Fourth Order Compact Formulation of Navier

Published in : International Journal forNumericalMethodsinFluids(2006)

Int. J. Numer. Meth. Fluids2006;Vol 50: pp 421-436

FourthOrderCompactFormulationofNavier-StokesEquations

andDrivenCavityFlowatHighReynoldsNumbers

E.ErturkandC.G¨ok¸c¨ol

∗

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EnergySystemsEngineeringDepartment,GebzeInstituteofTechnology,

Gebze,Kocaeli41400,Turkey

SUMMARY

Anewfourthordercompactformulationforthesteady2-DincompressibleNavier-Stokesequationsis

prented.TheformulationisinthesameformoftheNavier-Stokesequationssuchthatanynumerical

methodthatsolvetheNavier-Stokesequationscaneasilybeappliedtothisfourthordercompact

formulation.Inparticularinthisworktheformulationissolvedwithanefficientnumericalmethodthat

requiresthesolutionoftridiagonalsystemsusingafinegridmeshof601×601.Usingthisformulation,

thesteady2-DincompressibleflowinadrivencavityissolveduptoReynoldsnumberof20,000with

fourthorderspatialaccuracy.Detailedsolutionsareprented.

keywords:HighOrderCompactScheme;HOC;Steady2-DIncompressibleN-SEquations;FONS

Equations;DrivenCavityFlow;HighReynoldsNumberSolutions

1.INTRODUCTION

HighOrderCompact(HOC)formulationsarebecomingmorepopularinComputationalFluid

Dynamics(CFD)fieldofstudy.Compactformulationsprovidemoreaccuratesolutionsina

compactstencil.

Infinitedifferences,astandardthreepointdiscretizationprovidescondorderspatial

accuracyandthistypeofdiscretizationisverywidelyud.Whenahighorderspatial

discretizationisdesired,ie.fourthorderaccuracy,thenafivepointdiscretizationhaveto

beud.Howeverinafivepointdiscretizationthereisacomplexityinhandlingthepoints

neartheboundaries.

Highordercompactschemesprovidefourthorderspatialaccuracyina3×3stencilandthis

typeofcompactformulationsdonothavethecomplexityneartheboundariesthatastandard

wide(fivepoint)fourthorderformulationwouldhave.

∗

Correspondenceto:ercanerturk@

Downloadfigures,tables,datafiles,fortrancodesandetc.from.Allthe

numericalsolutionsprentedinthisstudyisavailabletopublicinthiswebsite.

Contract/grantsponsor:GebzeInstituteofTechnology;contract/grantnumber:BAP-2003-A-22

2

¨¨

¸OLE.ERTURKANDC.GOKC

DennisandHudson[1],MacKinnonandJohnson[2],Guptaetal.[3],SpotzandCarey

[4]andLietal.[5]havedemonstratedtheefficiencyofthehighordercompactschemes

onthestreamfunctionandvorticityformulationof2-DsteadyincompressibleNavier-Stokes

equations.

Intheliterature,itispossibletofindnumerousdifferenttypeofiterativenumericalmethods

fortheNavier-Stokesequations.Thenumericalmethods,however,couldnotbeeasilyud

inHOCschemesbecauofthefinalformoftheHOCformulationsudin[1],[2],[3],[4]and

[5].ThisfactmightbecountedasadisadvantageofHOCformulationsthatthecodingstage

israthercomplexduetotheresultingstenciludinthestudies.Itwouldbeveryuful

ifanynumericalmethodforthesolutionofNavier-Stokesequationsdescribedinbooksand

paperscouldbeeasilyappliedtohighordercompact(HOC)formulations.

Inthisstudy,wewillprentanewfourthordercompactformulation.Thedifferenceof

thisformulationwithReferences[1-5]isnotinthewaythatthefourthordercompactscheme

isobtained.Themaindifference,however,isinthewaythatthefinalformoftheequations

arewritten.Themainadvantageofthisformulationisthat,anyiterativenumericalmethod

udforNavier-Stokesequations,canbeeasilyappliedtothisnewHOCformulation,since

thefinalformoftheprentedHOCformulationisinthesameformwiththeNavier-Stokes

equations.Moreoverifsomeonealreadyhaveacondorderaccurate(O∆x

2

)codeforthe

solutionofsteady2-DincompressibleNavier-Stokesequations,usingtheprentedformulation,

theycaneasilyconverttheirexistingcodetofourthorderaccuracy(O∆x)byjustadding

4

somecoefficientsintotheirexistingcode.Withthisnewcompactformulation,wehavesolved

thesteady2-DincompressibledrivencavityflowatveryhighReynoldsnumbersusingavery

finegridmeshtodemonstratetheefficiencyofthisnewformulation.

2.FOURTHORDERCOMPACTFORMULATION

Innon-dimensionalform,steady2-DincompressibleNavier-Stokesequationsinstreamfunction

(ψ)andvorticity(ω)formulationaregivenas

∂∂

22

ψψ

+=−ω(1)

∂x∂y

22

1ω∂ψ1ω

∂∂ω∂ψ∂ω∂

22

+=−(2)

Re∂xRe∂y∂y∂x∂x∂y

22

wherexandyaretheCartesiancoordinatesandReistheReynoldsnumber.Forfirstorder

andcondorderderivativesthefollowingdiscretizationsarefourthorderaccurate

∆x∂

23

φ

∂φ

+O(∆x)−(3)

4

=φ

x

3

∂x6∂x

224

∆x∂φ∂φ

=φ−+O(∆x)(4)

xx

4

24

∂x12∂x

whereφ

xxx

andφarestandardcondordercentraldiscretizationssuchthat

HIGHORDERCOMPACTSCHEME

3

φ=(5)

x

φ

xx

φ−φ

i+1i−1

2∆x

φ

i+1ii−1

−2φ+φ

=(6)

∆x

2

IfweapplythediscretizationsinEquations(3)and(4)toEquations(1)and(2),weobtain

thefollowingequations

∆x

2

∂∂

424

ψ∆yψ

ψ

xxyy

+ψ−

−+O(∆x,∆y)=−ω(7)

44

44

12∂x12∂y

1

11ω1ω

∆x∂∆y∂

2424

−+O(∆x,∆y)=ψω−ψω+−

44

yxxyxxyy

ωω

44

ReReRe12∂xRe12∂y

∆y∆x∆x∆y

23232323

∂ψ∂ω∂ψ∂ω

−ωψωψ

xyyx

3333

−+++O(∆x,∆x∆y,∆y)(8)

4224

6∂y6∂x6∂x6∂y

Intheequationswehavethirdandfourthderivatives(∂

333344

/∂x,∂/∂y,∂/∂xand

∂/∂y)ofstreamfunctionandvorticity(ψandω)variables.Inordertofindanexpressionfor

44

thederivativesweuEquations(1)and(2).Forexample,whenwetakethefirstandcond

x-derivative(∂/∂xand∂/∂x)ofthestreamfunctionequation(1)weobtainthefollowing

22

equations

∂ωψψ

∂∂

33

=−(9)

−

∂x∂x∂x∂y

32

∂ω∂ψ∂ψ

244

=−−(10)

2422

∂x∂x∂x∂y

Andalso,bytakingthefirstandcondy-derivative(∂/∂yand∂/∂y)ofthe

22

streamfunctionequation(1)weobtainthefollowingequations

∂ψψ

33

∂ω∂

=−

−(11)

∂y∂y∂x∂y

32

∂ω∂ψψ

244

∂

=−−(12)

2422

∂y∂y∂x∂y

UsingstandardcondordercentraldiscretizationsgiveninTableI,theequations((9),

(10),(11)and(12))canbewrittenasthefollowings

4

¨¨

¸OLE.ERTURKANDC.GOKC

∂ψ

3

∂x

3

∂

4

ψ

∂x

4

∂

3

ψ

∂y

3

∂

4

ψ

∂y

4

=−ω−ψ+O(∆x,∆y)(13)

xxyy

22

=−ω−ψ+O(∆x,∆y)(14)

xxxxyy

22

=−ω−ψ+O(∆x,∆y)(15)

yxxy

22

=−ω−ψ+O(∆x,∆y)(16)

yyxxyy

22

WhenwesubstituteEquations(14)and(16)intoEquation(7)weobtainthefollowingfinite

differenceequation.

ψ

xxyyxxyy

+ψ=−ω−−−ψ+O(∆x,∆x∆y,∆y)(17)

∆x∆y∆x∆y

2222

ωω+

xxyy

12121212

󰀇

󰀈

4224

Wenotethat,thesolutionofEquation(17)isalsoasolutiontostreamfunctionequation

(1)withfourthorderspatialaccuracy.ThereforeifwenumericallysolveEquation(17),the

solutionweobtainwillsatisfythestreamfunctionequationuptofourthorderaccuracy.

Inordertoobtainafourthorderapproximationforthevorticityequation(2),wefollowthe

sameprocedure.Whenwetakethefirstandcondderivativesofthevorticityequation(2)

withrespecttox-andy-coordinatesweobtainthefollowings

∂

322223

ω∂ψ∂ψ∂ωωωω

∂ω∂ψ∂∂ψ∂∂

=Re−Re(18)

2322

+Re−Re−

∂x∂x∂y∂x∂y∂x∂x∂y∂x∂x∂y∂x∂y

∂∂∂ψ

3223224

ψψωω∂ψω∂ω

∂ω∂∂∂

=Re+Re+Re

24223

+Re

∂x∂x∂y∂x∂x∂y∂x∂x∂y∂x∂y∂x

∂∂∂∂ψ∂

3

ψ∂ωψ∂ωψ∂ω∂ωω

222234

−Re−Re−Re−Re−

322222

∂x∂y∂x∂x∂y∂x∂x∂y∂x∂x∂y∂x∂y

∂ω

3

∂ψ∂ω∂ω∂ψ∂ω∂ψ∂ω∂ψ∂ω

23222

=Re−

2322

+Re−Re−Re(20)

∂y∂y∂x∂y∂x∂y∂x∂y∂y∂x∂y∂x∂y

∂ψ∂ω∂ψ∂ω∂ψ∂ωψ∂ωω

2222334

∂

∂

+Re+Re+Re=Re

22342

∂y∂y∂x∂y∂x∂y∂y∂x∂y∂y∂x∂y

∂∂ω∂ω∂ψω∂ω

3

ψψ∂∂ω∂ψ∂

223422

−Re−Re−Re−−Re(21)

∂x∂y∂y∂x∂y∂y∂x∂y∂y∂x∂y∂x∂y

222322

(19)

IfwesubstituteEquations(18)and(20)forthethirdderivativesofvorticity(∂

33

ω/∂xand

∂

33

ω/∂y)intoEquations(8),(19)and(21)andalsoifwesubstituteEquations(13)and(15)

forthethirdderivativesofstreamfunction(∂ψ/∂xand∂ψ/∂y)intoEquations(8),(19)

3333

and(21)andfinallyifwesubstituteEquations(19)and(21)forthefourthderivativesof

vorticity(∂

4444

ω/∂xand∂ω/∂y)intoEquation(8),thenweobtainthefollowingequation

2222

∆x∆y∆x∆y

22

ω+ω−Reω+Reω+Reψω+Reψω=

xxyyxyxxxyyyyyxxxxyy

ψψψψ

661212

HIGHORDERCOMPACTSCHEME

5

∆x∆x

2222

∆y∆y

++ψψ

12121212

󰀈󰀈

222

Reψω−Reψω+Reω−Reω

yxxyxxyxxyyy

−Reψω+Reψω+Reψω−Reψω

2

2222

∆x

󰀇󰀇

∆x∆y∆y

ψψψψ

yxyxxyyxyxxyxxyy

12121212

󰀈󰀈

󰀇󰀇

∆x∆x∆x∆y∆y

2222

2

ψω−Reψω−Reω

yxyyxxxyxxxy

+Re++ψ

121212126

󰀇󰀈󰀈

󰀇

22222

∆x∆y∆x

∆y∆y

ψω

xyxyxyyyxy

ψω−Reω+Reψω+Re+−

2

612121212

󰀈

󰀇

∆x

2

∆y

2

ω

xxyy

+O(∆x,∆x∆y,∆y)(22)

4224

−+

1212

Againwenotethat,thesolutionofEquation(22)satisfythevorticityequation(2)with

fourthorderaccuracy.

AsthefinalformofourHOCscheme,weprefertowriteEquations(17)and(22)asthe

following

ψ+ψ=−ω+A(23)

xxyy

1

1

(1+B)ω(1+C)ω

xxyyyxxy

+=(ψ+D)ω−(ψ+E)ω+F(24)

ReRe

where

󰀈

󰀇

∆y∆x∆x∆y

2222

ψ−−

xxyyxxyy

−ωω+A=

12121212

2

∆x

2

2

∆x

−ReB=

ψ+Reψψ

xyyy

612

2

∆y

2

2

∆y

ψψ

xyxx

+ReψReC=

612

󰀈

󰀇

∆y∆x∆x

2222

∆y

ψ

xxyyxyxyy

−Re+ψψ+ReψψD=

12121212

󰀈

󰀇

∆y∆x∆x

2222

∆y

ψ

xyyyxxxxy

−Re+ψψ+ReψψE=

12121212

󰀈󰀈󰀇󰀇

∆x∆x∆x

22222

∆y∆y

ψψ

yxyyxxxyxxxy

ω−ω−++ψωF=

121212126

󰀈󰀈

󰀇󰀇

∆y

22222

∆x∆x∆y∆y

ψψω−ωω+ω+Re

xyxyxy

ψ+−

yyxy

612121212

󰀇󰀈

∆x

22

1∆y

ω−(25)

xxyy

+

Re1212

that,thefinitedifferenceEquations(23)and(24)arefourthorderaccurateWenote

󰀉

󰀊

4224

O(∆x

,∆x∆y,∆y)approximationofthestreamfunctionandvorticityequations(1)and

6

¨¨

¸OLE.ERTURKANDC.GOKC

(2).InEquations(23)and(24),however,ifA,B,C,D,EandFarechontobeequalto

zerothenthefinitedifferenceEquations(23)and(24)simplybecome

ψ

xxyy

+ψ=−ω(26)

11

ωω

xxyyyxxy

+=ψω−ψω(27)

ReRe

󰀉

󰀊

Equations(26)and(27)arethestandardcondorderaccurateO(∆x

22

,∆y)

approximationofthestreamfunctionandvorticityequations(1)and(2).Whenweu

Equations(23)and(24)forthenumericalsolutionof2-DsteadyincompressibleNavier-

Stokesequations,wecaneasilyswitchbetweencondandfourthorderaccuracyjustby

usinghomogeneousvaluesforthecoefficientsA,B,C,D,EandForbyusingtheexpressions

definedinEquation(25)inthecode.

InEquations(23),(24)and(25)insteadoffinitedifferencediscretizations,ifwesubstitute

forpartialderivativesweobtainthefollowingdifferentialequations

∂∂ψ

2

ψ

2

+=−ω+A(28)

∂x∂y

22

󰀇󰀈󰀇󰀈

∂ψ1ωω

∂∂∂ω∂ψ∂ω

22

1

(1+B)(1+C)+D−+E+F(29)

22

+=

Re∂xRe∂y∂y∂x∂x∂y

A=+−−−

B=−Re

C=+ReRe

D=

E=−Re+Re

F=++−

󰀈

󰀇

∆y∂ψ∆x∂ω∆y∂ω∆x

2422222

12∂x12∂y1212∂x∂y

2222

222

∆x∂ψ∆x∂ψ∂ψ

2

+Re

6∂x∂y12∂y∂y

222

∆y∂∂ψ∂ψ

ψ∆y

2

6∂x∂y12∂x∂x

󰀈󰀇

∂∆x∂ψ∂∂ψ∂

3222222

ψ∆y∆xψ∆yψ

+−Re+Re

1212∂x∂y12∂y∂x∂y12∂x∂y

22

󰀈

3222222

󰀇

∂ψ∆xψ∆yψ∆x

∂ψ∂∂ψ∂

∆y

+

1212∂x∂y12∂y∂x12∂x∂x∂y

22

󰀈󰀈󰀇󰀇

∆y∂ψ∂∆y∂ψ∂∆x∂∆x

23232222

ωωψ∂ω∆x

2

−

1212∂y∂x∂y1212∂x∂x∂y6∂x∂x∂y

222

󰀈󰀈

󰀇󰀇

∆yψ∂ω∆yω∆y∆x∆x

22222222

∂∂ψ∂ψ∂∂ω∂ω

+

+Re+−−

6∂y∂x∂y1212∂x∂y∂x∂y1212∂x∂y

2

󰀇󰀈

∆x∂

242

ω1∆y

(30)−

+

Re1212∂x∂y

22

WenotethatthenumericalsolutionsofEquations(28)and(29),strictlyprovidedthat

condorderdiscretizationsinTableIareudandalsostrictlyprovidedthatauniform

gridmeshwith∆xand∆yisud,arefourthorderaccuratetostreamfunctionandvorticity

equations(1)and(2).WeprefertocallEquations(28)and(29)FourthOrderNavier-Stokes

HIGHORDERCOMPACTSCHEME

7

(FONS)equations.TheonlydifferencebetweenFONSequations(28)and(29)andNavier-

Stokes(NS)equations(1)and(2)arethecoefficientsA,B,C,D,EandF.InfacttheNS

equationsareasubtoftheFONSequations.WenotethatFONSequations(28)and(29)

areinthesameformwithNavier-Stokes(NS)equations(1)and(2),thereforeanyiterative

numericalmethod(suchasSOR,ADI,factorizationschemes,pudotimeiterationsandetc.)

udtosolvestreamfunctionandvorticityequations(1)and(2)canalsobeeasilyappliedto

fourthorderequations(28)and(29).Moreover,anyexistingcodethatsolvethestreamfunction

andvorticityequationswithcondorderaccuracycaneasilybemodifiedtoprovidefourth

orderaccuracyjustbyaddingthecoefficientsA,B,C,D,EandFintotheexistingcodeto

obtainthesolutionofFONSequations.Ofcour,whenthecoefficientsA,B,C,D,EandF

areaddedintoacondorderaccuratecodetoobtainfourthorderaccuracy,evaluatingthe

coefficientswouldrequireextraCPUwork.Thismightbeconsideredasthecostofincreasing

accuracyfromcondordertofourthorder.

3.NUMERICALMETHOD

RecentlyErturketal.[6]haveprentedanew,stableandefficientnumericalmethodthatsolve

thestreamfunctionandvorticityequations.Thenumericalmethodsolvethegoverningsteady

equationsthroughiterationsinthepudotime.Inthisstudy,wewillapplythenumerical

methodErturketal.[6]havepropod,toFONSequations(28)and(29)andsolvethesteady

drivencavityflowwithfourthorderaccuracy.Fordetailsaboutthenumericalmethod,the

readerisreferredtoErturketal.[6].WhenweapplythenumericalmethodtoEquations(28)

and(29),weobtainthefollowingequations

󰀇

󰀈󰀇󰀈󰀈󰀇󰀈

󰀇

2222

∂∂∂

∂

1−∆t1−∆tψ∆tψ

2222

n+1nnnn

=ψ+∆tω−∆tA+∆t(31)

∂x∂y∂x∂y

Thesolutionmethodologyofthetwoequationsarequitesimple.Firstthestreamfunction

equation(31)issolvedintwosteps.Forstreamfunctionequation,anewvariablefisdefined

asthefollowing

󰀈

󰀇

∂

2

1−∆tψ=f(33)

2

n+1

∂y

UsingthisvariableinEquation(31)weobtainthefollowingequation

󰀈

󰀇󰀇

󰀈

n

2

∂1∂ψ∂

1−∆t(1+B)+∆t

n

+D

Re∂x∂y∂x

2

󰀈

󰀇󰀇󰀈

n

2

∂ψ

∂1∂

nn+1nn

ω

=ω−∆tF−∆t1−∆t(1+C)+E

2

Re∂y∂x∂y

󰀈

󰀇󰀇

󰀈

n

2

1∂∂ψ

∂

+∆t(1+B)−∆t

n

+D

Re∂x∂y∂x

2

󰀇

󰀈󰀇

󰀈

n

2

∂

∂∂ψ1

∆t(1+C

nn

)+∆t+E(32)

ω

2

Re∂y∂x∂y

8

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¸OLE.ERTURKANDC.GOKC

whereVisthespeedofthewallwhichisequalto1forthemovingtopwallandequalto0for

thethreestationarywalls.Forcornerpoints,weuthefollowingexpressionforcalculating

thevorticityvalues



•••

•••

1



V

1



•1

11

•−2

ψ+ω=−

(38)



22

3∆h92h

2

1

11

1

••

224

whereagainVisequalto1fortheuppertwocornersanditisequalto0forthebottomtwo

corners.ThereaderisreferredtoSt¨ortkuhletal.[8]fordetails.

UsingthisvariableinEquation(32),weobtainthefollowingequation

󰀈

󰀇󰀈󰀇

n

2

∂∂ψ∂1

g=ω

nnn

−∆tF+∆t+D)

1−∆t(1+B

2

Re∂x∂y∂x

󰀈

󰀇󰀇

󰀈

n

2

∂

∂ψ1∂

−∆t

+∆t(1+B)+D

n

Re∂x∂y∂x

2

󰀈

󰀇󰀇

󰀈

n

2

∂ψ∂1∂

nn

ω(36)∆t(1+C)+∆t

+E

2

Re∂y∂x∂y

Inthisequationtheonlyunknownisthevariableg.Bysolvingatridiagonalsystem,we

obtainthevalueofgateverygridpoint.ThenwesolveEquation(35)forvorticity(ω

n+1

)by

solvinganothertridiagonalsystem.

Inacompactformulation,thestencilhave3×3points.Thesolutionatthefirstdiagonalgrid

pointsnearthecornersofthecavitywouldrequirethevorticityvaluesatthecornerpoints.

However,thecornerpointsaresingularpointsforvorticity.Guptaetal.[7]haveintroducedan

explicitasymptoticsolutionintheneighborhoodofsharpcorners.Similarly,St¨ortkuhletal.[8]

haveprentedananalyticalasymptoticsolutionsnearthecornersofcavityandusingfinite

elementbilinearshapefunctionstheyalsohaveprentedasingularityremovedboundary

conditionforvorticityatthecornerpointsaswellasatthewallpoints.WefollowSt¨ortkuhl

etal.[8]anduthefollowingexpressionforcalculatingvorticityvaluesatthewall





•••

•••

1



11

V1



11

2

−4

ψ+ω=−

(37)



22

22

3∆h9h

2

1

1

111

1

44

󰀇󰀈󰀈󰀇󰀈

󰀇

222

∂∂

∂

1−∆tf=ψ+∆tω−∆tA+∆t∆tψ(34)

222

nnnn

∂x∂x∂y

Inthisequation,theonlyunknownisthevariablef.Wefirstsolvethisequationforfby

solvingatridiagonalsystem.Afterthis,whenweobtainthevalueoffateverygridpointwe

solveEquation(33)forstreamfunction(ψ

n+1

)bysolvinganothertridiagonalsystem.

Aftersolvingthestreamfunctionequation(31),wesolvethevorticityequation(32).For

this,similarly,weintroduceanewvariablegwhichisdefinedasthefollowing

󰀈

󰀇󰀇󰀈

n

2

1∂ψ∂

∂

ω

n+1n

=g−∆t(35))

1−∆t(1+C+E

2

Re∂y∂x∂y

HIGHORDERCOMPACTSCHEME

9

4.RESULTSANDDISCUSSIONS

TheschematicsofthedrivencavityflowisgiveninFigure1.Inthisfiguretheabbreviations

BR,BLandTLrefertobottomright,bottomleftandtopleftcornersofthecavity,respectively.

Thenumberfollowingtheabbreviationsrefertothevorticesthatappearintheflow,which

arenumberedaccordingtosize.

ForeveryReynoldsnumberconsidered,wehavecontinuedouriterationsuntil,inthe

computationaldomainboththemaximumresidualofEquations(23)and(24),whichare

givenas

󰀋

󰀇

󰀌󰀌

󰀈

󰀍

󰀌

󰀌

n+1n+1n+1n+1

󰀌

(39)

R=maxabsψ+ψ+ω−A

ψ

󰀌

yyxx

󰀌󰀌

i,j

󰀇

󰀌

󰀌

11

󰀉󰀉

󰀊󰀊

n+1n+1n+1n+1

1+B1+C=maxabs

ω+ωR

yyxx

ω

󰀌

󰀌

ReRe

arelessthan10

−10

.Suchalowvalueischontoensuretheaccuracyofthesolution.Atthe

residuallevels,themaximumabsolutechangeinstreamfunctionvaluebetweentwotimesteps,

(max(|ψ

n+1n−16n+1n

−ψ|)),wasintheorderof10andforvorticity,(max(|ω−ω|)),itwas

intheorderof10

−14

.Obviouslytheconvergencelevelsarefarmorelessthansatisfactory,

howeversuchlowvaluesdemonstratetheefficiencyofthenumericalmethodudinthisstudy

whichwasprentedbyErturketal.[6].

Usinganefficientnumericalmethod,Erturketal.[6]haveclearlyshownthatnumerical

solutionsofdrivencavityflowiscomputableforRe>10,000whenagridmeshlargerthan

256×256isud.Withagridmeshof601×601Erturketal.[6]havesolvedthecavityflow

uptoRe=21,000usingthenumericalmethodalsoudinthisstudy.Inordertobeableto

obtainsolutionsathighReynoldsnumbers,followingErturketal.[6],inthisstudywehave

udalargegridmeshwith601×601grids.Withthismanynumberofgridpointsweobtained

steadysolutionsofthecavityflowuptoRe=20,000withfourthorderaccuracy.

Figures2to6showthestreamfunctionandvorticitycontoursofthedrivencavityflow

betweenRe=1,000andRe=20,000.Thefiguresshowthevorticesthatareformedinthe

flowfieldastheReynoldsnumberincreas.Fromthecontourfigures,weconcludethatthe

fourthordercompactformulationprovidesverysmoothsolutions.

InFigure7weplotaveryenlargedviewofthetoprightcorner(wherethemovingwall

movestowardsthestationarywall)ofthestreamfunctioncontourplotforthehighestReynolds

numberconsidered,Re=20,000.Inthisfigurethedottedlinesshowthegridlines.Asitisen

inthisenlargedfigure,fourthorderstreamfunctioncontoursareverysmoothevenatthefirst

tofgridpointsnearthecorners.

TableIItabulatesthestreamfunctionandvorticityvaluesatthecenteroftheprimaryand

condaryvorticesandalsothelocationofthecenterofthevorticesforfuturereferences.

ThistableisingoodagreementwiththatofErturketal.[6].

UsingRichardsonextrapolationonthesolutionsobtainedwithdifferentgridmeshes,

Erturketal.[6]haveprentedtheoreticallyfourthandsixthorderaccurate(O∆xand

4

󰀌

󰀈

󰀍

󰀌

󰀉󰀊󰀉󰀊

n+1n+1n+1n+1n+1n+1n+1

−ψ+Dω+ψ+Eω−F(40)

yxxy

󰀌

󰀌

i,j

󰀋

10

¨¨

¸OLE.ERTURKANDC.GOKC

O∆x)streamfunctionandvorticityvaluesatthecenteroftheprimaryvortex.TableIII

6

andIVcomparestheforthordercompactschemesolutionsofthestreamfunctionandthe

vorticityvaluesatthecenteroftheprimaryvortexwiththefourthorder(O∆x

4

)Richardson

extrapolatedsolutionstabulatedinErturketal.[6].Theprentsolutionsandthesolutions

ofErturketal.[6]agreewitheachother.

5.CONCLUSIONS

Inthisstudyanewfourthordercompactformulationisprented.Theuniquenessofthis

formulationisthatthefinalformoftheHOCformulationisinthesameformoftheNavier-

StokesequationssuchthatanynumericalmethodthatsolvetheNavier-Stokesequationscan

beeasilyappliedtotheFONSequationsinordertoobtainfourthorderaccuratesolutions

(O∆x

4

).Moreoverwiththisformulation,anyexistingcodethatsolvetheNavier-Stokes

equationswithcondorderaccuracy(O∆x

2

)canbealteredtoprovidefourthorderaccurate

(O∆x

4

)solutionsjustbyaddingsomecoefficientsintothecodeattheexpenofextraCPU

workofevaluatingthecoefficients.

Inthisstudy,theprentedfourthordercompactformulationissolvedwithaveryefficient

numericalmethodintroducedbyErturketal.[6].Usingafinegridmeshof601×601,asit

wassuggestedbyErturketal.[6]inordertobeabletocomputeforhighReynoldsnumbers,

thedrivencavityflowissolveduptoReynoldsnumbersofRe=20,000.Thesolutionsobtained

agreewellwithpreviousstudies.Theprentedfourthorderaccuratecompactformulationis

provedtobeveryefficient.

ACKNOWLEDGEMENT

ThisstudywasfundedbyGebzeInstituteofTechnologywithprojectnoBAP-2003-A-22.E.Ert¨urk

isgratefulforthisfinancialsupport.

REFERENCES

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4

Type,JournalofComputationalPhysics85(1989)390–416.

2.R.J.MacKinnon,R.W.Johnson,Differential-Equation-BadReprentationofTruncationErrorsfor

AccurateNumericalSimulation,InternationalJournalforNumericalMethodsinFluids13(1991)739–

757.

3.M.M.Gupta,R.P.Manohar,J.W.Stephenson,ASingleCellHighOrderSchemefortheConvection-

DiffusionEquationwithVariableCoefficients,InternationalJournalforNumericalMethodsinFluids4

(1984)641–651.

4.W.F.Spotz,G.F.Carey,High-OrderCompactSchemefortheSteadyStreamfunctionVorticityEquations,

InternationalJournalforNumericalMethodsinEngineering38(1995)3497–3512.

5.M.Li,T.Tang,B.Fornberg,ACompactForth-OrderFiniteDifferenceSchemefortheSteady

IncompressibleNavier-StokesEquations,InternationalJournalforNumericalMethodsinFluids20(1995)

1137–1151.

6.E.Erturk,T.C.Corke,C.Gokcol,NumericalSolutionsof2-DSteadyIncompressibleDrivenCavityFlow

atHighReynoldsNumbers,InternationalJournalforNumericalMethodsinFluids48(2005)747–774.

7.M.M.Gupta,R.P.Manohar,B.Noble,NatureofViscousFlowsNearSharpCorners,Computersand

Fluids9(1981)379–388.

HIGHORDERCOMPACTSCHEME

11

8.T.Stortkuhl,C.Zenger,S.Zimmer,AnAsymptoticSolutionfortheSingularityattheAngularPointof

theLidDrivenCavity,InternationalJournalofNumericalMethodsforHeatFluidFlow4(1994)47–59.

u=1v=0

→

TL1

TL2

u

=

0

v

=

0

PrimaryVortex

u=0

v=0

BL1

BL2

BL3

u=0v=0

BR1

BR2

BR3

Figure1.Schematicviewofdrivencavityflow

1

0.99

0.98

0.97

0.970.980.991

Figure7.StreamfunctioncontoursforRe=20000,enlargedviewoftoprightcorner.

φ

x

φ

y

φ

xx

φ

yy

φ

xy

φ

xxy

φ

xyy

φ

xxyy

2∆x

2∆y

∆x

2

∆y

2

4∆x∆y

2∆x∆y

2

2∆x∆y

2

∆x∆y

22

10002500500075001000012500150001750020000

-0.118938-0.122216-0.122306-0.122060-0.121694

-2.067760-1.940547-1.918187-1.907651-1.900032

(0.5300,0.5650)(0.5150,0.5350)(0.5117,0.5300)(0.5100,0.5283)(0.5100,0.5267)

0.17297E-020.30735E-020.31896E-020.30022E-020.28012E-02

1.1182222.7390713.7564254.9653046.125275

(0.8633,0.1117)(0.8050,0.0733)(0.7750,0.0600)(0.7450,0.0500)(0.7200,0.0433)

0.23345E-030.13758E-020.16135E-020.16663E-020.16083E-02

0.3542711.5142922.1820432.4901682.885054

(0.0833,0.0783)(0.0733,0.1367)(0.0583,0.1633)(0.0533,0.1717)(0.0483,0.1817)

-0.49242E-07-0.14271E-05-0.14014E-03-0.34006E-03-0.46117E-03

-0.76887E-02-0.33647E-01-0.309618-0.456877-0.554507

(0.9917,0.0067)(0.9783,0.0183)(0.9350,0.0683)(0.9267,0.0883)(0.9300,0.1050)

-0.65156E-08-0.65524E-07-0.11017E-05-0.22287E-04-0.78077E-04

-0.29579E-02-0.12183E-01-0.30153E-01-0.140685-0.243259

(0.0050,0.0050)(0.0083,0.0083)(0.0167,0.0200)(0.0383,0.0417)(0.0583,0.0533)

-0.14459E-020.26237E-020.32865E-020.37247E-02

-2.1155442.3223102.4241362.497615

-(0.0633,0.9100)(0.0700,0.9100)(0.0767,0.9100)(0.0800,0.9117)

-0.42053E-100.37451E-080.11742E-070.27181E-07

-0.66735E-030.24926E-020.33420E-020.49718E-02

-(0.9983,0.0017)(0.9967,0.0050)(0.9950,0.0050)(0.9933,0.0067)

---0.57300E-090.23598E-08