Computer Methods in Applied Mechanics and Engineering(12)

FortheOseenvortex,wefollowasimilarsetupastheprevioussection,butplaceanOseenvortexattheoriginofthedomain.ÀWenowconsiderthenormalvelocityerroratthepointx¼L

;y¼0Á.Assumingthattheboundaryoftheinnermostdomainisoutsidethevortexcore,wehave

一些ME专业提升的论文。

T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462145

vexact¼

C

2pðL=2Þ

;

ð55Þ

whereasthesolutionwithDirichletboundaryconditionsis

uCimages¼X1X1aLiþL2paðÀ1Þiþji¼À1j¼À1aLiþLrjÞ2;ð56Þþðasothattheerrorcanbewritten(aftersimilarsimpli cationstoanalysisintheprevioussection)

¼uimages(

Àuexact

¼C p pX12pÀ2þp

arcsch2arþarðÀ1Þi½cschðcþiða;rÞÞi¼1

þcschðcÀÞÞ

'

iða;rð57ÞwherecÆiða;rÞ¼2arð1Æ2aiÞ.Again,wecanexpandthetermsforlargea.Inthiscase,however,caremustbetakeninevaluatingthesumforlargeasincethesumiszeroto rstorderina.Weobtainasimilarresulttotheprevioussection,namely

$

C

minðL;rLÞ

4ÀNg:

ð58Þ

Afurthercancellationbetweenthesecondandthirdtermsontheright-handsideofEq.(57)occurswhenr=1.Wecanthenshowthat $

CÀNL

16g

whenr¼1:

ð59Þ

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