2011 Hasmatuchi JFE Experimental Evidence of Rotating Stall in a Pump-Turbine at Off-Design...

Vlad Hasmatuchi1Doctoral Assistant

e-mail: [email protected]

Mohamed FarhatSenior Scientist


Steven RothDoctoral Assistant


Francisco BoteroDoctoral Student


Francois AvellanProfessor


Ecole Polytechnique Federale

de Lausanne (EPFL),

Laboratory for Hydraulic Machines,

Avenue de Cour 33bis,

CH-1007 Lausanne, Switzerland

Experimental Evidence ofRotating Stall in a Pump-Turbineat Off-Design Conditions inGenerating ModeAn experimental investigation of the rotating stall in reduced scale model of a low spe-cific speed radial pump-turbine at runaway and turbine brake conditions in generatingmode is achieved. Measurements of wall pressure in the stator are performed along withhigh-speed flow visualizations in the vaneless gap with the help of air bubbles injection.When starting from the best efficiency point (BEP) and increasing the impeller speed, asignificant increase of the pressure fluctuations is observed mainly in the wicket gateschannels. The spectral analysis shows a rise of a low frequency component (about 70%of the impeller rotational frequency) at runaway, which further increases as the zerodischarge condition is approached. Analysis of the instantaneous pressure peripheral dis-tribution in the vaneless gap reveals one stall cell rotating with the impeller at sub-syn-chronous speed. High-speed movies reveal a quite uniform flow pattern in the guidevanes channels at the normal operating range, whereas at runaway the flow is highly dis-turbed by the rotating stall passage. The situation is even more critical at very low posi-tive discharge, where backflow and vortices in the guide vanes channels develop duringthe stall cell passage. A specific image processing technique is applied to reconstruct therotating stall evolution in the entire guide vanes circumference for a low positive dis-charge operating point. The findings of this study suggest that one stall cell rotates withthe impeller at sub-synchronous velocity in the vaneless gap between the impeller and theguide vanes. It is the result of rotating flow separations developed in several consecutiveimpeller channels which lead to their blockage. [DOI: 10.1115/1.4004088]

1 Introduction

Major development of thermal power generators, either coal ornuclear, within a power generation mix requires the constructionof pumped-storage plants for compensating the random nature ofconsumption by storing energy in excess or delivering peakenergy to meet the demand. Therefore, reversible pump-turbinesprovide an efficient way to stabilize the electricity grid by movingwater back and forth between upstream and downstream reser-voirs. Such examples of pumped-storage power plants constructedin the 1970s are the Raccoon Mountain, Tennessee, USA (Adkins[1]), equipped with high-head pump-turbine impellers, or theGrand Maison, France, equipped with multi-stage pump-turbineimpellers (Courier [2] and Henry [3]). Nowadays, in context ofliberalized electricity market, pumped-storage power plants asGoldisthal, Germany (von Nessen-Lapp and Nowicki [4] andBeyer [5]) or YangYang, South Korea (Houdeline et al. [6]) arekey components for the development of new renewable CO2-freeprimary energies (e.g., wind or solar energy). Moreover, sincethe electricity network frequency must be maintained stable, thepump-turbines are subjected to a rapid switching between thepumping and generating modes with extended operation underoff-design conditions. In particular, during the start-up procedure,the machine is put in runaway operation (speed no-load condition)prior to its synchronization. Depending on the specific speed ofthe pump-turbine, the discharge-speed as well as torque-speedcharacteristics at constant guide vanes opening can be “S-shaped”featuring positive slope, see Fig. 1. The main issue with such a

characteristic curve is that at runaway, the machine may switchback and forth from generating to reverse pumping modes. More-over, it is well known that such unstable operation leads to a sig-nificant increase of structural vibrations and noise.

So far, the issue of pump-turbines operation under off-designconditions in generating mode was poorly addressed in the litera-ture. Nevertheless, early studies reported the technical challengesof pump-turbines operation under unsteady off-design regimes(Blanchon et al. [7], Casacci et al. [8] and Lacoste [9]). Moreover,the unsteady phenomena, which may develop in a pump-turbineduring a sudden load rejection, were addressed by few authors:Taulan [10], Pejovic et al. [11], Tanaka and Tsunoda [12], Oishiand Yokoyama [13]. Martin, Refs. [14] and [15] proposed a stabil-ity analysis to predict the occurrence of large flow oscillations foridealized machine. He argued that the unstable operation in caseof load-rejection with failed servomotor is mainly due to the pres-ence of a positive slope on the torque characteristics at runaway.Nicolet [16] stated that high head pump-turbines, which are com-monly of low specific speeds, are more subjected to “S-shaped”characteristic curves. Dorfler et al. [17] proposed an interestingmethod to avoid unstable operation of pump-turbines duringmodel tests: The inlet valve is partially opened and by-passedwith a second valve to adjust the flow rate. The resulting artificialhead loss improves significantly the hydraulic stability. Kuwabaraet al. [18] developed an advanced control of the guide vane inde-pendent servomotors, which was provided with an anti-S-charac-teristics control to be used upon load rejection. Klemm [19]reported the implementation of the Misaligned Guide Vanes(MGV) concept in the COO II pumped-storage power plant inBelgium to achieve an improvement of the machine stability forno-load and extreme part-load conditions. It consists in operatingseveral guide vanes independently from the rest of the guide vanemechanism in order to stabilize the machine. Recently, Shao [20]

1Corresponding author.Contributed by Fluids Engineering Division of ASME for publication in the JOUR-

NAL OF FLUIDS ENGINEERING. Manuscript received December 10, 2010; final manu-script received April 20, 2011; published online June 7, 2011. Assoc. Editor: EdwardM. Bennett.

Journal of Fluids Engineering MAY 2011, Vol. 133 / 051104-1Copyright VC 2011 by ASME

Downloaded 08 Jun 2011 to 128.178.182.153. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

proposed and validated empirical formulae of the pump-turbineinternal characteristics with MGVs, based on the internal charac-teristic theory of turbomachinery and the original model charac-teristic curves without MGVs.

In the present paper, an experimental investigation is carriedout in reduced scale model of a pump-turbine. Measurements ofpressure fluctuation in the wicket gates channels synchronizedwith high-speed flow visualization are performed for a wide rangeof machine operation. The analysis is focused on the onset and de-velopment of flow unsteadiness at runaway and low flow rateconditions.

2 Experimental Setup

A reduced scale model of a low specific speed radial pump-tur-bine featuring 9 impeller blades and 20 guide and stay vanes, Fig. 2,is installed in the EPFL PF3 test rig. Two 400 kW centrifugalpumps in series deliver 1.4 m3 s�1 under a maximum head of100 m. Off-design operating conditions, involving runaway andturbine brake, are investigated. Starting from nominal operation atfixed guide vanes openings (5� and 10�) and 20 m head, the rota-tion speed is gradually increased until occurrence of runaway con-dition, i.e., zero torque value. At this point, the operation becomesunstable and the machine may switch back and forth from gener-ating to reverse pumping modes. A specific procedure, commonlyused in model testing of pump-turbines, is adopted to stabilize themachine operation: once the runaway is reached, a butterfly valve,located in the main pipe upstream to the model, is closed. A by-pass over the specified valve, equipped with a second Iris dia-phragm control valve, is then used for a fine adjustment of the testhead value, providing simultaneously an additional hydraulic re-sistance. This procedure improves significantly the stability of themachine operation and allows exploring the entire “S-curve.”

Wall pressure measurements are performed using piezoresistivepressure sensors flush mounted in the spiral casing, stay vanes andguide vanes channels as well as in the draft tube. Pressure signalsare simultaneously recorded using VXI HP1432A digitizers with

16 bits A/D resolution, 51200 Hz sampling frequency and a mem-ory of 1 M-samples/channel, providing maximum 20 s recordlength. The entire experimental setup is illustrated in Fig. 3. Asshown in Fig. 4, the sensors locations are selected to cover the sta-tor channels from spiral casing up to rotor-stator interface in bothradial and circumferential directions. Moreover, high-speed flowvisualizations are performed in the vaneless gap between theimpeller and the wicket gates. To this end, air bubbles are injectedthrough a 1 mm nozzle located on the upper wall of a stator chan-nel. Bubbles entrainment visualizations are performed from thebottom through a transparent Plexiglas window and recorded at1000 fps with the help of a Photron SA1 digital camera.

3 Results

3.1 Pressure Fluctuations. According to IEC 60193 stand-ards [21], the speed, discharge and torque factors are defined byEqs. (1)–(3). The impeller outlet diameter D is 0.250 m, while E,the test specific energy value, is maintained at 200 J kg�1 for allthe operating points.

Fig. 2 HYDRODYNA reduced scale model

Fig. 3 Experimental setup for pressure measurements andhigh-speed flow visualization

Fig. 1 “S-shaped” characteristic curve of a pump-turbine ingenerating mode

Fig. 4 Pressure sensors location in the model

051104-2 / Vol. 133, MAY 2011 Transactions of the ASME


Speed factor:

nED ¼nDffiffiffi

Ep (1)

Discharge factor:

QED ¼Q

D2ffiffiffiEp (2)

Torque factor:

TED ¼Tm

qD3E(3)

The turbine characteristics at 5� and 10� guide vanes opening in(nED, QED) and (nED, TED) coordinates systems are presented inFig. 5. At 10� guide vanes opening, the turbine characteristicexhibits a positive slope after the runaway speed (see Fig. 5, oper-ating point OP. #3). When a pump-turbine prototype is brought insuch a situation, the operation suddenly switches to reverse pump-ing mode. The discharge as well as the torque is reversed with asubstantial increase of structural vibrations driven by flow insta-bilities. In our case, the use of the previously described stabilizingprocedure prevents such unstable operation and allows exploringthe positive slope part of the characteristic curve. Once in reversepumping quadrant, an increase of the rotation speed (OP. #6) leadsto strong cavitation occurrence on the blades at the impeller out-let, associated with significant increase of structural vibrationsand noise. Moreover, this unstable operation is even magnified bythe test rig control system, which fails in correcting the parametersfluctuation in real time.

A global view of the flow unsteadiness at 5� and 10� guidevanes opening is provided by superposing on the (nED, QED) char-acteristic curves, the standard deviation of the pressure fluctua-tions, defined by Eq. (4), in the spiral casing and guide vanes (seeFig. 6). The diameter of circles is proportional to the standarddeviation of pressure scaled by qE. The same scale is adopted forall graphs to allow straightforward comparison between differentoperating points and different locations. It can be easily observedthat the pressure fluctuations in the guide vanes channels, close torotor/stator interface, are by far more important than in the spiralcasing. Moreover, for any given location, the pressure fluctuationsare increased for low rotation speed, around the “S-shape” and inreverse pumping mode. At these conditions, a substantial increaseof the structural vibrations is observed. It should be noticed that inthe particular case of guide vanes location, the maximum pressurefluctuation is at least 25 times larger than in nominal conditionsand about 15% of the specific energy. The present analysis sug-gests that the source of flow unsteadiness at off-design conditionsis likely located in the impeller or in the vaneless gap between theimpeller and the guide vanes.

~c;p ¼1

qE

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XN

i¼1

ðpi � �pÞ2vuut (4)

Further analysis is carried out for six operating conditions (OP. #1to OP. #6) at 10� guide vanes opening, as marked on Fig. 5 andFig. 7. They are distributed as follows: normal operating rangeregion (OP. #1), runaway speed (OP. #3), low positive discharge(OP. #4) and high impeller speed with negative discharge (OP.

Fig. 5 Resulting “S-shaped” curves for the discharge and tor-que factors in generating mode

Fig. 6 Pressure fluctuations standard deviation (STD) Fig. 7 Investigated operating points

Journal of Fluids Engineering MAY 2011, Vol. 133 / 051104-3


#6). Two other intermediate points (OP. #2 and OP. #5) areselected in the transition regions between those already presented.

The pressure coefficient fluctuation is computed in accordancewith Eq. (5). Averaged spectra of pressure fluctuations in the spi-ral casing (sensor Sc1) and in the guide vanes (sensor Gv1) are pre-sented in Fig. 8 and Fig. 9 for previously selected operatingpoints. The plots scales are differently chosen in order to facilitatethe visualization. The pressure fluctuation in the guide vaneschannels is dominated by the blade passing frequency and its firstharmonic with lower amplitude (f¼ 9fn and f¼ 18fn), exceptingthe low discharge operating point. The spectral analysis showsalso a low frequency component (�70% of the impeller rotationalfrequency) arising at runaway (OP. #3), which further increases inamplitude as the low positive discharge condition (OP. #4) isapproached. At this point, it even represents the dominant fre-quency and is found to modulate the blade passing frequency. Inthe spiral casing (Fig. 8), as reported by Tanaka [22], the bladepassing frequency has lower amplitude than its first harmonic.The low frequency component is visible in this region as well andbecomes dominant in turbine brake mode (OP. #3 and OP. #4).Actually, this low frequency represents the signature of the flowunsteadiness at off-design operating conditions. Moreover, whenstarting from the nominal conditions and increasing the impellerspeed, a significant increase of the pressure fluctuations amplitudeis observed in the channels between guide vanes as well as in thespiral casing. At best efficiency point (OP. #1) the amplitude of

fluctuations is at least 10 times smaller than any of operatingpoints from OP. #2 to OP. #6. In the spiral casing, the pressurefluctuations amplitude is significantly lower (almost 10 times)than in the guide vanes region, except for the low discharge condi-tion (OP. #4). These observations confirm that the source of flowunsteadiness is located either in the impeller or in the vanelessgap between the impeller and the distributor.

Pressure coefficient fluctuation:

~cp ¼p� �p

qE(5)

Time history over 10 impeller revolutions corresponding to pres-sure fluctuations in the vaneless gap between the impeller and theguide vanes (sensor Rs3) are presented in Fig. 10 in comparisonbetween normal operating range (OP. #1), runaway (OP. #3) aswell as low discharge condition (OP. #4). As already observed inthe spectra (Fig. 9), the fluctuations amplitude at the normal oper-ating condition is insignificant compared to the low discharge con-dition. Surprisingly, the pressure fluctuations at this very unstableoperating point are not random but exhibit a remarkable periodic-ity. It can also be observed the way the rotor-stator interaction fre-quency is carried by the related low frequency (�0.7fn). An in-stantaneous image of the time-pressure fluctuations on the entirevaneless gap circumference is given in Fig. 11, with the help of atime-space-amplitude representation. Accordingly, one high

Fig. 8 Pressure fluctuations amplitude spectra in the spiralcasing, sensor location Sc1

Fig. 9 Pressure fluctuations amplitude spectra in the guidevanes channel, sensor location GV1



pressure instability source rotating with the impeller at sub-syn-chronous frequency can be detected. Moreover, the high pressureregion covers approximately 50% of the circumference. Thesefindings are consistent with those of Vesely et al. [23] who inves-tigated a medium head pump-turbine model and found that a rotat-ing stall with a frequency of propagation about 60% of the impel-ler speed arises at turbine brake and reverse pumping conditionshaving only one stall cell.

3.2 Flow Visualization. High-speed flow visualizations aremade with the help of air bubbles injection in the vaneless gapbetween the impeller and the guide vanes. A needle valve isused to control the amount of injected air. The injection pres-sure is maintained at a value slightly above the mean pressureat the injection location. We assume that air bubbles fairly fol-low the streamlines with almost no effect on the flow itself.Fig. 12 illustrates arbitrary instantaneous captures of the flowpattern in the guide vanes region for the normal operating range(OP. #1), runaway (OP. #3) and low discharge (OP. #4) condi-tions. To assess the flow trajectory in the guide vanes channel,we have summed the images captured during 3 impeller revolu-tions at 1000 fps. The result is given in Fig. 13 for the sameoperating points OP. #1, #3 and #4. So, at BEP (OP #1), thetrajectory of air bubbles is straight inside the guide vanes chan-nel, then becomes slightly unstable in the wake of the neighbor-ing guide vane. At runaway (OP. #3), the flow becomes moredisturbed, as evidenced by the scatter of air bubbles. At low

discharge (OP. #4) and during the passage of the rotating stall,part of injected air is found upstream to the injection site, sug-gesting the occurrence of backflow.

According to Fig. 11, at low discharge operating condition(OP. #4) one stall cell travels with the impeller at a constant sub-

Fig. 10 Time history of the pressure fluctuation in the vanelessgap for 3 operating conditions

Fig. 11 Time history of the pressure fluctuation in the wholevaneless gap at low discharge condition (OP. #4)

Fig. 12 Instantaneous air bubbles flow visualizations in oneguide vanes channel

Fig. 13 Superposition of air bubbles visualizations in oneguide vanes channel during 3 impeller revolutions



synchronous speed and induces the same pressure fluctuation am-plitude in the whole vaneless ring. Therefore, a time distributionof the phenomenon can be represented as a space distribution byplacing images for one rotating stall period with appropriate phaseshift to obtain a synthetic instantaneous view of the flow state inthe entire guide vanes circumference. A specific image processingwas applied on the high-speed movie at low discharge operatingpoint (OP. #4) in a previous work (Hasmatuchi et al. [24]).Briefly, the image processing procedure may be described as fol-lows: Firstly, the high-speed visualization movie for OP. #4 issplit in pictures frame-by-frame. Then, a periodic mask (for anangle of 18�, in accordance with the number of guide vanes)reported to the machine rotation axis is created in order to extractfrom figure only one guide vane and the corresponding adjacentchannel comprising the air injection aperture. The procedure isrepeated for each frame. Finally, 20 processed images areextracted from one rotating stall period and placed on the finalglobal figure to cover the whole guide vanes ring. Repeating theprocedure for each movie time step, a high-speed synthetic flowvisualization of the rotating stall evolution in the entire guidevanes circumference is provided. The result for one time step isillustrated in Fig. 14. Moreover, the impeller is placed in the cen-ter of the figure at its real position. A contour plot of the low-passfiltered pressure fluctuation (the rotor-stator interaction fluctua-tions are filtered) for the sensor RS3 placed in the vaneless gapdownstream the air injection aperture, is represented between theguide vanes and the impeller for one rotating stall period. Therotor-stator interaction pressure component cannot be appropri-ately spatially represented since its frequency is not a multiple ofthe rotating stall frequency. The pressure contour is in accordancewith the flow visualization for each angular position. The injectedair bubbles are visible only on a half of the circumference whenthe air pressure exceeds the pressure inside the flow. Therefore, in

accordance with the pressure fluctuation contour as well, no bub-bles are visible in the channels between the guide vanes 6 to 18.Channels between the guide vanes 18 to 14 show the state of theflow at the end of the stall passage. Vortices and backflow domi-nate the flow pattern. Once the rotating stall passed (channelsbetween guide vanes 14 to 6), the flow returns step by step to auniform pattern. In the channels between the guide vanes 9 to 6,the flow is even similar with the one in the normal operating range(see Fig. 12, OP. #1). The related visualization technique is inter-esting because it allows creating a global view of the flow patternin the entire guide vanes region without creating a transparentwall on the whole circumference, which is not realistic forobvious mechanical resistance reasons. Moreover, the draft tubepresence would restrain the visual field. The shortcoming givenby the air bubbles absence when the pressure is too high could beeliminated by correcting the injected air pressure level at eachmoment accordingly with the pressure in the flow. The challengeis given by the need of injecting the minimum volume of gas foran optimum visibility. The injection procedure must be performedquickly, because working at almost no-discharge condition, thegas introduced in the flow, remains inside the model.

3.3 Discussion. On one hand, flow separation occurs at theinlet of the impeller channels at runaway operating point. More-over, for the operating points in the S-region the flow separationdegenerates in blockage. On the other hand, due to the large vane-less gap between the impeller and the guide vanes at 10� opening,backflow cells are also developed with an alternate switchbetween generating and pumping modes of the impeller channels.Moreover, this explanation is in agreement with the results offlow visualization in the impeller of a small model Francis typepump-turbine obtained by Senoo and Yamaguchi [25] as well as

Fig. 14 Instantaneous synthetic flow visualization of the rotating stall evolution in the fullguide vanes ring and vaneless gap



Hayami et al. [26]. Furthermore, Staubli et al. [27] also concludedthat local vortices formed at the inlet of the impeller channels rep-resent the source of the unsteady inflow and outflow from theimpeller in the vaneless gap between the impeller and the guidevanes.

From a theoretical point of view, the mean velocity triangles atthe impeller inlet in generating mode can be represented as inFig. 15. The b angle is given by the relative flow velocity at theimpeller inlet, whereas the a angle is given by the absolute veloc-ity of the flow at the guide vanes outlet. At the best efficiencypoints (BEP) for 18� and, respectively, 10� (OP. #1) guide vanesopening angles, the mean velocity triangles are similar, while anenormous difference is observed between the mean velocity trian-gles for normal operating range (OP. #1) and runaway (OP. #3)conditions. Furthermore, at low discharge condition (OP. #4) therelative flow velocity angle diminishes even more. Figure 16 illus-trates a schematic representation of the flow pattern at the impellerinlet under off-design operating conditions. In agreement with themean velocity triangle at normal operating range operation, a uni-form flow at the impeller inlet is sketched. At runaway, flow sepa-ration occurs since the relative flow velocity angle, b, is stronglydiminished. Recirculation regions appear on the suction sides ofthe blades, leading to channels blockages. The difference betweenrunaway and low discharge condition is given by the accentuationof the flow separation phenomenon at the impeller inlet. The

superposition of the rotor-stator interaction over this situationrenders a very complex flow pattern. Brennen [28] provides an ex-planation about the rotating stall onset and development in a cas-cade of stator or rotor blades. Accordingly, three consecutiveimpeller blades operating at large incidence angle are considered.A perturbation of the incoming flow induces a stall in one of thechannels. Furthermore, the blockage induces an increase ofthe incidence angle for the follower blade as well as a decrease ofthe incidence angle for the forerunner blade. Consequently, thestall moves on a direction away from the incoming flow. Indeed,the stall cell can occur on several consecutive channels. In thecase of a rotor, the stall rotates in the same direction with theimpeller, but with 50–70% of its angular velocity.

4 Conclusions

The experimental evidence of rotating stall in a reduced modelcentrifugal pump-turbine at off-design operating conditions hasbeen presented. The experiment, carried out in the EPFL PF3 testrig, involves wall pressure measurements in the stator along withhigh-speed flow visualizations in the vaneless gap between theimpeller and the guide vanes with the help of air bubbles injec-tion. The focus is put on the off-design conditions in generatingmode involving runaway and turbine brake. Starting from the bestefficiency point, the impeller speed is gradually increased untilthe flow is totally reversed. The main conclusions may be sum-marized as follows.

• At the best efficiency point, the pressure fluctuation is verylow and mainly dominated by the blade passing frequencyand the first harmonic.

• As the impeller enters the “S-shape” domain, a substantialincrease of the pressure fluctuation is observed everywhere inthe stator. This increase is particularly pronounced in theguide vanes, while it is minimal in the draft tube.

• At runaway, the spectral analysis shows a rise of a low fre-quency component at about 70% of the impeller rotationalfrequency, which further increases in amplitude as the zerodischarge condition is approached. The time-pressure distri-bution in the vaneless gap between the impeller and guidevanes reveals one stall cell rotating with the impeller at sub-synchronous speed.

• High-speed visualization shows a quite uniform flow patternin the guide vanes channels at the normal operating range,whereas at runaway the flow is disturbed by the rotating stallpassage. The situation is even more critical at very low posi-tive discharge, where backflow and vortices in the guidevanes channels accompany the stall passage.

• A specific image processing technique is applied to create ahigh-speed visualization of the rotating stall evolution in theentire guide vanes circumference for a low positive dischargeoperating point.

To sum up, the flow in a pump-turbine operating under off-design conditions in generating mode, in the S-region, is domi-nated by one stall cell rotating with the impeller at sub-synchro-nous speed in the vaneless gap between the impeller and the guidevanes. It is the result of rotating flow separations developed inseveral consecutive impeller channels which lead to their block-age. The rotating instability is generating hydraulic unbalance andstrong structural vibrations.

Acknowledgment

The present investigation was carried out in the frame work ofHYDRODYNA II research project (Eureka N� 4150), in a partner-ship with ALSTOM Hydro, ANDRITZ Hydro, VOITH Hydro andUPC-CDIF. The authors would like to thank the Swiss FederalCommission for the Technology and Innovation (CTI), Swisselec-tric Research and Swiss Competence Center of Energy and

Fig. 15 Theoretical mean velocity triangles at the impeller inletfor BEP and off-design operating points

Fig. 16 Schematic representation of the impeller inlet flow pat-tern at BEP and off-design operation



Mobility (CCEM) for their financial support as well the HYDRO-DYNA II partners for their involvement and support.

Nomenclaturea ¼ absolute flow velocity angle

ao ¼ guide vanes opening angleb ¼ relative flow velocity angleq ¼ water density, kg m�3

~cp ¼ pressure coefficient fluctuation~c;p ¼ pressure fluctuation standard deviationf ¼ frequency, Hz

fn ¼ impeller frequency, Hzn ¼ impeller speed, rot s�1

nED ¼ speed factorp ¼ wall pressure, Pa�p ¼ time average wall pressure, Pa

D ¼ impeller outlet diameter, mE ¼ specific energy, J kg�1

N ¼ samples numberQ ¼ discharge, m3 s�1

QED ¼ discharge factorTED ¼ torque factorTm ¼ impeller torque, N m

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http://dx.doi.org/10.1109/TEC.1987.4765860

http://dx.doi.org/10.1051/lhb/1982030

http://www.hydroworld.com/index/display/article-display/351208/articles/hy=dro-review-worldwide/volume-15/issue-1/articles/goldisthal-pumped-storage-plant-m=ore-than-power-production.html.






http://dx.doi.org/10.1088/1755-1315/12/1/012059

http://dx.doi.org/10.1115/1.3262073

2011 Hasmatuchi JFE Experimental Evidence of Rotating Stall in a Pump-Turbine at Off-Design...

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