Computer Methods in Applied Mechanics and Engineering(12)

一些ME专业提升的论文。

T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462141

5.2.PropagationofanOseenvortex

Inordertoevaluateerrorsassociatedwithvorticityadvecting/di usingthroughthecomputationalboundary,weagainusetheanalyticalsolutionassociatedwithanOseenvortex.Thevortexisinitializedat(x,y)=(0,0)andadvectedbyanotherwiseuniform owwithspeedequaltothemaximumvelocityofthevortex.ThevorticityandazimuthalvelocityarestillgivenbyEqs.(39)and(40),respectively,r¼q

butðxÀUtÞ2

þy2 withtheradius,rrede nedwith.

Again,Candtheinitialtime,t0aresetsothatatt=t0,themaximumspeedassociatedwiththevortexaloneisUandoccursatr=R.AgainwesetRe=300.

Fig.8showstheerrorinthevelocityattheoriginforadomainthatnominallyextendsto±5RwithD¼0:05.Sincethevelocitydecayslike1/r,ithasalong-rangeR

e ect.Toachievelessthan1%errorwithoutcorrectedboundaryconditions,thedomainwouldneedtoextendto±100R.Theerrorisinitiallyzero(evenwiththeuncorrectedboundaryconditions)duetosymmetry.Astimepro-gresses,theerrorincreasesandreaches25%forNg=1.Thisoccursasthevortexpropagatesthroughtherightboundaryofthedomain.WithNg>1,thevortexispro-gressivelytransferredtothenextlargestmeshatintervalsoftime5·2nÀ1,n=1,...,Ng.WithNg=5,theerrorstaysbelow1%uptonon-dimensionaltime80,whenitleavesthecoarsest,largestmesh.Therearesmalloscillationsintheerrorevidentduringgrid-to-gridtransfertimes.Theassociatedtotalcirculationchangesbyatmost5%duringthesetransfers.WithNg=10,errorfromtheboundaryconditionisundetectableuptotime100andtheerroriscontrolledbythesecond-orderdiscretizationerrorandstaysbelowabout0.2%.Thesolutionattime100isshowninFig.9onthelargestmesh.Themagni edregionisshownasinaninsetandshowscontoursofthevorticityandnormalvelocity.Bytime100,thevortexwouldhavephysicallydi usedtoacoresizeofabout1.6R,whereasthegridspacingonthelargestdomainis12.8R!Theveloc-ity eldnearthecoreiscompletelywrong,butthecircula-

tionisnearlyconservedandthisinducesthecorrectpotential owfarfromthecore.Thephysical(smallest)domainisalsodepictedontheplotand,asisshowninFig.8,theerrorattheoriginisstilllessthanaboutoneper-centofthecorrectvalueatthattime.5.3.Potential owoveracylinder

Asa nalexample,weconsiderthepotential owinducedatt=0+byanimpulsivelystartedcylinderofdiameterD.Theimmersedboundaryuses571equallyspacedLagrangianpointsandthedomainisde nedsnuglyaroundthebody,extendingto±0.55DineachdirectionwithgridspacingD=0.0055D.Weinitiateauniform

ow