RANS Prop

RANS Computation of Propell er Tip Vort ex Flow for Stead y and Unst ead y Ca ses

by

@lVlcI . Shafiul Azam Mintu

A thesis submit te d to the Schoolof Graduate Studies in part ial fulfillment of the requirements for the degree of

Mast cr ofE ngineering

Faculty of Engineering and Applied Science Memorial University of Newfoundland

July 2011

Abstract

CFDsimulationswereconductedfordifferent marine prop ellers at steady and unsteadyflow condit ionsusing a commercial RANS solver ANSYS® CFX®.For stea dysimulation,a spiral-likedomain alignedwit hthevortex core was generate d wit hstructuredgrids. The simulat ionwas valida tedwith theDavid TaylorModel Basin DTl\IB 5168 propeller modelatopen-watercondition.

Variouseddy viscosityturbulencemodelsand Reynolds-st ressmodels were employed inthecomputat ions. The effect of the turbulencemodeling onthe solut ion wasinvestigated.The bladesurface pressur e andthe propeller performancewereals0 compute d. The simulation dat a werecompa redwiththeexperimentalda ta.

Theunsteady simulationwas conductedforpropelleratinclined flow condition.

A single domain was generatedwithstructuredgrids. Asimulationtechniquefor inclined flow conditionwas presented. The simula tionwas validated with themodel testdat a of DTMB 4718at design condit ion. Afully implicit coupledsolverwasused.

A segregate dsolverwit hIncompleteLower Upper(ILU) factorisationtechnique was employed inthe simulation. Algebraic Multi-grid(MG) mod el wasusedto accelerate theconvergence.Advect iontermswere descretiscd byhigh resolution schemewhile theviscousterms weretrea tedby employingcentraldifferencescheme.Thetransient termsweredescretisedwith Second OrderBackwardEuler scheme. The ShearStress Turbulencemodelwas employed inthecomputati on.

The effectofgridsensitiv ity anddomainsizewere investigate d.Theperiodic loadings onthepressureandsuct ionsidesofthe blades were comparedwiththe

Reasonable agreementwith the computedamplit udeof the pressur evari ati ons wasfoun d.Thepredictions ofthephase of thepressure variat ions

workisthefirst attemptof theCFOsimulationfor unsteadyprop eller flowinvestiga tion using a spiral like computationalgrid. Fur therimprovements and exteIllsionsoftl lliswo>rkcanbcmade.Suggest ionsare maderegardingfutur e workon

Acknow ledgem ents

I

would like to express my sincere apprecia t ion to

mythesis superv isor Dr.Wei Qiu,He gave me complete freedom to tryideas and

assisted me thro ughou t my

graduate

st udiesat

Memorial.

Iwould like to express

appreciat ion tomy manycolleaguesat the Advanced Marin e

IIyd rod ynamicsLaborat ory fortheir sincerc coop era t ion .

Special thanks to

BruceQuint on , aPhD

stude nt at Memorial

forhis sincere

cooperatio n and help in using STePS2's cluste r server.

Thank s arc

extende d to STePS2 for permit t ing me to

usetheirclust er server. Without this

the simulat ion experime nt would run forever.

Igratefully

acknowledge the suppo rt from t

he

Natura l Sciences and Engineerin g

Resear ch Counc il

of Canada (NSERC),

DefenceResear ch andDevelopm ent

Canada- At la ntic (DRDC- Atlan t ic),

Oceanic Consulti ng

Corporat ion and Memorial

University ofNewfoundland

(MUN).

Finally,I

would like to tha nk my

familyfor

all their

love and

encouragement.

Theyhave sacrificed

a

lot due to my

research abroa d . Especially,I

wouldliketo give my special

than ks to

my wife Moon . She helpedme to

concentrate on comp let ing

this thesis and supporte d me

ment allyduring the

course of

this work. With ou ther help and

encouragement, this st udy would not

havebeen completed.

Contents

Acknowled gemen ts Lis tof Figures

1.1BackgroundandMot ivation.. 1.2Objective ofthisResearch. 1.3Organi zation ofThcsis.

2Revi ewof Rel at edWork SteadyCases 2.1.1Exp eriment al Work . 2.1.2Numerical Work.. Unstea dyCases .. 2.2.1Experiment alWor k. 2.2.2 Numerica lWork.

3Propell erGeometr y and Grid Generatio n 3.1Propeller Ccomctry.

CoordinateSyst em I31adeGeometry.

I31adeSectionGeometr y . Hub Geometry . GridGeneration i\Iet hod.

GridGenera tionApproach GridGenerationon I31ad eSurface .

EllipticSmoothingTechnique ComputationalDomain for Unst ead ySimulat ion. 3.3.1 i\Iesh Transformation

4Comp utat.iona lMctho d GoverningEquations Discretisationof GoverningEquations.

Finit e Volume l\let hod.

SolutionStrat egyof theDiscretisedEqua tions Turbu lenceModeling

4.4.1 EddyViscosit yTurbulencei\Iodels 4.4.2 ReynoldsStressTurbulence(RSl\I) Models

Stead y Case . 5.1.1Simulati onTechnique.

ConvergenceTests Effect of TurbulenceMod eling ..

Effect of Bound ar y Condit ion.. Effect of Timestep Size.. UnsteadyCase ..

Valid ati on ofthe Unstea dy Dom ain. Grid Quality

Bound ar y Cond itio ns . Unstea dy SimulationTechniqu e. ConvergenceTests .

6Co n cl us io ns and Recommendations Stead yCase . 6.2Unsteady Case

I3ibliogr aphy

Appendices A Geometryof DTM13 4718 13 Addition alFigures

List of Figures

3-1Coordina te systemofprop eller. 3-2Cylindricalcoord inatesystem. 3-3 Definitionof rake

3-6Spirallike computationaldomain. 3-7DifferencebetweenH-typ e and O-typ e grid 3-8Grid generationon bounda ries 3-9Computat ional domain for unst ead y simulat ion.

5- 1Primary andsecondarycoordi na tesyste m . 5-2 Computa t ional domainofDTlv1l35 168 5-3Effectofgrid resolutiononcomputingtime .

5-4 Vx ,Vt,V,. across thetipvortexcenteratx/R=0.2386computed witheddyviscositytur bulencemodels

5-5 V:r,Vt,Vracrossthe tipvortexcenterat»[H=0.2386computed wit h Reynolds st ress mod els

5-6Comparison ofVx,Vt ,Vr acrossthe tipvortexcenter at:r/R=0.2386 with Reynolds stressmodels andeddyviscositymodels.. 5-7 Axialvelocity V,at:r/R=0.2386

5-8Tangentialvelocity

y,

atl;/R=0.2386 RadialvelocityV, atx/R=(J.2386

5-10Surface pressure coefficienton the

pressure side

5-11SlIrface pressurecoefficientont hes uctionside.

5-12 Effect of bound ary conditions

onVx ,V t,Vr across the

tip vortex

centeratx/R= 0.2386

5-13ConvergenceplotofJ(,andJ(qwithauto timest ep 5-14Convergence plotofJ(,andJ(qwith physicaltimcstep 5-15

Blade surface of DTMB 4718

5-16Computa tional domain

ofDT lvIB 4718

. 5-17Computa tionalgridsof DTi\lB

4718

5-18 Verifica tion of unst eady domainonVx ,Vt,Vr across thetipvortex centeratx/R=

0.2386

.

5-19y+distributionover the

blade surface

(pressureside)onDTMB4718

75

5-20Schema tic diagram of unsteady simulationset-up.

5-21Coordinatesofthe

unsteady simulat ion

set-lip. 5-22 Effect of gridsizeon

the comp uted

Cpatr= 0.5R 5-23 Effect

of domain size

onthecomputedCp a t r = 0 . 5R.

5-24 Convergence plotsofmass and momentum residuals 5-25Convergenceof propellertbrnstcoefficientwitbtime itera tions.

5-26 Convergence

of

propeller pressure

coefficientwith

time iterati ons .

5-27Computed firstharmonic

amplitud

e and phase of

the pressure

coefficienton DTlvIB4718 with .J=

0.751at

0.5R. 5-28 Computed first

harmonic amplitude and phase of the pressure

5-29Computedfirst harm onic amplitudeand phase of

the pressure

5-30 Computed thrust variation

with the angular position of blade

B-1 Additionalconto ur plots of axial

velocityVx

at

:f/R=

0.2386 .

B-2 Additional conto ur plots of tan gential velocity V,at

x/R= 0.2386

13-3Additional conto ur plots of radialvelocity V, at

x]

R=0.2386 .

List of Tables

PrincipaI Char acteristi cs ofDTi\II3 5168 SummaryofGrids

ComparisonofJ(,andJ(q. Principal Cha rac te rist ics ofDTIIII3 4718 NewPrincipal Characte rist icsof DTMB4718 A. I Modi fiedPropellerGeometr y of DTMB 4718 . A.2Blade Section Geometr y:Brocket .

A.3Blade Sectio n Geometr y, NACA G6(DT i\II3 I1Iod.),a=0.8 Meanlin e .

Nomenclature s

Non-di mensional distan ce oftheoutletboundary Chordlengthof aerofoil

Diameterofpropeller

Non-dimensionaldistanceof theoutletboundary htot Totalent halpy

Indicesofgridpoints on theleadin gedge Indicesofgrid pointsonthe trailingedge Tota lrake of propeller

Numhe rofgridpointsinthe stre amwise direction Number ofg rid points in t he hlade- to-blade direction

Numberofgrid points onthehounda ry in the spa nwisedirect ion Number ofgrid points onthe blade sur facein the spanwisedirection Turbulentkinetic energy

K, Prop ellerthrust coefficient,T/(pn^2UI) J(q Propeller torque coeflicient ,Q/(pn²D'\ ) LO.7R Chord lengthatO.7R

Rot at ionalspeedofpropeller Totaltorq ueon prop eller Radi aldist anceof propeller Hllhradiusofpropeller Radiusofpropeller

SM SE Energy source

Tot althrustonprop eller

Vx

Axial velocit ies in the tangent ial directi on across the

tip

vortex centr e

Tangen tialvelocitiesin the tangentialdirectionacrossthetipvortex

cent re v,.

Radi alvelocitiesin the tangent ialdirecti on across the tip

vortexcent re

II-rol Tot al velocity

y+ Non-dimensionalwalldist ance l3ladeindexangle

Densityof wat er ,9'J7kg/m-³ Dyn ami c viscosity

ofwater, 8.89x

1O-^4kgm-¹s-¹

Skew angle

of propellerblade

Any variableoffluid prop er t ies

Tensorproduct oftwo vecto rs Angularvelocit y Eddy viscosity

or turbul ent viscosity

Turbul

ent

dissipationrate Turbul

ent

frequ ency

Chapter 1 Introduction

1.1 Back gr ound and Motiva t io n

A marin e prop eller operat ing at the stern, within the wake of the ship, complicates the propeller hyd rod yn am ics. The blade

inflow

varies significant ly as the propeller rota t es,produ cing unst ead y for ces. Theseforcesresul t in additional unst ead y

loads

on the shaft and cause vibra tion on the hull.

\Vhen

the loadin g on the blades increases, the prop eller experienccs anot her

hydrodynamic

phenomenon called

cavitation,which degradespropellerperformance, erodes blade surfaces, produces

noise, and causes vibrat ion on the ship hull. Apart from the vibra t ion issues, the knowledge of the unst ead y load distr ibut ion is essent ial for the blnde fati gue analys is.

It is also

importa nt for proper shaft and stern bearin g design.

Addition al complexity arises from the propeller opera tin g wit h a shaft inclination.

This sha ft

inclinat iongenera tes unst ead yloadsall

the blades ur faces and consequent ly

creates vibratio n and cavitat ion, On the other

hand,

with the increasin g demand for

heavily load ed propellers, the occurr ence ofcavita t ion is linavoid able. Therefore, the

accurate predicti on ofcavitat ion is import ant.

The cavita tion phenomena

can be

revealed by

experimental,analyticalornumerical

techniqu es. Exp erimen tal methods provide valnable insights

into the cavitat ion physics

in various predetermi ned cond itions, bnti t is expensive tomanufactm ethe models and

theyare

vulnerable to slight flow condit ion

changes inside cavita tion tuimels(Rheeand Kout savdis,

2003).They alsoseriously

suffer fromscaleeffectsin viscousflowphenomena. Cur rentanalyt icalsolutionscan

not pred ict period icblade loads accurate ly, They underp redict the periodic propeller blade loads in

inclined

flow and

givepoor

prediction of time

3\'erage

propeller blade loads at substantia lly

off-designconditions(Jessup,

1982). On

theother hand ,

numerical meth ods can provide insight

into the

local flow characteristics

andareable to givesufficient information. The costand time

required forthe compnt ati on arc much lower than for

model tests and arc suita ble

for flow analysis.

Thenumericalmethods

based on potent ial theory arc widely

used inpropeller flow analysis.The potential flowmethods

arc

based

on the assum ptions of inviscid fluid aIHI irrotational motion. Using

thesemeth ods,prop eller performance

at

design conditions

can be predicted quite accurat ely,

forexample, Kinnas andHsin(1992),

Gaggero et

al.(2010).However,the

off-design propeller flow

phenomena,

which are domina ted by viscous effects,

cannot bepredicted accurately.Moreover, the

potentialflowmethods arc not able to predict the wake field wit h

a sufficientlevel of'accuracy, To take into accountthehighly

viscous effect

of the

propeller flow

field, it

is proven that the numerical

simulat ion based ontheReynolds-averaged Navier-Sto kes(fiANS) method is effective.The

unsteady fiANS meth od repr esents

thestate-o

f-t he-art in

computat ional prediction oftheviscous flow around propellers (Chenand Stern,1999).

The accurate

simulationof

unsteady propeller flows is a challenging

task due to

its complex physics and

geomet ry.

For example,

general

hub

shapescan produ ce

a

potentialflowperturbation and

will influence thc inflow (Jessup,

1989). Thc boundarylayer candevelopon

the blade surfaces and can cause Iargc gradicnts of

the vclocity field

(Qiu ct al., 2lllO).

Thetrailing edge flows

may

not maintai n thc const antpressure at thetrailingedge.Inthe caseof

tip vortex flows,

radialflows

can separa te at thc t ip thro

ugh the

adverse

pressure gradients

(Jess

up,1989).Tho

complexity of the

flowfieldand

gcomctry require special

techniques to

genera t e suitable grids for thccomputat

ioua l domain.

Anothercomplexity for thepropeller flow simulat ionarises from turbulcnccm odcling.

A

screw prop eller generates

non-equil ibrium regionsin

the boundary

layer with adversepressure gradients,andseparatio nofflowmay occur

(Krasilnikov

et

al.,

2ll09).Itindu

ccshighlyrot atin gfl owfindcnt ails a tip vortex, which causestur bulellce

ill the tip

vortcx rcgion.

ThelowReynoldsnumber (Re)flowsoccurat thenear wall

regions, while

highReflowsdevelop atthefar-field.Moreover,thepropellers operatingunder

off-design conditions develop strong H ow sepa ra t ion on the blades,

The simulat ion

of propeller

flow should

take

intoaccounttheseflowphenomena , Limitedst udiesweredoneto

find the effects of

turbulencemodeling011

thc prope llcr

tipvortcx compu t a tions.Thefluctuat ingflows should betreat

ed

with the appropriate

1.2 Obje cti ve of this Rese arch

Theobjectiv esof this resear ch workweretopredict the

viscous

flowspas tpropellers based on

the

RANS solutions.The commercial RANS solverANSYS-CFXWiU;used for this simulat ion. Bothunifor m an d inclinedinflow condit ionswere examined.A

gridgenera t ion program

f'ropGGM,develop edbyQiucta1.(2003) ,was employed

to

gcncratcas piral likccomplltationai domainforthestca dyand ullstca dys imulatiolls.

For the steady

simulat ion,

the propeller tip vortex flow was comp uted . The effect of turbu

lence

mod eling on the tip vortex computa t ion was performed to investigat e if the

Reynolds st ress mod elimproves

the tip vort ex comput at ions. For the unst eady

case,

the ANSYSCFX is used to predict the unst ead y blade load s.

1.3 Organizati on of T hes is

A review of the past works is summa rized in Chapt

er

2. Chapt er 3 describ es the process to develop the spira l like computa t ional domain for the numerical simulation. Chapt er 4covers th

e governingequa t ions

describin g the fluid flow and

their descreti sat on

meth ods .

Having explained the

analysis process in

Cha pte r 4, Cha pte r 5 describ es

the resear ch findings. The last chapte r

gives the conclusions.

Suggesti ons for future workare also pres ented .

Chapter 2

Review of Related Work

This chap ter focuses outheliterature review of workoust ead y and unst ead ypropeller bladeflows.A bricfsurnrnary ofso rneofthe maj or experimenta land numericalwork ispresent ed.The scope ofpredictingpropellerloads,forbothsteadyandunst ead y cases, using exist ing expe rimentaland numericalmethods are alsodiscusscd

2.1 Stead y Cas es 2.1. 1 Ex pe rime nta l Wo rk

Advancedflow visualizationaIHInon-intrusivemea..suremcnttechniquesdistinctl y impro vethe experimentalinvestigat ionsof flow fieldstudieson propellers.Thefirst LaserOoppler Velocirneter(LOV)measure mentsofmarinepropeller flowwerernad e byMin(1978)atMITandfur t heriusightiutothepropeller wakewasprovided . The viscouswakeswereidentified.Anexte nsionofMin's st udywas carriedoutby Kobayashi(1981)tomeasuretheviscouswake downstr eamofthe propeller with some det ails. Cendese(1985)and I3illet(1987) conducted LOV measur ements abo ut rnarine propellerswith theinclusion of turbulentmeasur ement s.Allof these experiment al investigationshave onlyidentifiedthewake as complicat ionsintheflow withhigh turbulence,butdetailedwakemeasurem entshad not beenmade.

Jessup (1998) used LOV

systemsto

obtain detailed velocity measur ements of

a

propeller wake

at downstr eamlocations.

OiFelice et

al.(2004) demonst rat ed

the

capabilityof

Par ticle Image Velocimetr y (P IV) for identifying the

flowst ruct ures in the

wake ofa propeller.

Inspiteofthesuccess in measurement. of propeller

flow feat.ures,t.h epressure field stillremainsuncleardueto thelimitat.ions of measur ement.

techniques. It. is desira ble to provide the detailed pressur e field by numerical

2.1.2 Numerical Work

Panel meth ods have long been applied

forthesolut ionof propeller

design and

analysisof flow

problems.

lie" and Valarezo (1985) made

the first. atte mpt.t.o analyse stea dy flow around

a

marin epropeller using 30 Bound ary Element Meth od . The classical Hess and Smit.h formula tion had been

used in this

paper. Kerwin et.al.

(1987)alsoapplied the

panel method t.o investiga te marin epropellers perform ance.

In recent years,numerous

research ers

have

used

RANSto simulatethe rotatin g

blade

cases,

For exampl e,Abdel-Maksoud et. al. (2004)

analysedt.hee lfect.oft. he hub capsha peon propeller

performan ce using commercial RANS code,

CFX-TASCflow.

Abdel-Maksoud and

Heinke (2002) predicted the

velocit.y distribution in

thegap

region of

a

elud

ed

propeller using the

samecode. Simonsen andStern(2005) computed t.helmll-ruelder-pr opellerint. eract.ion

by coupling

the

RANSand potential

codes. Rheeanel Joshi (2003) presented the

computa tions of marine

propeller flow

using the commercial RANS code, FLUENT.

However,

numerical

st udies on the tip vortex flow

of

open marin e

propellers are somehow limited except. for some

earlier st udies,

for example, Hsiao and Pauley (1999) and

Chenand Stern (1998). Hsiao and Pauley (1999)appliedaone-equat ionturbulence model on fine grids

to compnte

the tip

vortex flows.

The tip

vort.ex was

bett er predicted at. the locati oncloser t.o

tho prop eller while the wake

WI"

bett er predict ed at t.he fa r field location .

It.was concludedthat the eddy viscosity computed from

the Baldwin-Barth one-equation

turbulenc emodelmightbe toolargewithinthetipvortex andled toanoverl y diffusiveanddissipat ivetipvortex

Inthisthesis ,thesteady-sta t etipvor tex flow genera tedbya mari nepropellerwas comp uted using theRANSsolver ANSYSCFX.An investi gationwasdoneto show the effect of turbulencemodeling on the vortexflowcomputation.Vari ouseddy viscosit yandReynolds stress turbulencemodelswere employed intheinvesti gation .

2.2 Unst eady Cas es 2.2.1 Exp erimental Work

Theinvestigationonunstead yblade forcesrequiresadvanced ex perimental and numerical techniques. The totalunsteady and time-uvera gebladeloads were evalua t edbyBoswellet al.(1976and1978) andJcssupetal.(1977).However,these result s areuna b letorevealsignifica ntinforma tionon thedistributionof theperiod ic loadings over theblad eand underp redi ct time- average bladeloadsatsu bs ta nt ially off-design condit ions . Anexpe riment wasunder t akenbyJessup(1982) toobtain accurateandreliable measurements ofthepressuredistributionin uniformand inclinedflow.Hemeasur edunst ead ypressur edistributions on theDT l\lI34679 and DTI\!I34718mod elsin obliquenow. Single blad eforcesfor propellerDTMI3 4661in inclinedflows of10,20,and30degr eeswere alsoreportedbyBoswellet al.( 1981 and1984).To acquiresufficientunderst andingofunsteady propellerhydrodynamics, Jessup(1990) didanexperimentwith the symmet ric 3-bladed propellerDTl\l I3 4119, whichwas opera ted behind harmonic wakescreens with3,6,9and 12cyc1esper

2.2.2 Numerical Work

Pa nelmethodshavebeen em ployedto com putestea dyanduns tead yflowsaround prop eller s.Ker win and Lee (1978) proposed aliftin g surface method.Liuand I30se (1998)implementedinflow wa keand hyp erb oloidpane lalgorit h mtodealwiththe obliqueflow for highly skewedpropellers.Hsin(1990)andGaggero (2010)solvedthe unst ead y cav itatingflowby apot ent ialbou ndaryelement method .Polit is (2004) appliedtheboundar y eleme nt meth odtopredictunst eadytrailing vor texsheets ema nat ingfromeachblade. However,thepan elmeth od s arelimitedtoinviscidfluids only.Whenvisco usflowbecom esimport ant ,for exam ple,for tipvortexpred ict ions andleading and hub vorti cespredict ion s,thepanelmeth od s areunable topredict accurate ly.Toovercome theselimita tions,viscoussolvers mustbeused .

Most oftheresear cher shave simulate d theMITFFX (Massac huse ttsInstitu te of TechnologyFlappingFoil Exp erim ent)to revealthephysics ofunsteadyblade flow.For exam ple,Rhee andKoutsavd is (2003)presenteda twodimensio nal (20) simulationof unstead yflow aro undthe bla de sectio nembedded ina travelling wavefield.Intheirwork,an unstructureddy nami cmesh ingtechniquewasused.

Pa t er sonandStern(1997)validatedtheirtimeaccuratesolutionsoftheRANS equationsby simu la t ingtheNlIT FFX .Mostmarineprop eller s operate inahighly threedimension al andviscousinflow conditio n.Inthe case ofa prop ellerwithan inclin ed shaft,theprop eller opera tesinaprima rypot enti alflow field,butthesha ft inclinationcausesunst ead yloads on theblad es (Jessup, 1989 ).Thesepra cti cal configurat ionscausecomplex unsteadyffowphon om cna andneedto beaddressed

Gaggeroet al,(2010)usedRA NS solve rStarC C l\1+toinvestigate theobliqueflow phenom eno non modelprop eller OTl\lI34679 byusing slid ing meshtechnique.This techn iquetakes intoaccount therela ti vemotion of thepropellerblade aro undan inclinedaxisinside thefixeddom aininwh ichthepropellerinflowis gene ra te d

This approach isknownasmixingplan e approac h(Snchez-Ca jaetaI.,2008) .The advantageofthesliding meshtechniqu eis that it allowsfor timeaceurate simulation.

In their work,thewhole domain was generated byunstru ctur ed grids. An implicit unst ead y solut ionapproac h withalgebra ic multi-gridmodelwere employed.Superior capa bilitiesoftheHA:\S solverwere foun d over the poten tial solvel'inpred ictingthe unst ead y pressuredist ribu t ions and forcesat off-desig ncondit ions,alt houghthey alsodescribed theHANS solveras an immat ure toolforthesolut ionofthesteady and unsteadycavitati ng pro blem.l\rasilnikoveta l.( 2009)alsoemployedt hes liding mesh technique inanothe rCF Dcode,FLUENT. Intheirsimulat ion,the widelyused modelprop ellerDT:O-IB4679Willisimulate"intwostages. At thefirststage,the solut ionwasdoneby using MovingReferenceFram e (:O-IHF).This solut ionwas then used asiniti alcondit ionsfor timedepend en t simulat ion.Temp oraldiscreti zation wasdonebyfirst orde raccura te backwarddifferencediscret isation technique.The comput at iona l domainwas generatedby2.86millionunstructuredcells. From theircomputation,it wasfoundthattheHANS methodpredicted bett er inheavier loadingeonditions.Under lighter(.1=1.078) loading,wheretheinfluence ofviscosity islarger,theHANScalculat ionoverpre dictcd thepressure on the suct ionside. The samecase was alsoreportedbyLcras andHally(2010),butforheavier loadin g (.1=0.719).As explained byLems andHally (20 10), it maybeduetothe cavitat ion at thetip whichisignoredinthecalculation. They alsodidaprelimin ar y st udy with two modelprop ellers,DTi\1B4679 and DT:O-IB4718,to invest igateinclined flowphenomenonusing thecommercia lHANSsolver ANSYS CF X.Twodimensional structuredgridswere used on theblades,andtherema iningregions(hub andregions associated withthe blade)were made byunst ructur ed grids. The data for analysis were sampl ed on each blade O\'er ollly olle-t hird of a compl ete revolutioll andthen added togeth er to get theequivalentpressure on a singlebladeoverafull revolu tion . Good agreementswit htheexperime nta l da ta were found withthe meas ured average pressu res.The computedamp lit udesofthe pressur evari a tions werealso in

goodagreemcnt. However, thephase of the pressure variat ionswerepoorlypredicted.

Inthisstudy, the numerica lsimulatio nwas conducte dtopredict thcflow arounda propeller operatingwitha shaft inclination of7.5°.The commercia l RANS solver ANSYS CFXwasused.Asingle domain was genera tedwithstruct uredgrids. The simulat ionwas validate dwiththemodeltestdata ofDTlvlI34718 at design condition.

Theperioclicloaclings on thefaceancl back oftbe blade surfacewere computedand validate d wit h theexperimentaldat a.

Chapter 3

Propeller Geometry and Grid Generation

Thenumericalsolution ofBANS equations require'Sdiscreti zationof the fieldof interestintoa collectionofpointsorelementalvolumes. The efficiencyofaRANS solverlargelyd epcndson thequalit y ofgrid.Inthis chapt er ,the geometryofpropeller andthe gridgcnerat ionofthecomputationa ldomain arediscussed.

3.1 Propeller Geometry

Thc grid was genera te dbytheprogramPropGGM(Qiuet al.,2003).Theinput ofthisprogramonlytakes the 3-Dimensiona lcoordina tesoftheblade surfaceand docsnotinclude sectional pitch, chordlength,pitchdiameterratio , skewand rake distributions.A Fortran programwasthen developedto genera tetheCartes ian coord inatesoftheblad ebytakingthc basicprop eller geomet ry. Thc mathcmat ical formulationoftheprogram isdescrib edbelow.

Figur e 3-1:Coordinatesystem of propeller

3.1.1 Coordinate System

Acartesia ncoord inatesystem,Oxyz,fixed on thepropeller is applied.Thepositive z-axisdefined asdownstreamdirection and y-axislocated at any desiredangular orientat ionrelativeto thekeyblad e. The z-coo rd inateis deter mined by the right- ha ndedsystem(F igme3-1)

Acylind rica lcoordina tesystem is definedasfollows. Theang ula rcoordina te

e

is measured clockwise from the y-axiswhenviewedinthedirect ionof positi ve z-axis.

The rad ialandangularcoord inatesaregiven by

7'=

.J1li+ii

(3.1)

Figur e 3-2: Cylind ricalcoord inatesyste m

O=t an-1( y/ z)

3.1.2 Blade Geometry

A projected view ofabladefrorn upstreamis shown inFigur e 3-2.IntheFigure,ru isthehubradius,eistheskewHn glcmcasun..edfromthez-axis atradiusr.

The skewangle,IIm(r ),isdefinedastheangularcoordinateof thernid-chordline measuredfromthey-axis at radiusr wherethey-axis is alongthepropellerreference line.Asshownin Figure3-3(Carlton,I!J!J~),therakeof thepropelleris dividedint o twocomponents:genera tor line rake(ic)andskew ind ucedrake(i,) .Thetot alrake of the section withrespecttodirectrix(iT)is givenby

iT(r )=ic(r)+i,(r)

fExpandedview)

Figur e3-3:Definitionof rake

Thegcnera t or line rake,ic,is simply the x-coo rd ina te of the mid-chord line as shown intheFigur e3-2.The skew inducedrnke L;alsomeasuredin the xdirecti on,isthe com po nent ofhelicaldist an cearound the cylinder from thernid-chordpointof the sect ionto theproj ecti on ofthe directri x whenviewed normallytothe ya-pla nc. The skewind ucedrakeis givcnby as

i, = re,tan(e",)

Where,

e. ,

sectionskewangleandO"tis the nose-tailpitch anglc.

A hladeind ex angle,8k,is defined togencra lize theresultsto allblad es ot her than

thekeyblade:

J.=

21r(~(- I), k=

1, 2, ..,J( (3.2)

where[(isthennmberof blad es andkis the index of anyblade. Thekeyblad eis definedby th ek=I.The coordinatesofa point on thepressure andsuct ionsurface ofascc t iononthekt h blade can be writ tenas

:1',."=.em

+

c(s -

~) sin o!>

- I, .p cos o!> (3.3)

O,.P=Om

+

c(s -

~)~ + I,.,,~ + J.

(3.4)

Y.•."=-rs in O.•.,, (3.5)

z.,.,,=rcos O.,./, (3.6)

wherethesubscriptss andndcnotc thesuctionsideandthepressure side sur faces, respectively ;ls ,pisthesectionsurface andismeasuredina cylindricalsurfucc of rndiusrinadirection normalto the helical coordina te.

3.1.3 Bla de Scct ion Geom etry

TheNational AdvisoryCommit teeforAeronauti cs (i'\ACA)intheUSA, nowknown as:\ASA,de velopeda systematicseriesofaerofoil geometries.Someof these aerofoil shapes havebeen ado pted forthedesignofmarineprop ellers. Typical sect ion nsed for ship prop elleris;\ACA66 series with themeanline " =0.8.The sect ion geomet ry is given in theappendix.Themeanline orcamber lineis the locus of the mid-point sbet ween the pressur e side(upper]andsuct ionside(lower) when measur edperpendi cular tothe camber line, as shownin Figure3-4(Carlto n,1994).

Thetwo edgepoints ofthiscamber line arc known asleading andtrailingedges Thedist an ce between thesetwopointswhenmeasur ed alongthe

Figur c 3-4:Definition of anacrofoilscct ion

chordlineis thechord length,c, of thesection. The aerofoilthickness,t,isthe distan cebetween the upper and lower surfacesof the sectio n, usuallymeasured pcrp cndicularl y to th c chord linc.

Theupper and lowersu rfacesaremeasu redperp endicula rlyfrom thocambcrI ine. A pointPuon thc upper surfaceandapointPLon the lowersurface oft hc acrofoil,as shownin Figur c 3-5(Carlt on,1994),arc dcfincdby

Xu=xc -Ytsint/J Yu=Yc +YtCOS ,p .TL=.Tc+Ytsint/J YL=Yc-YtCOS,p

(3.7)

wherce,the slopeofthecamberline at the nond imensionalcho rda l posit iQn, xc ,is

(3.8)

Figur e 3-5: Aerofoil sect ion definition Since,p is verysmallfor marin epropellers ,Equations3.7ca n be simplified to

u«

=Yc +Yt

(3.9)

whercp,=t/ 2 isthesemi-t hickness of thelocal section.

Theleadin g edges areusuall y circular.Bnt in the simula t ion the circular edgewas repla cedby a sha rpedge for the ease ofgridgenera t ion process.

3.1.4 Hu b Geometr y

Some geomet rysimplificat ions weremad ein the gridgenera t ionofthe hub.

prop eller blades were assumedtobemount ed on an infinit e consta nt- rad ius hub cylinderand therefore axial variati oninhuh radiuswasignored.

fixed-pitchprop ellerswere also ignored .

3.2 Grid Generation Method

As statedearlier,theefficieneyofRANS solver lar gelydepend s onthequalityof the grid . For theeom putatio nof theprope llertipvortex flow, grid resolut ionwit hin the tipvortex eore hasprofound effectonthe physical solut ions(Hsiaoand Paul ey,1999).

Torepr esent the physicalsolut ionwit hsufficientaccuracyfor acomplextipvortex flow,the following issues must be conside red inthegridgenerat ionprocess (Qiuet aI.,2003) :

•Gri d Finen ess: A fine gridis necessaryto adequate ly resolvethe tipvortex.

Atleast15 grid points across thetipvortex core sho uld beusedto obtai na reliabl enear-fieldtipvor tex formarin e propellers(Hsiaoand Paul ey,1999).

•Grid Den s it y :The griddensityonthe prop ellerblade surfacemust be sufficientso that bound arylayer effectscanbe wellpredicte d.

•GridSmoot h ness: Grids must be smoot hthrougho utthecomputa tiona l

•Gr idOrth ogon ali t y: Thegridorthogo nalityat thesolid bound a ryis import an t ifthe zeronorm alpressur e gradien tapprox imat ionis applicd.

•Grid Effici eu cy: The gridgenera tor has tobe computa t ionallyefficientfor rout ineapplicat ions.

In this work,PropGGM wasused for gridgenera t ion purpose.Det ail descript ion of thestruc t ure,funct ionalit ies, implement ati ons and demonstrati on of thisprogram is given by (Qiuet aI., 2003).A brief descript ionof the gridgenera tion processis

3.2.1 Grid Gen eration Approach

Thecomputational domainwas createdasoneblad e-to-blad epassagewithtwo periodic boundariesbyfollowing theinletflow angle.Oneperiodicbounda rycont ain s thosuctionsideofablad ewhiletheothcrconta insthepressur e sideof theadjacent blade.This strategyresultedinaspiral-likecomputationaldomain. Theadvant age ofthiskind ofcomput ational dom ainisthatthe cluste redgridcanbe easil y aligned withthetip vortexandtheflow acrosstheperiod icboundaries canbeminimized (Hsiao and Paul ey,1999).Thedomain isenclosedbytheinlet boundary upst ream, the outlet boun da rydownst ream,the innerboundarylocated onthe hubsur face and theouter boun daryintheradialdirection. Thedomainis shownin Figure 3-6.The gridgenera tionwasdonebythree ste ps:

•Step1: Generationofthe surfacegridonthe blad eand hubsurfaces.

•Step 2: Genera tion of a twa-dimensionalgridin the fluiddomainbetween

•Step3: Smoothingtheiniti althree-dimensionalgrid .

A briefdiscussion ofeachste p is givenbelow.

3.2.2 Grid Gen erat ion on B lade Surface

After the genera tionof blad e surfacecoordinates , the sur face gridswcredistributed on theblade surface. Thepanels genera t edfromtheorigina l data shrinks to apointat thelas tradialstation.Thistype of panels or gridsareknown asO-type grid andare notaccept able for a struct uredgrid bas edRANSsolvcr.Tosolvet hisissne,thedata pointsfrom the originaldatawerefirstincreased.II-typegrids werethengenera ted fro man O-typ e grids. Details of theH-typ e gridgenerationcan befound in Qiuet al.(2003).Differencebetween H-type grids andO-type gridsarcshown inFigur e3-7.

Figure 3-6:

Spira l like computa t iona l domain

Thedistribution of H-typ e grid

can be cont rolled

by cha ngingthenumber of grid pointsin the spanwise and

chordwise

direction s as well

as

bythedistribution function s. Thedistributionfunctionsfor grid points concentrated

at two ends

in

the spanw ise or chordwise direction are defined

by

where

Q

and (3 are the gird distribution

fact orand

stretc hing

fact or ,respect ively.

When

a

=0.5, the

grid will

clust er evenly

at both

thetip

and root regions in the

spa nwise direct ion or

theleading and

trailing edges in the chordwise direct ion.

(a)a-type grid

(b)II-typ egr i d

Figurc 3-7:DifferencebetweenH-typ c alld O-typ c grid

The st retching factor , fl, should he greate r than one. The

larger the value,

the less concentra tio n ofgrid points at the end edge will be achieved.

In

anot her words, the grid will be mor e uniform ly distribut ed. The intermedi ate vari abl es, ¢ and

t/J,me

defined on the unit intervals,

E.and

(.

In the

progr am.E. = B and ( = B , where

I

or J is the orde r of the point in the chord wise directi on orthe spanwise direct ion, Nor

M

is the total number of points in the chord wise direction or t.hc spa nwise direction . The locat ion ofag rid point on the surface can be contro lled by adjusti ng the single va lued funct ion

¢(E.)in

the chord wise direction or

t/J

(() in the spanwise directi on. The contro lling funct ions for grid points concentrate d at one end in the spa nwise orchord wisedirecti on are defined by

Forhlade flow

simulat ion, clustered grids are requiredat the tip and root

regions

as well as the

lead ing

and trailing edges ofa blade sur face.

Forthisreas on,

the blade surface was subdivided into two regions in the spanwise direct ion.

The

first

region

was

froms=0.0

(root) to s

=

0.90 where s

is

the non-d imensional arc

length.The

two-end grid concentration was applied in this region . The second region was from s

=

0.90 to s

=1.0

(t ip). where the one-end grid concentrat ion was set.

In

the chord wise direct ion. the two regions were divided at s

=

0.45.

In

the region close to the lead ing edge, two-end grid concentra t ion was applied. In the other chord wise

region,

the one end grid concent ra tio n

was

used .

3.2.3 Grid Gen eration on Boundaries

To generate the spira l

like

bou nda ry, the domain wasdivided into

th ree

regions as

shown

in the

Figur e 3-8. The first region is above the tip from the leadi ng edge to

Fignre3-8:Gridgenera tion on boundaries

thetrailing edge,theregionfrom the inJetbonnd arytotheleading edgeisthe second region andthe regionfrom thetrailing edge to theoutle t bound ar yisRegion 3.

To generate the grids in RegionI,the following equations wereused (QiuetaI.,2003):

x(i,k)=x(i,I ( T I PNU M) r(i,k)=I'" p(i )

+

4>(1.0)

I'la~,:_~~:(i)

Ii(i,k)=Ii(i,KTI PNUM ) y(i,k)=r(i,k) sin[li(i,k))

(3.14)

z(i, k)=r(i,k)cos[O(i, k)]

where,i=ILE,lT E ; k=l\TI P N U M

+

I,K M A X. In the eq ua t ions,ILE andIT E aretheindicesofgrid points on the leading edge and thetrailing edge, resp ectiv ely,l\TI P NUM isthenumber ofgrid point s onthe blade surface in the spa nwise directi on ,l\MAX isthetot alnumber ofgrid point s on thebound ary in the spa nwise direction ,<I>(k)isthe cont rolling functionin theradi aldirection , x(i,l\T I P NU M), O(i,KT IPNUM ) and rtip(i )are thex,0andr-coo rd inatesat thetip,respectively.

To genera te thegrids intheRegion 2,thefollowingequationswere used:

(3.15)

y(i ,k)=r(i ,k )sin[O(i ,k )]

z(i,k)=r( i, k) cos[O(i, k)]

where,i=ILE-I,I,-I;k=I,l\M AX,<I>(i)isthe cont rolling functi oninthe helixlinedirection ,du:isthedist ancefromtheinletboundarytotheleading edgeon thetip,XLEisthe x-coord ina teof theinlet bound ary, Uooistheinflow velocity.u ,istheRPSof t he propeller,r'(k) istheradiusof the gridon theline n , x(I L E,k),O(ILE,k ) and r(ILE,k) are the x,

o

and r-coordin at es on theleadin g edge,respect h·ely.

ForRegion3,thefollowing equat ions wereused :

x(i, k) =:r(I T E,k)

+ ~[XTE

^-x(I T E, k)]

ro(k' )=[xn; - x(I TE,

k)]2"~~k)1l

lI(i,k)=II(ITE,k)

+ ~~

y(i,k)=r(i,k)sin[lI(i ,k)]

z(i,k)=r(i,k)coslll(i,k))

(3.16)

where,i=IT E

+

1, 1MAX;k=1,K MAX,q,(i)is again the cont rolling fnncti on inthehelixlinedirect ion,dTEisthedist ancefrom thetrailing edge totheont let boundar y on theti p,.LTEis th e :r-coordin at e of outletbound ary,r'(k)istheradius ofthegridon theline{J,.L(I TE,k),II(ITE,k)andr(I TE,k)arc thex,1Iand r-coordinateson t hetrailing edge,resp ectively.

3.2.4 Initi al Grid Gen eration

Afterthecomplet ion ofgridgenerat iononthe blade sur facesand periodicbouud ari es, a two-dimensionalgridwas createdoneachconstant radiussurface consideringthe blad e surfacegridandthefirst gridspaci ngs.Oneachconstant radius surface,a Bczicrcurve(Fauxand Pra t t,1979) was usedtodefineagrid linebetweentwo bound aries.ThisBeziercurve makes thegrid norm alto theblade surfacewhere the boundar ycond itionofzero norm alpressu re gradient is applied.

The gridpoints on theBezier curve were then distribut edbyusingthetwo-end cont rollingfunction.The two-d imensionalgridwas smoo thedby atwo-dimensional ellipt icsmoo t hingrout ineand by stac kingthcsesmoothed grids,theinitial3-Dgrid was genera te d.

3.2.5 Elliptic Smoothing Technique

The gridgenera tedbylinearinterpolationisnotsmooth.Anelliptic smoot hing routinewasused tosmooththe gird.Thisrout ine solvesa setofcoupledPoisson elliptic partialdifferential equationsandgeneratessmoothgrid.The equationsare givenbelow:

~xx

+

~YY +~"⁼P(~,^{7/, ()}

71xx

+

'1UY

+

'I"=Q(~,ll,()

These equations can betransformedintogeneralizedcoordinatesby

~=~(x,y, z) '1='1(x,y,z) (=((:r,y, z) TheJacobianof the tran sformationis computed by

where~x=~,

.7:,

_=~,etc.

(3.17)

(3.18)

(3.19)

[ ^~x]

^~y~,

⁼

^J

⁼

^J

^{[1'1 1'31 1'21 1]}

[

'I X ] [V (Z(- V (Z(] [1' 1 2]

Tly =J:r(z( - :r(z( =J

1'22

'J, X(Y(- :r(v(

1' 32

[

(x] [1'13]

(y = J =J

1'23

(, 1'33

(3.20 )

UsingtheJacobia n of the trans formationabove,Equation3.17can betransformed

i=1,2,3;j=1,2,3

Where,

P,Q

and

R

are theforcingfunctionsandarc usedtocontrolthe grid distri bu tion.Theyaredefinedas

(3.22)

whereaisa positi ve eonsta nt whichdet ermin esthedecayrate of the grid clust erin g andp , q and risd et erminedbyrewritin gEqua tion3.21at theboundary,i.e;at r/ =O

pl',+ ql',,+rl'<

=h

[ ill]

h= ii,=-J

2 [alll'" +

"221'""

+

a3:'I'«

+

2(a 121'" ,

+

"131'«

+

"231'"dJ h3

(3.23)

(3.24)

Thepartialderivati vesofI'withresp ectto ~,r/and(at theboundar y,i.e; atr/= O, aredet erminedbythefollowingthreerelations

r{·r,/ =O

(3.25)

1'<'1',, =0

wheresisthefirst gridspacingattheboundary. Thedesiredspac ingand orthogonality arespec ifiedhere.Expandin g theequat iongives

(3.26)

I ' """+~,,,+ ,,.= o

I Cmm"',m ,."" " Hoow"" H"ooH,O",, 'm,.,,,d,.,

I"=

-Z;~~:y~ :,::yd

⁼

-~;~~2

y"=

-~(.~;~;'--;;~;))

⁼

-~;~;

SlIbst itllt ing.T"andy"intot hescco ndequation inEquat ion 3.26givcs (3.27)

(3.28)

(3.29) .9/' 32

Z"=

\h~2 + /'i2 + /,g2

8/'12

.r,=

\h~2 + /'i2 + /, g2

.9/'22

u .

=

'h~2 + /'i2 + /,g2

Equat ion3.29 givesthelirstderivati ve1'".The second derivati ves1'"., can bederived fromtheTaylor series:

where jistheindexinthe rydirecti on.

A multipleblock,smoothing routin e wasused to smooththe initialgrids

I JlL"'"

on the schemedescribedabove,Asmoothgridcanbe obta ined inafewiterations by

3.3 Computational Domain for Un st eady Simulation

In

the previous sections, a computat

iona l dom ainfor steady

simu

lationcontainin g

only

a

single blade was

discussed.Forthe ste ady

(uniform)

flow condit ionitis

assumed

thatthepressur evaria tions over

all t

heblad es arc the

same and there

arcno

significant

flowinterruptions amo ng theblades. Thistyp e of dom ain is

computat

ionally

efficient since

it

requ

ires less memory and comput ing

time.

Butfor unst eady Inon- un iform)infiow condit ions,theflowis threedimensional and

requires consideratio n of all of

theblad es toaccurat cly investigat ethe unsteadyna t ureof prop ellerbladeloading.

Anattempt was then takento const ruct a dom ain

whichwouldconta in all

theblad es

ofa

propellerand t he hub.Thedomainwas created intwodifferent ways.Firstly,by modifyingProl' G GMforeach individu alblad eto generatesepara te

spiral like

domain

and combining them together to get the full

propeller. Secondly,byusing the mesh

transfor mati on feature ofANSYS CFX (CFX manual,

2005).The second option was

found much easier,

fast er and more convenient. Theprocedur eisdescribed inthe following section.

3.3.1 Mesh Tran sfo rm at ion

TheProl' GGM

genera tes one

blad e-t o-blade spirallikedomain

which is sullicient

for uniformflow analysis.For non-uniforminflow condit ions,allthebladesmust be considered. This was easily

done

by themesh tra nsfor ma tion

editor of ANSYS

CFX.

Theotherblades and

the full hub were

regenera t edby copying

androt atingthe

spira l like

domain .Finally, all

the domains wereglued togeth er

to crea te

aco

nti nuous mesh contained

in a single assembly from t

hemultiple copies.Asingledomainwas thus create dforthe ent ire

assembly wit hout the

need of ereat ingdomain

or periodic

Figurc 3-9: Comput at ionaldomainforunsteadysimulntion interfacesbetween eachcopy. Figure 3-9shows thecomputat ionaldomain.

Chapter 4

Computational Method

This cha pte r describ esthenumericalmeth odusedtosolvethe problem . governing equationsforfluid flowarc out lined first.Thediscreti sati onmeth od of the govern ingequa t ionsand the solut ionsstra tegyof thedcscretiscd equa tio nsare thcnexplained.Acomprehensivesummaryofallturbu lencemodelsisalsopresented.

4.1 Governing Equat io ns

Acommercialviscousflow code, ANSYS-CFX (V I l. OandVI3.0) wasusedfor the computa t ion.The setof equati ons solved byANSYS CF X arctheunst ead y Navier-Sto kesequat ions. Theunst ead y,three-dim ensional cont inuityequationfor compressiblefluid is

'f!f+'\l'(PU)=0 (4. 1)

For incom press ible fluid(e.g. wat er ) the density Pis consta ntandtheequat ion

'\l· (U ) =o (4.2)

TheMomentumconservationcallbepresented as:

0fjf

+ 'V.

(pU0U)=-'VI'

+

'V.{pv ['VU

+

('VujT-

~<5'V

·Un +S", (4.3) wherc<5is t he Kroneckcr dclta funct ion,

S.\I

is t he momcnt umsource, 0is t hc te nsor produ ct oftwovect ors,U0Visdefined as.

l

UxVx

u,

Vy

u. V ']

U0V=UyVx UyVy UyV, U,Vx U,Vy U,l',

Additionalsourcesof momentumarerequiredforflowsinarotatingfram e of rcfcren ce toaccountfor thc effects of Coriolis forceand thc ccntrifugalforceoIfthcfra mcrot atcs ata const a nt angularvelocityw,the sourcetermCHnbe expressedas:

S"'.,ol=-2pwxU-pw x(w x

r)

(4.4) wherethefirstterm

representstheCoriolisforc eaudt hesccondterm ist he

centrif ugal

force,

l'isthe

location vector

and Uistherelativeframevelocity,i.e.,the

rota t ing fra me

velocityfora rotatin gframeof reference.Thefina lform of themomentum equat ion becomes:

0fjf+'V'( PU0U)=-'Vp+ 'V.{pv['VU+('VujT

-~<5'V.Un -2pw XU-f", x (w xr)

(4.5) Thetot al energyequa t ioncan he expressedas:

D(p;~t"')

_

~ + 'V.

(pUiltod=

'V.

('\'VT)

+ 'V.

(U·r )

+

U·S",

+

SE (4.6)

whereh

tot

is the total enthalpy. The term

\7, (U' T)

repr esent s the viscous work duetothe,·iscous str esses andthetermU·

S.1/repre sent sthework

due to

exte rna l

momentum sources,

findSE

is the energy

source.

The remainin gunknown thermodyna mics

varia bles(p, P,i

and

T)are Iinked

togeth er through the assumpt ion

of hydrod ynami c

equilibrium

andareexpressed by only

two sta te vari ables. This express ion is known as the equation of

state.

For an incompr essiblefluid,where th edensityis const ant ,thereisnoneedfor th e st at e

eq ua t ion.

The flowfield can

often be solved by conside ringonly the

mHSS conse

rva ti on and momentum equa t ions.

The energy equati on only needs to be solved

alongside theot hers ifthe problem

involves any hea t transfer (Versteegand Malnlasckere , 1995).

For turbulent flows like propeller tip

vorte xflows,

the Navier-Stok es (N-S)

equat ions have

to be modified to produce the

Reynolds Averaged

Navier-Stok es (HA NS)

equations by employing

averaged and fluct uati ng qua nt it ies. The result ing equations

~ + \7' (PU) =O

^(4.7)

0jJf- ⁺

^\7.(pU<81U)=

^\7

^.T-pu<81u

⁺

^{5,11 (4.8)}

whereT

is the molecular st ress tensor and

pu<81uare the

Reynold s st resses. These

stresses are mod elled

by introducing turbulence model to enclose the governing

equati ons. Det ails ofthe turbulence mod els are described in Secti on4.4.

4.2 Di scr etisation of Governing Equations

Sofarwehave seen thattheflowphenomenaarcgovernedbypar tialdifferential equat ions. Analytical solutionslothese equationsarconly available for thesimplest offlows ,under idealconditions.Tosolve realflowproblems, anumericalapproach mustbe adoptedwherebythe eq uationsarereplaced by algebraicapproximations and theprocessisknown asnumericaldiscretisation.Thissectiondescrib es this descretisati onmeth odusedfor the simulation(VcrsteegandMalalasckcra ,1995).

Thereare significantcommonalit ies bet weenthe variousgoverning equa tionsandcan be writte nin thefollowinggeneral form:

(4.9)

where ¢isthe variableof int erest ,

r

isthediffusioncoefficient. TheEquation4.9 isknownastransportequat ionsinceitdescribesvarioustranspor tprocesses of dependentvariables.This equat ionrepresentsdifferentasp ect softhefluid mot ion.

The convectionterm representsthefluxof¢convect edbythe mas sflow ra te pu,thediffusiontermrepr csentstherandom ltlotionofparticles andthc source termrepresent sthe generationanddestruc tionof¢.Thenon-linearnatureofthe convect ivetermmakesit difficult to solvethe equationsdirectly,that is, as a setof simultaneousequations.An itorative solutionrncthodis theonlyway to solvethese equa t ions. The governing equationsarcdiscretised ,thatis, approximat elylincarised to obtainthe algebraic equations andare solvedat discretepointsthroughou t the

Avaricty oft echniques,includingandnot limitedtofi nitediff erenceandfi nite volullle mcthods ,are availabletopcrformthisnumericaldiscretisati oIlbut afiniteVOIUIIlC approachis adopted here.Abriefdescription ofthemethod is given below

4.2 .1 Finite Volume Met hod

The finite volnme meth od is probabl

y

the most popu

lar

descret

izati on

method used in

CF D. This

meth od draws on ideas from both finite element lind the finite differencediscreti sa ti ontech niqu cs.

In this approach the comp utat ional

domain is discret ized int o finite cont rol volum es.

also knownlIScells. The govern ingequat ions are integr atccl overeachcontro lvolume which

utilises the conser vatio n principl es directl y. The integr al form ofEqu lltion4 .90\·er acontro

lvolum e givcs

The

Equati on 4.10

repres ent s

the flux balan ce in a

cont rol

volum e where the left hand side represent s thc rat

e

ol

cha ngoof

e

andnetconvective fluxand righthand sidegives the

the net diffusive flux lind the generat ion or

destruction

of the

property

¢.Thesefluxes are evaluated by va rious numeri cal schemes

which

arcdiscussedin

the following sections.

Thema in adva ntage

of the

fini te

vohnne

meth

od

istha t thespa tial discretisat.ion is ca rried

out directly

in

the

physical space. Thus ,

the re arc

noprobl ems

withany transformation betw een

coordinatesyste ms ,likein

the case of the finite difference meth od . Compared to the finite differen ce

method ,anot he r advantage of the

finite

volume meth odistha t

it is flexible toimpl

em ent onboth struc t uredand unst ruct ured grids. This makes

finite volum e meth od suita ble for thetreat ment of flows incomplex

geome tr ies.

4.2.2 Discr eti sat ion of the Domain

Older CF D

codes

used a staggered mesh approac h where the

scala r

vari abl es wer e

calcu latedat t hecellcent resandt he vecto r \'aria blcs,i.e.

t he velocit ies,

at thecell

faces. But recent codes

,suchlISCFX,

usc a co-loca t ed

(no n-staggered)grid layout

where the vallies ofallvariablesarc calculated at

the centre

of each control volume.

This approachisbetter

than the

previous one

since it

requires onlyonemesh

to get the values of all

the variables. In

a complicated geomet ry, where curvilinear

(no n-rec ta ngular)meshesarc lIscd,applica tioIlo fco- locatc d

grid is easier because of

its simplicity.However,thismethodleads to adecoupling

of thevelocity and pressur e

fields givinga 'checkerboard'

effect.

Thisis overcome byRhie-Chow (Rhieand

Chow,

1982)interpolation

algorithm, which is

furthermodified bylvl ajumdar (CFXmanual,

2005) to remove thedependence

of thes teady-sta tesolutionon thetimest ep size.

4.2.3 Adve ctio n Terms

Advec tionterms can beapproxima t ed wit hdifferentadvectionschemes

available

in

CFX.

All

the schemes

have someadvantages and disadvantages.For

example, the

firstorderaecurate upwind

differencingsc hemegivesthemostrobust per formanceof the solver butitsuffers fromnumericaldiffusion.Onthe oth erhand,the second order

accurate schemeis free fromthediffusion problembut sometimes it givesnon-physical

results.A blend ofthesetwo schemes

can be

achieved by the use of thehighresolution

scheme, which was

implementedinthis simulation. Ablend factor

((3)

is sent ina range of0.0

forfullyfirstorder to

1.0for

fullysecondorderschemes.

Theblend

factor

valuesvary t hrollghoutthe domain based on the

local

solution field.

Ifthe variable

grad ients is

low

in a

flow region,theblend factor

will be

closeto1.0,but willbe closerto0.0 wherethe gradients

change sha rply. In other words, the scheme

is fully

second order as

long asthere arcnodiscontinuitiesin theflow.Itdropsto

first

order tokeep the

solution

bounded.A

centraldifferenceadvection scheme

is alsoavailable to

CF X

butit

is

reserved forlarge eddy

simulation tur bulence

modelonly. The third

order accurate

QUICK

scheme is also available

in

CFX

but notlistedin the main optiollsandpcrha psthis isllo trecommcudcd forgcneral use.

4.2.4 Diffu sion Terms

The reasonable way to discretise diffusion term s is to

employa cent ral

difference

scheme beca nse oft he physical nature oftheviscous

f1nx. Tlms,theirdiscretisati on on

st ruct uredgrids is st

ra ight forward. On unstru ctur ed trian gular

ortetrahedralgrids,

the visconsfluxes ar ebest appr oximat edby followingthe st and ardfinite element

approac h.CF Xalso uses t hisa

pproac h

by employingsha pe functi ons to evalua te the

derivati ves for

all the

diffusion terms.

4.2.5 Tran sien t terms

The tra nsient term of Equati on (4.10)

can be splitinto terms as

follows:

(4.11)

The

time

derivatives

of

Equ at ion(4.11)canbe approximatedby

eit her

the

first order

BackwardEulerschemeorbyII

second order

BackwardEulerscheme.The first

order scheme isrobust, fully

imp licit,bounded vconscrva tivein time,find docsnotcreatc n

tuuestc p

Iimitat ion.I3utsinccitis onlyfirstordcracc uratc,itsuffcrsfrolIl llumcrical diffusion in

time, similar to the numerical diffusion

experiencedwith

the Upwind

Diffcrencc

Scheme for discretising the advection term.

On

the other

hand,

the second

orderscheme is

also robust ,

implicit,conservat ive in

time,

and docs

not create a

timeste plimitation,

but it is

not boundedand mayhence crea

te

sorne nonph ysical overshootsorundershootsinthe solution.However,allthesimulations were

done with this Second Order Backward Euler

scheme,

4.3 So lution Strate gy of th e Discr etised Equations

There is no equati on (t ra nspo rt

orother) for

pressur e and

thisaspect is

therefore

treat ed differently.

Aconstra intis set on the solut ionofthe flow

field in

that when

theco lTect press ure issu bst it ute d inthemo mcnt umequat ions, t he res ult ingvelocity field sa tis fies mas s cont inuity. ANSYS CFX uses segrega t edsolverswh ichsolve themomentumequations usingaguessed pressureandobtainapressure correction relation.This'gu ess-a nd-c olTect' appro achofthelin ear solverrequires alar genumber ofiteratio ns,Acoupled solver solvesthehyd rod ynami c equa t ions(foru,v,lU,p)as asinglesyste mwit h afully implicitdiscreti sa tionmeth od.Thisredu ces the number ofit er ati on s to achievethe convergc llcccriter ia. Thediscret e syste mof Iineari sed equatio ns arc solved by allIncompleteLower Upper ([LU) factorisa tiontechnique accelerat edby algebra ic Multigrid (1IG) method (Ra w,laaG).

4.4 Turbulence Mod eling

Turbulencemodellin gis anot her importan tissue to consideresp eciallywhen the simulat iondeals withthefluctua tin gflowslikeprop ellerflows.Ascrew propeller induceshighlyrot a tingflow andenta ilsa tip vortex , wh ichcausestur b ulence inthe tip vort exregion .A viabl etool thenrequiredtorepr esent this turb ulence effect.

Therandomna ture of a turbulcn tft ow canbc cxpl aillcdbyintrodllCillgaver agedand Huctu a tin g compo nents.Forcxa mple ,a vclocity u(t) canbcdivid cdinto HIlHvcrage compone ntU and a fluctu a tin g com po nentu'(t)

u(t) =U+l/(t) (4.12)

Thus,it requiresmodification ofthe original unsteady Navier-Stokes equation s, conside ringtheaverage d and fluctua t ingquuntiti cs toproduce theReyn oldsAvera ged Na vier-St okes(RANS)eq uat ions. These equat ions areobtain edbyusin g the statisticalaveraging procedure. This averaging procedureintroducesaddition al st ress es intheflu id known as'Reynolds stresses'and needtobemod elledinorde rto Turbulencemodel sprovide the mod elfor the compu ta t ionof the

Reynold, st resses. There arc severalturbulence mod els available to CF X which can be categorized int otwo classes,eddy viscosit ymodels andReynolds st ress model s.

They are discussedbelow.

4.4.1 Eddy Visco sity Turbulen ce Models

These turbulence models arebased onthe hypothesisthatthe Reyn olds stresses arc prop or t ion alto meanvelocitygra dientsand Edd y (turbulent) viscosit y can berelat ed bythe gradientdiffusion hyp oth esis.

-p"i1®U=Pll,[V'U

+

(V'uf]-

~<5p(k +

/l'V"U) (4.13) where,<5istheKronecker'sdeltil lindIl,is theed dyviscosity ortur bulentviscosity, which hastobemod eled.Atthesame timethe eddy diffusivity,

r,

also hastobe modeledwhichis computed basedon the assumption that theReyn old,fluxesofa scalar arelinearly relatcdto thcmcallscalarg rndient.

- Pl/¢=

r,V'</> (4.14)

The eddyviscositymodels arc dist inguishedby themannertheyprescrib e eddy viscosityandeddy diffusivity.Thevariouseddy viscosit ymod els are describ edbelow.

ZeroEqua t ionModel

In this mod el theeddy viscosityis comp utedfromthemea n velocityanda geomet ric length seale using anempirica lformulaas follows:

Il , =

p!,.

U,I, (4.15)

where,!,.baproportioua lityeo nstantll nd l, i, th e length , cale prop osed hyP ran d tl andKolmogorov.Sincen otransport cquat ion isinvol\"ed,thisll1 odcl namcd as'zcroI

equation model. The advantage

of this

modelisits simplicity to implement and

cheaI'

in terms of

comput ing time.Butsince it is

based on

empiricalformuli

L"themod el

Tw

o Equation Turb u len ce Models

These mod els

solve

two separa te tran spor t equati ons , one for turbul

entvelocit y scaleandanot her for

turbnlent length scale. The product

ofvelocityand length

scale arc then nsed to model turbulent viscositY,II " The turbulent

veloc ityscale is computed from

the turbulent kineti c energy,

k,

which is provided from the

solut ion of its

tran spor t equation . And the the turbulent length scale is est ima ted from the turbu lent kineti c energy, k, and its dissipati on rate,<'

This model predicts the

tur bulence

viscosity from following rela t ion:

(4.16)

where C"isa constant. The values of e

and<are solvodfrom

the followingdifferential trnnsport

eq uat ions:

a~t) + \1.

(pU k)=

\1. [(II + ~

)\1k]

+

Pk -P< (4.17)

where the consta nt

coefficientsc,=

0.09,

Cd=

1.44,

C,2=

1.92,

Ok=

1.0

and 0,=

1.3.

Pk

is the

tur bulence produ cti on

dne to viscous and buoyan cy forces, which

Duetothe excellent performan cein manypracticalflows,thismod elis well establishedand widely validate d.However ,poorperform ances arc reportedin a varietyofimpo rta ntca.scslike confined flow»,flowwith larg eextrast ra ins,and in rota ti ng flows.

Thismodel shares the sametransportequatio nsasthosefor st andardk-Ecquation.

The onlydifferenceis the mod elconsta nts . The consta ntCi ,isreplaced by thestrain depend ent correct ionterm functi onC,IRNCwhere,

C<111NC =1.42-1"

whore,

Thedet ailsof theoth er constantscan befound inCFX manual(2005).

(4.20)

Thismodel,develop edbyWilcox (1986),assumes that thetur bulenceviscosity is linkedtothetur bulence kinetic energy,k, andturbulent freq uency,w ,viathe relat iou'

11'=p~

The tra nspo rtequa t ions for kan dware:

(4.21)

D~tk) ⁺

^{V'.(pUk)=V'.[(II}

⁺ ~)V'kl ⁺

^{Pk -rJ'pkw (4.22)}

Dr:::) +

V'.(pUw)=V'.[(II

+

;;:)V'wj

+ O:IPk -

(3pu? (4.23) with0:=5/9,(3=0.075,(3'=0.09,ak=2 andaw=2, andPkis calculate dasin the

it docs not requir e any non -lin ear dampin g fu nct ions.

The

near wall perf or m an ce of

this model

is also very att rac ti ve.

4.TheBas elin e(BS L)I.-wModel

This is a blended form of

WilcoxI.-wmode l

and modified

I.-emod el to eliminat e

the extreme sensitivity to freest ream conditions of the

Wilcox model. The1.-,

model is tran sformed toa

k-wformulation and multip lied

by

a

blend

ing

function

1-

F,. The

Wilcox mod el

is multipli

ed

by the function

F,

where F

I

is equal to

one near

the sur face and

switc hesover to zero insid e

the bound ar y layer.

corr esp ondin g

I.andwequati on s are then added to get the

I3SL mod el.

5.TheShearStre ssTran sp o r t (SST)Model

This model

can be

called as a improved

version of I3SL

model.

comb ines t

he

advantages of the I.

- wandthe1.- ,

model,

butit fails

to predict

flow separation

accurately,

The reasons

were revealed

byMen ter (1994)

and the

mainreasonis

that both models do

not account

for the transport ofthe t ur bulent

shea rst ress which results

in

an over predi cti on

of the ed dy-v iseosity, Xlent er then introdu ces limiters wher ethe ed dy viscosity is limited to

giveim

proved perf orm an cein flows with

adverse pressur e gra d ientsandwake region s.

The turbulent kineti c energy product ion is limit ed toprevent the build-up

of tur bulenceinstagna t ion reg ions. The

limitercan be express ed

as:

(4, 24)

whe re

F2is

a

blend ingfunction similar

to F

Iinl3SL model,"I

is a

const a ntand s=~,

K-epsilonOne EquationModel

This simpl e one equ ati onmod el,d

evclop c,l

byj\lent er (1994),i sderh 'eddirectl yfrom

thek-<

model and is therefore

namedthe(k-<)IE

mod el. The mod el

containsa

destru ction t erm ,which accollnt sfor th e stru ctur e ofturbulence. The ed dy viscosit y

is comput edfrom:

1'.

=pv;

(4.25)

where, v;

is the turbulent

kinemat ic eddy

viscosity. This

mod el includ es dampin g functi onsto ca tc h the low

Reynolds effect.

4.4 .2 Reynolds Str ess Turbulen ce (R SM) Models These

modelsdo notlise the

eddy viscosit y hypot

hesis,hutsolveall

transport eq uat ions

forall

components of the

Reynolds stress tensor

and the dissipa tion rate.

The transport

equa t ion for

theReynoldsstresses is:

iJPI~~

¹¹

⁺

^{V'. (pU0li0U) = (P}

^{+ q, +}

^V'.((/l

⁺ ~c'P~

^{)V'li0U) -}

~6p<)

^(4.26)

wherc o is the pressure-strai

ncorrelat ion andP

is the prod uct ion termvgiven by:

P =-p(li0U(V'u f

+

(V'U)li0U) (4.2i)

The pressur e stra in interaction is most impor t an t, hut most difficul t to mod el

accura te ly.

This intera ct ion

reducesanisot ropyof turbulent ed diesand therefore

it requires addit iona lcorrect ions.

The exact produ ction term and

thest ressanisot ropies theoreti cally

make Reynolds

Stress mod els more suite d

to complex flows.

But it requires large comput ing

time and in practic eit poorlypredict s some flows (forexa mple,axisasymmet ric jets,nnconfincd re-circulat ing flows)ducto th cid cnt icalproblemswith the e equat ion.

There arethree typ es of sta nda rd Reyn olds stressmod els available. Theseare known asLRR-IP,LRR-QIand SSG .Eachmod elhasdifferentmodel consta nts .Inthe LRR-IPandLRR-QImod els,dcvelop cdbyLannd er et al. (19i5),t hc pressurc-stra in correlat ion islinear.II'standsforlsot ropizationof Produ cti on,and QIsta nds for Quasi-Isotr opi c.On thc oth crhand,SSG mod el,dewl opcdbySpeziale (199 1),uses a quad ra t ic relati onfortheprcssu re-str uin correlat ion.

There are ot hertwo typ es ofReynolds stress mod els available,theOmegaReynolds Stress and Baseline (I3SL) Reynolds St ress mod els.Thesemod elsarcbased011w equationand areusedforamor eaccurate nearwall treatment.Thetwomodels relateto cachothcrin thc samcway asthctwo equation k- wandI3SLmodelsdo.