Computer Methods in Applied Mechanics and Engineering(15)

一些ME专业提升的论文。

2144T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–2146

AppendixA

Examplesofthepoint-operatornotationimpliedbysomeofthematrix–vectormultipliesinthepaperaregivenhere,forthe(relevant)caseofauniform,two-dimensional,staggeredgridwithequalgridspacing,D,inbothdirections(forwhichM=I).Theseoperatorscanallbesimplyderivedasspecialcasesofthoseusedforunstructuredmeshes[4],andthree-dimensionalversionsarestraightfor-ward.Notreportedherearetheregularization/interpola-tionoperators(E,H)whicharegivenby[36].Inwhatfollowsthesubscriptsiandjrefertotheithandjthcellsinthexandydirections,respectively,andthesuperscript(k),ifpresent,referstothekthcomponentofavectorquantitysuchasvelocityorgradientofpressure.ðDqÞð1Þ

ð1Þ

ð2Þ

ð2Þ

i;j¼qiþ1;jÀqi;jþqi;jþ1Àqi;j;

ð41ÞðGpÞð1Þ

i;j¼pi;jÀpiÀ1;j;

ð42ÞðGpÞð2Þ

i;j¼pi;jÀpiÀ1;j;ð43ÞðCsÞð1Þ

i;j¼si;jþ1Àsi;j;

ð44ÞðCsÞð2Þ

i;j

¼Àðsiþ1;jÀsi;jÞ;

ð45Þ

ðCTqÞ¼qð2Þ

ð2Þ

ð1Þ

ð1Þ

i;ji;jÀqiÀ1;jÀðqi;jÀqi;jÀ1Þ;

ð46Þ

ReD2ðLqÞðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

i;j¼qiþ1;jþqiÀ1;jþqi;jþ1þqi;jÀ1À4qi;j;k¼1;2:ð47Þ

AppendixB

Inthisappendix,wederivethetheoreticalestimatesfor

thevelocityerrorexpectedfromtheuseofmulti-domainboundaryconditionsforthePoissonequation.Wecon-siderthevelocityerrorsforpotential owaroundacylin-derandstationaryOseenvortexdiscussedearlierintheSection5.

B.1.Errorestimateforpotential owoveracylinderLetusconsideracircularcylinderofdiameterDsituatedattheorigininsidearectangulardomain[ÀL/2,L/2]·[ÀrL/2,rL/2].SidelengthLheredenotesthesizeofthesmallestDð1ÞandwerepresentNthesizeofthelargestdomainDðNgÞbyaL,wherea¼2g=2.Aspectratioofthedomainsisdenotedbyr.Toassessthevelocityerror,wecomputethevelocityerrorattopcenterofDð1Þðx¼

0;y¼r

LÞ.Otherpointsinthedomainscalesimilarly.Theexactpotential owsolutionatthispointforanunboundeddomainis

"u1þ D

2#

exact¼UL;ð48Þ

whereUisthefreestreamvelocityvalue.Thecorrespond-ingverticalvelocityviszeroatthispoint.

Toestimatetheerrorinducedbythemulti-domain

approach,thee ectofDirichletboundaryconditionsonthelargestdomaincanbeassessedusingthemethodofimages.Assumingthecylinderisplacedattheorigin,weobtain:X1uimages¼UþU

X1ðÀ1Þ

iþj

ðD=2Þ2ðy22

jÀxiÞ;

ð49Þ

i¼À1j¼À1

ðx2iþ

y2jÞ

wherexiandyjcorrespondstothedistancefromthecenter

ofÀthe(i,j)thÁcylindertothepointoferrorassessmentx¼0;y¼r

L.Substitutingxi=aLiandyandsubtractingtheexact(free-space)solution,j¼arLjþrLweobtain,theerror ¼uimagesÀuexact

1¼ÀUD2UD2XX1iþjr2Àjþ12a

2Ài2r2L2þ4a2L2ðÀ1Þi¼À1j¼À1r2jþ1222þi

UD2UD2p21

¼ÀXr2L2þ4a2L24ðÀ1Þjfcsch2½bjða;rÞ

j¼À1

þsech2½bjða;rÞ g;

ð50Þ

wherebjða;rÞ¼follows.Forthe4a

ð1þ2ajÞ.Thesumcanbebrokenupasj=0term,aTaylorseriesexpansionforlargeaisused.Forj50,1+2ajcanbereplacedby2ajforlargeaintheexponentials.Wethenobtain:

¼ÀUD2p2 1

L8a2

3

þCðrÞ!

þOðaÀ4Þ;ð51Þwherethesum

CðrÞ¼X

1ðÀ1Þj

csch2

prj þ2 prj !ð52Þ

j¼1

2sech2isindependentofaandcanbeevaluatednumericallyforagivenaspectratio,r.Thusweobtaintheestimate

$ÀUD2 1

L3þCðrÞ!4ÀNg;ð53Þwhichisthedesiredresultthatshowsthattheerrorde-creasesgeometricallywithincreasinggridlevel.Notethat

wecanbemoreprecisebynotingthatthesumC(r)goesrapidlytozeroforr>1,andincreaseslike1/r2forsmallr.Thuswecanalsowrite $

UD2½minðL;rLÞ

4ÀNg:

ð54Þ

B.2.ErrorestimateforstationaryOseenvortex

FortheOseenvortex,wefollowasimilarsetupastheprevioussection,butplaceanOseenvortexattheoriginofthedomain.ÀWenowconsiderthenormalvelocityerroratthepointx¼L

;y¼0Á.Assumingthattheboundaryoftheinnermostdomainisoutsidethevortexcore,wehave