Computer Methods in Applied Mechanics and Engineering(10)

一些ME专业提升的论文。

T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462139

~s¼SKÀ1Sc~

;ð28Þ

where~cisanarbitraryinputvector(withlengthequaltothenumberofdiscretecirculationvaluesonthegrid),~sisthesolution(withlengthequaltothenumberofdiscretestreamfunctionvalues),andtheoperatorSKSimpliesthefollowingoperations:

~cð1Þ¼~c8

;

ð29Þ>~cðkÞwherex2DðkÞnDðkÀ1Þ;~cðkÞ¼<

:

PðkÀ1Þ!ðkÞð~cðkÀ1ÞÞwherex2DðkÀ1Þ>;

ð30Þk¼2;3;...;Ng;~sðNgþ1Þ¼0;

ð31Þ~sðkÞ¼SKÀ1Sc~

ðkÞþbcs½Pðkþ1Þ!ðkÞð~sðkþ1ÞÞ ;k¼Ng;NgÀ1;...;1;ð32Þ~s¼SKSc~

¼~sð1Þ:ð33Þ

HereP(kÀ1)!(k)(k)!(kÀ1)isa ne-to-coarseinterpolationoperator

andPisitscoarse-to- necounterpartrestrictedtooDðkÀ1Þbybcs.

InconstructingP,itwouldbedesirabletopreserve(tomachineroundo )certainmomentsofthecirculationdis-tributionsothatthevelocitydecayratefarfromthebodyiscorrect.Inthepresentimplementation,weattempttopreserveonlythetotalcirculation.Switchingfrommatrix/vectortopoint-operatornotation,wewrite,forthetwo-dimensionalcase,

PðkÀ1Þ!ðkÞðc~

ðkÀ1ÞÞðkÀ1Þ

2i;2j¼~ci;jþ1ðkÀ1Þ1ðkÀ1Þ

2~ciÀ1;jþ2~ciþ1;j

þ12~cðkÀ1Þi;jÀ1þ12~cðkÀ1Þi;jþ1þ14~c

ðkÀ1ÞiÀ1;jÀ1þ14~cðkÀ1Þ1ðkÀ1Þiþ1;jÀ1þ4~ciÀ1;jþ1þ1ðkÀ1Þ4~c

iþ1;jþ1

:ð34ÞThe9-pointstencilleadstoaconservationofthetotalcir-culationandissecond-orderaccuratebasedonaTaylor-seriesexpansion.Wenotethatthecoe cientsinEq.(34)sumto4sincethecirculationinthe(dual)cellisthevortic-itymultipliedbythearea,andcoarsifyingthegridbyafac-torof2resultsinafactorof4increaseincellarea.Thethree-dimensionalversionofEq.(34)consistsofaveragingEq.(34)overtwoadjacent(i,j)planesofdatanormaltothevorticitycomponent,foreachofthethreecomponents.Forthecoarse-to- neinterpolationattheboundaryofthenext- nermesh,weusethevaluefromthecoarsermeshforthosegridpointsthatcoincide,andamid-pointlinearinterpolation(againsecond-orderaccurate)forthosepointsinbetween.

Wenotethatcirculationisonlystrictlypreservedifthereisnovorticityadvectingordi usingoutoftheoriginaldomain.Duringvorticitytransferfrom netocoarsemesh,circulationisonlypreservedtothelevelofdiscretizationerror,sincethediscretizationerrorisdi erentoneachmeshandadvectionanddi usionratesarethereforeslightlydi erent.Testsbelowcon rmthatchangesincircu-lationasstructurespassbetweenthedi erentdomainsareappropriatelysmall.

Utilizingthemulti-domaindescriptionofthecirculationandsolutionofthePoissonequation,wenowwritetheoverallsystemofequationstobesolvedateachtime-step.S IþbDt

K

ScðkÞ

ü 2IÀbDtCTC

cðkÞnþDtð3CTNðqðkÞnÞÀCTNðqðkÞnÀ1

22

ÞÞþDt

bckÞðcðkþ1ÞÃÞ þ½Pðkþ1Þ!ðkÞðcðkþ1Þn2

cð½Pðkþ1Þ!ðÞ Þ; k¼Ng;NgÀ1;...;1;

ð35ÞECSKIþbDtK À1

!SðECÞT~f¼ECSKScð1ÞÃþ1

2

ÀunB;ð36Þcnþ1¼cð1Þ

Ã

ÀS IþbDt2

K À1

SðECÞT~

f;ð37Þsnþ1¼SKScnþ1:

ð38Þ

Notethatinsolvingforthestreamfunctionatthenexttimestep,Eq.(38),wesavethecoarsi edcirculation eldsandstreamfunctionstouseontheright-handsideofEq.(35)atthenexttime

step.