noundary -layer TheoryMcGRAW-I4ILL SERIES IN MECHANICAL ENGINEERINGJACK

r. IIOT,MAN, Southern Methodist UniversityCo1tsu1lin.g Editor

Dr. HERMANN SCHLICHTINGProfresor J3rncrit.11~ Llrc ICl~gincrrirrgU ~ ~ i v r r ~ i t~. ~ ~ I I I I R ( .O cIrw ~ l~ ~ l, ~ nt, of ~ ~ ~ c nr Forrner 13ircctor of thc Arrodynnrninclre Vcr~rrclrsnnslnlt (:iittirrgc~~RARRONISC:KERT

. Cryogenic S y s t e m. hzlroduclion lo Heat and M a r Tran.fer AND DRAKE . Ana1y.ri.r of lien1 and Mos,r 7i-nnsfr.r- Ifen1 attd A4ass 7ian.fer

ECKERT

E C K E K . ~ AND DRAKE

HAM, CRANE, AND RODERS HARTENRERO AND DENAVITrrrNzE

. Mechatlies of Machinery. Kinen~nlicSynlhesis of I,inkages

. Turbulence. EtlGqineering Vihralior~~

J A c o n s m AND A Y R E

~ ~ v 1 N A l.. Ettgitleering i

Cot1.1idera1iotl.r n/.Ylrc.~r, .ylrci~,ntzd Slretzgth

Dr. J. KESTINI'rofe~sor at ljrown Urlivrmity in Providcr~cc., ltliodc Ialand

KAYS . Co~tveclir~e HealLICIIIY

and Mass Trcrtzsfir

. (~'o~nbt~slior~ Ettgine' P r o c e ~ ~ e s

M A R ~ N . I-IIEL.AN PIIELAN

K i l ~ ~ t t ~ aand ~ l i c D!jtian~Lsa/ machine.^

. I)!/~lan~ics Machinery qf. ~ l l ~ l d f l t l l e l lrfl ~ ln/fecharlim/ I)(rrigtl ~,~

RAVEN

. Arrlotnnlic Corrlrol En.gineerirtg

SOHP,N(:K

. 7'hroric.r ?f Ettgitteeri~lgExpcrir~~enlnlio~l. Iloundary-Imyer Theory

scrrLlcrr.rrNa si~io~.r:.u . srrlnr.Eu srrtol.eu

Dyttamic Analysir of Machines

. Kinenmlic Attaly.ris o f

iffechai~i.snzs

. Mccltnr~icolEr~~gineerir~ Desigrl

New YorkDiisselilorf

s ~ l i o r . ~ : .Sin~rrlnliot~ . ~ nf

Mcchnrriral Sy.clcnl.r S~IT)F(:KI;K . R~/j.igrro/inr~ Air (~ondilioni~~g ntld

New Uelhi

- Pa~iarno . Pnri~l . Siio I'nulo .

.

M c G R A W - H I L L BOOK COMPANY St. Louis . S a n Francisco Auckland BogotL . Johannesburg . London Madrid . Mexico . Montrenl

-

.

.

.

Singtrporo

Sydnoy

Tokyo

. . Toronto

Con tentsL i ~of 'Tnblcs tForc\vortl Alltllor'n I ' r r f n r r t o I.llr Sovclll 11 (I':II~~~P~I) I':[lil.i~ll l'rnnslntor's 1'1~elncot o t l l c Scvcl~t,ll (l211glisl1)ICtliI ion I'ro111 t l r r AIIIII~IV'S I'rcfncc t,o tho I " i ~ s(t( i r r l n n n ) IC(1it.ir111 Introdurtio~l xiiiXV

nvii xix sxi

I

I

A.

I'IBII~~;IIII~III~~ lrtws of n ~ o l i a ~ t a visrnrlr fluid for

rqltntiollrr of 111otio11 ntltl r o l ~ I . i l ~ r nl111lic.rl t o l l l ~ i t flon. ~~ty l n. I'IIII~:IIIII~II~~~ I,. (:vllrrnl nlrrss systclr~i t 1 n d r f n r ~ ~ l n hbotlg lr c. 'J'11r r ; ~ t c nl~irll I l ~ t i t r l r l n n l ~ is n t r n i l ~ r t ill I l o ~ v at ; I l t l rl. Ilrl:~t.i~,n I)rt\\nt.n strrsrr ittirl riitv o f ( l r f o r ~ t ~ a t ~ i o ~ ~ r. Slokm'n I ~ y ~ ) o t l i r s i n f. 131111~ viscosit.y n l l d l l ~ r r ~ ~ ~ o c l1)rrsnrlrr ~ ~ ~ i r y~~a g. 'l'lln N:rvior-Stokrs r l r l n t i o n n Iirfrrrt~c~ra

.,his I ~ o o k I w:rs sr.1 ill 1\111irj11n. 'Tllr r t l i t o r war 1'r:llrlc ,I. Crrrri nllrl (lie p r o t l r l r t i o ~ ~ ~ p r r v i s lvns s~ or,Jolt11 1.' l l : t r l r . 1 1)rincil)lc of s i ~ ~ ~ i l r i r fro111t l l r N:i\~irr.SI.okrs P~IIII~~IIIIR it,y n. I)orivnlion o f I1rynnltl~'o 1). c. rl. c. Frirtiol~lms flow 11s " n o l r ~ l . i o ~ ~ sf" tlrc N~cvics-Stoltcn q c ~ n l i o ~ ~ s o c 'I'llc Nn\,irr-Sto1cc.s eqt~nt,iol~s illt(\rl)rntrd as vort.irit,y lr1111~1)ort I I : I ~ ~ ~ I I R (.~ '1'11cen.tlsci t is t.11~ mt.io of t,l~c t,wo forces. I t is known a s t.110 Ilayitnk1.c ~slr.?t~bar, Tlir~st,wo flo\vs arc similar when t,I~c R. l 3.5 x lo6. 1 11e drng of R ~ ) J I P T C R hns rrcrnt,ly also 11ern invrst,ignf,rtl at, very high R.cynolds r~unlbers [I]. I t e r e too, a s wns t h e rase wit,l~h e cylil\clrr, thc tlrng coefficirnt increnses t n.pprecink~ly1)eyond its minirnrtnl at GD w 0.1 n t nbout R = 5 x lObn.t,t>nining CLI 0.2 n t Iteynoltls n l ~ r n b c r s close to R = 107. C:rit,ic,n.lrcvirws of drng mrnnurcnlrr~is n s p h r r r s RS it filn(*t,iorlof tJ1c Rry1101ds o nl~n~\)c,r t.11~ nntl Mn.c!h nnrnhcr wrre prrl)n.rrrl by A . 1%.Ilnilry n.nd ,T. Ilint~i. I n ] ns [ IV(*II a s 1)s A. I$. l$nilry an(1 Itrlrcrqnnnt.ity, thc dinplr~cernentthickness a,, is somct.imcs used, Fig. 2.3. I t , is dcfirletl I)y t.hc cqnnt,ion (2.6)

half frorn D t o E, and decelerated on the downstream half from E t o F. Ifcnce the pressure decreases frorn D t o E antl increases from i t o F. Wltcrl the flow is started ' u p the motion in t h e first insttarit is very nearly frict,ionlcss, ant1 remains so as Iqng as the bonr~tlnry lnycr remains thirl. O~rtsitlc I~onntl:~ry tho lrtycr tllcro is n tprr~l~s~ornlctl.io~ of prcsstrro in00 Itincl.ic energy i~long1 R, 1.110 rc:verso I.nlting pl:tc:o rilottg IC I(', so 1 IJtat IL 1)arliclc n r r i v o ~ 1' wiI,lt 1110H I L I I I ~ >vclociI,y 11s it, I t l i t 1 nl, J). A l I r c i ( l 1):~rI.iclt: 111, 1 wltich lrroves i 11 t,I~cimlnctlinlo vioi~til~y tatlo wtill in t,llc I)o~lntl:r.ry of I:~.yorrc:~rt:iitls under the influence of the same pressure field a s t h a t existing outside, I)crause the external pressure is imprcssctl on the bountlary layer. Owing tlo tlrc large friction forces in the thin boundary layer such a psrtic:lc consumcs so much of its kirtbtic

Fig. 2.4. Doundary-layerscpara-

tion ~indvortex forrnntion on a circular cylin(1er (dingra~n~natir)S

-

point nf 8cl1nrnI.lo11

'rhc displnccment tl~icltncss indicates l.llc tlistancc by which t l ~ external strcamc lines arc shift,cd owing to the ionnat,iorr of t,llc I,orlndary Iaycr. I n t.he case of a plate in parallel flow ant1 a t zcro incidcncc tlrc tlisplaccmrnt tVhickrressis about & of the bountlary-layer l,hicltncss 0 givcn in cqn. (2.1 a).

b. Srpamlion ant1 vortcx forrnntionllte bo11ntln.ry laycr near a fl:~,t plnLc in par:tllcl flow ant1 at, zcro incitlencc is ~jart,icrllarlysirnplc, Ijccausc tltc static prcssurc remains consCnnt in the whole field of Ilow. Sincc orlt,sitlc the ljo1111(1:~ry lnyrr t,hc vclocity rcn~aitrsconst,ar~t t,hc same n,pplics t.0 t.he p r c s s ~ ~ l~ecallscin t l ~ c re frictiorrlcss flow Bcrl~orrlli'scquat.ion remains valitl. F ~ ~ r t h c n n o r c , prcssnrc rcmnitls scnsibly cotrstnr~tover t.hc \vidth of t,hc tlrc \ ) o ~ ~ ~ r r l a r y a t a givcn rlist.ar~rc 1Icncc tlrc I)rossurc over thc widt.11 of the layer x. I~ountlary Iayrr has tlrc snmc mngnittrtlc ns ont.sitle t.hc bor~ndarylaycr a t the sanrc elist.ancc, ant1 t.lrc sarnc applies L cnscs of arbit,mry hotly phn.pcs whcn tJro prcssnrc o o~rt.sitlc l.hc I)o~tn(l:~ry I:~yt:r vnrics along t,lrc wall wit11 t , l ~ c Icngl.h of arc. 'l'llis fnct is cxprcsscd I)y saying t,h:~t, t,lrc cst.crnnl prcssnrr is " i ~ n ~ r c s s c t lon t,hc boundary " Inycr. Ilcncr in t.hc cnsc of tJrc ~not.ionp : ~ s ta plate l,hc prcssnrc rcmairls constant. t l ~ r o l ~ g l r ot,ltr: Lb o ~ ~ n t l n rInycr. ~~ y isi!rtinral~ly 'j'lrr ~hrnonrrnorr I,o~~rrtl:~ry of lnycrsrpnratiot ~nrt~tiot~c~tlprc~viously - - . c,ot~nc-clctlwrtll tlrr prcssurc t1istril)ution in ti16 orintlary layrr I n t,hc bountlary Inycr on a plate rro srpnmLion takrs pl:trr as no back-fldw occurs

..

\

111ortlcr to e~\plnitrtht. very ilnportntrt pl~rr~ornrrlon t)outrtlary-lnycr s ~ p a r a t i o n of let us rorrritlrr 1 l ~ Ilo\v :~l)orlI Ijlrrnt l)otly, r g abont, a rirrnlar rylirltlrr, a s shown r n i t 1 IClg 2 4 111 ft ic.1 inltlc~ssflow, t l ~ c f1111tlpnrl irlrs nrr : ~ r r r l r m l r t lon tllr npst,ream

energy on its pat.h from D t o E t h a t thc remaintlcr is too s ~ n a l l srlrmo~lntt.hc to "pressure hill" from E t.o F. Such a parLicle cannot move far into t,llc region of' increasing pressurc between lC antl P antl its moLio11 is, evcntnnlly, arrcst,ed. The external pressure callscs it t,lrcrl t,o move in tho opposite clircct,ion. The pllotograpl~s reproduced in Fig. 2.5 illnstrat.e the sequence of cvent.s near the downstrcarn side of a round body when ,z fluid flow is started. The prcssurc increases along t,Ile I,otly contour fronr left t,o right, the flow Ilnving been ma.tlc visil)lc by sprinltlirrg nlrtminirlln rlrrst on tho surface of thc water. Tlrc boundary layer can be casily rccognizetl by rcfcrcncc t o tlte slrorl traces. In Fig. 2.5a, Laltcn shortly aftcr the s t a r t of Lhc rnot,iorl; the rcvcrsc motmion has just bcgtln. 111 Fig. 2 . 5 b the rcvcrsc nrotion lrns pci~-t,r:.tctl a cot~sitlcrahlc distance forward :~ntll , l ~ c bountlnry Iayor lrns tllicltcnctl n.j)prcc:i:tl)ly. Fig. 2 . 5 shows how this rcvcrsc mot,iotr givcs risc t o a vortex, whoso sizc is incrc,iscd ~ still furt,l~cr Fig. 2.6tI. 'l'l~cvorLcx bccorncs scp:~mtctl in sllorlly n f L c r ~ : ~ rn.11c1 ~ I s rnovc!s tlow~~strearn tho fluitl. This circnn~stanccchangcs complctcly blrc fiolcl of flow in in tho waltc, and Lllc prcssnrc clisLrib~l(,ior~ suKcrs a rntlical cltnngc, ns cornparctl with frictio~rlcssIlow. 'L'llc final statc of nrotion can I)c irlrcrrctl from I'ig. 2.6. In t,he eddying region bclrind thc cylinder there is consitlcrable suction, as sccrl fro111 the pressure distribution curve in Fig. 1.10. This suction causes a large prcssurc drag on t.he body. 1 A t a larger distance from the body i t is possible t o discern a rcg111:lr patt,ern of vorticcs which move alternately clockwise and courrt~crclocltwise,and which is known a s a IGirmiin vortex strect [20], Fig. 2.7 (scc also Fig. 1.6). I n Fig. 2.6 a vortex moving in a clockwise direction can be seen t o be about t o detach it,sclf from the body before joining the pattern. I n a fnrtlrer pzpcr, von Icilrmhn [21] proved t h a t such vorticcs are gcncrally nrrst,nblc wit,h rcspcct to small t l i ~ t ~ ~ ~ r b apnrallcl ncrs

30

1I. Outli~rc bo~rndnry-lnyer or tlrcory

Fig 2.6. Jnslntltnnrnr~s~~lrotogrnpll of flow ~ r i t l r ronrplrte holrrlclnry lnyrr nrp:rratinrr i r r I I r r a.:~ltc: of it c.ircul:~r rylilrdcr, ~ ~ f l c r l'r~i~r~lll-'~irt.jrr~s 1271 Fig. 2.7. KhrmQ~i vortex strcct, fromA. Tirnlnc [38]

Fig. 2.8. Strmmlirrm in nvortrx strrrt (hll = 0 28). Thr fllrid i8 nt rest, nt t ~ infinity, a~itl h vortrx street move8 1pi~. . 5 ~ 2Fig. 2.5n. I,, r , tl.1271.

Src. nlal,

Ipip.

Fig. 2.5~1 Urvnloptnrnt. o l I)onr,dsry-In,yrrsrgnrntiotr with limr, nft.rr 1'rnrrclt.l-Tict.jcrrn 15.5

t o thr~ns~:lvc:s. 'I'lrc only nrmngnncnt wlridt sllows ncrrt.ral ctloilil,riurn is t,hat wit.11 0.281 ([Cia. 2.8). Tllc vort,ex sl.rcet rnovcs wit.11 n vcloc:it,y I L , which is slnallc\r I , \ ~ : I . I I t.Ilc no\\, vrIorii,y I I in front of t,he I ~ o d y I t cn.n l)c rcg:~rtletl s a highly idcalizctl . a pict,r~rrof t.hc rnot,ion in t.he wake of (,hc bocly. The kinetic energy cortt,ainetl in the vrlocit,y fit:ltl of tlre vort.cx street must be continually created, as the body moves t.llrongh tile fnitl. On the basis of this rcpresentrn.,tion it is possible t,o deduce a n exltrrssion for t.hc (1r:l.g from the perfect-fluid theory. I t s ~nngnit,rrtle nnit lengt,h per of tllr cylir~dric:~l hotly is given hy

.

Circ~tlur cylittder. 'l'hc frccluency wit,lr wl~ich vor1,irc~s r r shrcl in a I 3-2 x 10" very sharp increase is clearly visil)le, arid

an itlcnt,irnl plrcnomcnon is ol~srrvctlin a plot, of wall slrrari~rgstrcss. 'l'lrr srttltlcr~ incrcast? ill t,llrsc c1r1antit.ics tlcnot,cs t01:lt t.l~cflow Il;ts cl~arlgc.tl fro111 I:lmirl:lr t,o tr~rl)ulctll.. 'I'lrc Ttcynoltls tlrtrn1)cr R, 1):lsctl on t . 1 1 ~ ol~rrcnt,I(rtlgt,lr z is rc.l:tIrtl to t,l~(: ltry~toltls1111tl11)rr -:(I,?,,?/I, 11:1sr(I I I t.l~(, ~ o r ~ ~ t ( I : ~ ~ ~ y~ l : ~ y< ~ IrI ~, O IsI s I I R,, O l tl -i~:l ( ~~ ~ ~ I t,Itt, cvln:Ll.io~~

R,, . -

5

I/ R*

,

a s sprn from rqn. (2 1 a). l l r ~ l r c o tltr rriticxl Rrynoltls r~rtrnl~rr t

Fig. 2.23. Bor~ndnry-layer tllickness plobtedr against the Reynoltls number based on'the current lcngth z along a plate in pnrnllel flow a t zero incidence, ~s measrtrccl by llanscn [I61

there corrcspontls R g crlt = 2800. Tllc bor~~ltlary Inyrr or1 :I plate is Inrrlin:cr rrcnr t.l~t: lentling edge nrttl bcconles turbulent f~lrt.llcr tlo~vrlst,rca~n. nbscissn r,,,, of t l ~ t 'I'ltc poirrt of lrnt~sit~ion IICclctcrminctl f r o n ~Llrc k t l o w ~v:~lticof R, can ~ I n t . 1 1 ~caso of n plate, a s in the prcvior~slytliscussctl pipe flow, t.ltc rrrtntcricnl vaI11o of R,,,, dcpcntls t o a ~narkctl tlegrce on tlre arnorlnt of' tlist.~trl~ancc t.110 nxt,crn:tl flow, :1t1t1 in the value R, = 3.2 x 10%slrot1lcl be regartlet1 ns a lower limit,. With oxccpt.iorl:~Ily tlist~trrbnncc-frcc cxt.crnal flow, valrtcs of R, - 10%rlrltl I~igltrr 11:~vc been :~tt.ailrrtl. A 1):~rticul:trly rernarltable phcnorncno~l connccl,ctl with the transit.iotl from laminar t o trlrbrtlt:rrt flow occrtrs i r l t,he casc of blrtnt llotlics, sl~cll s circr~lar a cylintlers or spheres. I t will be seen from Figs. 1.4 arttl 1.5 t,l~aL tlmg coef'ficierlt o f a circrtlar the cylirrtlcr or a sphcre srtffcrs a s~~tltlctr :tot1 consitlcral~ledccrcasc 1lr:rr Itcynoltls rrri~nl~crs I)/v of bout 5 X lo5 or 3 x lo5 rcs~)ccLiv(~l~. fact was first, obscrvrtl 1' . 'I'lris on spttcrcs by G . 1I:iffrl 1141. It. is a conscquerlcc of t,ransition whicl~cnttses t.lle point of separation t o movc clownstacam, l)cca~rsc, t,hc case of a turbulcrrt 1)ountlary in laycr, tJtc accelerating influence of tthc cxt.crn:ll flow oxt,cntls fur1,her due t,o t.t~rbulrrrt. mixing. ~Tcncctile point of separation wllicll lies near the cqnator for a laminar I)o~rr~tlary I:~ycr nlovcs over a corrsitlcml~lotlislnr~ccit1 t,he downstream tlircct.ior~. In t,rlrn, the tlcad arca decreases considcmbly, anti t,ltc pressure dist,ribution becomes more like t,lrat for frict,ionless motion (Fig. 1.11). The decrease in tltc rlcad-wat,cr region consitlcmbly reduces t h e prcssrlrc tlmg, and t h a t shows itself as a jump ill the crtrve G, . f(R). L. T'mnrltl [26] provctl tlre corrcct,~rcssof t,lrc prrcccling = rcasoning 11y nlo~inl~ing I , l ~ i r iwirc ring III; a ~Itort, n (I~Y(~ILII(:C f r o ~ t t I,11occlri:lt,or ill or of a sphere. This car~scs boundary laycr to bccome art,ificially tur1)rtlcrlt a t n lower the Reynolds n l ~ m b c r arrcl the tlccrcasc in t,hc tlrng cocfficicrrt taltes place carlicr Lllatr would otherwise be the case. Figs. 2.24 and 2.26 reproduce photographs of flows which have been made visible by smoke. They reprcsctlt the subcrit,ical pattern with a large value of the drag coefficient and the supercritical pattern with a small (Iced-water arca and a small value of t h e drag coefficient. The sttpercritical pat,tern was achieved with Prandt,l's tripping wire. The preceding cxporimcnt sl~owsin a convincing nlanncr t,hat t h e jrtnlp in the drag curve of a rircular cylintlcr and sphere can only be interprctcd a s a borindary-layer phcnomcnor~.Otlror bodies with a blunt or rounded slcrn. (c. g. elliptic cylil~tlcrs) display :I type of rclat.ionship bctwcen drag coefficient ant1 Rcynolcls nttlnbcr wllicl~is sr~l)sta~~li:illy sinlilnr. \Vit,lt increasing slcntlcrness the jump in tfhc ctlrvc i)ccomcs ~ ' i r o ~ r c s s i vless pronor~nccd. cl~ For a streamline body, such ns tallat shown it1 Fig. 1.12 t.h(:rc is rlo jump, I)nc:~~tsc rto :lpprrci:r.l)lc srp:r.mt,io~roccltrs; t,llr w r y gmtlrt:tl 1;rc-ssrlrc irrc!rr:lso 011 I,l~c I~;lclt

.,,,.

,,

c. T~~rhulent in n pipe nr~d a hourldnry lnycr flow in of s11c.11 1)otlics csan I)e overcome I)y t l ~ c bor~ntla.rylayer w i t , h o ~ separat.ion. AS we ~t sllall also scc Int,cr in grrat,er tlct.ail, t,l~e pressure di~tribut~ion thc ext,ernal flow in t.xrrt,s a clct~isivcinfl~~c:rlce t,hc positmionof t.lle transition point. Thc bountlnry on Ia.yrr is I n t ~ ~ i n a r t.11~ in regiotl of prcssurc deereast, i. e. rollghly from t.110 leading ntlgc? to t.hr p i n t of minimr~tn pressure, ant1 becomes t,rtrhulent, i l l most cases, from t . l ~ : ~ t point on\\,nrrl t . l ~ r i ) ~ l g l ~.o ~ region of prcsslrrc inrrcn.sc. I n this corrnexion t l r t, ~ it is iml~ort,ant statc t,hnt, scparnt,ion can only bc nvoitletl in rcgiorrs of incrensing to prcssnrc nh?n the flotv in t,hc bountlnry layer is turlrulcnt. A laminar 1)ountlary layer,

43

'hhle 2.1. Tllickness of bormdary Inyer, 6, a t t.rniling edge oF flnt plate nt zero inridencc in pnrnllel t.nrb~~lent flow(, J= rrcr ntrenlll vrloclty:

I = lrnqtll or pla1.e:

r = kinrn>nl.le risrasily

IAirv =

(1,

~ r t / ~ * c l ifi,, 100 2002 0

I

1

1"

=.. .

U 110' lo7 lo7108

I

rill

"1

:I:J

150

X

10-e f t Z / ~ ~ v :

50 760 \VntOrv = 11

15 25 25R

2.0 x 4.0 x 2.0 x 8,:s x 1.25 x 2.3 x 1.35 x 3.4 x 2.3 x

0.73O+i42.30

10'

2.!)02.68

x lo-' ft2/4cc

5 10 25r,O

15 150500

10' lo7108 100

1.19 2-52 13.129.8

I?ig. 2.24. I i l o t v

I I : I S ~n

I ~ I ~ ~ I I n r ~~ \ ~ ~ , \ r n ~

~.~: rroll\

nL spl~rrc: n s ~ ~ h t ! r i I i ~ . : ~ l T'ig. 2.26. Flow past n ~plirre n n~~percrint, t.icnl I n \Virurlnl~c.rgrr \\lic.~rlnl,c.vNrr: \ ~ ( j [3!)]. Tlw nopcrcrit.ical flow pnt,tcrl~ nr.l~irin ved by tlle mounting of n thin wire r i ~ ~ g (I.rippingmire)

a s n.c shall see Int.er, can support, only n very slnnll pressure rise so t,l~at. scpnrat,iorr wonltl occur rven wit.l~ very slcndcr botlics. I n p,qrt.icular, t,his remark also applies t o the flow past nn aerofoil wit,li n pressure dist,rit)ut,iorlsilr~ilnrto t h a t in Fig. 1.14. 111 t.llis cnse scpamt~iotl most liltcly t,o ocrur on t.he sncI,ion side. A smoot,l~ is flow pattern nround n.n ncrotbil, contlr~civc ~ I I C creation of lift, is possihlr only wit.11a t,l~rhnlent t.o botlntla.ry Ia.ycr. Snrnming up it, ma.)i be st.at,rtl that, t.hc small drag of slencler botlies a s wrll &s t . 1 1 ~lift, of acrofoils are rna.tle possible t,l~~.ough cxist,enrc of n t,url)ulent, thc t)ountla,ry Inyer. y the Bour~clnr~-ln~er thickness: ( ~ c r ~ e r a l lspealc~r~g, thicknesq of a tnrbrllcnt 1,011ntlary leycr is larger than t h a t of n laminar boundary layer owing to grratcr energy losses in the former. Nenr a smooth flat plate a t zero incitlcnce the boundary layer incrcascs downstream in proportion to xoR(x = &stance from leading edge) It will 1-I(. R I I ~ W I IInter in Chap. X X I t h a t the boundary-layer tl~ieknrssvariation in (nrt)nlrnt flow is given 1,y the rqnntiond1

Methods for the preveatinn of separation: Sopnrnt,ion is most ly nn r~ntlcsir:~.I~lt! ~ ) l ~ r n o m c n bccnusr, it c l l t r ~ ilnrgo onorgy losscs. I'nr (.his rcnson rnc\t.llocls I ~ r ~ v o on l~ 1,(:t:11 tleviscd for the artificial prcvcntion of separation. T l ~ c sirnplest met.hotl, from t,l~c physical point of view, is t o move the wall with the stream in order t o rcdr~cehhc velocity difference between them, and hence t o remove the cause of boundary-layer formation, b u t this is very difficult to nchicvc in engineering practice. Ilowcvcr, I'rnndtl t has shown on n rolaling circ11.1ar cyli?zP.r tllat this metllod is very rfrcct.ivn. On the side where t h e wall and stream move in t h c same direction separnt.ion is cornpletely prevented. Moreover, on the side where t.11~ wall and st.renn~ move in oppositc tlircct,ions, separation is slight so t h a t on the whole i t is possible t o obt.ain a gootl experimental npproximat,ion t.o perfcct flow wit11 circulation ant1 a large lift.. Another very effective metliotl for tlic prcvcnt,iorl of scparation is ~ o ~ L ~ I ~ < I I . ! J Ltycr sudion. III this metl~otlt11c dccclcratccl fluid pnrticlcs in the bonntlary Inyrr are removed through slits in t,he wall into thc interior of the body. Wit,11 srrf'ficic.r~t.ly strong snction, sepxrat.ion can be prevented. Bo~tndnry-layersuct,ion was nsecl on a circular cylintler by L. PrantIt,l in his first. fnntl:~~nrntal invcst.igat,ion in1.o boundary-layer flow. Separation can be almost completely eliminated wit.11 suct,iot~ through a slit a t the back of t-he circ~llnrcylinder. Instrnnccs of t,he cffrct. of snc.I,ion can be seen in Figs. 2.14 and 2.16 on tile exnmplc of flows tlirougll n l~igllly divergent channel. Fig. 2.13 demonstrat,es t h a t witllout suction t.hcrc is strong separation. Fig. 2.14 shows how the flow adheres t o t.hc one sirlc on wlliclt srtction is applied, wherens from Fig. 2.16 i t is seen t h a t the flow complctcly fills the clrannel cross-sect.ionwhen t h e s ~ c t ~slits~ ~ pnt int.0 operation on botli si(Irs. I I I t . 1 1 ~ i o are latter case t,lie strea~r~lines assunlc a pat,t,crn whicl~ very sinlilar t,o l , l ~ ain liiet.j;)rllcss is t flow. In lat.cr gears sr~ct~ion sncccssfr~lly was used irt acroplanc wings to ill(-rcasc. (.11c lift. Owing t,o snc:t.ior~ the: rlppcr s r ~ r f a c : ~ t l ~ c on near t.ra.iling edge, t,l~e flo\\~ ntlllrrrs.

= 0.37

( )lJm,l

f

-'I5

*

= 0.37 (RI)-'l'

(2.9)

\vl~ic-llc:orrt:sl)ontls 1.0 rcln (2.2) for laminar flow. I'ahlr 2.1 gives vnlnes for tJrc 11o1111~l:r.ry-l:tyt~r i ( ~ k ~ ~ c s s tll~ o:~l(:~~laI.rtI ecln. (Z.!)) for several typical casos of air from : ~ 1 t t 1 watl~r flows.

t

Prnncltl-Tietjens: Hydro- nnd Aerodynnmics. Vol. 11, Tnl~lrn 11 and 9. 7,

44

11. Outline of boundary-layer theory

References

45

to t h e aerofoil at considerably larger incidence a n g l e s t h a n yonl(1 otllcrwisr b e t l r ~rase. stalling is clrl:cyetl, nntl m u c h l n r g t r m a x i m u m - l i f t values a r e achieved [3F]. Aft,er )laving g i v e n a s h o r t out,lino of t,he f n n t l a m e n t ~ a lphysical principles of f l ~ ~ i do t i o n s w i t , l ~v e r y snlnll friot.ion, i. c. of t l ~ c o u n d a r y - l a y e r t h e o r y , w c shnll m b proncccj t o clovc!lop n m t i o n a l tllcory of t l ~ c s opl~cnorncrlnf r o l r ~ 0110 o(1111~1.ions 01' m o t i o n of V ~ S C O I I Sfluids. Thf: (icscription will b e arr:~rlgctl in t h e followir~gw a y : Wt: shall begin i n I'art A by deriving Ghc g c n c r a l Navicr-Stjokes c c i ~ ~ a t i o nrso m w h i c l ~ , f i n t u r n , w e s h a l l d e r i v e l'randtl's b o u n d a r y - l a y e r e q u a t i o n s w i t h t h e nick of t h e sirnplificstions which c a n b e inl,rotlucctl a s a cor~seclucnccof tjhe s m a l l v a l r ~ c s visof c o s i t , ~ h i s will b e followed i n T'art I3 by a t l c ~ c r i p t ~ i o n t h e metjhods f o r t h e i n t e g m T of t i o n of t h e s e cqnat.ions f o r t h e caso of l a m i n a r flow. 111 P a r t C w e s h a l l d i s c r ~ s st h e p o b l e m of t h o origin of t,nrbulcnt flow, i . o. w e shall discuss t h o process of t r a n s i t i o n from l a m i n a r t,o t , u r b u l e n t flow, t r e a t i n g it, a s a problern i n t h e stabiliLy of l a m i n a r mot,ion. F i n a l l y , Pn.rt .D will c o n t a i n t h e b o r ~ n t l a r y - l a y c r t h e o r y for complet,ely tlcvclopcrl t ~ ~ r b u l r n to t i o n s . W h e r e a s t h e t h e o r y of l a m i n a r b o u n d a r y l a y e r s c a n m I)c trrat,ctl as n d c t l ~ l c t ~ i v e q u e n c e I ~ a s c t l n t,hc Nnvicr-Stolrrs tlifTerent,i~l s o eqllationx for viscous flrritls, t h o s a r n c is not,, a t prcscnt,, possible for t u r b u l e n t flow, t)ccnusc thc: m c c l ~ a n i s mor t n r h ~ l l c n tflow i s s o c o m p l c x t . l ~ a ti t c a n n o t b e nlastercci by p u r e l y t.l~rorct,icnl mct,hods. F o r t,his reason a t,rc;~iisc n t l n r 1 ~ n l c rflow must, tlraw 11e:~vily o ~t or1 e x p r r i m e r ~ t n l result,s ant1 t,llc s r ~ l ) j c r t n s t Ijc presented i n t,hc f o r m of a s e m i m cmpiriral throry.

References

1

[I] Acl~enbac:h,E.: J':xperilnent,s on the flow past spheres a t very l ~ i g lIZeynolds nr~lnbers. ~ J F M 54, 505--575 (1972). 121 Ilcrger, J':., ant1 Wille, It.: Perioclic flow p h e l ~ o ~ t ~ cA~ n I I ~ : I ~ l I I . Review of lpluitl hlcch. 4 , 3133--340 (1072). (31 nerger, 15. : Ucut.iln~nrl~~g I ~ y d r o d y n : ~ l ~ ~ (:riissen~einer I < h r ~ ~ ~ h n s r h c n dcr iwl~c ~ \I'irl)rlutransr aun t l ~ t z c l m l ~ t ~ ~ ~ e sbci ~ i g c ~ ~ I

s ~ I I C : I I tlicro is a free s~lrftlccor wllcrl l,lto tlrtlsily clisl.ril~trl.iotlis itll~orno~c:~~c:o~ts. 111 111,: ( : x s ~of a hornogcnrol~sfl~titlin tltc :l.hsenc:t? oS n Srrc s\tr(i~t!c t,Itrr(: i s c:tl~tiIiI)t.i~tt~~ I ) C ~ I V ~ ( : I I 1,110~vcigltbof'c:at:l~ p;l.r(.ivIo:LIEII it,u l~.yrlrrost inlt,llis port,ant oncs in n.pplicntions. Tltrls bltc Nnvicr-Stoltcs rqnations will now c:ortt,air~ only forces clue t o pressure, viscosity, and inertia. Unclor thesc asst~mptionsant1 ronvcntions i.hc Nitvier-St,olccs rcln:ttions for :In inromprcssiblc flr~itl,rcstrick:tl 10 s t c i ~ d y[low nncl in vcclor ftjrttt, si1111~lil:y o t

CHAPTER I V

General properties of the Navier-Stakes equationsReforc p:lssing on t,o thc int,rgrat,inn of t h e Navicr-St.okrs cq~rnl.ionsin t.11~ following ch:lpt,ers, it now sncms pcrtincnt, t o discnss sornc of their general properties. In tloing so wc shsll restrict ollrsclvcs to irrcornprcssiblc viscous fluitls.

R.

J)c.rivntion nf Reynolds's principle of sin~ilnrityfrom the Nnvicr-Stokes cquntiorla

TJr~tilt,Ilc prrscnt clay no gcnc,r:~la.~tn.lyt,ic n~rt,l~otl.s I~rcotnc 11:tvr availnltlc for tlle intc-gmtion of t,hc Nnvirr-Sl.okcs~clt~at,ions. u r t l ~ c r t n o rsolrtlion~ I~ c, wl~inlla1.r vnlitl for all values of viscosity are Irnown only for sonle particular cases, c. g. for Poiseuille flow tl\rorlgh a circular pipe, or for Couctte flow bet,ween two parallcl walls, onc of which is a t mst,, the other moving along its own plane with a constant vcloc:ity (set: Fig. 1.1). For this reason t l ~ c problcm of calculatir~gthe motion of a viscous fluitl was attaclrctl by first tackling limiting cases, t h a t is, by solving prohlcrns for very large viscosit,ics, on t h e one hand, and for very small viscosities on t,hc other, I)cmrlsr in t,ltis manner t.hc matllcmatical problem is considerably simpliActl. liowevrr, tlro casr of modcrat,c viscosit,ics cannot 1)c irlt,erpolat,cd I~ct~ween 1l:ven the limit,ing cases of vcry largc ant1 very small viscosities present great mathematical tlifficulties so t h a t rescarcll into viscolls fluid motion proceedetl t o a largc cxtcnt. by experiment. I n this conncxion t,l~c Navier-Stnlrcs equations ftrrnish vcry uscSttl 11int,s wtiicl~point t o a considerable rcduct,ion in the qnantity of cxpcrimcnt.al worlr required. I t is oft,cn possible t o carry out. expcrimcnts on models, which means t h a t in t h e e ~ p e r i m c n t ~ arrangement a geometrically similar al motlcl of t,hc not11a1 body, b u t reduced in scale, is investigated in a wirttl tunnel, or ol.hcr slritahlc arrangement. This always raises the question of t h e dynamic sim,ilnril?y of flclitl mot.ions which is, evidcnt.ly, intimately connect,cd with t h e quesobt,a.inod wit,l~ motlcls can Jlc nbilizcd for t h e prediction of tion of Itow far rcsult.~ tho I>t:haviollr of t h e FIIII-scaleI)ody.I

'J'llis clifl'crer~tinl cql~ationmust l ~ c indrpcnclent d the clloicc of t . 1 1 ~t~tli(.s t.lrc for various physical quantities, s11c:h as velocity, prcssnrc, clc., which appe:lr in it.

We now consider flows about two gcomctrically similar boclics of diKcrcrlt lincar tlimcnsions in streams of different velocitics, c. g., flows past two sptlcms in wllictl t h e densitics a n d viscosities may also bc different. Wc shall invcstigatc tllc corltlition for dynamic similaritfly with the aid of tho Navier-Sbokcs cqoat,iot~s.Evidently, dynamic similarity will prevail if wit11 a suitablc choice of t l ~ units of Icngf.h, c tirnc, ant1 force, the Navicr-Stoltes cqn. (4.1) is so t,mnsforn~ctl t h a t it, I)cco~ncs iclent.ica1 for the two flows with geomctric:ally similar botirltlarics. Now, it is [~ossil)lc: t o free oneself from (.he fortuitously selechcl r ~ n i t sif clirncnsiorllcss tlllntltitics n.rc introducetl into cqn. (4.1). This is acllievccl by snlcct.irlg ccrt,:~in sr~itnhl(: c:llar:rc:taristic mxgnitr~dcsi r l tllc flow a s our ~rrtil,s, ant1 I)y rcfcrring all othc:rs t,o t.11c:nr. ..Ll~its g., the frcc-slrcan~ , c. vcloci1,y anrltl tllc tlianlcI.cr of (,I111 spl~rrc:c:alt IJ~: st-l(:c:l.c:tl a s t h e rcspcctivc 11nit.sof vcloc:it.y nrltl Icrlgth. 1~r.tV, 1, and pl tlcnotc t l ~ c s ccltaractcristic rcfcrcnco magt:itrltlrs. II' we now introtltlcc into thc Navicr-Sl.olzcs eqn. (4.1) thc tlirrrrr~wionlrss ri~i.ios

As a1rr:ttIy oxpl~incrlin Chap. I, two fluitl nibtions are dynamically similar if, with gc?ornct,rit:allysirnilnr k)ot~ndn,ries, t.he velocity ficltls are geometrically similar, i. e., .if 1,ltc.v have gromct,ricnlly similar st~rcnrnlincs.

pressure

P = -p

PI '

This qucstiot~wits atlswcrrd in Chap. 1 for t h r case in which only inertia and visrntts fnrt*rs t:~Itc1)itrI.in tile proc,css. I t was found tllere t h a t for t h e two motions

72

I\'. C:rncr:tl prol,e~lirs tlrc Navier-Stokrs ecluat.ions of

c. The Navicr-Stoltes eqnations interproled as vortirit,~ t.rnnsporL e q r ~ n t i o ~ ~ s

73

1 ) or, c l i \ f i t l i ~ ~)g~ c)

172/18

(4.2) / ,I l ~ oIll~icl I r ~ o t , i o ~li~ n t l v rc-o~isitlcr:~t.io~l I,rt.omc similnr o i ~ l yif t l l ~ o s (,;in solntions t.xlr':Ss'.,I i l l {,(:rnls O[ (,II(: rcrsl~oclivc: t l i ~ n c ~ ~ ~ s i o ~v:iriiik~lcs nro itlc~~l,icnl. ricss 'Vliis rccltlily:s (,I1:Lt, I,ot,l~nlol,io~ls , l ~ rc,sl~c:c:t,ivc:t l i ~ n c n s i o l ~ ~ Nnvior-St,olrcs cc]r~:~.t.ions SO^ t c ess tlilTrr o ~ ~ \)y n fac't.or common t o all t.crms. 'l'l~c qnnntit,y p,/e V 2 rcprescnls tllc ly r:~t,ioof prc?ssltrc t.o 1.11c tlo~lblcof t.hc tlynanlic l ~ c a dant1 is u n i m p ~ r t ~ a n t t h e for tlynnrnic si~nilarit,yor t,hc t,wo motions I ) c c n ~ ~ s c i n c o n l ~ ~ r c s s i b lflow a clrnngc in c in 1)rcssIlrc cnrisrs n o c:h:irigc i l l v o l l ~ m c .l'hc sccond factor e V I / / L is, liowcver, vory irnl,ort,:inl, :ind m n s t nsslllne t,Ile s a m e valrlc for bot.11 motions if t h e y a r c t o I)e tlyr~n.n~ic::iIIy similar. Ilencc tlynnmic sil~lilarityis assured if for t h c t w o nlotions

(I1:gratl) ll

.

-

- I" grad P I?,

-t e " l V

"2

11'.

T l ~ r ~ s frictional terms in eqn. (4.1) vanish identically for potential flows, but ge~~erally the speaking both boundary conditione (3.36) for the velocity cannot tlrcn be satisfied sin~ulta~reor~sly. If the normal con~ponentmust ccsu~tn~u prencribed vnlucs along n bonntinry, t l ~ r n , potential in flow, 1.l1o t,iw ont.inl oon~pono~lt tl~orobyclolorr~~i~rnd I,llr\t l,l~n nlip oo~~cliI.io~~~ I I tO I ~ i~ no 110 I,III I 0 sdislicd nt Lf~oR I ~ I I I O l,i~no.Jd'or Ll~isrt!ason clno cnnnok regnril p>hnLinl ilowe a" p l ~ y s i t ; ~ d l ~ moatringfill nol~~tiona 1.110 Nnvicr-Stokon cqr~ntionn,bocnrrno tl~oy not nnt.inry tlic ~)r~:scril)rd of (lo boundary conditions. l'lrcro exist^, howcver, an i~nportantcxccption to t l ~ oprccccli~~g ~t.xtcmcnt wl~iclroccurn wl~ontho solid wall is in rnotion and when this condition tlocs not apply. The sin~plest parlicular case is that of flow ~mnta rotating cylinder wl~cnthe potential ROIIItion tloes constit,utc a meaningfnl solution to the Navicr-Stokcs cquntior~s,as explainctf i l l grcatcr detail on p. 80. The rcadcr tnny rcfor t,o two papers, one by G . 1InnieI [ 4 ] n~rdonc by. J. Aclteret [I], for fnrt.ller details. The following sect,ions will be rest,ricted to the consideration of plane (two-din\rnsional) flows because for such caocs only is it possible t,o inclicato son~e gcncral properties of Lhc NavicrStokes equations, and, on Mia oClrcr hand, plane flows roostituh by fir tho lnrgrst clans of prohlcrns of practical i~nportance.

c. The Navier-Stokes equations interpreted a s vorticity transport equatinns I n t,he case of two-tlimensional nori-stcatly flow in t.hc x,y - p l a ~ l c t,l~o vcloc:it.y vector b c c o ~ n c s

'J'his princil)lo was tlisc:ovrrctl I)y Osbornc Iteynoltls when ho i~lvrst,igxt.etlfluitl 1rlol.io1lt , l i r o ~ ~~'ipcsnn(I is, t IrrrcSorc, lznown ns t h e Reynolds prilrripla o/ si??rilnrib?/. gl~ 'I'hc r l i ~ n r t ~ s i o ~ ~ l r s s ral.io

e."Cc

=

vzv

_

a n d l h c s y s t e m of rquat,ions (3.32) a n d (3.33) t r n n s f o r ~ n s n t o i

R

(4.3)

is cnllctl t h e Itoynoltls rlrl~ii\)cr.JTere t h o ratio of t h e d y n a m i c viscosit,y 11, t o t h e clcr~si(,y tlcr~otctlb y e, = ,I./@, is t h e Itir~cmaticviscosity of t h e f l ~ ~ i tint.rotl~lccd l, cn,rlicr. S ~ ~ n i m i nn p we can s t a t e that, flows n h o u t g e o n ~ c t ~ r i r a l lsirrlilnr 1)odics g y a r e tly~~n.miaally similar whcn t h e Rcynoltls n u m b e r s f o r t11c flows a r c e q ~ l a l . T h u s Itcynoltls's similarity principle h a s been d e d ~ ~ c c once nlorc, t,his t,imc tl f r o ~ nt,he Navicr-Strokes c q ~ l a t i o n s ,having I ~ c c nprcvionsly dcrivcd first from a n c:st.irnnt,ion o r Sorccs :in(] sccontlly from dimensional analysis.

wl~iclifurnishes three equations for

u, v, artd p.

curl r r ~ , \rrl~icl~ rctlr~ccst o t.hc o n e W e now introduce t h e vector of v ~ r t i c i t ~ y , component a b o u t t , l z-axis for two-diniensional flow: ~

1,. I.'riclior~ienn flow a s LL801t~1io~~n" h e Navicr-Stokes equations of t not,i~~g, ~)arcnt.Irrt.ically, that, the .solutions for incomprcssil~lc/riclionless I t nay bo n.ort,l~ flown may also bc regarded as exact solutionn of 1.h~ Nnvier-Stokes cq~~nt,ionn, bcca~~sc s u c l ~ in rases t.ho frictional tcrnrs vanish itlont.icnlly. In t,llc case of incomprcssiblc, fricl.ion~csn flows tho vrloc.il.y vector can he rrprrscntn?tl an t,hc grntlicr~t f a potcnt.ial: o=

I~rict,ionlessmotions a r e irrotat.ionn1 s o t h a t curl minating prcssurc from eqns. (4.4a, b ) we o b t a i n

cct

= 0 in s ~ l c l lcascs. Eli-

grad di

,

whrrr t.he potcnt,ial @ R R ~ ~ S ~ t,hc L:lplacc cqllat.ion IOS

,'=0 .

or, in short,hand form

V2@=0.We t.11~11 l ~ o n have grad

(V2 @)

-V

2 (grad @)

=

0, that is, V 2 w

t

See foot.nota on 1,. 48.

This equation is referred t o a s t h e vorlicity transport, or transfer, equatzor~ I t stat(.s t h a t t h e subshantive variation of vorticity, wl~iclt consists of tllc l o r d a r ~ t r o t ~ v r c t , ~ c c ~ l

74

TV. Gencrol proprtic~ tho Novior-Stoke8 eqrlnlionrr of

C. The Nnvirr-Stoke3 rqrrol.ionfi intcrprrtccl W vorlicit,~ I Lrnnsport rqtlnt ion3

76

terms, is cqrtal t,o tlrc rate of clissipntion of vorticity t,lrro~~glr friction. Eqn. (4.6), togclher with tlrc equat,ion of contiiruit,y (4.4c), form n system of two equations y for t,hc two v ~ l o r i t ~components ?I, anti o. Finally, it, is possible to transform t.hcsc two cqriat.ions wit,lr two ~rnknowns ir1t.o one eq~iat.iorrwit4h one ~ ~ n k n o wby introtlrtcing tJhe st,rcam fnnc,tior~t(r(x.y). n Pltt,l.it~p

I n this form the vorticit,y t r n n s p o ~ tcquntiorr contains only one unknown, 11). 'Clre left-harrd side of cqn. (4.10) contains, as was the casc with the Navicr-Stokcs equations, the inertin tormu, whcrcas t2ho right-lrhrld siclo cont,aitts tho frirtionnl tcrms. I t is a fourtll-order 1tarti:ll dilfercrrtial cquatiorl in tile strcarr~f~irrctiorr '7, I t s solution in gcneral terms is, agnin, vcry clifficult, owing ho its bcirig iron-lirrr;rr.V. G1. Jcrrsorr 1 1 fount1 a s o l ~ ~ t i otn the vorticity transport cclllatio~~ 5 o (4.10) for the case of a sphere by numcricnl integration. The resulting pat,trcrns of s(.rcainlines for different Reynolds numbers arc seen plotted in Fig. 4.1 which also contnirls clingrnms of the distribution of vort,icit.y in t h r flow fioltl. Tltc smnllost. Itc:ynol(l~ number irrclrldcd, R = 5 in Figa. 4.1s anti 4.1~1,corresponds to thc casc w11c11tlrc viscous forces by far outweigh the inertia forccs and the resulting flow can bc described nu crecping motion, Scc. I V d ar~tl(~11nptr.r hr this casc the wholc flow fit-It1 VI. is rotatiorral and tho pattcrns of strcnmlirlcs forward and aft are nenrly ident.ic:nl. As thc ltcynolds number is incrcnscd the sphere dcvclops on its rcar a scparatcd region with back-flow and the irttensity of vorticity is progressively more concentrated near the downstream portion of the sphere, wherca.. in thc forward portioir tlrc flow becomes nearly irrot,ationnl. The flow patterns undcr consideration which have been dcdr~ccdfrorn the Nnvier-Stokes equation, allow us to rccognizc thc clrnractcristic changes which take place in the stream as the Reynolds number is made to increase, even i f a t t,he highest Reynolds number rcachcd, R =. 40 in Figa. 4.1 c and 4.1 f, tile boundary layer pattern has not yet had a chance to develop fully.

we see t,llaLtho cor~t,innil.~~ oclltation is s:tt,islictl aut~om:rt.irally. n ncltlit,ion l.lle vorLici1,y I from eclrl. (4.5) 1)c:conrcs w=+v2y,t (4.9)

Very extensive experimental inveutigations of the wake behind a circular cylinder in the range of Reynolds numbers 5 < R < 40 nre described in two papers by M. Coutanceau nnd R. Bouard [lc, ld] who coveretl both steady and unsteady flows. The development of very efficient elc~t~ronic computers in modern times has made i t possible to solve the Navicr-Stokes equations for flow past geometrically sirrrple bodies by purely numerical methods. In order to do this, tlrc differential equations are replnccd by difference equations. The numerical techniqucs used for this purpose will be explained in Sec. 1x1. Without discussing this matter here irr any ilcpt,h, we quote one irrteresting result. Figure 4.2 shows tlre flow past a rect,angular plate placed a t right anglcs t o the stream calculated by J. E. Fromrn and 1;". H. IIarlow 131. At the back of the plate there forms a vort,cx street similar to that bchintl a circular cylinder shown in Figs. 1.6 and 2.7. Figure 4.2ashowsan expcrirnentnlly detcrmined p n t t ~ r n slreamlincs, where*! Fig. 4.2b rcprcscrrts thc calculatcd ficld, both for n of Reynoltls number Vdlv = 6000. Thc agrconctrt bctwccn tflrc two patt,crrrs is rcmilrkably good, in spitc of thc fact that in this rnngo of 1tcyrroldanrtml~crsl.l1c aecluircs flow an oscillat,ory character, Fig. 1.6. Tltc earliest attcml)ts t,o ol)tain UIICII nl~rncrical sol~rt,iol~s t,lrn N:~virr-St,okc,s t,o c~clt~:ltions can I)c t.rnt:c,tl 1.0 A. 'l'ltotr~ 101 ~ 1 1 1 , ~ , c r formed such cslc~ilat.ions a circular cylindcr nt the low Rcyrrolds n~rrnl)crs -- 10 for R to 20. I,ater, the calculations wcre carried to R = 100 [ ] As the R.c:ynoltls numl)rr 2. increases, the degree of difficulty of such numcrical int.egrat,ions increases st,ccply. I n this conncxtion i t is worth consulLiiig the comprchensivc sr~mmnryby A. l'lrom and C. ,I. Apclt [7], as well the work of C. J . Apclt [I n] and I). N. tlc (2. Allerr and R. V. So~it~l~wcll nnd of If. B. ICcllcr nnt1 11. Takami L5nl. [ I I))

liiR. 4.1

Rn,

I':r.ltrrns or rlro(.iorr i r r :b visrorls llo\v 17ns11 n ~1111orc 11illi:rrtrt. 11c~y11old~I I I I I ~ O ~ S at II IT1)/v tlrrivcnct.ctl 1.0 s~r(,isfy I I O ~ t ~ ~ ~ ~ r t , i o n x ~ of I'ri(:t,iot~lt.:i?( flow, the potc~lt~ial llow theory bcing uscd for i h cvnlnaI,ion, wl~crcnsin tllc sc~c-otr(l region vorticity is inherent, and, t.hcrcfore, the Navicr-Stoltcs cq~int.iorlsm ~ ~ shn t. ~ l s e dfor its cvnluntion. Viscous forccs are import,ar~t~, c. of 1.11~ santc ortlcr 91' i. mngt~itt~tlc inertin forces, only in t l ~ c n 9 . scc:ontl region known :is t,hc bo~~.~~~lrtr?y Irr?yrr. This concept of a boundary layer was introduced into the scicrlcc of fll~irlmechanics by L. Prantit,l a t the beginning of thc present ccntury: it has provecl t,o hc very fruitful. The subdivision of the field of flow into tho frict,ionlcss oxtcrnnl flow i~ntl the cssentinlly viscous boundary-lnycr flow p c r m i t k d thc retluction of the mnt,llcmaticnl difficnlties inllcrent in the Nnvicr-Stokes cqnntior~s o sllcl~at1 c x t r ~ l tt, f l ~ n t t it, l)ccnme possible to integrals them for a large numbcr of cnscs. Tllc tloscril)t.ion of t,l~csc methods of integration forms t.hc subjcat of the boundary-laycr t.hcory prcscntctl in the following chapters. From n nt~mcrical analysis of the available solr~t~ions the Nnvicr-St,okc~s of cq~~at.ioris is also poasiblo t o show directly t h a t in tho limiting case of very lnrgc it Reynolds numbers there exists a thin boundary laycr in which t h e influcncc of viscosit,y is conccnt,ratcd. We shall rcvcrt to this topic in Cllnp. V. The previously discussed limiting case i n which viscous forcrs heavily outweigh inertia force3 ((creeping motion, i. e., very small Reynolds number) results in a considerable mathematical simplification of the Navier-Stokes equations. B y omitting the inertia terms their order is not rcduced, b u t they become linear. 'J'hc second limiting case, when inertia forces outweigh viscous forces (boundary layer, i e. very large Reynolds nrrmbew) present8 greatrr matl~rmaticaldifficulties than creeping motion For, if we simply substitute v = 0 in t h e Navior-Stokcs equations (3.32), or in the stream-function equation (4.10), wc thereby suppress the derivatives of Lltr highest order and with the simpler equation of lowcr order i t is i~npossiblct o satisfy sirr~ultancously all botlndary conditions of the cornplrto tliKrrcntial eclont~ous. However, this does not signify t h a t the solutions of sucll nn equation, sin~l)lificd by t.he elimination of viscous terms, lose their physical meaning. Moreover, it is posxil~lc t o prove t h a t this solut,ion agrees with the &mplete solutionof the full ~ s v i c ~ : - ~ t o k e ~ cq11nt.ions nlmost. everywhere in t,he limiting case of vrry large Reynoltls n r t n ~ b ~ r s . Tho exception is confincd t,o n 1.hin lnycr ncnr the wall - t.hc bountlnry 1n.yc.r. l ' h ~ l s , t h r complete nolution of t.hc Nnvicr-Stmkcs c ~ t r a l i o n s c:nn I)e t.llorrgl~l, nrc t:o~~rcist,ing of of two soirrt~ions,thc so-cnllctl "outcr" solution which is ohtninctl wit11 t11c ;lid of Eulor's ccluctions of motion, and a so-callcd "inner" or bonndnry-1n.yc.r solnt.ion which is valid only in the thin layer adjacent to the wall. The "inner" solut,ion satisfies t h c so-called boundary-layer e q ~ ~ a t i o n s which are dctlncctl from t,l~c NavicrStokes equations by ~oortlinat~e stretching nnti pwqsagc t o tho limit R + m, n.s will be shown in Chap. VII. The o ~ l t c and inncr solutions must he malchcd t,o each other r by exploiting the condition t h a t thcrc must, exist nn overlapping rrgion in which bbth solutions are valid.

Now, tllc t~cmpcratlrre dist,ril)l~tion nronncl the body may I)c pcrccivcd i n t ~ ~ i t ~ i v r l y , 1.0 n ccrtnin cxlcnt. 111l,hc limiting ca.sc of zero velocity (fluid a t rosl) the infll~cncc of tile I~ratccl11otly will extend ~ ~ n i f o r r n l y all ~ i t i c s . on With very small velocit,ics t h e fluit1 a r o ~ ~ r lttlc hotly will still he affectfed by i t in all directions. With irlcrcnsing tl vclocit,y of flow, howcvcr, i t is clcarly seen t h a t the rcgion affected by the higher tempcreturc of t h e body shrinks more and more into a narrow zone i n t h e immetlint,c vicini1.y of the body, ant1 into n tail of hcntcd fluid bchind it., 1Pig. 4.3.

. -

Fig. 4.3. Annlogy bet,weetl trnlperntuw and vorticity di~tributionill the neighb o ~ ~ r of~R dbody plnml in a strerrrn l of fl\lid a), b) I.lndCq of rrgion or it,crrhsrd trmprrstureflow

-.__--

n) n l lvclucitlrs - - _ _- - - - - - - - - _ _ . . _ _ for ~ r n _ _ uf Ir) fur _ vrlucitirn _ Inrge

'rllc so111t.iotl rqn. (4.1 2) nilrst-, n.s mcnt,ionkd, be of a chn.ra.cter sirnilar t.o t h a t of for vort,icit,y. At snlall velocities (viscous forces h r g e compared with inertia forces) (.here is vorticity in 1,tlc whole region of llow around the botly. On the other hnntl for' large vclocit,ics (V~SCOIIS forccs smnll compnrctl wibh ir~ctl~in forccs), we may rxprc*t,i firltl of flow in which ~ o r t ~ i c i is confined to a small Inycr along t h e surfacc ty of the I)otly and in a wake behind thc boily, whereas thc rest of the fcld of flow

80

I\'. C r n r t n l p r o p r r t i r s of the Nnvier Stolic~sr q r ~ n t i o n s

f. Mnt,lre~iinticnlillt~st,ration the procens of going t.o the l i m i t R of

-t

m

81

f.

M n t l t c ~ t ~ n t i c nl l c t n t r n t i o n o f t l t c process o f g o i r l g t o the l i n t i t i

R 4 oo t

P 7

Let

IU

rotinitlrr t.lie tlnmpr(l vihrntiorls ofnr-

ti

point-mass t l m r r i l ~ n t h y t,llc t l i f i r r n l i n l c r ( ~ ~ n t , i o n l

t1=x 11 12

4 k-

111.

tll

I- c,:r

-

I l ~ v n l l ~ c const.nnt. A z folloaw froin t.110 tnrrl.t:l~i~ip;o t11c " o ~ ~ l r rn"o I ~ ~ l , i orc111. (4.17). 111 1111 c of t ~r, o v r r l n p l ~ i n grnngr, t,lrnt. is for ~noclcr:i(cv n l t ~ r n t i l n r , t . l ~ ~ :o l ~ ~ l i on~nclnn. (4.17) : ~ n ( l(4.21) of n i ~n ni~~nl. ngrc!c*. 'l'lit~s\I.(; tnr~sl,I~nvc. lii11 zr(t*) = l i t n r o (1)I*

I).

-.

m

1

-.

0

H e r c i r r donolrn the vihrnlirtg ntnsn, c (.lie spring c:or~ntntlt., k I.l)c. tl:~nil)ing f:~c.lo~.. t.I~t:Irng(.ll r roordinnt.~ ntct~mcrcclfrom t , l ~ jiosit,io~l r i l ~ ~ i l i l ) r i i r nnrl I ~.III.i ~ n r '. '11c. i n i t i a l ron(lilions arc r of ~n. t 1 ILRRIIII~~C~ t,o be r - O at 1 - 0 . (4.14)In ~ l i n l o g y wit11 (.tie Nnvier-Stnlzo~ eqr~ntions for t.lie cnse \rlicn t l ~ lrr i t ~ r n i n t i c visronity, I*, very is sninll, we conflitlcr h c r r t l ~ litnitsingrnsc of v r r y stnnll mnss nr, h r r n ~ ~ this l o o rnrlnrs 1.11~e r m r nr l of t l ~ c Iiigllest or(1r.r it1 cqn. (4.13) t o brromt! very small. l'lir c o m p l r t r solrition of cqn. (4.13) u ~ ~ h j c t~ r the i n i t i n l c~orrclit.ion n t (4.14) lins I l ~form r

or, i n worcln: 'l'llc "or~t,nr" l i m i t of t.lic "innor" s o l r r t i o ~II ~ I~ , R " o ~ ~ t e rsolnlion. (:ondition (4.2:)) Icntls nt, oncc to "

1)r C~IIIRI"itrnt:r" l o the

l i n ~ i 01 tJ11' l

ntltl no t o the i n n r r solrltion .rr(t*) = A ( 1 - c x p (-- k t * ) } . (4.25)

x = A {c%xp (

I

)

--

c x p ( - k 11iri)): irr

-t

0,1 1

(4.15) s r r o i ~ (initinl cottl

'I'l~csnnle form r n n be obt.ninrtl from I,llr r.ot111)lclr40111t icm fro111r q n . (4.15) I I r * x ~ ~ : ~ n ( l(It(. g ~ in tirut t,rrni for s ~ n a l v n l ~ t r s I ntid r c t n i ~ ~ ithegfi~.nLtcr111 only, 1Iirit is I I1111t,ling l of ~l ~

w h r r r A in n f r r e [:onstrrnt \vl~osc~111110 n n I)(,l r t r r n ~ i t i r t \villi r r f c r c t ~ c r o r c l t tlit,iott. I f we put, i n 0 i n eqti. (4.13), we nrc l r t l t o I.lw si~nplifiedrqrltrt.ion

-

dx ktlt w i i i r l i is of first orclcr, nnrl \rliosc solrclio~iis

f

e:r =0,

7'11~ t\rfo iic!11tio11~, l i c onter s o l t ~ t i o nfrotn eqn. (4.17) nrirl L l ~ c t itillcr ~ o l t i t i o nIron1 r.qt~. (4.26). togct,l~er lorrn the ,:o!!iplcte s o l ~ t t i o n n contlition t l i n t c n r l i is 1 1 8 r t l i t 1 its p r o j ~ r I.III~~C of vnli(lity. o r ht finite 1, cqn. (4.15) tends ( c tlie outer solut~ionfor 711 + 0. whcrens a t constant t* eqn. (4.15) t o t ~ t t,o tlie inner nolutiolt. 'l'lle pnrtinl solilt,io~is l~ give Illc cori~plrtc,co111posit.cn o l ~ ~ t . i o n \\.l~iclris vnlitl it, t l ~ c cnl,ire rnnge o f v n i ~ c s f t 1)y :l(lditig t h r n l t o g r t l ~ r r , c m e t n h r r i ~ i g o r t,l~ntI h r ronlli)on tcr111from eqn (4.23) ~ n n s the included only orrre, tlint, in sul~t,r:rtrl.l~ c~tscvitcl~tl t,o t h e linrnr vrlocity tlistril)l~t.ion (!oucstt.(, flo\sr, a s it, oceurrctl of 1)cI.wc~c~n flat l)lat,cs in t h r rnsc rrprc~srr~tctl Pig. 1.1 . T h e cc~nnt~ion Cnsr J l,\vo in of' yicltls tho sntilc, lin~if fi)r r1 -- 0, i. C. fn1. x = O \v11c11110 i ~ l ~ l rylintlcl is p r r s c ~ l t . n cr I (.Itis c,nsr, I,II(, Ilr~itl~.o(;tlt>s insitlt: IIlc out,cr rylintlrr a s n rigit1 I)otiy. Ilcncc il. is seen 1Iln.t ('nso I1 yicsltls n lineal vcloril~y tlislril)~~t,iotl llir t,\\'o sy1111)1.ot.ic ~ S C x -- 0 POI. C S n11t1x -- 1. 'l'l~isI ) r h n v i o ~ r ~ a l t cts rn.sy t o rintIrrst,e.r~tl ~ i why t,llc vclocit,y tIist.ribut,iol~s for 1Ilr. ot.l~er, inl.crn~etliatc vnlllrs of x tlilTc,r so lit.t,lc from n st,rniplit line. oS :I singlr (-ylintlrr rot:~t~ing a11 infinit,e fluid (r? -+ m, in ( Q ,- - - 0) t:tln. (5.15) givcs i r = r12 (ol/r. ;lntl 1.11~ t.ort111ct.rn.11~11lit~trt1 t ~ c fluid t o t.he t ) h c:ylit~tl~t~. I)c.c:on~rs A l l -- 4 rr 11 11 r I 2 ' 1 ' 1 1 ~ vc~lorit,y tlist,ril)r~I.io~~t h e flr~itlis t h o in s:lntc :IS t.11;1(. :1ror111(1 lint> v o ~ . t t ~ x S ~ I . I ~ I I ~. I- J2I n: r,",), i l l f'rialio~~lrss :I ~S Ill Ilow, o r111 t , I ~ t ; pnrticr~I:tr(:;isr

-

Pig. 5.5. Vclocit.y distrib~ltio~lt vaaryir~g a times in fllo ncigl~bo~~rl~oodn vortex of filament cnl~srtlby tho action of viscosity1; circulnlinn or l l ~ c vortox nnrncttl nt 11 nio 1 w1:c.n vircoslly Itrylnr lo ncl: -i . t I ; / ? n r.

-

= .

0

90

V. ICxnct sol~tl~iot~~ Nnvior-StOkcs c q i ~ : ~ l i o t ~ ~ of tho

a. Parallcl flow

9I

a s derived by C. W. Osecn [21] and G. 1Ia1nel [I]]. This velocity distribution is represonbd graphioally in Pig. 5.5 Here 16 dcnotcs t,he c i r c ~ ~ l a t i oof the vortex n filamolt, a t time 1 0, i. c. a t t l ~ o morncr~twhcn vi~cosit~y nssun~edt.o I)cgit~it* is . ~ c t i o ~An cxpcrimenLal investigation of this procoss was 11ntlcrt,nlta11 A . Tirnmo l. 11y [40]. K. Kirdc 1171 mndc an nnnlytio s t ~ ~ tofythe caso when the valooity distribution f in t,l~o vortcx tlilT(:r~from I.hnt irnposctl hy pot,cnt,inl t.hcory.= 1

t.hc compleme~dczry error /u?tdion, erfc q, 11.w been t,abulatedt. The velocity distribution is rcpresontcd in Pig. 6.0, and it may bo notctl t h a t tho vclocity profilcs for varying tinies arc 'si1nilar', i. e., they car1 bc rctl~lccd the sanlo cllrvc by changing t,lrc to scalc ttlong the axis of ordinates. TIIC cornplcmcntary error f ~ ~ r ~ c twhicl~ i o r ~ appcnrs i r eqn. (5.22) has a valuo of about 0.01 at 7 -- 2.0. %.killg into accorir~t l ~ o ~ t tlcfir~ition of t,l~c: t~ltic:ltnossof the: I ~ o t ~ n t l ~Inyor, 0 , wc: ol)t.r~ir~ try 6 = 2 q a J Z x 4

4. The sudder~ly necclernted plane wall; Stokes's first problem. We r ~ o wprocccd t o calculntc somo non-steady par;rllcl flows. Sirlee the convcctivc acccleratior~terms vanis11itlcr~tic:ally,t,l~c frictior~ forcos int,rmct with t,ho local nccelcrnt.iori. Tho si~nplcst flows of this clam occur when motlion is stnrtcd i r n p r ~ l s i v a lfrom rest.. We s l ~ a l l ~ begin with t h e c,wc of t h c flow near a flat plntc which is s ~ ~ t l d c n accelcr:~t.cdfrom ly rost and n ~ o v c s it,s own plnno with a const,:~ntvclocitty [lo. in This is onc of the proI~lcmswhioli wcro solvctl by (2. St,okos in his colcbr:rtccl memoir or1 p e r ~ t l ~ ~ l u r r ~ s [3ri]t. Seloct8ing t.ho z-axis along the wall i r ~the direction of U,, we obt,air~trhc simplifiotl Navicr-SOnlccs oqt~nt.ion

JZ.

(5.23)

I t is seen t o be proportional t o the sqnnrc root of tho ~)rotIl~ot, kir~ornnLiovisc:osiOy of silt1 time. This problem was generalized by E. Decker [3] t o ir~clr~dc: more genrml rat.rs of nccclnraI.ior~a s well a s the cqses involvii~gsrict~iorior blowing or tho c f i c t of compressil)ility.

'rho prrssuro in tho wI101o space is constant,, and Ll~e bol~nclirry conclit,iol~s arc:

Tho cliIT(:rcnt.iel ccl~~at.ion (5.17) is ictcr~t.icalwith the equntior~of h e s t contlaction which clcscribcs tllc p r ~ p n g n t ~ i n r ~ of Itoat, irl tho space y > 0, whcn a t time 1 = 0 t h e wall y = 0 i s sr1dt1c11l.yI~catcd o a t,cmpcr;~t,nre t which oxccecls t h a t i n the surroundings. 'l'l~e pnrl,i:~ltliffcrcnt.ia1 oq~iat,ion(5.17) can be retlucctl t.o a n ortlinary diTcrt:irt