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The Evaluation of a Waterjet System using Computational Fluid Dynamics Validated by Wind Tunnel Tests

by CoavidMurri n,B. Eng.

A thesissubmitted to theSchoolofGraduate Studies inpartial fulfilmentof the requirements for the degree of

Master of Eng ineering

Faculty of Engineering and AppliedSc ience Memorial University of Newfoundland

Februa ry2002 St.John' s, Newfoundland,Canada

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Acknowledgeme nts

Acknowledgements

I wouldlike to thank my supervisorsDr. Neil Bose andDr. ShinChin for their time and valuable suggestions throughoutthe tenure ofthis degree program.Iwouldalsolike to thank MemorialUniversity of Newfoundland(MUN)for providing the experimenta land numerical facilities required for thisprojectand I am gratefulfor the financial support provided by theInstituteforMarine Dynamics(lMD)and MUN , Furthermore, the technicalassistance of BernardHealy,TomPike, andJimGosse for the preparationof the model tests is gratefullyacknowledged.

FinallyIamindebted to mywife andfamily for their continuedsupport, patience and understanding.

Abstract

Abstract

Traditionalmethodsof marinepropulsion have been limited 10 screw-type propeller arrangements butin recent yean effortstoimprovevessel speedhaveled10the developmentof practicalwaterjet systems. Aswaterjettechnology continues to grow, methods of testingand evaluatingwaterjet propulsion systems have emerged.

Conventional methods of testing propellerdriven craft have been applied to warerjets and these haveincluded self-propulsion tests usingtow carriages or waterjetsystemrests in water tunnels.Implementation of these testshas beenproblematic due 10thesmallsize of models.the speed required during modeltesting ofhigh speed craft at equivalentspeed, andIhe difficultyinobtaining detailed flowinformation through thejet.This study investigatesthe applicabilityof largerscale testing ofa waterjetsystemusinga wind tunnel.

In additio nto physical lesting,computersimulatio nshave emergedas a validmethod for evaluating the behaviourof fluids and perfonnance of equipment.Physical experimentation forms anintegral pan of any CFOsimulation as the accuracyof simulationresults is obtainedIhroughvalidationagainst experimentaldata.Once validated, however,thenumerical codeis capableof providingengineeringquantities such asforce,velocity and pressure,at a level of detail notpossiblethrough physical experimentation.

Thefocus ofthis researchwas to studytheapplicabilityof CFDanalysis 10 warerjet testing and to evaluatethepropulsion perfonnance of a waterjetunitusing

ern

Abstract

simulationvalidatedbyexperimenta lresults. A full-scalewarerjet wastested atthe MemorialUniversity of Ne....foundland windtunnel, andnumerical analysis was achieved withCFX5.6® CFDsoftware.Once validated,the CFDsimulatio nwas used10pred ict the propulsion performance ofthe waterjetunit using the momentum fluxmethod.This thesis presents a compa risonof theCFDpredictionsand the windtunnel tests.

Table of Contents

ABSTRACT

ACK~O\\'IEDGHfENTS

TA8IEOF co~n:m

LIS!OF FIG UR ES

LISTOFTABLES

INTR OIHJCTIO N

LlTF.RAT1JRE REVIEW

Sr\ II L1Tt: D EA!Il.UYStS J.l SI:\lILAJUTY 3.1.1 GEOMETlUCSL\IILARITY 3.1.2 KL''E.,'1 ATICSIMILARITY 3.1.3 DYNAMICSIMILARITY

3.2 ISDICIAL ApPROACH&:\IATRlxMETHO DS 3.2.1 T1IEINDIO ALAppROACH 3.2.2 MATRIXMETHODS J.J WATERJETDI:-'lENSIOS AL ANALYSIS 3.4 DISCl:SSIONOF NON-Ot\IENSIO:"'ALTE R.\I S 3.4.1 ADVA."ilCECOEFFIOENT

3.4.2 REYNOLDSNUMBER 3.4.3 FROL'DE NUMBER 3.4.4 CAvrTAn o :",NL"MBER

TableofContents

II

III

IV

VII

IX

10

,.

10

"

II 12 12 14 16 21 21 21 23 24

Tableof Contents

3.4.5 WEBER:-O'UMBER 25

3.4.6 MACHNL'MBER 25

3.4.7 Po\"'ER.SH....fTTORQl.c.THRUST.

""" "D

VOLWolE Ftow R....raCOEff1CIE..'TS 26

30S SCALr.'iGLAws 27

3.6 Snl~(ARY 29

::!THE~1O\I Exr U~1t ]I'X~IETHO D 31

4.1 I:-OT RODt' CTIOSTO TIlE~IO~I£." l.;!o1Ft.uxMETlI OD 31

4.2 STATIO:"'} 33

4.3 I~'TER.\I EDI ""TEST....TIOSS 38

·

U \'EI'"A CO' iRACTA 39

4.5 PROPL'lSIOSPEIU'ORMA:"'CECALCl:LATlO:-O S 40

4.6 PREDICTEDFULLSCALEPOWER 43

4.7 SWoIMARY 44

~ INSTRUMENTATION 46

5.1 HOT-W IREA~EMOMETRY 46

5.1.1 AOV....:-O'TAGES""''iO DISADVAl'.'TAGESOFHar- WIREANEMOMETRY 49

5.1 .2 CAUBRATlON ANOExPERlME.'lTAL$ET-UP 51

5.2 Wt'"DTt;:-';.(EL 53

5.3 I""Dt:CTIO~MOTOR ASDISVERTER 53

5.4 DYSAMmtETER 54

50SDATAACQUIsmos 55

5.6 L"SERI~"TER FACE 57

f! MODELTESTISG 59

6.1 i:'o.IRODl"CTI O:'" 59

6.2 THE~ loDEL 62

6.2.1 BACKGROUND 62

6.2.2 i'dPROVISOTHEl\.lPEUER 63

6.2.3 1!>IPROVL>';QTHESYSTEMSTIfP.'ESS 67

6.2.4 IMPROVlNGTIlE VElOCITYMEASUREME.'lTSCHE.\1E 71

6.3 ~IODELTEsTs 77

6.3.1 DATA ACQl1lSITION 77

6.3.2 TEsTMATRIX SI

6.4 UNCERTAINTY OF1I0T_WIREMEASUREM ESTS 91

6.4.1 ANEMOMETER 92

6.4.2 CALIBRATION A..'<DCONVERSION 93

6.4.3 EXPERIMENTAL CO:'lDITIOSS 94

6.5 SUM!>IARY 97

2I~IRQDrCTIO~TOCO " PliTA T I O ~AIFLl'1DDYNA ~IICS 99

TableofContents

7.1 I~TROoUcnOS 99

7.2 THEE LE.\IE."'iSOr CFDcooE 99

7.2.1 THE PRE-PROCESSOR 100

7.2.2 SOLVER 100

7.2.3 PosT PROCESSOR 100

7.3 GRIDDEfI~ITIO~A:-o :\IISHI:-C 101

7.4 SOLLTJ O:-M ETHODS lOot

7.5 PRoBLDlso Ln"C CSlsCCFD 106

7.5.1 SI.'MERICALD ISCRETlZATIO!'l QFTHEFl:'.on VOU ,:MEM ETllo D 108

7.5.2 SEGREGATEDA-....1>COUPlEDSOLVERS 109

7.5.3 PRESStJRE·VELOCITYCotJPl.lSG Il2

7.5.4 MULTIGRJDM ETHODS 115

7.6 nooxr uarcoxnrnoxs 115

7.7 TI:RBL"LENCE 118

7.8 THEAp PLICATlo No r C FD 121

.!!: NUI\fERICALSIM111ATION 125

8.1 AI\hTHEMATICALDESCRIPTIO:'liorTHEWATERjET·THEBOUl'iDARY,V ALUE

PROBLEM 125

8.1.1 GoVERJI,1NQEQUATIONS 126

8.1.2 BOUNDARYCONOmONS 130

8.2 FLowDo~tAL"; 133

8.3 :\I IS HISG 138

8•.& BOL"fl;DARY COSDITIOSSA SD SO Ln:R PARAMETERS 14-.1

8.5 SoL\"ERPARA~IETERSASDIS lT lALlSATIOS 1.&5

8.6 GRIDl:O'll EPE....1>E.'iCE 1.&7

8.7 VALID....rros 149

8.8 Smu u no:-lREsULTS 156

8.9 APPLICAnosOFTHE:'olo~IE.~tFLux:\IETHOD 162

8.10 Sl~IUnY 176

.2CO:"CLl:S IOSSA:"DRECO) I:'oIE:" PA Tl OXS 178

9.1 PHYSICALEXPERIMENTS 178

9.2 XliMERICAL SIML1..ATION 180

9.3 REco~nIE""llATIO:o.'S.·ORFurt;REWORK 181

1!! REFERENCES 184

List ofFigures

List of Figures

Figure1-1.Arh~naHigh-Speed Catamaran 2

Figure4-1.Momentum FluxMethod SIm ianDefin itiom 12

Figure4-2.Watel'jet Streamlines 14

Figure4-3.Capture Areafor TypicalWaterjet Jj

Figure4·4.CaptureArea Dimensions ss

Figure5-1.Whearston l!'Bridge 48

Figure5·2.DataAcquisition andtnsmon emanon 56

Figure5-3.Data Acq uisitionGUI 57

Figure6-1.WalerjetDefin ino ns 60

Figure6·2.Typical Set-up for Testinga\Vafa j erSystem usingaWindTunnel _ _ 61

Figure 6-3. Computer GeneratedImpellerModel 64

Figure6-4.LaminatedObjectManuf actur e System 66

Figure6·5.Closing the Walla/theWindTunn el 69

Figure6·6.OriginalBaseplate andBrack et 70

Figure6·7.StiffmedWalerje! System 7/

Figure6-8.MountingBracker 7J

Figure6·9. VelocityTemplate 7J

Figure6-10.Templa te Suppon 74

Figure6-JJ.TopView a/TemplateSupport 75

Figure 6-12. WindTunnelBracket Assemb ly 76

Figure6-13.Testin gAppa ratus 76

Figure6-14.StationLocoiions 81

Figure6-15.Free StreamVelocityProfi le 8J

Figure6-16.Stationla VelocityPrOfi le S4

Figure6-17.Contourplotorientation 8.S

Figure6-18.Station 2 VelocityCom our 86

Figure6-19.Station3VelocityContour 87

Figure6-20. Station5Velocity Contour 88

Figure6·21.Station6VelocityContour 89

Figure6-22.Station7VelocityContour 90

Figure8-1.Impeller Coordinat eSystem 110

Figure8-2.Watel'jetBoundaryCond itions HI

Figure 8-3.Componentdescription 1J5

Figure8-4.Jet.impeller,andexit /J7

Figure8-5.Infl ated boundary H9

Figure8-6. Meshofwaterj et system 140

Figure8-7.Compon entVolumeContribution 141

Figure8-8.Compon ent Nodal Contribution /42

Figure8-9.Isometricviewof waterjermesh UJ

Figure8·10.Topview o/watel'jet mesh UJ

List ofFigures

Figure8-11.

Boundarycondjt j'O~ ",~~~~~~~~~~~~~i44

Figure8-12.Gridrefinement /48

Figure 8-13. Orientation of contourplots ISO

Figure8-14.Station2contour plot(CFD) lSI

Figure8-l 5.Station2contour plor(model tests) lSI

Figure8·/6.Station6 contourplot(CFD) 151

Figure8-17.Station6contourplot(modeltests) lS3

Figure 8-18. Station7contour plot (CFD) 153

Figure8-19. Station7contourplot(model tests) 154

Figure 8-20.Comparisonof velocityprofiles Ij j

Figure S-2I.Centeriine veiocuy 1$7

Figure8-22.Centerline velocitycontours lSS

Figure8-23.Centerlinepressurecontours Jj IJ

Figure8-24.Station2-YVelocity Contoun(CFD) 1$9 Figure8-25.Station2-2Velocity Contours(CFD) 160

Figure8-26.Station6-YVelocity Contours (CFD) 160

Figure8-27.Station6 -2Velocity Contours(CFD) J61 Figure8-28.Station7-YVelocity Contours (CFD) J6J

Figure8-29.Stmion7 -2VelocityContours(CFD) J62

Figure8·30.Stationlocations(CFD) 163

Figure B-3/.Stationvelocity contours(CFD) 164

Figure B-32.Inlet streamlines J6j

FigureS · ]3.Normalited energyjlux 171

List ofFigures

List of Tab les

Table 3-1.Dimensions associatedwithengineeringphysical quantities IJ

Table3-2.Waterjet systemvariables /6

Table3-3.Dimensional analysisof waterjetsystem 29

Table5-1.AnemomelerPan iculars 52

Table5·2. DynamometerTechnical Data 55

Table6-1.Confidence Levelsfor GaussianProbabilityDensityFunction 79

Table6·2.VolumeFlowrateThroughWeuerjet 9/

Table6-3.Unc:enainryfor Hot-...ire Anemom ettr 96

Table 8-1.Solverparameters /45

Table8-2. Mesh Staristics /49

Table8-3. Loss coefficientsforvariouscomponents 168

Table8-4.Waterjetlosscoefficients 169

Table8-5.Waterjet headlosses /70

Table8-6.Windtunnel momentum fluxcalc ulations /72

Table8·7.Propulsion performancecalcularions 17)

Table8-8.Full· scalemomentumflux calc ulations /74 Table8-9.Full·scalepropulsion performancecalculations /75

ListofFigures

Appendices

Appendix A:Fabrication Drawings A- \

AppendixB:Matlab@Routines 8·1

AppendixC:Experimental Results C·l

Introd uction

1 Introduction

Traditionalmethods of marinepropulsion havebeen limitedto screw-type propeller arrangements but contemporary efforts to improve vesselspeed have ledto the developmentof practicalwaterjet systems.Suchdevelopments challenge theheretofore- acceptedtheorythatwaterjetsareinherentlylessefficie nt thanscrewpropellersand in recentyears there has beena remarkableincrease inthe numberof warerjet manufacturersand vehiclesequipped with warerjet propulsion systems.

1be historyof warerjet technologydates back tothe timeof Archimedes,when he was creditedwith inventinga deviceused forpumpi ngout flooded ships,the Archimedean screw (Allison, 1992).Technologicallimitati ons,coupl ed withalack ofunderstanding of theprinciplesof propulsion beforethe 19^thcentury, however,stuntedwaterjet developmentwhilepaddle wheel and propeller technology flourished.

During the 1960' sand 1970' s, somehigh-speedhydrofoils wereequippedwithwaterjet propulsionsystems bUIthe high cost of design,outfining,and operation limitedtheir applications10 military endeavours.In the 1980' s,however,lowerfuel costsand increased highway congestionwere catalysts in an effort towards viable transportation alternatives.Theresult was the developmentof waterjettechnologycapableof competing with trad itional screw propellers .High-speedalumi niumcatamarans,for example, were relati vely easy to designand buildcompared to other dynamically supportedcraft, and vessels propelled by waterjet systemsbecame feasible surrogatestopropeller craft, especially in thehigh-speed market.The pioneering workofHamilton ledtothe

Introduc tion

developmentof the modemwaterjetunit andFigureI- I showsa typicalaluminium catamaranusingHamiltonwaterjets:

Figu r eI-I. Athena High-S peed Cata m a r a n

Presently,thou sand s of waterjetsare produced eachyearfor the recreationalmarket.to be used in waterscooters and small fishing boats. At the conunercial levelhigh-speed passenger ferries equipped with multiplejets having installed powers of more Ihan 70 MW are commo nplace.Thehigh-speed tra nspo rtatio nof cargo. andcontainerizedgoods has yettoberealised.butintimeitislikelythatwaterjet technolo gy willdominate most high-speedmarine applications.

Some advantages of waterjet propulsion arelisted below:

Elimination ofappendages Impro vedmanoeuvra bility

Improved braking especially at speed Red uced fuelconsumpt ionathigh speeds

Introduction

Greatlyreducedunderwaternoise

Reduceddraft (depending on hull type)

warerjetsandpropellerspropelboatsdifferentlyandinthepast traditiondictated the manner in whichwaterjet performance was evaluated.Varioustestingmethods have been proposed10determine theperformancecharacte risticsof warerjets and the momentum flux method.recommendedby the '96IITChasemergedasthe industrystandard.

Con ventional methods of testingpropellerdrivencraftwerealso applied 10 waterjets, withoutsuccess,andself-propulsiontestsusing lOWcarriages have givenwayto large scale testing of waterjetsusingwindrunnels. Although Reynoldsnumbers aremuch smallerwhen using air asthe workingfluid,it hasbeenshow nthat testing of waterjets using windtunnelsproducesresultsthat areapplicabletoreal-worldapplications (Griffuhs- Jones. 1994).

Inaddition 10physicaltesting, computer simulationshave emerged as a validmethod for evaluatingthebehaviour offluids and performance ofequipment.thanks in part to advancesincomputingpower inrecent years.Numericaltreatments are generallyless costly than physical tests andproducepractically unlimited level of detail intheir results.

Computational fluid dynamics (CFD)isbased on the analysisof fluidsystemsby means of computersimulationand has beenused fora wide range ofindustrialandnon- industrialapplications.Physicalexperimentationformsanintegralpartofany CFD simulationasthe accuracyofsimulation results is obtainedthroughvalidationagainst experimental data.Once validated.however,the numerical code is capable of providing

Introduction

engineeringquantities suchas force.velocityandpressure,at alevel of detailnot possiblethrough physical experimentation.Used inconjunctionwithgood experimental data,computersimulation representsan extremely powerful1001forengineering analysis.

The focus ofthisresearch wastostudythe applicabilityof CFDanalysisto waterjet testingand to evaluate the propulsionperformance ofawaterjet unit using

crn

simulation validatedbyexperimentalresults.A full-scalewarerjetwas tested atthe Memorial Universityof Newfoundlandwind tunnel,andnumericalanalysiswas achieved withCFX 5.l@ernsoftware.Oncevalidated,the

crn

simulationwasused topredict thepropulsionperformanceofthewaterjetunitusingthemomentumflux method.

LiteratureReview

2 Literature Review

Renewed interestin waterjetpropulsionoverthelast 20years hasled 10 a better understandingof the principles of waterjet propulsion ,more efficientpumping units,and theevolutio nof the modem waterjet.These advancesare the result ofresearch into both modeltestingtechniques andthemanner inwhichwaterje tperforman ce isinterpre ted.

Traditional testing method shave givenwaytospecifictests tailoredto theunique properties of waterjetsand advance s innumericalmodellingtechniquesand highspeed computinghave made compute rsimulationmore feasible .Asthenumerical modellingof waterjetsystemscontinuestoevolve,modeltestingplaysanimportantroleintheir validation.Thefollowingchaptersummarises thepublishedresearchonexperimentaland numericaltreatmentsof waterjetpropulsion.Morespecifically,it highlightsimportant work related to the testingof waterjetsin windtunnels,andtheirsubsequent computer simulationandvalidation.

Griffith-JonesandBowen (1992)discussed modelling oftheflow throughthe intakeofa waterjetpropulsionunit and aplaninghull.Usinga wind tunnel,theyobserved flo w separation from theintakeroo f ofthe waretjetunit. Acknowledgingthat the turbulence levels intheflow wouldbereduced. the impellershaf t wasremovedfrom the intake 50 that numerical simulatio nwouldbesimpler. Thesidewallofthe windtunnelwas angled inwardsto simulate theangle ofincidenceofa typicalplann inghull. Theirresults showedthat there wasa significantpowerlossdue to non-uniformityand flow separati on,

Literature Review

Widmark and Gustafsson(1997 ) performed3-d imens ionalcomputationalfluiddynamic s (CFD)calculatio nson acomple te waterje tunit with two differentcodes, SHIPFLQW and FIDAP.The pressure andvelocitydistributionthroughout the waterjet unit was studied in ordertodeterminethe losses at the i.nletandoutlet,The rotational veloc itycomponent normallyassociated with rotor shaftswas omittedinthesimulation because the waterjet that wasmodelledwa s equippedwith a shaft prot ectionhub.Furthermore , guidevanes werenotmod elled at the outle tsince auniform volumeforce wasusedto modelthe impellerand didnot account for theswirli ngof therotor.Results indicated that a capture width 70Cl>larger thanthe inletwidth shouldreplace the30Cl>recommendationof Kruppa etaI.(1996)formomentumflux calculatio ns.

Tumock andHughe s(1997)undert ooktheevaluationofa CFDcode forinvestigatin g hull-watetjetflow interactio n.Aphysical model was builtfrom faired strips ofplywood attached toa base plate by a seriesof ribs to definethe outlineshape. The front face was transparent toallo w flowvisual isation withwooltuftsand pressure distribu tio nwas monitoredbya number ofstaticpressure tapsalon gonehalfof thejet unit, at a number ofradialandlongitudinalsectio ns.Themod el was attache d tothe sideofthewindtunnel to simulat ethe flowto the warerjetunit.Theydetermined mass flow through the duct exit as theproduct of the speed at the midpointandthe cross-sectionalarea,andthe flow throug htheexitplane of the workingsectio nwas obtainedby mass continuity.For simp licity,aconstan tmassflow ratethough the duct exit wasdefinedforall the

crn

models.Itwas acknowledged. however,that waterjet -impe llersoperati ngat constant speedsdoDOl:nec essarily experience constantmassflow rate through theduct.The

LiteratureReview

simulationconverged with residuals ofIx lO')after700 cyclesand itwas determinedthat a flat plate doesnot accuratelymodelthepressure changes thatoccuraswater enters the inlet.If a waterjetduct is to be designedforaspecific application.they concluded.the influenceof the surrounding hull mustalso beconsidered in addition to theflowthrough theduct.Animportant conclusion wasthat

em

workcouldbeextre mely beneficialat the design stage.Theresultsof aCFD simulationcan provide engineers with velocity profiles,pressuredistributionsandsubsequentviscousforce distributionsinorder to betterunderstandthe resistance andpropulsionaspects ofwaterjets.

Verbeek andBulten (1998) used theresults ofwind tunnelexperiments to valida teCFD resul ts.Itiswell known that acurvedpipe withuniform flow leads tonon-uniform flow due10secondary effects,and Ihal thevelocityincreaseis caused bycentrifugal forces thatlead10a maximum velocityatthe top of pipeduet.Theopposite,theyconcluded.

happensinwaterjetsdue the boundary layerunder thehull. Theuniformvelocityinthe boundarylayerresultsin the entrainment of high-speedwater at thebottom of the duct, andlow speedwateratthetop.Results showedthat7-9%ofthetotal installedpowerwas lostat the inletdue10thisnon-uniformity, andthat moreuniformvelocityprofilesresult from increasedturbulencein the flow.

Allison eral. (1998)investigated theparalleldevelopmentof computationalfluid dynamics(CFD)with the Reynolds Averaged NavierStokes (RANS)equations. Results indicatedthatthe blade force sand pressuresyieldedbynumerical softwarecompared wellwith thosefoundfromconventionalmethods.Simulationresults.theyconcl uded, can beusedto:identify potentialproblemareas suchasre-circulationandflow

Literature Review

distortions,provide fluid loading onsolidparts,predict overallpetfonna nce of devices, andcorroboratethe results obtainedfrom other designmethods.

Roberts andWalker(1998) studied the ingestioneffectsof a waretjetinletand statedthat currentdesignpractisescouldlead to the under prediction ofthrust for flushwaretjet intakes.The experimentswere based ona1:7.67 scale waterjetmountedto aclosed circuit wind tunnelandequipped witha secondary fan exhaustingto theatmosphere. The driveshaft wasnot modelled, butthe shaftand fairing were expectedto increasethe outlet distortionandflow lossesin a real intake.It was concludedthat wind tunnel tests provide a convenientandeconomicalmeans of obtainingthe detailed flow measurements neededto understand the physicsofintakeflows andvalidatecomputationalprediction methods.A major limitationof thetest,however, was the inability of air measurementsto provideinformation pertaining10cavitation.

Mununga,Huntsman.andHothersall(1998) reported on the testing ofa waterjetunit using a wind runnelto investiga te the effectsof a splitterplateandscreengrid. Thenon- uniform loadingdue10flow separationwasinvestigated andrevealed unbalanced loading ontheimpeller.Theyundertoo k thedesign ofasplitterplate and screengrid toimprove the quality of flow through the intake,andhence improvetheperformance of the waterjet unit.Results showed a dramaticimprovement inflow uniformityusingthesplitterplate, andmarginalimprovementusing intake screens

Many ofthepapers ofthethirdRINAWaterjet Conferencein 2001 investigated hull- propulsorinteraction using RANS codes.Allisoneral(200 1) used the UNCLE codeto understandtheflow behaviour arounda shipwith andwithout waterjets.Results indicate

LiteratureReview

that. forweterjets.alarge portionoftheupstreamflowisdrawnintotheinlet.The behaviour ismuch differentfrom thatobserved with thebarehull.wher e streamline passed downstreamrather benignly.

Seil(200 1) validated simulation results with experimentaldatafor thevelocity distribution at the ductexit and foundthemto be ingood qualitativeandquantitative agreement.Usin gFLUENT® codewiththe k-eturbulencemodel.theeffect of theshaft.

shaftro tationand scaleeffect(Rey noldsnumber) onthe wate rjet inletflow was investigated.Itwas determined that shaft rotationhad a significanteffect on distortingthe wakeat the ductexit.

Hu and Zangene h(2000used differentcommercialCFDcodessuch asFLUENT.UNS.

RAM PANT. andTASCflowto calc ulatewaterjetimpellertorque .Thepredictedtorque valueswere comparedwith measurements and the prediction accuracy was seento be very good. They concludedthat the shaftgreatlyinfluencestheflow fieldinthe waterjet andshouldnotbeneglected in CFDcalculatio nsoftheintakeduct

Similitude Analysis

3 Similitude Analysis

3.1 Similari ty

Whenusingphysicalmodels,caremust be taken toensurethat resultsare transferred from modelscaleto fullscalecorrectly.Itisoftenthecasethatcompletesimilarity between the twoscalesisnotphysicallypossible. anda systemoflawsthat maintain similaritybetweenthe mostsignificantelements ofmodel scale andfullscaleisrequired.

The followingconditionsmustbe satisfied inorder for specificforces on themodel and full-scaleobjecttobesimilar:

Geometric similarity

Kinematicsimilarity

Dynamic similarity

3.1.1GeometricSimilarity

Geometric similarityreferstomaintainingcorrectlength scaleratios between prototype and model.Thisisgenerall y straightforwardinterms of physicaldimensions suchas the length to breadth ratio.but canpresentsome interesting challengeswhen dealingwith difficult factorssuch assurfaceroughness.In ship modeltesting, forexample, evenif the modelsurface isan exactcopy oftheprototypesurface,flowalongthe surfacewillnot besimilar dueto theflowcharac teristics ofwater over large andsmallscales.In the case of alarge-scalefactor.modeldimension smay be extremelysmall, andstructural

Similitude Analysis

limitations can make ildifficult to maintai ngeometricsimilarity.Suchisthecasewhen workingwithmodelpropellers.asthetrailing edgesof the blades have10 be made relatively thickerthan their full-sc alecounterparts.forpractical reasons (Harvald,1983).

Geometric similarity, then, cannot alwaysbe maintained between individ ual components of themodel andprototype,and caremustbe takento ensure thaicorrectionfactorsarein place.or the effect isminimal.

3.1.2KinematicSimilarity

Inorder to maintainkinematicsimilarity. the ratios betweenvelocities in the modelmust be equal totheratios betweencorrespondingvelocitiesintheprototype.al corresponding positions.Thiswillbe discussedin more detail in Section3.3. asitis releva nttothe waterjet system,inparticular.

3.1.3Dynamic Similarity

Dynamic similarityrequiresthat force-scale ratios arethe same formodelandprototype.

Inorderto achievethis.forcepolygons(vectors) mustbesimilar(i.e. the directionofthe force s,andthe ratio oftheforce scalesmustbethe same).Achie ving completedynamic similarityisnot alwayspossible,and theexperimen teris charged with theresponsibility ofselecting the forces that dominate,and thosethaiarerelevant to both the modeland prototype.Further detailsare suppliedin the sectiononwaterjet dimensional analysis.

Similitude Analysis

3.2 IndlcialApproach&Matrix Methods 3.2.1TheIndlclalApproach

Rayleigh's indicial method consistsof determiningthe variablesrelevanttoasystem and writingtheminterms of'fundamentaldimensions' ,Thechoice offundamental dimensions canbesomewhatarbitrary,butit hasbeen generallyacceptedthat mass, length. andtime,aresuitableunits fordescribingthebehaviour of engineeringsystems.

These dimensionsare familiarto most people.and because they have physicalrelevance.

it is easytovisualise oneobjectbeinglonger than another.for example.Thefunctional relationshipcanthenbewrittenin terms of the mass [M], length [L],and time[T]

dimensions,and the exponents of each dimension equatedto ensuredimensional homogeneity (Sharp,1983).

Solvingfortheconstantsin the exponent ofeachvariableleads to aseriesof dimensionlessgroups,or1tterms.The1tterms fonn thebasisofsimilitudetheory,since two geometricallysimilar systemswillbeboth kinematicallyanddynamically similarifit

terms in onesystem are equaltothose of the other.Buckingham developed a method of identifyingthenumber of releventnterms based onthenumber of variablesand dimensions,Hismethodstates:

Ifan equation involvingkvariables is dimensionall yhomogenou s,itcan bereduced toa relationshipamong k·rindependentdimensionless products, where r istheminimum numberof referencedimensionsrequiredtodescribethe variables(Munsoner al. 1998).

SimilitudeAnalysis

Determiningaset of1tterms is accomplished by firstselecting from the originalset of variables. a setofrepealingvariablesequaltothenumberofreference dimensions.The repeating variablescan then be combinedwiththe remainingvariablesto form the necessary1tterms.For a given system. ofparamount importanceistheway in which one variable behavesasaresultofchanges10theothers.Thesevariables aretermed dependemvariables, and it behoves one to limit theirappearanceto asingle1tterm.Itis impo rtant.then.toexclude the dependentvariables fromthelistofrepealing variables.A 1ttermisformedbymultiplyinganon-repeatingvariablewith theprod uctofthe repealingvariables, eachraisedtoanexponentthatwillmakethe combination dimensionless. Repeatingtheprocedure for the remaining non-repeating variables forms subsequent1tterms.Someconunonengineeringunitsexpressed in terms oftheM.L.T system are showninTableJ-L

Table3-1.Dimens ions associated withengineeri ng ph}'sical qua ntities PhstcatQuantlt ~ Svmbol pjmenalon Jor,.M.lm Sleni~{

Mass M M

Len th L L

Time T T

RPM N

.,

Area A L'

Mass Densitv 1MLl~

Force F MlIum'

Toraue

a

^MHL2T·2

Dvnamic Viscosity

"

^IMIILr'ITr

SimilitudeAnalysis

3.2.2MatrixMethods

The indicial equationinherent to the Rayleighmethod can be solvedusing elementary matrixalgebra. The equationmaybewrittenas a dimensional matrix withtheinfluencing variablesoccupying columns ofamatrix,and rows signifying the M,L,Tsystem.The values atcorresponding locationsin the matrixaresimplytheexponentof theM,L,or T dimension,forthe variableinquestion. The solutionof asystemoflinear equations is possiblebyreducingthe first threecolumnstotheunit matrixandobtainingtherankof the matrix.Therankofthematrix specifiesthe numberofindependent equations that are necessaryto describetheexponentsofthevariablesinthesystem.Buckinghamtheory is thensatisfied when the first three columnshavebeenreduced to the unit matrix, sincethe total number ofdimensionlessquantities requiredisequalto the numberofvariables minus therank of the dimensionalmatrix (Sharp .1983).

Echelonin a matrix exists when thenumberofzerovaluesin rows reading from leftto rightincreasesfromtopto bottom.Matrices exhibitingthis characteris ticcanbe manipulated by rowandcolumnoperations.andthevariablescanberelatedto one anotherwithgreat freedom.Aset of repeatingvariables equaltothenumberof fundamentaldimensions canbeforced to theunit matrix,and theremaining dimensions can thenbe written in terms ofthe others. If the variablesare written in theM,L.T system,forexample,the unitmatrix will bea threebythreematrixmadeupof3 repeatingvariables andtheremaining columnsprovidetheindices ofthe1tterms.

SimilitudeAnalysis

The situationmay arise, however,whereitis not possibletowritethe repeatingvariables of choicein echelon formand onemust rely on linearalgebra.Ithas beenshown thatany matrixcan partitioned into:

I)A unitmatrix consistingof aset ofrepeatingvariables

2)A matrixmade upof theremainingvariables

Repeating the operationthat transfonned therepeatingvariablesinto the unit matrix formsthesecondmatrix.Consider,for example, an eight bythreematrixthathasbeen panitioned intoa threebythreematrix(A),anda five bythreematrix(8). Inorde rto form theunit matrix,matrixA mustbemultipli edbyit'sinverse(I=A·^I),soA=A*I.

MatrixB,then must undergothesame operation(0:::I• 8),andthe finalmatrix canbe writtenas thecombinationofmatrix Aand D.

Thematrix method isa veryquickandpowerfultool for manipu lating the variables of interestintodimensionlessform,When faced withalarge numberof indepe ndent variables,thematrixmethod canbeused withsimple computerprograms to providea veryfastsolutionfor thenon-dimensionrelationship between variables.Thesimplicity of theapproachalsoallowsone torepeat the operationswithdifferentsetsofrepeating variablesuntilthe desired set of7tterms is obtained.As with other method s,the finalset of1ttenns canbe the result of compoundingthe resultsof the matrix analysisin orderto provide convenientsolutions.Theapplicatio nof thismethod withrespectto theanalysis of wa terjets isdescribedinthe nextsection.

Similitude Analysis

3.3WaterjetDimensional Analysis

The variablesnecessaryto describe thewaterjetsystemaregiven inTable3-2:

Table3-2.Wat erj etsystem variables

P arameten s

S mbcl Fund ani ent aIUrilt s""

Shaft sceee N

. ,

Characteristiclen th l l

Fluid density IM][L]~

Velocity V Lm -¹

lbvoamcviscosity [M][l)"'rrr' Gravitationalacceleration [Llrrr'

Pressure

IrMlrl -

^1m

-

^Z

Surface tension [MJrrr'

Dependent Variables

Thrust T IrMlfl1m'

Power P IrMlfl1'm'

Shafttorque Q. [M][ll [TJ'

Volumetric flowrate Q [lJ'[TJ"

Thrust.torque.power,and volumetricflowrate are dependent variablesandthe behaviourofthe waterjersystemcan be describedby:

TorQ,orPorQ

=

4l(N.L.p,V.!J.,g.P.<P) [3.1J

In order to begin the dimensional analysis, a matrix ismadefromthe indicesof these variables.

~' g ]

2 3

-2 -I

o

2 I -I I -I

~

^[:

~

^·3

T -I 0 0 -2 -I -I -2 -2 -2 -3

Similit ude Anal ysis

The firstthree variablesare chosen as repeatingvari ables, and the sub-matricesaregiven by:

[

0 0 ']

A

-=

0 -)

.J 0

[

I 0 , 0 , , 0

3 ]

B:;::.I I -I I -I 0 2 2

-2 -I -I -2 -2 -2 -3 -2 -I

Theinverse of the firstmatri xbecome s:

o . 1 ]

I 0

o

0

and bothmatrix AandmatrixBare multipliedby the inverseof matrix A.Matri xA multiplied by it's inversegivesthe identitymatrix (A ):

[ 1 0 0 ]

A3:;::. 0 I 0

o

0 1

andtheresulta ntforAI.Bis givenbyD1:

SimilitudeAnalysis

[

2 I 1 2 2 2 3 2

~ l]

01:= 4 1 2 1 2 3 5 5

I 0 I 0 1 I I I

Finally,thematricescanbeaugmented to form asinglematrix:

N L

~ . [ ~ ~

^{0 4 1 2}

P 0 0 1 1 0 I

p r,p P Os

a

2 2 3 2 ~ l]

I 2 3 5 5

o

I

Itisthen clearthatBuckingham's theoryhas been respected,and nine non-dimensional termscan nowbedeterminedfrom theresultingmatrix.The system,thencan be written asfollows:

Thefrrsttermcan bere-written suchthatthegeometric parameter(L')isreplacedby [3.2J

impeller diameter(D).andtheresulting termis recognisedasthe thrust coefficient(KT).

K,

= (j>V~D' )

^[3.3)

The second term can alsobe slightly modifiedto resembletraditionalnon-dimensional terms.Replacingthe'V'termwiththe advance velocity(V,ot) ,and thegeome tric

Simili tudeAnalysis

paramet er with the impellerdiame ter(D),the term is recognised as theadvance roefficient( l).

Recognising that

J = (~)

[N).\B

[3.4)

[3.5)

andthatthe dynamic viscosi ty (jJ)isrelatedto thekinema tic viscosity(11accord ingto:

. : «

p

wecansubs titute for'N'and'p'in the thirdtermandarrive attherecipr ocalofthe Re ynoldsnumber.

., p

v (v)

Re ={XVL¹=

( i)t2 ^{= VL}

[3.6)

[3.7)

Similarly, wecan substitutefor'N'in thefou rth term,invertand arriveatthe reciproca l oftheFroudenumber:

Similitu deAnalysis

Re placi ngthe pressureterm'p 'with the changein pressure'L1p' ,thefifth term can be writtenasthe cavitationnu mber( o):

[3.9)

Subsutu tlng for'N"inthefifthterm resultsintheWebernumber(WII!')

w,= pI..;N'

⁼pLI

( ~V

_

J' = (P:,J

L

[3.10]

The remainingthreeterm s arerecognisedasthepowercoefficie nt(K,),sha ft torque coefficient(KQ»,andthevolume flowrate coefficient(KQ).

K,

= _P

fX'/JD'

-

[3.11)

[3.12]

[3.13]

Similitude Analysis

3.4Dis cussionofNon·Dimens ion alTerms

Itisnotoften physicallypossible.or necessary.tosatisfyallof thel'ttermsin any particularsystem.Insuchcases.the most important terms are respected andother. less significant. terms canbeneglected. provided certain assumptionscanbe made.The followingisa discussionof therelevant non-dim ensionaltermsfortesting a waterjet systeminawind tunnel atfull scale.

3.4.1Advance Coeff icient

Kinematic similarityisaccomplishedwhen thevelocitiesatcorresponding points of the model andprototype havethesame direction.andhence the angleof attack ofthe impelleris similarbetweenmodelandfullscale.Forthis reason.theratio of thespeed with whichthe fluid flowsintotheimpeller(i.e.thespeedofadvance).andthe velocity of theimpeller(circumferentialvelocity)mustbethe same for boththemodeland the prototype.The advancecoefficientcan be thought ofastheratioof theaxial veloci ty of flow into theimpeller.to thetangential velocityofflowrelative to the impe ller.

Kinematicsimilarity.then,can be accomplis hed ifthe advancecoefficientfor the model and prototype are the same.

3.4.2Reyno lds Number

How regimescangenerallybeclassifiedaseitherlaminar, turbulent. or transitional.The significance ofthe Reynoldsnumberisthatitis very usefulin determiningflow regimes forspecific fluids. ata given velocity.Itcanbethoughtofastheratioof inertial forces to viscousforces.andis importantin mostproblemsinvolvingfluid dynamics.Inspectionof

SimilitudeAnalysis

thevariablescomposingtheReynoldsnumber showsthat,in manycases, matchingthe Reynoldsnumbers inmodelandprototype isnot possible.Insuch cases it is important10 ensurethatthe flowregimes aresimilar. The high-speedflowsthatcharacterise waterjets existintheturbulent regimeanditis importantto ensurethatflowregimesin model waterjet systems also behavein a turbulentmanner.AccordingtoMunsonetal.(1998), scaleerror is negligible providedtheReynoldsnumbersforthe flowin themodeland prototypearegreaterthan5 x10'.

For the modelwaterjet system,flat plateboundary layer theorywas appliedat the wall of thewindtunnel.Thein letwas locatedapproximately9.5metres fromtheleading edgeof the windtunnel. Assumingthat me distancefrom tile forwardperpendicular to the inlet of the prototypewaterjetis at least9.5metres,thevelocity inthe windtunnelisa limiting case.Thekinematic viscositiesofairandwaterare1.46e·~m²/s,and1.17e-f.m²/s, respectivelyand it followsthaiany speedgreater than 0.77mlsprovides sufficiently turbulent flow.

V,(Re.v....

)=(5 e '

.1.46,rIO-'m^1/S)=o.77mI S [3.14]

L 9.5m

V,(Re.v..

)=( 5 e

^{l.l.17Xl0-6ml/s)}=0.062m/ .l'

[ 3. 15J

L 9.5m

From thisitmaybeconcluded that the flow regime in the boundarylayerofthe tunnel wall islikelytobeturbulent forboth the modeland prototype,providedthe velocity is greater than0.77mls.Inadditiontothis,lhe velocity profile in thetubular section oftile

Similitude Analysis

waterjet may beexaminedbycomputing theReynolds numbe rforviscousflo w ina pipe. Itis importa ntto ensurethat thevelocity profil esaresimilarbecause it isthen possible to conclude that theboundar ylayersin themodel and prototypewill be similar.Usingthe sectional diameter(D

=0 .3.5

rn)as therefe rencedime nsion:

Re_•

=(~)

_v....

=( 7.6~-lm ls .o.3Sm) = L85e~

^(3.16]

1.46e-'m^J/s

Re

=(~). (7.68e·lmls.o.3Sm )= 2.30e'

^(3.17]

p v, 1.17e-⁶m^Jl s

Accordi ng toMunson eral. (1998),theflo win apipeisturb ulent providedthatthe Reynolds number for theflowis greaterthan 4000.Itis therefore likelythat the flow regime,and velocityprofile forthe modeland prototypewillbeapproximatelysimilar for the assumed, minimum. velocity.The speed in thetubularsectionofthemode l waterjet was expected tobemuchlargerthan0.77mis,andturbulenceinbothmodel and prototypewas ensure d.

3,4.3 Froude Number

\Vaterjet systems perform work onwaterby liftingit thro ughanelev ation and expelling it abovethe watersurface. TheFroudenumber canbe thoug htof as the ratio ofinertial force sto gravitationalforces,andalthoughit isimportant for testin gof waterjet sin wave tanks,orwater tunnels,ithas no real sig nificancewhe n testin ginair.This is dueto the fact thatthe model waterjetisnot expellingthe flow from onefluidinto another.In

SimilitudeAnalysis

addition10this, the waterjerwas attachedto the wind tunnel at a ninety-degreeangle.

such thatthehull was effectivelyon its side.There isno lifting componentin!he model.

Froudescaling,therefore, wasneglected.

3,4.4Cavitationnumber

Cavitation isthe processofformation ofthevapourphaseof aliquid whenit is subjectto reducedpressureatconstantambient temperature (Harvald,1983).The occurrenceof cavitation can be detrimentalto the effectivenessof a propeller.as wellasphysically destructive. Upon formation.cavitation bubbles can erode propellerblades.parts ofthe jet ducts and stators.and causeabreakdownin flowandsubsequentloss ofthrust.The situation.therefore,shouldbeavoidedatallcosts. watenets. fortunalely,areless susceptibletothephenomenasince theintakeslows thewaterbefore deliveringit tothe impeller,anddecreases the chancesof cavitation(Allison. 1992).Furthermore, the "Final ReportandRecommendations the23

rnc'

submitted bythespecialist committee on validationof waterjertest procedures

(m e,

2002) assumesthatanycavitationin the pump or intake duringoperation does not affect the poweringcharacteristics of waterjets.

Tbeexperimentalset-upin thewindtunnelwas nOIdesigned to measure cavitation.but pressure taps canbeplacednear the impeller 10determine pressurevarianonsat high speeds.Should detailedtests regarding the likelihoodofcavitationbenecessary. a cavitation tunnel shouldbeused.In anyevent.the system wasnotsetuptomonitor,or considertheeffectsof cavitation. and thecoefficientwasthereforeignored.

Similitude Analysis

3.4.5Weber number

TheWebernumber is theratio of theinertiaforce tothe surfacetensionforce.Itis often importantwhen consideringthesurfacestressesfromcavitationbubbles. Surface tension, however,isnota property of gasesand hasnosignificancewhen performingexperiments in air.Similarity ofthe Webernumberisneglectedforthepurpose of this analysis.

3.4,6Mach Number

When dealingwith air at highspeeds.the assumptionofincompress ibilityisnotalways appropriate. According toMunson et al, (1998).a fluidcan beassumed to be incompressibleif theMach numberislessthan0.3.TheMach number istheratio of the inertiaforcetothecompressibilityforce and isexpressedas theratio ofthevelocityof interest(Vj)withrespect tothevelocity of soundin air(c) :

Ma : ~

c ^(3.18J

The velocity ofinterestis made up of theimpeller speed.and theaxialvelocity(V.) :

(3.19)

[3.20)

Itfollowsthat foranyvelocitylessthan 99mis,theassumptionofincompressibilityis valid.The maximumspeed of thewindtunnelis15mlsand inorderto approachthe

Similitud e Analysis

boundsofincompressibility,a shaftspeedgreate rthan5000RPMwouldbenecessary.

This is wellbeyond the operatingspeed of mostwaterj et systems.Sincethe Mach numberisinvariablyless than 0.3,we may concludethatthefluidiseffectively incompressib le.as is normally the case in low spee dwindtunnels .

3.4.7Power, Shaft Torque,Thrust,and Volume Flowrate Coefficients

In order to maintain dynamic similitude,the directionof the forces andthe ratioof the force scales must be the same.The remaini ngcoefficients,then,are extremely important inorder forusto assess the performa nce of thepropeller.Withsimilitude assumed,it is possibleto determinethepower,shaft torq ue, thrust,andvolumetlowrateofthe model and prototype.To summarise,thenun-dlm ens ic nalcoefflcientsof importance are:

AdvanceCoefficie nt:J'"

(~

⁾

ThrustCoefficient:

s, = (pN~D4)

ShaftTorque Coefficie nt:KQ, '"

~sD5

VolumeFlowRateCoeffi cie nt:K_Q'"

N~J

SimilitudeAnalysis

3.5 Scali ngLaws

Scal inglaws permit the magni tude ofa vari ableinonescale 10becalcul ated fromits valueina differentscale.1benon-dimension altennspresent ed earlier provide a means of determi ningthe full-scalevalues of several important variab les forthewaterjet system.

Theratio ofamodelvariable 10ir s corres pond ingprototypevariableiskno wn asthe scale forthatvariab le.Thelen gth scaleis definedasthe ratioofalineardimension ina prototype .to thecorres pond ing dimension for themodel.and is denoted byA:

[3.211

where thesubscripts pand mreprese nt the modeland prototype,respectively.

Equa tingthe advanc ecoefficientsfor the modeland protot ype satisfies thecond itionof kinema tic similitude:

J

•

~(~)~

N.D.

(~)=

N,D,

J

, (3.22J

Scalingtheshaftspeed. orad vancevelocity.isthenaccomplishedthrou gh the follo wing relationship:

ForPowerwehave:

~

_V..

⁼ ( '!.L)(!:L)

_{N.. D..}

⁼ ( '!.L)(A)

_N.. ^[3.231

SimilitudeAnalysis

[3.24J

where

( _lp ELl _.. .( _l p~", l.

⁸⁶⁶

P",. Similarlyforthrust:

[3.251

Shafttorqueis scaledaccordingto:

Finally,volumetricIlowratecanbescaled according to:

[3.26)

[3.27J

[3.281

[3.29J

[3.301

Simil itudeAnalysis

[3.31)

Inthis study,the IIlOlkIwas full scale and hencethe scalefactor(}.)isequal to unity.For a given advance velocity, then,the shaftspeeds forthemodelandprototype wereequal, since the impeller diameters wereme same. The power, thrust,andshafttorquethen scaled accordingtotheratio ofmedensityofwatertothedensityofair.

3.6Summary

The testingof warerjetsusinga windtunne l isasimple, andeffective alterna tiveto traditionaltestingmeth odsat smallscales.A seriouslimitation,ho we ver, is the inability of air measurements toprovideinformationpertainingtocavitation.

Thedimensionalanalysis, summari sedinTab le 3-3 revealed thaiiftheadvance coefficient forthemodeland prototypeare equal andthe scalefactorisunity,thenthe velocityof air throughthewindtunnel isequaltothespeed ofthefullscale protot ype travelling inwater,ata given shaftspeed.Based on thisinformati on ,the thrust,shaft torque, and power are allscaled by theratioof thedensity ofairand the density ofwater.

Table3·3 .Dimensiona lana lystsofweterjetsystem Parameer. Ratio:~f.,;Scale-;.1i·~'~~''''M~.f:tc-t!

length L,IL. II.I

Shaf lspeed VlVm (Np/N",)(A)

Power P_Pm (p_emIlN_N.,)(AI'

Thrust T,!Tm (p_PmlIN_N.,)IAI

Shaft torque 0..,10- (WPmlIN_NmIIAI' Volumetricflowrate 0JOm INJNmllAI

SimilitudeAnalysis

When a large Dumber ofvariablesmust be considered .matrixmethods are usefulfor determi ning thenon-dimensionaJtermsrequired 10 sufficiently describe thebehaviour of the system. In ordertodeterminetheimportance ofeach.thetermswere manipulated as requiredand transformedinto physicaJlymeaningfulnon-d imen sionaJterms. Acomplete analysisof eachtermand itsrelevance on thesystem wasundertake nsuchthatsimilitude was satisfied forthemostimportant aspects of theexperimentalendeavour.

TheMomentum FluxMethod

~

The Momentum Flux Meth od

4. 1 Introd uction to the Mom entum Flux Method

Aswaterjettechnology continues togrow, methods oftestingandevaluatingwarerjet propulsion systems haveemerged.Inthe past,these efforts hadbeenbased on traditional methodsforevaluatingscrewpropellers.butrecent work has shownthattheunique characteristicsof waterjetsystems requireunique testing methods.The waterjet is an integral pan of a vessel's hull and as such,traditional concepts suchas thrust deduction do not apply to weterjets in thesame physical way as they do forconventionalscrew propellers(DyneandLindell,1994).Moreo ver.the evaluation of some basic physical quantities suchasthrust,forexample, requires anindirect methodofrneasurernentbased on flow rates.In responseto thisissue, the momentumfluxmethod wasdeveloped, andis thefocusofthis chapter.

Elementary momentumtheory canprovidevaluableinsightconcerningmarine waterjet propulsionand the momentum-fluxmethod canbeused 10evaluate the power,thrust.and efficiency characteristics of the waterjet. This method, describedin the21-International Towing Tank Conference(lTIC'96),isthe resultof an initiative broughtforthbythe ITIC SpecialistCommittee onWaterjetsasking forcomments on possible power predictionmethodsfor waterjets.Thismethodspecifiesthat thrustbecomputedfromthe changein momentum fluxthroughou t thewaterj etsystem.The vesselis considered tobe stationaryinamoving flow, and all flowvelocity measurementsusedin momentum and energy calculationsaremade relativeto the vessel(Kruppaet al.•1996).

TheMom entumFlux Method

Momentumflux canbedefined as a measureof themo mentum influidpassingthrough a unit area of asurface ina givenunit of time.Similarly,theenergy flux is ameasure of the amountofenergyinaquantity of fluidcrossingaunitarea ofa surface ina give nunit oftime. The locationsof momentumandene rgy fluxmeasurement s foratypicalwaterjet aresho wn inFigure4·1,below.

Stationnumber Location

0 Free Stream

..

InlelVelocity Protile 1 InletPoinl01 Tangency

2 Inlel Throat

3 PumpFace

4 Internet Pump Point

5 ^PumpExil

6 Nozzle

7 Vena Contracta

Figure ...·1.Moment umFluxl\lethod Sta tion Definiti on s

TheMomentum flux Method

4.2Sla/lon1

The fluidmomentumat theintake ismeasuredat Station I to accountforthefluidforced throughthejet units due to theforward motionofthe vessel , withoutpower.The velocity distributionof theflow is necessaryforcalculatingthe intake momentum flux.

Momentumand energyfluxes aredeterminedbyintegration over a properlydefined capturearea with ameasuredorcalculatedvelocityprofile. Withthe velocityprofile and flow ratekno....'n,the geometry of the capturearea mustbe determined.The locationof the inletsurvey plane (Station 1) and the resulting effect of the proximity ofthe inleton velocity measurementsis a concernanda potential source of errorin themomentumflux method.Inaddition to this, the shape andsizeof the capture area must beinvestigated.

Inaneffort to standardise testing practises andreducepotential biaserror,thelocation of reference stations hasundergone considerable refinementA major resultof this effort has been the development of Station la,locatedoneinlet width forw ard of StationI (ITIC,2002).Thewidth of the inletis definedas the maximumwidthbetweenportand starboard transversepoints of tangencyand Station Ia is thereforesubstituted in place of StationIforallmomentum flux calculations.

In theory,in order to determinethe shape of the capturearea, the locationofstreamlines enteringthewaterjetmustbeknown.Thisis difficultinpractice, sincethestreamlines separateneartheintake,as shown inFigur e 4-2.Whilesome streamlines continue along

The Momentum Flux Method

thehull. others enterthewaterjelunit.result ing ina somewhatcomplicatedcapturearea.

or volume.

Figure4-2.warerjeeSt reamlines

Variousstudies havebeen undertake ntodetermine theinfluence oftheshapeof the capture area on powerprediction and ithas been concluded by the 215 l l l T CWaterjet Comminee (Kruppa eral.,1996) that both power andthrustestimatesareinsensitive to capturearea andshape.

The recommendation of the 21^5lITTC\VaterjetCommittee is to use a rectangular capture area with a widthb., 30% widerthan the inlet width The inlet height is then obtained by computingtheheightrequiredto obtainthe given flowrute, by continuity.Figure 4·3 and Figure 4-4.show the capture area at stationla foratypical waterjet unit. The area begins at the hull surface. and as a result contains both a portion of the free stream. and the vertical height distributionassociated with the boundary layernear thehull.

The Momentum Flux Method

Wal er JelUn lt

Figu re 4·3.Capt u reAn aIcrTypicalwaterj et

Figu re44.CaptureAreaDimension s

TheMomentum fluxMethod

Although the three-dimensional behaviourof the flow is recognised bytheIITC Committee.the flow is assumed tobeconstant across the width of the inlet,due to a lack of knowledge and expertise inthis area.In order10 obtain a betterunderstanding of this, thelITerecommends a sensitivity studybe used to determine the effect of various intake shapes.

Concernshavealso been raisedinregard10the stateoftheintake openin gindetennining the velocityprofile.Ideallythe effectivewakeingestedbytheintake, i.e.the flowfield includingthesuctioneffects on the flow aboutthehull, shouldbemeasured. The effective wakeisdifficultto measureand itistherefore suggested bythe 23^rdITTC Special istCommitt ee ontheValidationofwaterjetTestProceduresthat theboundary layer velocity profile shouldbemeasur ed with closed intake openings(IITC,2002).

In order to calculatethe size ofthe intake area hiand Alare determined implicitly from

where,

~-volume flowrate ofthe watetjet

Al- intakeareaatstatio nla

Ul. (Z) -velocityprofileatstationla

Theassumptionoftwo-di mensionalflow yields the followingsimplifica tion [4.IJ

TheMomentum flux Method

[4.2)

[4.31

where,

w_ - width ofinlet

b_l-maximum widthofthe capture area hI-heightofthecapturearea

The momentumandenergyflux for Station1a arefunctionsofthe intakevelocityprofile, and therefore sensitive to the limitationsdescribed above.Further, frictionalforces along the hull reducethetotal headinsidethe boundarylayerandthelocalenergy velocity accountsforthisby consideri ng bothkineticandpotentialenergy(Kruppa et.al.,1996):

[4.4)

where,

VE-localenergyvelocity v -shipspeed

u-component ofvelocityinthedirection of motion

TheMome ntumAuxMethod

c,-

staticpressure coefficientgiven bythestatic pres sureatStationla(PI) and thestatic pressureinthe undisturbedflo w(Po):

Themomentumfluxat Stationlaisgiven by:

where,

The energyflux at StationIaisgivenby:

4.3 Intermediate Stations

(4.5]

[4.6]

[4.7]

[4.8J

Ingeneral .the momentum and energyflux canbe determinedateach of thelocations between Station la andStation7to accountfor the lossesalongthewaterjetunit.An accurat edescriptionof thevelocityprofLiesat theintermed iate stationscanbe difficult, especiallyneartheimpeller ,or when small modelscales areinvolved.Ithasbeen suggestedthat numericalsimulations usedinconjunctionwithlarge or full-scale model testsmay beusedto developagreater appreciation of thedynami cs of thewaterjet syste m (Thornhill,1999).

TheMomentumFluxMethod

Theenergyfluxat the intermediate stations is detennined byintegrating thelocalenergy velocityatstatio n

T.

and is given by:

EJ

==1·

^p,

fV~ ' dQJ

a,

Theenergy flux fortheundisturbed flow aheadofthevehicle.StationO.is:

4.4 Vena Contracta

[4.9]

[4.101

The crosssectional areaofthe waterjetisdecreasedatthenozzlein ordertomaximise thethrust. Streamlinesfrom the outletnozzlecontract afterthe orificetoaminimum valuewhenthey allbecomeparallel.at thispoint.theveloci ty andpressure are uniform across the jet.This converge nce is calledthevena C01l/racta.fromthe Latin'contracted vein'.Ifthe exitisnOIa perfectl y smooth contour,thediameterofthe jetwillbeless than the diameterof the hole (Munsonet al., 1998)anditisnecessarytoknow the amo unt of contractiontocalculate themomentum flux.At the venacontracta, thestatic pressure coefficientiszero andthe energy associated with thefluid iskinematic.

If the flow rate throughthewaterjetisknown.the momentumflux can be determinedas follows:

The MomentumFluxMethod

M,=-p,

J l4h

·dQ/+J(PJ-Po)·dA7

Q, •

[4.1\)

The pressurereduction(P7-Po)caused bytangen tialvelocitiesofthe jet(Ul0),is found from:

tt' u¹ p, -Po=--p.

r -;-dr

where,

A,-crosSsectional area of the jet

RJ - radius oflhe jet

The Energy Flux at Station 7 iscalculatedfrom :

E,

=-~,p ,

^{JV; ,·dQJ}

Q,

[4.12)

[4.131

The local energyvelocityatStation 7,VF:l'acco unts for thetangentialand rotational componentsofthejet flow:

4.5PropulsionPerformance Calculations

(4.14)

The values for themom entum energyfluxthroughout thewaterjetsystemcan be usedto detennin ethepropulsionperfonn ance characteristics of the waterjet .

Change or Momentu m Flux

The changeofmomentum,.:1.\1,canbewrittenas:

TheMomentum FluxMethod

[4. 15)

where.

(I -angle betweenthecentrelineof thejet andthehorizontalplane.

According totheKruppa etal.(1996).the change ofmomentumis equal to thesumof theforces on the pumpandtheinternalducnng,plus the changeofhullresistance dueto the action of the waterjet.This isalso equalto the effective modelresistanceminusthe tow-ropeforce.and the effective fullscale resistance is computedfrom:

[4.16)

where.

A- scalefactor Pm-fluiddensityatmodelscale Ps - fluiddensityat fullscale [fTect!nJetSystemPower

The effective jet systempower is computedfromthe IncreaseinenergybetweenStation 1a and Station 7:

[4 .17]

Elevation Power

The powernecessarytolift thewaterabovetheundisturbedwatersurface toaheighthJis computed from:

TheMomentum FluxMethod

{4.18l Internal Losses

The losscoefficientsfortheintake,~13,and diffuser,~51,are computedfrom:

[4.19J

r •E,-E,

!on £, ^[4.20]

Inmost situations, thevelocity distributionat Station 3will benon-uniformwilhlarge varia tion,and difficultto obtain.Ingenera l,itisdifficulttomeasurethevelocity distributionatanyposition insidethe waterjetsystemduringself-propulsiontests, and onemayconcludethat internalloss coefficientsmay be obtainedthroughseparatetest rigswith large scale factors,oran accuratenumericalmodel(Thornhill1999 ).

"Thepower Deededto overcomethe inlet and outletlossescan thenbedeterminedfrom:

[4.2 1J ErrecnvePumpPower

Theeffective pumppoweristhesum ofthepowercontributions described previously:

[4.22]

Ifthe increase of meantotalheadacross the pumpisexpressedas;

TheMomentumFluxMethod

Theeffectivepumppowercan alsobeexpressedas:

[4.24J

:'tlod elShaft Power

If the inflow non-uniformities are accountedforbythepumpinstallationefficiency,llinsh and the pumpefficiency.IIp.isknown, the powerneededtopropellhe modelcanbe expressedas:

[4.25)

Themodel shaft powercanalsobedeterminedfromtorquemeasurements.IftherOMis notequal to:

2·Jr ·Q·n

thentheinternalloss coefficients orefficiencyvaluesshould be reconsidered.

4.6Predic tedFullScalePower

[4.26J

Inorder todeterminethe full-scalepowerofthewaterje tsystem.the volumeflowrate, size ofintakearea. and energy velocities at StationIandStation 7must beknown.Scale effectsoftheboundarylayer profile do notpermitadirectconversion ofthesequantities, andilisnecessarytofollowtheprocedure outlinebyKruppaet al.(1996).

TheMomentum fluxMethod

The full-scaleboundary layerthicknessandvelocity profileare predicted accordingto boundary layertheoryandthehullroughnessis considered. The static pressure coefficientis considered tobethe same for themodel and prototype.

Momentumtheoremisusedto computethevalues ofQJ,MI,hl.andM7usingthe full-scaleveloc ityprofileandmaintaining thechangeinmomentumflux:

Full-scalevalues ofE1andE7,~IlS. ~7S.

nes,

and 11...sare estimated [4.27J

Ifa large.or full-scale modelisused10determin ethesequantities,theresultscanbe converted withsome confidence (Thornhill,1999).

Thefull-scaleeffectivepump powercan thenbedetermined as describedinsections 0 throughO.Thepumpshaftpower isthen:

4.7Summary

Themomentumflux method , initially proposedin the "FinalReportand [4.28)

Recomm endations tothe21¹¹IITC:WaterjetsGroup"(Kruppaet al.,1996 ),has been regardedasa stepin therightdirectionasfaras waterjet testing is concerned.The methodhas manyadvantages overconventiona ltesting methodswhenappliedto

The Momentum AuxMethod

waterjets.andhas undergone significantrefinemenu.particularly those ofthe 23^rd

rrrc

(2002).

Themethod relies heavily onan accuratedescriptionof the flow rate. This can be accomplished with a reliable flowmeter ,butscale model warerjets donotoften allow spaceforsuch a device.tosay nothingof thesettlinglength requiredprior tothe meter inlet.In addition to this. assumptions of theflow behaviour necessary to simplifythe analysis at keylocations of the waterjet system,coupledwithIhe estimation ofmodel efficiencies leaves room for improvement inthe method.Large-scalereodel tesung and/or numerical simulation may improveconfide nceinfull-scale predictions.

The workdiscussed in thisthesis has beenundertaken arfull scale. andthe momentum flux calculations benefitfromnumericalanalysis.Thenumericaldata allowsintegration over thousandsofdata pointswhichwould provenear impossible to mea sure experimentally.

Instrumema tion

5 Instrumenta tion

5.1 Hot·Wire Anemometry

Theoriginsof practicalhot-wireancmometrycanbeattributedtheworkof Ziegler (1934).He developedaconstant temperatureanemometertomeasureflowfluctuations using afeedbackamplifier thatmaintainedconstant tem perature across aheated wire.

Hot wire anemometrymakes use ofthethennallossof heated resistancesensorsin order to determine velocityfluctuations.Asensoris placed ina gaseous flow,and the convective heattransferfrom aheated wire is measured.The magnitudeofthe convectionis influenced by changesintemperature , pressure, and velocity andthe sensor willimmediatelydetectany change inthe fluidcondition that affects the heattransfer fromtheheatedelement.If onlythevelocityoftheflowchanges,ortheinfluenceof otherchanging parametersiseliminatedby suitablecircuitry, then the instantaneousheat loss of the sensoris adirect measurementofthefluid velocityat thatpoint intime.

Hot wire anemometry canbedividedinto the followingflowregimes:

Subsonic incompressibleflow

Subsoniccompressible,transonic,andlow supersonic flows Highsupersonic andhypersonic flows

Theseflowregimescanbe fun herseparated into continuum flow, slip flow,andfree molecu larflow.For thepurposes ofthis discussion, onlysubsonicincompressible continuumflow willbeconsidered.

Instrumentation

In subsonicincompressibl e flow,the heattransfer froma wire is afunct ion ofmass flow, totaltemperature,andwiretempera ture .Forconstantdensity,the massflow variat ions depe ndonlyonvelocityfluctuati ons. In mostcases. the meanfreepathof the panicles is muchless manthe diameter of the wire sensor,and the continuummodel isvalid;

conventio nalheat transfermethodstherefore apply.

Neglecting conduct ionand radiation,theheat balan ceforan electricallyheatedwire is given by (Stainbacket al.,1997):

Heat Storea»ElectricalPowerIn-AerodynamicHeatTransferOut

~T", =P -Q

where:

c. -specific heatof wire T..- cremperature ofwire

T. .-adiabatic wall temperature lc-current

Rw•resistanceof wire L-characteris tic length d.. -diameter of wire

[5.1)

[5.2)

Instrumentation

h - coefficientofheat transfer

Selling thehealstorageterm10 zero resultsin thefoll owing:

[5.31

Thereareseveralcircuitsthatmaybeusedto measure thethermallossacross a sensor.

Usingrelati vely simplecompensation circuitry,theConstantTem peratureAnemometer

(cr

A) is capableofmeasuring veryrapidvelocity fluctuations.Theinstrument suppliesa sensorheating currentthat varieswith tbefluid velocityto maintainconstantsensor resistance and constantsensortemperature .

In it' s simplest form,the

cr

A consistsofaWheatstonebridgecircuitanda servo amplifier.

Ftgure5-1.Wheatst oneBridge

TheprobeandIWOtopresistances occupy theactivebridge arm,whilethepassivebridge armcomprises theothertopresistance, the comparisonresistor, andvarious