research article flow modeling in pelton turbines...

Research ArticleFlow Modeling in Pelton Turbines by an AccurateEulerian and a Fast Lagrangian Evaluation Method

A Panagiotopoulos12 A Cidonis1 G A Aggidis1

J S Anagnostopoulos2 and D E Papantonis2

1Lancaster University Renewable Energy Group and Fluid Machinery Group Engineering Department Engineering BuildingBailrigg Lancaster Lancashire LA1 4YW UK2School of Mechanical Engineering Laboratory of Hydraulic Turbomachines National Technical University of Athens9 Heroon Polytechniou Zografou 15780 Athens Greece

Correspondence should be addressed to G A Aggidis gaggidislancasteracuk

Received 26 June 2015 Accepted 21 October 2015

Academic Editor Robert C Hendricks

Copyright copy 2015 A Panagiotopoulos et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The recent development of CFD has allowed the flow modeling in impulse hydro turbines that includes complex phenomena likefree surface flow multifluid interaction and unsteady time dependent flow Some commercial and open-source CFD codes whichimplement Eulerianmethods have been validated against experimental results showing satisfactory accuracy Nevertheless furtherimprovement of accuracy is still a challenge while the computational cost is very high and unaffordable for multiparametric designoptimization of the turbinersquos runner In the present work a CFD Eulerian approach is applied at first in order to simulate the flowin the runner of a Pelton turbine model installed at the laboratory Then a particulate method the Fast Lagrangian Simulation(FLS) is used for the same case which is much faster and hence potentially suitable for numerical design optimization providingthat it can achieve adequate accuracy The results of both methods for various turbine operation conditions as also for modifiedrunner and bucket designs are presented and discussed in the paper In all examined cases the FLS method shows very goodaccuracy in predicting the hydraulic efficiency of the runner although the computed flow evolution and the torque curve exhibitsome systematic differences from the Eulerian results

1 Introduction

The design of Pelton and other impulse type turbines wasbased on existing know-how and any design improvementswere mainly conducted after extensive experimental testingby the trial-and-error method In recent years significanteffort has been directed towards a better understating ofthe details of the complex unsteady flow in the runnerwith the aid of Computational Fluid Dynamics (CFD)and modern numerical modeling [1] Commercial softwarehas been developed and used based on Eulerian mesh-type codes like Ansys-CFX [2 3] and Ansys-Fluent [4 5]that are designed to simulate the flow in various physicalproblems Also a number of noncommercial software toolshave been developed like the Open FOAM [6] as alsovarious Lagrangian approaches like the Smoothed Particle

Hydrodynamics method [7] or other in-house codes like theFast Lagrangian Simulation (FLS) [8 9] that were developedto solve specific flow problems

At first the Eulerian mesh-type methods were used inorder to simulate the flow in Pelton turbines Several papershave been published investigating the flow in the injector[10ndash12] in the stationary buckets [4] and in the rotatingrunner [2 5] as well as analyzing specific flow mechanismslike cavitation [13]

In most cases commercial CFD software was used asit can simulate the flow with adequate accuracy in orderto calculate the hydraulic efficiency and evaluate the geo-metrical characteristics of the turbine Nevertheless thecomputational cost is too high to be used formultiparametricdesign optimization where thousands of flow evaluationsare required to obtain an optimum design solution Other

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2015 Article ID 679576 13 pageshttpdxdoiorg1011552015679576

2 International Journal of Rotating Machinery

meshless simulation methods based on the Lagrangian ap-proach have also been developedThemost popular one is theSmoothParticleHydrodynamics (SPH)methodology [14 15]the computational cost of which is however comparableto the Eulerian methods while further development andvalidation are needed to improve its accuracy

In order to reduce the computational cost a particu-late method based on the Lagrangian approach has beendeveloped by the Laboratory of Hydraulic TurbomachinesNTUA [8 9] This Fast Lagrangian Simulation (FLS) methodintroduces appropriate adjustable terms in the flow particleequations to approximate the various viscous and pressureeffects of their trajectories

The aim of the present paper is to evaluate the perfor-mance and accuracy of the FLS method by comparing theresults against the corresponding ones of the Ansys-Fluentcommercial software [16 17] that uses the Volume-of-Fluid(VOF) technique to simulate the jet and free surface flowin the runner The accuracy of the latter has been validatedagainst experimental results in Pelton runners [18ndash21]

In the present study the appropriate computationaldomain mesh density and settings of the VOF techniqueare at first investigated to achieve the best compromise ofaccuracy and computational cost Then an initial test case isdefined including the exact geometrical characteristics of aPelton model runner installed in the laboratory The FLS andFluent software are applied to various operating conditioncases of this turbine and for runners of different design inorder to compare their numerical performance based on thetime history of torque development and on the hydraulicefficiency of the runner

2 Numerical Modeling of the Flow

Two computer codes were used for the simulation of thecomplex flow in a rotating Pelton runner the commercialsoftware Fluent and the in-house software FLS The Fluentcode [16] is capable of solving multiphase flow problems withfree surfaces which are highly relevant to impulse turbines Itis based on the spatial discretization of theReynoldsAveragedNavier-Stokes equations on a computational mesh usingcell-centered numerics (finite volumes) and offers flexibilityin choosing between segregated based and coupled basedsolver

The two-fluid-flow (air and water) problem with freesurface like the Pelton runner case is being solved usingthe established Volume-of-Fluid (VOF) method VOF is anEulerian-Eulerian method based on tracking and locatingthe fluid-fluid interface An additional factor the volumefraction is introduced which represents the percentage ofeach fluid volume in every cellThemethod canmodel two ormore immiscible fluids by solving a single set of momentumequations and tracking the volume fraction of each of thefluids throughout the domain

For the two-phase (water-air) flow in Pelton runnersthe air is defined as the primary phase and the water assecondary so as the tracking of the interface between themis accomplished by the solution of a continuity equation

for the volume fraction of the secondary phase that has thefollowing form

1

120588119902

[120597

120597119905(119886119902120588119902) + nabla sdot (119886119902120588119902 120592119902)] =

119899

sum

119901=1

( 119898119901119902 minus 119898119902119901) (1)

where 119898119901119902 is the mass transfer from phase 119901 to phase 119902 and119898119902119901 the opposite and 119886119902 is the volume fraction and 120588 the

density The present application of Fluent-VOF software forflow simulation in Pelton runners is analyzed in Section 4

21 Fast Lagrangian SimulationMethod TheFast LagrangianSimulation (FLS) [8 9] is a single-phase flow simulationmethod and it is based on the tracking of a representativenumber of fluid particles in order to model and calculate theflow pattern and the energy exchange in the rotating runnerof impulse hydro turbines The exact water-air interfacepattern is not required for such computations though theflow width on the bucket surface could be estimated from theproperties of the tracked particles

The water jet is being separated into discrete particlesas shown in Figure 1 and their equations of motion arenumerically integrated until they exit from the inner bucketsurface The jet is considered ideal that is with uniforminitial velocity and the flow frictionless Also due to theperiodic symmetry conditions only two consecutive bucketswere modeled

The fluid particle equations are solved in a rotatingorthogonal systemof reference and are expressed inCartesiancoordinates as follows (bucket rim is on the 119909119910 level)

1198892119909

1198891199052= 119891119909 (119909 119910)

1198892119910

1198891199052= 119891119910 (119909 119910) + 120596

2119910 + 2120596

119889119911

119889119905

1198892119911

1198891199052= 119891119911 (119909 119910) + 120596

2119911 minus 2120596

119889119910

119889119905

(2)

where120596 is the angular rotation speed of the runner and119891119909119891119910and 119891119911 are the additional terms functions of the local surfacegeometrical characteristics

The particle motion equations do not contain particleinteraction ormechanical losses terms and hence they cannotreproduce the real flow picture in the bucket For this reasonthe FLS model introduces a number of additional terms inorder to account for the various hydraulic losses (impactfriction and change direction) as well as for the pressureeffects that control the spreading of the surface flow in thebucket which are activated after the impact of a particle onthe bucket surfaceMore specifically the friction losses on thebucket surface are modeled as reduction of particle kineticenergy by a factor analogous to the square of particle velocityand to sliding distance hence the particle velocity magnitudeafter a time step Δ119905 becomes

1198811015840

119901asymp 119881119901 sdot (1 minus 119862119891 sdot 119881119901 sdot Δ119905) (3)

International Journal of Rotating Machinery 3

Rotation axis

Δ120579

1205790

(a) (b)

Figure 1 FLS modeling jet discretization and jet-bucket interaction starting angle (a) surface flow simulation and representation (b)

The energy losses at the jet impact on the bucket are takenanalogous to the square of the normal to the surface particlevelocity component and this gives

1198811015840

119901asymp 119881119901 sdot (1 minus 119862119894 sdot cos

2120593119894) (4)

where 120593119894 is the particle impingement angleThe progressive change of particlersquos path direction (and

momentum) as it slides along the curved bucket surface alsocauses minor energy losses which are modeled using a termsimilar to the impact losses

1198811015840


2Δ120593) (5)

where Δ120593 is the angular change in direction of the slidingparticle during the time step Δ119905

The adjustable coefficients 119862119891 119862119894 and 119862119901 are introducedin the above modelling equations (3)ndash(5) so as they can betuned in order to match the results with the correspondingones obtained by more accurate CFD methods or by exper-imental studies This constitutes an important feature of theFLS model that can significantly improve the reliability andaccuracy of its results at almost no additional computer cost

Finally in order to model the pressure effects on thesurface flow spreading and evolution a particle acquires atthe impact point an artificial ldquospreadingrdquo velocity componentperpendicular to its impacting plane (119878 Figure 2) whileits main velocity component is correspondingly reduced topreserve kinetic energy The magnitude of this spreadingvelocity depends on the radial and circumferential positionof a particle in the jet cross section and it is determined bytwo corresponding additional adjustable coefficients [8]

The first tuning of the above coefficients has been carriedout with the aid of available experimental data in a Peltonturbine model so as to minimize the squared differencesbetween the measured and the computed characteristiccurves of turbine efficiency [8]This problemwas solved withan optimization software based on evolutionary algorithmsconsidering these coefficients as free variables the values of

Splitter line

Jetflow

rarrVS

rarrV2

rarrV1

rarrV2998400

120593r

Figure 2 Sketch of surface flow spreading modeling

which are iteratively optimized from generation to genera-tion

It must be noted that some secondary flow mechanismsdeveloped during the jet-runner interaction like the jet cutor the attached flow at the back face of the bucket are notmodeled by the FLS method However their effect on theenergy transfer and hydraulic efficiency of the runner aretaken into account in an implicit manner through the aboveadjustable terms More details on the model formulation andits regulation procedure as well as on the parametric designof the runner can be found in [8]

The application of the FLS model for the simulation ofthe jet-runner interaction in Pelton turbines requires onlya mathematicalnumerical description of the inner bucketsurface and since it does not use a computationalmesh it canbe very easily adapted to any design or operation data of therunner Also a specific postprocessing algorithm is developedand used for the presentation and comparison of FLS results(eg Figure 1)


Figure 3 Pelton model turbine and its runner installed at the LHT

3 Test Case

The geometrical characteristics of the Pelton model turbineused as a reference case correspond to a Pelton turbineinstalled in the LHT at the National Technical Universityof Athens (Figure 3) [22] The pitch diameter of the runneris 400mm and the axis is horizontal with two injectors of36mm nozzle diameter The net head the rotational speedand the nozzle stroke can be adjusted for experimental rea-sons according to the IEC standards [23]The runner contains22 buckets which were designed and constructed in the labbased on old literature guidelines [24] as shown in Figure 4Therefore the achieved efficiency of the turbine is smallerthan that suggested in more recent bibliography [25] whichrefers to the state-of-the-art turbines Nevertheless all geo-metrical characteristics of modern runners and buckets areincluded and hence the flowmechanisms that take place dur-ing the interaction of the free jet with the runner are the same

Although the Pelton turbine is installed at the labo-ratory test rig and the total efficiency of the turbine hasbeen measured experimentally a direct comparison betweennumerical and experimental results is not easy The mea-sured efficiency represents the total efficiency of the turbineincluding all losses according to the IEC standards [23]while only the hydraulic efficiency of the runner can becalculated numerically So the comparison would requirethe estimation of minor losses like the losses in the nozzlethe mechanical losses and the windage losses caused by themovement of the runner and its interaction with the mistyenvironment into the casing These losses highly dependedon the quality of the turbine construction the geometricalcharacteristics of the casing and the type of the bearings Sotheir estimation would introduce considerable uncertaintyto an experimentally derived efficiency of the runner Forthis reason the present work focuses on the comparisonand evaluation of the performance and accuracy of the twosoftware tools when used for Pelton turbines analysis anddesign The provided detailed geometric data of the runner(Figure 4) can be used as benchmark for the validation andevaluation of other numerical modelling tools and methods

The comparison of FLS and Fluent results was initiallyperformed for the reference operating conditions of thePelton turbine one injector is operating with nozzle stroke12mm which corresponds to the best efficient point (BEP) ofthe runner with one injector according to the experimentalresults The rotation speed was 1000 rpm the diameter of thejet was 12mm and its axial velocity was uniform and equal to4445ms For modeling purposes the velocity profile of thejet is taken uniformly

4 Accurate Eulerian Flow Evaluation

The basic steps for the numerical simulation with a mesh-type Eulerian method of Fluent software are the design ofthe appropriate 3D geometry the construction of the gridcovering the whole domain and the solution of the systemof flow equations The simulated geometry is designed usingthe SolidWorks 12 commercial software and it was essentialto avoid very small edges and narrow faces or volumesThe geometry was separated into two domains the rotatingdomain which contains the buckets and the stationarydomain from where the free jet is injected Due to theheavy computing requirements and thanks to the periodicsymmetry of the runner only two consecutive buckets weremodeled (Figure 5) Also only the half symmetric part ofthe domain was considered a common practice in modelingPelton turbines

41 Torque and Efficiency Calculation The hydrodynamictorque and the hydraulic efficiency of the runner are com-puted after completing the evaluation of a jet-bucket interac-tion flow starting from themoment of impingement until theevacuation of the bucket

During the unsteady flow simulation the total torque ona bucket is calculated at every time step (or its tangentialposition 120593) by adding the torque on the inner surface ofthe first bucket to the torque on the backside of the secondbucket as shown in Figure 5 The latter is developed due tothe adherence of a jet portion after it is cut by the bucket(Coanda effect) and also due to possible interference between


B

AA

B

55

25

97 67 Dj

1625

45

107

900∘R31

150

R288

(a)

100∘535

50

295

50

200

100

5080

R5

Section A-A

(b)

Section B-B

1742

100∘

100∘

53

50

422

64∘

empty120

empty160

R2367R2000 R253

R130

R100

R146

R90

(c)

Figure 4 Details of the runner design [22]

the back surface of the bucket and the outflow water from thefirst bucket

119879119903 (120593) = 119879in (120593) + 119879119887 (120593 +360

119873119887

) (6)

where 119879119903 is the torque in a single bucket 119879in and 119879119887 are thetorque at the inner and backside surfaces respectively and119873119887 is the number of buckets on the runner (22 for this case)

The torque curve of one bucket against the rotating angleis repeated periodically every 360∘119873119887 where 119873119887 is the totalnumber of buckets (22 in this case) The sum of the torquevalues of all curves for every angular position represents thetorque curve of the runner So the total mechanical energytransferred to the shaft during one rotation can be calculatedfrom the equation

119882 = int

360

0

119879 (120593) 119889120593 (7)


Second bucket

First bucket

Surfaces wherethe torqueis calculated

Figure 5 Two consecutive half buckets used to calculate the totalrunner torque

where 119879(120593) is the torque on a single bucket and 120593 the angularposition of the runner (120593 = 120596 sdot Δ119905 Δ119905 is the time step and 120596is the angular speed)

The runner power is then calculated as

119875 =119882 sdot 120596

360(8)

while the power of the water jet is

119875119908 =120587120588

81198632

jet1199063

jet (9)

where 120588 is the water density and 119906jet119863jet are the velocity anddiameter of the jet

Finally the hydraulic efficiency of the runner is defined as

120578 =119875

119875119908

(10)

42 Computational Grid and Accuracy of Results The com-putational grid was made using the Ansys Workbench com-mercial software package after an extensive investigation dueto its strong influence on the final speed of the simulationand the accuracy of the results The elements around therotating buckets and the cylindrical jet were tetrahedral andhexahedral respectively as shown in Figure 6 In addition aninflation layer was adjusted at the inner and backside faces ofthe buckets in order to reduce the 119910+ value below 50 whenthe 119896-120596 turbulence model is used The inflation consists of 5layers of hexahedral cells with very small thickness placed atthe region where the boundary layer is being developed

The quality of the mesh is acceptable as the minimumorthogonal quality is 015 while the average is 087 Sincethe position of the jet changes transiently it is not possibleto refine the shear layer therefore the entire mesh had tobe fine enough to have minimum impact on the simulatedefficiency In addition as the unstructured tetrahedral mesh

type can decrease the accuracy of the results a very densemesh was used at the area of high importanceTheminimumandmaximumcell volume are 11times 10minus12m3 and 15times 10minus8m3respectively

Moreover a study about the appropriate mesh size wascarried out in order to achieve independency between the sizeof the mesh and the accuracy of the results The same casewith the same settings was simulated for four different sizesof tetrahedral mesh The densest mesh consists of about 516million elements and it was found that themaximumpossibleaccuracy in terms of the calculated efficiency was achievedThe efficiency dependence on themesh density is representedin Figure 7

From the above results it can be deduced that theindependency was achieved formeshes withmore than 24Mcells For the present simulations a mesh with approximately28M cells was adopted to ensure the accuracy of the resultsThe simulations were carried out using a four-core IntelXeon 34GHz and 16GB RAM The CPU-time was almostproportional to the size of the mesh and for the 24M cells itis about 32 days

The torque variation curves on a bucket obtained usingdifferent density meshes are presented in Figure 8 where theangular position of the bucket is taken zero at its verticalor normal to the jet position (Figure 1) At the start of jet-runner interaction (minus40∘) a negative torque can be observedcaused by the interaction of the jet with the back surfaceof the bucket At this stage the jet is starting entering thebucket moving almost in parallel with the back surface thuscausing there a high pressure region as shown in Figure 9responsible for the negative torque Just afterwards the torqueis increasing as more water interacts with the inner surfaceAt an angle of about minus22 degrees the second bucket beginsto interact with the jet leaving less water to move towardsthe first bucket This reduced amount of water that movestowards the first bucket is not smooth due to divergence ofthe jet caused by Coanda effect (Figure 10) This divergedportion of water hits the first bucket at around minus10 degreescausing an irregular increment of the torque (Figure 8) Afterthis point the developed torque decreases smoothly as thewater is leaving the bucket until evacuation completes at +30degrees The total jet-runner interaction period lasts about70∘ rotational angle of the runner

43 TurbulenceModeling and Computational Details Duringthe simulations using Fluent many different settings weretested some of which had strong influence on the resultsThe 3D single precision solver was selected in order toreduce the computational cost as no important differencewas observed compared to double precision simulation Theappropriate model for the simulation is the Volume-of-Fluid(VOF) with pressure-velocity coupling solutionmethodThepressure-based coupled scheme increases the computationtime but provides a much more stable and reliable solutioncompared to the SIMPLE orPISO schemesThe transient flowis simulated both as inviscid and as turbulent using the 119896-120596 SST model in order to assess the impact of the frictionand turbulent losses on the efficiency The time step wasconstant and equal to 5 120583s which corresponds to 003 degrees


Stationary domain(structured mesh)

First bucket

Second bucket

Rotating domain(unstructured mesh)

Figure 6 Computational domain and structure of the 3D mesh

Nor

mal

ised

effici

ency

()

1 2 3 4 50Number of cells in millions

998

100

1002

1004

1006

1008

101

1012

Figure 7 Runner efficiency response to mesh refinement

of runner rotation angle while the maximum number ofiterations was set to 10 as suggested in the literature [17] Thesolution was not influenced by the time step as soon as thecontinuity residuals were under 10minus3 even if most of the timeduring simulation they were close to 10minus4 where the residualstarget was setThe continuity residuals target for convergence

minus45 minus35 minus25 25minus5 5minus15 15Bucket angular position (deg)

516M cells mesh243M cells mesh


minus5

0

5

10

15

20

25

30

35

40

45

50

Torq

ue (N

m)

Figure 8 Individual torque curves of mesh refinement simulations

was set to 10minus4 while the rest residuals were always below10minus5

The highest flow velocities in the computed field cor-respond to the free jet velocity which is about 45ms Inaddition the coupled implicit schemewas used and therefore


Pressurecontour 2

0

15000

30000

45000

60000

75000

90000

105000

120000

135000

150000

(Pa)

Figure 9 High pressure development at the backside while the jet is entering the bucket

Figure 10 Diverged flow moving towards the first bucket

theCourant number had aminor effect since the pointGauss-Seidel scheme used is unconditionally stable according to thelinear stability theory Consequently the default value of 200was used while different values have negligible influence inthe solution

Finally the possible effect of surface tension was consid-ered by using the ldquocontinuum surface forcerdquo model whichis available in Fluent platform However and in agreementwith the literature [2] the implementation of this model hasnegligible effect on the simulated flow in a Pelton runnerMoreover the Weber number of the relatively high velocityflow in the bucket is large (gt50) and only at the lastevacuation stage where the remaining water forms very thinand separated films on the bucket surface theWeber numbermay be considerably reduced But the contribution of thisstage to the energy exchange and the developed torque isminor Consequently the surface tension effects are ignoredin the simulations

A comparison between the torque curves calculated withinviscid and 119896-120596 SST turbulence model is shown in Figure 11As can be observed in this figure the developed torqueis slightly lower for the turbulent flow simulation whilethe corresponding efficiency is about 35 reduced (848compared to 883 of the inviscid flow)This is mainly due to

Inviscid 883k-120596 SST 848

20100 30 40minus20minus30minus40 minus10minus50Bucket angular position (deg)

minus100

102030405060708090

100

Torq

ue (N

m)

Figure 11 Comparison of inviscid and turbulent flow numericalresults

the friction losses at the flow boundary layer along the innerbucket surface and the impact losses of the entering jet flow


5 FLS Method Application andComparison with VOF

Tracking of about 5000 fluid particle trajectories using a timestep for integration of 2 times 10minus5 sec was found to produce sta-tistically accurate results for all cases examined in the presentstudy A complete flow evaluation requires about 10 CPU secin a modern PC which is almost 4 orders of magnitude lessthan the corresponding evaluation time using Fluent Thissignificant advantage of the FLSmethod allows its applicationfor multiparametric design optimization studies of impulseturbine runners in conjunction with modern optimizationsoftware

After completion of the fluid particles tracking the FLSpostprocessing algorithm computes the runner performancein terms of the developed torque on the blades and thehydraulic efficiency of the runner while it can also calculatethe local forces exerted on the blade during the energyconversion procedure The mechanical energy transferred toeach bucket is obtained from the equation of conservation ofangular momentum [8]

119882 = 120588119876119906(119877run119906jet minus1

119873sum

119894

119903119894119908119894) (11)

where119876119906 is the cumulative flow that enters each bucket 119877runis the runner pitch radius and 119908119894 is the tangential velocitycomponent of particle 119894 at the moment it exits the bucketat radial distance 119903119894 119873 is the total number of fluid particlesthat interact with a single bucket The hydraulic efficiencyof the runner can then be obtained as the ratio of thedeveloped mechanical power divided by the correspondingnet hydraulic power at the inlet as in (10)

The performance and accuracy of the FLS method areevaluated by comparing its results with the correspondingones of Fluent software for a number of different cases Thecomparison is based on the computed values of the runnerhydraulic efficiency the pattern of the torque curve and thedistribution of the surface flow on the inner bucket face Atfirst the comparison is carried out for the design point ofthe laboratorymodel turbine for which the coefficients of theFLS model are calibrated

Next two cases with different flow rates one lowerand one higher were simulated maintaining the remainingparameters constant An additional simulation took placefor different head and rotating speed of the turbine thatcorresponds to a smaller specific speed of the runner Alsoin another test case the geometry of the runner was changedby reducing the number of buckets from 22 to 18 Finally acomparison is made for another Pelton runner with differentbucket design

51 Comparison at the Reference Test Point Firstly the com-putational results of the reference test point (BEP) using FLSand Fluent are being compared with each other In Figure 12the resulted total torque in one bucket is compared It canbe observed that the agreement between the curves is goodas the general shape and the maximum torque are the sameThe small differences can be explained due to the inability

VOF method 883FLS method 885

minus100

102030405060708090

100

Torq

ue (N

m)

minus40 minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 5 10 15 20 25minus45Bucket angular position (deg)

Figure 12 Comparison of the computational results for the refer-ence point

Waterdeviation

Jetspread

Figure 13 Water deviation close to the solid surface calculated withFluent

of the FLS method to model secondary mechanisms like thepreviously mentioned negative torque at the beginning of theinteraction with the jet and the Coanda effect thus resultingin a smoother torque curve Another minor phenomenon isa small radial deviation of the jet caused when it is cut atthe bucket tip as shown in Figure 13The amount of deviatedwater hits the next bucket at a slightly different time (earlier)and this may explain the slightly higher torque values of theVOF in the increasing part of curve of Figure 12

In addition to the torque results the spreading rate of thesurface flow on the inner bucket surface is also consideredin order to adjust the spreading model coefficients of theFLS method In Figure 14 an isosurface is being representedwith green color as it is calculated by Fluent whereas theblack lines are the orbits of the particles obtained by FLS Itcan be observed that the surface flow pattern on the solidsurface calculated from Fluent and FLS is quite similar butthe spreading rate shows certain differences After an analysisof the particlesrsquo orbits and the velocity fields obtained by thetwo methods it was ascertained that two main reasons causethis difference


Figure 14 Water phase comparison of FLS (black lines) and Fluent (green areas) simulations

Firstly the jet flow as computed by VOF exhibits a signif-icant spread just before it reaches the bucket (Figure 13) dueto the high pressure field developed in the impact area belowConsequently the flow path lines follow a smooth curvedorbit before reaching the bucket surface with the curvatureradius being analogous to the radial position with respect tothe jet axis On the other hand all flow particles of the FLSmethod move straight until they hit the inner bucket surfaceand then change direction instantly to flow parallel to thebucket surfaceTherefore during the first interaction stages aflowparticlemodeled by FLS travels a longer distance to reachto a particular position on the surface than the correspondingone simulated by Fluent Moreover the small impact losses ofthe flow are accounted for at the impact point thus reducingthe particlesrsquo velocity when they hit the bucket whereas thisis taking place more progressively with Fluent simulation

For the above reasons the spreading rate of the surfaceflow is computed lower of the FLS at the first jet-runnerinteraction stages while the outflow as it is being calculatedfrom Fluent starts earlier (Figure 14) To compensate forthis difference the variation limits of the coefficients of thespreading model are properly regulated by using properconstraints during the optimization procedure of the set ofFLS coefficients for the reference case in order to matchalso as best as possible both the torque variation curveand the hydraulic efficiency As a result the subsequentinteraction stage becomes faster with the FLS simulation andthe evacuation of the bucket happens a little earlier (about 7∘in Figure 12)

In spite of the inability of FLS method to simulateaccurately all flow details and mechanisms the resultinghydraulic efficiency of the runner is almost equal to that ofFluent (883 compared to 885 Figure 12) This confirmsthe capability of the regulated FLS model to reproducesatisfactory main flow characteristics (eg torque curve) andalso to implicitly take into account the effects of all secondaryflow mechanisms on the efficiency

52 FLS Performance for Different Turbine Loading In orderto evaluate further the FLS method performance two test

VOF 135 load 867FLS 135 load 864


Bucket angular position (deg)

minus100

102030405060708090

100110120130140

Torq

ue (N

m)

minus45 minus35 minus25 25minus5 5minus15 15

Figure 15 Comparison of VOF and FLS method for off designturbine operation conditions

cases at different operating points of the turbine are sim-ulated by changing the flow rate (loading) of the runnerAll other design and operation characteristics remain thesame except of the jet diameter which depends on the nozzleopening (or spear valve stroke) of the injector The first caseis for smaller nozzle stroke which corresponds to 50 of thedesign point flow while the second is for higher nozzle strokeand 135of the design flowThe torque curves and the runnerefficiency values computed by VOF and FLS methods arepresented in Figure 15

Τhe shape of the torque curves shown in this figure issimilar to those at the reference point (Figure 12) but themaximum torque is as expected analogous to the loading ofthe runnerThe agreement between the FLS andFluent resultsis also good in both cases with the differences of the FLScurves being again at the same regions (nonnegative torque atthe beginning smoother curve slightly displaced to the rightand faster bucket evacuation)

Moreover in both cases the calculated efficiency by theFLS method is again very similar to that of the VOF method



minus200

20406080

100120140160

Torq

ue (N

m)


Figure 16 Comparison of VOF and FLS methods for higherhydraulic head

with the differences being 01 and 03 (Figure 15) namelywithin the order of numerical solution accuracy of Fluentsoftware It is noted that the increased efficiency of the runnerat smaller flow rates is due to the inviscid simulation of theflow in which the increased friction losses that have a thinnerfree surface flow along the bucket are not accounted On theother hand the obtained reduction of the efficiency higherthan the design flow rate is due to a different mechanismduring the evolution of the free surface flow on the bucketsurface a portion of water may leave the bucket through thecut edge area as shown in Figure 14 but not with optimum(minimum) outflow velocity thus causing a small drop inoverall efficiency of the runner This phenomenon becomesmore intense for higher flow rates due to the wider spread ofthe surface flow on the bucket

53 FLS Performance for Different Hydraulic Head In thiscase the inlet water pressure was increased by 56 (from100m of the reference point) which corresponds to exit jetvelocity of 557ms The higher jet velocity requires higherrotating speed of the runner which is calculated equal to1250 rpm in order tomaintain the same jetrunner speed ratioas in the reference point The torque curves calculated byFLS and VOF methods are shown in Figure 16 The shapeof the curves and the calculated efficiencies are similar withthat corresponding to the reference point while the onlydifferences are the higher values of torque caused due to theincreased kinetic energy of the jet The agreement betweenFLS and Fluent results is again very good with the samesmall systematic differences in the pattern of torque curveand runner efficiency that differs only 03 percentage units(Figure 16)

54 FLS Performance for Modified Runner Design Finallya modified runner design was obtained by reducing thenumber of buckets from 22 to 18 In this case the angulardistance between two consecutive buckets becomes 222larger while the amount of water jet that interacts with eachbucket is also equally increased This affects mainly the latest


minus100

102030405060708090

100

Torq

ue (N

m)

Bucket angular position (deg) minus45 minus35 minus25 25 35minus5 5minus15 15

Figure 17 Comparison ofVOF andFLSmethod for different runnerdesign of 18 buckets (from 22)

stage of jet-runner interaction which becomes longer As aresult the last part of outflow leaves the bucket lips at lessoptimum velocity and in addition more water exits from thecutout area For this reason the hydraulic efficiency of therunner becomes substantially lower almost 25 percentageunits as computed by Fluent (from 883 in Figure 12 to859)

This efficiency reduction is again reproduced well by theFLS method as shown in Figure 17 As can be observed inthis graph the shape of the torque curves is also similar to theprevious cases but the deviation of FLS and Fluent values inthe higher torque area (minus20 to +5 degrees Figure 17) becomesmore pronounced

These results support the previous discussion accordingto which the irregularity of the VOF curve in this area wasattributed to the Coanda effect and the impact of the backflow from the previous bucket The peak of the torque curveis now displaced to higher angular positions of the bucketat about minus3∘ compared to about minus10∘ in the reference case(Figure 12) On the other hand the FLS does not simulatethis mechanism hence its torque curve remains smooth andreaches maximum at about minus13∘ as in the reference case

55 Models Performance and Comparison for Different BucketDesign TheFluent-VOF and FLSmodels were finally appliedto simulate the flow in a modified runner design obtained in[5] The new runner is drastically different in terms of thebucket shape including changes of the main dimensions thescheme of the cut and the exit angle as shown in Figure 18In addition the number of buckets was 20 and their positionwas changed in terms of the radial distance from the centerand their inclination

The higher attainable efficiency of this new runner isconfirmed by both Fluent and FLS computations while itsvalue obtained by the FLS exhibits again close agreementwith the Fluent value being only 05 higher (95 comparedto 945) However the differences on the torque curvescompared in Figure 19 are more pronounced than in thereference runner especially in the beginning of the jet impact


Table 1 Comparison of FLS and Fluent computational results

Case description Discharge ( of BEP) Net head (m) Runner speed (rpm) Number of buckets Eff VOF () Eff FLS ()Reference point 100 100 1000 22 883 885Low flow rate 50 100 1000 22 912 913High flow rate 135 100 1000 22 867 864High hydraulic head 100 156 1250 22 882 885Modified runner 100 100 1000 18 858 859Different bucket design 100 100 1000 20 945 950

(a) (b)

Figure 18 Comparison of the reference bucket (a) and the modified bucket design (b)

minus100

102030405060708090

100110

Torq

ue (N

m)

VOF method 945FLS method 950Backside torque (VOF)

Bucket angular position (deg) minus45 minus35 minus25 25minus5 5minus15 15

Figure 19 Comparison of VOF and FLS methods for a newmodified runner design

(at about minus35 degrees) in the area of the maximum torque(minus15 to minus5 degrees) and to the end of the interaction (after+10 degrees)

These discrepancies were investigated by examining thecharacteristics of the flow in the new runner as computedby the VOF method and it was found that the attachmentof the flow at the backside of the new bucket is muchmore pronounced A much larger portion of the jet flowremains attached there causing substantial torque due tothe Coanda effect as shown in Figure 19 (backside torquecurve at minus35 degrees) Afterwards this diverged portionof water hits the leading bucket at an average angle of

about minus15 degrees in agreement to available experimentalresults [19] but at reduced impact velocity causing reducedpeak torque values and delayed evacuation of the bucketAlthough a different tuning of current FLS coefficients couldmitigate the above deviations the present results indicate thatfurther development of the FLS model may be required inorder to take into account the effects of this important flowmechanismThe structure and philosophy of the FLSmethodfacilitate the introduction of such new relations

56 Summary of Runner Efficiency Results The results forthe hydraulic efficiency of the runner obtained from thevarious test cases carried out in this study are concentratedin Table 1 As it can be seen the agreement is in all cases verysatisfactory with maximum deviation of only 05 percentageunits Considering themuch faster flow evaluation by the FLSalgorithm these results support its important advantage forbeing used formultiple evaluations as required in parametricperformance studies and numerical design optimization ofPelton turbines

6 Conclusions

This work aims to validate the capability of a particulatenumerical method the Fast Lagrangian Simulation (FLS)algorithm to reproduce in a reliable and accurate way thevery complex flow created during the jet-runner interactionin Pelton turbines

Themethodwas compared with amore accurate Eulerianmesh-type VOFmethod which has been validated in variousprevious works and proved to provide satisfactory resultsThe FLS model is tuned based on the numerical results at areference operation point of a laboratory model turbine and


then it is applied together with the Eulerian method to anumber of different test cases by modifying the flow rate thehydraulic head and the number of buckets and the bucketdesign of the runner

In all these cases the hydraulic efficiency values obtainedby the FLS method showed very good agreement with theEulerian method results while the predicted evolution of thefree surface flow in the bucket in terms of time variation ofthe developing torque was also satisfactory

The present results are very encouraging towards theimplementation of the FLS tool to perform multiparametricand multiobjective design optimization studies in modernPelton runners at very low computational cost Furthervalidation tests are needed using different runner and bucketshapes for a better and more general adjustment of the FLScoefficients while further development in order to take intoaccount the effect of other important mechanisms like thebackside flow could enhance the accuracy of its results

Finally the presented results along with the provideddetailed geometric dimensions of the Pelton model runnercan constitute a benchmark set of data for the validation andevaluation of other numerical modelling tools and methods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Zidonis and G A Aggidis ldquoState of the art in numeri-cal modelling of Pelton turbinesrdquo Renewable and SustainableEnergy Reviews vol 45 pp 135ndash144 2015

[2] L F Barstad CFD analysis of a Pelton turbine [MS thesis]Norwegian University of Science and Technology TrondheimNorway 2012

[3] A Santolin G Cavazzini G Ardizzon and G Pavesi ldquoNumer-ical investigation of the interaction between jet and bucket ina Pelton turbinerdquo Proceedings of the Institution of MechanicalEngineers Part A Journal of Power and Energy vol 223 no 6pp 721ndash728 2009

[4] B Zoppe C Pellone T Maitre and P Leroy ldquoFlow analysisinside a pelton turbine bucketrdquo Journal of Turbomachinery vol128 no 3 pp 500ndash511 2006

[5] A Zidonis A Panagiotopoulos G A Aggidis J S Anag-nostopoulos and D E Papantonis ldquoParametric optimisationof two Pelton turbine runner designs using CFDrdquo Journal ofHydrodynamics vol 27 no 3 pp 840ndash847 2015

[6] J R Rygg CFD analysis of a Pelton turbine in openFOAM[MS thesis] Norwegian University of Science and TechnologyTrondheim Norway 2013

[7] J C Marongiu E Parkinson S Lais F Leboeuf and J LeducldquoApplication of SPH-ALEmethod to pelton hydraulic turbinesrdquoin Proceedings of the 5th International SPHERIC Workshop pp253ndash258 Manchester UK June 2010

[8] J S Anagnostopoulos and D E Papantonis ldquoA fast Lagrangiansimulation method for flow analysis and runner design inPelton turbinesrdquo Journal of Hydrodynamics B vol 24 no 6 pp930ndash941 2012

[9] J S Anagnostopoulos P K Koukouvinis F G Stamatelos andD E Papantonis ldquoOptimal design and experimental validationof a Turgomodel hydro turbinerdquo inProceedings of the ASME 11thBiennial Conference on Engineering Systems Design and Analysis(ESDA rsquo12) vol 2 pp 157ndash166 Nantes France July 2012

[10] T Staubli A Abgottspon PWeibel et al ldquoJet quality and Peltonefficiencyrdquo in Proceedings of the Hydro International Conferenceon ProgressmdashPotentialmdashPlans Lyon France 2009

[11] R Fiereder S Riemann and R Schilling ldquoNumerical andexperimental investigation of the 3D free surface flow ina model Pelton turbinerdquo IOP Conference Series Earth andEnvironmental Science vol 12 no 1 Article ID 012072 2010

[12] D Benzon A Zidonis A Panagiotopoulos G A Aggidis JS Anagnostopoulos and D E Papantonis ldquoNumerical inves-tigation of the spear valve configuration on the performanceof Pelton and Turgo turbine injectors and runnersrdquo Journal ofFluids Engineering vol 137 no 11 Article ID 111201 2015

[13] A Rossetti G Pavesi G Ardizzon andA Santolin ldquoNumericalanalyses of cavitating flow in a pelton turbinerdquo Journal of FluidsEngineeringmdashTransactions of the ASME vol 136 no 8 ArticleID 081304 2014

[14] J-C Marongiu F Leboeuf J Caro and E Parkinson ldquoFreesurface flows simulations in Pelton turbines using an hybridSPH-ALE methodrdquo Journal of Hydraulic Research vol 48 no1 pp 40ndash49 2010

[15] P Κoukouvinis Development of a meshfree particle methodfor the simulation of steady and unsteady free surface flowsapplication and validation of the method on impulse hydraulicturbines [PhD thesis] National Technical University of AthensAthens Greece 2012

[16] ANSYS ANSYS Fluent Theory Guide Release 15 ANSYSCanonsburg Pa USA 2013

[17] ANSYSANSYS Fluent User Guide Release 15 ANSYS Canons-burg Pa USA 2013

[18] A Perrig Hydrodynamics of the free surface flow in Peltonturbine buckets [PhD thesis] Swiss Federal Institute of Tech-nology Lausanne Switzerland 2007

[19] A Perrig F Avellan J-L Kueny M Farhat and E ParkinsonldquoFlow in a Pelton turbine bucket numerical and experimentalinvestigationsrdquo Journal of Fluids Engineering vol 128 no 2 pp350ndash358 2006

[20] E Parkinson C Neury H Garcin G Vullioud and T WeissldquoUnsteady analysis of a Pelton runner with flow andmechanicalsimulationsrdquo International Journal on Hydropower and Damsvol 13 no 2 pp 101ndash105 2006

[21] L E KlemetsenAn experimental and numerical study of the freesurface Pelton bucket flow [MS thesis] NorwegianUniversity ofScience and Technology Trondheim Norway 2010

[22] F G Stamatelos J S Anagnostopoulos and D E Papanto-nis ldquoPerformance measurements on a Pelton turbine modelrdquoProceedings of the Institution of Mechanical Engineers Part AJournal of Power and Energy vol 225 no 3 pp 351ndash362 2011

[23] International Electrotechnical Commission ldquoHydraulic tur-bines storage pumps and pump-turbinesmdashmodel acceptancetestsrdquo Standard IEC 60193 International Electrotechnical Com-mission 1999

[24] M Nechleba Hydraulic Turbines Their Design and EquipmentArtia Prague Czech Republic 1957

[25] JThakeTheMicro-Hydro Pelton TurbineManual DesignMan-ufacture and Installation for Small-Scale Hydro-Power ITDGpublishing London UK 2000

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



meshless simulation methods based on the Lagrangian ap-proach have also been developedThemost popular one is theSmoothParticleHydrodynamics (SPH)methodology [14 15]the computational cost of which is however comparableto the Eulerian methods while further development andvalidation are needed to improve its accuracy

In order to reduce the computational cost a particu-late method based on the Lagrangian approach has beendeveloped by the Laboratory of Hydraulic TurbomachinesNTUA [8 9] This Fast Lagrangian Simulation (FLS) methodintroduces appropriate adjustable terms in the flow particleequations to approximate the various viscous and pressureeffects of their trajectories

The aim of the present paper is to evaluate the perfor-mance and accuracy of the FLS method by comparing theresults against the corresponding ones of the Ansys-Fluentcommercial software [16 17] that uses the Volume-of-Fluid(VOF) technique to simulate the jet and free surface flowin the runner The accuracy of the latter has been validatedagainst experimental results in Pelton runners [18ndash21]

In the present study the appropriate computationaldomain mesh density and settings of the VOF techniqueare at first investigated to achieve the best compromise ofaccuracy and computational cost Then an initial test case isdefined including the exact geometrical characteristics of aPelton model runner installed in the laboratory The FLS andFluent software are applied to various operating conditioncases of this turbine and for runners of different design inorder to compare their numerical performance based on thetime history of torque development and on the hydraulicefficiency of the runner

2 Numerical Modeling of the Flow

Two computer codes were used for the simulation of thecomplex flow in a rotating Pelton runner the commercialsoftware Fluent and the in-house software FLS The Fluentcode [16] is capable of solving multiphase flow problems withfree surfaces which are highly relevant to impulse turbines Itis based on the spatial discretization of theReynoldsAveragedNavier-Stokes equations on a computational mesh usingcell-centered numerics (finite volumes) and offers flexibilityin choosing between segregated based and coupled basedsolver

The two-fluid-flow (air and water) problem with freesurface like the Pelton runner case is being solved usingthe established Volume-of-Fluid (VOF) method VOF is anEulerian-Eulerian method based on tracking and locatingthe fluid-fluid interface An additional factor the volumefraction is introduced which represents the percentage ofeach fluid volume in every cellThemethod canmodel two ormore immiscible fluids by solving a single set of momentumequations and tracking the volume fraction of each of thefluids throughout the domain

For the two-phase (water-air) flow in Pelton runnersthe air is defined as the primary phase and the water assecondary so as the tracking of the interface between themis accomplished by the solution of a continuity equation

for the volume fraction of the secondary phase that has thefollowing form

1

120588119902

[120597

120597119905(119886119902120588119902) + nabla sdot (119886119902120588119902 120592119902)] =

119899

sum

119901=1

( 119898119901119902 minus 119898119902119901) (1)

where 119898119901119902 is the mass transfer from phase 119901 to phase 119902 and119898119902119901 the opposite and 119886119902 is the volume fraction and 120588 the

density The present application of Fluent-VOF software forflow simulation in Pelton runners is analyzed in Section 4

21 Fast Lagrangian SimulationMethod TheFast LagrangianSimulation (FLS) [8 9] is a single-phase flow simulationmethod and it is based on the tracking of a representativenumber of fluid particles in order to model and calculate theflow pattern and the energy exchange in the rotating runnerof impulse hydro turbines The exact water-air interfacepattern is not required for such computations though theflow width on the bucket surface could be estimated from theproperties of the tracked particles

The water jet is being separated into discrete particlesas shown in Figure 1 and their equations of motion arenumerically integrated until they exit from the inner bucketsurface The jet is considered ideal that is with uniforminitial velocity and the flow frictionless Also due to theperiodic symmetry conditions only two consecutive bucketswere modeled

The fluid particle equations are solved in a rotatingorthogonal systemof reference and are expressed inCartesiancoordinates as follows (bucket rim is on the 119909119910 level)

1198892119909

1198891199052= 119891119909 (119909 119910)

1198892119910

1198891199052= 119891119910 (119909 119910) + 120596

2119910 + 2120596

119889119911

119889119905

1198892119911

1198891199052= 119891119911 (119909 119910) + 120596

2119911 minus 2120596

119889119910

119889119905

(2)

where120596 is the angular rotation speed of the runner and119891119909119891119910and 119891119911 are the additional terms functions of the local surfacegeometrical characteristics

The particle motion equations do not contain particleinteraction ormechanical losses terms and hence they cannotreproduce the real flow picture in the bucket For this reasonthe FLS model introduces a number of additional terms inorder to account for the various hydraulic losses (impactfriction and change direction) as well as for the pressureeffects that control the spreading of the surface flow in thebucket which are activated after the impact of a particle onthe bucket surfaceMore specifically the friction losses on thebucket surface are modeled as reduction of particle kineticenergy by a factor analogous to the square of particle velocityand to sliding distance hence the particle velocity magnitudeafter a time step Δ119905 becomes

1198811015840

119901asymp 119881119901 sdot (1 minus 119862119891 sdot 119881119901 sdot Δ119905) (3)


Rotation axis

Δ120579

1205790

(a) (b)



1198811015840


2120593119894) (4)



1198811015840


2Δ120593) (5)





Splitter line

Jetflow

rarrVS

rarrV2

rarrV1

rarrV2998400

120593r







3 Test Case









B

AA

B

55

25

97 67 Dj

1625

45

107

900∘R31

150

R288

(a)

100∘535

50

295

50

200

100

5080

R5

Section A-A

(b)

Section B-B

1742

100∘

100∘

53

50

422

64∘

empty120

empty160

R2367R2000 R253

R130

R100

R146

R90

(c)



119879119903 (120593) = 119879in (120593) + 119879119887 (120593 +360

119873119887

) (6)



119882 = int

360

0

119879 (120593) 119889120593 (7)


Second bucket

First bucket





119875 =119882 sdot 120596

360(8)


119875119908 =120587120588

81198632

jet1199063

jet (9)



120578 =119875

119875119908

(10)










First bucket

Second bucket



Nor

mal

ised

effici

ency

()


998

100

1002

1004

1006

1008

101

1012






minus5

0

5

10

15

20

25

30

35

40

45

50

Torq

ue (N

m)





Pressurecontour 2

0

15000

30000

45000

60000

75000

90000

105000

120000

135000

150000

(Pa)








minus100

102030405060708090

100

Torq

ue (N

m)







119882 = 120588119876119906(119877run119906jet minus1

119873sum

119894

119903119894119908119894) (11)






minus100

102030405060708090

100

Torq

ue (N

m)



Waterdeviation

Jetspread













minus100

102030405060708090

100110120130140

Torq

ue (N

m)








minus200

20406080

100120140160

Torq

ue (N

m)







minus100

102030405060708090

100

Torq

ue (N

m)











(a) (b)


minus100

102030405060708090

100110

Torq

ue (N

m)








6 Conclusions










References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Rotation axis

Δ120579

1205790

(a) (b)



1198811015840


2120593119894) (4)



1198811015840


2Δ120593) (5)





Splitter line

Jetflow

rarrVS

rarrV2

rarrV1

rarrV2998400

120593r







3 Test Case









B

AA

B

55

25

97 67 Dj

1625

45

107

900∘R31

150

R288

(a)

100∘535

50

295

50

200

100

5080

R5

Section A-A

(b)

Section B-B

1742

100∘

100∘

53

50

422

64∘

empty120

empty160

R2367R2000 R253

R130

R100

R146

R90

(c)



119879119903 (120593) = 119879in (120593) + 119879119887 (120593 +360

119873119887

) (6)



119882 = int

360

0

119879 (120593) 119889120593 (7)


Second bucket

First bucket





119875 =119882 sdot 120596

360(8)


119875119908 =120587120588

81198632

jet1199063

jet (9)



120578 =119875

119875119908

(10)










First bucket

Second bucket



Nor

mal

ised

effici

ency

()


998

100

1002

1004

1006

1008

101

1012






minus5

0

5

10

15

20

25

30

35

40

45

50

Torq

ue (N

m)





Pressurecontour 2

0

15000

30000

45000

60000

75000

90000

105000

120000

135000

150000

(Pa)








minus100

102030405060708090

100

Torq

ue (N

m)







119882 = 120588119876119906(119877run119906jet minus1

119873sum

119894

119903119894119908119894) (11)






minus100

102030405060708090

100

Torq

ue (N

m)



Waterdeviation

Jetspread













minus100

102030405060708090

100110120130140

Torq

ue (N

m)








minus200

20406080

100120140160

Torq

ue (N

m)







minus100

102030405060708090

100

Torq

ue (N

m)











(a) (b)


minus100

102030405060708090

100110

Torq

ue (N

m)








6 Conclusions










References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











3 Test Case









B

AA

B

55

25

97 67 Dj

1625

45

107

900∘R31

150

R288

(a)

100∘535

50

295

50

200

100

5080

R5

Section A-A

(b)

Section B-B

1742

100∘

100∘

53

50

422

64∘

empty120

empty160

R2367R2000 R253

R130

R100

R146

R90

(c)



119879119903 (120593) = 119879in (120593) + 119879119887 (120593 +360

119873119887

) (6)



119882 = int

360

0

119879 (120593) 119889120593 (7)


Second bucket

First bucket





119875 =119882 sdot 120596

360(8)


119875119908 =120587120588

81198632

jet1199063

jet (9)



120578 =119875

119875119908

(10)










First bucket

Second bucket



Nor

mal

ised

effici

ency

()


998

100

1002

1004

1006

1008

101

1012






minus5

0

5

10

15

20

25

30

35

40

45

50

Torq

ue (N

m)





Pressurecontour 2

0

15000

30000

45000

60000

75000

90000

105000

120000

135000

150000

(Pa)








minus100

102030405060708090

100

Torq

ue (N

m)







119882 = 120588119876119906(119877run119906jet minus1

119873sum

119894

119903119894119908119894) (11)






minus100

102030405060708090

100

Torq

ue (N

m)



Waterdeviation

Jetspread













minus100

102030405060708090

100110120130140

Torq

ue (N

m)








minus200

20406080

100120140160

Torq

ue (N

m)







minus100

102030405060708090

100

Torq

ue (N

m)











(a) (b)


minus100

102030405060708090

100110

Torq

ue (N

m)








6 Conclusions










References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










B

AA

B

55

25

97 67 Dj

1625

45

107

900∘R31

150

R288

(a)

100∘535

50

295

50

200

100

5080

R5

Section A-A

(b)

Section B-B

1742

100∘

100∘

53

50

422

64∘

empty120

empty160

R2367R2000 R253

R130

R100

R146

R90

(c)



119879119903 (120593) = 119879in (120593) + 119879119887 (120593 +360

119873119887

) (6)



119882 = int

360

0

119879 (120593) 119889120593 (7)


Second bucket

First bucket





119875 =119882 sdot 120596

360(8)


119875119908 =120587120588

81198632

jet1199063

jet (9)



120578 =119875

119875119908

(10)










First bucket

Second bucket



Nor

mal

ised

effici

ency

()


998

100

1002

1004

1006

1008

101

1012






minus5

0

5

10

15

20

25

30

35

40

45

50

Torq

ue (N

m)





Pressurecontour 2

0

15000

30000

45000

60000

75000

90000

105000

120000

135000

150000

(Pa)








minus100

102030405060708090

100

Torq

ue (N

m)







119882 = 120588119876119906(119877run119906jet minus1

119873sum

119894

119903119894119908119894) (11)






minus100

102030405060708090

100

Torq

ue (N

m)



Waterdeviation

Jetspread













minus100

102030405060708090

100110120130140

Torq

ue (N

m)








minus200

20406080

100120140160

Torq

ue (N

m)







minus100

102030405060708090

100

Torq

ue (N

m)











(a) (b)


minus100

102030405060708090

100110

Torq

ue (N

m)








6 Conclusions










References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Second bucket

First bucket





119875 =119882 sdot 120596

360(8)


119875119908 =120587120588

81198632

jet1199063

jet (9)



120578 =119875

119875119908

(10)










First bucket

Second bucket



Nor

mal

ised

effici

ency

()


998

100

1002

1004

1006

1008

101

1012






minus5

0

5

10

15

20

25

30

35

40

45

50

Torq

ue (N

m)





Pressurecontour 2

0

15000

30000

45000

60000

75000

90000

105000

120000

135000

150000

(Pa)








minus100

102030405060708090

100

Torq

ue (N

m)







119882 = 120588119876119906(119877run119906jet minus1

119873sum

119894

119903119894119908119894) (11)






minus100

102030405060708090

100

Torq

ue (N

m)



Waterdeviation

Jetspread













minus100

102030405060708090

100110120130140

Torq

ue (N

m)








minus200

20406080

100120140160

Torq

ue (N

m)







minus100

102030405060708090

100

Torq

ue (N

m)











(a) (b)


minus100

102030405060708090

100110

Torq

ue (N

m)








6 Conclusions










References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











First bucket

Second bucket



Nor

mal

ised

effici

ency

()


998

100

1002

1004

1006

1008

101

1012






minus5

0

5

10

15

20

25

30

35

40

45

50

Torq

ue (N

m)





Pressurecontour 2

0

15000

30000

45000

60000

75000

90000

105000

120000

135000

150000

(Pa)








minus100

102030405060708090

100

Torq

ue (N

m)







119882 = 120588119876119906(119877run119906jet minus1

119873sum

119894

119903119894119908119894) (11)






minus100

102030405060708090

100

Torq

ue (N

m)



Waterdeviation

Jetspread













minus100

102030405060708090

100110120130140

Torq

ue (N

m)








minus200

20406080

100120140160

Torq

ue (N

m)







minus100

102030405060708090

100

Torq

ue (N

m)











(a) (b)


minus100

102030405060708090

100110

Torq

ue (N

m)








6 Conclusions










References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Pressurecontour 2

0

15000

30000

45000

60000

75000

90000

105000

120000

135000

150000

(Pa)








minus100

102030405060708090

100

Torq

ue (N

m)







119882 = 120588119876119906(119877run119906jet minus1

119873sum

119894

119903119894119908119894) (11)






minus100

102030405060708090

100

Torq

ue (N

m)



Waterdeviation

Jetspread













minus100

102030405060708090

100110120130140

Torq

ue (N

m)








minus200

20406080

100120140160

Torq

ue (N

m)







minus100

102030405060708090

100

Torq

ue (N

m)











(a) (b)


minus100

102030405060708090

100110

Torq

ue (N

m)








6 Conclusions










References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119882 = 120588119876119906(119877run119906jet minus1

119873sum

119894

119903119894119908119894) (11)






minus100

102030405060708090

100

Torq

ue (N

m)



Waterdeviation

Jetspread













minus100

102030405060708090

100110120130140

Torq

ue (N

m)








minus200

20406080

100120140160

Torq

ue (N

m)







minus100

102030405060708090

100

Torq

ue (N

m)











(a) (b)


minus100

102030405060708090

100110

Torq

ue (N

m)








6 Conclusions










References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















minus100

102030405060708090

100110120130140

Torq

ue (N

m)








minus200

20406080

100120140160

Torq

ue (N

m)







minus100

102030405060708090

100

Torq

ue (N

m)











(a) (b)


minus100

102030405060708090

100110

Torq

ue (N

m)








6 Conclusions










References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus200

20406080

100120140160

Torq

ue (N

m)







minus100

102030405060708090

100

Torq

ue (N

m)











(a) (b)


minus100

102030405060708090

100110

Torq

ue (N

m)








6 Conclusions










References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












(a) (b)


minus100

102030405060708090

100110

Torq

ue (N

m)








6 Conclusions










References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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