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A MASS-WEIGHTED UPWIND-BASED C O N T R OL VOLUME FINITE-ELEMENT

METHOD FOR STEADY TWO-DIMENSIONAL VISCOUS

C O M P RESSIBLE FLOWS

BY

HAO ZHOU

A thes issubmittedtothe Schoolof Graduate Studies in partial fulfillmentof the requirements forthe degree of

Mastel'ofEngineering

Faculty ofEngineering&.AppliedScience Memo rialUniversityofNewfoundland

April,199 7

St.John 's Newfound land Canada

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AB ST RACT

The formulationand implementationofa mass-weightedupwind-basedcontro l volume finite element method(C VFE)"I) for steady.two-dimensional.viscouscom- pressibleinternal80""5isrepo rtedin this thesis.In thedevel opme nt of this method.

aCVF E),I for steady.quasi-o ne-d imensionaJ .viscous com pressibleBows wasalso formu la ted. Theproposedmet hodisa colocated.shock capt uring form ulat ion.

Polygonal controlvolumesareconstructedaroundeachnode inthe finite-element mesh, anddiscretized forms of thegoverningequations are obt ainedbyderivingal- gebraic approximat ions to integr al conservatio n equations foreachcontro lvolume.

Theproposedmet hods areformulatedusingthevelocity com ponents,pressur e, and tempe rature as thedepend entvaria bles:dens ity is cal cula t ed. from an equat ion of stat e.Linearinterpo la tionisapplied topressure and diffusedscalars.and amass weighted upwind functionisapp liedto tbe convectedscal ars. An interpolation functionincorporating a pseudo-veloci tyand a pressuregradientisusedtorepresent mass conserving velocities;this allows the development of a ccloceted method. valid forcompressibleSows.The discretizedforms of the governing equations are solved usinganiterat ive algorithm. In this algorithm,linearized forms of the disc:retized moment umand continuityequat ionsaresolvedin a segrega tedmanner by using a trid iagonal mat rix algorithm.

The prop osedQuasi-one-andtwo-d im ensionalCVFE M's are appliedto several inviscidandviscous compressiblefluid flowproblems ,and the solutio ns generated atecom pared withtheor etical,num erical,andexp eri mentalresultsavailablein the lite rature.This comparisonshows that tbeprop osed CVF E M'scan generate solutio nstbatare inagreementwiththe expec ted physical beha viourofsomecom-

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pressible flows and with theavailab leresults.The results suggestfurtherresearch isrequired in the evaluationandenhan cement of themess- weighted interpolation functions cu rrentl yused.however.astbeshock smeari nginthe proposedmethod is excessive.and the accuracyof solutionisnotsatisfact ory.

iii

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ACKNO~EDGEMENTS

I wouldlike to takethis cppo rt unlty[0tbankmysupe rvisor.Dr.X.A.Hookey.

fo r the patience.guidance.andsupport tha the basextendedtowards me during the course ofthisresearch program.His criticalreviewsof myworkandhisdedication toteachin gmade my graduat e studiesatMemo rialUnive rsit yofNewfound landan enjoya bleandre war dingexpe rie nce.

[wouldliketothan kthe staffof theCente rforComput er Ai dedEngineeringfor providingtechnicalassistanceandCreecomp uting time.The useofthese computers greatlyfacilita ted thecomputationalworkundert akeninthis thesis.

Finall y, theday-eo-daysupport.loveand encouragement tha t my wifeXiejun providedfor mekeptme onaneven keel.andallowed metoremainfocussed00 myresearc h.

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n

2.4 Integral Conservat ionEquationsfor aControlVolume. 23

2.5 Interpola tion Functions. 23

2.5.1 Interpolationof/-l,k,Cp,andS 2.5.2 Interpo lationofArea . 2.5.3 Interp ola t ion ofPressure .. 2.5.4 Int erp ola t ion of Velocity 2.5.5 Interpolat ion ofTemperat ur e 2.6 Derivat ionof theDiscretizedEquat ions. ..

2.6.1 Moment umEquat ion. 2.6.2 Energy Equat ion 2.6.3 Conti nuity Equatio n 2.6...l Boun dary Condit ions. 2.7 Solut ionoftheDiscretizedEqua t ions. 2.8 Conclusion.

24 24 24 25 26 27 27 28 28 32 34 34

3 Formulationof the Prop osedTwo-DimensionalCVFEM 36

3.1 Int roduc ti on. 36

3.2 Governing Equations 36

3.3 DomainDiscret ization 39

3.4 IntegralConservati onEquat ionsfor aCont rol Volume.. 40

3.5 Interp ola t ion Funct ions. 40

vi

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3.5.1 Interpolat ionofp.k.

e,.

andS 3.5.2 InterpolationofPressur e. 3.5.3 InterpolationofVelocities 3.5.4 Interp olati on ofTempe rature 3.5.5 Interpo lation ofDensity .

3.6 Derivat ion ofthe Discreuz edEquations... . . .. . . . . 3.6.1 .r-~[omentumEqua tion.

3.6.2 EnergyEquat ion .

3.6.3 ContinuityEquat ion 3.6.4 BoundaryConditions.. ..

3.6.5 Summary . . . .

3.7 Solutionofthe DiscretizedEquat ions

51 55 55

3.7.1 Relaxation oftheDiscretized Equations 56

3.8 Conclusion.. 58

4 Results 64

-1.1 Iarroduceion..

-1.2 Quasi-One- DimensionalTestProblems .

-1.2.1 Int rodu ction .

64 65 65 -1.2.2 Shock-Free Flowthro ugh a Converging-D ivergingXozzle 67 4.2.3 Converging-Diverging NoeeleFlow with aNormal

Shock . .. -1.2.4 Summary

4.3 Two-Dimens ional Test Problems. 4.3.1 Intr oduction.

69 72 72 72 4.3.2 Inviscid Flowthrough a Channelwith aCircularArcBump 74

vii

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InviscidFlow through a PlanarConverging. Diverging Nozzle 80 4.3.3

4.3.4 Shock-Laminar Boundary LayerInteraction 83 5 Concl us ion

5. 1 Cont ributi ons OfTbe Thesis. 5.2 ProposedExtensionsOfThisWork Refe re nces

Appendix

A Inter polation Functions for theProposedOne- Di me ns ional CVFEM

A.l Interp olat ion of Pressure. A.2 Interpolation of Area ..

.-\.3 Interpolation of Velocity .

:\..3.1 Int er polation of aConvect ed Scalar. :\..3.2 Interpolationofa Diffused Scalar .-\.3.3 Interpolationof Mess Conserving Velocity :\..4 Interpolat ion ofDensity

B Int egr ation ofFluxesin thePr o posed One- Dimensional CVFEM

B.l Introductio n.

B.2 Momentum Equation .

B.3 Energy Equation B.4 ContinuityEquation

8A.l Introduction.

viii

113 113 114 116

120

120 120 120 121 121 121 121 122

123 123 124 126 126 126

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8.4.2 AlgebraicApproximationof

90 .

8.4:_3 AlgebraicAppro ximationof

go ..

B.4.4 Algeb raicApproxi ma tionof

9 0.

8.-1.5 Comp leteAlgebraicApproxima ti on ofthe Linea rized.~Iass 126 12; 127

Flow. . ... . 12;

8.4:.6 OutflowBoundaryCond it ionfortheContinuityEquation 128

C Asse mblyof the DiscretizedConservationEquationsinthePro- posed One-DimensionalCVFEM

c.i

Introduction.

C.l.l Assemblyofthe Moment um Equation C.2 Assem blyoftheEnergy Equation C.3 Assemb lyofthe Cont inuityEqua t ion

DInter p o la t ion Functions forthe Propo sedTwo-Dimensional CVFEM

0.1 Inte rpolat ion of Pressure. 0.2 Interpolationofa Convected.Scalar 0.3 Int erpol a t ionofa Diffused Scalar

129 129 129 131 132

133 133 13< 136 0.4: Inte rpo lationof~Ia.ssConservingVeloci ty

0.5 Interpolat ion of Density

... . . ... .. 136 .. .. . . .. . 137

E Integrated Fluxesin the ProposedTwo-DimensionalCVFEM 138

s.:

Intr oductlcn. 138

E.2 Convec t ion- Diffusion Equati onforScalertil. 140 E.3 Modificati ons forthe Momentum and

Ener gy Equatio ns. .. 141

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E.3.1 Modificationsfor the Moment umEquations l-U E.3.2 Modificationsfor theEnergy Equations. 1.t2

EA Continuity Equation 1.t3

EA.! Int roduct ion EA.2 Integrationof

9

EA.3 Integrat ion of

9.

E.4A Integrationof

9.

E.4.5 Integrationof

9.

143 1.t4 1-15 145 145 F Assemblyof theDiscreti zed Conservat io n Equations in the Pro-

pos edTw o-Dimens ional CVFE M 146

F.! Introduction.. 146

F.2 Assemblyof the MomentumEquations. ^.^_

.. . . .

147

F.2.! x-MomentumEquation 147

F.2.2 y-MomentumEquation 151

F.3 As sembly ofthe EnergyEquation. 154

FA Assembly of the Conti nuityEquation.

.. . . . . . . . .

157

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List of Figures

2.1 Discretization of a variable area ductby theproposed one-dimensional

CVFEM. 35

3.1 Discretizationof anirregular-shapedcalculation domain by thepro- posedtwo-d imensiona lCVF E M:(a) three-node triangular elements:

and(b) polygonal control volumes. 59

3.2 Details of the domain discretization, and related nomenclature: (a)

an internalnode; and(b) a boundary node. 60

3.3 A typical three-node triangular element. 61

3...1 Types of elementsused in the discretization equation.. 62 3.5 The nodeduster involvedin the discretizationequation fora node

(i.j). 63

91 4.1 Problemschematic for quasi-one-dimensionalinviscidflow through

aconve rging-divergi ng nozzle.

~.2 Shock-free Bow through aconverging-di vergi ng nozzle(coa rse mesh):

distribution ofM.plpo.andTITo. 92

4.3 Shock-free flow throughaconverging-dive rgi ng nozzle (fine mesh):

distributionof1'4,plPo,andTITo_ .... . .... 93 4.4 Converging-diverging nozzleflow withashock:distribution ofM,

plpo ,andTITo..

xi

94

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97 4.5 Two-dimensionalinviscid Bow throughachan nelwit h acircular arc:

bump:proble m schema t ic... . .. . .... . ... 95 4.6 Two-dimensionalinviscid Bow through achann el with acirculararc

bump: (a) grid 1(uniform61x21),and(b) grid 2 (nonuniform 61 x 21)... . ... . .... ...• .. . .. .. .. . • . 96 4.7 Subso nic tnvtscld Bowthrougha chan nelwit h acircula r arc: bum p:

distribution ofMachnumb e r on theupperandlowerwallsof the chan nel.

-l.8 Subsonicinviscidflow through achan nel with acircular arcbump: isoMach lines genera t ed on(a) grid L.and(b) grid 2. 98 -l.9 Transonicinviscidflow thro u gha chann elwit ha circulararc:bum p:

dist ri bu t ionof~fachnumberontheupperand lower wallsof tbe channel. .. . . . .... .. . .. .. ... . . . ... . . .. 99 -l.10 Transo nicinviscid llow thr ough achannel withacircu lar arc: bump:

isoMachlines genera ted on(a) gridI.and(b)grid2.

-l.11Supersonicinviscid Bowthroughachan nelwithacirculararc bump:

distributionof).-Iacb numberontbe upper and lowerwallsofthe channel.

100

101 -l.12Sup e rsonic inviscidflowthro ugba channel withacirculararc:bump:

isoMachlines generatedon(a)gridI,and(b)grid 2. .. . . 102 4.13 Invisc:idBowthrough a plan arcoovergin g-divergi ng Don ie:prob lem

schematic... . .

4.14Inviscidllow through aplanarconverging-di verging nozzle:(a) 31x11

node grid,and (b)61x21node grid .

xii

103

104

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105

lOB 4.15Inviscid flowthrough aplanarconverging-diverging nozzle:distri bu-

tionofstatic to stagnationpressureratio,P!PO.along the lowerwall

of the cceete. . .

-U6 Invisc id flow througha planarconverging-divergingnozzle:distribution ofsta tictostagnat ion pressure rat io.p!Po.alongthesymmet ry plane of the nozzle. . .. .. ... .. . 106 -1.1Tlnviscid Bowthrough a planar converging-divergingnozzle:iso\lach

lines generatedon(a) 31xllnode grid,and(b) 61x21node grid. . 107 -1.18Schematicrepresentationof theinteractionofanincid ent shockwith

alaminarboundarylayer [9J. .. ... ...

4_19 Shock-lam inar boundar }"layer interact ion:(a) 61 x41node grid;and

(b) 121x 81nodegrid _ ..

-1.20Shock-lam inar boundarylayerinteracti on:problem schemat ic.

-1.21Shock- lam inarboundary layer interac tion: variationofthe static pressureratio,P!Pl .along the surface ofthe flat plate. . .. .. . -1.22Shock-laminarboundary layerinteraction:variationof the skin fric-

tion coefficient,c/'alongthesurface of theBat plate. . .. . .

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109 110

111

112

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Li st of Tables

-l.I Shock-freeflow through a converging-divergingnozzle:Accuracyof thepredictedthroat and outletMach numbers. 90 ..l..2 Converging-d ivergin g nozzle llowwit h a shock:Predictedvalues of

'\;[~hktthe position, and the thickness of the shock. 90

xiv

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Nomenclature

Symbol Description

.4 .-1,

a,b,c

det ds

9 h

he

', )

ma trixof asyst emof linear equation s areaof atrian gularelement

coefficients in the discret izedform s of theX,ymoment umand cont inuityequa tions

coefficientsinthe interpolationfunctio ns forpand~d const an t pressure specificheat

coefficientsin thealgebraicapproximation of theint egrat ed coovect ion-d iffusion fluxacross acont rolvolume face pressure coefficient derived.fromthe momentu m equa tio ns determinant

different ialarea

coefficients in thealgebrai c rep rese nt ationof tbe pressure gradie nts(~)and(~) .respectively coefficientsin thealgebraicapproximationof tbeintegr a t ed massflux acrossacont ro lvolumeface

constan tused inthe MAWscheme massfluxvecto r

heigbtofa duct in quasi-one-dime nsionalproblems;

belf-beigbtof theplanarconvergi ng-di verging nozzle heightof thebump in achan nelwitha circul ar arcbump ononewall

unit vecto rsinthex andydirec t ion s ,resp ect ively

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r

L L.

L.

M

N

R S S

T

v,v

<lV,av I,y

com binedconvect ion-d iffusion lIux therm al conductivity lengthdimension

lengt hof the convergentportionof the planar convergin g-divergin gnozzle

lengthof thedivergent portion of aconverging-divergingnozzle :\lachnum ber

mass80wrate

totalnumberof nodesina discretized domain outwardunitvecto r normaltothe differentialareads, orto aboundary

pressur e gasconstant volu me t ricsour ceterm

generalizedsource term that includesSand pressure gradie nts coefficientsin the linearized. expression forthesource terms area ofacont rol volumesurface

tempe ra t ure

pseudo-velocities(deri ved from themomentumequations) velocitycomponents intheIand11directions,respectively velocityvector

cont rolvolum e.andsurfaceofthecontrol volume.respectively axesandcoordinat esin theCart esian coordinatesyste m

GreekSymbols

under-relaxaricnparamet er xvi

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.

8 angleof thewallin the divergent portion ofa converging-d ivergingnozzle

§z,61 boundary layerthickness at a positi onz,and at theendof a

r

¢

n

Subscripts at t

calc

e,E,w, W,P

ipl,ip2, ip3 M max,min

cal cula t ion domain,respectively diffusion coefficientfordJ

normalizedchangeinIt.,Vi,PiandT,.respectivel y.

betweentwosuccessive iterations

oblique shockangle in a shock-boundary layer interaction;

angle ofthe wallinthe convergentportion ofa planar co nverging-divergin g nozzle

dynamicviscosity nodal valueof ¢ massdensity scalardependent variable ca lcula t ion domain

post-shockcondi t ions in quasi-one-dimensionalproblems averagevaluewit hinan element

calculated value denotes an element

values atlocat ionsina one-dimensionalfinite-volumegrid value at node i

integr at ion points1,2and 3 usedin theMAWscheme,respectivel y value atthe midpoint of a side of a triangular element maximum and minimumvalues,respectively

xvil

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neighbournodeof nodei

th

o

x,y spec u,v .p .T,p,rb

stagnation conditions value at the centroid of an element condit ions at a shockinthetest problems specified value

identify the coefficients inthe u,v,p, T.p;¢interpolation functions.

respecti vely

value at the throatof a converging-divergingnozzle componentsinthexandydirections,respect ively condit ions after a shockin two-dimensional problems 1,2 inlet andexitcondit ionsinthe test problems 1,2, 3 valuesat nodes1,2and 3,respectively

Su perscri p ts

control volume facek,wherek:=1,2,3 calculat ed value from aprevious iteration denotes terms whichresult from thelineari za t ion ofthe mass Bow,orBux,with respect tovelocit y denot esterm s which result fromthe linearizati on of the mass Bow,orflux,with respectto density denotes termswhichresult from the evaluat ionof the mass Bow,orBux,using availablevalues ofdensityand velocity

denotes a vectorquant ity

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Chapter 1 Introduction

1.1 Aims and Motivation of the Thesis

The primary purpose of thisthesis is to developa mass-weightedupwind-base d Control Volume FiniteElementMethod(CVFEM)for steady, two-dim ensional, viscous compressible int ern a l fiuidBowproblems .Themethod will be valid for use over a wide rangeof Mach numbers,from subsonicto supersonicflow speeds.

Theproposed methodistberesult of an effort to enhance theca pab ilities ofsome of the earlier CVFE:\t['s[I l-{I31 develope dfor two-dimensionalcompressible and/ or incompressibleBows. The proposedCVFEMisdevelop ed fromthe worksofHookey [91 and Saabes[101.Thesecondary goal of thisthesisisto testthe"mass weighted"

interpolation functionproposedbySchneider and Raw{I llandim plement ed. later bySaabas[101inthe context ofcompressibleflows.

The computersimulationof the Bows ofinte rest requires:(1) a mathematical mode lof the physicalproblem,whichin itsgenera lform.includes partialdifferential equationsandbounda ry conditionsthat govern thedist r ibutio n ofthe unkn own depende ntvariablesin the regionofint erest;(2) a numericalmethodforthe solution ofthe mathematicalmodelinregu la r- andirregular-s ha ped calculatio ndomains;

and(3 ) a computerim plem ent ation ofthenumericalmeth od. Themathematical

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modelin conjunctionwit h a givennumerica lsolution proced ureisreferredto as anumerical model. Inthis thesis.emphasisis placedon thedevelopmentof a nume ricalsolu tio nproce d ure. The mathematical modelused is notnew.Briefly.

it assumes the fluidis an idea lgas thatiscompressible and Xewtonian.and the two-dimensionalNevter-Stokesequationsandthe continui tyequation governthe moti onofthe fluid.

The CVFE:\rIproposedin thisthesis employsthree-nodetria ngu la r elementsto discretize two-dimensio nalcalcula t iondomains.Eachelement isfur t her discret ized insucha waythatupon assemblyofallelements.cont ro lvolumes areform ed around each node inthe calculationdomain.Element -basedinterpolationfunctions for thedepende ntvariab lesand the subdomain-typeMetho dofWeighted Residu- als(MWR ) is used toderi ve algebraicapproximat ionstothe gove rningequations.

In con ventional finite elementmethods(FE).I's),tbe Gale rkin me t hod of weighted residualsisused.where theweightingandinte rpol a tion functions are equ ivale nt.

InCVFE~I's,however,foreachnodein the calculationdomain. the weightingfunc- tionis set equalto one overa suitablychosencontrolvolumeassociatedwit hthe node,andzero elsewhere.Controlvolume finite elementmetho dsforfluidflow and heat transfer,therefore,involve theimpositionof physical conse rva t ion principles on finitecontro lvolumesinthecalculationdoma in.whichmakesthem amenable to easy physicalinterpretation,andtheirsolutionssatisfyglo ba lco nserva t ionre- quirements,even forcoarse grids.

Thepropose dmet hodisrestricted to thesolut ionof ste adytwo-dimension al Bows,however,itsbasicformu lat ion can be exten ded touns tea dy flows usingthe procedu resdescri bedin [14].Inadditio n.extensionsto three-d imens iona lproblems can beperform ed byincor porating theideasfrom the CVFEMprop osed.by Saa bas [I Oj.Inthe formul a t ion ofthepropos edCVFEM,thefluidis assumed.to bea

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perfectgaswit hconstantspecific heat.AlthoughrealisticcompressibleBows are usually turbu lent .the proposedCVFE~[has beendevelopedinthecontext ofthe wellunderstoo d physicalbehaviour of laminarNewto nian fiuidBows.mainlyto facilitatethe fonnulation.implementa t ion.and testi ng of the proposed method.

TheCVFE~{,sin[l l- {13]areconserva tive':theirformu lationsensure ccnserva- tionof the appropriatescalarvari ables evercontiguousand non-over lappingcont rol vo lum es.andconsiste nt calculationof fluxes across theboundari esof the control volumes. which togeth erensureconserva tio noftheappropriatedepen dent variab les overthe controlvolume andthe calculation domain (221.The proposed CVFE).t whichis anextension of those in[IJ-[13J,is alsoconserva tive.In addition.itmay beclassified as ashock captu ringor smearingmethod :alldiscontin uities.such as shocks, areauto maticallycaptured by themethod, butthese discontinuitiesare smear edoverseveralgridpoints.Discontinuousor extremelyrapid changesin the depende ntvariablesareapproximatedbyfinitegradients .The degreeofsmeari ng ind uceddependson themannerinwhichtheintegralconserva tionis enforced.and thetypeofinterp ola tion functionsused.Shock ca pt uri ng methodsareincont rast toshockfitti ngmet hods.where shocksaretreat edasinternalboundaries inthe calculationdomain,andare.therefore,main t ain ed as sharpdiscontinuities: the Rankiue-Hugoni orrela tions(16)andsatis fact ionofflowcompa tibilitycondi t ions are used todetermine conditions acrosstheshock.Inshock fitting met hods . the presenceofshocks must beknowna priori,however.shock capturing methodsdo notrequire suchknowledge,as all discontinuitieswit hintheflowfield are auto- matically captu red.Thecompressible ftowCVF E~[proposed byHookey19J was also ashockcapturi ngmet hod, butemployedadaptivegrids to red uce thephysical lThose finite-difference schemes which maintain the discretizedversion oftheconservati on sta t ement exactly(except forround-<lff'errors)for any mesh size overan arbitraryfiniteregion containing anyDumberof gridpointsare saidtohavetheCOru ennilillflpropertll{30J_

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smeari ngofshocksinherentto such met hods. TheproposedCVF E\Jattempts to address problems experiencedatincomp ressible outB.owboundaries when using the CVFE).(of Hookey[91.The interpolation schemesusedby Saabas(101were successfulin avoiding these difficulties ina colocated CVFE).Iforincompressib le Bow.TheproposedCVFE).{will,therefore,attempttoincorporatethesuccessful port ion ofbothCVF E)"('sint o one method.

Sincethe proposed numerical methodwas developed ina CVF EMcontext.no review willbegiven tothe othernumerical methods for compressible or incom- pressibleBuid Bow,for exam plethemethods developedinthe contex toffinite- difference methods (FDM's),finite-volumemethods (FVM's) ,andfinite-element methods(F EM's).A rather thorough review of these methodscanbefoundin[91.

1.2 A Review of CVFEM's

Cont ro lvolume finite element met ho ds offera combinationof the geomet ric flexi- bility offiniteelementmethodsand theease of physica linterpr etationassociat ed with finite volume meth od s.Theformulation ofa CVFEM involvesthe following steps(41'

1.Discret izat ion of thecalculation domain intoelements,and afurth er discretization into control volumes associated with the nodes of each element.

2.Prescrip tionof appropriateelement-based interpolationfunct ions for the depende ntvariables.

3.Derivationofdiscret izati onequations, which arealgebraic approximations to thegoverningdifferent ialequat ions.Thesu bdomain meth odof weighted residualsisused.toderive algebraicforms oftheintegral conserva tio nequatio n for each control volume.

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4. An element by elementassembly of thediscretizationequation.

5. Prescription of a procedureto solve theresult ing discretization equations.

The main featu res that distingu ishthe CVFEM'sthat have been developedfor thesimulation offluid flow prob lemsaxe the type of discretization,interpolation.

and solutiontechniquesemployed.The choice ofinterpolationfunctionis influenced by the dom a in discret izationemployed,as only certain functions are applicable to the desiredequal-order(or colocared)discretization.The followingsubsections review theeVF EM'sthat have been developedfor thesimulationofincompressi ble and/or compressible fluid Bows,andthe typesofinterpolation schemes used in these methods.

1.2.1 CVFEM'sfor Fluid Flow Problems

Ithas been demonstrated by Patankar[14J that anunr eal ist ic and checkerboard numericalsolut ioncan result when a linear interpolation scheme is used for pressure. Similardifficultiesaxe also encounter edin the discretizatio nof the continuity equationifbot h velocity and density axe interpolat edlinearly [9J.In the proposed method, density is calculated using the equation of state and is expressed in terms ofpressure. To avoid unrealistic solutions,specialpreca uti ons arenecessa ryif a colocated storage schemeis to be used.

One way to avoid such difficultiesinorthogonalgrid FDM'sis theuse of staggered gridsto store velocitycomponentsandpressur e atdifferent loca t ion s in the calculationdomain [14J. An advantage ofgrid staggering is that pressures are stored at thefaces of themomentu m contro lvolum es,which ensurestha tpressures at adjace nt,notalt ernate, nodes areused to evaluatethepressuregradients in the momentum equations. Similar ly,checkerboarddensityand velocity fieldswould be prevented due to the staggered storage [9J. Thelimit at ions of this stagge red

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storagearetha t twodifferent fam iliesofcontrol volumes areusedforthemomen- tumandconti nuityequations.and massconservationisnot stric tly satisfied o'..er themomentu m controlvolumes. Further,twodifferent cont ro l volume sets lead totedious and com plicatedbookkeeping,and thesedifficult iesarecompoundedin three-dimensional 80wproblems.

•-\.nunequal-o rder CVFE).{forincompressi bleflowswas pro posed byBaliga

and Patankar[I).Inthismethod,the domainisfirstdiscret ized by six-node tri- angularelements. These -macroelemects"are thendivided intofourthree-node

"subelements" by joining the midpointseach sideofa macroeleme nt.Pressureis storedattheverticesof themacroelements,andvelocity is sto redatthevertices ofallsubelements,whichresults in an unequal -orderscheme thatavoidspressure harmo nics intbe solu t ion.Amixed orderinterpolat ion scheme is appliedtotbis formulationinwhich pressureisinte rp olatedlinearlyin the macroelements,and thevelocitycom ponentsareinterpolated byBow-orient edupwind typefunct ions wit hinthesubelements.Polygonalcontrolvolumes used todiscretize the continuity equa tion areconstruct edaroundthe vertices of themacroelements,whilethose usedtodiscretizethe moment umequenoes are constructed around the verticesof the su belements.Thisform ulat ionbasbeenusedinconjunc t ionwit hSIMPLE, SI~I Pl ER.andSIMP LEC[2,6,8} solutio nalgorithms.Several disadvanta gesare associat edwit hthisformula tion,becauseoftheuse oftwotypesof elementsand controlvolumes:(l)massconservati onisnotstrictly satisfiedover themomentum controlvolumes and thevelocitiesatthevert icesofthemacr oelementsdo notex- plicitly enter the discret izedcontinuity equati on ; (2)thediscretiz edequat ionfor pressu reis quit edifferentfromthatof theotherdepende ntvariablesand asepa- rate equation solverisrequired;(3)the two differentcontro l volume setslead.to tedi ous and complicat edbookkeeping,andthesedifficulties wouldbecompounded

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in three-dimensions:(-I)excessivelyfine grids forvelocity,andthus an expensive computat ion.areneededin problemswithrela ti velylargepressuregradientssince the velocitydiscretizationis finer thanthe pressurediscret ization inthismethod;

and (5)given anequal numberof velocitynodes. an equal-orderdiscretizat ion wouldprovidea much betterapproximation ofboundaryirregu laritiestha n this unequal-orderform ulation.

To circumventtheproblems associated with the unequal-orderstoragemeth - ods,Prakasb and Patankar [31 introduceda CVFEM usingan equal-or de rvelocity - pressure interpolat ion,where pressureandvelocity nodes arecolocated.\Vithin each element .the velocitycomponents (whentreatedastransport ed scalars)are interpolatedusing flow-oriented(FLO)upwindfunct ions(withou tsourcetermef- fects), andthe pressureis interpolatedlinearly.The mass flow field is obtained. from linear interpolationof a pseudo-ve locityfieldand pressu re.The domain discretizatio nagaininvolvesthe formation ofpolygonalcontrol volumes abouteach node in the calculationdomain.Thevelocities inthe conti nuityequationare interpolated byusing a modified formof thediscrerieedmomentum equatio nderivedfor tha t control volume.The velocityateachnodeis splitintotwo parts:one is a func- tionoftbe nodalvelocitiesand anyvolumetricsource term(excluding the pressure gradient):andthe otherisrela t ed to the appropriatepressuregradientover that controlvolume. Conseque ntl y, at each node withinthe domain,a pseudo-velocity fieldand a pressurecoefficientcan bedefined.These two values are thenint erpo- lated linearlywithineach element,to provide aninterp olation sche mefor tbe'mass conserving'velocities.Whenint er pola t ing the'mass conserving'velocitieswithin anelement,the elementpressure gradientisused inconjunc tionwitha pressure coefficient.In this way,spuriousharm onicsin the pressure field,commonly enco un- teredin early equal-ord erFEM'sare avoided.A SIMPLERtype algorithmis used

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to solvethe resultingdiscretizationequat ions.This method has beenextendedto three-dime nsionsby LeDa inMuirand Baliga[15].

Inanattempttorefine his previousequal-ordermethod. Prakash[4] proposeda second method.Here.the Interpolationof v.elocity within an elementis performed usingflow-orientedupwindtypefunctions thatinclude the effects of pressuregradi- entsand othersourceterms in the strea mdirection(FLOS).The sameinterpola t ion function is used to derive algebraicapproximationsto both the moment umand continui tyequations.Itis the explicit inclusionof tbe appropriate pressuregradientsin the velocityinterpol ation functi ons tbatpreventsthe appearanceofspuriousoscil- lati ons ofvelocityandpressur e. Pressure is interpolatedlinearly.Butconvergence difficulties have beenreported by Prakash[4] whe na SIMPLEC solutionschemeis applied to solvethe discretizedmomentum and continuity equations.

Hookey andBaliga[7,

81

modifiedthe interpolationfunctionofPrakash[4] to includesource term effectsin directionspar allel andnorma lto the mean Howwithin eachelement.Furt herm ore. theyintroduceda SIM P LECtypesolutionalgorithm inwhich theyderivedvery completepressure-co rrec t ionequations[91.The result- ing pressure-correctionequation.however,invo lve d up to twenty -five neighbouring nodesintwo-dimensional problems,and this wouldmake extensions tothree- dimensionalproblemsimpractical. In [9j,Hookeysuccessfully reducedthe number of neighbouring nodes in the discret ized equations. Themaximumneighbouring nodesfor two-dimensionalproblems wasredu ced to eightfor bothpressure and velocity equations.This methodwas,therefore,bettersuitedfor three-dime nsiona l formulations.The discretizedmomentumandconti nuity equations were solvedin acoupledmannerusinga CoupledEquation Line Solver (CELS).The CELS solves the cou pledequatio nssimultaneo uslyalong agrid line in tbe calculationdomain, and iterativelyimproves theoverallsolutionby successivelysweepingthe doma in

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line byline,in alternatingdirections,until a desiredlevel of convergence is obtained [91·

In[111-[131, Schneiderand Raw have also proposedan equal-ordercolocated CVFEM. Based on quadrilateralelements rather than triangular elements. this CVFE),.{ also uses different interpolation functions: linearint erp ola t ion is applied to the diffusionterms. a :VIAss Weighted(?fA\-V) upwind scheme is used forcon- vection terms,and theconvect ion terms in the momentum equations explicitly include the pressure gradients to couplethe velocity and pressurefields to prohibit harmonic pressure fields. The use of triangular elements isbelieved to be more efficientfor irregular-shaped. domains and adaptivegrid methods, because triangular elements allow more freedomintheplacementofnodes within the calculation domain. Schneider and Raw used a direct banded solver for the solution oftheal- gebraic discretization equationsof two-dimensional problems.In three-dimensions, the cost of sucha solutionmethod is prohibitive.

Combininga modified version of the~IAWupwind scheme in triangular elements with the use ofa pseudo-velocityfield and thepressure gradientto define 'mass conserving'veloci ties. Saabas [10] developed an equal-orderCVF EM forthe sim ulationof three-dimensional tu rbulentincompressiblefluid Row.This method overcomes the problemswith outflow boundaries inherent with the method developed byHookey[91, and forms the basis forthe CVFEM developed inthis thesis forthesimulationofcompressibleRows.

1.2.2 Int erpolationSchemes

As discussedin the previoussubsection, there are some basic considerations invo lved in the solution of fluid How problemsto prevent spurioussolutions in colocated CVFEM's. The equal-orderCVFEM'sthat have been developed have used some

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JU

form of couplingbetweenpressur e andvelocity,usuallyatthe int erpolat ion function levels.toprevent solution harmoni cs.Thedevel op me nt ofappr opriate interpola tion functions mustalso considerthefollo wing:(l)preve ntion ofnega ti ve coefficients in the discretizedequations:(2)mini m izationoffalsediffusion tba tarisesdue to locallyone-dimensionalupwindi ng;and(3 ) suitable coupling betweenvelocityand pressu reto permit colocatedmethods.This subsec tio ndiscussesthe interpolation functions usedintheprevious CVFEM's.andprese nts theinterpolation functions usedintheproposed CVF EM.

Linearinterpo latio n was applied successfullyinearlyFDM's forbeattransfer problems . Butitgavephysical lyunrealistic oscillat orysolutio ns fo r convection- diffusionproblemswhenthe grid Peclet numberwas greaterthantwo [141.This difficultyoccurred,because linearinterpola tionofthecocvecrion term gave rise tonegative coefficients in thediscretization equa tions and this led todivergence.

Attempts to overcomethis difficultyled to thedevelopmentof upwinddifference schemes(91.

In theCVF E" l' s of Baliga etal.{il-(IO).thedependent variablesare inter- polatedbyBow-orien tedupwindtypefunctions. These functions eliminatethe oscillatory solu tionstbatoccur wit hlinea rinterpolationof conveceionterms,and provideamorereal ist ic approxim atio nof thephysical behaviourofthe dependent variablewit hin an element.To reduce false diffusion,wbichmaybe incurred when locally one-dimeusioualupwindingisused.tbeinterpolation functi onsare defined withrespectto alocallySow-oriented co-ordi natesystemspecific to each element.

These functionsarederived by solvingasimplified. versionofthe appropriategov- erningequationwithi nthe element ,written in terms of flew-orientedcoord inates.

In[41, a source term wasincluded in theinterpolat ionfunction toprovidefor a source-relatedvariationof thedependent variablein a directionparallel to the10-

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cal flow direction.In [6.i,81,this source-rel atedterm was modifiedto accountfor a sourceinfluenceindirections bothparallelandnormalto thelocalHow direction.

The inclusionof thissourceterm was shown to improvethequali tyof solutionin convection-diffusion problems involvingsources[6, 7J.

It was found that formulat ions thatincludedsourceeffects,especiallythe pressure gradient. in the element-basedinterpolationfunctions forthe convected scalar. allow for equal-ordercolocated formulatio nsinflow problems[6.7J-Thus.this approach appearsattractive: the sameinterpolat ion function can be used.for velocities in the moment umandcontinuityequatio ns,and spurious oscillationsof thepressur e orvelocityfields can be avoided.

It is not necessary,however, to use thesame interpolation functions inboth continuityandmomentumequations. It may actuaUybe disadvantageousto do so.Thebasic philosophywhen developingan interpolatio n function shouldbe to choose the appropriate.andphysicallymean ingful, type of interpolationforthe terms inwhich the inter polated scalarappears. This was done inPrakash and Patankar

131,

wherethe velocitywas interpolat edusing an upwindscheme for the convective term,and a linear scheme for the diffusion and'ext ra viscous terms', and the'mass conserving'velocitywas linea rly interpolated. by a pseudo-velocity andpressuregradient.

In thecontext of quadrilateralelements, Schneider and Raw [111 introduceda positive-coefficient upwindi ng procedure, in whichthecoefficients thatarise dueto the algebraicapproximat ionof theconvectio n fluxes are assured to be positiveat anelement allevel, andhenceata cont rolvolumelevel. In[12J.an upwinding formulation was presentedthatwas similarinform to the positivecoefficientscheme, however,thedirection alityof the flowwas moreclosely approxima ted,reducing false diffusion,butallowing forthe appearanceof some negativecoefficients.For

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tbe prob lemstested.itwas roundthat themagnitud eor the negati ve coefficients were such that they didnot poseanydifficulti es.Tbe effects ofdiffusion.bot hpar- allel and norm al to the meanBowdirection,andsourceterms.werealso explicitly accoun tedforin the interpola tio nfunction for theconvectedscalar.Inthismethod.

thepressure gradientswereincludedintheinterpo lation funct ionsforvelocitiesat an elemental leve l.thisinterpo lation scheme allowedforthedevelopmentoran equal-ordercclocatedformulatio nforincompressi bleBowproblems.

In thecontextortrian gu lar elements , Prakash [51in t rod ucedthedonor-cell GVF EM scheme as a meansor ensuringpositivecoefficients. This approach sta ted tha t the valueora dependentvariable convectedoutof a control volumemust bethe valueor thedependentvariableat thenode within thecont rolvolu me.

Altho ughit guarantees positive coefficients,this approach takes littleaccountof the directionality or tbeflow.and takes no accountor the effectsor diffusionand sou rce termson theinterp o lation or the convectedscala r. The positi veinll.uence coefficient schemeproposedby Schneiderand Raw[I llis a more attra cti ve approach toeliminate negative coefficients.even though it involvesmorecomputations.

Inthis thesisthe~lAWscheme developedby Saabas[10Jwillbe used fer the first timein the developmentof aCVF E),lappliedto compressibleandincompressible flows.Thisinterpo la tionwillpermitthe develcpmeeroracclocared met hod.and should eliminat etbeoutflowdifficultiesin the CVFEMorHookey[91whenitis usedtosolve purely incom pressible Rowproblems.

1.3 Key Featu res in the Formulation o f t he P r o- posed CVFEM

For reasonsor economyand simp licity,manyideas used inthe proposed two- dimensionalCVF EM werefirsttested inaquasi-one-d imensi onalformula tion.The

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13

derivationofboth the one- and two-dimens ionalCVFEM 'sare prese ntedin chapters 2 and3. respectively.Toavoidrep et it ion,the keyfeatures in both formulations arediscussed in thissection.The topicsincludedhere are as follows: (I) choiceof thedepe ndentvariables;(2)thenecess ityof specialprocedures for thecoloc a t ed methods;(3)appropriateupwindingof the dependentvariables:(4) lineari za t io n proceduresfa. the governingequations; (5) physical boundaryconditionsandtheir numericalim plem ent a t ion;and(6)the solution methodsused.

1.3.1 DependentVariables

The proposedCVFE~['sare based on the so-called primitivevariables.i.e.the velocity components(u,v ),temperature(T).andpress ure(p).Thecontinuity equation is used as an equatio nfor pressure,themom ent um equationsareused to solvefor the velocitycomponents,theenergyequatio nissolved fortempera ture.and density iscalculatedfrom an equationof state.

1.3.2 ColocatedDe p endentVariableStorage

Acoloca eed storageis appliedtoall the dependent variablesin tbe proposed CVFEM.Toprevent spurio usharmonicsolutionsthe How-orie ntedupwindtype functionproposed by PrakashandPa t an kar[3]. in whichthe mass Howfieldin each elementis determinedfrom linear interpolationof pseudo-ve locityand pressure fields, is employed.The use of the pseudo-velocity andpressu rein the discreti za- tio nof the continuityequation provides the necessary couplingbetweenve loci ty and pressure.This method was formulatedin the context oftheSIM P LERsol u- tion algorithm[14J- Saabas[10] employed a similarmethodto colocateddepende nt variablestorage in a three-dimensionalincompressibteCVFEM.

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1.3.3 Upwinding of the DependentVariables

The ~lAWscheme first introducedbySchneider andRaw[111 and later imple- mented bySaabas[10]in thecont extof a triangular elementCVFE~Lisusedas theinterpolationform ula forthe convectedscalarsin this thesis. .-\Spreviously mentioned,the evaluation ofthe),[.-\Wschemein tbecontext of compr essibleBows is asecond arygoal of this thesis.

1.3.4 Linearization Procedures

In CVFEM 's,tbediscretieed forms ofthegover ning equat ionsare obtained.by formulat ing suitablealgebraicapproximations of integral conservation statements applied to controlvolumesconst ruct ed around each nodeinthe calcula t iondomain.

In steady state problems,the conservation of a particular scalar variablerequires a balance between the net transportof thisvari a ble out of thecont rol volume by convectionand diffusion fluxes,and the net generation of the variable withinthe volume.Volume integra t ionofsource termsisused to determinenet generat ion, and nettransport is evaluatedfrom the integration ofconvect ion-diffusionfluxes over thesurfac e of the controlvolume. These fluxesmust belineari zedin an appropriate manner withregard tothe dependentvariables inorderto obta inthe discretizedequations . Thesolution toa fluid flow problemisobtai ned using an iterati veprocedurein whichlinearized discretizationequations are solvedduring eachit erat ion. The scheme used to linearize the mass flux is discussed inthis sect ion.

The derivation of the discretizedform of thecontinuityequation involvesinte- grationofthe mass flux.pii.over the facesof a control volume.Forcompressible flows,the mass 8uxisnonlinear,andifitis linearized using densities from a previ- ous iteration and treating velocityasthecurrent unknown,convergencedifficult ies

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10

couldarise.In thesteadyaccelerationofa supersonic Bow througha divergingduc t.

whenthedensity isinit ially assumed constantthroughoutthe duct,thevelocity would decrease to satisfymass conservat ion. Thedecreased velocitywouldcause pressuretoincreasethrough momentumconsiderations. The increased pressure would leadto an increased density,when theequation ofstate isused toupdate density,whichwhensubstitutedintothecont inuity equationfurtherreduces the velocity.Thisnumerical behaviouris oppositeto thatof the physical flow,and causes divergence.

Linearization of the mass Bux can also beachievedusingavailab levelocit ies.and treating densityimplicitly,or as tbe currentunknown.This approach,however,is viableonly at highsupe rsonicBow speeds,where changes in velocitybecome negligi- ble. Fora steadydeceleratingsubsonicBow through a divergingduct.if the velocity isinit ia llyassumed to be constantthroughoutthe duct.density woulddecreaseto satisfy mass conservation. Thischange in densitywouldca use pressuretodecrease, throughtheequationof state.andwhen this decreasedpressure is su bst it ut ed into the momentum equationsthe velocitywouldincrease.Theincreas ed velocity,when substitutedinto thecontinuityequation ,would furtherdecreasedensity,and this physicallyunrealist ic behaviourcont inues unt ildivergence.

Toovercomethese difficulties,a lineari zationtha t takesinto accountthe roles of bot hdensityand velocity in thesa t isfact ion ofmassconservati onhas to be formulated. This is accomplished by recognizingthe dual roleof pressure in compressible Bows: it acts00bothveloci ty and density,through momentum and state equations,respectively,to ensure mass conservation. A suit ableformulationinvolvesa fullNewt on- Repbsoa linearizationofthe mass flux,

g:

(1.1)

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where,

9

islinearized.wit hrespectto velocity,

9

islinearize d withrespect to density.

and

9

isevalua tedfrom availab levaluesof bothdensity andvelocity.

1.3.5 Boundary Conditions

Thenecessary boundary condit ions for a numericalsolutionofafluid flow prob- [em arediscussedinthis section. Furtherdetails on the implementationof these boundary conditionswillbe given inlat er chapters.

The boundarycond it ions appliedto compressibleflows dependuponthe type of flow,subsonicorsupersonic,thatexists at the boundary in questio n.In thissu bsec- ric e,boundaryconditions for inviscidsubsonic and supe rsonic Bows are discussed first,and then tbe boundaryconditions forviscous Bows will bepresented.

Invis cidSubsonicFlows

Atsubsonic inflow boundaries,thevelocitycomponents , u,v,pressure,p,and temper ature,T,are all specified. To define the mass enteri ngthe calculationdomai n it isnecessary to specify thepressure,which,along with the specifiedtemperature, is used todeter mine the density atinlet . This specifieddensity,theinlet geometry , andthe specified velocit ies toget her determine the inletmassflow.

In subsonic inviscidBows,theinfluenceof pressur e is ellip tic,as a pressure disturbance willeventuallybefelt everywhere in thecalculation domai n. The proposed numericalmethodaccounts for thisellipticbeha viour byinterp olat ing pressurelinearly.Withthis elliptic treatment,a downstream boundary condition for pressu rebecomes necessary. In theproposed method,thestatic pressure at outflowboundariesis specified.

The proposedmethod has beendeveloped for viscouscompressible flows,how- -ver,it maybe used tosolve for inviscidBows by settingviscosityequalto zero.At solid boundaries,atangency conditionisdemandedby definingthe dot prod uctof

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thevelocity vectorand aunitnormal to the walltobe zero:

(\.2)

Atout flow boundaries.toensurethat nodiffusive effects areencounte red.the diffusion transport ofmomentum andenergy iseffective ly set equal tozero using:

Vo ·ii = a

wheref/Jisa general scalar dependent variable.

Inviscid SupersonicFlows

(\. 3)

Insupersonicflows.asin subsonic flows.thespecifica t ionoftheinletmassflow rateisrequit ed.andthis is done by specifyi ngu, u,P. andTat inflowbou ndaries.

lnviscidsuperso nicflowsarehype r bolic,and out flow bound arycondit ions are notphysicallypropagatedupstream into the calculation domain.therefore.no boundarycondit ions are appliedor needed atout flow boundaries. To allowso- luti onsforinviscid flowswiththe proposedviscousCVFEM.the gradients ofu, u, andTare set to zeroatcurdcwbound ariesusing Eq.(1.3).

ViscousCom pressib le Flows

ViscousBows alwayshaveelliptic influencesdueto the presenceofthe viscous terms.however, tbe boundaryconditions areverysimilartothoseusedforinviscid flows,excepttbat the no-slip condit ion is alwaysusedon solidwalls.Wallbound- arycondi t ions fortheenergy equationare eitherthe fixed tempe raturevalueor adiaba tictype, Eq.(1.3).Inflow boundarycondit ions are thesameasforinviscid Bows, buttheelliptic beh aviourthatisnow present requir es specialtreat mentat outflow bou ndari es.This specialtreatmentisthe sameasthat usedinthesolutio n

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ofiuviscid flows with the proposed vriscousBowCVFEM:it involves the specifica- tion of zero diffusion at the outflow boundary for velocity and temperature,using Eq. (1.3).

1.3.6 Solutionof the Discretized Equations

In the proposed GVFE:\iI,a coupled set of nonlinear algebraic equations for velocity, pressure, and temperature are obtained as approximations of the corresponding integral conservation equations. The nonlinearities in the equations are resolved by Picard iteration, in which the coefficients in these equations are evaluated using the most recent field values. A segregated solution method is used, rather than a coupled method,to solve the final algebraic equations. The segregated solution method is believed to have the following advantages:(I) less storage is required, as coefficients do not have to be stored simultaneously;and (2) the advancement inthe solutions of each dependent variable do not have to be matched in each iteration. With regard to the second point, itisnot necessary to solve for allthe dependent variables to the same level of convergence in each iteration, emphasis can be placed on the solution of the pressure field in order to ensure that mass conservation is enforced more completely at each iteration. As the momentum equations are inherently nonlinear, and are solved through an iterative process, it is not necessary to solve the nominally linear algebraic equations exactly in each iteration. More detail about the solution procedures will be given in a later chapter.

1.4 Su mmary

The main ideas and motivations behind the development of the proposed CVFEM were presented in this chapter.The derivation of a CVFEM for quasi-one-dimensional CVFEM is presented in Chapter 2,the formulation of a two-dimensional CVFEM

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isdiscussedinChapter3.and thetesting ofthe proposedCVFEMformulationsis presented in Cha pter-I.

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Chapter 2 Formulation of the Proposed Quasi-One-Dimensional CVFEM

2.1 Intro duction

The main purpose oftbisthesis istoformulate and im pleme ntaCVFEM for steady, two-dimensio nal,viscous com pressibleRuid Bows. For reasonsofecon- omy and simplicity,however,many ideaswereinitial lytest ed inthecontextof quasi-one-dimensionalproblems. The followingtopicswillbeused. to presentthe form ulationof theone-dimens ionalCVFEMin this chap ter: (1)defini tion ofthe governingequa tions;(2)dom ai n discretizationdetails; (3) deriva t io nofcontrol volumeintegral conservationequations;(4) specificationof element-based interpo- laticn functions forthe dependent variables;and (5)deri vation of discreti zed.forms ofthe conservationequationsandboundaryconditions.

2.2 Governing E q uat io ns

The governingdifferentialequationsCorsteady,quasi-o ne-di mensio nal,viscouscom- pressibleBowof a perfect gasthro ugh a duct of variablearea,W,are:

20

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Continuity :

1;

(pwu)::: 0 Moment um:

d dp d ( dU)

¹

d(d )

- (pw uu) :::-w- +- IJ.W- +S" w+- - IJ.-(uw)

dx: dx dx dx 3dx dx

Energy:

d d (k aT) sr

^I ⁽

dP)

dX

(,uwuT) :::

dX

;;;w;jX +

wq;

⁺~ uW(j;

State:

p =

;T

(2.1)

(2.2)

(2.3)

(2.4) whereuis the velocity.p isthe pressure.pisthe density.IJ.is the dynamic viscosity.

Tisthe absolute tem pera ture.kis the thermalconducti vit y.e"isthe specific heat at const a nt pressure.R isthe gasconsta nt . andS" and

sr

are the volumetricsource.

or generation,terms forthe momentum and energy equation.respectively. The proposed CVFEM is formulated forvariab lefluidproperties ,bute" was assumed constantmainlyto keeptheene rgy equat ionrelativelyuncomplicated.It should be notedthat ,if required.the viscous dissipationterm may be included insT.

These differential equationsmay becast in the following conse rva t ive forms [91:

dJ Sw (2.5)

dx

⁼

dg

~ 0 (2.6)

dx

where .Jisthe combinedconvection-diffusiontrans por t.Sis a source te rm,and9

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22

isthe mass Sow.pwu.Equation (2.5) representsthe momentum equation when:

J

=

pwuu-

t-tw~

andthe energy equatio n when:

"dp Id(d)

S

=

S - - +- - t.l- (uw )

dx 3wdx dx (2.7)

J=pwuT-~wc.!!..

'" dI

S=ST=

~ + .!:.~

Cp c;.dx ^(2.8)

Applyingtheappropri a t e censervationprincipleto a control volumeV.whichis fixedinspace, integralformsofEqs.(2.5) and(2.6)can be obtained:

(2.9)

f

s;

⁼ ⁰

Javds: (2.10)

where

av

isthesurfaceof thecontrol volume,andVisthe volume ofthecont rol volume.

2.3 Domain Discretization

Inthe proposed equal-orderone-dimensionalCVFEM.thecalcula t ion domainis discretizedby a distribution of nodes inthei:direction.and alldependentvaria bles arestoredat allof the nodes.The variationin the area of theduct is approximated byapiecewise linearcurve. Controlvolume(cv)faces are placed midwaybetween the nodes inthe calculationdomain.Asam p le discretizationofa variablearea ductis illustrated inFig.2.1,withthe duct shapeshown bythe solidline.

Piecewise linear approximationof theduc t boundary does not provideas ac- cura te a formulationas would result if the functionalrepr esenta t ion ofthe actual duct wasusedtodefine the area variationof thecont rolvolume. The tinear area vari at ion used here was chosen foritssimplicity and generality,and also tobe con- sistent with the proposedtwo-dimensional CVFEM,in whichboundarysurfaces

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are ap proximatedwithpiecewiselinearcurves.Theplacementof controlvolume facesmidway betweennodes wasalsochosenforsimplicity.Tbe controlvolumes employedfill tbe entirecalculati ondomain.do not overlap,and theirbounda ries do notcoincide with nodes,exceptattbe domain boundary.Thesechar acte ris t ics helpto formulateaCVF E"i tbat possessestbeconservative property

1 301.

2.4 Integral Conservation Equations for a Con- trol Volume

Withreferencetothecontrolvolume surro und ingnodeIinFig.2.1.Eqs.(2.9) and (2.10) canbe writ tenas:

r (

Similar cont ributionsfrom other) Jo -JvSwdx + elemen ts associated wit hnodei

+ (~ound~contributio ns,) = 0 (2.11) Ifapp hcable

g, + ₍^Similarcontributi ons from other) elements associat edwit h nodei

(Bound arycontribut ions, )

=

0

ifapplicable (2.12)

Wheretbesubscri pt0ind icat es tbecentro idof tbe element1·2.whichbydefinit ion isthe posit ionof the control vo lumeface.

2.5 Interpolation Functions

Toderivealgeb ra ic approxima t ions totbeintegralconserva tionequations,interpolat ionfunctionsforthe fluid properties, sources, duct area,and depend ent varia bles mustfirs t bedefined.The funct ionsusedinthisthesisaredescri bedinthissection.

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2.5.1 Interpola t ionofp,k,Cp,andS

Val ues of11.kandGoare supplied at thecent roidof each elemen t.and thesecen- troidalvalues areassumed toprevail overthe corres pondi ngelement.The source terms.S-and

sr.

inthemomentum and energy equations.respecti vely.arelin- earized usingTaylor's expansion114].ifrequired. and expressed asfollows:

S"=

s:

+

S;u..

ST= S;+S;T.."

(2.13) (2.14)

whereScreprese nts term sindependentofthe supersc ri pt ed variable, andSppro- vides fora lineardependen ceof S on the corresponding variable.Thevaluesof ScandSparecalculatedatthecentroidofeach element,andassumedtoprevail withinthe element.The averageof theappropriatedepende ntvariable.used inthe linearizedsourceterm ,isassumed to be the arithmeti c mean ofthecorres ponding valuesat thetwonodes oftheelement.

2.5 .2 Int erp ol a ti on of Area

The variableareaof a duct,w.isinterpo lated linearly withinanelement:

w=~z + b., (2.15)

this resu lts in a piecewise linearapproxima tionto theduct sha pe.asshownbythe dashed lines inFig.2.1.The coefficientsa.., and b. inEq.2.15are definedby the ductareas at thetwonodesof the element.andthexcoordin ates ofthenodes.

2.5.3 Interpolation of Pressure Withinan element,thepressureisint erp olated linearly:

p=a,.x + bp (2.16)

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The coefficients^GpandbpinEq.(2.16 ) aredefined by the nodalpressures andx coordinates,as shownin AppendixA. UsingEq.(2.16) .and thederivations in AppendixA.thepressure gradientcan be expressedas follows:

(2.17) Sinceapis constantwithin an element,(-dpj dx )can be included inS;.Eq. (2.13).

2.5.4 Interpolationof Velocity

In the propose dCVFE~I.different functionsareused to interpo latevelocityin the convectionanddiffusiontermsinEq.(2.1), andina mass conserving velocityused to evaluatethemass flow termpuns-

Interpolationof aCo n vec t ed. Velocity

"be~[AWint erp ola t ionscheme[101is used toeval ua te velocity when it is a con- ected scalar.In the contextof thisone-dimensionalCVFEM.the~[A\Vscheme reducesto a pureupwindscheme,therefore.the value of theconvected velocityon the cv faceis simplythe valueofvelocity at theupwindnode.

if Ui>O. U"= Ui

if u,<O. U"= U1+1 The ),-IAWscheme willbe discussed in Chapter 3.

Interpolation of Mas s Con se rvi n g Velocity

(2.18)

The mass conservingvelocity,u'",whichis used in the massBow,pwu,is int erpo- la te dby assuming alinearvariation of a pseudo-velocity,tt,and pressure coefficient, d".The mass conservingvelocitycanbe writtenas followsfor each nodewithin thedom ain :

m • ~(dP)

Ui =t.li-U.;

d;;:"

^(2.19)

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wherethe subsc ript.e.on thepressure gradientindicates thatthe elementalpres- suregradientisbeingused.The pseudo-velocity andpressurecoefficients areeval- uatedfrom thedlscrenzed form of the momentum equation . Ifonedivides the discretizedmomentum equatio nfor nodei,Eq.2.2i.by the coefficientmultip lying

u"

thenU,is thesum of allterms on theR_H_S.of theequation. except forthe pressure terms,dividedby this coefficient:the pressur e coefficient

eli

is the volu me of thecont rol volu me divided bytheu;coefficient.

The pseudo-velocities,Ui,and the pressurecoefficients.

cr:.

are knownatthe nodalpoin ts.In orde rto dete rminethevaluesofthesequantitiesonthecont rol volumefaces withinan element,linearinterp ola t ion ofthe nodalvalues is used:

d't =

adZ+bd

The evaluationof the

u ,

andd'tterms willbe discussed in Section 2.6.3.

Interpolati.on ofaDiffusedVel o city

(2.20) (2.21)

wbenvelocityistreatedas a diffusedscalar,for exampleinthe visco usterm ofEq.

(2.2). itisint erpo lated linearly withinan eleme nt. 2.5.5 Interpolation of Temperature

Temperatu reis interpo latedin a similarmanner tovelocity.Inconvec t ion terms. itis interpolatedusingtheMAW.orpure upwindscheme:

if u.>O , To=T;

if il;<0, To=Ti+1

andin diffusion terms,the temperatu reisint erp ola ted linearly:

T

=

^aTI+br

(2.22)

(2.23)

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27

2.6 Derivation of the Discretized Equations

To obtainalgebraicapproximationsof tbe integra l conservation equationsfora cont rolvolu me.Eqs.(2.9)and(2.1O).theelementcontributions toEqs. (2.11)and (2.12)are derivedandassem bledin an approp ria tema nner.Algeb raicappro:cima- tions oftheboundarycont ri butio ns ace then eval uated.ifapplicable.andaddedto the elementcontributions.Theproceduresinvolvedinderivingalgebraic approx- imationsofthecontrolvolumeintegralconservation equationand thebound ary conditionsare described in thissection.

2.6.1 MomentumEquation

Withineach element, thecombinedconvect ion-di ffusi ontrans port,Eq.(2.7),is evaluated at thecont rolvolume

I eee.

orcentro idoftheelement,indicated bysub- scriptO.Subst itu t ion ofthevelocity interpolation funct ionsintoEq.(2.7), and evaluatingthe expressionatthe contro lvolumeface gives:

(2.24)

RearrangingJoⁱⁿ^{terms of}^thenod al velocities givesthefollowing:

(2.25) Integrat ionofthesource termoverthevolumeenclosedbythe elementisgivenby:

Detailedderivati ons of thecoe fficientselfC2.dllanddl2in Eqs.(2.25)and(2.26) areincludedinAppendix B.

Expressi on s similartoEq.(2.25)ca nbe derivedfortbe integrate d flux across thecontrolvolumefaceineachelement.andwhen these expressionsare added,

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appropriately,with the volumeint egr at ed source,Eq. (2.26) ,and any applicable boundarycont ribution.thediscretizedformof the integral equation representi ng conservat ionof momentumis completed. Theresulti ng equation can becastinthe followingformfor a node i:

(2.27) wherethe summatio nis over the two nodesthatneighbou r nodeiin Fig. 2.1.The assembly ofEq.(2.27) is demonstratedin Appendix C.

2.6.2 Energy Equation

The integralequa tionrepresent ingconservation of ene rgyisdtscretizedin a similar manner to the momentum equation, with the dynamicviscosity,IL,replaced by k/ep,andS" replacedbyST.Theint egra ti on ofthe convection-diffusionduxat the control volumeface in an element.and thevolumeintegr ationof the source term is describedin AppendixB.The resu lti ngalgebraicapproximation of the integral conservationequ ationcan be writt en in thefoUowingform:

(2.28)

The assembly of Eq.(2.28) is described in AppendixC.

2.6. 3 Continuity Eq uat ion Introduction

Thediscretizationof the continuityequation will be discussed in thissection.The approachtaken hereisto use particul ar forms ofthediscretized momentumequation,Eq .(2.27),to definetwo new nodalfieldswhichwill beused inthe prescription ofsuitableinterpolationfunct ionsfor the mass conserving velocities.Equation

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(2.27)canbe rearranged asfollows:

(2.29)

Inthis express ion.(ap/ar.)istheaverage ofthe pressur egrad ient act ingevera controlvolumecenteredatnodeiof vo lume,j,\r0.DefiningU.andd':as:

u. = Ea::;;

⁺

c:

d;'~~~V

tbenEq.(2.29)canbe rewrit t enas:

Lli=

Ui-cr:{J~

_X_,

(2.30) (2.31)

(2.32)

Similar expressionscanbe writtenfor eachnod ein the computa tion aldomain.It should be not ed tbat tbe pseudo-velocity,

u ",

and tbepressurecoefficient.

et:.

fields should be e..'aluatedbeforethe discretizedmoment um equati onisunder-relaxedas willbe consideredin sectio n3.7.1.Furthermore,tbeDirichletboundary conditions on tbe velo city components(ifany)areincorpo ratedintothepseudo-velocityfields as follows:atthepoints wheretheve locityisspecified.the appropriate11, isset equal to the specified velocity,and thecorres pondingtr:issetequaltozero.Inthis way,the velocityinform ation av-ailableonthebound ary isincorporated directly intothe discretized equationfor contin uity.

Whe nthecontinuity equatio nisdiscret ized ,themass Bow islinearized wit h resp ecttodensity,P,and velocity,u:

(2.33)

- -

where

9

islinear ized wit hrespecttovelocit y,

9

islineari zedwith respect to density,and

~

^isdetermined fro m availablevalues of density andvelocity.Togenerate

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an algeb raicapproxima tionto thecontrolvolume mass conservationequation.Eq.

(2.10),it is necessaryto evaluate the mass flowing acrossthecont rolvolumeface ineacheleme nt.Theseapproxima teexpressionsforthe massflows are then assem- bledinanappropriatemanner to givethefinaldiscretizedform of thecont inuity equa tio n.Thefollowingsub sections briefly illustratehow eachofthe termsinEq.

(2.33 )is evaluatedat the controlvolumefacein an element,indicated by subscript O.Thisdiscussio nuses thenomenclature in Fig.2.1.

Algebrai c Ap p ro xi mationof90

The mass conservingvelocity,Eq.(2.19),isusedto evaluate the velocity intbe90 term.Linear interpolationis used to evalu atethepseudo-v eloc ityand the pressure coefficientat the controlvolu meface.and the algebraicapproximationof90can be defined asfollows:

- (_ 1'>

-P,)

90

=

(pw )o uo-d;)--

X2-XI

wherePois evaluatedexplicitlyusingthe~IA\Vscheme.

AlgebraicApproximation of 90

(2.34)

To evalua te90,the equa t ion of state, Eq. (2.4).is used implicitly to definetbe mass Rowra te.andtheresultingexpressio nis evaluatedatacont rolvolume face as follows:

• Wotto

90

=

KT

1PI (2.35)

In theproposedCVFEM, tbe density is interpolatedin anupwind manner, by upwindingboth thetemperature andpressur e in this equatio n .ThevalueofUo

in Eq.(2.35) is thelinearlyinter polatedmass conservingvelocityat the control volume face.

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31

Alge brai c App roximationof90

Theremaining term tobe approximatedinEq. (2.33) is 90, which iseval ua t ed with available densit iesand velocities :

9 0

=PfJIt'OUo

wherePoandUoareinterpolat edlinearly.

Co m p leteAlge braic Approxi mati on of theLin earizedMas s Flow (2.36)

The co mplete fonnof thealgebraic approximation of themass flowingacross a controlvolumefac e is assembledby addingEqs.(2.34),(2.35) and (2.36 ). The result ingexpression canbe writtenintermsof the nodal press ures:

(2.37) where thecomple te derivationof EllE2and 80isincluded in AppendixB.

FinalFor mof theDiscr etize d Continuity Equ ation

Expressions similar to Eq.(2.37) can bederivedfor thecont rolvolume facesin the two elements which makecont ri bu t ionsto thecont rolvolume for nodei, When these expressionsareadded appropriately,alongwith any boundary contri butions.

the discretized form oftheintegr almass conservation statement for a control volume iscomplet ed.Theresulti ng equationca n be cast in the followingform:

(2.38) wherethesummation isover thetwo nodesneighbouring nod e iinFig.2.1.The assemblyof Eq.(2.38)isdemonstratedin Appendix C.

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2.6.4 BoundaryConditions

During thediscussionofthe discret izedforms ofthe governingequations.mention wasmadeof theadditionofapplica bleboundarycontri but ions.Ifanodeiswith in the calculatio n dom ain.no boundarycont ri butionsare requir ed.and the equation as derived fromthe elementcontributions aloneiscomplete. For the nodes on inflow and outflow boundariesof aquasi-one-di mensio naldomain.the integrated flux outofthedomain,acrossthedomainboundary ,must be calcula ted andadded tothe discretized equa tion. Thefollowingsubsectionsdescribethe evaluat ion of this integrated Bux,and tbeimplementationofapplica ble boundaryconditions.

Specified Value Boundary

The specified value bound aryconditionisveryeasytoapply. Whenthevalueofa dependentscalar at nodei.

»;

isto be givenaspecified.value.r;q«'the discretiz ed equatio nfornode i is writteninthe followingtrivial form:

(2.39) This equationisimpleme ntedinthecontextof the discrerizedequat ionsby set ti ng allcalculated coefficientsequalto zero.and thenredefining:

a~

=

1 ; bf

=

d>.,.",,; (2.40)

This procedureisused tooverwrite coefficients in tbediscret izedmoment um,energy and conti nuityequationstospecify boundaryvalues of u,T,andp.respect ively.

Inftow and OutflowBo und aries

Atinflowand outflowboundaries,ifthe valueofd>isnotspecified.the gradien t of d> is setequaltozeroat the boundary:

~ =

0 (2.41)