades,
have a sharp drop in efficiency and severe pressure fluctuations at off-design operating
regimes. When operating at part-load, in the turbine draft tube, it appears a high-level of
residual swirl, as a result between the mismatch of the swirl configuration at the runner inlet
imposed by the guide vanes, and the angular moment extracted by the turbine runner [2]. The
helical vortex (which itself is part of the flow rotation) appears due to the decelerated flow
rotation and it is the source of flow unsteadiness with associated pressure fluctuations, that
can lead in breakage of the runner blades.
Evaluation of the turbine efficiency for the full turbine operating range, depending on
the discharge coefficient and energy coefficient with respect to the IEC regulations [3], is
still the standard experimental investigation on model turbines in order to predict the
performance of the real size machine, and the resulting efficiency hill-chart usually displays
a peek efficiency, wich is called the “best efficiency point” (BEP). Due to technological
advances in recent decades, numerical simulation can be made on all components of a
Francis turbine, which allow a high accuracy analysis of the spiral, the guide vanes, the rotor
and draft tube, respectively [4].
For this reason in the last 20 years, it appeared numerous programs dedicated to
investigating the hydraulic losses wich occurs in the draft tube, such as:
● GAMM Workshop – „3D - Computation of Incompressible Internal Flows” [5]
● Turbine – 99 Workshop on draft tube [7]
● FLINDT Project – Flow investigation in draft tube [8]
Better understanding of the particularities of the swirling flow at the runner outlet / draft
tube inlet, is the key in developing new techniques for controlling the swirling flow
downstream the Francis turbine runner. A very good example is the water injection method
from the runner crown along the axis machine, in order to eliminate the helical vortex and
the associated pressure fluctuations, developed by (Susan-Resiga, et al. 2006) [9]. An
important, and still open, issue in Francis runner design is the blade geometry at the trailing
edge, wich directly determines the swirling flow at the runner outlet and further downstream
the pressure recovery and hydraulic losses in the turbine draft tube. As a result, is essential to
correctly estimate the hydraulic losses and the pressure recovery in the turbine draft tube in
the early design stages, if possible even before acctualy designing and computing the runner.
This is possible only if one has the capability of predicting the swirling flow at runner outlet
within a wide range of operating regimes, in order to establish the best swirling flow
configuration at runner outlet at the design operating regime.
Susan-Resiga, et al. [10], developed a mathematical model that enables calculation of
the swirling flow at the runner outlet for the whole operating range of a Francis turbine
runner and showed that this model is able to capture the main features of the swirling flow.
In this paper i will present the methodology of computing the swirling downstream the
Francis turbine runner for variable operating regimes, in order to validate the theory. In this
theory the dimensionless discharge φcalc, dimensionless flux of moment of momentum m2
and the swirl-free velocity vsf, respectively, is computed.
2. EXPERIMENTAL DATA
The experimental data further used in this paper corresponds to three operating regimes of
the GAMM Francis turbine model [5], wich is a turbine with medium specific speed ν = 0.5
and it is characterized by the following main parameters:
 Q = 0.376 m3/s;
 H = 5.957 m;
 ω = 52.36 rad/s.
The turbine geometry, in meridian plane is given in Figure 1, with the reference runner
radius of R2 = 200 mm.
The hill-diagram of the the turbine model is presented in figure 2, represented by the
discharge coefficient φref and energy coefficient ψref, respectively, with respect to the IEC
regulations [3].
From the hill diagram it can be seen that operating point 4 at minimal discharge,
operating point 1 wich is the best efficiency point and operating point 5 at maximal
discharge, respectively, are at constant head.
Fig. 1 – Meridional cross-section through the
GAMM Francis turbine [6].
Fig. 2 – Hill diagram of the GAMM Francis turbine [1]
In table 1 is given the corresponding dimensionless discharge of the three points at
constant head.
Table 1 – Dimensional discharge and head
φexp
P4
P1 (BEP)
P5
0.22
0.286
0.33
ψexp
1.07
The experimental available data wich i will use are in order to compute the swirling
flow, are the velocity profiles downstream the runner, from the measurement section S2 (see
Fig. 1), wich has a measurement radius of Rw = 218.38 mm. Further i will present the
necessary steps in order to compute the swirling flow at the runner outlet.
3. COMPUTATION METHOD
3.1 Dimensionless velocity profiles
The dimensional velocity profiles used are from survey section S2, for three operating points
at constant head (see Fig. 2). The survey section has a corresponding dimensionless radius
of rw = 1.092 and it is made dimensionless by the reference radius, as follows: rw = Rw / R2.
All the measurement radius are made dimensionless by the reference radius and all the
experimental velocity components are made dimensionless with respect to the reference
velocity Vref = ωR2 = 10.472 m/s.
In figures 3, 4 and 5 it is represented the dimensionless velocity profiles, from the
minimum to maximum discharge, respectively, at the runner outlet, for the three operating
points at constant head. In figure 3 at part load operating regime, the axial velocity it is
decreasing until it develops a stagnation region with helical vortex breakdown and at full
load operating point the axial velocity has an excess near to the axis, or with other words we
have an axial jet.
Fig. 3 – Experimental dimensionless velocity components measured at Part load operating regime
Fig. 4 – Experimental dimensionless velocity components measured at Best efficiency operating regime
Fig. 5 – Experimental dimensionless velocity components measured at Full load operating regime
A very important fact is that the following analysis is performed only for the dimensionless
velocity profiles.
3.2. Computation of the discharge
The discharge can be computed by integrating numerically the axial velocity profile over the
survey section. By introducing the dimensionless velocity v and dimensionless radius r,
respectively, the dimensionless discharge can be computed by using the simplest integrating
approach, as follows:
Trapezoidal rule:
 calc   v z 2rdr  i 1 (ri v z ,i  ri 1v z ,i 1 )(ri  ri 1 )
rw
N data
(1)
0
where i = 0…Ndata , with i =0 coresponding to the axis and i = Ndata to the wall, respectively.
The computed dimensionless discharge is given in table 2, for the three operating points
at constant head, along with the errors between the dimensionless discharge from the
velocity measurements φcalc and the corresponding value from the measured overall
discharge φexp.
Table 2 – Computed dimensionless discharge
P4
P1 (BEP)
P5
φcalc
0.215
0.308
0.363
φexp
0.22
0.286
0.33
ε [%]
2
7
10
With the computed dimensionless discharge, it can be computed the discharge fraction
(normalized streamfunction),wich is the fraction of the volumetric flowrate discharged
through a disc of radius r ≤ rw. As a result, the discharge fraction can be computed for the
experimental data, from equation (1), as follows:
q01 
r
1
i

(r j v z , j  r j 1v z , j 1 )(r j  r j 1 )

 calc  calc j 1
(2)
where i = 1…Ndata.
3.3 Swirl-free velocity profiles
The swirl-free velocity vsf is used in order to describe the flow kinematics, instead of the
relative flow angle β2, at runner outlet, because it remains unchanged at different operating
regimes. Introducing the dimensionless radius and dimensionless velocity, it can be written
in dimensionless form the swirl-free velocity, as follows [10]:
v sf 
rv z
r  vu
(3)
In order to have an average swirl-free velocity independent of the operating regime, we
have to express the swirl-free velocity versus the discharge fraction.
From figure 6 it can be seen that the swirl-free velocity profile collapse on the same
discharge fraction (with blue lines), but this was expected since the swirl-free velocity is
related with the runner geometry at the trailing edge, wich is not changing with the operating
point for fixed pitch blades as in Francis turbines.
In figure 7 it can be observed three different regions. The first 2 regions with blue lines,
corresponds to the runner crown wake and wall boundary layer, and it is a result of the
viscous effects developed by the runner crown surface and wall surface, respectively.
Fig. 6 – Computed dimensionless swirl-free velocity profile at constant head
Fig. 7 – Linear fit for dimensionless swirl-free velocity at constant head
The data of these two regions, shown with hollow circles are discarded since the
mathematical model is an inviscid flow model. The third region corresponds to the main
swirling flow, shown with filled circles. The data shown with filled black circles from figure
7 are fitted with a linear fit, having the equation:
vsf ( y )  vsf _ ct   vsf [q01 ( y )  0.5]
(4)
where vsf_ct is the constant value of the swirl-free velocity and δvsf is the slope of the swirlfree velocity profile. These two coefficients are going to be the optimization parameters for
the swirling flow at the runner outlet in order to obtain the best performance of the draft tube
within a given operating range.
It can be seen that the experimental data from the third region are reasonably clustered
around the constant fit (red line).
3.4 Computation of the flux of moment of momentum
From the fundamental equation of turbomachines written for hydraulic turbines:
 ( Q)( gH )   (RV )Vr dS1   (RV )V z dS 2
S1
S2
M1
(5)
M2
, we have the hydraulic power written as the product of the mass flow rate and the specific
energy multiplied by the hydraulic efficiency, and in the right side of the equation we have
the rate at wich the fluid does work on the runner, given by the difference between the flux
of moment of momentum upstream the runner and downstream the runner, respectively.
Because in this paper we are investigating the swirling flow at the runner outlet it is
necessary to compute the flux of moment of momentum downstream the runner. By
introducing the dimensionless radius and dimensionless velocity, respectively, the flux of
moment of momentum downstream the runner can be written in dimensionless form, as
follows:
m2   (rv )v z 2rdr  i data
(ri2 v ,i v z ,i  ri 21v ,i 1v z ,i 1 )(ri  ri 1 )
1
rw
N
(6)
0
The computed dimensionless flux of moment of momentum with equation (6), is given
in table 3.
Table 3 – Dimensionless flux of moment of moment of momentum
P4
P1(BEP)
P5
m2 , Eq. (6)
0.029
0.0157
-0.002
m2   calc  rv
0.02
0.01
-0.002
Fig. 8 – Moment of momentum versus the discharge fraction
In figure 8 it is represented the moment of momentum, with filled circles, versus the
discharge fraction, fitted with cubic fits, for all three operating regimes at constant head.
If we multiply the average values of the dimensionless moment of momentum, given in
figure 8, with the corresponding dimensionless discharges computed with Eq. (1) (table 2),
as follows: m2   calc  rv , we obtain the dimensionless flux of moment of momentum,
with the corresponding values from table 3. This is done mainly to verify if the
dimensionless flux of moment of momentum obtained with equation (6) is correctly
computed. If we compare the values from table 3, obtained with the two approaches, it can
be seen that the values are in a very good agreement, especially at full-load.
4. COMPUTED SWIRLING FLOW AND RESULTS
With the computed integral quantities that characterized the swirling flow, the dimensionless
discharge φcalc, and dimensionless flux of moment of momentum m2, respectively, and in
addition the swirling flow satisfies the kinematic constraints corresponding to the runner
blade at trailing edge described by the swirl-free velocity, we can compute the axial velocity
by solving the variational problem [10]:
yw
f (v sf )   v z2 dy 
0
v
1 yw yw 
1 z



0
y
2
 v sf

dxdy


(7)
Given the swirl-free velocity, wich in this case we chose a constant swirl-free velocity
of vsf_ct = 0.278, and by minimazing the functional (7), to the integral constraints (8), (9):

yw
0

yw
0
v z dy   calc

1  v z
 v
sf


v z ydy  m2


(8)
(9)
, we can compute the axial velocity vz at the runner outlet, and with the axial velocity
computed, we can compute the circumferential velocity vθ, as follows:

v
v  r 1  z
 v
sf





(10)
The numerical results for the the three operating regimes at constant head investigated in
this paper are represented in figure 9, togheter with the experimental data.
Fig. 9a – Computed versus measured velocity profiles at Part load operating regime
Fig. 9b – Computed versus measured velocity profiles at the Best efficiency operating regime
Fig. 9c – Computed versus measured velocity profiles at Full load operating regime
Fig. 9 – Dimensionless axial and circumferential velocity profiles at S2 cross-section, (Fig.1).
5. DISCUSSION
In the above figures it is represented the computed axial velocity and circumferential
velocity, computed with φcalc, m2 at a constant swirl-free velocity of vsf_ct = 0.278.
Figure 9 shows that the mathematical model correctly captures the swirling flow
evolution at different discharges. At part load P4, Q > QBEP, one can see that the velocity is
gradually decreased until a stagnant region develops in the axis neighborhood and the flow
co-rotates with respect to the runner. On the other hand at full load P5, Q < QBEP, we have an
axial velocity excess near the axis and counter-rotating flow with respect to the runner
rotation.
The agreement with the experimental data are quite good except the wake of the crown,
but the lack of correlation is expected since the model does not account for the viscous
effects which lead to the crown wake. One can see from figure 9a that at low discharge, the
flow model correctly captures the stagnant region extent.
Figure 10 shows the maps of axial and circumferential velocity components versus the
radial coordinate and discharge coefficient, at the reference section S2 (Fig. 1).
Fig. 10a – Axial velocity versus dimensionless
radius and dimensionless discharge
Fig. 10b – Circumferential velocity versus
dimensionless radius and dimensionless discharge
Fig. 10 Axial and circumferential velocity profiles for variable discharge coefficient φcalc = 0.16…040
at S2 cross-section
From figure 10 we have an overall perspective on the swirl evolution with the turbine
discharge. In figure 10a, it can be easily observe the development of the central stagnant
region (lower-left corner) when the discharge decreases. Figure 10b shows the development
of a counter-rotation when the discharge is increased and at lower values of the discharge
(under φcalc = 0.308), the whole flow rotates in the same direction with the runner. The
boundary of this stagnat region is an unstable vortex sheet which evolves in the precessing
helical vortex known as vortex rope [11].
6. CONCLUSION AND PERSPECTIVES
This paper presents in detail the necessary steps required in order to compute the swirling
flow downstream the Francis turbine runner. It is shown that for computing the swirling flow
at the runner outlet we need to compute the dimensionless discharge φcalc, dimensionless flux
of moment of momentum m2 and the swirl-free velocity profile vsf. This are the only three
quantities required for computing the swirling flow (i.e. axial velocity and circumferential
velocity) at the runner outlet. The computed velocity profiles are in a very good agreement
with the experimental data, excepting the runner crown wake region, because the viscous
effects are not captured by the inviscid flow model, this is a limitation of the mathematical
model. In the near future, the next step is the computation of the swirling flow at the draft
tube inlet in order to evaluate the draft tube hydrodynamics, followed by the optimization of
the swirling flow at the draft tube inlet, by optimizing the swirl-free velocity in order to
achieve the best draft tube performance that will improve the turbine efficiency. The final
step is to actually design the turbine runner, through an inverse design method [12], such that
we have at the runner outlet the prescribed swirling flow.
7. ACKNOWLEDGMENTS
This work was partially supported by the strategic grant POSDRU 107/1.5/S/77265, inside
POSDRU Romania 2007-2013 co-financed by the European Social Fund – Investing in
People.
Dr. Sebastian Muntean and Prof. Romeo Susan-Resiga would like to thank
the support of the Romanian Academy program.
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August 29, 2011