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e 9 9 9

SPECRA MEHOS INFU YM

Hu

nue or ompuer Applaon n ene n ngneerngNAA Lanle Reearh Center, apton Vrna 66

T A Zag

NAA Lanle Reerh Cener, aton, Virinia 66-

ITRODUCTIOn ern aea o ompuonal u n pera meho havebeome he prevlng numerl ool or argeale alulaon. Th erinl he e or uh hree-menonl applaon a reulon of oogeneou urbulene puon f nn n her ow, an global wether oeling. or an other appliaion uh he traner bounr lr reatng ow, opreiblew, n magneohronm, perl meho hve proven o be

a vable alernaive o he raonal ne-erene an eelemenehnueperl meh re hrere b he epnn he lun n

erm of global an, uuall, orthogonal polnomial. ine he inineeenth entur h ha been a tnar anaial ool for linereprble ierenl euon. Nnlnere peen nerble lgebr ule, even on a oern ompuer. Thee iutie were uroune eeivel in he erl 970, n onl hen perleho beome ompeve wh lernav algorhm B he preen

tie however peral eho have been rene an extene o the

I Th US Gnn h h ight o rein nonexluie oltee lien n nd ony pight onening thi pper

9

Annu.Re

v.FluidMech.1987.19:339-367

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340 SSAN & ZANG

pon w an pobl n u an a onl aabl b nqu

Nual peal o fo paal nal quaon w ognall velope b oolo Tou appoa wa popob Blnova 9) an Hauwz ag (952) nueal opuaon w onu b lbean (19) T epn o opungnonlna e an a v awbak unl Oag (1969) anlaen a (19) evlope e ano eo a ll o ebakbone of an lag-al peal opuaon

Te eo an oe ue n u ean po o 90 ae now pa Galekn o e funaenal unknown ae e panon on an quaon o a v b

nqu ue n aa anal T avn of opu ae fable an alenave aon ee e peal olloaon enque n w e unanal unknown ae oluon valu a leooaon pon an e ee epanon ue olel fo e pupoeof appoang evave T appoa wa popo b Ke Olg (971) an Oag (197)

Man uful von of pa o av bn veop n9 an epeall ung e 1980 evew ue an of e

en nnovaon an foue on olloaon enque n

veon o eal applable o nonlnea poble We veapplaon o bo opeble an nopble ow o vou awll a nv ow an alo o all eang ow In no bev we o no ove e applaon o eeoolo agneoona aop an oe elae el Moeove we e ouelv o e e-enonal applaon o wlleabealgo wle ung oe wo- an vn one-enonal applaon ofo novl al o

W non e o o al fo o ne aonal

oal efeene applaon n oe l an eoeal evlopen n e neal anal of peal eo Te onoapb Golb Ozag (19) b o an applaonvlop po o 9 I wll be fene a a GO e followng v ea a ov n e poeng e b Vog e al(9) lu-naal applaon peall ulg nqu a

e b an Han (985) Copebleow ppaonae ovee b Huan e al (985a) Te ole of peal eo noolog plan b aau

Baee (85) e boo b anuo

e al (198) onan a al pon of an pea agoan peen an eauve uon of e eoeal ape of nual o I eene eae a H

Annu.R

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UAMTA

SPCTRL MODS 4

T moiaio or u o pra mo umria auaio

m rom ara approxmao propr o orooa pooma xpao Suppo or xamp a a uo u(x) xpa a ruat C o []:

NUN(X L anT.(x), )

wr Tn(x) o aro x) aa om o paoo or pra)

a = (2/c)f1 u(x)(x) (l_X)-1 dx (wr C 2 a Cn or 1. uuox o ori io a ourir o ri A imp iraio--par arumGO C ra a

na a n , or a > 0 )proi a U i ii ria Coqu approx

maio rror ra ar a agraa T rap org r o a orr aura xpoa or or praaura Our pma or rw o umra mo orpara rta quao ta x pra aura o drta outo

T appoxmaio u ri i pia o pra Garimo. A ara approximaio rm pra ooaio io o rpoao ra xpaio ) u rpa oo(2) or xpao o w odo

wr j ar pia o-a ooao poi i ] or moprom opma o o ooao po

o 0, (5)T o o ooao pot a xrm aura approxmato C C 2) o t tga appag Equato 2)

N

a (2IN) ; 1 U(j)T (xJ (6)j

wr o 2 a orw o mar wr 2) or (6)

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42 SSAIN & ZANG

ud for h xpanion oin h xpanion (1) i dirniadaayay o fom e appomao o wavr deae aeruird for robl a and

A grapial diinion bwn radiional approxiaion and pralon provdd n gur for h pl a of iang drvaivo e uo +s 2+4) o [ ] om e alue oe uoa a ni nubr o grid poin. A ni-dirn or nilnmeod ue loal omao o emae deae wee a pel hod u global inforaion. n i gur a ond-ordr (nra)

RROR RRR

0 0 ' R

' RRR RR

Fgr oron o nederence ( nd hehev ecrl rig dier-enion. Th oid curv rereen he ec uncion nd h dhed crv hernuerc poon. he olid ne re he ec ngen 0 nd he dedlne he oie ngen The error n oe noed i he nuer o nerv N

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SPECTRAL METHODS 343

e-derece med cmpared w a Cebye pecral cllcamed Te e-derece apprxma emae e derae aay fm e parabla a erplae e fuc a = ad

e w adjace grd p. A eparae parabla ued a eac grdp. Te pecral apprxma e er ad ue all e aalablenfoation about th funtion If th a + g oint thn therpa pyal frm wc e derae exraced a degree ad e ame plymal ued fr all e grd p Ne a elcal med prduce a ecd-rder accurae derae w e errrang a l / wha th o fo th global tho aexpeally

An ntial at of an tal tho i th hoi o anion

funton. Con r e cae a buded area oan ouiere are e m famlar expa fuc bu ey are ly apprprae r prbm w perdc budary cd e apprpraeolloaton ont on [0,2n] a

j=nI j , l (7)I e geeral perdc cae rmalzed 1 1 ] e apprpraecla f fuc e Jacb pymal. Te prper cllca pare geerally and e xa of th la plymal reaed(CHZ C. 2 h ot oonl u aobi olnoal a hCebye ad egedre e.

O a ubuded dma e bu cce f aguerre r Hermeplymal are rarely adable N ly are fa rafrm aalablebu ee epa fuc ae relaely pr relu prpere(GO C A beer apprac cmb a mappg w a Furero Chbhv n th a vaiabl o (19) a hown thatpecra accracy ca be aceed fr () (0 w e appg =* c ad a full Furer ere prded a () exb a

lea algebrac decay a Mreer f() a expeal deay ea Furer ce ere wll uce Te laer cae equale ahbhv ere n w *() palar (194) ed a th o (o ee) Chbhv olnoal wok wll on 0 whnobn wth an onntal ang ovi that ua fata expeally.

Te prce f umercal derea parcularly mple we eexpa fuc are rgmerc plymal arg frm Uj' ealue f a j e cmpue

lak = I) L exp ()

(8

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344 INI & ZNG

nd en usesL ika exp (ikxJ ()

o pproxmedujdx Xj. Te Fs Fourer Trnsorm (FFT) cn beused o elue bo o e sums gen boe Te ol cos o compunge dere n s mnner s 5log rel operons ll operon couns gen n s reew presume or smplcy s powero nd e complex FFT s used oweer FFTs llow prme

cors o 3 nd 5 re jus s ecen nd re wdely lble nd rel olcomple FFTs oer 0% sngs (Temperon ). ] Te FFT cn lso be used o derene uncons re expnded n Cebyse

seres snce expnsons n ese specl Jcob polnomls reduce o cosneseres. Moreoer n erms o e Cebyse coecens deres reobned by smple recurson relons (CHQZ C. ). For Cebyseseres e ol operon coun or derenon s 5 log +

For clsscl xpnson uncons e mrx rprsens derenon edqujdxq Dqu,s nown n closed orm (oleb e l . 98). Unlke e derenon mrces or lerne locl dscrezonsese mrces re ull Hence e mrx-ecor mulplcon produces e dere e collocon pons coss operons se

operon couns sugges or 1 rnsorm meods re ser or derenon n mrxecor mulplcon On mode sclr nd ecor compuers e rnsom meods become ser n emrx-ecor mulply meods or beween nd 3 (CHQZ C. 2).

n mporn ssue n mny pplcons o Cebyse specrl meodss e mnner n wc e boundry condons re enorced Drcleboundry condons re generlly srgorwrd. Neumnn boundrcondons b nocd b lng bound lus o nsu desed nol dere or by buldng e boundry condon no derenl operor (ree e l 5) For yperbolc sysemscrcersc boundry condons re rul necessy (Goleb e l Hussn e l 5) Cnuo Qureron (1) dscuss ow

o mplemen crcersc boundry condons or mplc me dscrezons Cebyse specrl meods e e dngs (oer sndrdne-derence scemes) ey requre e sme number nd ype oboundry condons s e nlycl oulons o e problem nd no specl derence oruls re requred e boundry.

Te specr o e dscree d erenon operors

Dqre n mporn

crcersc o numercl meods. For Fourer pproxmons o perodc problems ese re obous purely mgnry nd growng s or Dl nege rel n growng s j 4 or D2. Indeed or perodc

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ET ET 4

prbem c a 0 e Frer eenvae are exacy eqa er anayc cnepar T mean a Furer pecra medprpaae e numercal lun w zer pae errr T llraed n Fre r e prbem we n

x n

x

Te an pae f e ne-derence n apparen werea eFue lun ndnuable fm e ue ne Of cue nreal prblem varale cecen r nnnear erm wl nrducennzer (b l eavely mall) pae e.

Fure play e eenvale f a Cebyev apprxman djdxn 1 ] w meneu rce bndary cndn a = + 1 .Te eenvale are predmnany manary bu d ave neave reapar. Te ablue value f e are eenvale rw a Tee

eenvalue may be rprn a Ae al e pedc dcreeprble a prely manary eenvale werea e nnperdc cnnuu prem a n dcree eenvaue Neveree Fure decnvey e nare f e eenvale f e dcree prblem and ee arecrcal fr b mederencn med and erave ceme Teeenvae f Cebyev appxman d/d meneuDcle bnday cndn a = 1 and = ae eal andneave ad e lare eenvale rw a (Gleb man19

pace wen e vn a evun prbem c a Ut Lwere e perar L cnan al e paal dervave ne cmbne apecra dcean a andard ne-deence ecnque e me devave Te eap Fr Adam-Bafr Cran-Ncn

1.5,-

'-_

c cc c on prod o ipl v uon ho xct ouion pnd th curv

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46 AN & ZANG

ad uge-utta cee ae e oe mot como ued (HQC. 4) Te talt ego of tee ceme deped upo te patalopeato. e talt p opete of oue metod ae ualtatel te ame a toe fo ecodode cetaldeece pata opeato.

Hoee te ece talt lmt tpcall a facto of (l!) malefo oue appoxato ee n te ode of te get pataldeate tat appea

Te talt popete of ee metod ae moe utle oexample eap og ucodtoall utable fo adecto polmuc a ut+ux 0, ce te dcete egeaue o te patal opeato ae egate eal pat O te ote ad ecodoe daafot and RugeKutta metod ae tctl tale (ad ot eaklutale le te oue coutepat) fo te ame eao

Te tpcal tmetep lmtato o Cee metod ae j fot-deate opeato ad j o ecod-deate oe. Tee aefa moe tget ta te aalogou etcto fo formgd tedeece appomato Te ae fom te codg of te colocatopot ea te oudae ee gue ). Altoug t codg ecee m me e rered fr he hgh re hmetod ad qute adatageou fo polem t ouda lae.

2

6

E

6

I

-

2

Fgre genvlue he hehev deivtve oertr fo " 6

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SPETRA TDS 34

wever a ubtatal dadvatage r prblem wt very lttletructure ear te budare. It ca be allevated t me degree bymappg but a mappg t a um grd cterprductve becauet detry te patal accuracy.

Cebyev tme-tep lmtat dappear we mplct te dcretzat are emplyed e prcpal diulty btag ecetlut te reultg mplct equat ce te matrce tatrepreet te deretat peratr are ull. I me pecal cae atdrect lut metd are avalable. ee typcally requre lw-rderplymal cecet ad multdmeal prblem at mt eperdc drect (GO C. ad Mer et al )

e ue mplct tecque mre geeral tuat requre teratve metd. a bee e te majr develpmet te curretdeade (CHZ C. ). et u dete a typcal lear mplct ytemarg rm a pecral dcrea a u e mple eravecemeRcard metd ut

(1)

were a accelera paramee. e mar wll be ll ad wllave egevalue tat grw rapdly a te umber grd pt creae.e ulle te matrx de t preclude teratve metd ce

tram tecque r deretat permt te matrxvectr prductu be cmpued lg pera raer a 2. elw vergee tat reult rm te large egevalue a beamelrated by precdtg. I t cae te bac teratve ceme

( 1 1 )

were H a precdtg matr wll accelerate cvergece H a gd apprxmat t , ad t wll be relatvely expeve H readly verted. e rmer dt met by lwrder te

deece (Orag 9) ad teeleme (Delle Mud 9)apprxmat t ltug te latter cdt certaly ldr e-dmeal prblem tee partcular precdtg becme reagly expeve t vert a te dmealty te prblemreae

e mt attractve apprac t very large prblem t cmbe ale accrate bt mre readly verted precdtg wt multgrd tecque. pectral multgrd metd take advatage te act tatmt teratve metd are gly eectve reducg te errr cm

pet crrepdg t te upper al te egevalue pectum butare very ecet te remag lwrequecy cmpet. u a multgrd metd e cmbe teat te dered grd wt (muc

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34 AINI & ZANG

eape) eaons on sessely oase gds. e deals of s meodae admedly sble b ey ae been aefly desbed n a sees ofpapes (Zang e al. 192 194 ee e al 95 Pllps e al. 9).Band e al (195) ae demonsaed a many peod poblemsan be sessflly soled n s manne o e need fo any peondonng

noe een nnoaon as been e deelopmen of speal mldoman enqes. ese allo speal metods o be appled o geomees fo a sngle global epanson s ee mposble o elsenadsable bease of esolon eqemens a ay dely oe edoman. In a mldoman enqe e fll doman s paoned no(no neessaly dsjon) sbdomans ese may be paed ogee

a nefaes o else ey may oelap. e al pa of e paedmlt doman meods ae e nefae ondtons. ese may be epessedeplly as onny ondons (Oszag 19 Kopa 9) may asefom a aaonal pnple (Paea 194) may onss of negal onsans (Maaaeg & ee 9) o may e enfoed by a penalymeod (Deles & Hall 99). e spealelemen meod of Paea so dae e mos gly deeloped of ese. Many enqes s as sopaame elemens (Kozak Paea 19) ae been booedfom onenonal neelemen meodology. ndeed ee ae many

smlaes n s appoa o e -eson of e ne-elemen meod(Babska & Do 91) Fge 4 llsaes a speal-elemen gd as ell

Figure A petllnt gid p) nd epndng nuel lun bm)f w iul lind (ute G. . nd nd A. T .

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SPECTRAL METHODS 49

as h coud souion fo ow as a cyind (Kaniadakis a 98 .n ss onvn is chvd wh d nub of subdoainswh h nub of id ons on ach sub doan ncass Th scaovansubdoain hods w dvisd by Mochoisn (19) and

a cuny bn nvsad xnsvy n uo.

IVICID FLOWhas h sis uid-dynamca obms a hos ha a sadyinviscid incossib, and ioaona In s of h vocy ona

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0 SSAIN! & ZANG

whr h dnsiy is a quadrai funion of " or subsoni owhis proble is ellipi Sree e al ( 198 have demonsraed he reainy ha sra uird ehods ahiv for his ase Thy havapplied hs hniques o he wodimnsional ow pas a iular ylin

der. Usin a r 2000 rid poins hey have obaind an sia for hfree-srea Mah nuber a whih he ow rs beoes soni areeso six diis wih he resuls of Van Dyke & uann (198 based on aRayleih-ensen expansion.

or ransoni ow h ponial quaion is of mixd yp wih asuprsoni pok bddd in a subsoni ow. Thre wil b a soni linand usually a shok ha rinas h suprsoni reion. Th hallninnuerial ask is o obain a onvrd soluion o h disr nonlinearpoenial equaion Speral muliid mehods have proven ompeiivewih nidirn hods and hav ahivd subsanial onois insora (Sr al. 198.

Sill wihin h onnes of invisid ow, on an obain h s ofvoriiy b y resorn o h Eulr quaions

o V- (pq)

oq +qVq = p V,sS=Oat

' (1

whr q is h vloiy is h prssur Sis h nropy and = /As is he ae for all nuerial ehods, he real deliay is he reaenof soni lines and shok wavs. Th disoninuiis arisin fro shoksare espeially roublesome for speral mehods The lobal naure of hese

approximaions indues osillaions in h soluion ha are ssenially ofa ibbsphnomnon yp Th hihfrquny omponn of h soluiondays vry slowly. This par of he sperum mus be lred o produa prsnable approiaion. daild ahaial analysis of lrin hniques n Fourir spral mhods for linar hyprbol problswih disoninuous soluions has bn prsnd by Majda a. (978.A posproessin proedur ha involvs ahin h ompued soluionwih simpl disoninuiis has bn disussd by Abarbanl al. (98.

The rs appliaions of speral ehods o opressible ows foused

on h ramn of shok wavs in on-dimnsional probls (olb al. 198 an Hussaini 198 1 Taylor al. 981. As is h as wih

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SPECTRA METODS

ntedeence ehods seca ehods fo obes non shockseque soe tye of exc o c nuecal dssaon. In souonso aa deena equatons the exc dssaon ay ake he foof a nea secta te o ay conss of an aca-scosy te hats dded to te uer equtons. s rt vsosty my be nonner.oxaons based on Chebyshe oynoas ay be sabe wthouany exc dssaon snce he Chebyshe deae oeao contans ct dssaton (otteb e a. 1981.

Mos nesatons hae conned hesees o obes whose souon (een n wo denon; Sake l 98 wee ehe ecewse onsnor ese eewse ner. No one s yet exbted setr souton to obe wh boh shock waes and coex ow sucue n whch

seca accacy was aaned (Hussan e a. 98b.he dcues tha shoc-catun seca ehods exhb ae nodue to any ntnsc d cuty n eson tansonc and suesonc owsKova et a (198 soved te Rneb ow robem w s smoottwo-denson tansonc ow wth a cosedfo souon by a Chebyshe seca ehod They wee abe o ehb he usual seca accuacyon hs cas of obes.

The shoc-tn aoach ouazed by Moe (98 fo nedeence schees was adaed o seca dscetzatons by Saas e a1.

( 198. s tenque vods te bbs enomenon by tretn te soas a bounday ahe than s n nteror reon of te ow. t s beto ows that conan a few eoetcay se shocks Soe obesfo whch hh-esouton esus hae been obaned by hs ehod aethe shockotex nteacon (Saas et a 982 the shockubuence ne ton (n et a 198 nd te bun-body robem (Hussn e 198c.

BUNDARY LAERIn any aeodynac acaons he bounday-aye equatons ae neonom nd usefu mode of vsous eets esey wen ouedntertvey wt n nvsd mode for te outer ow (RD 1981 Insay aabes the wodensona bounday aye s descbed by

8 ( 81) 81 81 v -v-(f ) ' '

(6)

whee1s the nondenona seawse eocy s he noa eocty' s the noa coodnate s the steawse coodnae vs he kneac

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2 SSAIN! & ZANG

viscosiy and is he ressure-radien vecor. The boundary condiionsare 0 a 0 and as n inow condiion is requireda soe =

Chebyshev secra aroxiaions o a siiar version of his syse(a/a 0) ae fairy aihforad o obai by ie econdiioedieraive schees Sree e a. 984 This work deonsraed ha acobinaion of doain runcaion yicay a = and rid srechino ack rid oins near he soid boundar is quie eecve. A ere 0 coocaion oins wi usuay ied vaues for he wa shear anddiacen hicn ha hav hdii accuacy whi 0 oi

roduce ve-dii accuracy.The fu nonsiiar equaions are ore chaenin since here is a

Chebyshev aroxiaion in wo direcions. Sree e a (984 used anaeaindirecion ye of econdiionin o obain a souion ornon siiar ow rouhy 0 oynoias in coued wih 2 in arerequred for ree-d accuracy Sree e a found ha he Chebyhearoxiaion in roduced a subsania irovemen over a sier ed schee ha used nie dierences in oeher wih hebyshev

8

S

4

atan

.

-.2

-4

10 14 l 1

Fig 5 Stemles (tp d ski fio (bttm fom Cheyshe spl solutoof the oudyye equos outesy of C. Steett.

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S S 5

ooon n ob nur of srws roxon ssy usf for ndn srd ow In s s, rn nqus n r n for ndrn roxons Fur 5dsys srns nd skn fion fro fy sr soon of

srd boundryyr ow T rrow rks ron of ow s os sns o nur rsouon To obn fourd uy n skn fron r qurs 40 ooon ons n nor dron nd 26 n srws dron orrsondnrqurns for sndrd sondordr ndrn od r20nd 200ons, rsy Moror, sr souon rqurs ony10% of PU kn by ndrn od.

NA VERSTOKES FOWMu of urrn nuss for sr ods s rbub o

r suss on s y ouony nns robs n sousdndn norssb ow. T onrn suons of rdnson oonous soro urbun by Orsz Prson(972) wr rury nun. Subsqun uons of rdnson rnson nd urbun n s wboundd ows so bn rsus ors for s robs r subsny ordu nd onsun n os for oonous ows T

rsn o nonrod boundry ondons ks ury Fourr ods nror, nd dd suons of rnson robsyy rqur n ordr of nud or ss n do urbunrobs T ss ss of su robs onsss of ows rssud o b rod n wo drons, . Posu ow nd yorCou ow for yndrs of nn n n s ss on nds bys dsrzon n ony on dron Sr ys of don sr nqus r urrny bn xord o xnd furr ss of sous robs r nb o sr ods

In ny ons rfrrd rson of NrSoks quons s

Voq = 0, (17)

wr q s oy, s rssur, u s ory, + (1)lqIs rssur d nd vs kn sosy Ts

sod roon for s ford bus, s nod by Orsz (1972) us of roon for urns Forir ooon odsons kn nry. On n sy sow onu s onsrd

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AN & ANG

a h onrvation of inti nr i a imorant fornumra raon In ra, man ha f h mdrnn hm orad a m bow aby m, hn nonnar nab no our.

Homogou Tuulc

Homonou, ioroi urbun i rha h on uid-dynamia robm for hh r rod boundar ondon n a aadron ar juab Hn, ourr ra mhod ar day udfor h a of robm Morovr in h noninar ofh NavrSo quaon a a wo quada, ou Gan mhod ah mo naura and n ra hnqu for hi robm (Orza

aron 192 Roao (198 dvod a inar oordina tran

fomaon ha rm muaon of o h onan ra, har, androaon hn h onn of od bounday ondon Roao (1981and Badvan (98 hav dud hniqu for mnmzn hoa, CP m, and I/O o of uh aorhm

Th orna muaon of Orza aron r on 2 rd Bh ary 9 6 muaon wr far roun Roao (91, Krr(1985, and L Rynod (1985 hav fomd numou 128 mu

aon. B fuy xon h a mmr of h Tayor-Grn

vorx Braht a. (198 ahivd a muaion of h o a a Rynod numbr of ih an v ouon of 25ourr ooaton aroxmaon o h robm ar ao ob.

or h aroxmaon u of h roaion form of th Navr-So quaon rua. (Gan aroxmaon o h nvd fom ofh quaon auomaa onrv momnum and n nryn h abn of mdrnn rror

h rvw b Roao Moin (198 du man aiaon ofh thnqu o robm n homonou urbun. Hr, nd

mnon on o n aaon A rimar oa of mot of hmuaon of oo urbun ha bn o abh numray h xn of an nra ran Th nra ran ha, of our, bn abhd xnay bu on for Rynod numbr xdn,. vn houh h hih-rouon auaon of Brah a( 198 r rformd a a Rnod numbr of , hh unomfortaby ow by xrmna andard hy dd ahiv h r auib na an n a numa muaon of ubun. Badna a.(95 and an Ro (9 hav muad h vouon of urbun

nny n roan o Wu a. (98 hav rfomd auaon ofomrd urbun L Rnod (985 hav anad h ruuof urbun n axmmra onran and xandn o Moin

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PC MO 55

et . 198 e se nuer sutons to extrt te reseortc structures of soe turbuent ser os Kerr 198 s exinehihorder orretions nd ssce structure in isotroic turbuencenon ss scaas.

A fw acaons usn h coocaon chnq ha bn e to oressbe, ooeneous turbuence. eeresen et 1981 suted subsonic turbuent os t unfo sher Tey used cooctonhod n a bcaus a Gakn hod s uch ore cubersoe ndcosy for robes it ore thn qudrtic noninerities. Coressibetoienson turbuene s been inestite by eort et 198and by Deore 198 fo wh a fay sandad sch and htter it n it tederencin etod bsed on the ies of ert

et . 198Linear Stability

Mos nsaons o saby and anson n wa-boundd ows yt est n rt uon te resuts of nerstbty teory. Te OrrSoerfed equton hs been the bss for ny nesttons of the stbtyo ncoressibe re os Drzin & Reid 1981. Ti eienuerobe is descrbe by fororer orinry ierenti equton. TheCebyse roxton eeoed by Orsz 1971 or te teor

saby rob as bn adod and ndd by any inestitors[A serte eeoent of Cebyse etods for ornry erentieienue robes s been onucte by Ortiz or furter ets, tereder shoud consut Ches & Ortiz 198 nd ore recenty, Ortiz &Sr 98 Leonrd & Wry 98 deeoed Gerin hodo ow ha uss sca acob oynoas. Saa 198 donstrte tt for exteror os su s te re bounry yer teuse of ony f te usu Cebyse bsis s sbe. Boy 198 sdeeoe etos n te coex ne tt re usefu for os n ic

h cca ay s w saad o h wa on Kcz 198 hasuse Cebye oynos for ssessn te stbity of oscaoy nePoseue o. Mc Go Muiris 198 s use te Gerkn tenueo an h na saby of so asyc ows ha a anto te ortebrekdon robe

Te ststbity erons of tese roes re ore cutbecuse the eienue enters noninery Chebyshe ethods for tie ndndn bu saay own ubaons o osu ow adscussd by Bay

nns 198 Bds

Mos 198,b usd

sectr eto to soe te ore dicut ener stistbityrobe of sefsr boundry yers.

Tese etos e been extened, in te nner of Foquet teory,

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56 UAl & AG

to inld waly nonliar t. In addition to a Chbyh dirtization th dirtio normal to th wall o ild ral orirharmoni i th tramwi dirtion. Orzag & Patra (98) and Hrbrt(983a ha d thi approah to dtrmin th ntraltabilty rfaof nit-amplitd twodimnioal ollminShlihting wa in hannel ow In trn th lnar ably of ee neurl nie-mpiude wen be emined u th er b o ome peil emporll ndpatially ayig ow a b itigatd. Orzag & Patra (98) had th thnq to tdy th trato of two-dmnoal ad thrdimiona Tollmi-Shlihting wa in hannl ow Hrbrt(983ab 984) ha prformd a dtaild tdy of hannl and bondarylayr ow. H ha nrald th dtail offndamntal and bharmoi

inbiliie in prllel owTranston

Tranition to trbl i highly nonlinar and a fll imlation of thNieroe equion i required for it ineiion Te primrdiul o lorim or inompreible ow i e imulneounform of th iompribility ontrait ad th no-lip bodaryodition. Thi otrait i mot aily bt lat rigoroly atid inpitting mthod of whih th Orzag & ll (980) algorithm i th

prototyp Th plitting rror of thi mthod ar ( ) nar th bondaryfor th ormal prr gradit and diion trm (Dill 985). Thyappar to a no rio error in e nnelow problem. ee eummr poided by Golieb e al. (984) o h yt npblihd worby S Oag Dill Irali Howr ar (984a)diily dmontratd that th bodary rror prod ro in urie n Taylor-Cott ow bo e pil nd emporldireiion re rened e lorim pper o onere o nwer that diagr with xprimt i th third digit. ar (984a) and

Klir & Shmann (984) did an inmatrix thi thatompltly liminat th plitting rror at a modt xtra ot. arfond that t rlt of thi algorithm agrd with th xprimntal rltto th fll for digit that wr aailabl H aribd th tiity of throtatingylndr problm to th fat that it dyami ar din b thmotio f th boundr rathr n by a man prr rdien.

prodr for rding althogh ot ntirly liminatng th plittigrror at th bodary wa did by Fortin t al. ( 9) for itlmntmthod (rdiord latr b Kim

&oi 985) and applid to ptral

algorthm y ang aini (98) It oit of modiin e boundr ondiion or e inermedie ep o e lorim o bo e

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ECTRA METHOD 357

noli nd dirgncfr condiion r id h n of h fulli o highr ordr in h iz of h im .

Th big dvng of h liing hniu i h y ruir h oluion of only Poion uion (for h rur) or Hlmolz

uion (rom Crn-Nicoon dicriion of h vicou r).Th oiiv-dni lr uion r muh ir o olv numriclly hn indni could uion h ri in un li hod.Th OrgKll rcu nd KlirSchunn lgorihm ror odir oluion mhod of h y diud in h ion on inviidow. Th ZngHuini lgorh loy iriv hniu o h ii licbl o widr cl of roblm. Th mo ohiicd nd owrful of h iriv chniu i h crl muligrid mhod. m h o of ingl im of ordr

3log

,n for robl

wih vrib gori r nd rnor cocin. n onr rlll-ow roblm vn wih uniform rnor oi ruirordr 4 orion r by dir mhod

On wy o voi h liing rror i o ing incomriblNvir-So uion in ingl h coul h divrgncfronrin wih h momnu uion. Th nuricl diculy of hiroh i h on u invr lrgr of uion (i nvolv hrur wll h hr vlociy comonn) which i indni. n fw il c dir hniu r vibl (oin

&Kim 980). Th

rondiiond iriv h of li l. (985) h bn lid ochnnl ow (ng & Huini 985) nd o h hd boundry lyr(ng & Huini 985b) roblm h involv vribl rnoroin nd h lo bn ud in vriion of wly nonlinrbiliy hory for gnion oin ow (Hll & li 98

ny of h numricl roblm cud by h incomribiliy conrin cn b voidd by n xnion in funcion h r divrgncfr (Ldyhny 99 Tm 977). onrd & Wry (98) r

lid hi id o crl hod Thy dvid of bi funionfor i ow h r boh divrgn fr nd ify noli boundryondiion. Siilr bi funion hv bn dvlod for righ ndcurvd chnnl (or l 983) nd for h rlll bondry lyr(Slr 4) Thi cl of mhod cn b ui conomicl of orginc only wo vribl r grid oin r ruird o cify h ow ld (owvr in ul imnion i y b mor in inr of CPU i o or vrl ddiionl unii r grid oin.)Th incy of h mhod dnd uon h bndwidh of h m

ri h ri from h imlici rmn of h viou rm. n hxml cid bov h bndwidh i ui mll roughly of ordr 0

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8 AIN & ZANG

Thi rirmnt ha ditatd th o pial Jaobi polnomial, rathrthan Chbhv on in pip and bonda-la ow A a onntranform mthod ar not appliabl in th nonpriodi dirtion. Hn,th ot o valating th nonlinar trm inra a rathr than a log oov, n vn lightl mo gnal a, th mati anb ompltl ll.

Orzag & Pata ( 8) pfomd a paamti td of th ondaintabilit in hannl and pip dmontrating that britial intabiliti xit at Rnold nmbr a low a 00 Klir & Shmann(8) rpliatd man o th fatr of th Nihioa t al. (80) xpimnt on hannlow tanition Both gop alo obtain good antitativ agmnt with th pdition of wal nonlina tho. Rozhdtvn

Simain (8) hav xhibitd a vait of onda ow

in plan hannl. Th bharmoni intabiliti that wr prditd bHrbrt (8b, 8) wal nonlinar anali [and that ar alo in vidn in bonda-la xpimnt (Sai t al 8)] w podd b Spalat (8) and arin (86) or th bonda lar anb Zang & Haini ( 8a) and Singr t al ( 86) o hannl ow Thxitn of a imila nonlina intabilit of nt mod in hannl owwa novd b ang & Haini (l8a). A dtaild omparion ononlinar t on th laminarow ontrol thni o pr gradi nt, tion and hating in bonaar ow wa mad b ang

&aini (8b) Kit & Zang (86) hav pod a dtaid tdof th oltion imnt fo imlation of th lat tag o tan ition to trbln in hannl ow Th panwi dirtion pla thgatt dmand on th roltion ba of th v hap panwigradint that o nar th tip of th haratriti hairpin vortx. igr6 whih i xtatd fom that wo illtat th tt

a (8a,b) ha pomd a afl nmial td of nonaximmtri intabiliti in laial Talor-Cott ow H ha pro

dd ordigit agrmnt with th wav pd mard b King t al(8) or both th on wav-vortx and th two wavvortx tata & Tman ( 86a,b) hav imlatd aximmti phialCott ow nl pvo wo, th dd not am atoial mt Thi wa a ia ato in thi in poding thtanition btwn 0 and 2 votx tat obvd b Wimm ( 6).

Ihomogous Tuulc

Invral a th algorithm hav bn d to imlat rbln in

wall-bond ow Ozag & Pata (83) pormd a 6 imlationof tblnt hannl ow that podd th tblnt vloit pol,inlding th lawo-th-wall bhavior. or & oin (8) omptd

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PECR MEO 359

bln ow n a cd channl on a 8 x 4 gd. Thy odcdo of h daa on low-od blnc ac nd xhbd o ofh c of ca ala Lonad 985) ha don o analyof-gadn c n bln bonday lay

0

-1/4

Y /2

-3/4

1

i4

Y 1/2

3/4

-1

0/4

Y /2

-3/4

0

1/4Y /2

3/4

/4

xlL = 4X

/ 3/Z

z

Fgure 6 Seme (left) nd pe (rgh) oty ou teme oto o hipi oex loReyoldumbe helo to. y the loe hl o hehnnl hn

Annu.R




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360 SSN & NG

More Reaisi Geomeries

A noed above here i a banial inreae in o when here i morehan one inhomogeneo direion in he roblem. The Kleier-Shmann

inn-arix hni ha ben exendd o wo nonriodi dire ion by eQere d Rofor 85) who ed i o dy hralonveion in a are aviy Sree Haini 8) imilaly exended he li algorihm of Zang & Haini (986) and ed i o dy heee of ne-lengh ylinder in TaylorCoee ow K e al. (1987)have develoed an algorihm for hree noneriodi dirion Thi ehod reenly rea only he rere erm imliily. Th here an be a evere imee limiaion ariing from he vio erm. orhoine8) ha dvloed a nmber of mehod for roblem wih more han

one nonerodi direion. n general ieraive ehnie are ed forolving he reling imlii eaion. There ha no ye been any yemai omarion of hee mehod Leonard (984) ha derived a e ofdivrgnefree bai fnion for wo noneriodi dirion b aneien olion ehnie for he imlii eaion ha no ye beendevied

Several o he mlidomain eral ehod have been alied o vi o roblem. orhoine 8) ha erformed oe amle al

laion of hannel ow The eral elemen ha been ed o allaeha ranfe in a wodinional grooved hannel Ghaddar e al 8)and o inveigae abiliy and reonane henona in bddd aviiin hannel ow Ghaddar e al 86ab). Oher aliaion inlde

wo-dimenional ow a a ylinder and ow a hreedimenionalroghne eeen Karniadai e al. 1986).

Spera/Fiie-Dferee ad QuasiSpera Mehods

Hereofore, hi review ha been onned o nmerial id-dynamial

wor ha emloyed eral direiaion in all oordinae direionThere have of ore been nmero omaion ha ed mixed ral/niedierene ehod, i.e. algorih wih eral dire iaion in ome direion and nie dierene in he oher Thearallel bondarylayer raniion allaion ofWay aini (984)fall ino hi aegory They ed a orier eral mehod in wo eriodidireion and eond-order nie dierene in he normal direionThey demonraed ha deie he negle of nonarallel ee heeilaion old rerod fear obrvd xrimnally by Kova

nay e al. 62) o he o-alled woie age of raniion. A lighlydieren eral/niedierene mehod wa ed by oin m 82)in heir large-eddy imlaion of rblen hannel ow and by iringen

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PECTRA METHOD 3

( 95) in a sdy of aive onrol in hanel ows. ore reely idsoe al. (96) have sed a simlar algorihm i a highresolion diresimlaion of a rble RayleighBnard ow

Anoher alernaive o re seral mehods is wha migh be ermed asi-seral mehods. Sh algorihms emloy Forier exansios inall direions b iieorder aray is no aaied as a esl ofnoneiodi hysial bondary ondiions in a leas oe direion. hesimlaions by Riley ealfe ( 90) of a ime-develoing miig layerfall io hi aegory I his idealized ow he mea veloiy is solely afnion of he ransverse oordinae y Alhogh he ow eends oy iley eale oed on a nie domain in y and sedsie or osie exasios o efore freesli bodary odiios i y

Qasiseal mehods have also bee sed by Crry e al ( 9) o sdyBnad onveionre sera mehods have been deveoed for he me-deeong

mixing layer. Cai e al (9) sed a oangen ransformaio i yombied wih a orier mehod ealfe e al ( 96) alied hyerboliangen or algebrai ransformaions ombied wih a Chebyshev mehod.

iley ealfe (90) have fod ha large-amlide wo-dimensional disrbanes have a rononed ee on he evolion of ablen ming lae ealfe e al ( 96) hae obsere ha he miing

layer exhibis hree-dimensional seondary isabiliies simiar o hoseha or i wallbonded ows hese isabiliies aear o ao forhe mshroomshaed feares ha are observed exerimenally. Cain eal. (9) have erformed large-eddy simlaions of his roblem.

RACTIG FLOWS

An emergig aliaio eld for seral mehods is reaig ows. Theseows are eseially hallegig bease hey oai shar gradiens inboh sae and ime and bease mos real ows involve doens or evenhndreds of seies Flame fros and sho waves ae a addiionalomliaion Some of he imora feares are mixig raes igiioand ame holding.

here are a nmber of simlifyig assmions ha lead o more raable b less realisi modes of reaig ows. he mos drasi of heseis ha he reaions roeed wiho hea release ad ha he ahnmbe is so low ha he ow may be reaed as iomressible. Riley eal (96) have erformed some hree-dimensional simlaions of a woseies ime-develoig mixing layer. hey sed a asiseral mehodand obaied good agreemen wih boh similariy heory ad exerimenal daa.

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32 SSAN & ZANG

McMurry e al. (198) employed a low-Mach-number approximaionha include ome mild hea-releae eec bu neglec he acouic mode.They performed ome wo-dimenional calculaion ha indicae ha he

rorder eec of hea releae i o reduce he rae of mixingDrummond e al. 198) applied a Cheyhev pecral mehod o auperonic quai-one-dimenional diverging nozzle ow wih a imple buquie i wo-pecie hydrogen-air reacion. The pecral mehod provedo e quie economical compared wih a benchmark niedieence reul.The hebyev grid-poin diribuion wa quie well adaped o e hapgradien a he nozzle inow b le well ied o he fairly unifomouow region.

PRPCTA decade ago pecral mehod appeared o be well uid only o prolemoveed y ordinary dierenial euaion or by parial dierenial euaion wih periodic boundary condiion. And o coure he oluion ielneeded o be mooh Some of he obacle o wider applicaon of pecralmeho were a eniiviy o onar coniion reamen oiconinuo oluion c) reoluion and ime-ep limiaion impoedy he andard pecral grid and d draic geomeric conrain.

Subanial progre ha been made on he implemenaion of Neumannboundary condiion on characeriic boundary condiion for hyperbolicyem and on he ue o preure and inermediae boundary condiionin incompreile ow. There have een ome heoreical advance onlering echnique for diconinuou oluion o linear proem. Moreover he developmen o hock-ing echniue ha opened a new eldof applicaion o compreible ow wih hock wave. Some eciendirec oluion echniue have een devied ha enale evere vicouime-ep limiaion o e overcome in cerain pecial geomerie. Thedevelopmen of precondiioned ieraive mehod and in pariclar pecral muligrd echnique have radically expaded he cla of prolemha can e handled ecienly y pecral mehod. Moreover hey lendmuch greaer eiiliy comined wih mappin echniue) o he ridpoin diribuion. inally variou mulidomain echnique have expandedhe range of pecral mehod o many prolem of real pracical inere.

Lieraure Ced

nl . Gtt . Td .8. l hd f dnnu 77 ngle Re en. pon V.

GR. 8 Compuaon o Vscos-Invscd Ineracos, Conerece ProceedngsNo. 9

Buk o R. 8 o et

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es o he oned nd p esonso he ne eeent ehod NumerMath. -

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ene opton nd odeng Jlu h 4 : 2 6Bsden C. 8 ehn mproe-

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Bngen, S. 8 Ate ono o tns-on y peod sonong. Phys.uds 44

Bno, . N. 44 Hydodyn heoyof pssr nd mp s ndener o on o he mosphee. aslN 11 eg Conto O Seond

Wethe Reg tteson ed DyonhoBoyd . 8 Copex oodne

ehods fo hydodyn nsesnd Stoe egenpoes hn neo ngy. J Comput. Phys. 4

Boyd . 86 Sper mehods sngron s nons on n nnener Cmput. hys n pess

Bhe . eon D. s S. A.Nke B. G. o R H Frsh U8 Smse sre o he Tyo-

Geen oex J Fuid Mech 0 4 -

Bey, J. S Denns, S. C R. 8 Theon of egenes fo he s-ony perrtons o osee o. .put hs 4 : 8

Brnd A on S R Tyo D. poed spe tgdehods o peod ept poes. Comput. hs. 8 -

Bdges T , os, . 84 Df-eren egene proes n hhhe pee ppes nonney. .put h 4-

Bdges, T. os, . 84b

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ud Dyamc Bern SprngeVeg.In pesChes , O E . 68 n he nme-

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Dng, Roy, . 8 D nd ge-eddy smon o homogeneos ene smed o sod ody oon

oc. Symp. ubu Shea ows 5ththaa, N Y, pp Deoe . 84 Nme smon o

hoogeneos soop odensontene n opesse o. Rech.Asp. 84( -

Dees H C A . An ptmhng proede or go eemenons st ath ts ppl 2 -4

Dee . nd E. Cheyshepsedospe soon o seondodeep eqons th ne eeent pe-

ondonng . Comput. Phys.60

Dn . G, Red, W. 8 Hydodym Stty Cdge C-dg Un. ss. 2 pp

Dmmond ., Hssn, . Y Zng, T.A 6 Spet mehods o modengspeson hey eng o edsA 4 466

Edson, T . Hssn, . Zng T A86. Ston o he en Ry-eghBnrd roem sng spe!ne deene ehne. NASA CR8 NASA ngy Rs. Cn.,Hmpton, V.

Esen E hene E, Rsmssen E0 n ne ehod fo nte-gton o he hydodyn eqtonsth spe epesenton o he ho-zon eds. Rep. No. , Dep eeoo.Copenhgen Un Den.

Feeesen, W. , Reynods W. C Fege, H. 8 . Nmer smo of om-pesse homogeneos en sheo ep N F- Dep eh. EngStnod Un C.

on eye R Te R. Rsoon nmrqe des ons deNeSokes po n de nopresse. . Mec. 0 -0

hdd N e, A. T., k, B. 84. e nsfe enhnemen n osoy

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. ng . . . ul

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smuaon of chann ow ransiionhsrucur of hapn vor NAA TPASA Langly Rs n Hampon Va

u H ayo D Hsch, R S Psudospcta hods fo soion of

th incopssib NaviStoks aions Cmt uids In prssLayhnskaya, O A 6 Te ate

matia e f isus Inmessibe w Yok Goon & Bach 4pp

Laun, 86 umsh Smuaonu avn nussung s amn-ubunn bgangs in Pan-gnschichng FFB 60

L, J Rynods, W 185 axprimns on h scu of homo-gnous urn P m Tubuea

Fows Ihaa N. Y., p 7-

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Qu P d Roufo A 8Compuao of naua onon nwomnsna as wh Chbyshvpoynomas J Cmpt P : 08

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SPECTRAL METHODS 35o ow Par umrca sus fowavyvox ow wih on aving wavJ Fuid e 46 6- 3

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Mu, P A u WH Ry, J J. Maf R W 86 D numasmuaons of a ang mng ay whchc hat as AIAA 60

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Ms R. D Mn, P 4 Dc nu

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Orsag S A 6 mica mhos fo

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3 HSSI &

th ton of bnc hy. lud 0 (Spp II)

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Wier 1976 xpeient on icouud ow btween concenic ottinphere J Flid Mech. 7 7

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prien on ubcriil rntonechn AAA Pap No 8596Zn T A Hun . Y 95. Nuer-

c i o bii of o

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n T A Husini 9c Recentpplction of pecrl ethod n uddnc Le App Mah 22 : 7909

n A uini . 196. n

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n A. Won S. uini 19 Spcl utii ethod o llip-tic eution. Copt Phy 4 4O

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n T. Wog 'S uni . 194 Spectrl ulrd ehod for ellpc euon II J om Ph :-

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