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European Journal of Mechanics B/Fluids 29 (2010) 321–335

Contents lists available at ScienceDirect

European Journal of Mechanics B/Fluids

journal homepage: www.elsevier.com/locate/ejmflu

Experimental analysis of the swirling flow in a Francis turbine draft tube: Focuson radial velocity component determination

Sylvain Tridon a,∗, Stéphane Barre a, Gabriel Dan Ciocan b, Laurent Tomas ba LEGI-INP Grenoble-CNRS, BP53, Grenoble, 38041, Franceb ALSTOM HYDRO France, 82 av. Léon Blum, Grenoble, 38000, France

a r t i c l e i n f o

Article history:

Received 20 July 2009

Received in revised form

15 February 2010

Accepted 19 February 2010

Available online 26 February 2010

Keywords:Francis turbine

Swirling flow

Draft-tube

Efficiency drop

Pressure recovery coefficient

Radial velocity

a b s t r a c t

The draft tube of a hydraulic turbine is the component where the flow exiting the runner is decelerated,thereby converting theexcess of kinetic energyinto static pressure.In thecase of machine refurbishmentof an existing power plant, most of the time only the runner and the guide vanes are currently modified.For financial and safety reasons, the spiral casing and the draft tube are seldom redesigned, even if thesecomponents present some undesirable behaviour. In some cases, the installation of an upgraded runnerleads to a peculiar and undesirable efficiency drop as the discharge is increased above the best efficiencypoint value. It is found to be related to a corresponding sudden variation in the draft tube pressurerecovery coefficient at the same discharge.

The swirling flow exiting the runner is complex and highly turbulent. The radial velocity is rarelymeasured because a quite complicated measurement setup is needed. However, this velocity componentis greatly needed in order to properly initialize the numericalsimulations, andits influenceis important inspiteof its small magnitude. Velocity measurements downstream of the runner include radial componentmadeat CREMHyG (Grenoble) by LDV,and PIV techniquesare presented. An analytical formulation for thisvelocity component based on the formulation for the conical diffuser and on the three vortices structure

is proposed and compared with measurements. © 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

The draft tube is an important component of a reaction turbine(i.e. Francis or Kaplan turbine). It is located just under the run-ner and allowed to decelerate the flow exiting the runner, therebyconverting the excess of kinetic energy into static pressure. As aconsequence, it increases the global efficiency of the turbine. Theclassical shape of a draft tube can be decomposed into three el-ementary components: the conical diffuser, a curved divergingdiffuser and the last straight divergent diffuser with one pier for

structural reasons — see Fig. 1.The role of the draft tube can be pointed out by consideringthe head balance between upstream and downstream water levelrespectively noted 1 and 2 in Fig. 2.

This balance may be written as follows:

H 1 − ∆H t − K U 2in

2 g = H 2 +

U 2out

2 g (1)

where H is the turbine head defined by the difference between theupstream and downstream level (H = H 1 −H 2). K is the draft tube

∗ Corresponding author. Tel.: +33 476827027; fax: +33 476825001.E-mail address: [email protected] (S. Tridon).

losses coefficient and g is the gravitational acceleration. U in andU out are respectively the inlet (outlet) draft tube velocities (seeFig. 2). ∆H t is the turbine power (per unit volume flow rate). Thispower is then deduced as:

∆H t =U 2in

2 g

H

U 2in2 g

− K − A2in

A2out

(2)

where Ain and Aout arerespectively theinlet(outlet)drafttube area.

This value is to be compared to theone obtained without a diffuser:

∆H t =U 2in

2 g

H

U 2in2 g

− 1 . (3)

Then the relative power gain obtained by the addition of a diffuseris therefore:

∆P = P with diffuser − P without diffuserP without diffuser

=1 − A

2in

A2out− K

H

U 2in/2 g − 1 . (4)

0997-7546/$ – see front matter© 2010 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.euromechflu.2010.02.004

http://www.elsevier.com/locate/ejmfluhttp://www.elsevier.com/locate/ejmflumailto:[email protected]://dx.doi.org/10.1016/j.euromechflu.2010.02.004http://dx.doi.org/10.1016/j.euromechflu.2010.02.004mailto:[email protected]://www.elsevier.com/locate/ejmfluhttp://www.elsevier.com/locate/ejmflu

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322 S. Tridon et al. / European Journal of Mechanics B/Fluids 29 (2010) 321–335

Fig. 1. Scheme of a Francis turbine.

According to the Eq. (4), the optimum draft tube efficiency isobtained by minimizing both the draft-tube outlet velocity U outand the draft tube losses coefficient K , with special attention beingpaid towards maintaining the pressure under the runner above thewater vapor pressure to avoid creating a cavitation phenomenon.This relative power gain can be roughly estimated for the present

study using: H = 20 m, Ain

Aout = 1

4 , K = 0.2, U in = 6 m/s.With these values one may obtain a relative power gain of about7.5%, which is considerable. It has to be argued that the draft tubegain decreases with an increasing turbine head (H ). So for higheraltitudes, the difference between 1 and 2, the diffuser gain tends todecrease quite sharply.

The swirling flow exiting the runner is complex and highly tur-bulent. Numerical simulation of swirling flow in a draft tube needsboundary conditions at the inlet to be performed. Zhang et al. [1]have shown that except in the near-wall shear layer the averagedvelocity profiles in the cone can be approximated by the Batchelorvortex family. Paik et al. [2] used mean streamwise and swirl ve-locity profiles measured by a laser velocimeter to initialize theircalculations. Several experimental studies have been devoted to

the description of the flow structure under the runner. Most of thetime, only the axial and tangential velocity components are mea-sured by the LDV technique. The radial velocity is rarely measuredbecause a complicated measurement setup is needed — see [3].However, this velocity component is greatly neededin order to ini-tialize properly the numerical simulations and its influence is im-portant in spite of itssmall magnitude. Often the radial componentis neglected or barely approached with the analytical formulationfor a conical diffuser (Eq. (5)) — see [4].

C r = C r 0 tan

θ r

R

. (5)

Other studies present methods to calculate the radial com-ponent by integrating the continuity equation — see [ 5,6]. Some

usable results have been obtained using these methods but somedifficulties still remains. The use of the conical flow assumptionmay be inadequate if for example the flow is non-axisymmetricor if the overall vorticity structure emanating from the runneris perturbed and distorted. In most practical cases, some distor-tion effects initiated by downstream perturbations like some flowrate asymmetry as experienced in the elbow and the draft tubemay change the global structure of the flow in the conical zone

just downstream of the runner. These effects may be amplifiedby the presence of a pressure recovery break-off in the draft tubeas experienced in some particular flow configurations — see [ 7].In these conditions the conical flow assumption which is axisym-metric do not hold, therefore the radial velocity component mustbe measured or estimated in a different way in order to obtain

a sufficiently accurate flow description for the initialization andvalidation of numerical RANS or LES simulations.

Fig. 2. Sketch of a hydro-electric power station.

In this paper, radial velocity component measurements ob-tained at CREMHyG (Grenoble) with two different PIV setups willbe presented. An analytical formulation for this component basedon the three vortices structure presented by Resiga et al. [8] willalso be proposed.

The obvious practical importance of predicting the complexflow downstream the turbine runner, in the draft tube, led usto the FLINDT project (Flow Investigation in a Draft Tube) initi-ated in 1997. Experimental measurements and numerical simula-tions were performed at the EPFL (Ecole Polytechnique Federale deLausanne) with a draft tube that presents an efficiency drop in or-der to understand it. The aim was also to establish a data basefor project participants. The velocity profiles and six independentcomponents of theReynolds stress tensor were measured at differ-ent locations in the geometry for three different operating points.From an engineering perspective, the numerical simulations madeby Mauri et al. [9] on a 300 000 nodes structured mesh gave quiteacceptable results when compared to experimental data. However,some discrepancies still remain and Avellan et al. [10] attributesthese differences to the fact that the k–ε model used in these sim-ulations is well known to be unadapted to this particular flow

configuration. In terms of inlet conditions, they noted that the bestagreementwith theexperimental data wasobtainedusing a turbu-lent length scale of 0.2% of the inlet diameter. The velocity profilesimposed on boundary conditions are derived from experimentalmeasurements.

Mauri et al. [9] carried outa numericalstudyof the FLINDTdrafttube and attributed the efficiency drop to a global instability un-derscored by Susan-Resiga et al. [8] (see next paragraph) leadingto a Werlé–Legendre vortex [4]. The flow in the draft tube wassimulated using the CFX-TASCFLOW 2.10. The inlet profile is spec-ified using a cubic spline. A linear interpolation is applied to thecircumference. The radial component is imposed as the analyticalformulation for a conical diffuser (Eq. (5)). The inclination of thevelocity vector in the radial direction is, therefore, determined by

the geometry of the cone. This theoretical profile gave better re-sults than the one extrapolated from the relatively few and wigglymeasured values of the radial velocity component.

Resiga et al. [8] have developed a theoretical model based on ananalysis of the velocity profile at the draft tube inlet. The authorsshow that an analytical representation maybe set on experimentaldata. It is achieved by superposing two counter-rotational Batch-elor vortices and a solid body vortex in the center. A non-viscousstability analysis, applied to these profiles, assuming a fully devel-oped flow hypothesis, shows that there is a critic discharge fac-tor leading to a flow instability well-correlated with the observedefficiency drop.

Some important work in the field of draft tube flow simulationconsists of the whole data set displayed during the three Turbine-

99 [11] workshops that occurred in 1999, 2001 and 2005. Theobjective of these researches conducted in Sweden was to assess

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and advance tubine flow simulation state of the art. The draft tubestudied in this context does not present an efficiency drop near thebest operating point, but modeling it remains a major challenge inpractical industrial design. One of the main conclusions of theseworkshops was that a precise specification of the input velocityprofiles was essential to obtain an accurate numerical simulation.It was also noted that a too simplified expression for the radial

component (i.e. conical flow assumption) may introduce too manyuncertainties in the initialization process of the simulation. Hence,due to the high sensibility on initial conditions, the resultingsimulation did not converge towards the right flow solution.

Alongwith the workshops Turbine-99, Cervantes presents in hisdoctoral thesis [12] a numerical study on the impact of the bound-ary conditions on the calculation. The results are obtained usingthe k–ε turbulence model. A factorial design approach was usedto determine that the parameter having the greatest influence onthe determination of the draft tube recovery factor is the initial ra-dial velocity field. He also concluded that the surface roughnessand the turbulent characteristic length, which is assumed constantthroughout the conical diffuser initial section, do not have a ma-

jor effect. However, Cervantes acknowledges that, due to a poor

grid resolution in his simulations, the minimum y+ values foundin these computations are much too large to correctly describe theboundary layer dynamics. Another interesting contribution is theway in which the author considers the radial velocity under theturbine runner [5]. An iterative process based on the Squire Longequation was used to estimate the radial velocity. This approachrequires experimental measurements for axial and tangential ve-locity at inlet and outlet of the calculation domain. It is also dif-ficult to judge the outcome because Cervantes did not succeed inconverging some simulations where the deduced velocity profileis imposed as an upstream boundary condition. This work is inter-esting, but the fact that it may be affected by a mesh dependenceresults problem cannot be ignored.

Gyllenram [6] presents an analysis of the swirling flow in a

pipe based on the time-averaged quasicylindrical Navier–Stokesequations andits relevance to steady, unsteady andturbulent flow.This method is developed to determine the critical level of swirl(vortex breakdown), but it can also be used for an estimation of the radial velocity profile if the other components are given ormeasured along a single radial line.

The present paper is organized in four parts. First, the in-vestigated case and velocity measurement instrumentation arepresented. Then, results will be presented and commented. Thethird part is devoted to the presentation and discussion of the ana-lytical representation of axial and tangential velocity profiles fromLDV measurements. In the fourth part, the particular case of theradial component is treated. Conclusion and discussion follows.

2. Methodology

In order to compare turbines of different dimensions, one de-fines dimensionless coefficients of energy ψ and discharge ϕ (seeEqs. (6) and (7)), with runner angular speed ω , head H , dischargeQ and runner outlet radius Rref ,

ψ = 2 gH ω2R2ref

(6)

ϕ = Q π ωR2ref

. (7)

Specific speed υ (see Eq. (8)) is a dimensionless parameter de-

fined with ϕ and ψ which do not depend on Rref . On a nominal op-erating point, corresponding to the best efficiency operating point,

Fig. 3. Efficiency break-off obtained by increasing the discharge coefficient ϕ.

specific speed is a classification criterion for turbomachinery.Francis turbines have a specific speed ν between 0.14 and 0.65,

v = ϕ1/2ψ 3/4

= ω√ Q /π(2 gH )3/4

. (8)

The global ‘‘efficiency’’ of the draft tube is quantified using itsstatic pressure recovery coefficient χ (see Eq. (9)), defined withthe discharge Q andthe differentialpressurebetween the inlet andoutlet sections of the draft tube,

χ =( P

p + gz )outlet − ( P p + gz )inlet

Q 2/2 A2inlet(9)

where Ainlet is the area of the inlet section.

2.1. Francis turbine model presentation

The investigated case corresponds to the scale model of a Fran-cis turbine of a high specific speed, ν = 0.55 (nq = 86). The scalemodel supplied by ALSTOM HYDRO FRANCE is installed on the testrig of the CREMHyG and the tests are carried out according to theIEC 60193 International Standards [13]. The turbine model has aspiral casing with a stay ring of 24 stay vanes, a distributor madeof 24 guide vanes, a runner with 19 blades of 365 mm outlet di-ameter, and a symmetric elbow draft tube with one pier. The drafttube is composed of three different parts, the cone in Plexiglas, theelbow in fiberglass and the straight diffuser in sheet steel.

By increasing the discharge (see Fig. 3) the efficiency presentsan important drop of more than one percent close to the best op-

erating point. It is found to be produced by a corresponding dropin the draft tube pressure recovery. The perfect similarity betweenthe two profiles on Fig. 3 confirms the idea that the ‘‘accident’’ oc-curred in the draft tube. The uncertainty on efficiency and pressurerecovery coefficient are calculated from the individual and inde-pendent uncertainties of the considered parameters according tothe IEC 60193 International Standards [13]. Therefore, for the ef-ficiency, the total error is the quadratic sum of errors on H , Q , T and N , ∆η/η = ±29.5 × 10−4. For the pressure recovery coeffi-cient, the total error is the quadratic sum of errors on ∆P and Q ,∆χ/χ = ±28.4 × 10−4.

The four operating points where full velocity measurementsareperformed on the survey section are selected around the efficiencydrop. They are detailed in the Table 1 and marked by a square on

Fig. 3. OP2 is the closest from the best operating point and OP3 justfollows the efficiency drop.

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Fig. 4. Druck sensors positions.

Table 1

Four operating points selected near efficiency drop for full velocity measurements.

Operating point Efficiency drop Discharge coef. ϕ Energy coef. ψ

OP1 No 0.368 1.18

OP2 No 0.374 1.18

OP3 Yes 0.380 1.18

OP4 Yes 0.390 1.18

Table 2

Druck sensors details.

Desi gnation Position Acquisition mode Bandwidth (Hz)

DK1 Cone upstream DC 0–200

DK2 Cone downstream DC 0–200DK3 Cone right side DC 0–200

DK4 Cone left side DC 0.4–200

DK5 Spiral casing inlet (HP) AC 0–200

Table 3

Main characteristics of the LDV optical system.

Optical characteristics Under the runner

Wave lengths (nm) 488/514.5

Focal length (mm) 400

Probe diameter (mm) 60Beam spacing probe (mm) 38.5

Number of fringes 40

Fringe spacing (µm) 5.13/5.41

Beam half-angle (◦) 2.725Measuring volume d x, d y (mm) 0.21/0.22Measuring volume d z (mm) 4.35/4.58

2.2. Fluctuating pressure instrumentation

Measurements of fluctuating absolute pressure are obtained onthe wall using Druck sensors placed on the draft-tube inlet sectionat a distance Rref under the runner. The sensors positions arepresented in the Fig. 4 and detailed in the Table 2. The sensor DK4is mounted on the opposite side fromthe sensor DK3 and is used in

AC mode to measure only the fluctuating part. The acquisition timeis 60 s and the time-series are sampling at a frequency of 500 Hz.

2.3. Laser doppler velocimetry instrumentation

The 2D velocity profile survey is performed by the LDV mea-surement method along three diameters at 120◦ in the cone at 0.63Rref under the runner — see Fig. 5 — and one diameter at 0.26 Rref under the runner. The experimental data used in this paper wereobtained with a two-components probe Laser Doppler Anemome-ter (LDA), operating in backward-scattered light on-axis-collectionmode and transmission by optical fiber, with a 5 W argon-ion lasersource. Bragg-cell shifting at 40 MHz is used to resolve directionalambiguity of the velocity. The main characteristics of the optical

system for the measurement under the runner are detailed in Ta-ble 3.

Fig. 5. Sketchof theFrancisturbinemodeland LDVflowsurvey section in thecone.

Optical access

Fig. 6. Three optical access at 60◦ for LDV measurements in the cone.

Spherical silver-coated glass particles are introduced in the testrig flow. These particles are hollowed in order to match the wa-ter density and are able to follow a flow fluctuation frequency of

up to 5 kHz [14]. The mean diameter of these particles is 10 µm.Three cylindrical glass optical windows with plane and parallelfaces are placed at 60◦ with their axes 0.63 Rref under the run-ner (see Fig. 6). They are used as interfaces and exempt rotate thecone between each azimuthal direction. The geometrical referenceof the measurements is obtained by positioning the laser beams onthe window’s faces with an accuracy betterthan 0.05 mm. Both ax-ial and circumferential components of the velocity are measured.Accuracy in the velocity measurements is estimated to be 2% of themeasured value; see [15].

2.4. Particle image velocimetry instrumentation

2.4.1. Measurements on three vertical planes

During the first measurement campaign, the 3D instantaneousvertical velocity field in the cone is investigated with a Dantec

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Calibration target

Cameras

Scheimflung

Fig. 7. Calibration setup for 3D PIV measurements.

M.T. 3D PIV system, which consists of a double-pulsed laser, twodouble-frame cameras, and a processor unit for acquisition syn-chronization and detection of vectors by cross correlation. Theilluminating system is composed of two laser units with Neo-dynium–doped Yttrium Aluminium Garnet crystals (Nd–YAG),each delivering a short impulse of 10 ns and 15 mJ of energy ata frequency of 1 kHz. Thus the time interval between two succes-sive impulses is fixed to 100 µs in order to have approximately

a 10 pixel displacement in the axial direction. The output laserbeam of 532 nm is transformed into a sheet of 4 mm width and 25◦

divergence.

TwoNanosense cameras with a resolution of 1280×1024pixelsoperated in dual-frame mode are used for 200× 200 mm2 investi-gationarea.The cameras areplaced in a stereoscopicconfiguration,focused on the laser-sheet with the scheimflung setting, synchro-nized with the two pulses. Positions of seeding particles are cap-tured by detecting their scattered light.

For the optical access, the cone is manufactured in Polymethylmethacrylate (PPMA) with a refractive index of 1.493, equippedwith a narrow window for the laser’s access and two large sym-metric windows for the cameras access, having a flat externalsurface for minimizing the optical distortions. The corresponding

two-dimensionalvector maps,obtainedfrom each cameraby a fastFourier transform-based algorithm, are combined in order to ob-tain the out-of-plane component, characterizing the displacementin the laser-sheet width.

The correlation between the local image coordinates and realspace coordinates is realized through a third order optical transfermatrix, which includes the correctionof distortions due to differentrefractive indices in the optical path and to the oblique position of the cameras. The calibration relation is obtained acquiring imagesof a calibration target with equally spaced markers, moved in fivetransversal positions in order to have volume information. The tar-get displacement in the measurement area, with accuracy withinthe narrow limits of 0.01 mm in translation and 0 .1◦ in rotation,insured a good calibration quality; [16,17]. The overall uncertainty

of the PIV 3D velocity fields is 3% of the mean velocity value (seeFig. 7 who shows the calibration set-up). The measurements areperformed for three vertical planes at 120◦ which correspond withthe diameters of the LDV measurements survey.

2.4.2. Measurements on four horizontal planes

During the second measurement campaign, the horizontal ve-locity field under the runner is investigated with a LA VISION PIVsystem, which consists of a double-pulsed laser, one double-framecamera ProX with a resolution of 1600×1200 pixels, and a proces-sor unit for the acquisition synchronization and the vectors detec-tion by cross correlation. The illuminating system is composed of two laser units with Neodynium–doped Yttrium Aluminium Gar-

net crystals (Nd–YAG), each delivering a short impulse of 10 ns and20 mJ of energy at a frequency of 20 Hz.

Calibration target

Scheimflung

Camera

Fig. 8. Calibration setup for 2D-PIV measurements under the runner.

Fig. 9. Circular calibration plate under the runner.

For the optical access, the above part of the elbow is equippedwith a window directing to the runner — see Fig. 8. A special circu-lar calibration plate of diameter 355 mm is designed for the mea-surements – see Fig. 9 – and held under the runner with a magnet.The elbow is dismantled for calibration plate positioning. Then theelbow is reassembled and the test rig filled with water. The cam-era is installed under the elbow and focused on the laser-sheet. Af-ter the acquisition of the calibration images, the calibration platetouches it by an LDV window in the cone and takes off the drafttube by the large window near the draft tube outlet. The measure-ments are performed for four horizontal planes at 0.13, 0.26, 0.50and 0.77 Rref under the runner.

3. Results

Throughout this paper the velocity is made dimensionless bythe runner outlet velocity (runner angular speed multiplied byrunner outlet radius, ω Rref ), and lengths are made dimensionlesswith respect to the runner outlet radius Rref (Fig. 5).

The Reynolds number under the runner for the considered op-erating points is about 6.6 million. It is calculated with the run-ner’s peripheral velocity and outlet diameter and the tests are car-riedout velocity similarityaccording to the IEC 60193 InternationalStandards [13]. The flow behavior for high Reynolds numbers isasymptotical. Despite this, the scale effect due to a Reynolds num-ber influence on hydraulic efficiency must be taken into accountfor a correct scalability from the model to the prototype. The effi-

ciency scale-up for a reaction turbine is clearly defined in the IEC60193 International Standards [13].

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Fig. 10. Axial and tangential velocity under the runner obtained by LDV — OP1 to OP4.

Fig. 11. Tangential velocity spatial repartition obtained by PIV — OP4.

3.1. Velocity profile under the runner

The LDV velocity measurements provide the axial and tangen-tial velocity component in the cone under the runner on threediameters at 120◦ or six radiuses at 60◦, described in Fig. 5. Byobserving these velocity profiles it is obvious that the flow underthe runner is not axisymmetric because of the small extent of thedraft tube cone and thus the elbow influence. Therefore, we calcu-late the velocity field on a complete section by a cubic spline inter-polation between thesix azimuthalvalues corresponding to the sixradiuses. PIV measurements are made under the runner for threevertical planes orthogonal to each LDV measurement axis visibleon Fig. 5. After post-processing, one obtains the whole 3D instan-taneous velocity fields at 1 kHz or 10 Hz depending on the experi-

ment performed. Very good agreement is found between the meanvelocity profiles derived from 1 kHz or 10 Hz acquisition.

Fig. 10 shows a 3D visualization of the axial and tangential ve-

locity obtained by the interpolated LDV measurements under therunner for the four operating points (OP1 to OP4 as defined in Ta-ble 1). The axial velocity is represented on the vertical axis and the

tangential component is described by the colorscale. The positionof the spiral casing tongue is represented on the axes.

Fig. 11 shows the juxtaposition of three tangential velocitymean field averaged from 3000 instantaneous 3D fields acquiredby PIV for OP4 and azimuthal direction no. 3 (V3). The vortex core

axis tends to the left side which corresponds to the elbow side.

Fig. 12 shows the evolution of the three components of the ve-

locity field obtained by PIV and LDV for operating points OP1 toOP4 as described in Table 1. Results are displayed for z = 0.66 Rref .The overall aspect of the mean velocity profiles does not dependmuch on the operating point chosen. Each operating point exhibits

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Fig. 12. Radial, tangential and axial velocity component obtained by PIV— OP1 to OP4.

a two counter rotating vortices structure. One is centered close tothe cone axis and extends roughly until 0.5 Rref . The tangential ve-locity repartition for this vortex shows that it is counter rotatingwithrespect tothe runnerand thatit seems tobe close toa Burger’svortextype.The other vortexconcerns theexternalpart of the flowwhere the velocity evolves quasi linearly with the radius above 0.5Rref . This one is co-rotating with respect to the runner and corre-sponds to the swirling rate introduced by the runner tangentialvelocity field. It has to be noted that for the majority of the flow(±0.2Rref < r

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Fig. 13. Axial and tangential phase velocity obtained by LDV.

break-off. One of these, is a possible apparition, for operating con-ditions corresponding to the break-off, of a global 3D large-scaleinstability implicating the entire flow in the diffuser. A simplifiedanalysis based on a parallel and symmetrical flow have been done

by Resiga et al. [8] in the context of the FLINDT project. We startfrom this analysis and adapt the analytical representation of thevelocity components to our case. Then, a simplified stability anal-ysis can be done, which is presented here. This result will pave theway for a global3D stabilitycalculation which can be performed asa future extension of the present work.

4.1. Resiga’s formulation

We introduce in this paper the analytical representation of thevelocity components previously used by Resiga et al. [8], whichconsist of the following Eqs. (11) and (12),

V z = V z + V z 1 e− r 2

R21

+ V z 2 e− r 2

R22

(11)

V θ = V θ 0 r + V θ 1R21

r

1 − e

− r 2R2

1

+ V θ 2 R22

r

1 − e

− r 2R2

2

. (12)

These equations correspondto the superposition of threeBatch-elor vortices. One Batchelor vortex with an infinite characteris-tic radius which corresponds to a solid body rotation, V θ = V θ 0 r and V z = V z 0 , and two Batchelor vortices, one co-rotating and theother counter-rotating with respect to V θ = V θ 0 r , and co-flowing/counter-flowing with respect to V z = V z 0 , respectively. Faller andLeibovich [18] have previously used these axial velocity functional

forms with only two vortices to fit their experimental data for aradial guide-vane swirl generator.

If R0 is the dimensionless survey section radius, then the dis-charge coefficient ϕ can be obtained by integrating the axial veloc-ity profile with the following Eqs. (13) and (14),

ϕ = Q π ωr 2ref

= 1π

V z (r )r dθ dr (13)

ϕ0 = V θ 0 R20 + V θ 1 R21

1 − e− R

20

R21

+ V θ 2 R221 − e−R2

0

R22

. (14)

Axial andtangential velocity profiles of Eqs. (11) and (12) definean eight parameter set π = {V θ 0 , V θ 1 , V θ 2 , V z 0 , V z 1 , V z 2 , R1, R2} tobe determined by fitting the experimentaldata. For each operatingpoint, with a set of experimental data (r j, v z j , vθ j ) j=1...N , the vectorerror is defined in the Eq. (15),

ek(π ) =

V z (r k, π ) − v z k for k = 1, 2, . . . , N V θ (r k−N , π ) − vθ k−N for k = N + 1, . . . , 2N . (15)

A special least-square procedure has been developed in theMatlab environment in order to fit the equations to the experi-mental data minimizing the term Σ 2N

k−1e

k(π )2. It used an initial

set of eight parameters, Π given by the user and furnished afterminimization, a converged value of Π . This analysis was maybeperformed in a quasi axisymmetric reference with respect to thevortex itself and then, when done, translated into the draft tubereference. Fig. 15 shows the analytical profiles obtained with theseequations. One can observe that these equations are inefficient tofit properly the axial velocity profile.

4.2. Blade-wakes adapted formulation

Following the results shown on Fig. 14, the blade-wakes needto be added in the both equations. Two anti-symmetrical Batchelorvortices are added in the both equations, it means four parametersmore: blade-wakes radius R3, vortices center along the blade R4and the axial and tangential amplitude of these vortices, V z 3 andV θ 3 respectively. We obtain the following Eqs. (16) and (17),

V z blade-wakes = V z + V z 3R23

r − R4

1 − e− (r −R4)

2

R23

− V z 3R23

r + R4

1 − e− (r +R4)

2

R23

(16)

V θ blade-wakes = V θ + V θ 3 e− (r −R4 )

2

R23 − V θ 3 e

− (r +R4 )2

R23 . (17)

Fig. 14. Axial and tangential velocity phase averaged obtained by LDV — OP4.

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Fig. 15. Axial and tangential velocity profiles — OP1 to OP4.

Table 4

Swirl parameters — OP1 to OP4.

OP ϕexp V θ 0 V θ 1 V θ 2 V z 0 V z 1 V z 2 R1 R2 V z 3 R3 R4 V θ 3 ϕ0 er.%

1 0.368 0.276 −0.247 −0.624 0.333 0.067 −0.010 0.55 0.05 0.207 0.25 0.63 −0.032 0.373 1.462 0.374 0.287 −0.272 −1.499 0.332 0.081 −0.006 0.57 0.05 0.216 0.26 0.63 −0.031 0.378 1.073 0.380 0.278 −0.229 −3.237 0.324 0.129 0.016 0.58 0.05 0.247 0.29 0.61 −0.044 0.386 1.534 0.390 0.286 −0.232 −4.930 0.318 0.163 0.038 0.61 0.05 0.273 0.29 0.61 −0.051 0.395 1.24

Then the discharge coefficient can be obtained numerically byintegrating the new axial velocity profile, but this integration istaken error function erf and exponential integral E i, so it is notdetailed here.

By observing the phase-averaged velocity contours in Fig. 14,R3 can be estimated to 0.2 Rref and R4 to 0.65 Rref . Swirl parameters

found by the least-squared solver for operating points OP1 to OP4are listed in Table 4. The last two columns contain the values of the discharge coefficient ϕ computed with Eq. (14), and the corre-sponding relative error with respect to the measured value shownin the first column. This error is a good indicator of the accuracy of the fit, as well as for the measurement’s overall accuracy.

Fig. 16 displays the data as well as the curves fitted for the fouroperating points in Table 1 and the first measuring axe V1. Thequality of the fit can be assessed byobserving that most of the timethe curves approach the experimental points.

By fitting the different parameters on a large band of discharge,a suitable parametric representation for the swirling flow wasfound at the Francis runner outlet. Fig. 17 shows the variation of characteristic vortex core radii, angular and axial velocities with

respect to ϕ, together with the corresponding linear fits. V θ 2 isthe characteristic angular velocity of the centered vortex which

increase in intensitywith thedischarge. Moreover,one shouldnotethat V θ 0 and V θ 1 are almost constant over the investigated oper-ating range. Finally, Fig. 18 presents the variation of the differ-ent characteristic blade-wakes parameters. A first conclusion fromFigs. 17 and 18 is that swirl parameters have a smooth, generallylinear, variation in ϕ over the investigated range. As a result, one

obtains the velocity components as C ∞ functionals V θ (r , ϕ) andV z (r , ϕ), further employed in a parametric study of the flow stabil-ity or other properties.

4.3. Stability analysis

The results of a simplified stability analysis of the flow underthe runner are presented in this section. Indeed, it is useful to firstexamine the swirling flow at the draft tube inlet before perform-ing an analysis of the flow in the straight cone or even in the whole3D geometry.Thus, several assumptions must be admitted,and theresults must be interpreted accordingly. Such results may be quiteuseful if there is a correlation (even qualitative) with the overalldraft tube behavior over a certain range of discharge variation. A

steady mean flow with axisymmetric axial and tangential analyti-cal velocity profiles derived from experimental data is considered.

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Fig. 16. Axial and tangential velocity profiles — OP1 to OP4.

An inviscid incompressible fluid is considered. The diverging coneangle of 6.7◦ lets us assume that the bulk flow is parallel. As far asthe mean flow is concerned, the radial velocity is one order of mag-

nitude smaller than the axial velocity since V r V z

= tan 6.7◦ ≈ 0.12.Within these assumptions, the mathematical model used here cor-

responds to the theory of finite transitions between frictionless

cylindrical flows originally developed by Benjamin [19]. The equa-

tion of continuity for axisymmetric incompressible flows conduct

to a generalized eigenvalue problem. The eigenvalues are com-

puted numerically once the problem is discretized.

The first four eigenvalues were then obtained and presented

on the Fig. 19.One can see that for values of ϕ between 0.355

and 0.395 all the eigenvalues are negative. The flow is supercriti-

cal and cannot sustain axisymmetric standing waves. However forϕ < 0.355 and ϕ > 0.395 the largest eigenvalue becomes posi-

tive followed by the next eigenvalues. The flow is subcritical with

standing waves described by the corresponding eigenvectorψ .The

critical state occurs according to our computations for values of

ϕ = 0.355 and ϕ = 0.395. These discharge values are quite closethe value of ϕ = 0.378 and ϕ = 0.390 where the sudden drop indraft tube pressure recovery respectively appears and disappears.

The values cannotbe exactly in agreement because the assump-

tion of axisymmetricflow is notrespected in ourcase. Furthermore

the conical part of the draft tube is quite short (1.18 × Rref ) whichimplicates parasitical effects to the stability approach used here.

This stability analysis is quite well correlated in terms of dis-

charge coefficient ϕ’s range with the measurements of fluctuating

pressure in the cone presented in the Fig. 20.

5. Focus on radial velocity component measurements

In this part we will compare the conical flow formulation forradial velocity with azimuthally averaged data. Then a new analyt-ical formulation based on the three Batchelor vortices superposi-tion will be presentedand compared with phase averaged data andnon-phase averaged data on a single diameter.

5.1. Non-antisymmetric characterization

The 2D PIV velocity measurements in a horizontal plane underthe runner provide the radial and tangential velocity componentson a near full section for one phase (i.e. for one angular position of the runner). The focal lengthof theused lens andthe dimensions of

the elbow do not permit the covering of an entire flow section. Thepresented figures are top views which means the runner and theflow are anticlockwise oriented and the axial velocity is orientedin the figure. The inlet ofthe spiralcasing is onthe right sideof thefigure and the draft tube is on the left side. Indeed, the horizontalaxis of the figures corresponds to the downstream–upstreamdirection of the global machine. Fig. 21 shows respectively radialand tangential velocity contours for operating point OP3 whichcorresponds to the maximum of the efficiency drop amplitude.These phase-averaged velocity contours are obtained by averaging2000 instantaneous velocity fields at the same phase angle.

It is clear that both velocity components exhibit highervalues intheexternalpartof thestudied zone especially near theelbowinte-rior side. By observing these velocity contours it is obvious that the

flow under the runner is not axisymmetric. Part of the explanationmay be the small extent of the draft tube cone and thus the elbow

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Fig. 17. Characteristic vortex core radii, angular and axial velocities versus discharge coefficient.

Fig. 18. Characteristic blade-wakes parameters versus discharge coefficient.

influence. A non axi-symmetrical criterion has been used to char-acterize clearly this state. Considering a pair of axi-symmetricalpoints A and A, the criterion C non-axi is defined in the Eq. (18),

C non-axi = 100 ×|V A − V A |

V ref (18)

where V A and V A are the velocity component values (tangential orradial) for the two considered symmetric points. The Fig. 22 showsthe C non-axi map for radial and tangential velocity componentrespectively. Obviously these figures are symmetric from theircentres by definition of the non axi-symmetrical criterion. Thenon-axisymmetry of the runner with 19 blades made the blade-wakes still visibles. Then, the major differences for radial veloc-

ity component are left–right oriented, whereas for the tangentialvelocity component it is upstream–downstream oriented.

5.2. Comparison of azimuthally averaged data with conical flow formulation

For both the radial and tangential velocity components, the az-imuthally averaged velocity profile is calculated on a complete ra-dius and compared with analytical formulations for the operatingpoint OP1 to OP4 on the Fig. 23. Good agreement is obtained butit should be noticed it is in terms of azimuthally averaged dataonly.

Mauri et al. [4] choose the conical flow assumption to initializetheircalculations. This barely approachedformulation givesquite agood agreement with experimental measurements for azimuthallyaveraged data only, butis stronglyerroneouswhen we consider thenon-axisymmetry of the flow — see Section 5.4.

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Fig. 19. The first four eigenvalues function of the discharge coefficient.

Fig. 20. Fluctuating pressure in the cone — OP1 to OP4.

5.3. Analytical formulation for radial velocity component

Even if the radial velocity component is obviously non-axisymmetric, an axisymmetric analytical formulation for thiscomponent is proposed, based on the three Batchelor vortices su-perposition. As for axial and tangential velocity components in theSection 4, this analysis maybe performed in a quasi axisymmetricreference with respect to the vortex itself and then, when done,translated into the draft tube reference. To manage it, the conical

Table 5

Swirl parameters for radial velocity — OP1 to OP4.

OP ϕexp V r 0 V r 1 V r 2

1 0.368 0.198 −0.028 0.0042 0.374 0.204 −0.029 −0.0393 0.380 0.235 −0.032 −0.0544 0.390 0.238 −0.033 −0.210

flow formulation fora solid body rotation (Eq. (5)) isusedand com-pleted with the two counter rotating Batchelor vortices with fournew magnitude parameters V r 0 , V r 1 and V r 2 . We obtain the Eq. (19),

V r = V r 0 tan(θ r /R0) + V r 1 e− r 2

R21 + V r 2 e

− r 2R2

2 . (19)

Blade-wakes canbe neglected forthe radialvelocitycomponentbecause of their small magnitude on this velocity component. Aspecial least-square procedure developed in the Matlab environ-ment is used in order to fit these equations to the experimentalmeasurements. The radial velocity component is obtained experi-mentally only by means of the PIVtechnique so we are constrainedto using it. Swirl parameters found by the least-squared solver for

operating points OP1 to OP4 are listed in Table 5.Fig. 24 shows the variation of vortex characteristic radial ve-

locities with respect to the discharge coefficient ϕ. Linear leastsquares fits accuratelyrepresent V r 0 , V r 1 while for V r 2 a parabolic fitseems to be quite satisfactory. V r 2 is the characteristic angular ve-locity of the centered vortex which increases in intensity with thedischarge.

5.4. Comparisonof analytical formulations with radial velocity profilemeasured by PIV technique on a single diameter

The 3D time-resolved PIV velocity measurements in a verticalplane under the runner provide the radial, tangential and axial ve-locity components. For all radial, tangential and axial components

the mean velocity profile is calculated on a complete diameter andcompared with the analytical velocity profiles for OP1 to OP4 onthe Fig. 25. These mean velocity profiles are obtained by averaging3000 instantaneous velocity fields. The relative difference to finalmean for the 3000 instantaneous velocity fields is under 3% withonly 500 fields. Hence, the quality of the averaged profiles is reli-able. In this case, it could be noticed that the analytical formulationfor radial velocity component fits well the experimental measure-ments on a single diameter, contrary to spatially averaged data.Indeed, the strong radial velocity component peak increasing withthe discharge of the centered vortex is well predicted.

Fig. 21. Dimensionless radial and tangential velocity phase-averaged — OP3.

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Fig. 22. Non axi-symmetrical criterion C non-axi for radial and tangential velocity component — OP3.

Fig. 23. Radial and tangential velocity profiles – Measurements vs. analytical profiles – OP1 to OP4.

In the case of spatially averaged mean values of the previoussection this peak was flattened by the averaging process becauseit is not centered on the geometry. It is now clear that the spatiallyaveraging process destroy the flow asymmetry making possible aglobal agreement between spatially averaged experimental mea-surements and the conical flow formulation. This kind of approachcan and was used to initialize numerical simulations of the wholedraft tube flow. However, the radial velocity field is clearly non ax-isymmetric and it also evolves strongly from operating points OP1

to OP4. It is important to note that the effect of the accident itself is then clearly identifiable on the radial velocity profile even at the

runner exit which is located as far as possible from the elbow and

the diffuser.

6. Conclusion

Two experimental investigations including LDV and PIV mea-

surements have been performed in the conical diffuser and the

draft tube of a Francis turbine. The measurementswere obtained at

four operating points near the ‘‘best efficiency point’’. Particularly,3D time-resolved velocity fields were obtained by means of the PIV

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Fig. 24. Characteristic radial velocity versus discharge coefficient.

technique at the runner outlet (draft tube entrance). The accuracy

of the measurements are determined.

TheLDV measurementspermit us to obtainanalytical represen-tation of axial and tangential velocity component of the swirling

flow under the runner by the superposition of one solid body ro-

tation and two counter-rotating Batchelor vortices — see [8]. To

catch the blade-wakes influence on the velocity profiles, an evo-

lution of this formulation is proposed to have a better agreement

with unsteady experimental data. By adding two anti-symmetrical

Batchelor vortices on the classical formulation, the wakes of therunner blades are represented on the analytical velocity profile.

Obviously this representation is axisymmetric so it is basedon velocity profiles on a single diameter. For this turbine, theflow under the runner is far to be axisymmetric as was shownby the experimental results. A simplified stability analysis is thendone and gives quite good agreement with the drop of draft tubepressure recovery.

Starting of the 3D time-resolved PIV in vertical plane underthe runner and the classic PIV in horizontal plane under the run-ner we accede to measurements of the radial velocity component.Like the two others velocity components, the radial velocity is nonaxi-symmetrical. Anyway we propose a new analytical representa-tion based on the formulation for the conical diffuser and the threeBatchelor vortices superposition adapted from [8]. The compari-son of the analytical formulation for a conical diffuser (Eq. (5)) –see Mauri et al. [4] – with azimuthally averaged profiles give quitegood agreement but is far to consider the strong non-axisymmetryof the swirling flow under this runner. The new formulation for ra-dial velocity component fits well the experimental data for radialvelocity profile on a single diameter.

Our measurements and proposed model permit an analyticalaccurate representation of the unsteady 3D velocity profiles atthe runner outlet, catching the blade-wakes and the radial ve-locity component. This analytical representation is usefully usedas boundary condition for unsteady numerical simulations. Thestrong non axisymmetry of the velocity profiles, observed experi-mentally,is to be included in themodel, as a futurestep,to increasethe analytical representation accuracy.

Fig. 25. Radial, tangential and axial velocity profiles – Measurements vs. analytical – OP1 to OP4.

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7. Nomenclature

H head mH nom nominal head m

Q turbine discharge m3 s−1

T available torque N mN rotating speed rpm

ω runner angular speed rad s−1 f 0 runner frequency rotation s

−1

Rref runner radius mRsec measuring section radius mϕ discharge coefficient 1ψ energy coefficient 1ν specific speed 1χ pressure recovery coefficient 1

ρ fluid density kg m−3

Aref static pressure reference section area m2

η efficiency 1R1 vortex 1 core radius 1R2 vortex 2 core radius 1R3 blade-wakes radius 1

R4 vortices center along the blade 1V r 0 characteristic radial velocity vortex 0 m s

−1

V r 1 characteristic radial velocity vortex 1 m s−1

V r 2 characteristic radial velocity vortex 2 m s−1

V θ 0 characteristic tangential velocity vortex 0 rad s−1



V θ 3 characteristic blade-wakes tangential velocity m s−1

V z 0 characteristic axial velocity vortex 0 m s−1



V z 3 characteristic blade-wakes axial velocity rad s−1

Acknowledgements

The authors would like to acknowledge ADEME, ALSTOMHYDRO FRANCE and DGR for their financial support.

References

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