Linear Stability of Plane Couette Flow at

Moderate Reynolds Numbers

Sirod Sirisup

Department of Mathematics and Statistics

McGill University Montreal

January 2000

A thesis submitted to the faculty of Graduate Sludics and Researcli in partial

fulfilrneiit of tlie requircriicnts of tlie dcgree of Master of Science.

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Abstract

Tlic liiiear stability of plane Coiicttc flow is stiidicd 1)y iisiiig ;r syiiil>olic corri-

piitatioii package (Metlicniaticii). Tlic powr scrics iiictliod is iisccl t,o ol>t;riii tlic

soliitiori of the Orr-Sommerfeld erli~;ttioii. This soliitioii slio~vs tliat plaiic Coiicttc

How is iiiist;rble.

Résumé

La stiililiti! liii6airc di1 Hus de Coiiettc siir 1111 pliin est Utii<li6e ]>ils I'iiit,~riii61liiii1e

d'un oiitil de cnlciil syiiil>olic (14iitlieiiintiai). La iiii!tlir~ds (le sCric: de puissiuicc ost.

iitilis6c potir obteiiir In solut.ioii dc I'6qiiatioii de On-Soiiiiiicrfcld. Ccttc soliitioii

tiioiitrc que le Hus (le Couette siir i i i i pliiii est, iiistii1)la.

Acknowledgment

1 woiild likc to express iny grat,itiitle to Professor 1C.K Xiiii for iiitrotliiciiig iiit!

to this iiitcrest,iiig problciii iii fiuid dyiitiiriics, Iiis cuiisistciit. cuiisidcr;it,ioii tliiriiig iiiy

stiidies and Iiis lielpful siiggcstioiis iii coinpletiiig tliis tlicsis.

1 aiii alsu ixitlclitc<l tu Mr. Duiisliciig Yu fur iiiy first iliscussiuii iilioiit. kltrtliii-

rtiatica, tu Miss Lucy Cainpbcll for Iiclpirig inc witli BWY ;iiitl iilso for I>ciiig ;r

rcsoiirccful pcrsori to rile doriiig iny kltistcrs a t McGill. 1 also tliiiiik Miss \'croiiiqiic

Guiliii Cor tniiisliitiiig tlic aùstracl. iiitu Frciicli.

1 tliaiik the Royal Tliai gowriiinciit for tlicir firiiiiicial support diiriiig iiiy illtwt,i!rs

dcgree at 4IcGill aiitl also tlic pcrsoiiiiel a t IPST (cspcciiilly iii tlic DPST ilcpiirt-

meut) in Tliailaiid aiid OEA irt \Vi~sliiiigtuii D.C. for ùeiiig cfliciciit iirid ilepr!iirl;rl~lc.

L,~st I~iit, rmt Ii!;ist, 1 aiii gr;it,cfiil t,o iiiy hmily for t,lii!ir last,irig Iovr! iiiitl si.pport,.

Contents

1 Introduction 4

1.1 Hy<lrotlyn;iniic stal>ility . . . . . . . . . . . . . . . . . . . . . . . . . . -1

1.2 The Plaiic Coiicttc Flow . . . . . . . . . . . . . . . . . . . . . . . . . (i

1.3 Tlie Iiy<lrotlytiainic stability tlicory . . . . . . . . . . . . . . . . . . . i 1.3.1 Hy<irodyniuriic stiibility for case of t,lie Coiictte flow . . . . . . 10

1.3.2 The analytical solution . . . . . . . . . . . . . . . . . . . . . . 11

2 The Computat ion 13

2.1 Forniulatiori and Solutioii . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Grapliical solution of the cliaracteristic cqu;rtioii . . . . . . . . . . . . 17

3 Conclusion and Summary 32

3.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Conclusioii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

List of Figures

1.1 Tlie Plaiie Couette flow . . . . . . . . . . . . . . . . . . . . . . . . . . 8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Crossi~ig ciirves 23

2.2 Tlic lirst iiiiigiiificiitioii . . . . . . . . . . . . . . . . . . . . . . . . . . . ?.1 2.3 Tlie secoiid iiiagiiificiit.ioii. . . . . . . . . . . . . . . . . . . . . . . . . 2.1

2.4 A Ixttcr viem of a grapliiall solutioii . . . . . . . . . . . . . . . . . . . 23

2.5 Tlie grapli for the iieutral c iw . . . . . . . . . . . . . . . . . . . . . . ?G 2.6 Tlic grapli for t. lit! diirnpcd caw wlieii n = 9 . . . . . . . . . . . . . . . 27

2.7 The grapli for t. lie ariiplifictl case mlicii tu = 9 . . . . . . . . . . . . . . 28

2.8 The grirpli for the daiiiped case wlieii a = 10 . . . . . . . . . . . . . . 20

2.0 Tlir grapli for the nniplified case wlieri (Y = 10 . . . . . . . . . . . . . . 30

2.10 Tlie grapli [or Llie daiiiped, iicutral, aiiil)lilicd ciws wlieii cu = 0 irud

10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

List of Tables

2.1 A table of b1 nritl its first tlerivtitivc tit y = 1 lis rt = 5 iiiitl R = 2Xi 1G

2.2 A table of b2 arid its first tIerivat.ivc at y = 1 as ru = 5 aiirl R = 250 17

2.3 A table of the cjbi aiid its first tlcrivativc at y = 1 tis tu = 10 ;ilid R =

200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 A t,al>lc of tlie cjb2 aritl its first tlcrivativc at y = 1 41s tu 10 aiitl R = 20O 17

Chapter 1

Introduction

1.1 Hydrodynamic stability

- 7 1 licire ;ire t~vo t,ypo of Hui11 iiiotiori: laiiiiiiiir inotioii ;incl t,iirl)iil~!rit iriotioii. \\'r

kiiow tliat Iiirniiiar iiiotioii occiirs oiily wlicii the Rcyiiol<ls iiiiiiilicr is vcry siii;ill. If

iiii iiifiiiitesiiual disturbaiice is iiitrotliiced in tlic 1aiiiiii:ir flow iiiid it (Icc;iys, t h flow

cari inaintaiii its laininar iiiotiori. Ho~vcvcr, if :lie ~listiirl>;riicc grows, tlic riiaiii How

will bc distiirbed and turbulerice niay rcsult. Iii tliis c u c we say tlic laiiiiii;ir iiiotioii

is unstable. If tlic tlisturbaiice decays ~ve say tlic fiow is st;rblc.

The problern of ~Icterminiiig if the lsminar iiiotion is stalde or unst;ildc witli

respect to tlic iiifinitcsiiiial disturbaiicc is a stabilily prol>lciii. Tlic st;dility Ilicury

t,hat. usrs t,lir idea of irifinit,esirrial clist.iirhaii~:es aiid docs riot go Iiiy)iid t,lici first,

approxiination is called the liiicar stal)ility tlieory.

Wc are, rio\%*, iiivestigatirig in tlie Iaiiiiiiar inotioii of tlic siniplest foriii: tlir pliiiic

Coiiet,t,c flow.

Viscosity plays an iiriportiiiit sole in tlie How. Evcii vcry siiiirll viscosit,y cirii

inake Huid How uiistable. Accordirig to Yih [5], Case aiitl Orr Ii;ive provctl t h t t,lie

iiiviscid plane Couette How is stal~lc. Howcver, the stnl~ility or iiist;il>ility of viscous

plane Couette llow Iiiu iiot bceii firiiily estal>lislied.

Because of the iiifiiiite estent, it is impossible to iiivt!stigat(? tliis kiiitl of How

csperinieiitally. Howver, I>y iieglectiiig infiiiite estent, one cati do tlic esperitiiciit

for the plirric Conette How in tlic Iiilioratory, 11s L.S ïiickorniiiii iintl D. Rnrldcy [l(i],

F. Diwiaiid ct ul [13], N. Tillinark itiid P.H. Alfrc<lssoii [12], D:iiicli»t iiiid Diiviiirl [14]

aiid S. 13ott,iri et (11 [la]. Homever, witlioiit, iicglecting iiifinit,c c!st,c?rit,, WC (:;III ;ilso

still study t h stahility of tlic flow frorn a. miitlicin~~tical poiiit of vicw.

Tlie ana!yticiil solutioii for tlic goverriirig cquatioii for t,lic stal>ilit.y ~iroliloiii ( TIia

Ors-Soniincrfcld cqiiatioii) for tlic pl;rii Coiicttc flow is kriowii. 'l'lie ;rccoiripiiiiyiii~

riiiiricriçal coiiipulalioii for tlic soliitioii of Llic problciii is iioii-triviiil. TIicrc! ;ire ir

mirnbcr of numerical corripiitations for tliis probleiii, siicli iis C:.i3. Davis iiiid A.G.

Morris 1191 and Soutliwell and Cliitty [20]

To stiidy tliis problciii, DcardorfC [21] iiscd a conpliiig fiiiitc dilfcrcncc inci.lio<ls

with trial and crror iiiirncrical ~net,lintl t,o st.iidy th! st,al)ilit.y of t,lic! ~iliiiie Coiic~ttc!

How. Iii 1973, Dwcy user1 ;r "coniplctc" ortlionoriiiiilizirtioii witli ii p;rr;rllcl sliootiiig

procedure to invcstigiite instability of the How. Stndics i i ~ i to i,liis <Iirt,r Ii;i\v iiut,

sliowri instability of the plane Coiicttc Ilow I)ut tliis lias iiot I)c!cii rliscoiiiitcd.

Tliere is also aiialytical work for the special ctws of iiistirbility of pliiiic Coiii:ttc

How for large valiies of a non-dimensiorid pariirnct,cr iri tlic prolilcni. 1-lopf [17] us(d

asyrnptotiç metliods of approxirnatioii to stiidy tlic proldeiii. His stiidies clid iiot

sliown iiistability of tlic plaiic Coiicttc flow. \\'iisow (181 sliowcil iii Iiis worlt i.!iir~

for a given \wvr riiimbcr tlie plane Couette flom is stable if t,lir prodirct, of t,lw miivc~

iiiiinber and the non-diinensional paraineter is siifficieiit,ly Iiirgc.

In numerical works for studying tlic instability, we Iiiive to denl witli p;ir;rriieters

wliose values range over a wirle iiit,erval, wliicli iiiakc tlici r:o~iiput,nt,ioii c!scc!otliiigly

intensive. Tlius, synibolie coinputatiuns iiiay offer soinc I>eiiefits in solviiig sucli

problciiis. 144th the awilability of Iiigli performance persona1 coiriputer in rcc(!nt

yciirs, symbolic conil~utntioiis Iiave facilitatcd tlic i~ivestigiitioii of otlicr types of

flows; for exaiiiple, Taiii [22].

In tliis tlicsis, WC \vil1 show that tlie plane Coiiettr How is inistablc liy iisiiig tlir

power series solutions olitained from a syniholic coinpiitntioii.

1.2 The Plane Couette Flow

Rom Roscriliriad [Il, tlic govcrnirig ccliintioiis for iricoiril>rc~ssiI~l<! viscons flow,

witli rcfcrcnce to n gcncral Certcsiiiii coor<liri;itc systciii :~.,y,r with thi! \rl!iIJcity

field U = (u, u, vu) wlicrc u, u , rit ;ire tlic vclocity coiiipoiiciii,s in :r, 11, z tlir(!cl.ion

rcspc?crivrly, iirc

Tlicse arc the Navier-Stokes cqiiatioii and

is the Continuity cqiiatioii.

Hcrc, 11 is tlic dciisity of tlic fluid, i/ = f; is tlic kii1ciii;itic viscosity; 11 is sliitic

viscosit,y, p is pressiire, niid V2 = & + 6 + -. \\le nrc iiit,ercst(!~l iu tlic HOW

urlled the Couette How.

Siipposc that t h e are twvo pxollcl iiifiiiitc plates fillecl iii tlic g;lp bct\vrcii t~liciii

witli a viscous fluid (Figure 1.1). Wliile tvc keep the lowr plate statioiiary . wc triorc

tlie iipper plate witli a coiistaiit velocity U iii tlic z-directioii. Tlie frictioii I~ci.wcieii

the plate surface iirid the fluid will inducc a fluid flow; iiiid tliis flow is rallcd thv

plaiie Coiiette flow.

To find tlic vclocit>* distributiori for tlic Iniiiiiiar plaiic Coiictlc flo\\l, suppose

the flow is stendy (independent of tirnc) and tliere is no vclocit,>, in t,lie y, z <lircction,

i.e. ü = TL,^, w ) = (u,D,O). Thcn eqiiations (1.1),(1.2) eiid (1.3) ;riid c<liiiit,ioii (1.4),

Figure 1.1: The Plane Coiicttc How.

WC cari concliide frorii eqiintioii (1.G) and (1.7) t,liat p = p(:c). klorcovc!r, siiiw t,li(!rc,

is no prcssiire griidiciit in tlie Coiiette How, wc Ir;we

Oiie c;iii casily solve t,liis ordiii;try ~.lifïcreriti~il eqiiütioii, cqnatioii (l.S), to ol)t;iiii

wliere A, B are constant deterinincd by the following conditions: 1. At y = 0,

u(y) = O and 2. At y = 1, u(y) = W . Fiiially Ive get the vclocity distribiitioii for t h

plane Couette How to l x

,u(y) = vy.

1.3 The hydrodynamic stability theory

In tliis section rve providc a I~rief introduction to tlic Iiydrotlyiiaiiiic stiil~ility

tlieory iu tlie case of viscous iiicoiiipressible parüllel laiiiiiiar Ilow iii the iilisciice of

csternal forces.

F ~ I I I I I Roxonl~c~~~I [I], WI! C I L ~ (I~rivi! t,I~t! ~ O ~ I I I I I I I I ~ ~ O I I for tlw I ~ y d r o t l y ~ ~ i ~ t ~ ~ i ~ st,ill~il-

ity IIS followx. Lt!t

.I: tltw~tc! tht! tlixt.rn~t:c! alol~g t,lw flow,

11 tlwot,t! t,lw C ~ ~ S ~ I I I I C I ! I ~ ~ ~ , I v ~ c I I t,l~t! ~ I I L ~ C S ;i11tI

z cltw~tt! t,lw clistit~~cr tl~iit is pc~pontlicnlnr t,o 2: ii~ltl !I.

r\ncl (,n, 11, r u ) i ~ ~ direction of :I:, 11, z , rc!spcctivc!ly.

111trotluci11g 21 cl~i~ractcristic: Icl~gtlr, L, i~~ l t l a cl~;;r;~ctcristic velocity, U,,, ; I I I ~

tlt~fini~~g

\V11tw, we define the Reynolds ~nunber, as R = %. Supposc that there is a two-tlimensional, stently solntio~~ of equ;~t,io~~ (1.9) -

(1.12):

U = U(!/), C' = 0, \I' = 0, pU = cor~stant

Let an infinitcsi~nal perturbations (u', u', ~u ' ,~ ; ) I)e introrlocecl. We ttllcn 11;1vt!:

For wl~icl~ wc set the direction of the flow to be along the s-;~sis.

Now, siil~stitiitc cqiiiition (1.13) iiito cquutioiis (1.9) - (1.12) iiiitl liiiciiriao. CVc

ctin obtiiin tlic lincar equatioiis:

au' au' ai; - + ,y- = -- + 'V'r; a7 82 O!/ R

\Ve let the perturbation fiinctioris,u', /,lu< ;iiitl p), tiikc the hrni ei(<'z+~i-l*"),

wlicre,

n is a rcal wave iiurnbcr dong x dircctioii,

/3 is ii rcd waw numbcr dong z direction,

c is ir coiiiples iiuinber , c = c, + ic,.

WC I1avc

1. If ci > O, the tlisturl~ance grows witli tiuic and the flow is iiiistal~le (aniplificd

case).

2. If c, < O, t.lw distiirbaiii:e (lccays wit,li tiine ;inil t,lii: How is st,:il~l(! (r1;iiiipoii

case).

3. If ci = O, the distiirbance iicitlier grow iior decays (iicutriil ciwc).

Tlieri we cari tvrite :

= ( .2 / )c i in~+i i : -~rr) (1.18)

Now, we siibstitutc equation (1.18) - (1.21) into cquatioii (1.14) - (1.17) and

cliiuiiiatc pi and obtain :

Wliero y' = cu' + P Z , D = 2 iiiid the boiiiitliiry coiitlitioiis for cqiiiitioii (1.22)

;il'(!:

For the two-diiiieiisioiiiil tlic!ory IVL! Iiiivc! [i = O iiiitl it is siiiiiliir if wr, 1li4iiici:

4. 111 = -

11y ' (1.25)

III = -icr+, ( 1 .?G)

wliicli provitles the strcarii fuiictioii:

q, = *(y)c'"('-'5)

Tlieii equatioii (1.22) ciiii I)c witteii as:

'The ditferc!nt,ial witli rt!spc!t:t to !/ is cIsiiot~t!tl Iy priniw ;riid io. Ihliiiitioii (1.27)

is calletl tlic Orr-Surnmerjelil cq~iutiotr witli the I~oiiiitlnry coiitlitioiis:

1.3.1 Hydrodynamic stability for case of the Couette flow

Froiu section 2.1 , the reloçity profile for the 1~1;iiic C:oucttc Hoir is i1(!1) = fi!/.

it cati Iw seeii tliiit tlie Ixisic ilow for the plaiie Coiictte flow is U = I/. Siiict.

tlicii wc caii writc tlic Orr-Suniiiicrfcld for ;i ciisc of tlic plaiic Coucttc Llow ;L.

\Vc riiii coiicl~idc tlint tlic I)otiiiclnry coiiditioiis for tlic Orr-Soiriiiit~rL!ltI for tlir wsc!

of tlic pl;riic Coiicttc Ilow :ire

witli boiiii(l;iry coiitlitioiis:

to irivt?Agate tlir Iiytlrotlyn;iiriie stnliility of tliv plane Coiirt,t~! How.

1.3.2 The analytical solution

III tliis part we will solvc the rqiiatiori (1.27) iiti;ilytically witli tlic Iiciiiiiel;iry

coiiditioiis

First, eliange the variable 11 to x ;iritl tlien wc w i t e

(D' - u2)4(z) = q(+)

11

TIicii froiii (1.27) iiiitl tlic iil>ovc cqiiiitioii wc! c m gct:

iiiitl tlioii

Tliis is tlic Airy cqiiatioii wliicli lias tlic solutiori kiiowii iis tlic Airy fiiiictioii

.%(y) iuicl Bi(y).

Tlic!ii tvi! (:ail get. t h fiinct,ion 4 ;w:

(131)

'iYiis is t,lic iui;ilyt,ic;iI solutio~i of t,Iw c;isc of t,Ii(! pl;iii(! Coii(!t~tc llow, hi priiiciplt!.

iiripositioii of tlic I)oiiri(lary coiiditioris oii ;i liiieiir coiiil>iiiiitioii of tlicsc four soliitioiis

will providr the cliaractcristic ecliiatioii witli wliicli to c:ilculatc t,lic cig(!iiv;iliirs.

Howve\w, tlic difiiculty witli coiiipiitatioiis iiivolvirig iiitcgrals of tlic Airy fiiiirtioiis

iiiakcs tliis approacli proliibitivc.

Chapter 2

The Computation

2.1 Formulation and Solution

For tlic phric Coiit!tt,c flow the Imsic How is U = !/ wlit!rt! O < 11 < 1, \Vc It!t rlic

pert.iirbatioxi strcani furictiori I>e

Froiti the liiicar Iiytlrorlyiiaiiiic stability tlieory (iii tlic ~)rcviotis cliiiptcr) \vib Iiiivr

+ govcriicd by tlic Orr-Soiiiiiicrfcld cquatioii

Wlicrc , c = c, + %<:, is a t:oinplex paranict,er,

a is a real piraineter,

aud R is the Reynolds iiuiriber bmed on tlic wicltli bctmceii tlic pliites iiud tlic

maximum value of U

For tliis equatiori me foiirid tliat tlie appropriale bouiidiiry coiiditioiis al y = (1

and g = 1 are

+ = & = O (2.2)

Equatioii (2.1) h a a set of four fuiidamexital solutioiis satisfyiiig the four iiiitial

coriditions a t y = 0:

(+,d',&',<bu') = (1,[),0,0);

rcspcctivcly.

'Jow, wve coiistriict two liiioarly iii~lcprridciit, powvcr sories soliit,ioiis t,o ~iqiiirt~ioti

(2.1) iii view of the coiiditioii (2.2). Let $1 aiid 91 siitisfy t,lie coiiditioii

a t y = O respectively. We set nr

41 tu) = E h!lk; k 2

wliere bk iiiid rlk depeiid on the pariiiiietcr obtiiiiied by iisiiig tlic syiiil~olic calciil;itioii

package. For &!(y) the condition (2.3) givrs 62 = ? iiiitl I,:, = O. Likrwiw!, for $,(!/)

the coiiditioii (2.4) wc set (1, = %. \Vc corisidcr first tlic iicutral case wlicrc s = O

iiicrcases wvitli iiicrciisiiig iiuiiil)er of ternis. But we iilso Iiiivc to iivoid cnisliiiig t h

PC.

Fiist, me calculate the solution &,&2 mitli as inany terrils ;xi caiii bc »I)taiiictl on

a pcrsoiial coiiiputcr witli a Pciitiuiii II proccssor. Aficr tliat \ire coiiil~iitc tlic viiliic

at y = 1 and plot a grapli on O < y < 1 iii doing tliis wc try cornpiiting wit,li a riingr!

of Reyriolcls nurnber, frorn R=0 to 300 a range of cu froin O to 30 iirid a raiigc of c,

froin O to 1. \Vc scck a suitablc ainouiit of tcrnis tliat iiiakcs tlic fiiiictioii coiivcrgc.

However, we have t,o impose t,he Iioiiri<lary at, y = 1 t,o ol)t,;iin t,lw cli;iia<:t,i-rist,ir:

equation, whicli is:

III tliis operation, WC liavc to iiiiiltiply the fiiiictioiis iiiid tlwir first elcriviitiws.

Soiiictiiiics, tlic coinpntcr cr;mlics if tlic iiiinibcr of t,criiis iis<!d t:xcac!<ls I.lio c'oiiipu-

tat,iorial c;ipncit,y of tlie porsorial compirtcr.

We Iiiive foiiiid tliiit if w use iil>oiit 95 tcriiis we ciiri ;iccoiiiplisli th! coiivc!rgcbiicc

of the soliitioiis obt,aiiictl aiid ;it t h sninc t,iiiie tlic coiiipiit;itioiiiil ciip;ir:it.y c i l t,iir-

pcrsoiial coiriputer is iiot cxceedetl.

Il is iiotcd tliiit tlic coiivergciicc of Llie soliilioii tiietiiis lliii~ t.lic iiiiiiicric;il oiill~iils

of tlic soliitioiis converge to ;i ccrtaiii v;iluc wlicii ;i syxific v;iIiit' of t,cmris Iiiis I~ivii

iised. Evcii if the latter value is csccctlcd tlic iiurricrical soliitioii still coiivcrgo. WC elo

iiot waiit to convcy tlic idcu t h llic coiivcrgciicc is provcii iii tlic iiii;ilyt.icirl wiiy For

riiotler;rt,e R viiliir (0-300) npproxini;it,ely 95 t,c!riiis are rrqiiired ori riiinit!ric:;il oiitpiit,~

corivcrgciicc. For R greiitcr tli;in 300 rriorc tliiiii 9.5 tcriiis ;ire riectlctl on coiivergciicc

of the iiuiiierical oiitputs. Bcciriisc of tliis, we c;ui use soliitioiis oljtiiiiicd oiily for

rriocleratc Hcyriolds niiinbcrs. i.e. froin 0 to 300.

Tlie followiiig arc csiiriiplcs of the value of tlic fuiictioiis ;iiid tlicir first doriviit~iws

iit u = 1 for r r = 5, R= 250 and tu = 10. R= 200, resp~ct,ivcly. \\'c iiotc. tli;it h i ( l ) ,

d,(l), 441 ) aiid &1) arc coiiiples riiiiril>crs.

4'1 26501.1 - 18102.2 1

23502.1 - 18209.4 1 *

25505.1 - 18215.4 1

23505.4 - 18215.5 1

23505.4 - 18215.5 1

T<:~IIIS

80

85

90

95

100

Table 2.1: A table of $1 and its first dcriv;rtivc a t y = 1 iis ci= 5 niid R = 260

$4 139.171 - 950.443 1

139.184 - 951.777 1

139.219 - 951.847 1

139.222 - 951.849 1

139.222 - 951.849 1

Table 2.2: .4 table of $2 aiicl its first clcrivative nt y = 1 ;LS ru= 5 niicl R = 250

Ternis

1 100 1 256900. - 687621. 1 1 2.90889 * IO7 - 1.33998 * 10' 1 1 Tablc 2.3: A tablc OS tlic $1 iiiicl its first dcriv;itivc a l y = 1 ;is tu= In ;rii<l R = 200

80 1 83.5883 - 109.156 1 1 4453.89 - 501.806 1

$2

Tcriiis

1 Terms 1 $2 d:,

di

80 1 269990. - 769201. 1 1 3.02114 * 10' - 2.00039 * 10' 1

8 1 4'1

TIic fiiiictions g i , g1 linvc very vcry lengt,lily cxprcssioiis. Tlio followirig is 4, , 4 ~ 2

st,atcd in ;i M;itlicriiatica "Short" forrii :

Tlie iiumbcr in tlie p;irentliescs slio!vs t h iiiinibcr of tcriiis tlitrt fullow t,lic! Lirst. t,oriii.

Tlicse ternis arc polyiioiiiitil of t u aiid R.

III tlic iiest section we will show Iiom tlic soliitioii of tlic cli;iractcrist,ic cqiititioii

ciiii ùc obtaiiicd grapliically.

2.2 Graphical solution of the characteristic equa-

tion

(using equ;itiori (2.5)). Becaiisc tlic cliaracteristic (:qii;itioii is iii coinples foriii tliv

rcd part niid the iiiiügiriary part of tlie cliaracteristic eqiiatioii Iiaw to bc zero. M'1,

iiotc tliat 110th of tliesc parts of the cliaractcristic eqiiatioii arc iri;iilc II I ) of niany

iii;rriy teriris. This cüii bc ubserved iii tlic "Sliort" forrri of k1;itliciii;rtic;i tis fdlowiiig:

Tlie first 152 tcrms of real part of the cliar;ictcristic equatioii arc:

61338689 c RIU cr12 082759 $ RNI ru~l

80617071415975781~I815391744000000 - 131~~868802~08~9085~ll2884~l80llll0+ 17616?'$ R1" ru" 1313371 C" RIU tu"

41073120150738875213414400000 - 821~16240301477750-12~8288OO0llll+ 13 f? RIU ml2 331 cG R1" n "

3217121599896207360ll0Oll0 - 47184450131811041280Ollll0ll+

6175381 c RI2 tu" 204101608339551507773479342571521100000+

497671û3 2 RI2 u12 ~@JJIJJ c3 RI2 Lp12

13606773889303433851565289504768000000- 291~329457097512813353541027841lllOllOO +

14922773 cl RI2 rux2 2356643 <!' RI2 <y1'

11613178283900512533341~~11113~000000~537~~171~12773171~7~5~1359~1~196000000O~

tiiiel tlic first 117 t,erins of tlic iiiiagii1;iry ptirt, of tlic clitir;ictoristic cicliiathii ;ire!:

(; R 2 29 R"fJ 269 c f i l tr"c' R3 a 5 $ RQrfi 307 R" ru"

18900 726485760 + 10897286.10 1995840 + + 80029~71321600- 643 RQt6 29 62 RfJ n"~ 6-6 RQru" c5 RQr>

+--- R ru' 16005934264320+17~1356582400~261534873600 2905943040 726485?600+956861)-

c R a 7 19RJa7 5 9 c R h 7 62R%t7 6%3a7 --- 467776 7783776000 + 3891888000 - + .18648600+

317 R h 7 1117cX"a7 7 % R h 7 1516JR'cr7 r:.lR"tr7 543058483968000- 181019494656000+27'2209766400~28582025400 1SüS10(i9~100-

cQR" a7 10723 R7 a' 13cR7ai 41 % R7 tu' 46702656000- 6924647340772761~0000~56070018953625600 27877002177021000+

857c3 R7 tr7 23 f i R' a 7 47 c' 12' ni 167262013062144000 - 2172233936872000 + 3620389893120000-

41cR30e 1?R3ae ËiR3,8 541 R? tr" + 75785673600 908107200 + 1362160800 15929715529728000- 13eR?mg r? R%t9 281 c"I)h9 c , ~ ,p r r !~

36040080384000 + 663075072000 - 90509747328000 + 3175780~0800- 2 R b n " 4289 R7 O" 25903 r: Ri n"

793945152000 - 1800408308600918016000 + 7201633234403~720~~100- 7067 cZ R7 a!' '7.121 c" R7 a" 223 c'l R7 rr"

30776210403434496000 + 9232863121030348800 - 133809610449715200~ 137 c"' a" ni @! <!7 R? ,!l

66904805224857600 - 72.1077978624000 + 2534272925184000+ 142019 Reûe 347639 c R\?

5920822763664978987417600000 - 740102845458122373427200000+

R c P c R d L R3 a" <: R3 (ri' 2 R\di 1702701000 851360500 179976932800 + 77138989200 - 37892836800'

'rlit! ~riipliiciil soliitioii of tlic clitrriict,eristic cqiiiitiori is iiccoiiiplislietl Ijy iisiiig

rlii! spc~ciiil piickagc of Lltitliciiiiiticii ccillcrl "IiiiplicitPlot". This will giw ;i coiitoiir

plot, of tlic iinplicit. fiiiict,ioii ~irovirlctl. TIic soliit,ioris art! foiiiitl ;LS li~llows: I3c!c:~iiisc~

wc! kiiow tliiit c, slioiild lie Iictw(ic!ri zero iiiitl oiie, WC! fix ii viilnc for fi, tlicii solvc! tlicn

r(viI ;riid iiiiiigiiinry piirts of tlic cli;iriictr!ristic c!qiiatioii t,o ol>tciiii tlic cr>rrc!slioiidiiig

v;iliics of c, niitl II'. 'I'licsc iirc tlic poiiits wlicrc tlic ciirvc!s cross. 'lb ol)tiiiii ii Iic!t,tc~

vicw of tlia crossiiig ciirvcs, w clivirlc Llii! rciiigc ol c, iiiicl R iiiLo iiiiiiiy iiil.i!rv;ils.

Tci iIrt,c*riniiic t,lic! prcSs(!iicc of crossiiig ciirvcs, wc! lirst plot, t.lic! citrvc! of t.Iiv rriil

piirt of t h cqiiiitioii, tirid tlicii wc plot tlic! ciirvc of tlic iiiicigiiiiiry piirt 'ïlicii, wc!

<:oiiiljiiic tlictii oii tlic siiiiic sciilc, ciiid cxiiiiiiiic tliciii lor Llic prcsciicc or iibsoiicc~ 01'

;III iiil.crsc!i:tioii.

Aiiy crossing curves oliscr\,erl tire iiiagriifietl to obtaiii a iiiorc iicciirate soliit,ioii.

Tlitw ~irociisses tire sliowii iii figures 2.1- 2.4 ( t h figiircs sliowii ;ire obtiiiiiccl witli

<r = lG, tlic vcrticiil axis is c ;ruri tlic Iioriïoiital u i s is R).

Figure 2.1: Crossiiig curves.

Figiirc 2.2: Tlic first inagnificatioii.

Figure 2.3: The sec:ond niagiiificatioii.

Figiirc 2.4: A Ixttcr vicw of :i gr;ipliic:il soliitioii.

\V(? r q m t t,liis procciss iisiiig n iiicro;aiiig I)y 1 frorii O to 30. 'TIio rc!siilt of

tliis calciilatioii is presciite(1 iii Figiirc 2.5. \ik ciiii sec tlic loop striict,iiri- sliowiiig

iiistal~ility of tlic plaiie Coiicttc Row.

Now, wc coiisitler t,lir (:lac! of ;i <I;iriipcid soliit,ioii; M'c! c:oiiipiitc t,liin progr;iiii iigiiiii

vit11 cu = 9, 10 aiid c, = -A aiid solvc the eigciiviiliic prolilciii iigiiiii to liiid tliiii

t h e arc solutioris sliowii in Figures 2.6 and 2.8.

We &O coiisitler a cnsc of aiiiplifietl solution; SV(! rliii tlic prograiii iigiiiii witli

u = 9,10 aiid ci = &. Tlic soliitioiis arc prcsciitctl in Figurcs 2.7 iiiid 2.9.

Tlir case of arriplified, (laiiipcid and neiit,ral c:asrs arc! c:oiril>iried \vl~c!ri n = !), 10

iiiid are represeiiterl iii Figurc 2.10.

We iiiote tliat the syinbolic coinputatioii \vlieii c, # O lx-coiiies iiiucli iiiore difficiilt

aiid very tinic consuming. Tlius, \c Iiave to set tlio value of n first Iieforc cloiiil: tliv

c;tlculatioiis, or the prograiri will terriiiii;ite Iiebre the spccific iiiiiiil~cir of tciriris arc

obtaiiied. Silice our objective is to sliow tliat the plaiic Couette How is iiiistalilr,

we Iiaw performetl coinputatioii for a iiori-zero ci oiily for the two valiirs ;il>ovo. If

wc csaiiiiric Figurc 2.10, iii coiijuctioii witli Figurc 2.6, wc scc tliiit u disturbiiiicc

witli wave tiiiniber ru = 10 adinits a <lairiperl soliitiori witli (( = -- ;it R = 250; ;i

iieutral solution a t R = 232; aiid ;in amplificd solution ;it R = 253 witli c, = A.

5 CO .- LI.

Chapter 3

Conclusion and Summary

3.1 Discussion

\V(? Ii;rvc? st,iitlied thci st,al>ility of the plaiic Coiir!t,t,c! How l>y iisiiig t,lic Coiiipii-

tatiorial approaclies. Havirig atialytic coefficients, tlie Ors-Soriiinerfeld cc~ii;itioii lias

tlic power serics soliitioiis tliat coiiverge to tlic nrialytic;rl soliitioiis (Scc [G]). To

ol>taiii the Iiigh acciiracy p o \ ~ series soliitioii, \ve Iiavc t,o iisi! as iiiiiiiy t,eriris as

possible. h4orcovcr, iii tloiiig tlic ~.oiril~iitatioii, wc do iiot use Iluütiiig ~ioiiit opcr;i-

tion (i.c. cliaiige tlic rational riiiriil>er to ii floatirig-point. iiiiiiibar ;riid t,lirii do tlics

coriiput;ttioii) tliat iricniis wc do iiot liavc a rouiid off crror. So the soliitioiis olit;riried

Iiavc vcry Iiigli acciiracy. Usiiig a vcry Iiigli pcrforiii;iiicc coiiipiitcr witli vcry large

atnoiirit, of RAkI. riiorc! t,c!rrns I:;UI b~ nl>tairicd.

3.2 Conclusion

Tlic calculatioii lias bccii doiic oii pcrsoiial coiiiputcr witli Pciitiuiii II 350 MHÏ

CPU and 128 klegabytes of RAhI. Frorii the resiilts obtaiiied, me ctiii coiicliidr t,li;rt

the Plane Couette flow is unstalh (we cari sec the loop structure iii Figure 2.5,

in wliicli WC Iiaw joiiicd poiiits to Coriii ;i coiitiiiiioiis ciirvc. \Ve Iiwc oiily rloiic

t,liis for n fnr. ciirvcs so as riot t,o rriaka th! tigiirc t,oo c:rowlccl). hlorc ~ir~~c:iscly,

fsorn Figure 2.10 we caii see the regiori iii whicli tlic flom is <l;iiiipcd, iii:iitr;il ;uid

tlioii iiriiplifictl. Tlic rcsiilt ol~tiiiiictl is i i i iigrceiricrit wit,li cspt!riiiiciitid worlc froiii

Tillinarlc aiid Alfredçoii ciiicl F. Dnvicriid el (11. Froiii tlicir cxpcriiiieiitiil rcsiilt,~, t h y

Iinve foiiri~l tliilt tlicro is tml>iilciico iii plaiic Coiit,tc llow wlit!ri t,lit! IZi!yiiolds iiiiiiil)i!r

is about 360 or greiiter.

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