A third-order-accurate upwind scheme for Navier-Stokes solutions at high Reynolds numbers

A tliinl~i,riler-;icc~lratc upivintl sclicnic i s prcscntcrl tor solw tion of tlic s tcx l y ?-I> Navicr-Stokcs equations in strciim-fiiiictionl vorticity foinii. TIIC S C I I C ~ C i s found lo hc accu ra t~ and st:hlc ;at high Kcynolds noinhcrs. A series o f tcst computations is pcrlhnnctl on flows with largc rccirciiliiting regions. In partiwli ir, highly i l ~ ~ ~ r a t i : s o l u t i o n s arc oht;iiocil for flow in a drivcn S~II I ITC cwi ty (whicli l ias hccotnc a s tamlan l tcst case for cvaluating thc iiccw racy, si;ibility, iind cificiency of Navicr-Stokes algorithms) iip to Rcynolds aiimhcrs u t I 04. 'Tlrcsc computations arc iiscd to critically cvalii;ilc tlrc accuracy of othcr cxisting firstand scconilortlcr-accur:itii upwiod schcn~cs. I n addition, computalions iirc ciirrictl out f o r Ilow in a ch:innel with symmetric suddcn crp;insion, flow in ;I ~ l i a n n e l with n symmctric;illy placcd blunt basc, ;mrl thc Ilowiicld of a n impinging jet. Good agrccmcnt i s ohtiiincd wi th the coinpiitations of other investigators as wcl l as wi th tlic aviiiliil>lv cx~~criincnliil d;ita. ~ * ' I b i s rcscarch war condiictcd undci McDonncll Dougkts Inrlcpcndcnt Research and Dcvelopmcnt Program. Lahoratorics; Member AIAA. 4' **Kcscaiclr Scicntist, McDonncll Douglas Kcsearch and whcrc Q i s a transport propcrty of tlic fluid. n lc i l iain criticism ofcmploying tlic rliffcrcncc opciators [ I l q . ( 1 ) ) is tlic introduction of a largc zcrtificial i l i f fw ion in thc dircction of thc hias, thcrcby rcsulting iii II consiilcmhlc loss of nccriracy. Furthcrmorc. wlii lc thc c c n t r ; l l i l i f f ~ ~ c n ~ c ~ l diffitsion opcr;itor in tlic Navicr-Stokcs equations i s t l i i id-ordci ~~cc i t r : i tc , tlic convcctioi i oI)criitor is only first-onlcr i icc~mte, residling in the ovcrnll accuracy of t h c algorithm hcing first-orrlcr in imcsll s i x . A t i q Wolf\htcin, ant1 1sraeli3 liwc lproposei~ i/ recond-oiiler-;icc~Ir;ltc upwind sc~miie, which is written Although this sclicmc has riot becn tcstcd ns cxtcnsivcly :is t hc first-onlcr-accur;Itc schcmc, t hc rclal ivc accuracy uf the convcction operator is i in ortlci Icss than that uf the diffusion ~ p ~ r a t o r . Forthcrmore, the schcnic tcnds 10 enhihit instability 511 high Reynolds niimhcrs. On tlic hasis o f a yon Ncumann typc stability analysis for thc linciirizctl cqualions, Alias, Wolfshtcin, and Israeli3 find that a Gauss-Seidel solution o f the second-order upwind scheme is stablc i f t h e mesh Rcynolds number is Icss than 2 t.$ (this value i s 2 for a centrcil-difference scheme). A third-ordcr-accurate upwind scheme is proposed in this paper. In this scheme, both the convection and diffusion opcrators arc thirtl-orrlci iiccuiatc. Thc resulting matrix rcmains diag onally dominant a t all Rcynolds numhcis. Heleared Io A M A Io publish in PII forms. R C 24 1J A 7 ( i J ) = [ai; la-. I ] ,