... -layer Theory. Dr. HERMANN SCHLICHTING .... a. Exprrin~cntnl restllts for
811100th piprs h. J
noundary -layer Theory McGRAW-I4ILL SERIES IN MECHANICAL ENGINEERING
r. IIOLMAN, Southern. Methodist University
JACK
Co1tsu1lin.g Editor
Dr. HERMANN SCHLICHTING Profresor J3rncrit.11~ nt, tlrc Ihgincrrirrg U ~ ~ i v r r ~ofi t~. ~ ~ ~ I I I I R ( .O ~ rIrw ~ ~c l~n~r l, ~ Forrner 13ircctor of thc Arrodynnrninclre Vcr~rrclrsnnslnlt (;iittirrgcn
. Cryogenic S y s t e m
RARRON
. hzlroduclion lo Heat and M a r Tran.fer AND DRAKE . Ana1y.ri.r of Jim1 and Mos,r 7i-nnsfr.r
ISC:KERT
ECKERT
E C K E K . ~ AND DRAKE
- Ifen1 attd A4ass 7ian.fer
. Mechanics of Machinery
HAM, CRANE, AND RODERS HARTENRERO AND DENAVIT
. Kinen~nlicSynlhesis of I h k a g e s
rrmm . Turbulence jmonsm
AND A Y R E
. EtlGqineering Vihralions
~ ~ v 1 N A l.. Ettgitteering i
KAYS . Co~tvecliueHeal
M A R ~ N.
Proce~~es
Kil~~tnalics and Dytian~ksa/ machine.^
IVIELAN
. I)!/~lan~ics qf Machinery
PIIELAN
. ~ l l ~ l d f l t l l e t lolf ~ ln/fecharlim/ ~,~ h.rigtl
RAVEN
I'rofe~sor at ljrown Univrmity in Providcncr, Rliodc Ialand
and Mass Trcrtzsfir
. f h ~ b t ~ s / i oEngine' n
LICIIIY
Dr. J. KESTIN
Consideraliot~.~ n/.Ylrc.~r, .ylroi~,ntzd Slretzgth
. Aulotnnlic Corrlrol En.gineerirtg
SOHP,N(:K
. 7'hroric.r ?f Engitteering Expwir~lenlnlio~l
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Con tents xiii XV
nvii xix sxi I I
L i b r a r y o f Congress G t n l o g i ~ ~ing 1'11blirntio11Data
Virsl p ~ t l ~ l i s l ~i nr otn.cklc Inany ~ ~ ~ . t ) l ) l v t r ~ s thnt wwc c:onsitlerccI nnsolvnble in tho past,. These inclrttlc rlumcric:n.l solr~t~ions of 1.111: Navier-St,okes cqr~at~ions for moderntely large Reynolds numbers (Chap. IV), nutnoricsl integration of tho boundary-layer equntions for laminar nntl turbulrnt flows (Chap. ZX),n8 well RS t,he expLicit nntnerienl int.rgr:~t.ionof the 0rr-So1111t~~l.r~ltI equation of the theory of ~ t a b i l i t ~of y laminar boundary lnyrrs (Chap. XVI). Another sultjcct newly t,nlcrn into nccount nrc cxnct s o l ~ ~ t ~ i oof l l s the Nnvier-St,olcc~a cqu:~t,ionsfor the non-st.c:~tlyst,ngr~nlionflow (Cl~np.V), t31~c tltcory of t.111. I:ln~i~~:t.r Itonr~tlntylngcr o f scrontl ortlor (Chap. V I I nntl LX). 7'110scc:tions o n t,I~t:(::~lrr~l:~I io1101 two-tlinletlsion:~I,inconiprrssiblc, t,urbulcnt boundary Inycrs (C1in.p. S X I I), on t l ~ c st,abilit,y of Inn~inarboundary layers wit.11 compressibility and heat-t,rnnsfbr c:flccl.s (See. X V l I e ) , nntl on losses in nnscn.tle llows (Chnp. X XV) Ilavc: bee-n c ~ ) ! ~ ~ l ~ l r t c l y revised.
xviii
Ar~t,l~or's Prefncr
t.o tho
Seventh (Englisl~) Er1it.ion
Along with this ncw material, I fee1 t,hat I ought, t o niention the topics which I spcoifioally omit,t,ctl l,o include. I d o not, discuss t,he effect of chemieal reactions on flow processes in boundary laycrs a s they occur in the presence of hypersonic flow. The sarnc applios t.o I)onndnry Inycrs in rna.gncto-fl~~itl-clytin~~tics, low-dcnsitty flows and Rows of non-Nowt,onian fluids. I still t.11onght t h a t T ought t o refrain from giving a.n rxposit,ion of t,lir st,at,ist,icalt,heory of t,ttrl)~~lenrc in this etlit,ion, as in t,hc prcviolls OIICR, hrrnusc n o ~ ~ ~ d t.l~crc n y s arc avnilnblc otlrcr, good prcscnt,nt.ions in I,oolr form. Oncc again, t,hc lists of refcrenccs have bcen expanclcd considcrahly in many rhnpt,crs. The nurnl~crof illust,rations increasctl by about G5, hut 20 old ones havc been omit,t.cil; the number of pages increased hy about 70. I n spite of t,his, I hope t h a t t,he original character of t,liis book has becn retained, and t h a t it, still can provide tlie reader wit,l~a bird'.?-e?yview of this important branch of the physics of fluids. As I worlrrd on the new manuscript I once more enjoyed t,hc vigorous assistance that I rcccivetl from scvrral of my professional collcagues. Professor K. Gersten cont.tihutctl sect,ions on boundary layers of second orrlcr t,o the part on laminar boundary lnycrs (Seas. VIIf ant1 I X j ) . This is a special field which he successfully worked out in rccent ycnrs. l'rofcssor T. K. Fnnneloep contributed the completely reformulated sc-ct.ion on the nurncrical inkgration of t,hc boundary-layer equations included in Scc. I X i . In t.hc part on turbulent boundary layers, Professor E. Truckenbrodt provitlcrl me witall a new version of the largest portion of Chapter X X I I on twodimensional and rot~ationallysymmetric boundary layers Dr. 1,. M. Mack of the California Institute of Technology was good enough to contribute a new section on t h e stability of boundary layers in supersonic flow, Sec. X V I l e . Dr. J. C. R o t t a tliorougl~lyreviewed P a r t I) on turbulent boundary layers and made many additions to it*.For the Russian litcrxtnre I rccrivstl nlurh help from Professor Milrhailov. The translation was once again cnt,rustctl t o Professor J. Kestin's competent pen I express my sincerc thnnlrs l o all tliose gcntlcmen for thcir valuable cooperation.
I should also like t o rcpcat my aclrnowlcdgemcnt of thc hclp I rcceived from scvernl professional friends whcn I worked on the fifth (German) edition Nat.urally, their contributions havc now bcen rctaincd for tlie seventh edition. This is the extcnsivc contribution on comprcssiblc hminar bountlary layers inChapter XIIT written by Dr. F . W . Rirgcls, Profcssor Iurbation rnct,horls in flnid mechanics was prepared by M. Van Dykt:t. The basis of these rnat,hotls can be Itraced t o 1,. J'raritlt.I's early co~lt~ribut~ions. 'I'lic 11onntlar.y-layer tlicory finds its application in the nnlcnlxtion of t,he skinfriction d m g whic:h ac1.s on a body as i t is moved t,hrongh a fluid : for example t h e rlr:lg cxpcricncctl by a flat p1n.t~at,xcro incitlcnce, tilo t1m.g of a ship, of an aeroplane wing, aircraft, t ~ a c r l l ror , t,rrrl)ineI)latlc. 13o11ndnry-layerflow 11:~st,I~c peculiar property t.ll:~t,untlor ccrt.airl conditions l h e flow in t h e imnictliat,c ncighbonrhood of a solid wall 1)ccomcs rcvcrscd causing the I~ountlarylaycr t o separate from it. This is accompnnirtl 11y a morc or lrss pronouncctl fonnat,ion of eddies in t h e wake of t,hc body. 'J'hus t.hc prcssnrc distribution is rltangcd and differs marlrctlly from t h a t in a frict.ionl(\ss strcnm. ?'hc tleviation in prcssurc tlist,ribution from t h e ideal is t h e canse of form drag, antl its cniculat.ion is t h l ~ made s possible with the aid of bouriclarylaycr t.llnory. 13ountlary-hycr t,heory gives a n answer t,o t h e vcry i r n p ~ r t ~ a question nt of' w11n.t shape mnst, a hotly t ~ ogiven in orclrr t o avoid t.llis dct.rimarital scpn.ration. Scpnr:rI.ion m n also oc.c:ltr in l.llc int.crn:tl flow t.hrorrg11 R (:11nnnc1 ant1 is not, confined t o rst,rrnnl Ilows past solitl I~otlicx.I'rol~lrms conrrcct,crl with l.11~How of fluids throilgl~t,hc c l ~ m ~ n c fl st m ~ ~ cby t l t.hc blntlcs of t,urhomachines (rotary compressors ant1 t,url)inos)ran also he 1,rrntrtl wit,ll tho n.itl of 11ountl:~ry-hycrt,Jlcory. I'r~rt.llcrmore, ~~lw~lon~ wllic:l~ c n n occur at, t,llc point of rnn.xirnnm hft, of nn acrofoil and which arc ~ 1)c 11ntlcrsI.oot1only on t h r 11n.sis of I~onntlary-layer assocht,t:tl with s t . : ~ l l i t l (:;I.II
theory. I h d l y , problrms of l ~ c a ttransfer I)ctwcwl n solitl hody ant1 n fluitl ( p s ) 1)11vnoflowing past i L also bclong t o thc class of problems in wltic41 l)o~~t~tl:~ry-l:~yc.r m r n n play n dccisivc pnrL. At, first the l~ountlary-layertheory was devclopotl rnn.inly for t l ~ ctwo of 1:~nlin:lr flow in a n incon~prcssil)lcfluitl, RR in 1.llis c:uw t h ~ ) l ) ~ t ~ o ~ n t : t ~ o I o I~j,~)oI.I~t-sis ~it::l.l for shr:~ringst.rrsscs a1rr:ttly cxistctl in thc form of Sto1tt.s'~ I:\w. ' l ' l ~ i s t,t,l,it: W:IS sul)scqucntly tlcvclopctl in a 1:Lrgc ~lurnhcrof rcsonrclt p:Lpcrs :LII(I rt::1(:11vtl s1tt~11 a stagc of pcrfoct.ion I h t a t prcscnt tltc problem of Intninar llow c:1.11III: consitlt~rctl t o h:lvc hccn solved in its main oullinc. 1,:llcr the Ll~coryw:ls cxl.ot~clt:tl1.0 int:lurlc turl~ulcnt,incornprcssil)lc bountlary layers which are morc irnportzmt from (ht: poitlt, of vicw of practical applications. I t is true t h a t in tltc cnsc of t ~ t r l ) ~ ~ l flows c n t . 0. Iloynolds introduced t h e fundamentnlly important conccpt of nppnrcnt, or virt,~tnltltrl)ulent stresses a s far back as 1880. IIowevcr, this conccpt was in it,sc.lf itisuffioirnt tso mn.ke tltc theoretical analysis of turbulent flows possible. Great progress was acllicvecl with the intmtlnction of I.'randtl's mixinglcrtgt.l~thcory (1025) which, t,ogol,hrr wit,li systematic cxperimcnt.s, paved the way for the thcorctical ttrcntmcnt of turl1111c1tt flows wit,l~the aid of boundary-ln.yrr t.hcory. llowevcr, a rational theory of fully developed turbnlcnt flows is st,ill noncxist.cnt,, antl in vicw of the cxl,rtmc complexit,y of sucll flows i t will remain so for a consitlcmhlc time. Onc cannot even be ccrtain t h a t science will cvcr be successfnl i n this t,aslr. Tn modern times tho phcnomena which occur in t h e boundary laycr of R cornprcssiblc flow have becomc the subject of intensive investigntions, t h e impulse having Iwcn provided by thc rapid incrcasc in tllc spcctl of flight of motlcrn aircmft,. I n atltlition t o a velocity 1)oitntlary laycr suc:h flows dcvclop a tllcrrnal bonntlnry h y c r ant1 its cxist~cnccp h y s :I.U irnportant part in t h e process of heat txansfcr bctwceri the Iluitl and the solitl body past which i t flows. At vcry high Mach numbers, the surface of Lhc solid wall bccornrs heatetl t o a high t,cmperature owing t o the protlnct.ion of frictional heat ("tllcrrnnl barrirr"). This phenomenon prcscnts a tliffic:nlt analytic problem whose ~ o l ~ t t i o n is import.ant in n.ircmft tlcsign antl in the ~~ritlcrsl~anding of the motion or satellites. r
7
1 he phenomenon of tmnsit,ion from liltninar t o turbulcnt flow which is ~ ~ I ~ ~ : L I I I B I I t.aI for t,he science of fluid tlynamirs was first investigated a t thc end of t l ~ cI0t.11 cent,nry, naniely by 0. 12eynoltls. I n 3914 1,. 1'm.ndtl cnrrictl out, his fnmous expcrimrnts with sphcrrs antl ~ u c c c c d c ~inI showing t h a t the llow in Ihc 1)ountlnry layer car1 also I)c either laminar or turbulcnt and, furthermore, that, tltc problem of separnt,ion, ant1 hence the problcm of the calculat~ionof dmg, is govcrnctl by this t r a n ~ i t ~ i oY'hcoretin. rat invest,igations into t,he process of transilion from laminar t o tnrbulcnt flow are basctl on t.110 acceptlance of Iteynoltls's 11ypot11o~is l,liat tohe l a t t m occurs ns a conscclucncc of a n instability dcvolopcd by Ihc 1nminn.r 1)onntlary layer. 1'rnntlt.l ittif.int.ctl his thcorc1.icn.l investigntion of trnnsition in tllc ycar 1921 ; after marly v:rin cflort.~, succcss came in the ycar 1920 wlicn W. Tollmicn compntrd theorct,icnlly t,hc crit,ic:aI Reynolds numbor for transition on a flat plate a t zero incidence. Ilowcvrr, nlorc t.lran ten years werc to pass 1)efore l'ollmicn's theory coialtl ho vtdficd throngl~tho vcry carcful experin~enLsperformed by 11. 1,. 1)rytlcn antl his coworltcrs. Tho stn1)ilit.y tltcory is capsblc of taking into account the cKcct of a nurnhcr of parnmctcrs (pmssurc gradient,, suction, Mach numlter, transfcr of heat,) on tmnsition. This theory has found m ~ n yimportant applications, among them i n t l ~ cdosign of scrofoils of' very low drag (1aminn.r ncrofoils).
Modc:rn invcstigalions in i d ~ cficld of fluitl dynamics in general, a s well as i n t(11c ficld of bountlary-hycr rcscarch, are characterized by a vcry close relation bc!twcen theory ant1 cxpcrimcnt,. The most important steps forwards have, in most cases, barn t,nltcn as a result of a s m d i n u m l ~ c rof f i ~ n d a m c n t dcxpcrimcnt,~bacltetl by t,hcorot,icnl considcrat,ions. A rcvicw of t J ~ c tlcvclopmcnt of boundary-layer t.Ileory wllich st~rcsscstllc rnuf,nal cross-fertilization bctwccn theory and cxpcrirncnt, is containctl in a n n.rliclc writtrn 11y A. l k t z ? . Vor about, twenty years aft,er its inccption I)y T,. I'randtl in 1904 t h c bonndnry-la.ycr tllcory was being developed nln~ostexclwivcly in his own institute in Goettingen. One of t h e reasons for this st.nt,c of nffnirs may well havc been root,cd in the circum~t~ancc that, J'randtl's first pnblionthn on boundary-layer theory which appeared in 1904 was very dimcult t o understantl. This period can be said to have ended with I'randtl's Wilbur Wright Meniorial I,ect,ureo which was dclivcrcd in 1927 a t a meeting of the Royal Aeronautical Society in 1,ontlon. I n later years, roughly since 1930, other research worlters, particularly t,hosc in Grent nrit.ain and in tllo U.S.A., also took a n active pn,rt in its tlevrlopmcnt. Toclay, the study of boundary-layer theory has spread all over thc world; together with o t h r branches, it constitutes one of t,he most import,ant pillars of fluid mechanics.
Tho first survey of this I ~ m n c hof science was given by 1%'. Tollmien in 1931 :. S11orl~I.vaftcrin two short articles in the "llan~lbnch dcr ISxpcrirncnt~alpl~ysiIr" a cotnprnl~cnsivcpresentation it1 "Aerodynamic wartls (1936), Prnrdtl p~~l)lishcd 'J'hcory" ctlitctl I,y W . I?. Durands. lluring t.he intcrvcning four dccndcs tllc volume of rescarch into this subject has grown cnorrnonsly$. According t o a review published by 11. I,. Drydcn in 195.5, t,hc rate of publication of papers on boundary-layer theory reached one hundred per a.nn7r.ma t t h a t time. Now, some twenty years later, this rate has more than tripled. Like several other fields of research, the t,heory of ho~rntlarylayers has reachetl a volume which is so enormous t h a t a n individual scientist., even one working in this field, cannot be expected to master all of its specializctl subtlivisions. I t is, tl~rrcforc,right that, the task of describing it in a nlotlcrr~Ilanclboolt has been cnt.rustcd t,o several authorst. The hist,orical development, of bountlary-layer theory has recently been traccd by I. Tani*.
. . " 1,. J'mllrlI,l, Tho goncmlion of vortiron ill fluirlsofatn.zII viscosit,y (15td1Wilbilr Wright Memorial
Jfir(llr% 1!J27).J . Jtoy. Aoro. Soc. 31, 721-741 (1!)27). : (!/. tho bildiogr.zl~hyon 11. 780. : I,. l'r:~n(ll,l,T11c 111ecl1a.11irs ol' v i ~ c o u nfluids. Arrodynamii~tl1oory (W. I?. I h r m ~ drd.), , \'ol. 3, 34 208, I%crlin,1935. 6 11. Schlirh~ing.So~netlrvcloprncn(.sof I~oundnry-layerrcsearch in the past thirty years (The 'I%ird L ~ t l c l ~ r ~Metnorin.l lcr J,rcture, I!W)). J . h y . Aero. Soc. 64, 03- 80 (l%U). Srr nl?lo: 11. Srlilicl~ling, Rccrrtt progress in houndn.ry-lnycrresearch (The 37th Wright. Brothers I ~ ~ i t r 1i t :t 1 1 r 1 , ! 7 ) \ I . J t t i r i : ~ l 1 427 440 (1!)74). * I . 'I':\t~i.Ilislory of I~o~~nrlnry-lilyor rmrnrcl~.A n ~ ~ r i Itrv. n l rrf Izluid Mwhnnirs 9, 87- 11t (1977). -
Part A. Fundamental laws of motion for a viscous fluid CHAPTER I
Outline of fluid motion with friction
Most t.Ileoret.ica1 invcst,igat,ions in the ficld of fluid dynamics arc based on the concrpt of a perfect,, i. c. frictlionlcss antl incompressible, fluid. I n t h e motion of s u c l ~a perfect flnid, two cont,act.ing layers cxpcricnrc no tnngcntinl forccs (sl~caring st,rcssrs) b ~ ~a cl t, on tach other wit.11 normal forccs (j)rcssums) only. This is cqr~ivalcnt, t.o s t a l . i ~ ~tl~nf, g a pcrfvct, fluitl olrcrs no inl.crria1 rc~isI.antxt o a c11angc in S I I : I ~ O .The t l ~ c o r ydescribing !,hc motion of a pcrft:cl. lluitl is ~ n a t l ~ c ~ ~ ~ ~ n vcry t . i c : far ~lly tlnvclopctl ant1 supplies in many cases a satisfactory dcscril;t,ion of real motions, such a s e. g. tlle motion of surface waves or the formation of liquid jets in air. On the ot.her hand t h e theory of perfect fluids fails completely t o account for the drag of a body. I n this connrxion i t leads t o thc statement t h a t a I~otlywllich moves uniformly t,llrongh a fluid which cxt.ends t,o infinity experienccv no drag (tl'Alcmbcrt.'s pamtlox). 'Pliis unacceptable result of thc thcory of a pcrfect Iluid can be traccd t o the fact that. t.11einner layers of a real fluitl tmnsmit t,angent,ial as well a s normal stresses, this lxing also the case ncar n solitl wall wetted by a fluid. Thesc tangential or frict,ion forccs 111 a rrxl Ilnitl arc conncctctl with a propertry which is callctl the viscosil?/ of thc Ilnid. IZccai~scof tho a l m n c e of t,angcnt,ial forccs, on the 1)oundary bctwccn a perfect llnitl : ~ t ~ ta.l solitl wnII Lhcrc cxist,s, in gcnt~rnl,:I. tlilrrrcncc in rc~l:~l.ivc t,:~ngrnl.i:il of int.crvrloc.it.ics, i. c. t.11crc is slip. On t,hc other hi~ntl,in r(::11 l l ~ ~ i (the l s cxi~t.cn(:t~ molecular att,ractions causcs thc flnitl t o a d l ~ c r ct o a solitl wall antl t,his gives risc l,o slrraring stmsscs. 1,hc exist,cncc of tangcnlial (sl~caring)s,,rcssc:s n r ~ llhc condiliols 01 ,to d i p n(::~.r
.
solitl walls const.itut1e the essential tliffcrcnccs bctwccn a perfect and a real fluid. Clert,ain fluids wl~icharc of great, practicd imporl,ance, such as water and air, havc vcry smnll coefficients of viscosity. I n many instances. t l ~ cmotion of such llwids ol sn~nllviscosity- a.grccs vcry well wit.11 t h a t of a perfect Iltritl, bccausc in most cases the shearing stressc?~ arc vcry small. For this reason the cxist,cncc of viscosit,y is corrlplctcly nrglcct.cd in the t,heory of perfect fluids, ma.inly bcca.11sethis introdnccs a far-reacl~ing ' ext.cnsivc niathesimplificatiott of the equations of mot,ion, a s a result of matical theory I~ecomcspossilh. 11 is, I~owcvcr,i s l p m ss the & fact that, ,
even in fluitls wit,lt vcry srnall viscosit,ics, unliltc in pcrfwt. fluiels, t.he rontlit.ion of no slip near n, solill I~oundaryprevails. 'l'l~isc:ot~dil~ion of no slip int,rotlures in many (::~sos very h r g c tliscrcpar~cicsin t,hc laws of moLiorl of perfect an(\ ronl fluids. In pnrt.icular, t h vcry largc tliscrcpel~cy1)ctwccn Llle v d r ~ oof' drag in a rral ant1 a pnrkct, Iltti(1 I1:w its pl~ysicalorigin in the contlil,ion o f no slip n w r :L wall. of t.l~cgrc:~L 'I'l~is11oolct1r:rls wil,l~1 . 1 1 ~rnot,ior~of l l r l i t l s of'sm:~IIvisrosil,y, I)(-r:~llsr I ~ : ~ c t , i citnporl.ance al of' the problcln. Ihrirtg 1,llc course of lhc s t ~ d yi t will l~cconlc clear how this p:trtJy consistent ant1 p:l,rl.ly tlivcrgcnt I)cl~aviourof pcrfrct and real fluids can be cxpl:tinotl.
T l ~ quantity r p is
n propertry of thc fluid and depcntls t o n great cxl.cnt on it,s ternpcrnt,rlrc. It is n rneasuro of tho i)i.~co,qit~y OF t h e fl~iid.'1'11~ I:LW of' friction givrtl by cqn. ( I .2) is 1znow11:LS Nrwtotc's 1rr.v~of friction. ICqn. (1.2) cnn bc rrg:~relvrl :I.R t,llc c1rlinil.ion of visc:osit.jy. It. is, Ilowevcr, nccxssary t o st.ross t h a t the cxnrnplc cot~siclcrc:d in IGg. 1.1 (:onstit~~t.rs :L p:~rt,ic~~larly simple case of fluit1 motion. A gnncr:~liz:~l,it,r~ of this sitn111ve:rsc is cont,:~.inc(l in Stolccs's I:IW of fridion (cf. (!II:L~. I I I ) . ' 1 ' 1 1 ~~ l i m c ~ ~ ~ s i ~ o f visrosi1,y c:all IIC tlotl~rc:c:tlwit,hol~t,diFlicull.y from cqn. (1.2)-I-.'1'110 sl~c:nritlgs1,rcw is ~ncnsurcdin N/m2 =I J'n nrld tltc vcloc:it,y grntlicnt du/tl?y in ~ o I .c ~ I V I I ( Y *
h. Viscosity 'I%(: I I : L ~ , I I ~ of' C v i ~ r o s i t ~can y 11cst I)c v i ~ r d i z c dwith the :lid of t,ltc following cx~wrimnnt,:Consitlcr the ~ n o t ~ i oof n a fluid l)cl,\vccrt two very long pn.rallnl ~)latcs,one of wl~inhis a t rrst, the other moving wit,l~n, constant velocity pnrallcl t,o i t d f , a s sl~owuin Fig. 1 . l . 1,ct tJ1o clist.anco h c t w c c ~thc ~ plates bc h,, the prrssnre Iwing const,nnt
wllcre tho square 1~r;~(:Iccts arc I I S C ( ~ to ( I c r ~ o 11ni1.s. t~ ' 1 ' 1 ~ : L ~ ) O V C is not. 1hc o ~ ~ l or y, even the most, witlcly, employctl unit of viscosit,y. l'riblc? 1. I lists t,he various t ~ n i t s togct.lrcr with t h i r conversion factors. .15qn. (1.2) is rc1:rtctl t.o IIooltc's law for all c~l:ist,ic: solicl I)otly in w11ic:h rasc: t l ~ c shearing sCrcss is proport,ional t o the strain
I l r r c (: denotes lhe n~oclnlusof shear, y the change in anglc bct.wc.cn tfwo linrs wlliclt were originally nt right anglcs, nntl 6 tlcnotcs t.110clisplr~ccmcnt,in t11c tlircc:t.ion of a1)scissae. Wllcrcas in thc cnsc of a n elastic solid t h : sl~caringstrcss is proporl.ional t,o the nw~gniturleof the strain,, y , expcricnrc tcacl~cstll:~tin t l ~ ccase of fluitls it is proport,ionnl t.o the vale of chnnrlc. of strain tly/tll. If' we put t . l ~ r o l ~ g l ~tllc o t ~fluid. t Exprrintcnt t.c~:rcltcst.l~:rtt.11~fluitl atll~rrcsl.o l)ot.l~~valls,so I,II:II, it,s vclovity :rI, the lownr p1:~t.cis zero, : I , I I ( ~ t,11:1t3.t Lltc ltplwr ph1.c: is rt111al to t . 1 1 ~vcloeit,y of the plate, I J . Ir'rtrt~l~ermor~, I.llc vclocit.y tIist,ril)r~t,ionill t,llc fluid I)ct,wccn the pIat,cs is linear, so that, the fluid vclocit,y is proport,ion:ll tto t.ltc tlist,ancr ?/ from t 11c. lowvr platr, :~ntlwe h:tvr
we s1r;~llobtain, as bcforr, t
In ortlnr 1.0 s ~ l p p o r tt,l~emotmionit is necessary to apply a I~n~~gc.nt,ial forcn t,o thn tlpprr l)lnto, tho force 1)cing in cc~t~ilibriurn with t l ~ cf'rict~ionalforces in t,l~cfluid. I t is Icnown from expcrimont,~t , l ~ a ttJtis forcc (ta.l~cnper unit a w n of t,l~cplal,c) is proprt.ion:~.Ito t,hc velocity 1J of the 11l11)erplat.c, ant1 invcrsrly proport,ion:~l to l h c tlist,:r.nrc~h. 'l'llc 1'ricI.ion:ll force por nit, area, tlcnotctl by t (Srict.ional shearing sl,rcw) is, t,licreCore, proport.ionn1 1.0 lJ/h, for which in general we may als? ssulist.itr~t,c tlii/tl?/. 'l'ltc: 1)ro1)01.t~io11:rIil.y far:l.or I)ct,wcnn t ant1 d71 tly, wl~iclrwe s l ~ a ldl c ~ ~ o tI)y , c u,, tlc11~1(1s or1 tho r~al~llrc of 1.110 l l ~ ~ i (11, l . is ~rna.llfor. "lhiri" fluids, s11c11nk wal.cr or :~l(:ol~ol, I ~ u It : q n in the case of vcry viscous liquids, srtclt a s oil or glyccrinc. 'I'hl~s rclnt,ion for fluid frict.ion in t,lte form wc 11;tve ol)t,:~inctlt,llc ftl~~tl:rrncnl,al
I
= fL
du ~ I Y.
>
(1.2)
'f l
a11
?I!/
bccausc 5 = XI. Jlowcvcr, this analogy is not, complctc, I ~ c c a ~ l st.llc: c st,rc:ssas in :r flt~itltlepcntl on one const,atlt., t,l~cviscosit.y ti, wllcw-:is tllose i r l :tn iso1,ropic vI:~sLic: solicl tlnpcntl on two.
I. Ont,line of fluid lnotion with friction
8
Table 1. l . Visco~ityconversion factors n. A l d l ~ t eviscosity 11
Numerical values: In t,lrc case of liquids the vi~cosit~y, / t , is nearly indcpcndent, of pressurc and tlccreascs a t a high raLc with increasing tcmpcrat,urc. 111 thc case of gascs, t o n first npproximat,ion, thc vi~cosit~y cnn be talrcrr t o be intlcpcntlcnt of prcmitrc bi~t,it irrrrcnscs wil,lr l,cmllcrnl,rtrc. 'I'Iio Itinc?~nal,icv i ~ c o s i l , ~ 1 1 , , for litl~~itl.q has t,hc s m w type of t,cmpcrrat.i~rotloj~otttlottc:oas p, I)ct.n.~tso0l1r tltwsit,y, 0,( - I I J L I I ~ ~ S only ~liglrtlywith tcnrpomI,urc, Ilowcvcr, in t h caw of gn.scs, for whiclt C, tlcc:ro:t..qc~ consitlcrsbly with incrc:~sitig tc1npcrn1,11rc, 11 incrcascs rnpitlly willit (,cmpcmt.urc. Table 1.2 contains some numerical values of Q, p and v for water and air. Table 1.3 contains some additional lisefitl tlat,a.
I kp soc/m2 kp hr/m2
I'n see kg/m lir J Ibf sec/ft2 Ihf hr/ft2 Il,/ft scc
l'ahle 1.3. Ivclocil,y 11s it, I I I L ~nl, J). A l I r c i ( l ~~:~rl.iclt: of tho wtdl in I,llc bo~lntl:r.ryI:~.yorrc:~n:iit~s wltich lrroves in IJICi~nmctlinlovioi~til~y 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 boundary layer. Owing tlo tlrc large friction forces in the thin boundary layer such a psrtic:lc consumcs so much of its kinbtic
(2.6)
Fig. 2.4. Doundary-layerscpara-
tion ~indvortex forrnntion on a circular cylinder (dingran~n~atir) S
' I l c displnccment tl~icltncssindicates l.llc tlistancc by which the external strcamlines arc shift,cd owing to tire fonnat,ion of t,l~cboundary Iaycr. I n the case of a plate in parallel flow nntl a t zcro incidcncc tlrc tlisplaccmrnt thickness is about & of the bountlary-layer IJ~icltncss0 givcn in cqn. (2.1 a).
b. Srpamlion and vortcx forrnntion
..
llte bo~~ntln.ry laycr ncnr a f h t plate in par:~llclflow and al, zcro incitlencc is part,icrllarly sirnplc, Ijccausc t h e static prcssurc remains c o n s h n t in the whole field of Ilow. Sincc orlt,sitlc the 1m11ntI:~rylnyrr tho vclocily rcnmins constant t,hc samc qjplics to the p r c s s ~ ~ l~ecausc re in the frictiorrlcss flow Bcrl~orrlli'scquation remains vnlitl. Furthcnnorc, tlrc prcssnrc rcmnins scnsibly constnnt over thc width of t,hc \ ) o ~ ~ ~ r r l alayer r y a t a givcn rlist.ancc x. 1Icncc tlrc prossurc over thc widt.11 of tlrc 1)ountlary Iaycr has tlrc snmc mngnittrtlc ns out.sitle t.hc boundary laycr a t the samc tlist.ancc, ant1 the same applies l o cnscs of arbit,mry body pl~n.pcswhcn tho prcssnrc o~rt.sitlc1 . h ~I)o~ln(l:~ry I:~yt:r vnrics along t,lrc wall wit11 t , l ~ c1cngl.h of arc. 'l'his fnct is cxprcsscd by saying 1,h:~Lt,lrc cstcrnnl prcssnrr is " i ~ n ~ r c s s c t lon " t h c boundary Inycr. Ilcncc in the cnsc of the motion p s t a plate l,hc prcssnrc rcmains constant. througIrouL t,llr: bountlnry Inycr. isi!rtinral~ly 'j'lrr phrnonrrnon of 1murrtl:~ry InycrsrpnraLiot ~nrt~tiot~c~tlprc~viously - - . c~onnrclctlwrtl~tlrr prcssurc t1istril)ution in ti16 orintlary layrr I n the boundary lnycr on a plate rro srpnmlion takrs p h r r as no back-fldw occurs
\
of bountlary-lnycr s ~ p a r a t i o n In ortlcr to r\plnitr t IIV very import nrrt pl~rnornrr~on let us rorrritlrr 1 hr Ilow :~ljouIn Ijlrrnt hotly, r g abont, a rirrnlar rylintlrr, a s shown i t 1 IClg 2 4 111 ft ic.1 inl~lcwflow, t l ~ cflu~tlpar1 irlrs nrr :~rc.rlrmlrtlon tlw npstmam
-
point nf s c l ~ n r n l l o ~ ~
energy on its pat.h from D t o E t h a t thc remaintlcr is too slnall to srlrmount t.hc "pressure hill" from E t o F. Such a parLicle cannot move far into t,hc region of' increasing pressure between lC antl P antl its molion is, evcntunlly, arrcst,ed. The external pressure causcs it t,lrcrl t,o move in tho opposite clircction. Tlrc p l ~ o t o g r a ~ l ~ s reproduced in Fig. 2.5 il1nstrat.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 from left t,o right, the flow Ilnving been ma.tlc visil)lc by sprinltlitrg nlrtminirlm drrst on tho surface of thc water. Tlrc boundary layer can be casily rccognizetl by rcfcrcncc t o tlte short traces. In Fig. 2.5s, Lakcn shortly aftcr the s t a r t o f lhc rnot,iorl; the rcvcrsc motmionhas just begun. In Fig. 2 . 5 b the rcvcrsc nrotion lrns pci~-t,r:.tctl a consitlcrablc distancc forward : ~ n dl , l ~ cboundary Iayor lrns tllicltcnctl n.pprcci:~l)ly. Fig. 2 . 5 shows ~ how this rcvcrsc mot,ion givcs risc t o a vortex, whoso sizc is incrc,iscd still f u r t h x in Fig. 2.6tI. 'l'hc vorLcx bccorncs scp:~mlctlshortly a f L c r ~ : ~ rn~. tIds rnovc!s tlow~~strearn in tho fluid. This circnn~stanccchangcs complctcly blrc fiolcl of flow in tho waltc, and Lhc prcssnrc clisLrib~lI,ionsuKcrs a rntlical change, as cornparctl with frictio~rlcssIlow. 'L'llc f i n d statc of nrotion can I)(> inrcrrctl from Wig. 2.6. In t,he eddying region bclrind tlic cylinder there is consitlcrable suction, as sccri 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 rcgul:~r patt,ern of vorticcs which move alternately clockwise and courrt~crclocltwise,and wllich 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 further pzpcr, von Kilrmhn [21] proved t h a t such vorticcs are gcncrally nrrstablc with rcspcct to small tli~t~urbancrs pnrallcl
Fie. 2.511
Fig. 2.5b
Fig. 2.7. KhrmQn vortex strcct, from A. Tirn~nc[38]
Fig. 2.8. Strrnmlinm in nvortrx strrrt (hll = 0 28). Thr fluid i8 nt rc~t,nt
infinity, and t h vortrx ~ street move8 Fig. 2 . 5 ~
Fig. 2.5d
t o thr1ns14vt:s. 'I'lrc only nrmngnncnt which shows ncnt.ral cqoilil,rium is t,hat with vort,ex sl.rcet moves with n vcloc:it,y I L , which is slnallc\r 0.281 ([Cia. 2.8). I,II:I.II t.Ilc flow vrIorii,y I I in front of t,ho body. I t cnn l)c r c p d e t l a s a highly idealized p i c t , ~ ~of r r t.hc mot,ion in the wake of (,hc body. The kinetic energy cont,ainetl in the vrlocit,y ficltl of the vortcx strect 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 exprrssion for t.hc drng from the perfect-fluid theory. I t s ~nngnit,utleper nnit lengt,h of tllr eYlindric:~lhotly is given hy
.
C i r c d u r cylittder. 'l'hc frequency wit,lr which vor1,irc~sa r r shrtl in a Is u f i ~ o t lby tho fluitl Ijcforo c:nt.c-rilig t h e pipe. With a n arrangernwt, which is as f'rcc from dis111rl)ancrsas possil,lc rrific.:~i ,~ 10"nn I)c att.ainrrl (ii, = dCnot,cs the mratl Rcynoltls nr~mbers( i i d / ~ ) , , exceeding velocity averaged ovcr the cross-sectionnl n.rea). M'i(.l~a sharp-ctlgctl nrtt.mnc~c1 . 1 1 ~ criticnl Iteynoltls numhcr becomes a.pproximnf.rly
(c%ll
=
R,,,
2300
(pip).
,I his v x l w can be rcgnrtlctl as t,ltc lowrr lintit. Ihr 1,11(y
t
rit.ic:nl 1tc:ynoltls
1i11111l)t'l.
I)c.low whir11 even st,rong distmrbrtncrs (lo not, rnusc t,l~cflow to I~c.c>ornc* t,~~rl~r~lnr~l..
I n t,hc t r ~ r l ~ u l o region nt the pressure tlrop becomes approximately p r ~ p o r t ~ i o n a l t,o the square of the mean flow velocity. In this case a consiclerably larger pressure tliffcrencc is requirctl in ordcr t o pnss a fixed quantit,y of fluid t.hrol1g11 the pipc, ns corrlparocl with laminar flow. l'his follows from t,ho fact t h a t t.ho plrcnomcnoll of t.url)ltlrr~tmixing dissipat,cs a largc q~t:tt~t,it,y of' enorgy which c : ~ ~ ~ sthe c s rcsist,:tnc:c? 1.0 Ilow t.o incrcasc considcr:tl)ly. lrurl,llcr~norr,in Ihc casr? of Lurl~ulcrlt,llow t,hc volodistritlu(.ion over the cross-scct,ior~alarca is much tnoro cvcn t h r l in hminnr flow. 'rhis circumst,ance is also t,o be explained by turbulent mixing which causes a n cxc:hangc of momcntum bctwecn the layers near t h e axis of the tube and those near t,hc walls. Most pipc flows which are encountererl in engineering appliances occur a t such high Reynolds numbcrs t h a t turbrllcrlt motion prevails a s a rule. Thc laws of t u r b ~ l l e n tmotion through pipes will be discrlssed in detail in Chap. XX. 111 a way which is similar t o the motlion through a pipe, the flow in a boundary laycr along a wall also becomes turbulent when the extcrnal velocity is sufficient,ly largc. ISxpcrimental investigations into the transition from laminar t o turbulent flow in the I,ollntlnry Inyer were first carried out by J . M. Burgers [GI and I3. G . vnll (lcr licgge Zijncrl 1171 as wcll as by M. IIansen [lG]. The t,ransit.iorl from laminar t o turbulent flow in t h e boundary layer becomes most clearly discernible by a sutltlcn a.nd largc increase in the boundary-layer thiclrncss ant1 in the shearing stress near the wall. According t o eqn. (2.1), with 1 replaced by the current coortlinatc s , the dimensionless boundary-layer thickness 6/1/1'27~; becomes constant for laminar flow, and is, a s seen from eqn. ( 2 . l a ) , approximately equal t o 5. Fig. 2.23 contains a plot of this tlimcrlsiorllcss boundary-layer thickness agairlst the IZcynoltls number IJ, z / v . At R, > 3-2 x 10" very sharp increase is clearly visil)le, and
Fig. 2.23. Boundnry-layerthickness plobtedr against the Reynolds number based on'the current lcngth z along a plate in pnrnllel flow a t zero incidence, ~s measured by llanscn [I61
a s sprn from rqn. (2 1 a). l l r ~ l r ct o t h r rritiral Rrynoltls r ~ u r n l ~ r r
there corrcspontls R g crlt = 2800. The bountlary Inyrr or1 :I plate is Inr11in:cr near t.l~t: leading edge and bcconles turbulent f~lrt.llcrtlowr~st,rca~n. 'I'llc nbscissn r,,,, of t l ~ t point of lrn~lsit~ion can be clctcrminctl from L11c k t l o w ~v:~lric ~ of R, I n t.llc caso of n plate, a s in the prcviot~slydiscussed pipc flow, the nun~cricnlvaI11o of R,,,, dcpcntls t o a ~narkctldegree on the amount of' tlist.~lrl~ancc in tho nxt,crn:tl flow, :111tj the value R, = 3.2 x 10%hot1lcl be regartlet1 ns a lower limit,. With oxccpt.iorl:~Ily - 10%rlrltl higllrr 11:~vc been :~tt.ail~rtl. (list-rrrbnncc-frcccxt.crnal flow, valrlcs of R, A 1):~rticul:trly rernarltable phcnorncnon c o n n c c l d with the transit.ioll from laminar t o trlrbrllt:nt flow occurs in tJle casc of blunt llotlics, s11cl1a s c i r c ~ ~ lcylintlers ar or spheres. I t will be seen from Figs. 1.4 a r d 1.5 t,llaL the tlmg coef'ficierlt o f a circrtlar cylintlcr or a sphcro suffcrs a sutltlcn :d consitlcral~ledccrcasc Ilr:lr Itcynoltls n ~ i m l ~ c 1r.'s I)/v of bout 5 X lo5 or 3 x lo5 rcspccLive1~.This fact was first, obscrvrtl on sphcrcs by G . 1I:iffrl 1141. It. is a conscquerlcc of t,ransition which causes t.he point of separation t o movc clownstacam, l)cca~rsc,in the case of a turbulcr~t1)ountlary laycr, the accelerating influence of the cxt.crn:d flow e x t m d s f u r l h r due t,o t.t~rbulrr~t. mixing. ~Tcnccthe point of separation whicll lies near the equator for a laminar I)o~rr~tlary I:~ycr nlovcs over a cor~sitlcml~lo tlislnr~ccin the downstream tlircct.ior~. In t,urn, the tlcad arca decreases considcmbly, anti thc pressure d i ~ t ~ r i b u t i obecomes n more like t,hat for frictionless motion (Fig. 1.11). The decrease in t h c rlcad-wat,cr region consitlcmbly reduces t h e prcssrlrc dmg, and t h a t shows itself as a jump in the curve G, .= f(R). L. Pmnrltl [26] provctl t l ~ ecorrcctncss of t,hc prrcccling reasoning 11y nlo~inl~ing n Ihiri wirc ring III; a ~Ilort,( l i ~ I m ( : cin f r o ~ t tor IJIOccl~i:ll,or of a sphere. This car~scsthe boundary laycr to bccome art,ificially turl)~llcrlta t n lower Reynolds n l ~ m b c rand the tlccrcasc in t,hc drag cocfficicr~ttaltes place carlicr Lllar~ would otherwise be the case. Figs. 2.24 and 2.26 reproduce photographs of flows which have been made visible by smoke. They reprcscnt the subcritical pattern with a large value of the drag coefficient and the supercritical pattern with a small dead-water arca and a small value of t h e drag coefficient. The supercritical pat,tern was achieved with Prandt,l's tripping wire. The preceding cxporimcnt shows in a convincing manncr t,hat t h e jump in the drag curve of a rircular cylintlcr and sphere can only be interprctcd a s a borindary-layer phcnomcnor~.Othor bodies with a blunt or rounded slcrn. (c. g. elliptic cyli~~tlcrs) display :I type of relationship bctwcen drag coefficient and Rcynoltls number wllicl~is s ~ ~ l ) s t a ~ ~ l isimilar. :illy \'Vit,h increasing slcntlcrness the jump in t h curve bccomcs ~ ' i r o ~ r c s s i vless c l ~ pronor~nccd. For a streamline body, such ns t h a t shown it1 Fig. 1.12 t.h(:rc is rlo jump, I~nc:~usc no :lpprrci:r.l)lc scp:~.rnt,io~~ occ~lrs;t,lw w r y gmtlrr:~l Iyssrlrc ir~c!rr:lsoon I,l~cIl;lclt
.,,,.
,,
c. Twhulent flow in n pipe nr~din a hourldnry lnycr
11. 011tli11e of Iw~~ntlnry-Inyrr theory
42
Tnhle 2.1.
of suc.11 I~otlicscsan I x overcome by t l ~ cbor~ntla.rylayer w i t h o ~ ~ separat.ion. t AS we sl~allalso scc Int,cr in grrat,er tlrtail, t,he pressure di~tribut~ion in thc ext,ernal flow t~xrrt,sa clet~isivcinflut:nce on t,hc positmionof t.11~transition point. Thc bountlnry Ia.yrr is laminar in the region of prcssurc deereast, i. e. rol~ghlyfrom t.l~eleading ntlgc? to t.hr p i n t of minimum pressure, ant1 becomes t ~ ~ r h u l e n in t , most cases, from t . l ~ : ~ point t onward t h r o u g l ~ o ~t .~l ~t ,region r of prcsslrrc inrrcn.sc. I n this corrnexion it is iml~ort,antto statc t h t , scpamt,ion can only bc nvoitletl in rcgiorrs of incrensing prcssnrc n h the ~ flow in thc bountlnry layer is turlrulcnt. A laminar 1)ountlary layer,
a s wc shall see Int.er, can support, only n very smnll pressure rise so t,hat. scparat,iorr would occur even wit.l~very slcndcr botlics. I n prt.icular, this remark also applies t o the flow past nn aerofoil wit,li n pressure dist,rit)ut,iorlsimilnr to t h a t in Fig. 1.14. I n t.llis cnse scpamt~ionis most liltcly t,o ocrur on t.he sncI,ion side. A smoot,l~flow pattern nround n.n ncrohil, contlucivc t.o ~ I I C creation of lift, is possihlr only wit.11a t , ~ ~ r h n l e n t bountla.ry Ia.ycr. Summing up it, ma.)i be st.at,rtl that, t.hc small drag of slencler bodies a s wrll &s t . 1 1 ~lift, of acrofoils are ma.& possible 1,111~ough thc cxist,enec of n t,url)ulent, t)ountla,ry Inyer. y the thicknesq of a tnrbulcnt Bounclnry-lnyer thickness: ( ~ c r ~ e r a l lspealc~r~g, Imr~ntlaryh y c r 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 incidence the boundary layer incrcascs downstream in proportion to xoR(x = distance from leading edge) It will he ~ l l o w nInter in Chap. X X I t h a t the boundary-layer tl~ieknrssvariation in (nrt)nlrnt flow is given by the rqnntion d 1
= 0.37
( ) lJm,l
-'I5
f
*
= 0.37 (Rl)-'1'
(2.9)
whic-ll c:orrcspontls 1.0 rqn. (2.2) for laminar flow. I'ahlr 2.1 givns vnlnes for thn I~o~~~~~l:r.ry-I:tyc:l. t11i(~Ii11ns~ o:~l~:uIal.r~I from eqn. (Z.!)) for several typical casos of air : ~ 1 dwatl~rflows.
43
Thickness of bormdary Inyer, 6, a t t.rniling edge oF flnt plate nt zero inridencc in parallel t.nrlwlent flow U,
= rrcr ntrenlll vrloclty:
Air v =
150 x 10-e f t Z / ~ ~ v :
I = l r n q t h or p1al.e:
100 200 2 0 5 0 750
3 3 15 25 25
r = kinrn>nl.le risrasily
2.0 4.0 2.0 8,:s 1.25
x 10' x 10' x lo7
x lo7 x
108
Methods for the prevention of separation: Sopnrnt,ion is mostly nn r~ntlcsir:~.I~lt! pl~rnomcnonbccnusr, it c l l t r ~ ilnrgo l ~ onorgy losncs. I'nr thin rcnson rnctllo~lsI ~ r ~ v1,cm o tleviscd for the artificial prcvcntion of separation. Thc simplest 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 circdar cyli?zP.r tllat this method is very rfrcct.ivn. On the side where t h e wall and stream move in t h c same direction separnt.ion is oornpletely prevented. Moreover, on the side where the wall and s t r e n n ~move in oppositc tlircct,ions, separation is slight so t h a t on the whole i t is possible t o obtain a gootl experimental approximation t o perfcct flow with circulation ant1 a large lift.. Another very effective method for tlic prcvcnt,ion of separation is h m ~ d < r n ~ Ltycr sudion. I n this method thc dccclcratccl fluid pnrticlcs in the boundary Inyrr are removed through slits in t,he wall into thc interior of the body. Wit11 suf'ficirnt.ly strong snction, separation can be prevented. Boundary-layer suct,ion was nsecl on a circular cylinder by L. PrantIt,l in his first. fnntl:~~nentalinvcst.igat,ion in1.o boundary-layer flow. Separation can be almost completely eliminated wit.11 suct,ion through a slit a t the back of t-he circular cylinder. Instnnccs of the cffrct. of snc.I,ion can be seen in Figs. 2.14 and 2.16 on the exnmplc of flows tlirougll n highly divergent channel. Fig. 2.13 demonstrat,es t h a t witllout suction thcrc is strong separation. Fig. 2.14 shows how the flow adheres t o thc one sirlc on wlliclt suction 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 snct8ionslits are put int.0 operation on both si(Irs. I I I t . 1 1 ~ latter case t,lie streamlines assunlc a pattmn wlticl~is very similar t,o l , l ~ ain t liiet.j;)rllcss flow. In lat.cr gears suction was sncccssfr~llyused in acroplanc wings to in(-rcasc.(.l~c lift. Owing t o snc:t.ior~on the rlppcr surface nnar the tmiling edge, t,l~ef l o \ ~a , l l ~ r r r s -.
t
Prnncltl-Tietjens: Hydro- and Aerodynnmics. Vol. 11, Tnldrn 7, 11 and 9.
11. Outline of boundary-layer theory
44
to t h e aerofoil at considerably larger incidence a n g l e s t h a n y o u l d otllcrwisr b e t l r ~ rase. stalling is clrl:cyetl, nntl m u c h l a r g t r m a x i m u m - l i f t values a r e achieved [3F]. Aft,er h a v i n g g i v e n a s h o r t out,lino of t h e f n n t l a m e n t d physical principles of f l ~ ~ imdo t i o n s w i t , l ~v c r y snlnll friot.ion, i. c. of t h c b o u n d a r y - l a y e r t h e o r y , w c shnll proneed t o clovc!lop n m t i o n a l t h e o r y of t l ~ c s op l ~ c n o r n c n nfroln 0110 oq111~1.ions 01' m o t i o n of V ~ S C O I I Sfluids. Thf: description will b e a r r : ~ n g c t lin t h e following w a y : Wt: shall begin i n P a r t A by deriving Ghc g e n e r a l Navicr-Stjokes e q u a t i o n s f r o m w h i c l ~ , i n t u r n , w e s h a l l d e r i v e P r a n d t l ' 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 sirnplificntions which c a n b e inl,rotlucctl a s a consequence of t h e s m a l l v a l u e s of visn t h e metjhods f o r t h e i n t e g m cosit,~ T h i s will b e followed i n Part I3 by a t l c ~ c r i p t ~ i oof 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. 111P a r t C w e s h a l l discuss t 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 p r o b l e m 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 bountlary-laycr t h e o r y for c o m p l e t e l y tlcvclopcrl t u r b u l e n t m o 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 I)c trcat,ctl as n dctlnctlive s e q u e n c e Imsctl o n t,hc Nnvicr-Stolres tlifTerent,i~lequationx for viscous fluids, t h o s a m e 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 u r b u l e n t flow i s s o c o m p l c x t.hat i t c a n n o t b e m a s t e r e d by p u r e l y t.l~rorct,icnlmct,hods. F o r t,his reason a t ~ r c ; ~ i i socn t l n r l ~ n l c n flow t must, d r a w 11e:~vily o n e x p r r i m e n t n l result,s ant1 t,llc s u b j c r t m n s t Ijc presented i n t,hc f o r m of a s e m i cmpiriral throry.
References
[I] Acl~enbach,E.: J':xperilnent,s on the flow past spheres a t vcry high Reynolds numbers. J F M 54, 505--575 (1972). 121 Ilcrger, J':., ant1 Wille, It.: Perioclic flow p h e n o n ~ c ~ ~ Annual n. Reviow of Fluid hlcch. 4 , 313--340 (1072). w I ~ c ~einer ~ Iins. (:iiltingcn 1!)07; L. Math. u. I'h~.s.Mi, 1-37 (1908); Engl. trnt~sl.in SAC'\ Thl 1250. 151 Ulenk. H . . I~urlis,I).. and Licbcm, I,.: uber die 3lcssung von \\'irbelfrequer~zcl~.1,uftfnhrtforsrh"ngl2, 38--41 (1935). [O] Burgers, J. M.:'The motion of n fluid in thr houndnry lnycr nlong n plnne s n ~ o o t lsurfnce. ~ R o c . First lnternationnl Congress for Applied Mcchnnics, 1)elft. 11J-- 128 (I!)24). 171 (;h:~ng, P.K.: Sep:~rntionof flow. l'ergnn~ot~Press. \Vnshington I>.C., 1!)70. of fluid n~rchnnicsto \r-intl enginrering -- t\ Frcetnnn Scholnr [R] C,rrn~nk,J . E.: Ap~~lirntion Icrture. Trnns. AhNlC F h ~ i d sEngineering 97, Ser. I, 9--38 (1!)75): sre nlso: Lahor:~tory sin~rtlntiot~ of the ntlnosphcric houndnry Inycr. t\lA,\ .J. 9 . 174(i-1754 (1!171). He icw of Fluid Blrch. 8 , 75-- 100 (1970). 18111 Cerlnnk, .I. E.: Acrodynnn~icsof h~tildingn.r\n~~ilal ,!I] Crnnnt. J E . . and S ~ ~ C w.z.: II, \\lint~.tunnci s i l n 6 t i o n of wincl ionciing on structures. Jlrctinp: I'reprint 1417, r\SCIC Sntionnl Structural 1Sngineering Alreting. 13nltit11orr.hfnryInrd, 171--2 j April, 1971. [I01 j)nvenport, ,\. G . : 'rhc rclntionship of wind structure to wind Ionding. Pror. Confercnrc on \ \ . i ~ ~ 15llrrts tl on I3uildi11~unnd Str~wtures,Sntionnl I'hynirnl I,nhor:~tory. 'Trtldingtol~, Jlid(llrnrx. (:rrnt Itritnin. 26--28 ,111nr I!l(i:l. Ilcr Mnjmty's Stationnry 0flic.r. I,ontlol~. \'()I. I , 54 -- I I2 ( I !N\5). %
-
,
References
45
[ l l ] Dotnm, U.: Ein Beitrag zur StabilitAtstheorie der Wirbelstmssen u n b r Berucksicht,igu~~g endlieher und zeitlich wachsender Wirbelkerndurchmesser. Ing.-Arch. 22, 400 - 410 i 1954) -,. [12] I h h u , W.: Uber den Einfluss 1aniinn.rcr und t ~ ~ r b u l c n lSt.riimrtng cr nvf dnu Riinlg~nbildvon Wnssor untl Nit,rol~cnzol.Ilclv. p h y s act,:^ 12. 100--228 (I!):)!)). II3] Ihrgin, W.\\'., nncl l eqnnl to zrro; rrniforrn exlcnnion in the zdirection
X
ex
=
-
IGg. 3.4. 1,ocnl tlislorliori o f llr~rtlclrnrr~~t whcn au/ay > 0 with a11 oll~rrterms bring cq11n1 to zcro; r t t ~ i l t r r n sl~onr ~ dcfor~~~:~lio
vr1ocait.y. Thus 2, r e p r c s c n t ~t.he rate of rhn,gdion in the x-dircct,ion sufli?retl by the clcmctit. Similarly, the atlclit,ivc terms C, = a11/'11/ant1 C, = aio/i)z drscri1)e the rate of rlorig:rl.io~l i r ~t,llc y- xntl z-tlircctions, rcspcctively.
( ~ Ir e -
, dy " l'r I H _ az
kz :
dy < ~ t )
t I r dy & (it-
4
r1.z dl) --
Fig. 3.5. Local dintortion of when - dx -
dy d i
-PA
fllli(1
clc:~ncr~~,
E,, = E,, = ((&lay) -t (atqa~)) >o with nll other terms bcing equal to zrro; distortion i n ul~npe. ( T h diagram has bern drnwn for &1/8y = au/a.r )
1.0 iirsL ortlcr in the tlrrivxt.ivcs. Ihlring this tlist.ort,iorl, howcvcr, tdre shxpc of t11e rlrmcnt~,clcscril)cd by the angles a t its vertices, remains ~lnchengcti,sitrcc all right nnglrs cont,inue t o be that, way. Thus k tlescribcs the Iocd, i n s t a n t m ~ e o ~~~Os I I L ~ I C I Y ~ C rlilrr(dio?~of a fluid elenlent. Whcn the fluid is incompressible, d = 0,a s must be e ~ p c c l ~ c tIn l . a co~nprrssil)lcfluid tllc cont.inuit.y equation (3.1) shows t h a t
t,Ilat, is L11:1t, t.llr voll~nirt,rictlil.il,xt,ion, t.11~rcla.t.ive cliangc in volume, is equal t o 1,hr n r p t i v c of t0ic rc:l:~Livc ra.l.c of change in the local densit-y. ; The rcI;ltivc velocity ficlcl presents n tljlfcmnt appearance when one of the olr-diagonal t,erms of n~at.rix(3.13a), for example au/av, has a nth-vanishing, say pasiCive, value. The corrt-spontling field, skctchcd in Fig. 3.4, is one of pure shear stmin. A rectangular clement of fluid ccntrctl on 11 now distorts into a parallelogram a s indicnt,ccl in the diagram. l'hc original right angle a t A changes a t a rate measured Ity the angle y,, = [ ( ~ I L / @d!/I ) tll]/tly, t h a t is a t a r a t s &lay. When both au/ay
Fig. 3.6. Local distort,ion of flnid element when
5 = t ((avlaz) - (&lay)) + instnntnticous rigid-body rotation
o;
11 I . 1)t.rivntion of t.lm cquntionrr of n ~ o l i n nof
56
n
rotnprcasibln viscous tlnitl
arltl r?n/8z 11avc posit,ive nonvanishing vnlnrs, the right, angle a t A will distort owing t,o t,l~esl~pcrposit,ionof t.wo n~ot,ions,t,llc st.:~tcof affairs bcing illr~st,ral,etlin Fig. 3.5. 1 L is c l r : ~ rthat, 1.hr right n.tiglr at. A now distorts a t t,wiac the m t c
t , r r ~ n sof matrix (3.15%). In general, t,hc thrcc tlcscrilwtl by I wo of the orf-tliagon;~l . . ~ f f - d i i ~ g ot.rrms n ~ l Ex!, - F,/,, F,, = d,,, : L I I ~ E,, = Fyr tlcsrxibc t h e rate of dist,ort,ion of a right, nnglc locatrd in ;L plane nornmnl t,o the axis the index of which does not nppt'ar ns n srll)script.. 'l'hr tlistort.ion is volume-preserving and affects only the shape of t,hc rlcmcnt . (lirrr~mstanrrsnro ilgain tli!fcrcnt in the pzrticulxr case when a u / a y = - a v / a z illrrwt.r;ct,ctl in Kg. 3.6. k'roni t . 1 1 ~preceding considerations and from t h e fact that. t ~ o w2,, t) \\.e ran infsr n.t oncc tllat, the right angle a t A remains undistorted. 'I'his is also rlrar from t h r diagram which shows t h a t t h e fluid element rotates with rcsprrt, t,o t.llr rcfkrencr point A . I n s l a ~ r t n n e o ~ ~ sthis l y , r o t d o n occurs without dist.ortion ant1 call Iw dcsoril)rtl as a rigitl-l)otly rot,ntion. The instant,ancons nngulnr vrlorit,y of this rot,at,ion in
-
m). 111 this case the [lroccss of m;~t,h(~lrl:~l,ic::~I sin~plificntionof the tliffcrcntial cqn. (4.10) requircs a consitlcrablc amount of rart:. I t is not pcrlnissiblc simply t,o olnit (,lie viscous tcrrns, i. c., t,lle right,-h:~ntlside of (:(in. (4.10). This woultl rntlnco the ortlcr of t,llc oqu:~t,iot~ from four to two, :LII(I t.110 solution of the simplifictl cquat.iou could not be made to salisfy the full bountl:lry condit.iol~sof t.Iw originn.1cquat.ion. The problem wl~icllwas ontlincd in tllc prccctling scr~t~cnccs belongs essentially t o t l ~ ercdrn of hou~cdnr?/-kr!/erlheory. We now proposc: to tliscuss briefly the genrral st.at,r~nc~lt,s which can t)c made about the solutions of the Nnvicr-St,oltes cqnnl,ions for t,ho special mse of small viscous forccs a s cornpared wit.ll t,hc incrt,in forccs, thnt is in t,11r limiting case of very Inrgr: 1tt:ynolds n11m1)crs. The following analogy rnny scrvc to illust,rate tl16 c:llsr:~cter of the solut.ions of t,hc Navier-Stokcs cquat~ionsfor the litnit.ing c;~seor vt:ry small viscosil,y, i. c., of vrry small friction terms, as compared with t,llc inertia terms T l ~ etcrnprrat1lrcb distribution O(r,y) aboul, a hot, I)otly in n fluid strrarn is clcscrihtl 1)y the following tlilTrr'rrrntial rqrlation, Chap. XI 1 .
c.
Ilcre v , c , :tntl k tlrnol,c 1.I1t: tlrnsity, sl>rc:ilic Itc::rt, :mi contlucl,ivit.y of Ihc lluitl rc!spwlivrly; 0 in tltc! tlilli~rcnc:c?I)ct,wco~~ Ihc loonl t,t:llll)onrt,r~rc:ant1 (,hat a t :t vory 1;rrgc tlisI.:~t~(:c f r o n ~t11c I)otly, wl~orc:IJlc: l.c:tnpcr:~l,~~rc:, 7', is c:onsl,ant ant1 cq~~:rl lso ' / , i. c . 0 - - '/ - '/',,,. 'I'llt: vcloc(ity lic:ltl w(z, y) :rt1!1 ~ ( zy), in oqn. (4.12) is ;~.ssurnctl t o I)c known. 'L'hc t,ernpnrat~~rc: distribution on t h e I~ountlaricsof Lhc botly tlcfinetl b ? ~'/I,, 3 7', is prrsc:ril)ctl nrttl in the sirnplcst cnsc it is constant wit11 respect Lo sl)nct: and t.imc 1)111., gcncrnlly spcalting, it varies wit11 both. I'rom the pl~ysioalpoinl. of view cqn. (4.12) roprosents the 11c:rt 1):~lanc:cIhr an clcn~cnt,ary v o l u t ~ ~'l'hc e . IcfhIlantl sitlc represents t . 1 1 ~qu:mt,it,y o f heat, c:xcl~:~~~gotl I)y convcc:tiorl, whereas the rigl~t-ll:rndside is the ~ ~ ~ ; ~ n of ( . i11r:lt t . v t:xt:I~:~n~cd 11y con(I~t(:tion.T11c frit:l.ion:~lhcatl gcneratcd in tile fluid is ncglcetctl. Tf 7',,> T,,tho prol)lom is t h a t of detcrrnining 1.ltc tcmperatl~rcfield around a hot body which is cooled. B y inspection it is scrn t h a t cqn. (4.12) is of the same form a s eqn. (4.6) for the vorticity w . I n fact thcy hccomc itlentiral if the vorticity is replaced by thc tcmpcraturc tliffercncc and t.hc kincmatic viscosity v by t,hc ratio k i p c known a s thc thcrmal diffusivity. 'l'hc bountlary conclit,ion 0 0 a t a largc distance from thc body corresponds t o the condition tr, = 0 for the undisturbed p,nrnllcl stmam also a t a largc distance from t h e body. llcncc we may expect t h a t thc solutions of the two equations, i. e. the dislribntion of vorticity antl t h a t of tcmpcraturc around t h e body will be similar in chnrnctcr.
-
Now, tllc tcmpcratlrre dist,riI)ution around the body may be pcrccivcd intnitivrly, t o a ccrtnin cxlcnt. In t h e limiting ca.sc of zero velocity (fluid a t rosl) the inflncncc of tilo I~eatccl11otly will extend ~ ~ n i f o r r n lon y all ~ i t i c s .With very small velocit,ics tho fluid arountl t h e hody will still he affectcd by i t in all directions. With incrcnsing velocity 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, antl into a tail of hcatcd fluid bchind it., 1Pig. 4.3.
.-
Fig. 4.3. Annlogy betfween trnlperntuw and vorticity di~tributionill tho neighb o l ~ r h dof
R
body plnml in a strerrrn
of fl\lid -.__ -- -
a), b) IAndCq of rrgion or iw.rrhsrd trmprrsture
n) for ~ r n n l lvclucitlrs - - _ _- - - - - - - - - _ _ . . _ _ Ir) fur Inrge vrlucitirn uf __ _ _
flow
'rllc so111l.ionof eqn. (4.1 2) ni~rst-, a.s mcnt,ionkd, be of a chn.ra.cter similar t.o t h a t for vorticit,y. At snlall velocities (viscous forces h r g e compared with inertia forces) t.hcro is vorticity in i h c whole region of llow around the body. On the other hand for' largc vcloeitics (V~SCOIIS forccs smnll compnrctl wibh ir~ctl~in forccs), we may rx-prct,ifield of flnw in which vorticity is confined to a small Iaycr along t h e surfacc of the I)otly antl in a wake behind thc boily, whereas thc rest of the fcld of flow
'l'ho limiting caw nf vory ~nlnllV ~ R C O I I R li~rtm
79
remains, practically speaking, free from vortioity (scc Vig. 4.1). 11, is, I.l~ercforc, to be cxpected t h a t in the limiting case of very small viscons forces, i. e. nt 1;rrgc: Itcynolds numbers, the solutions of the Nevicr-S(.okcs c q ~ ~ : ~ t ~arc i o n SO s ('otlsti(.~tt.rtl a s 1.0 permit a suldivision of the ficld of flow inl,o a n cxtcrn:~I rrgion wlti(*ll is f r ~ o from vorticity, and a thin layer near the I)otly togcthcr with a wakc I~(:llit~tl it.. I11 1.11~:first, region tho Ilow mny Im oxpnctctl 1.0 sntisfy OIOc t l ~ ~ ~ r t i oofn sI'ri(:(,iot~ltw flow, the potcr~t~ial llow theory bcing uscd for i h cvalnation, whcrcas in tllc sc~-otr(l region vorticity is inherent, and, thcrcfore, the Navicr-Stoltcs cq~iationsm ~ ~ shn t. uscd for its evaluation. Viscous forccs are i m p o r t m ~ t ,i. c. of thc santc ortlcr 91' mngnitt~tlcn9. inertia forces, only in t l ~ cscc:ontl region known :is thc bou~tdrtr?yIrr?yrr. This concept of a boundary layer was introduced into the scicncc of fluid mechanics by L. Prantitl a t the beginning of thc present ccntury: it has proved t,o he vcry fruitful. The subdivision of the field of flow into tho frictionless oxtcrnnl Ilow iwtl the cssentinlly viscous boundary-laycr flow p c r m i t k d thc reduction of the rnnt,llcmatical difficnlties inllorent in t h e Nnvicr-Stokes cqnatior~st o snch an extent, t h a t it, l m m n e possible to integrate them for a large numbcr of cascs. The tloscript.ion of t,l~cscmethods of integration forms t.hc subject of the boundary-laycr thnory prcscntctl in the following chapters. From a nt~mcrical analysis of the available soluteions of the Navicr-Stokc~s c q ~ ~ a t i o intsis also poasiblo t o show directly t h a t in tho limiting case of vcry lnrgc Reynolds numbers there exists a thin boundary laycr in which t h e influcncc of viscosit,y is conccntratcd. We shall rcvcrt to this topic in Chnp. 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 numbem) present8 greatrr mathematical difficulties 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 impossible t o satisfy sirr~ultancously all boundary conditions of the cornpleto tliKrrcntial equnt~ous. However, this does not signify t h a t the solutions of sucll a n equation, sin~ldificdby t.he elimination of viscous terms, lose their physical meaning. Moreover, it is possil~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 m b ~ r s . Tho exception is confincd t o n thin lnycr near the wall - the bountlnry In.yc;r. l ' h ~ l s , t h r complete nolution of t.hc Nnvicr-Stmkcs cqtralions c:nn I)c t l ~ o r r g l of ~ l nrc t:onrcisI,ing of two sointions, thc so-cnllctl "outcr" solution which is ohtninctl with the nid of Eulor's equations 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 houndary-layer eqmtions which are dctlncctl from tho NavicrStokes equations by ~oortlinat~e stretching nnti pwqsagc t o tho limit R + m, n.s will be shown in Chnp. VII. The outer and inncr solutions must he malchcd t,o ench other by exploiting the condition t h a t thcrc must exist nn overlapping rrgion in which bbth s&tions are valid.
f. Mnt,lremnticnl illust,ration of the procens of going t.o the l i m i t R
I\'. C r n r t n l p r o p r r t i r s of the Nnvier Stnlzcs r q r ~ n t i o n s
80
f.
M n t l w n ~ n t i c n li l 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
limit R
4
oo
t
-t
m
81
In sl>il.c o f thn sirnl~lificatiorl, I l ~ e diflixentinl rqrrnt,ion (4.20) is o n r of noco~irl~ l c g r r r :i t c.nn 1)r mntlc? t.o e:rt.idy LIIO i n i t i a l rondit,iot~(4.14) h y t . 1 c~l m i r r
P 7
Let
IU
rotinitlrr t.lie tlnmpr(l vihrntious of n point-mass tlmrrihctl h y t,hc t l i f i r r n l i n l c r ( ~ ~ n t , i o n
I l ~ vc n l l ~ cof const.nnt. A z folloaw froin t.110 tnnl.ol~i~ip; t o t11c " o ~ ~ l r rn" n l ~ ~ t , i rt111. o ~ ~ ,(4.17). 111 1111 ovrrlnpping rnngr, t h t . is for ~noclcr:i(cv n l t ~ r nof t i i n r , t.hc n o l ~ ~ l i o ni n nqnn. (4.17) i ~ n d(4.21) nit~nl.ngrr:r. 'l'lit~s\VO tnr~sl,I I ~ v ~ ~
H e r c i r r donolrn the vihrnling mnsn, c (.lie spring c:or~ntnnt., k I.l)c. tlnniping f:wto~.. r t.I~t:Irng(.ll r o o r d i n n t . ~nlcrlmcrccl from t,lw jiosit,ion of r i l ~ ~ i l i b r i n nnrl ~ n . I t.lw t i ~ n r '1'11r . i n i t i a l ron(lilions arc ILRRIIII~~C~ t,o be r - O at 1 - 0 . (4.14)
or, i n wortln: 'l'h "or~t,nr" l i m i t of t.lic " i n n r r " solntion IIIR~, "outer" solnlion. Condition (4.23) Icntls nt, oncc to
In ruinlogy w i t h (.tie Nnvier-Stnkon eqr~ntionsfor t.lie cnse \rhcn t h r lrinrnintic visronity, I*,is very sninll, we c o n d c r h c r r t l w limitsingrnsc of v r r y smnll mnss nr, h r r n ~ ~ this n r l o o rnrlnrs 1.11~l e r m of thc Iiigllest o r d r r i n cqn. (4.13) t o brromt: very small. l'lir c o m p l r t r solrition of cqn. (4.13) s ~ ~ h j r trnt the i n i t i n l rondit.ion (4.14) hns the form
nntl no t o the i n n r r solrltion
x = A {exp (
I
)
--
c x p ( - k 11iri)): irr
-t
0,
(4.15)
where A in n f r r e constnnt \vl~oscv11111e r n n h r ( I r t r r n ~ i n r (w l i t h r r f c r c t ~ c rt o 11 srron(l initinl contlit,ion. 0 i n eqn. (4.13), we nrc l r t l t o t.lw simplified r q r ~ t ~ t . i o n I f we put, i n
-
dx ktlt
f
e:r =0,
w i i i r l i is of first orckr, nnrl whose solrclion is TO(/) = A r x p ( - c t l k ) .
(4.17)
This solrrtion is idcnt,irnl wit,h the first term of the aomplctc s o l r ~ t i o nd u r t o the feliritous choice of t.lw ndjuntnhle co~~utnnt,. However. this solution r n n n o t he ~ n n t l ct.o satisfy t,lie init,iol coridit.ion (4.14); i t thus reprc~entRa eolut,ion for 1n.rge values of t h r time, t ( L c o ~ ~so111t.ion). l~r" 'l'hr8oIntion for smnll vnlrtcs of t i m e ("inner" s o l t ~ t i o n )snlisfies n.noLhrr diflerentinl equation \rliirlt can also t is "stret.cllcd" be dnrivwl from eqn. (4.13). 111 order to n c h i c w this, t.hr i n ~ l o p c n t l r n vnrinl~lr: t i n t,hnt a now "inner" vnrinhle t* = t/m (4.18)
.rr(t*) = A ( 1 - c x p (-- k t * ) } .
I 1.1
t
*
(I*) = A , r x p (
-
kt*) 1 A,.
1 n l n i ~ ~ d c l ~ f .1.0 c vProfrssor l Klnns Grrnten for (I1c rosisrtl vrrnion of t,liin section.
1,. I'r~l.wlt.l, Annrhnr~liche1t11t1~ ~ w t z l i r hhlnt,henintik. c I,rrt,ures d r l i v r r r d nt. (;oet,t.ingrn U n i vrrnil.y ill t.hr \Yint.rr-Srmcnt,rr of 1!):11/:12.
l i n ~ i 01 l thr
(4.25)
'I'l~csnme form r n n be obt.ninrtl from Lllr r o n ~ p l c l rsoI11t icm fro111r q n . (4.15) b y r * x ~ ~ : ~ n ( l(It(. ing tirut t,rrni for small v n l w s of I nnd r c t n i ~ ~ ithe ~ l gGrnL tern1 only, I h n t is b y p ~ t , l i n g
7'11~t\vo iic!ution~, t l i c onter so111tion from eqn. (4.17) nrid L l ~ cinner ~ o l u t i o nfrom r q n (4.26). togct,l~erform the m!!iplcte solution o n condition t l i n t c n r h is 1 1 8 r t l ill its projwr I.III~~C of vnlidity. ht finite 1, cqu. (4.15) tends ( c the outer solut~ionfor nt + 0. whcrens a t constant t* eqn. (4.15) t o l ~ tt,o l ~ the inner nolution. 'l'lle pnrtinl solut,io~isgive Illc cornplrtc, cotnposit.~n o l ~ ~ t . i owhich n is vnlitl i n the cnl,ire rnnge o f v n i ~ c so f t i ) y ridding t h r n l t o g r t l ~ r r ,rcmemhrring t h a t I h r ronlmon nr.rorrling t o tcr111from eqn (4.23) ~ n n s the . included only once, tlint, in sul~t,r:rclrrl from the RIII tho prcsrription
-.
+ r t ( t * ) - II *i m xm: ( l * ) =
x(1) = ~ " ( 1 )
TO
(t) I r t ( t * )
-.
-- l i m xn(1). 1
tJ
(4.27)
A graphical roprencntation of the complete .soIr~t,ionfrom eqn. (4.15) i~nhown i n F i g . 4.4 for the cnse when A > 0. Curve (a) corresponds t o t , l ~ eouter solution (4.17). Cnrvcs (I)), ( r ) nntl (d) represent solutions of t h e c o t n p l c t , ~t l i f i r r n t i n l equation (4.13) \vitlt vr c l r r r m s i r ~ gfrom ( h ) t o ((I). I f wc now cor~ipnrcthis r x a m p l o w i t h t h r Navier-St,okcs cq~mt,ions,we COIICIU~O l.liat. t.Iie r.o~nplctccqrt:~tion (4.13) is nn:~Iogonn1.0 t h r Nnvicr-Stokes cq11a1.ionsfor n v i ~ o o n nIluicl. w l w r m s I l w sirnpIiliv(1 t q w t t , i o ~(4.1(;), ~ t ! o r r c s ~ ~ o n ~I lns l h ~ l c r ' sr q u n t i o m for n u i ( l r a l l l ~ t i d .WIP i11iti:11
iv int,roduccd. 111this manner, cqn. (4.13) is I r a ~ ~ s f o r m et,o tl
i o nsoI111io11 . now is w l i i c l ~p ) v r r n n l l ~"ri n n r r " ~ o l ~ ~ t ~WII:
he C~IIRI l o the "innt:r"
Fig. 4.4. SOIIII~OIIS o f t h r v i O r : i t i o ~r~q ~ ~ : i t i ( ~ n ( t . I:!). (a) Sol111io11 of t l ~ e s i n ~ p l i f i t dr q n : ~ t h ~ (s!. 14). 111 -- 0 : (11). (c), ((I) rcprrsent so111tions o f 111rvo111111rt(:tlil'li~rcntin1 cquntion (4.13) \I it11 v n r i o l ~ sV:IIIIIY o f i11. JVhcn i r l is w r y s~nnll. soI111io11((I) a r q ~ ~ i r rI~nun(l:tr,yn layer 141:1rnrtrr
conclil.ion (4.14) plays n part wl~ichia ein~ilnrtn 1 . 1 1 ~no-slip condit.ion of n r e d f l ~ ~ i 'Chc d . latter cnn be saLislic4 by Ihc solutions of 1.11~Nnvicr-Stokcs e q ~ ~ e t i o nI s~ u tnot by those of Euler'a In) tile frictionlcsn solution ( p t r n t i n l flow) cquntio~~s. 'I'lw slowly-varying solnt,ion is trnnlngrr~~s whicl~f:rils to satisfy the no-slip contlil.ion. 7 ' 1 1 ~f~rst-vnryingsolntion rcprcscnts Lhe counlr?rpart of tho bonndnry-lnycr ~oIuLionwhicl~ia delcrn~incdby t.ho prcscnm of viscosit.y; i t clin'cn fron~ zero only in n narrow zone near tho wall (boundary Inyer). I t is to bo n o k d that the second bonndnry condition (no slip a t tho wall) can only be sal,inficd if this bountlnry-layer solution is a.dclwl, t,l111smnking tho whole sol~~bion phy~icnllyred.
C H A P T E R V
This simple rxarnplc cxl~ibit,sthc sarnc matbcmstical Aaturcs M t.l~oscch?usscd in 1,110 prcrcding cl~u.plcr.I t is, nrrrnrly, not pcrn~iasil~lc s i ~ nrly In onlit t l ~ cviscou~tern18 i l l tlw Nnvicr-Stokm equation, wlmn performing the process or going over to t.he limit Tor very small viscosit.y (vrry I;irge llcynolrln n ~ ~ r n b r rThis ). w n only bc: clonc: in tile intrgrnl solnl.ion itxlr. W e sha.ll t l c n i o n s t r a t c l a t m in g r c a t c r c1cl:iil t h t if, is n o t t1cc:c:ssary t o rctain l h c Cull Navirr-St.olrcs eqnnt,ions f o r t h e process of finding t h e l i m i t for R -+m. F o r l h c salte of n ~ a t h r m a t i c a lsimplification i l will p r o v c possible t o o m i t c e r t a i n t . r r n i ~ in i t , pnrticnlnrly c e r t a i n s m a l l viscous tcrnls. It is, h o w e v e r , i m p o r t a n t t o n o t e that, n o t a l l viscous t r r m a c a n b c ncglrctrtl. ns t h i s w o r i l ~ ldepress t l l c o r d r r of t h o NavierStolrrs r q n n t i o n s
[l] Ackcrct, J . : Ubcr cxnkte J5sungen dcr Stokes-Navi~rGlcicl~ungeninkomprrmihler 1Pliiimigkciton bci vcriin~lcrbn(:rr~~r,l~c~li~~~rtngc?~~. %1\;\11' 3, 259--271 (1952). [In] Apeelt. C. ,i.:'l'hc ~ t r n r l yI l r t i v of a viscous flnid p a t n circulnr cylindcr a t Reynolds numbers 40 and 44. Ikitisl~A l t C ItM 3175 (IWil). (Lh] Allen, D.N. 1)c G . , m~ciSo~~t.hwcll, 1t.V.: ltclaxation methods npplicd to deternline the mot,ion, in t,wo d i ~ n m ~ i o nof a , R viscons flnid pnat n Bxetl cylinder. Q~mrt.J . Mecb. Appl. MnLIl. 8 , 12!)-145 (1!)55). [lo] Coutnnccau, M., nnd Uo~lnrd,R.: 15xpcrirncntnl dckr~ninnt.ionof t.lw main fcnt,nrrs of the vinrn~~n flow in of R circular cvlinder in 11uifor111 tra~~slation. Par1 I. Stendy now. . . tho . ~wakn . JFM 78, 231 -256 (1977j. [Id] ( h u t a ~ ~ c e n M., u . ~ 1 r I1h ~ n r d It.: , RxprritnrnLal detcnnination of thc mnin fcnCuren of thc f l r ~ win tbe wake of R circulilr cylinder in uniform trnnslation. Part 2. Unsbndy flow. visco~~s .11W 79, 257- 272 (15377). [2] Ihmnis, S.C.K.. and GRII-ZIIChang: Nn~ncricalsoI~~t,it)ns for stcarly f l o ~past x c i r c ~ ~ l n r cylintlcr nt, I R,,, the flow pnttcrn is entirely tliPFcrenl, and bccomca lurhulc~tt.Wc shall d i s c ~ ~ this s s type of flow in greater detail i n Chap. XX. 'I'hc rcl:rtion bchwecn t h c pressure graclicnt and thc mean velocity of flow is ~ ~ o r ~ n arcI~roscntcd lly in cnginccring applicat,ions by introducing a resistanc~coe//icient o/ pipr /low, l . 'l'his coc:fticicnt is tlofincd by setting tho prcwlrc gr:ulicnt proportional t,o the clyn:~.rnicIicatl, i. n., 1.0 tlic square of t h c moan vcloc:itsy of flow, aocorcling t o tho
Jlcro R tlcnotes Lhc Roynolds n u m l m calculated for thc pipe d i n m c b r and moan vclocity of flow. The laminar equation for prewuro loss in pipcs, cqn. (5.1 I ) , is in cxcellcnt ~ g r c c m c r l twith experimental rcsvlts for t h c laminar range, a s SCCII from Fig. 6.3 which rcj~rotfuccscxpcrimentnl p i n h m o a s ~ ~ r cby d (2. 1I:i.gc:n [I()]. From this i t is possible to infer t h a t t h e Ilagcn-I'oiscuillc parabolic vclocity distribution represents a solution of t h e Navicr-Stokes equations which is in agrcemcnt with expcrimental results [22]. I t is also possiblc t o indicatc a n exact solution of the Navier-Stokes equations for thc case of a pipe with a circular annular crosssection 1201. T h e problem of laminar and turbulent flow through pipcs with cxcentric annular crow-scctions was discusscd theoretically in ref. [38] which also contains experimental results. 3. The flow between two concentric rotating cylinders. A f ~ ~ r t l icxamplc cr wliieli leads t o a simple exact solution of t h e Navier-Stokes equations is affortlcd by the flow between two concentric rotating cylinders, both of which move a t tlifTvrcnt but steady rotational spccds. Wc shall dcnotc thc inner and outcr radii by r , , and r2 rrspcctivcly, and similarly, t h e two angular velocities by w , , and w,. Thc NavicrStokes equations (3.36) for plane polar coordinates r e d w e t o
lr~trotlr~cing the cxprcssion for d p / d z from cqn. (5.9) we ol)tain
with dcnobing the circi~nrfcrcntialvelocity. The lmundary contlit,ions arc: u - r l r 1 1 ~ for r = rl a n d u = r, (0, for r = r2. The solution of (5.14) which satisfies tticsc rrquirements is
with
t
- - -
- -
This qndrntic Iaw which nnmrnra dp/dz 12' fitn t.urbulcnt flow vcry well. It is r e b i n d fin Irmminnr flow, although in Lhnt rnngo dp/dz 12. Thus lor Innliner flow A was to bo a mnnbnt.
n. Pnrallrl flow
V. Rxnct uolutions of tltc Nnvinr-Stokcs rqr~ntions
88
H!)
x' - r-r, 5 - 5
(1)
x A,-.I--x2 11,
1 - 1 . 2
x
( ~ n n e r o t n t i n p , outcr a t ITS^).
(5.lGn)
Fig. 5.4. Vclorily dist~ributinl~ it1 thr n n n ~ ~ l rl)c~t\r.c~r~~ ~n t,\\.o.c-o~lc.c.~ltri~.. ~ . o t ; t t~i ~ ~J l~i l l ~> Iln ~~ ~~; lrl sl ~ l l I:tlrtl with t l ~ ci~itlof cqtm ( 5 . l 5 n , b). n) Cnsr I : irinrr cyli~ltlerrot.nt.ing;orct,rr cylitlticr at, rcst, r o 2 - 0 h ) ( h e I I : inner cylitlclrr a t rcst, tol = 0 ; o~tt,rrcylitlclrr r o t a t i ~ l ~ r, - r;uIius or i1111l.r 1.~1i11dcr.r, = r:uIim oI'o1111,rcyliwlpr =
,J?
r .
--.I.
2nr
It, is sccn, therefore, t,ll,zt, t,he case of fric:l.iot~lrssflow i l l t,hc r ~ e i g l l l ) o ~ ~ r h oof o t la vort,cx line constitput.cs a. solut,iorl of t.hc Navirr-Stokes c q u n t i o r ~ s(c/. Scc. I V b ) . In t,llis connexion i t m a y be i n s t n ~ c t i v et,o n~cnt,iotla11 cxnrnple of a n cxnct n m slendy solut.ion of t h e Navier-Stokcs cclunt,iotls, rlnmcly t h a t which tlescribcs t h e process of t1cca.y of n vortex t,hrough bhc a c t h l of viscosity. T h e distribr~t~ioll of t . l ~ e t,angct~t~ial vrlority component 7~ wit,lr respect t o t . l ~ eradial tlista~lcer a n d tirnc t is give11 b y
-
T I is ~tol.t.worIhyIItnt, t,ltr vclocit.,y vnrirs s t r t m g l ~ rwil.l~t,hc rnt,io x rl/rz of t,hc I ~ W O radii i l l Cnsr I, w h r r r a s for ( h s r I1 it is almost i n t l c p r n c l c ~ of' ~ t it. MThcnx = rl/rz + 1 , Iwt.l~c~tscvitclttl t,o t h e linrar vrloeity tlistril)l~t.ionof (!oucstt.(, f l o ~ a, s it, ocrurrcd 1)cI.wc~vnl,wo flat plat,cs in t h r rase rrprc~srntcclin Pig. 1.1 . T h e cc~nnt~ion of' Cnsr J yicltls tho satrtr linlif fiw r1 -- 0, i. C. fnv x = O \vhcn 110 i n ~ i e rrylintlcl is p r r s c ~ l t .I n (.Itis c,nsr, I J I t ( ~I l ~ ~ i1~)(;11t>s tl insitlt: I I I C out,cr eylintlrr a s n rigit1 I)otiy. Ilcncc il. is seen lIln.1 ('nso I1 yicsltls n lineal vcloril~ytlislril)~~t,iotl POI. llir t , \ \ ~ sy111pI.ot.i~ C ~ S Cx S -- 0 n11t1x -- 1. 'l'ltis I ) r h n v i o ~~rtaltcsi t rn.sy t o ~intirrst,e.rltlw h y t,hc vclocit,y tlist.ribut8ions for lhr. ot.lter, inl.crmrtliatc valurs of x tlilTc,r so l i k l c from n s t m i g h t line.
Pig. 5.5. Vclocit.y distribution a t varying times in tho ncigltbo~~rhoodof n vortex filament cnl~setlby tho action of viscosity
-
1; circulnlinn or l l w vortox nnrncttl nt 11 nio 1 w1:c.n vircoslly Iwylnr lo ncl: -.it I ; / ? n r.
=.
0
90
V. ICxnct sol~ttion~ of tho Nnvior-StOkcs c q i ~ : ~ l i o t ~ ~
a s derived by C. W. Osecn [21] and G. 1Ia1nel [I]]. This velocity distribution is represented graphically in Pig. 5.5 Here 16 dcnotcs t,he circulation of the vortex filamolt, a t time 1 0, i. c. a t tho moment whcn viscosity is nssumed t o bcgit~it* a c t i o l ~ An . cxperimenLal investigation of this procoss was ~ ~ n t l c r t ~ n11y l t aA ~ .~Tirnmo [40]. K. Kirdc 1171 mndc an nnnlytio study of the caso when the velocity distribution in t,ho vortcx tlilTcr~from I.hnt irnposctl hy pot,cnt,inl theory. =1
4. The suddenly necelernted plane wall; Stokes's first problem. We now procccd t o calcuhtc somo non-steady par;rllcl flows. Since the convcctivc acceleration terms vanish itlcr~tic:aIly,the frict.ior~forcos i n t r m c t with tho local ncce1crnt.ion. Tho si~nplcst flows of this clam occur when motion 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 r ~ acce1cr:~tcd ly from rest and n ~ o v c sin it,s own plane with a constant vclocitty [lo. This is onc of the proI~lcmswhich wcro solvctl by (2. Stokes in his colcbr:rtccl memoir on p e r ~ t l ~ ~ l u r r ~ s [3ri]t. Selecting tho z-axis along the wall in the dirnction of U,, we obtain t h e simplifiotl Navicr-SOnlccs oqmt.ion
a. Parallcl flow
9I
t.hc complemenhry error /um%ion, erfc q, 11.w been tabulatedt. The velocity distribution is rcpresontcd in Pig. 6.0, and it may bo notctl t h a t tho vclocity profiles for varying tinies arc 'similar', i. e., they can bc rctl~lccdto the same curve by changing t,hc scalc ttlong the axis of ordinates. T h c cornplcmcntary error f ~ ~ n c t i whicl~ o r ~ appcnrs ~ t tlcfir~ition iu eqn. (5.22) has a valuo of about 0.01 at 7 -- 2.0. %.king into a c c o r ~ r tho try 0 , wc: ol)t.r~ir~ of t,l~c: t~hic:ltnossof the: I ~ o t ~ n t l ~Inyor, 6 = 2 q a J Z x 4
JZ.
(5.23)
I t is seen t o be proportionnl t o the sqnnrc root of tho protll~ot,of kir~ornnLiovisc:osiOy silt1 time. This problem was gcncralized by E. J?ccker [3] t o ir~clr~dc: more genrml rat.rs of nccclnratio~~ a s well a s the cqses involving suctior~or blowing or tho c f i c t of compressil)ility.
'rho prrssuro in tho wh01o space is constant, and Lhe bol~nclirryconclit,iol~sare:
The cliIT(:rcnt.ial equation (5.17) is icIcnt.ical with the equation of h e s t conduction which clcscribcs the propngatrinrl of Itoat, in tho space y > 0, whcn a t time 1 = 0 t h e wall y = 0 i s sudtlcnly hcatcd t o a t,cmpcr;~t,nrewhich oxccecls t h a t i n the surroundings. 'l'lle pnrl,i:~ltliffcrcntial oquation (5.17) can be retlucctl t.o a n ortlinary diTcrt:ntial cqu:~t.ior~ 11s tho sul)st.il,nt.ion Y (5.19) 2 1/ If wn, further, n.ssumc x = Uoj(r]), (5.20)
"--3'
wc o11I.air1t h e followi~~g ordinary tliITorcntial equation for
Fig. 5.6. Vclocity dishibution above a snddenly accelerated wall
/ (q): 5. Flow forn~ntioni n Cmuette motion. The s~~bstiLuI.ion (5.10) which Imds to eqn. (5.21) d m not, in general, lend to a sol~ttionof 1.hc so-cnllcd lwnt conduction cquntion (5.17) i r morc complicntod boundnry contlilions aro itn~~osctl, Sincc cqn. (6.17) i~ linear, solution^ (i)r il, OILII be obtained by the use or the 1,nplncc t,mnsfor~nationnnd by tnoro direct nlcl.hods dcvclopc:tl kt c. g., in conncxion with tho study of the conduction of hcnt i n solids. Mnny r c ~ ~ l obtni~~ctl, for the tcmperaturc vnriation in nn infinite or semi-infinite solid, cnn be tlircctly transposed and uacd for the ~oIut,ionof problems in viscons flow. Thw the prcccding problem in which the formation of tho boundary layer noar a suddenly accclcrakl wall has bwn invwtigntrcf can flat w:dl at. also be nolvcd for tlw CDSC when the wall movur in a direction parallel to ar~otl~or n ~ and t a t a distantx, h from it. This is the problcm of flow forn~ationin Couettc motion, i. c.,
t Soe
c. g. Shoppard. "The Probability Tnbgrnl", Rritish Atwoe. Adv. Sci.: Math. Tsblea vol. vii (3039) and Works Project Administration "Tables of the Probability Function", New York, 1041.
a. of how the velocity profilc varion with tirne tcnding nsyn~ptotically to the linear the diutribtrtion nlrown in Fig. 1.1. The diITcrcntinl cqriation is the same en before, cqn. (5.17), lmt with modified I)o~~r~clary conditions which now are:
'rllr s o l u t i o ~of ~ eqn. (5.17) which sntinficn tho bor~ndary :d initial r o ~ ~ d i t i o rran ~ s Iw oht.:~inrdin t.l~cform of a ucrio~of c o n ~ p l c n ~ c ~ ~error t n r y fun~l~ions
-
11
('0
=
x erfc
'y'
,,-I,
rrfc
x crfc [ 2
7x1
r2 n
-
4-
711
-
,, -.I3
(71
rrfc (2 q1 - tl)-1- crfc (2
-1 I ) ?I# -
?I]
-1- 71) - rrfc (4 11,
(5.24) --
11) 1- rrfc (4 71, .1-
71) -
. . . 4- . . .
I';L~IIVI
flow
lnyor ncnr tho wall. 'The influonce of vi~cosityrcnrhcs the pipe ccnf.rr only in the 1:rt.c.r st,:~grn of motion, antl tho velocity profile tonds asy~npLoLicallyto tho p t ~ r o l d i ctlistribt~l.io~~ for ste:rtly flow. The corresponding solut,ion for an nnn~tlarcircul~rrcross-section was given 113. W. Murller 1201.
,1,IIC n c c r l c r ~ ~ of t h 11 I111idovrr ~ , I I c wl~nlcI C I I ~ I , I I of pipe d i ~ c ~ ~ s s11~re t % ( lI I I I I H ~ , ,.41rcsrlllly ~ 'I& d i n ~ i r ~ g ~ ~ i nfrom l ~ c tt.lic l acrcIornt.io~~ of n fluid in t l ~ cilllet j ~ o r t i o ~or ~ aa pipe in ~ I J * : L ~ I , IIOW. rechngnlar ve1ocit.y profile whicl~exists in the entrance ucct.iol~is grncl~lnlly t r a n s f o r ~ ~ ~as ed t.he fluid progresses through the pipe with x increasing, antl tends, ~ ~ n dthe c r influence of viscosity, to nssnine the Hngen-Poiscnillc parabolic diaI.ribntion. Since I~c?rca@z :t 0 tho flo\rs is not onc-rli~nensiond,nncl the vdocity depends on x , nu \vrll ns on t.ho rndi~rs.Thin p r o h ~ nwak rlisrusricd by 11. Srl~lichl.ing[DO), who gave t,l~osolrlliolr for L\vo-tlin~c~~sio~~nl I l o ~tl1ro11~11 n st.r:~igl~t. rhannel, antl by I,. Srhiller 1291, ; ~ n dB. 1'1111nin1241 for nxinlly symrr~rt.rir,nlIlow ( r i r c ~ ~ l a r pipr): s r r nlno Sew. IX i nnd X 111.
wllerc 71, := h,/2 1/ F i (lot~~t.cn the cli~nenniol~lcsn tlistancc between t,l~ctwo wnlllr. 'Tho solut,ion is represellted in Iiig. 6.7. 'rlw corly profiles nre &ill aplwoxi~nntelysimilar and rc~nainso, an long nn t,llr bolllldary layer l ~ n snot sprcad to the stationary wall. The s~lcceedingvcloc:ity r)rofilcn :).re no l o ~ ~ g "similer" rr a ~ ~t tcl~ ~ nsynptotirnlly cl to t,lre linrar distrihrrt~ionof tile s k n d y st&?.
Fig. 5.8. Vclocit,y profilc in n rircrrlnr pipe d ~ ~ r i ~ ~ g ncc~rlrration,art given by 1'. Szgtnnnski [87]; T .- v //I12
E x a c t solrtt,ions for n o n - s t r a t l y Coric.l.t,c flow werc rlcrivcd I)g .I. S t . r i n l ~ c ~ i(331 rr for 1I1r ( x s r W I I I ~ I I O I W ol' 1,110 w d l s is ILI, ITS^, in ;I, s h n ( l y flow I I , I I ( ~ is 111v11S I I ( I ( I ~ I I ~ . S wc:r~lv~.at,c:clt o R givc.11, c:onstnut, vcloc:it,,y. 'l'o (lo t,his, il, is Iicbvc:ssal,y 1.0 solve! ['(lit. (5.17), whirli is itlcnt,ical w i t h t h o one-dirnrnsior~nl Iicat conrluci.ion cqtlat,ion, l)y lncnrls of n l~otiric~r. srrirs. A spccic'll CR,SC i t , t.llis class of soluf.iotls is t . h d w h r n t ' l 1 ~ moving wnll is sutltlrnly st.oppctl s o t , l ~ a li,t rcprrscnt,s t h o d e c a y of ( h ~ o t , t , cf l o ~ .
n:!
95
b. Other oxnct solulionn
V. Exnct sohltiono ol' tho Nnvicr-Slnltcq oqr~nt,ions
94
If we now prescribe a cnnutRnt vclocity v, < 0 a t thc wall (suction), wo notice that cqn. ( 5 . 2 7 ~ ) is satisfied in~mediatelyhy a flow for whicl~o = v, and that the prc.wuro p bcaorncs indcpnndcnt, of uirnultrmco~~sly. Accordingly, we put - (l/e) (+/ax) = tI(J/cll,, whom 11(t) donotes bile frwstrrnm vrlocity n t jr very largc d i s h n c : ~from t . 1 ~ w:rll, nncl I~cnccobtain 1l1c followir~gclilTc.rcl~l.itrl cqtmtion for u ( y , 1):
Tho velocity profile u (!y,t) thus has thc form of a damped harmonic oscillaLion, thc
amplitude of which is
in which a fluid layer a t s distance y has a phase . motion lag y l/;t% with respect to the motion of the wall. Fig. 5.9 rcprcscnts -this for scvcral instants of time. Two fluid layers, a clistanco 2 n / k = 2 n d 2 v/n apart, oscillate in p l i ~ c This . distancc can be regarded as a kind of wave length of the motion: i t is somctimcs called the depth o/ penetration of tho viscous wave. The layer which is carried b y tho wall has a thidrncss of t11c order d Jqand dccrcasos for decreasing kinematic viscosity and increasing frequcncyt. I/, c w i ? ; ,
-
au 3t
1
l,,,
b
- .
ag
d .U. - -1 dl
azu ay2
'
(5.28)
According to .I. 'r. Stuart m2] thorecxista an oxnct soluI,ion ofccln. (5.28) for tllo arld,r:rry o x k r r ~ a l vclocity 'lll~isso1116ionis whcro
Sllh~tituLingthc I.wt t h r w c q u ~ t i o n si n k cqn. ( 5 2 9 , we a m led ID n psrtinl diffcrrntial oq11st.ion for the unknown function g(!/. 1 ) = g(7. 1); thin hnn 1110 forrn
Tllc following non-di~ncnsionnlvarinhlcs hnvo been i r l t r o d u d in the prccocling: I'ie. 5.9. Vrlocit,y rlistrihution in the neighbourhood of an oscillating wall (Stokes's second problem)
Solutions of (5.32) hnve hccn ohtaincd by J. W n h n (411 who crnploycri Lnplaoo transformations and who restricted hirnuclf to severnl apecinl forms of the functior~/(1). (:cncrally speaking, the following cxternnl flows, U(1). hnve been incIudw1: a ) dnrnped nnrl undampcd oscillations,
h) stop-likc chnngc from one vnluo of vclocif.y to xnot.lwr, c) linear incre.nuc from ono vnltlc to anoll~cr. of non-steady solutions. A general c1:rss of no11-stcntly sol111ionsof the Nnvinr-Stnltw ,scq~latio~ls which possran bor~ndnry-lcycro11arnctr:r is ol~tainrdin the sr)ccinl m m when tho velocity componcnta arc indopcndcnt of Lho longitudin:~lcoordinnl,c, a. 'rhc s y s t c n ~of rrlr~nt.ions(8.02). writlnw for plnno flow. nasun)cs 1.11~form rlnm
8. A
n In the upncial c.wc whcn the cxlcrnnl flow is indcpcnclcnt, of time, /(t) - 0, c q ~ ~ a t i o(5.32) I-~ds to the uirnple solution '(7, 7') = 0. This CDIIRP* v01oi~il.yprolilo from oqn. (5.30) to I~oromciclont.ic:rrl wiI.11 Llw nuyrnptoLic s11c1io11 prolilo givcw IILIAW i l l WIII. (14.l;).
The preccding examples on one-tlimcnsional flows were very simplc, I)cca~~se tho convective acceleration which renders thc equations non-linear vnnishcd idontically everywhere. WG shall now proceed to examine sorno exact solutions in which thcsc terms are retained, so t h a t non-linear equations will havo to t)o considcrcd. We shall, however, restrict oursclves to steady flows. ----.
t
.-
Tltc ROIIIL~OII in cqn. (5.2(in) roprcscr~t.salso t,l~ctcmporat~~rn c1intril)ution in Lhc rarth which is m u w d by t.hc pcricxlio Iluc*t.r~at.ion of I.ho k ~ n p c r a t u r con tho surfncc, my,from clay 14) d:ly o r over t,hc scnfu~nsin a yc::w.
9. Stagnation in plane flow (Hiemenz flow). Tho first simple examplc of this t,ype of flow, represented in Fig. 6.10, is that lending l o a shgnc~tionpoint in plane,
1). 0t.lwr exact volr~tio~~n
97
'I'hc l m ~ n r l a r yeonclit~ionsfor / and F arc obt,i~inedfrom 11 v -- 0 at. t l ~ ewall, wl~rrc. ?/ =-. 0, n.nd 2) : : po :tt the stagnation point, a s wrll a s froin 11. ==(J = n. x a t a Inrgt: tlisl,ancv: Sroni t,ho wall. '~'IIIIS -2
whcrc n tlcnoks :L cot~st,nnl.This is a n cxa.~nplcof a. plane polcnt,ial flow wl~icharrives from thc !/-axis and impinges on a flat wall placed a t y = 0, dividrs into two streams on the wall and Lenvcs in bot,h directions. The v i s c o ~ ~flow s m w t adhere l o t,he wall, wl~crcastho potential flow slides along it. I n pot.entia.1 flow the pressure is given by Rcrnoulli's cqr~nt.ion.Tf pa, dcnotcs the stagnat,ior~pressure, and p is t.11~ p r c s s ~ ~ rnt. c a.n arbitrary point., wc have in pot,cnt.inl flow
For v i s c o ~ ~Ilow, s wc: now ninkc t,hc n s s ~ l m p t ~ i o n ~ ?I
=x
Po - p
/' (71) =
;
?I
=
Q (L ":r2
-
1(?I) ,
-1- F (y)1 .
111 this way t,hc cquat,ion of cont,inuit,y ( 4 . 4 ~ is ) snt,isfietl idcnt,icnlly, : ~ n dthc t.wo Navicr-Slnltcs cqr~at.ionsof plane flow (4.4n,l)) n.re snfliciont t o dctmminc l l ~ cf ~ ~ n c lions i ( y ) and F ( y ) Substituting cqns. (5.34) an(\ (5.35) i1it.o eqtl. (4.4a.,b) wc 01)tain t,wo o r d i n 9 . r ~tlifTercnt.inl eqantions for / and F: i'z
aqtl
/ /'
-
=
0) can be found in [20]. Whcn the rotations arc in opposite scnsps (s< 0). physically meaningful solutions can bc obtained for s < - 0 2 only iTunifc)rrn suction :it right, : i ~ ~ g I rto s the dislr is n(lniittc(1.
C(m) = s
The prol~lemof a rotating dislr in a housing is discussed in Chap. X X l . It, is particularly tiotcwor(hy t h a t the solutior~for tlrc rotating disk a s wcll a s 1.llc solutions obtained for the flow with stagnation are, in the first place, exact solutions of the Nnvicr-Stokes cquntions anti, in t h e sccond, t h a t thcy are of a houi~drcry-la?/rr kyps, in tho scnsc discussrd in the prccctlirlg chapter. 111 t,hc l i ~ r ~ i t i r ~ g ( m e of vcry small viscosity t,hese solntions show t h a t tho influence of viscosit.y rxl.rntls over a vcry small lnycr in tllc ~ ~ c i g h l ~ o t ~ r lof. ~ o the o t l solid wnll, ~ 1 1 c ~ t . c : : ~ ~ i l l 1,llc wl~oloof 1.l1c rcmnining region t.hc flow is, j)rnct,ic;llly spcalring, i(lrnt.i(~:ll \ v i t , l ~ (.he corrcspontling itlcal (potcnti:~l)cnsc. ' ~ h c s ccxamplcs show Surthor l . l ~ ; ~ t . The one-dimensional examples t h e b ~ ~ n n i l a rhyy e r has a thickness of the order of flow discussed previonsly display tho snmc bonntlnry-layer character. In this conr~cx-ior~ the rcatlcr may wish to conwit a pnpcr by G. I loG, the drag bccomcs considerably grcatcr than t i , i ~ tgivcn in cqrt. (7.34). Rou~dnry-lnyerthickncss: 11, is impossible to int1ic:~t.ca hor~ntlary-layerl.l~ic:lzncss it1 a n ~lnamhiguousway, because tlic influence of vi~cosit~y in the bonndary laycr clccrcascs asymptot,ically out,wards. 7'110 parallel component,, u,tends asymptotically t o the valuc [Im of thc potcnLiaI flow (thc function / ' ( ? I ) tends asymptotically t o 1). If i t is tlcsircd t o define thc boundary-layer thickness a s t h a t distance for which IL --- 0.99 [I,, thcn, a s scon from l'ahlc 7.1, q 5.0. ITcnec t01c bonnt1:~ry-laycr t,lliclrness, a s tlcfinctl Ilcrc, becomes
A physically meaningful nrcnsurc for t.hc 1)ound:wy layer t.hiclcness is tJro rlisplnrxmnt lhickmxs (TI, whit:li was dreatly i~~trotlucntl in eqn. (2.0), JGg. 2.3. 'l'llc tlisplaccnlcnt thickncss is t h a t distance by which t h c external p o h n t i a l field of flow is displaced ouLwards a s a conscquencc of thc decrease in vclocily in tho 1)ountlxry m
layer. Tlic dccrcasc in volumc flow d u e to tlie influence or fricl.ion is j ((I,, --I&) cly, so t,hnt for 0, wc havc thc definition
Wilh 1r./17, from cqn. (7.26) we obtain
-
0
..
l l t c clislnncc y =; dl is s l ~ o w ni n K g . 7.7. '1'11is is t11c distnncc by wlucl~t l ~ cstrcarnlines of t h e external potential flow are displaced owing to t h e effect of friction near the wall. T h e boundary-layer thickness, 6, givcn in eqn. (7.36), over which t h e potential velocity is attained to within 1 pcr ccnt. is, i n round figures, three times ' larger t h a n t h e displacement thickness.
a2
which will be used W c may a t this point cvaluate t h e momcitli~mthicknms latcr. T h e loss of morncntum in t h e boundary layer, a s comparcd wilh potential flow, m
is givcn by ,g J IL(TJ,
- u ) dy,
so t h a t a new thickness can be defined by
0 m
e~
~ ~ b , = ~ ~ u ( ~ ~ - - ~ ) d y , u-0
aZ
=I&
(1 -
&) d y .
Y-0
Numerical evaluation for t h e plate a t zero incidence gives:
1 / ~ -
4 =0
. 6 ~
(momentum thickncss).
(7.39)
It is necessary to remark hcre t h a t near t h e leading edge of t h e plnte t b c boundm y - h y c r theory acascs t o apply, sincc thcrc t h c assumption a2u/8x2 a2u/8y2/ is not satisfied. Tho boundary-laycr theory applics only from a ccrlain valuo of the Rrynolcls numbcr R = lJ, x / v onwards. Thc rclntionship near tho Icatli~rg edge can only be found from t h e full Navier-Stokes equations becnusc i t involves a singularity a t t h e leading edge itself. An a t t e m p t t o carry o u t such a calculation was made by G. F. Carrier and C. C. Lin [5] a s well a s by B. A. Bolcy and M. B. Fricdman [3].
1
1 lo5 nnequivocally dcmonstratcs thc valitlity of t,he bountlnry-ln.yer n.pproximntions from the physicnl point, of vicw. TII spit.(! t,I~is,
144 VTI. Bonndary lnycr equations for two-dimensional flow; boundary layer on a plate cartrain m a t l ~ e m n t i c j n n sh a v e axpenclod m u c h effort t o c r e a t e R. "mnthemnf,icel proof" f o r t,ho v a l i d i t y of t h e s o simplifications; i n thiw connexion c o n s u l t t h e work of 11. S c h m i d t a n d I 0. Ileferenro [ G 3 ] drrnonnt,rntn~t,hnt t.lw limiting cnne of l l ~ m ouolut,ions, oI)tnin~dW ~ I I,IIc I tnnxirnun~velocity cxccns tend^ 10 in fin it,^, trnnnforms illlo tllr wr-ll-known dl-sitnilnr nolttl.ion of n plro wnll-jet in t,lm absence of nn cxtrrnnl vclociLy -- n cnnc trcr~tcclhy R I . 11. (:lartt:rl (ucc 1401 in (Illr~p.X I ) - ~ when we put, p = -2. A pnrtirulnrly drt,niletl n~onogrnpllon exnrt., self-nirnilnr solt~t.ionsfor lnminnr Imlndary lnyeru in two-din~cnsionalnnd rot,ntionnlly symmetric nrrangemcnt,~,inrl~lsiveof the nssocintrtl thcrinnl boundnry lnycrn (am Chnp. XTl),wns prlhlinl~cdh y C . 1'. J>cwey nntl J. F. Grosn [141. Their consitlcrnt.ionn inclntlc t,lle elTt:ct.s of con~presaibilit~ (nee Chnp. XIJI) wil.11 and mitl~out,hcn,t tmnnfer, relate Lo vnryitlg vnlnes of t.he Prar~tlt,lnumber, and incJ~tdesome rases of suction and blowing. K. 1(. Clien nnd P. A. Libby 191 cnrried out nn cxtx?nsivc invcst,ignlion of b o ~ ~ ~ ~ rlnycrs lnry which are el~ornctorizcdby ~ m n l lclcpnrtnrcs from t.11~nelf-ui~nilnr\vctlge-flow boutltlnry lnycrs of tho I'nlknrr-Sltan type. Rvidcnt,ly, R I I ~ I I1)ounrlnry Inyerrr nre n o longer nolf-~in1iln.r.
.
,I ltr I ) ~ I I I I I ~ * ~L o ~ .n~~ l i 1 , i rollow o n ~ Prom c(ln. (V.3) n l ~ t lnrc?: /' : 0 nl. o, 0, I / I 1tt1(1/" = 0 a(* 17 == w . 'I'lris i s nlso :I j)nrl,icrrlar caso of I,llo clasa of 'similar' s o l ~ l t ~ i ~ t t ~ consitlcred in C h a p . V111. ISquntiot~(9.12) is o b t n i n c d f r o m 1 . 1 1 ~m o r e gcncral tlifli~rc~tl-
-
First,, u p o n ~ n u l t ~ i p l y i ncgq n . (9.12) b y
1"
a n d i n t e g r a t i n g o l ~ c e ,\vc? I1:1vc
w h e r e n is a r o n s t n n t of i n t r g m t i o u . 1t.s v a l u e is zero, a s
7 b.
.-.
0, ntld tial e q u a t i o n (8.15) f o r t h e case of 'similar' b o u n d a r y layers, if we p u t a 4- 1. T h e e x a m p l e u n d e r consideration is o n e of t h e rare cases w h c n t h e s o l ~ t t i o rof'~ tllc botrndary-layer equation c a n b e 01)tdncd a n a l y t i c a l l y i n closcd f o r m .
f 00.
'1'1111s
T = d
Flow in n convergent channel
71
v;
-(,I
- --
/'
.- 1 ant1
/"
--
0 lor
--
- 1 1 2 (I* + 2)
T h e c a s e of p o t m t , i a l flow g i v e n by t h c e q l r n t h n
U ( s ) = -2L x
>0 is related to flows pt~t a wedge, a n d also l e a d s to 'similar' solutions. W i t h i t rcprcscnt,s two-dimengional mot,ion in n c o n v e r g e n t ohnnncl w i t h flat, walls (sink). T h e v o l u m e of flow for a frill o p e n i n g a n g l e 2 n a n d for a s t r n t , ~ ~ of n l ttnit I ~ c i g h tis ($ = 2 n ?I,, (Fig. 9.2). Int,rodncing t.he simi1nrit.y t,ransformat.ioti
w h r r e t h e a d d i t i v e c o n s t m i t of i n t r g r a t i o n is s e e n t o b c c q ~ l n lt o z e r o in v i r w of tile I m ~ n d a r ycondition /' = I at 17 = oo . T h e int,egral r a n b e rxprcssctl ill closrtl f o r m a s follows:
o r , solving f o r
1' = w/11: /'
Fig. 9.2.
1 % ~in n ronvrrgrnt rhnnnrl
=
= 3 t8anh2
168
c. Flow past a cylinder; syrnrnet,ricol case ( nle~iusnerir~)
IX. Exnot, soI~rt,ionnof tho ntrady-~t~atc boundary-layer equations
l6fl
rmmbrr of tcrms is rcq~rirctl;in fact,, h i r nr~rnbcris so 1:~t-g~ t h a t i t oeagcs t,o be ~rnctira1)loto t,al)ulaf,c them all with a reasonnl,lo :tnlollnt, of r~rtmcdcnlworl (30" possess a point of inflexion because they lie i n t h e region of increasing pressure. The di~tribut~ion of shearing stress ro = ( h l ay)o is plotted in Fig. 9.6. The position of t,he point of separation rcsnlts from the condition that. TO = 0, nnd is given by # s = 108.8"
4
Fig. 9.6. Variation of sl~earingstrew nt the wall over the circumference of a circular cylindor for a lnrnirinr boundary lnycr
0
172
d. Ronndnry layer for the potential flow given hy U(x) = Uo - ax"
1X. Exact m111ti0118 of tlw st.rady-state 1)oundnry-layerequations
wcrc t,erminated a t ~ 9 t1he , point of separat,ion would tllrn ollt LO If. tJle power be at, +s Iog.oo. Iktt,er accuracy can nowadays bc obtainetl with numerical
.-=
mct.l~ods,sco Sccs. JXi antl Xc3. 'rhc nccllracy of t,his r.ale~~lnt.ion I)nsod on 11 powcr scrirs can I)n t,cst,od for spced of convcrgcncc of t,hc omit,t,ctl Imrtlion of t,hc serics by invoking t,hc co?adilions o/ com.pnbiOility at, the wall. I\ccortling 1.0 ctln. (7.15), wc ~ n t ~ sIlnvc: t,
173
were mado the bitsis of his boundary-lnyer calculations. f l i s tne:~snrcment~s sltow&l s ~ p a m t ~ i oa nt 4, == 81°, whereas t,he calculation intlicnted!(bs = 82O. 1,ater 0. l~lnellsItnrL publisl~cdextmsive expcrimcr~t,nlt1nt.a on the l~rcsd+~rr d i s t s i l ~ ~ ~ t l'ig. , i o ~ 1~. ,1 0 , wl~ichpoint t o :I l:~.rgcinfl~tenccof tho Itryt~olrlsnurnl~cr.'Vor vnlrtrs of t.11(,Itc:~~t~oltls mlrnbcr hcloto Me crilicrd tJtc prcssI1rc r n i n i t n ~ loct:tIrs ~ ~ ~ i ~ . I r . t - : ~ (nrnr l~ (b - T o 0 , ~ I , I I ( I tho prr-ssurc is nrn.rl.y corwI,:~.t~l, over t.110wholo I I O I V I I N ~ , ~1o1.1ioo ~ ~ ! : / I I 01. I ~ J I (:ylitl~lt,l.. V For 1tc:ynoltls numbers above the critical t.hc pressure rni~tiln~rtn shifts tm (b -- go0 n11~roxi1nat.t ly, in agreement with t h e potential-flow thcory and, on the wl~olc,t,l~e prrssure distribution tleparts less from t h a t given by the potcttbial theory thnn in tho previous case. nct,wcen t,l~escvaltlcs, i. e. n m r a critical Rcynoltls n ~ ~ l n bof e rn~pproxi~nnt.clyTJ, I ) / v = 3 x 10" t,he drag cocflicicnt, of l.hc t:irc~ll:~rcylintlcr tlrc~rc~ascs n h r ~ ~ p t ' l(Fig. y 1.4-), and this phcnorncnon ir~tlicat~cs t01at t,l~cbor~ntlary Ia,ycr I ~ n s I)ccon~ctrlrbulent (see Sec. X V [ r I f ) . T l ~ elaminar hountlnry layer on a circt~larcylir~dcrwas rllso ir~vcst,ignI.c~rl I)y A. Thorn [G7], a t a Iteynoltls ~lrrmbcrZ J , I l / v = 28,000 m t l 11g A. F:~gc[lOlit1 the mngc U , L)/v = 1.0 t o 3.3 x 10% A ppncr by L. Sclrillcr nntl W. Lirlkc [54] cont,ains some considcrat,ior~sconcerning prrssurc drag ant1 skin f r i c t i o ~in ~ t.110 rcy$on of Iteynoltls m ~ m h c r sbrlow the crilicnl. In the r:lngc of l t c y r ~ o l t l11111111)(:rs ~ from about GO t o about 5000 there exists b c l h d the cylintlcr a vortex stxcct w l ~ i c l ~ sl~owsa regular, periodic structure (Figs. 2.7 and 2.8). Tllc frcqucr~cya t which vortices arc shed in this so-callcd von I 0, decelerated flow). 16 is now --
.-
t,o stipulate a powcr srries
-
When cqnation (0.22) is written in the form U ( T ) -- l J , ( l -x/L) for 71 = 1 , it can also be intarprcted as represcnt,ing bhe potcntinl flow along a flat wall wl1ic11~tn.rtsat x = 0 and which a t ~ n on t ~ to ~ anothrr infinite wnll a t right nnglcs In it at, r L.Jt is of the snnlc ~JIIIC as the cnsc of decelcratrd stngnat,ion flow shown in Fig. 2.17, thc stngnnt,ion ]mint bring nt z .-: L.
-
174
TX. Exnct ~olut,ionuof tho stondy-state boundnry-lnyrr eq~latioris
e. Flow in tho wake of flat plxtc nt zero incidoicc
in r* for the ~ t r c n mfitortion in n mnnncr ~irnilnrto thc enao of the cylinder, Scc. TXr, the cocfficirnta being functions of y:
1 I ~ n c cthe vrlocit,y of flow becomes
175
and tlorolcrnt~edflow are uenn plot8t~cd in Fig. 9.8. 11. ~ h t m l dbe noted t,Iint.nll p r o f i h in tlccclcrntcd flow have n, point of inRexion. D. It. IInrtroo [38] repcntcd tl~csocaloulntior~snrid obtninctl good ngrcemc~lt,wit11 L. Hownrth. The case for a / i J , 0.125 wns rnloolnted more ncc~tvnt.clyby 1). C.F. Ileigll 1441 who ~ 8 nn~ clecl.ronic d digit.al computer for t,hc purpose nnd who pnitl ~ l ~ c c i a l nL,tentio~~ to tlic region of scpnrntion. TIICvaluo of the form fnctor a t l.110 point of sepnrat,ion il.srlf wns founrl t,o ho x* = 0.1198. ,llic nictliod ornploycd by L. JIownrtl~was cxkt~tlnrlby I. 'l'ntli 1001 t.o ir~cluclcI.lw caws corrcspontling to n 2 1 (with a > 0). tiowcvcr, I. 'l'nni did uot publisl~nny t.nbles of the furwLionnl roeflicicnts but confined liitnsclf to reporling lho l i d rosnIL for n = 2. 4 nnd 8. 111 Iiis cnsc, (no, MIC poor ronvcrgcnro of tlic ~ c r i c sdid not pcr~nithim 1.0 dotcr~ni~ic the poiut of sqi:witIIIIWT~V:~ , tion wil,l~unfficic:nf, ncrurncy and 110 formrl himnrli rowpell~cl Lo I I U C I,. Il(~wnrl.l~'s f w ~ ~ l i ~ ~ ~R ~ tCi Il Ii W o nI I ~ ~ .
-
.
Tnl.rothring t,I~rsevnlnes i n h t.hr rqunt.ions of motlion (9.2) and comparing coefficients we obtain a ~yutcni01 ordinary dilTcrontia1 equ:tt.ionu for t.ilo F I I I I C ~ ~ O I IfRg ( r l ) , lI(11), . . . . Tho first t h o of Ll~cucare: lof"-1- I0 10 " 0 ,
-
/['" -1- 1,
- 2 lo' -k 3 /,>" = - 1 , fz"' -1- 1, 12"- 4 1,' 1', -1- 5 1,'' 1, = - 4 + 2 1,'"
3 I,
I,",
e. Flow in the wakc of flat plate nt zero iucidence
Only tho first cqnnt,io~~ is non-linmr, nntl i t in idcnt.ical wit.11 tlint for n flat plats nt zero incidence:. All rornnining equations are lincnr nnrl contain only t,he function f, in the homogeneotls portion, wherons t,he non-liomogeneous b r n w arc for~nrdwit,lt t,Iie nid of the remaining funct~ions 1.. I,. flowarlh solved trho first. scven tliKcrcntinl eqnations (up tO and including I,), and calctllatod t,zblcs for Llicm. 'rim ucricn (9.25) converges wrll with t.hcso valnrs of I,, in t.he rnngr - 0.1 _< x* 5 -1- 0.1. Jn tllc casc of decclorntctl flow (x* > 0) t,l~cpoint. of scpration is a t z* = 0.12 npproxi~nntrly, I)ut for thc sliglit.ly cxhndetl rangc of valnns t.ho convcrgor1r:o of the scries (9.25) is no longer wsured. 111ordor to roach t.lrc p i n L of separnt.ion,-I,. 1Iownrlh used a nninericnl proccdum for tho ronl.innnt.ior~of the no111t.ion.V ~ 1 ~ r i t .profilrs y for sevcrnl vn111csof r* for hot,h a c ~ ~ l e r n t r d
The application of the boundary-layer equations is not rcstrictml t o rcgions n m r a solid wall. They can also he applied when a stratum in which thc infltwncc of frict,ion is rlominating cxists in the interior of a fluid. Such a case occurs, among ot.l~crs,w h c n two laycrs of fluid with tliffcrcnt vclocitics mcct, for instnncc, iri tho wake bcliind a body, or whcn a fluid is tlischarged through a n orifice. We shall consider three examplcs of t,his typo in the prcsent ant1 in t h e succccding scctions, and wc s l ~ a l lreturn to them whcn considcririg turbulent flow. As our first examplc we shall discuss the case of flow in the wake of a flat plate a t zero incidence, Fig. 9.9. Behind the trailing edge the two vclocity profiles coalesce int,o one profilc in t h e wake. I t s widt,h increases with increasing distancc, and its mean velocity decreases. Tlie magnitude of the dcprcssion in t h e vclocity curvc is dircctly conncct,cd with tho drag on tAc bocty. On thc wholc, howcvcr, a.s wc shall see later, the velocity profile in t h e wake, a t a large distancc from t h c body, is intlrpenrlent of thc shape of the body, cxccpt for a scale factor. On the other hand thc vclocity profile very closc t o thc body is, evidently, detcrmiricct by t h e boundary layer on tho hody, and its shape dcpct~dson whcther or not thc flow has separated. The momcntum equation can be used to c.alculatc thc drag from the vclocity ~wofilcin t.hc wnlro. For this j n ~ r l ~ o swc c draw a rcrtarigr~larcontrol snrfacc AA, 13113,
Fig. 9.9. ~\pplirnt.ionof the niomcntun1 equation in tho calculation of the drag on a flat p l a k nt zero incidence from thc velocit,y profilo in the wake t
r .
J ho in&ycntlrnt varin1)lo
in Lhr nhovc r q u n t h s difkrs from that in Chap. VIT by
R
factor
1.
176
e. Flow
I X , Nxnrt ~ o i ~ ~ l iofo t~ ~I I sC~~cndy-state boundary-Inyer rqrmtiona
as shown in Fig. 9.9. The bonndary AIBl, parallel to the plate, is placed a t such e distancc from the body t h a t i t lies ovcrywhere in the region of undisturbed velocity,
Croswxxtion
ill
l l ~ ewake of flnt plntc at mro incitlrncc
I
I
Rnte of flow
177
Dlonient~~ni i n dircclion r
I/,.
Purthorrnorc, t,hc pressnrc is constant over the whole of t,he control surface, so t01at j~rcssurc forces (lo not contribute t o the mornenturn. When c a l c ~ l a t ~ i n g the flux of momontunl across the contml surfacc i t is necessary t o remcmber that, y owing t o ront,innity, fluid nu st loxvc t,l~ronghtho hountlary A I B l ; tho q ~ l a n t i t ~of A 1 l l l is cqu;rl t.o tho tliffcrent:c I)clwccn t h t ontcring Lhro~rglt fluid leaving Ll~rongl~ A I A and loaving through BIR. 'rho boundary AT3 contribntcs no term t o t.hc nom men tam in the x-diraction becanso, owing t o symmetry, t h e transverse velocity vanishes along it,. The momentnm balancc is given in tabular form on the next page, and in i t the convc~ltionis followed t.11:bt inflowing masscs are considcrcd positive, and ontgoing masscs arc taken t,o bc negative. The width of the plate is denoted by b. 'l'hc tot,al flux or morncntnm is cqnal to t h e drag D on a flat plate wetted on orlo sitlc. 'l'hus we have
C -- Control srlrfnrc
2 Rnte of flow
=
0
::Mornctit~~m flus
-=
Drng
03
D
=be/u(~,-u)dy. ic nntl highrr t~crtnsin is stnall rotnparctl wit.11Urn,so thnt q~~n.tlrn,t
v-0
Intrgration may bo prrformctl from y = 0 t o y = oo instcad of t o 2/ = It, because for ?/ > h thc intcgrantl in eqn. (9.26) vanishes Ilrnce thc drag on a plate wetted on both sides bcromrs
+
/
2D =be u(u,-u) -m
dy
.
(9 27)
This cqnat,ion applies t o a n y symrnet,rical cylintlrical body ant1 not only t o a flat plat,o. Tt is t,o bo rcrncmbcrctl t h a t in t h e more general case thc intcgral over the profile in t,he wake must be t,aken a t a sufficiently distant sect.ion, and one across whirh t,ho st.at.ic pressure has it.s undisturbed value. Since near a plate there are no pressure tlill'crrnccs cit,hcr in t,l~elongit~ldinal or in t h e transverse direction, ccln. (9.27) npplins t,o any tlist.ancc brhintl the platc. Furthermore, eqn. (9.27) may 11c: nppltc(i t.n a n y section x of tlhc I)o~lntlarylayer, when i t gives t h e drag on t h e portion of t-l~c plate between the leading ctlgc and tlltat sect,ion. The physical meaning of tho ir~t~cgml in eqn. (9.20) or (9.27) is t h a t i t rcprcscnts tho loss of momentum due t o frict,ion. I t is itlcntical with the intcgral in eqn. (8.31) which dcfir~edthe mome?ltum thickness a,, so t h a t eqn. (9.26) can he givcn tllc alternative fbrm
Wc shall now proccrd to calculate tthc velocity profile in the waltc, in particular, dist.ance x t)ehintl the trailing edge of the flat plate. The calculation must bn p(:rformcd in t,wo sLcps: 1. Through an expansion in thc downstream direction from I.he Irntling t.o t,hr tmiling ctlgr, i . c. I)y n ~ : ~ l c u l a t i owhich n inv?lvc:s thc cont.inu:~.t.ionof t J ~ oIllilsius profile on thc plalo near d.hc tmiling cclgo, antl 2. Through a n expansion in t,hc nl)st,rrarn direction. 'fhe lattw'is a kind of asymptot,ic'int,egration for x Inrgc tlistancc behind t h r plate and is valid irrespective of the s h p e of the 1)orlp. It. will 1)c nrrrssnry hrrc 1.0 n ~ n k clhc nssrrmpt,ion t,llat t.he vc1orit.y difference in t.11~wn kc (0.29) 711 (", !/) ' U , - - u ( z , y) 9.1, a. large
711 IIIIIY
hr 11t~gltv~t.rt1.
,I ,l ~ c~ ~ o c c ~ lmnltrs u w 11xc o f n nict,l~otlol' c:o~~l,inuir~g n. Iznown solul.ioii. 'l'ltc~ (:ILI(:uInt,ion st,arts with t.11~ p~viilea t the t.miling ctlge, calculnt.ct1 with 1 . 1 1 ~aid ol' Jllnsius's
~ncthotl,and we sha.11 refrain from furthrr disrussing it hrre. 'I'hc asympt,ot,ic cxpmsion in t.he upst,rraln direction was calcnlatcd by W. Tollrnicm 1091. Sinrt: i t , is t,ypical for problems o F flow in t,hc wake, antl since we shall m d t e nse of it in t,hc more ilnport,nnt, tmbulcnt case, we propose t o devot,c some t,itnc t,o an account, of it. As t h r prrssnre trrm is r q r ~ a to l zero, the bonntlary-layrr cynntiot~(9 2)rombinetl wit11 rqn (9 29) gives
'I'he partial tlilli:rr~~t.inlcqunt,ion call, here 1.00, be tmnsformctl into a n or(li11iir.y tliffcrcnlinl ecpat,ion by n snit,a,blc?t,mnsrormnt,ion. Sirl~ilnrlyto 1 . 1 1 ~assuml)tion (7.24) in 13lasirrs's mct.l~odfor t,hc 11x1 plate wr put.
antl, in adtlit.iot~,wr assnme t.hxt( u, is of' the forin tl1
= U-c
(-;)-kg(,]),
whew 1 is the lrngt,ll of t h r platc, Fig. 9.9. Tho power -- .j for 1:i n eqn. (9.31) is just.ifict1 on the ground t h a t the ~no~nent.urn , intlrpondrnk of r . int,cgrnl whicll givrs t,hc drag on tho plnt,c i l l oqn. (!1.27) I ~ I I S ~I)r
IX, Exact solutions of tlrc steady-statc bonndary-layer equations
178
Hence, omit,t.ing quadm.t.io terms in 1 5 , the drag on a platc wetted on h t , h sidca, a s givrn in eqn. (9.27), is transformed t.o
difference in t l ~ cwakc of a flat platc a t zero incidonrr becomes
+m
2n=beCJ,/u,dy. y--m
.
,I Ito volocit.y clist.ril)~tt.ion given Iby this n.syrnplotio cclllnLion is rr:prrsc:nt,otl i t t I'ig. !I. 10.
lnt,rodurir~g,fttrt.llcr, t . 1 ~assumpt,ion (9.31) into (9.30), ant1 dividing t,hrough by (I,2 . (x/l)--lIzz-1, we obt,ain the following tliffcrenti:tl cquation for g(t1):
C
JI"
1-
4
71
JI' -1-
hq
=7
(0.33)
0
with the lmnntlnry conditlions 0' = 0
at
11
=0
and
Integrating onre, we have 0'
I
:
71g
JI
-- 0
at
71
= co .
f. The two-dirnensionnl larnir~nrjet
--0,
\
whoro flrc rorlstnnt of integration vanrshes on account of tho t w ~ n d n r ycondition a t q = 0. Rcpcatcd integration gives the solution
g = exp ( llerc ma& from from
'1 ?12).
I t is remnrkablr t h a t the vclocitty distxil)nt.ion is identical with (::~uss'.s c:rror-tlistribntion function. As assumed a t the boginning, cqn. (0.35) is valitl only a t grcnt, distances from the platc. W. Tollmicn verified that. i t may bo nscd a t about z -- 1 . ]pig. 9.1 1 corlt,nins n plot, from wllirh tho wliolr vc:locit,y.lit*ltlrnn IN! ittliv.r.t~tl. Thc: flow in tllc \ d t o of n platc as wc-ll a s in t l ~ a tbc:l~intl any othrr body is, in most cases, turbulent J5ven in the case of small Itcynoltls n n n ~ h r r s say , R, < 106, w11en the bountlnry laycr rcrnains laminar a s far a s tho tmiling cdgc, the flow iri t,Ile waltc still bccomes f u r b ~ ~ l c nbecause t, the vclocity prolilcs in the wnltr, all of which posscss a point of inflexion, arc c ~ t ~ r c m c l~~nstnI)lc. y I n o t h r r wortls, cvcn with c ~ m p a r a t ~ i v esmall ly Rcynolds numbers tho wakc 1)ecomes turbulent.. 'l'ur\)ulent wakes will be discussed in Chap. XXIV.
(9.34)
The efflux of a jot from nn orifice affords a furtllrr oxample of motion in tho abscnco of solid boundaries to wliich it is possible t o apply the boundary-layer theory. We proposc t o discuss the two-dimensional problem so t h a t we shall assume
the constn.nt of int.cgrat,ion n.ppcaw in Lllc form of a cocffcicnt and can be cqr~nlt o unit,y without loss of generalit,y, a s the velocity distxibution function u , eqn. (9.31) st.ill contains n free coefficient G . This constant C is determined the condition t.hat thc drag calculated from the loss of morncr~t~um, eqn. (9.32), ,. (7.33).
I~nntl,from cqn. (7.33) we cnn w h c tlown tho skin fric+t,ionon n. plntc I wct,t,otl o n I)oll~sitlcs in the form:
011t.11r: ot.llnr
vclocitydistribrttion Fig. 9.10. Anyn1pLot.i~ in tho laminar wake bohind s flat plate, from erp. (9.35)
Fig. 9.11. Velorily distribution in t l ~ cla- t minar wake l)cl~intla flat platc at zcro innidenco
180
I X . Exact solrltions of tllc stcntly-stato boundary-layer equations
t h a t t , l ~ ejet cmcrgcs from a long, narrow slit and mixes with the surrounding fluid. This 1rol)lom was solved by 11. Scl~lichl.ing[60] and W. Biclrley [3]. I n practicn, in this case, ns in t h e previous ones, tho flow becomes tl~rbulent,.We slinll, howevcr, discuss hero the laminar c:we in some tlet,nil, since the turbulent jet, wlticll will be oonsidcretl later, can be analyzed mntllcmaLically in a n identical way. Thc emerging jet carries with i L some of the surroutttlit~g Iluitl whicli wns originally a t rest becauso of the fridion developed on its periphery. The resulting patt.ern of strcarnlines is shown in Fig. 9.12. We shall adopt a system of coordinates wit.11 i1.s origin in Lhe slit and wit,l~ita axis of abscissae coinciding with the jet axis.
I . 'Chc flux of t n o n ~ c n t , t ~inr ~fhc ~ z-tlirwt,io~lis i ~ l t l o p ~ t l c noft , r, at:c.orcli~~g to rqu. (936). 2. HI^ :~ccclrrnl,iot~ t~(~rttis :IIIII LIIC I.rivl,iott Ltvm in C I I I I(U.2) . nrt: 01' n~:~gnitu(lc.
of'
1,111:
WIIIII*
0r111.r
(:or~sr:cl~~c~~t.ly, the assumptions for the iritlrpcr~tlcr~t~ vari:ll)lt: a t ~ t l for the st,rcntn func.t,ion can be writtcri as
if s ~ ~ i t ; t b lc:onstxtlt c fa,c:tors arc i t ~ t : l u ~ l r'I'l~rrrlnrc., ~l. t,lt(. vc,l1)c.it.3rt ~ o m ~ ~ o ~a r~r t ~ r ~ t s given I)y 1,llc f'ollowing expressions:
The jet spreads outwa.rtls in t.hc tlowr~stmamdiroct.ion owing t,o the influence of frict,ion, whcrc:w its vc?locit,y in t , l ~ ecetrtm decrcascs in t h e same direction. For the saltc of simplicit.y we sllall assume t h a t the slit is infinitely small, but in order t o rt!t.:lin a finite volnrnc of flow as well as a finite m o t n o ~ t u m ,it is necessary t o nssumo a n infir~itcfluitl vel6oit.y in t l ~ cslit. 'l'l~cprcssurc gratlicnt tlpltlx in ~ J I C direction can Iicrc, as in t.he previous cxan~plc,be neglected, bwnose the constartt pressure in 1 . 1 1 surrounding ~ fluid irnprcsscs itsclfon the jet Consequel~tly,the total n i o m e n t ~ ~ m in t.he r-tlirrct,iou, clcnot,c:tl I)y J, must, remain const8ant arid intlcpo~~dnnl of 1,he distance r from tho orifice. Ilcnco
11, is ~)ossil)lct,o tnnlzc a snit.;~l)lcassumption regnr(ling blic velocity distribution if i f , is ror~sitlcrctlt11:tL the velocity profilcs ~i(r,y), jnst :IS in the oasc of a flat plate a t ZWO inci(lcnce, arc most prol)al)ly sinli1a.r. 1)ecnuse the problem as a wltole possesses I I O ch:~mct,nrist,ic! l i ~ ~ e ntlimcr~sion. r \Yo shall ass~!mc:, t,hercfore, t h a t tho velocity u is a fi~nntiot~ of ylh, where h is the \vitlt,li of t,he jet, suitably defined. We shall also nssumc t,l~:tt.h is proportional 1.0 x*. Aacortlingly we can write the strcam fut~ction i t 1 t,hr lorm
I whcrc OI is n lice constant,, l,o be clot~crmittctlInter. 'l'hus t01c a.lmvc c q ~ ~ : ~ t 1,r:lnsfolms .iot~
K. Pnrnllcl &reams in hminar flow
and t,llc?clnsh now clet~otostlilTrrcntia.tion with rcspcct, t o (. T h c boundary contlitions art? ( ~ 0 F: = O ; t=oo: (0.41)
and, hencc, for tlio volocit,y distribution
r=O
whcro t . 1 ~consl;~rtl~ of inl.rgr:it,iotr was m:itle cqunl to I. 'l'llis li)llows if we p t Ff(0)- 1 , wlticlt is prrnlissihlo wil.ltottt losx ol'gc~lcrnlil,yImnnsc of I,llo frcc cotlst.:inl. a in t,l~c rrlat.icin Iictwoen f ~ n P. d 1Cq11atio11(9.42) is n clill:rrnt.ial cqn:tt,ion of 1tic:t::~t.i'~ typc ancl can Iir int.rgrat.ctl in closctl t,rrms. \Ye oli1.ni11
r
I h c transvcrsc: vclooil,y a t thc bountlnry of Iht. jet is 7
F
-1
00
ant1 the volume-mtc of discltxrgc per unit height of slit bocorncs Q = e J v (I!/, or -m
I 11vr1t ing this rqnnt ion wr obtain
I. =t,anh Since, furt,llcr, tlP/tlE q n . (9.37) and is
-
I
-
1
E=
1 - exp(-BE) - - 1 4- cxp
(3s)
'
1:in11~ E, the vc1ocit.y ( l i ~ l r i l ~ ~ t l(:all ~ i o nI I tloclucctl ~ from -
r
(I
t . a n 1 16) ~ .
(9.44)
. I
1.he vrlorily tlisl.rilwtiorl from cqn. (9.37) is soon plott.ctl in Pig. !).Is. 1L now rcn1:tins t,o dc(.crtninc. t81rc const:tn(. a , :LWI this ciln be (lone wit.11 t h e aid of condition (!).:3R) wl~ichs h t c s t h a t t,l~crnomcnl,um in 1 . 1 1x-tlirrcl ~ ion is ronst,nnt.. ( h n b i n i n g rqns. (9.44) :111(1 (0.36)we o b h i n
w e shall assume t h a t tho flux of momontum, J, for thc jet is given. It is proportional t o Lhr excess in pressure with which the jet leaves t h e slit. lrrtrodricing t h e kinematic mo~nenlmt.I/@ = K, we have from eqn. (9.45)
Tlic volumc-rate of tlisclmrgo increases in the tlownstrcam direction, bccai~sc:flnid particles are carried away with t h e jet owing Lo friction on its boundnrics. I t also increases with increasing momcnt~um. The corre,sponcling rotationally symmct.rica1 casc in which the jet cmcrgcs from n small c i r c ~ ~ l aorificc r will be tliscussed in Chap. XI. The problem of t,hc twonarrow slit was solvctl Iiy dirne~~sional laminar compressible jet cmcrging from S. 1. P.zi [4!)] nntl M. Z. JZrzywo1)locki [42]. 1n.rnina.r Moasurcmrnts performed I)y TI:. N. Antlrntlo [I] for tho t,wo-tli~ncnsiot~:~I jct confirm t.he preceding thcorct~icdargurncnt vory well. 'l'llo jct r c n ~ a i l ~laminar s n p t,o R 30 a p p r o ~ i r n a t ~ r l where y, the Ibynoltls number is rcfcrrctl to thc cfflrrx vclority and t o the widL11 ol' tho slit. Tho casc of a Lwo-tlinlensional ant1 t.llat of .z circular trtrl~ulentjct is discusscd in Chap. X X I V . A comprchensivc review of all probloms involving jets can be found in S. I. Pai's book [49].
-
g. Pnrnllel streoms i n laminnr
i n x t,\\o-rJimrt~. Fig. 9.13. ,VrIcirit,ydist,ril~ulio~~ sion111nn(J cire~ll~wfrcc jcL fro111 cqns. (9.44) :md (11.16) icspect~ivcly.For tho two-tlirner~. / ~ for , the xionnl jct [ = 0.275 KIP y / ( v ~ ) ~and y/vz. I< and K' circnlar jct. C 0.244 t1twot.c: Ilir kincrnat.ic monwnt.um J / e
-
(!).48)
Q = 3.3010 (I< V X ) " ~ .
h w
Wo shall now 1)rirfly cxnminc the laycr 1)ctwccn two pnrallcl, Inminnr sl,rcnms which move a t tlifTercnt vclocitics, xntl so provitlc a h t r t l ~ c rcxnrnplc of the npplicability of the bountlnry-laycr equations. Thc forrn~~liition of thc problctn is scot1 il111sLraLctlin Fig. !).14: Two it~il~ially scp:ir:~Lc(l,u t ~ d i s O ~ ~ r Iprnllcl ~ ~ x l , H L ~ ~ ! I L I I I S whith move with t h e vclocit.ics TJ1 nncl (I,, rcspcctivcly, l~cgintm intcrc& thro11g11frit:l.iorr. I t is possi1)lo Lo assurnc t h n t the transition from the vclociLy U , t o vclocity ( I , talccs in n narrow zone of mixing and t h a t t h e transvcrsc vcl&ty component, v , is everywhere smalc oomp,zrcd with t h e longitudinal velocity, 11. Consequently, t h e boundnry-layer equation (9.1) can be usctl to describe the flow in thc zoncs I and 11, and t h e pressure t1crm may be omitted. I n n manner analogous t o t h a t employed for thc boundary layer on a flnt platme (Scc. VIIe), i t is possible t o obtain t h e ordinary tliffcrentinl equation
184
IX. Exact solutions of the steady-skate boundary-layer equations
11. Flow in the irlet lengt.h of n strnigl~tc h n n ~ ~ r l
185
1/
by int'roduring t,hc dimensionless t r a n s v e r s e coordinat,e 9 = y lJl/v z a n d tlte t = / I , w e a r e led t o t,Iic b o u n d n r y s t r e a m f u r ~ c t . i o sy~ = v V 1z /. A s s u m i n g t , l ~ nIL/U contlit ions
1/
IICCXIISC Y) =- 0 t,l~cre.T h e s o l ~ l t ~ i oofn t h e dilTerential e q u a t i o n (9.49) s u b j e c t to t h e b o u n d a r y contlitior~s (9.50) a n d (9.51) c a n n o t b e o b t a i n e d i n closed f o r m , a n d a numerical m c t h d n i m t b e employed. It is possible t o o b t a i n e x a c t n u m e r i c a l solut h s I)y t h e IISC of a s y m p t o t i c e x p a n s i o n s f o r 77 + - co a n d 17 -+ -1- cro togetfher wit.11 a series e x p a n s i o n a b o u t r] = 0 ; s e v e r a l s u c h solutions were p r o v i d ~ dby R. C. 1,oc:Iz 1451. 'f'hc prthlcrn w a s first, solved by n ~ l m e r i c a li n t e g r a t i o n by M . 1,essen [44a] st,art.ing with a n nsymptot.ic expansion for r] -+ -00.
.J.l w tlia.gr:~rnin
Pig. 9 . 1 4 prc.scnt,s v o l o ~ i l ~profiles y for I = U , / U 1 = 0 a n d 0.5. A n irnprovcd ~ ~ u m e r i c solution al w a s p ~ ~ h l i s h cby t l W. J. Christian [lo]. T h i s special cnsc of t h e int.eract,ion l ~ e t ~ w e enn wide, l~olnogericousj e t ancl an adjoining m a s s of quiescent, a i r is o f t m tlescribcd b y t h o t e r m "plane half-jet".
R. C. J.oclr [45l s t u d i e d , i n atltlit.ion, t h e case wl1e11 t,hr t.wo half-jets differ in t h e i r clensit.ics ancl viscosit,ies, a n d riot o n l y in tllcir velocities. A n e x a n l p l c of stlch :I case is t,lrc flow of a i r o v e r a wnt,cr srlrf:~.cc.T h e solution n o w tlrpcntls o n t , l ~ cp:lr:Ltnctcr x -- I,, p2/p1p1 in atltlit.ion t.o I. Lock provided sevcrnl c x n c t solut.iotls n s wcll a s solutions which were l)ascd 0 1 1 t h e rnoment.um int.cgral r q u a t ~ i o n An . approsim;~lc mc.t.llotl w a s also conccivc.tl 1)y 0 . It~~[~r(~~sil~Ic*, nry-lnyrr Ilnw nt the nt.ngnnlion point, of n grnerzl t~ocly.Archiven of Mecl~anics(\Vatsaw) 26, 46% 478 (1!t74). ofirl.s O I I t.11~ 1.l1rcr-din~rnsior~nl second order Im~nclary1481 I':q~rnft~ss.H . I).: Mnss-t.rn~~sfer I:tycr flow nl. 1 . l ~ :s t q ~ ~ i hl~oint m of hlunti bodicn. Mecl~.Rcs. (!OIIIIII. I , 286 - 2!)0 (1974). [4!)] I'ai, S. I.: I'l~lid dynnmics of j r t , ~ .1). \'a11 Nost.rnntl ( ! o I I ~ ~ INew ~ I I ~York, , 1954. 1601 I'ol~ll~nuwn.I wl~ena ~ l v ~ r ps rcc s s ~ ~gr:dicnf.s rc exist, the flow is aln~ostalways turbulent l ) r c n ~ ~ sin c , atlclilion, f l ~ ocxistcncr: of an nrlvorsc prrssure gradient favours the transition from laminar 1.0 L11rh111c:nt. llow. 1t is, ~ ~ c v c r t l r c l11sef111 ~ ~ ~ s ,to clarify some of t,he f~mdamcntalrelations :wso~~i:~lcd o.it.11 tho p r o v o ~ t l i oof ~ ~ srp;;rntin~~on f l ~ ccxarnplc of I:m~itrnr flow, in particular, 1.0 n~atlrcmai,icnltreatment than is tho 0cw11sr I : I I I I ~ I Il10ns : I ~ i ~ r c1 1 1 1 1 ~ 1 1Inore rrarlily :I~II~II:II)IC t, wsc wit11 I ~ ~ r h l r nIlnws. 'I'l~vrr arc. svvrr:tl ~nrt.l~ocls of prcvcnt,ing scpr:tl.ion. The simplcst of t,lrenr consists in v g r i ~ d i r ~lo~ ir.r~n ~ i ~ ihrlow n the limit for wlrirl~ncpnrat,i~n : r r r : ~ n g i ~for ~ g I I I V ; ~ ~ l v r r s1)rrss11r(:
22 1
I'olloaing 1,. 1'randt.l [I61 we sllall show how it is possil)lo 1.0 cni.imaLc t.lw pcrn~issil,lc rl~ngnilntlcof the :~dverscprcssure grn.dicnt for wl~irlrscjmrat.ion is jnst. prcvcnt.crl. 'l'llc a r g ~ ~ m r n t will he I):~.srdO I I ~ , I I I ?von I ~ & ~ ~ I I ~ ~ I I - I ' ~ I I I I I ; L IaI ~~~ I~I ~ ~ r o x i disc~~ssrd ~ ~ ~ : ~ tinSco. i o t ~ XI). 11, will h c : ~ w ~ ~ n i r ~ l l l ~ : i t .t l ~ rI ~ o I I I I l~ i: i ~y ~~ in ~iwlwl 1111nt1 I1.y t.11~: 111~.xwrc: didril~ution(Irt(.r~ni~~c.(l I),y t110 11.(~.51 ~I.:IIII poLr111i:rI flow 111) 1.0 :h point, which lirs w r y clclsr in t,lie point, of scp:wation, sue11 as point80 in Fig. 10.14. St:rr.Ling with this point, it will I)c assumed that i.hc pressure gratlicnt is srtoh that t.110 s11:1pcof I I I C vnlnrit,,y profiln T O I I I : I ~ I I R I I I I C I I R I I ~ ( ~~)rocrc~ling ( I ~ W I I R ~ ~or~ trIu~t,, : L I I Iin, oI,11rr \ v ~ r d s , - 10 will tho f i l q " : C:~f:lorA rc:mainn ronst:~nt.;fiincc aL sc:lr:~ral.ionA .- - 12 a valnc of A be cl~osrn.As seen from 'I':tl)le 10.2 this lcadn t.o a definite value for the second slrnpr factor, narncly I( = - O.l:169, so L11:lt Il'(K) = 1.Tr23. Using tl~csevalurs it is sccn from rqns. (10.28) and (10.29) that prevention of separation i~npliesthe following relationsliip between the vclocity U(z) of potential flow and the momentum thickness d,(x): -7
8,' =z= -0.1369 -
-V ( x )
v
I t follows that dZ/dz
=
.
0,1369 U"/U'2, or
Fig. 10.14. Devclopnrent of boundary I:~yerin thr! case when laminar separation is prevented
Fig. 10.15. I'otcntial velocity fi~nction for n laminar boundary layer with and without separation
On the o t l w hand the succeeding vclocity proflcn are given by tho ~non~entutn rquation (10.36) for
3.
=. 0, or
(le U --dz
I'j s
-V
R
5
I
Fig. 10.13. Scpar:~t,ion bul)l)le in a laminar bountlsry li~yornfiar I. 'l'nni 123). a) Shape of bubble (nchcnratir): b) l'rcssl~rodistrib~~tion in hnbhle along t l ~ ewall (w!w matic). 'l'hc nrc'snurc hetwoen S and V in tlro I)r~hhlo
= F ( I < ) = 1.523,
(10.40)
-
~ r l ~ c the r e nnmeriral value for F f K ) which corrcsnontls to A - 10 Itas been irrscrlctl. From cqn" (l0.38) ant1 (10.40) i t follows that the value of the sh:tpc fart,or rrrnains constant a t A -: - 10 if 0.1369 n = 1.523, or if U U" 0 = = 11.13 z 1 1 , (10.41) \
r
- -
U'=
a > 11 : no separation;
n
< I l : separation .
(10.41: I )
.Llrc . preceding argument slrows that t l ~ eboundary I:lycr can support, Llre atlvcrsc prrssurc gra-
t 11, n.it,lr 11ienl.q if o > 11, wlrereas n < 11 in~pliessrparation. If a rrnrainn r o ~ ~ s l a t~ n~ = A -- -- 10, t.11~ I)ou~~tlary lnyrr rcm:lins ~ I t.lw I vrrgc of scp:~r:rt.ion.
222
X. Approximate methods for steady eqnations
Qnalitnt.ivcly i t is a t once possil)lc to mnkc the following nt~atcmcntregarding the shape of [,he potential velocity function U(x) which Icatls to no xcpnrntion. I n viow of cqn. (10.41)
U" > 0 is a nrccnmry condition for n rctnrtlrd flow (IJ' < 0) t o xrlhcro t,o the wall. I n other words, t.11~ ~nn.gnitudeof the advcrsr pressure grntlicnt, n ~ r ~ tlccrrnsr st in t,lie flow direction. Ii'ig. 10.15. ,J hns scpnrntion will nlwiiys occllr il' the f~~r~t:l,ion I/(%)iu cx~rvctltlownwnrtln 1)chinrl its mnxi~nnm (11" < 0). In the opposit.e rase, whim tho vc1ocit.y function c i ~ r v r si~pwnrds(U" > O), srjmration tnny he ol>viatetl. 15vcn the limiting c,we of IJ" = 0,i. e. the rase of a velocity which tlccreanea lir~carlywith the length of arc, always Icatls to separation. Thin latter remark agrees with the rmitlt fonnd in Src. I X d ; i t was conccrnrd with the boundary lnyer aaaoriatcd with a potentinl cquntiona wm q u o k d flow vc1ocit.y which dccrcnwd linearly, and the solution of thc tlil~rret~tial from a pnpcr by I,. Ilowarth. The su//icient condition for the absence of aepnration iu givcn by
lly way of n further cxam~)leof retarded flow we shall ronnider the flow t.hrorrgh a divergent chnnncl whose walls nro straight. This ca.w in corollary to the cum of the houndn.ry layer in a divcrgmt chnnnol trentcd in Sec. IX b. The flow is nccn sketched in Fig. 10.16, where x tlcnotrs tIw rndial tlislnncc frorn t.11~ nourrc a1 0. The wall is nsann~ctlto I q i n nt, x .- n \vhcrc the entrnncc vcloril,y of the potrntinl strrarn is put cqnnl to U,. The poknlinl flow in givcn by
%
Computing thc v n h c of the qnantity a from cqn. (10.41), which is decisive for separation, we obtnin here o = 2. Applying the criterion given in eqn. (10.4111) we cor~cludethnt, scpnration occurs in :ill cnscn irrrnpcctivc of the m e g n i t ~ ~ dofe t.ho nnglo of divrrgence. This oxnmplc RIIOWR very clrnrly thnt c lnminnr nt.rm~nhas only n vcry limitctl cnpncity for nnppwting nn ntfvrrsn prcrrsnrc grntlirnt without ncpration. Acrording t,o a c:alculation pcrforrnctl hy K. Pohlhnuscn [In] tho point of ~cpnrnt.ionoccurs n t xr/rl = 1.21 nntl is sccn to be indcpentlcnt of tho anglc of divcrgence.
-
Wo RIIRII now procrcd to cnlcnlntc tho potential flow and the varintion of hountlnry-lnycr 1imit.ing cane of o I I, whrn tho bonntlnry lnycr re~nninu thickncus which are wnorinktl wiLh t,I~r on t,ho verge of sepraLiot~.I k w r cqn. (10.41) we Imvo
U' U" u, -- I 1 x -
....
u
or, npon intrgrathg: In U' -. 1 I In 11 -I- In ( - C,'), i. r . IJ'/CJ1I tho constant of integrat.ion. ltcpcntccl i n t a g r n t h ~~ i v c s
=
- C,', whero 6,' denotes
Fig. 10.16. J,nniinnr honnclnry layer in n tlivrrgent chnnnc4. SrpnmLion occnrs a t r,/n = 1.21 intlcpendcnt,ly of the nnglc of tlivcrgence -x
p--
1 U-lo -
= C,' z
+ C, .
-
For z 0 wo uhould hnve lJ(r) .- IJ,. no that C, = $6Nn-'O. Putting furthcr C,' UOl0= C,, we obtnin from cqn. (10.41) w.
Eqnat,ion (10.43) reprcsrn1.s the pot.cnlia1 vslocit,y for wl~ichcloparation can jnst be nvoidwl. T h r constm~tC, can IIC tlctnrminccl from the vnlnc of the bountlnry-layer I.hickness do a t the origin z = 0. We hnvc A U' P / v = - 10 or d 1/10 ;q/(--D7). From eqn. (10.43) we ohtain
-
-
,
: -I
rc----,ys
):
The prcrrding concl~~sions npply only n.s long as t,he displaccnwnt clTect of the Iioimdary lu,yrr n ~ n ybe nrglcrlccl. Ilo\vevcr, this is not the cnsc u h c n the angle of divcrgcnco in small. Whcn thin nnglc is small, the boundary laycrs fill the whole channel cross-scction aflcr a certain inlet length ( r / .Scc. XI i) and the flow gorn over nsytnptoticnlly to that discussed in Scc. V 12 undrr the heatling of channel flow. When the included angle does not excced u certain valnc which drprnds o ~ the ? Reynolds number, there is no separation. n nuninlnry rrvirw on nrpnrntior~in Ilcccntlg, S. N. l3rown nnd I(. Stewnrtnon [I] ~)~~l,linhrtl whirh the rnathematical qucst.ion renbrcd on thr ning~~larity which occllrn in 1.hr tlifl'~:r.rlll.ial eqmt.ionn a t tho critical point has been ernphnnized. Sccilso tho work of S. (:oldstein 141. 11 Inore physicnlly inspircd review of thin ]~rob~cln nrm h ; r~c c c ~ ~ l lbeen y puh)inhrd by J . C. \villi:lln. 111 (291, n.nd by P. IC. Chang [2c].
and hrnrc
From 6 - ,!r n t x = 0 we hxvr C , - 10 rl/lJ, doZ,which gives the final solution for thr potcntinl flow nntl thr vnrinl,ion of bo~~ntlnry-lnyrr thirknrsu
It, in srrn that, t.hc n,ngnil.~~tlr of I.hr prrmissil)lr dcrrlornt.ion (tlcrrmsc in vclorit,~)is very small, Irring ~)roporlion:ilto .I: (1 1. I t s v n l ~ ~isc vrry nearly rcnlizctl for t,hc cnsc of constant vc1ocit.y nlr~natho IInh p1:ilr at. zrro i w i t l r ~ ~ rJn r . the prcsrnt. cnnr the incrrnsc in hound:wy-lnyrr thirltncsn, 0, is ~~roporlionnl to 3:I'.5" :his vnlnc also tli1h.m h n t lit,t,lc from lhe rase of n fln.1. plate nt zrro i n ~ ~ i ~ I'cw l r ~whi~,Ii ~ ~ ~0t - 2+5.
-
45-72 (1969). 121 I3ussn1ann, I0 (cooling). Hence it follows t h a t for a flat plate a t zero incidence in psrallcl Row and a t small velocities the temperature arid velocity distributions arc idcr~tical provided that the Prandtl number is equal to unity:
Analogous, simple asymptotic formulae can also be established for the case of frcc convect,ion on n verlical flat plate, [73], see also eqns. (12.118a) and (12.1181)).
g. Thermnl bounilnry layers in forced R o w
I n the present section we shall consider several examples of thermal boundary layors in forced flow. I n solving thcso problems, uso will bo made of tho simplified thermal boundary-layer equations. J u s t as in the case of a velocity boundary layer, the general problem of evaluating tho thermal boundary layer for a body of arbitrary shape proves t o be extremely difficult, so t h a t we shall begin with the simpler example of the flat plate a t zero incidence. 1. Parallel R o w paet a Bat platc at zero incidence. We shall assume that the x-axis is placed in the plane of the plate in the direction of flow, the y-axis being a t right ar~glcsto i t and to the flow, with the origin a t the leading cdgc. The boundarylayer equations for incompressible flow and constant properties (i. e. independent of temperature) have been given in eqns. (12.61 a, b, c): assuming t h a t the buoyancy forces are equal to zero as well as t h a t dpldz = 0 [18, 941, we obtain
T h i ~result corresponds to eqn. (12.52) which Icd us lo thc f~rrnulat~ion of Llw important Iteynolds analogy between heat transfer and skin friction. 11. Blasius introduced new variables for thc solution of the flow rquat.ions, sce oqns. (7.24) and (7.26). (y) is 1110slrcnm fnnclion):
'rhe diffcrcntial cquation for /(q), cqn. (7.28) bccornos f f"
+ 21"'
=0,
with thc boundary conditions: rl = 0 : f = f' = 0 ; 11 =- cm : 1' of these equations was given in Chap. VII, Table 7.1.
L-
I . 'I'l~esolution
Including the eflect of frictional hcat, as seon from eqn. (12.63c), the temperature distribution T ( 7 ) is given by the equation
'I'hc: 1)ountlary ronclitions arc: 11 = 0
:
u =v =0 ;
T == T,,,
or
aT/8g = O
'I'he vrlocit8y ficld is it~tlcpcndrnt,of t,hc tcmprraturc firltl so that tlrc two Ilow equations (12.03a, b) can be solved first and the result can be employed t o evaluatc the tscmpcmtnrc field. An important rclatior~shipbetween the velocity distribution and thc temperature distribution can bc obtained immediately from eqns. (12.63 b) may be neglected in eqn. (12.63~). :md ( I 2.fR c). Jf l h c hcat of friction p the two rquat.ions, (12.03b) and (12.63c), become identical if T is rrplaced by 76 in the sccond cqr~at~ion a.id if, in addition, the properties of the fluid satisfy the equation
I t is convenient to represent thc general solution of eqn. (12.65) by tho superposition of two solutions of the form:
ITcrc O1(7) dcnotes the grncml solution of thc hornogcncous cqu:~tion and 02(t7) denotes a particular solution of the non-homogeneous equation. It is, further, convenient to choose the boundary conditions for 01(7) and O , ( q ) so as t o rnakc 01(7) the solution of the cooling problem with a prescribed temperature diKcrcnce betwecn the wall and thc external stream, T , - T,, with 0 2 ( 7 )giving the solution for the adiabatic wall. Thus 01(7) and Oz(v) satisfy t h e following equations:
with 0,
1
1
nt
7
0
1
arid
O1 -- 0 nt 11
==
oc> , r
~ l
g. Thcrrnnl bonntlary layrrs i n forcrtl flow
2!15
Table 12.2. Dinicnnionlcss coefficient of heat trensfcr, a,, nntl din~cnnionlcnsndi:hnt,ie wall ternpcmturc, b, for R flat plate nt zero incidence, from eqnn. (12.70) and (12.76)
-
0 and 0, = 0 a t q -= co . 'l'lic value 0,(0) pcrmit,s us with 0,' -. () at, r] t.o rvalnntc tJie constnnt C from cqn. (12.66) in n manncr to satisfy t.lie boontfnry nondition 1' - T,,, for 7 - 0 . This yidtls
( h d i n g prohlcnt: T h e solution of cqn. (12.67) was first given by 13. l'ohlhnnsen [94]. I t rnn Iw writtcn a s
IIct~cofor P = I : 0,(q) = 1 - / ' ( q )= 1 - u/U,, and for P = 1 the temperature dint.riltc~t,io~~ bccomcs itlont.icn1 with tJlc vclociLy t l i s t r i b ~ ~ t i oin n accordn.nce with cqn. (12.64). The t.cmpcmt,ure grntlicnt at, t.hc w;dl, a s calculated from eqn. (12.60), wil.11 /"(O) = 0.332, bccomcs:
- ("'I)
= a, (P) = (0.332)'
dtl
0
Tltr coristnnt rrI is s c o l t,o tlcpcntl solcly on thc PrantltJ numbcr, a, (P). Some valucs c:clct~l:~t.rd hy 1.: I'ol~ll~nr~srn nrc rcproclnccd in 'l'ablc 12.2. They can bc interpolated with goofl nrrltracy from t,hc formula
Fig. 12.9. Tempc.ratnre ~listril~ntiot~ I\ t t 11 sn1:i11 vrlo(.tty on :I I~mtrtlflnt plate a t zcro iw~tlrnrt~, plotted for various Prantltl nnn~hcrsP (frictional 1lr:it ~~rglrrtctl)
For very sm:tll I'rantll,l numbcrs, cqn. (12.59~1) givcs rs,
-
0.564 ;/-P
(P -- 0 ) ,
'I'llc t.cnil)crnt.urc wllicll is assumctl by t,llc w;111 owing t o fri(:l.ionnl Ilml. 1 . 1 1 ~rrrlirthrtlic ~ c ~ nf~ntpern/?ur. ll 'l',, is thus, by cqns. (12.00) nntl (12.72): l'?,,, -
r
= I
l
- T,,
--
Urn? b (P)
(12.74)
rP
/Irlinhnfic ~twlb: '1'11~ so11tl.ionol' w ~ n (12.(i8) . cmi I)c ol)t.:einctl by t.hc mcthotl of ol' t,lle 1)nrnrnrt.rr'. Itzis 'v:lri:lt,io~~
from cqn. (12.72). I'or a const.ant 1'rantll.l nnml)cr t,ho :~tli:~,l~nl.ic: \v:tll l,c-tril,rr:\.I,t~r(~ is C rl, \ v l t i t ; l ~ W:I.S ~ ) l o l , t . i ~i~l ll prol)orI,ionnl t o l , l ~ cn(lin~l)ntict . c t n p ( ~ r : ~ t nI r ~~ 1!,v,2/2 Icig. 12.3. Some n~lrnc.ric:alvalucs of IJIC I;wI.or h ( P ) ;we givc.11 i l l 'I':IIIIv 12.2; 1'01. motlrmt,c: Prnr~tlt~l ~ i u m b c r sthem vn.lucs may 1 ~ :int,cl.pol:rt,od wit,l~sr~lIinionb;I(:(;II1 / P . T h e valncs for Iargvr l'r:~t~tlt,ln111111)t\rs ~ : I III(> I inracy from thr. forniul:~,11 f'crrrd fronl l'ig. 12.10. In t.11~bmit.ing m s r , wc! I1:1vc (841
-
X I . Thcrtnnl boundt~ryl a y r n in lnminnr flow
296
g. l r l ~ c r t dbountinry lnycrs in rorcctl flow
I t is remarliablc t h a t for P = 1 wc havc cxactly b = 1. Thus, for a gas with P = 1 flowing in a parallel ~ t ~ r c a with m velocity U, past a flat platmea t zero incidence the ternperaturc rise due to frictional heat is equal to the adiabatic tempersture, i. e. t o t h a t which occurs from velocity U, to zero. The adiabatic wall temperature [16, 201 measured st various Reynolds numl)ers U , x/v is seen plottcd in Fig. 12.11. Thc agreement, is vcry good in thc laminar region. A t the point of transition from laminar t o turbulent flow in thc boundary layer the temperature increases suddenly. The temperature distribution for a n adiabatic wall represented non-tlimcnsionally is
207
I'ig. 13.10. Adiabatic wall tctnpcrnturc 7'" of a flat plate nt zero incidrnce with velocity Ua for v:irioun v n l u r ~of tho I'rmrltl 1111ni1)cr;a h r 1':. 15ckcrtanclO. llrcwitz [IB] as wcll RR D. Moksyn [84]. Vor large I'rnndtl n~ttnl)rrn, according to D. Meksyn [84], we havc 0 = 1-0 PI13
and is seen plotted in Fig. 12.12 for various values of the Prandtl number. From eqns. (12.74) and (12.75), we obtain that the constant C from eqn. (12.68a) is
c = (T,
- T,)
- (T, - T,)
= T, -
T,
.
The general solution for a p x c r i b c d tempcraturc difference between the wall and the free stream, T,#, - II',, eqn. (12.66), is thus T(7)
- T,
= [(T, - T,)
- (T,
- T,)]
01(7, P)
uz 02(7, P) + ----
(12.76)
cP
with T, -- II', comes
from eqn. (12.74). Thc dimensionless temperature distribution be-
Fig. 12.1 1. Measurenlcnt of ndiabatic wall ternpcrnturo on n flnt plate in n parallel air strenm a t zero incidence in a lntnittar nntl trtrbnlcnt boundnry Iayor, nfter Eclzert and Weise [20]; theory for laminar flow and P = 0.7
It is shown plotted in Fig. 12.13 for various values of the Eckert number E = Um2/c,(TW- T,), from eqn. (12.28). For b x E > 2 the boundary layer near the wall is warmer than the wall itself owing t o the generation of frictional heat. I n such cases the wall will not be cooled by the stream of air flowing past it.
Heat transfer. As scon from eqn. (12.2) the 11c11.t flux from plate to fluid at, station x has the value q(x) = - k(t3T/ay)v-o or
Thc rate of heat transfcr per unit time for both sides of a plate (length 1, width h ) is 1
Q
= 2b
/ q(x) dx,
so t h a t
I
o1
a ) Neqkding /rictioml heat : I n this case T ( q ) - T m = ( T , - T,) (7) by eqn. (12.69) with ( ~ l l ' / d q )=~ - a , (T, -- T,). With nl from cqn. (12.71 a ) we have
Fig. 12.12. Temperature excess in the laminar boundary laycr on n flirt plate a t zero incidence in s parallel stream with high velocity in the absence 01 hrolinq for vnrions l'rnndtl numbcra (ndinbntic wnll)
2!)H
XI[. Thcrn~albor~nrlnryInyrrs in Inn~innrflow
The cnsc of turbulcnt flow can bc approximatcd by l h c equations
N, = 0.0296
N , = 0.037
'fi. R , O ' ~ t u r b u l c n t ), ;/F . R,O.R
(turbulent) ,
(12.79~) (12.7!)d)
which we quotc here for complctencss, b u t without proof. 'l'hc preceding forrnulim for t h e rate of Iicat transler a r c in good ngrccmcnt with t h c n~rasnrcmontnt h o t o 1'. Elias [31], A. Rrlwarcls n r d B. N. I'urbor 1271 nntl .J. J(c:slin, ID.1'. M~wclc*rILIHI 1%.E. Wang [66J. b) With frictional heal: I n this case with T ( q ) from cqn. (12.76) we obtain
elinlri~~ntion i n a I:in~innr1)ounrlary Iaycr on a Iicnbrl ( E > 0) and IGg. 12.13. 'rrn~l)cr:~t~lrc. cooled ( E < 0) llat plntc: nL zcro incidcncc i n a parallol stmnn~for the case of a laminar hoi~tidnry layer and w i l . l ~frict.ionnl IIC:I.L accorrntctl for nn calc~thbdfrow rqn. (12.70). I'mntlt,l nntnber P 0.7 (air). 'l'llr lcn~pc!rat,nreof the wrdl is n~aint.air~ctl constnnt a t y,.. (hrvc: h x E = 0 for zrro rrict,ictn:~IIlr-nt,;C I I ~ V Ch x E = 2 corrcspo~id~ to an ntliabalir wall; E = 1Jm2/c,,(7',,- 7>,); b .- 0.835. I'or h x E > 2 (.he hot wall ccb:tsrs I*, I?c cooled by t,hc strf:n~~i of coo1er air, RIII(.C tJ~o'hral, c:nnliion' providctl I)y frichnal I~rnl.prc!vrtit.s cooling
wherc T, is t h e adiabatic wall temperature. It is identical with the wall tcmpcrature in t h e thermometer problem a n d follows from t h e equation Urn2 - Urn2 T,-T,=b(P) - - - - - z i p ---. (12.80) 2 c, 2c l i e r e b(P) can be takcn from Table 12.2. Iritrotlucing t h e Mach number M = U,/cw from (12.27), T, may also bc taken from
T h u s we o b h i n the following expressions for t h e local and total h e a t flux from cqns. (12.77) and (12.78) respectively
I ~ ~ [ r o t l ~ r c itlitnr.nsionlcss rrg coefficients in the form of thc Nlrssclt ~ l u r n t x rfrom (12.31) instoatl of t,l~c?locnl and total Iicat flux, rcspcctively
WIII.
I t now ceascs t o bc useful t o basc t h e cocfficicnt of h c a t transfer a(x)on thc t.cmpcraturc diffcrcncc (Tw- T,) from eqn. (12.29) o r t o clcfinc thc Nnssclt numl)trr :rx i ~ t eqn. (12.31) because t h e heat flux is no longcr proportional t o t h a t tcmpcmturc diffcrencet.
t E. Eckcrt and W. Wcisc [17] hnvc, thcrcfore, ~t~ggrnLed to introclr~coa N I I R Rnnrnl)cr C ~ ~ N*
based on the difference (T, - T,). Wc mighL tlicn cxpcct to obtain rrs a 8rnt approxiina t.'Ion, even in compre~ibleflow, the mmc forrnu1.w for N* a9 in eqn. (12.79a. b). If, on tho otllcr b a s d on (T,- T,) thcn eqn. (12.81) Icnds to tlic following hand, wc retain the N u ~ c l number t cxprcasions instead of (12.79a):
XI1. Thrrrnnl borrndary layers in lan~inarflow
300
The cooling action of a stream of fluitl on a wall is considcm1)ly rctluccrl because of t,hc h r a t gencmtrd by friction. I n t h r nbsrncc of frirtionnl heat, heat will flow from the platc t o thc fluid ( q > 0 ) as long as T, > II', but in actual fact,, if frictional hcat is prcsont, a flow of hcat persists only if T, > T,, eqn. (12.81). Taking into acronnt thc valuc tlcducctl for T,, we obtain the condition t h a t heat flows from wall to fluitl (nplwr sign) or in tho reverse tlircction (lower sign), if
g. Thcrmal boundary layers i n forced flow
ant1 thc solution must satisfy thc boundary contlitions 1 7 ~ 0 :
A numcricnl cxamplc may serve to illust~mtcthe signifirancc of cqn. (12.82): 111 a stream of air flowing a t TJm = 200 m/scc, P = 0.7, cp = 1.006 k.J/kg dcg wr obtain P 1JW2/2c, = IF tlcg C. The wall will begin Lo be cooled whcn
1/
If tltc tenipcrat,urc difference bctwren wall and stream is snlallcr than this value the wall will pick u p a port,ion of thc hcat generated by friction. I n particular this is tho case whcn thc tcmpcraturc of the wall and stream arc equal. An equation for thc rate of hcat transferred from n flat platc a t zcro incitIcnce but with variable material properties was derived by H. Schuh [110]. The temperature field on a platc placed in a stream with a linear temperature distribution w54 studicd in ref. [128]. 2. Additional sinlilar s o l ~ ~ t i o nof s the equations for thermal boundary laycrs. I n the casc of a flat platc a t zcro incidence, the velocity and the temperature profiles t~lrnctlout to bc similar among themselves. This means that the distributions a t tliffcrcnt clistanccs z along thc platc coulcl hc mn.tlc congruent by a sniLahlc stretching in the y-direction. Since i t is lcnown t h a t there cxist velocity boundary layers other i.han those on a flat platc for which this is true (e. g. the wedge profiles discussed in Chap. IX), it, appcnrs useful to stucly the possibility of tho cxistcncc of additional similar solutions of the energy equation. This problem was investigated in detail in ref. 11351. At the prcscnt time, we sha,ll start with the class of velocity boundary leycrs on wedges and will awnme t h a t t,hc cxtcrnal flow is of the form U(z) = tc1 x"'. 111 an analogous manner, we stipulate that tho wall-tcmpcraturc distribution also sa1,isfies n power law, say one of thc form T,(x) -,'Y = = T Ix". Walls of constant t,rmpcrat,ure are inclutlctl as thc casc n == 0, and t11c valuc 12. = (1 --711,)/2 h r a t flux q . 1nt.rotlucing tlic sitnilarit,y variable corrr~pondsto :I, crn~stc~nl,
wr ol~tnintlic f'ami1i:ir rquations (0.8~1)for the bdocity u = i J ( . z ) . /'(q), or
lIcrc E =. 71,2/c,, prol~lcm.
a:=];
17=03:
O-:()
rrprcwnts the nppropriaLc form of 1I1r I':ckorl. t~ttrnl)crli)r illc
'/I1
It is clear from cqn. (12.84) t h a t its right-ltantl sitlc vanisltcs in tltc al)scncc of frictional heat and that all solutions arc thcn of tlrc similar typc. IIowcvcr, if frictional hcat is includcd, similar solutions arc rcstrictctl to tltat combinatio~~ of pnramctms for which thc right-hand sido becomes intlcpcntlcnt of z. 'rl~isocc:urs whcn 2 ~n - 7 t = 0 , tltat is, whcn thcrc cxists a firm cor~pling1)ctwcon tltc vclocity distribution in thc cxtcrnal flow ant1 tho tcmperatarc tlistril~utionalong t.110 w:ill. According to this result., thc casc of a c o ~ ~ s t ~tcmpcmt~ure ant lratls to similar solut,ions only on a flat plate (1i1=1t =0). 011thc olhcr hnntl, if tho contlil,ion 2 111, - - 11. - 0 is satisfied, thcn for every pair of values of m and P thcrc cxists one tlcfinitc valur: E, for which there is no flow of ltcat (O'(0) = 0). Jn this rasc, the tcmpc:mt,~~rc distribution along the wa.11, once again lcnown as tfhc atliabalic wall-tcmparat,urc: distribution T,, is given by
Numcricnl valucs for thc function b(m,P) havc bcnn romput,rtl by 1%A. llrun 171. I n the particular case whcn m = 0, the numerical valucs of l'ablc 12.2 arc rccovcrcd. Wlicn thc cffcct of dissipative hcat is ncglcctcd, wc obtain the simpler cquation
whose solutions for different valucs of thc parameters m, n, and P have bccn published by a number of authors [79, 121, 32, 33, 89, 1401. E.1E.G. Eclrcrt [I91 has dcmonst,ratcd that for n = 0, the local Nusselt number is given by thc equation
Eerc
N %
x = -a = k
v---y U ( x ) . 1:
O' (0) = -
id,
0' (0) .
( 12.88)
The function F ( m , P) is seen plotted in Fig. 12.14 on thc basis of the numcriral data provided by 11. 1,. Evans [33]. Jn addition, thc asymptotes for very small and vary large l'randtl numbers, cqns. (12.57) arid (12.01 n), rrspcctivrly, have also
302
XJI. Therninl boundary layers in laminar flow
F ~ R12.14. . 1,ocnl Nu.sselt nwnher as a fnnction of tho Prandtl number and of the flow parameter m for flows wliosr fire-stroam velocity is distributed according to the law U ( z ) = u, zm = = v , z P l ( 2 - 0 ) (wwlge Ilow) hut for a constant wnll kmperature and in the absence of dissipation Asymphtic approximations for P -t 0
I\riylilpt.r)tinapproximations for P .+ ca and P P + u, and p = - O.I!)O:
+ - 0,100 according to eqn. (12.61a), and for
Approximation for inbrmcdinto PrnntitI nurnbcrs and /i'= 0 , according to eqn. (12.71a).
g. Tliernial bonndary Inycrs in forced flow
303
been indicated (see also [119]). F o r thc Rat plate (m =0) the earlier rclntions from eqns. (12.69a) and (12.62a) are, naturally, recovered. The caso of stagnation flow ( m = 1) leads to eqns. (12.69b) and (12.02b). I n tho special cnsc of a separation profile (m = - 0.091) i t becomes necessary to adopt a different asymptotic approximation for P -t oo,as shown in 1321. The thermal boundary laycr associntcd with tho thrcc-dirnc:nsional vc:locity boundary layer on a rectangular corner a t zero incidence is also of thc sclf-similar type when the external velocity distribution is of thc IIartrcc class given by U ( r ) -C x m . The velocities a s well a s the temperature distributions for this case havc bccn worked out in a thesis by Vasanta R a m (ref. 1921 in Chap. XI). Figure 11.I!) givcs an idea of thc vclocity distribution for dilTcrcnt vnlrics of tlic ~)rr..ssurc-grntlicntparamctcr m. I'hc diagram in Fig. 12.16 supplcmcnk tlic procctling o m in t,l~rtt,il, c : c ~ ~ ~ t . r i i t i x an example of the associated temperature distribution. For a uniform cxtcrnal Ilow with U(x) = Um = const and in the case of a hotter (i. e. cooled) wall for which T, > Tm the solut,ion nevertheless exhibits a zone near the corner itsclf, shadcd in the figure, in which ( T - Tm)/(Tw- Tm) > 0, that is in which T > T,,,. 'l'his zonc occurs when dissipation is included and corresponds to a condition where thc local fluid temperature exceeds the wall tempcraturc. Thus, locdly, the heat flux is reversed and proceeds from the fluid t o the wall in spite of the fact t h a t a t a largc distancc from the wall the temperature of the fluid is lower than t h a t of the wall, Tm < T,. The physical reason for this seemingly anomalous behavior is rootad in the increased local ratc of heating due to dissipation which occurs near thc corner. Phcnorncnn of this kind are important in the hypersonic flow rogimc. 'l'hc rcsulling Inrgc inc:rcnscs in temperature which occur in such cases can cause burning of the surface of the body in the stream (ci. Sec. XI11 e). 3. Thetmal boundary layere on i~othermalbodies of nrbitrary shape. N. I'' rocssling [39] carried out calculations on the tcmpcraturc distribution in thc laminar boundary layer about a body of arbitrary shapc for tlic two-climcnsionnl and axially symmetrical cases. I n his calculations, in which friction and cornprcssion work wcrc neglectccl throughout, he assumed a powcr serics for the potential vclocity clistribution around the body expanded in terms of the length of arc (Blasius serics), similar to Sec. I X c , i. e. of the form:
U = u, x 4- u, 2" -1- u, x"k
... .
The velocity distribution in the boundary laycr is ass~~mctlto have the form: Fig. 12.15. Tornpcrntnre distributioli , along a lienkd wnll (T,,,> T,) i n a right-angled corner in n laniinar boundary layer with a constant external velocity Urn(inclusive ofdi~sipntion),aftnr ~n.qnndRani (1441. 1,irictn of const.nrit bmper:~.tnrc:for P = 0.7 arid E = 2.4. The local temperature exceeds the wall ).,'l' i n the hntcl~r:drcgion; conseqnent,ly,in that region heat flows fluid + wall t~rnlwraf.nre( 7' ; i n spite of the fncl, t,liat the wall tmnperatnm exceeds the free-stream t.aniporntnre. Tlie reason for t . I ~ i nproccm l i c ~in tlis~ipntion.Rckert nnnibcr E -- IJ&/r, (T,,, 7',)
Correspondingly, the assumption for thc tcmpcrature distributio~iwas of tlic form :
I n a manner similar t o t h a t for the velocity boundary laycr in Scr. TX c it, is found t h a t the functions T I (y), T3(y), . . . satisfy ordinary diffcrcntial cqnations which include thc functions f l , i,, . . . of the vclocity distribution. I n this case, howcvcr, thc functions T,, T,, . . . also depend on the Prandtl number. The first auxiljary functions TI(?/)
304
g. Thermal bonndnry lnycra in forccd
X11. Tl~ortnalboundary Iaycra in laminar flow
for t,hc two-tlitncnsional and axially syramotricnl case were evnluntcd numerically for n Prantltl number of 0.7. ' L ' h met,hocl under consideration is somewhat cumbersome by its natnrc, ns was the case with the velocity boundary layer, particularly for slcntlrr h t l y forms when a large number of terms in the power expansion is rrqnirrd, as shown i l l 1281. Nurnrrous solut,ions for self-similar thermal boundary layers inclusive of the elTwt,s of blowing and suction can be found in [34, 44, 134, 101. In t,lw sprcinl case when P = 1, antl when the heatf due t o friction is neglected, the tfiKcrcnt.inl cquat,ion for the temperature distribution in the boundary layer around an xrbit,r:sry cylinder is itlcnt.ica1 with that for the transverse vclocity component (vdocity component in the direction of the generatrix of the yawed rylintler). This r a n be seen upon comparing cqns. ( 1 2 . 6 3 ~ and ) (11.58). The relation, which has already t)ecn discusscd in Scc. X I d , was ut,ilized b y I,. Golancl [46] for t,ho e v a l ~ ~ n t ~ iof o n the temperatlure distribution in the boundary layer around a aylintlcr of sprcinl form. I n the ~~eigltl~ourhootl of a stagnation point, where t h e velocity distribution is r ~ ~ ~ r c s o nbyt ~IJ~ (z) d == x with nz =.. /? = 1 , thc Nusselt number dcfincd in eqn. (12.87) a n be rcprescntad by t h c cquation
on contlition thatf oncrgy dissipation is neglected. T h e character of the function A ( P ) emerges from Fig. 12.14 and Table 12.3. I n thc former, the curve for /? = 1 corrcsporlt1.s t o t l ~ cfunction A . For .z circulnr cylinder we p u t U(z) = U, sin ( x / R ) ,so that. 91 I 4 fJ,/D. lrcncc
The above rxprc.ssion agrcrs reasonably wcll with t h e measurements performed by R. St.llnliclt antl I 101° the flow is turbulent. The agreement hetween theory and experiment is exccllcnt.
E. Pohlhausen's ralculations have been extended by IT. Schuh [I091 to the case of Iwge I'mncttl numbers such as exist in oils. The casc of very small Prandtl numbers is treated in a paper by F.M. Sparrow a r d .T. I,. Gregg 11261. The limiting cascn whcn P + 0 and P -zoo were exambled by E. .I. Lo Fcvre 1731, according to whom we may write
Some numerical values for intermediate Prandtl numbers are contained in Table 12.6. Calrulntions with n hmpemt~~rc-clcpcn(lctitt viscosity were performa11 1)y 'l'. Tllirn [50]. The olI'ccL of suction or blowing on tho rate of llrnt Lrnnshr from n vcr1,irnl plat,e in naturul cortvrction is tlcscribctf in refs. [29, 1241. Atltlitionnl rlassrs of similar solutions in natural {lows were discussed by I 10'
n P 4 '
m7
ma
W'
on
mn GxP
w*
mn
The laminar thermal boundary layer around heated bodies in natural convect.ion can bc convenicntJy madc visible with t,he nit1 of a Schlicrcr~ninl.liotl tlcviscd by E. Schmidt [l05]. A parnllcl bcarn of light is pr~uscdt.hrol~glttht: l)our~cl:~ry s on a screen plnccd layer in a direction parallel to the plate and produces s h a t l o ~ a t a large distance from the body. The density gradient in the air a t right anglcs t o the surfacc causes the rays of light to be deflected outwartls. The dellexio~iis largest a t point,s where the densit.y gradient is stecp, i. e. ncar the body. Wit.11 n sufficiently large distance between screen and body thc space taken u p by tho heated layer remains dark so t h a t in the Schliercn picturc the shadow of tho body is surrounded by a sliadow due to the thermal bountlary layer. Tltc rays of light. zouc n.ro111it1 which are deflecled out of tho tempernturc ficld crcatc 2.11 ilh~rnin:~lntl thn dark shadow. 'l'he oritcr cdgc of this zoric of light is forn~rclby 1.ltn r:r:) s whidt just skirt the surface; consequcr~tlythcir dcflcxion is proportional to thc tlwsity
320
X11. Thrrn~nlbonnclnry layers in Inminsr now
= 2 x 10' t o 4 x 101°): N =0.726 ( P x G)1l4 f o r l a m i n a r flow, arltl ( P t o 9 x IOIL): N = 0,0674( G x P1'20)-11Vor t u r b u l e n t flow.
x G -:4 x
10J1)
, = 0.420 Gl14, whirl^ n-ns c o n F o r t h e s p h e r e J.I. Shell [115] calculated N f i r m e d by m e a s u r e ~ n e n t sin a i r . S u m m a r i e s of r e c e n t w o r k o n nntur:rl r o ~ i v w t i o n a r e containctl in refs. 165, 961.
g r a d i e n t at (.he surface, i. e. t o t h e local coefficierlt of h e a t transfer. F i g u r e 12.28 represents a Schlieren p h o t o g m p h t a k e n o n n heated v e r t i c a l f l a t plnto. T h e c o n t o u r of t h e p l a t e is s h o w n by a b r o k e n w h i t e line. It i s easy t o recognize o n the s h a d o w t h a t t,hc b o u n d a r y - l a y e r t h i c k n e s s increases as d4. T h e e d g e of t h e z o n e of l i g h t sllows (,It& the local roaflicicnt o f h e a t b m n s f c r is p r o p o r t i o n a l t o z-'I? T h e p i c t u r e in Fig. 12.2!) gives a n i n t c r f e r o g r a m f o r t h e s a m c t y p e of bounclary l a y e r ; i t w a s o b t a i n e d hy E. R. G . R c k e r t a n t l E. S o e h n g e n (13J.
,/I I] Allw, 1l.J.. ILINI Im~lz,1s.C.: A ~ n c l , l ~ for r d rvtlvr~Ii~t,ing ht:nt, trmnsft~in I,IIc: I ; I I I I ~ IfIl o: Iw~ regions of Iiotlics. NIICA Itep. 704 (1!)43). 121 An~hrolz,G.S.: 'l'h rffrct, of surfnco t,cmpcratrrre vi~ri:~hiliIy on Ircnt, C X I : ~ I R I I ~ ( i:l l l i r ~ ~ ~ i t r i ~ r flow in n ho~rncliwy I:~,yer.Soviet I'hys. 'l'ecl~n. 17hys. 2, 758. 748 (I!)57). r l ' r ; ~ ~ ~ o(. ~l:~.~i Zh. Tekh. Viz. 27, 812- $21 (1957). c.ylit~(lrrs. I:!] I3j0rkl1111d.(:. S., and ICnys, W. M.: Ilnnt t,r:itlsfor I)ct\ww~c.onconl.ric~rot:rti~~g J . Jleiit 'l'rnnsfer 81, 1 7 L 186 (1!)59). [4] Brnn, 1'3. A,, I h p , A., antl Kestin, J.: Sur un nouvcnrl t,ype d r s t,or~rhillonslongitrltlinnnx d n n ~I'i.cortlenrcrtt a~rt.onrd'un cylindm. C. It. Ar:ntl. Sci. 263, 742 (1!)0fi). d gr;i(lipl~t [R] Iliiyiikt,iir, A.lt., ICcnlin, J . , antl Mi~etlor,1'. 17.: 1nllucnc.e of c o n ~ l ~ i n rprcssrltx and turbulence on the transfer of heat from n plate. Int. J . Iloat Mans Trnnsfcr 7, 11751186 (1!)04). [(i]Ten I l o ~ c l ~M.: . Die WBrrnriil)ert,ragnng. Borlirr. l!Xl(;. [7] I%rrrn.15. A , : Sclcrkcl ro~nhr~sI.ion prohlrn~s.Vol. I I , 185 I!M, A(;A ltl), ~'l!~K:~lllf~ll I+(*ss, 1,011~l~~11, I!Wi [8] ~ h n p n ~ a nI).Jt., . and I t ~ ~ h e s i nM.W.: , ' l ' r n ~ p r n t u r rnnd vrlorit,y profilrs i n thr ('OllIp'r4slhlc:, Inlninnr borrndi~ryInyrr wiLh nrl)ilr~rrytlist.rihl~tionof s~~rfnc.o tc:tnl~rr,tlItrr.. .I .\S I / ; . 547 --505 (1!)49). [!I] Jhvico, T1.R.. nntl I3ourne, D.E.: 011tho cnlculnlion of hont, ant1 nlirns trnnsfrr in I:tn~it~nr and t,urbulrnt boundary layers. 1. 'Chc Ia~ninarcme. Qrritrl,. .I. Mcctl. Appl. Mirth. I), 457- 407 (1956); ~ o nlao e Qrtnrt. J. Mech. AppI hInt,l~.12, 0 3 7 XW ~ ~ (1959). [10] I)ewey, C. F., and (:~oRR,J. I?. : 1t:xnct nitnilar aolnt.ion of 1 . 1 1 ~lnrnin:ir bonnclnry-l~iyr~ eqrtn. tiona. Advnnros in Heat l'rnr~afer4, 317--440 (lNi7). [I I] I)ienemnnn, W.: Uerechnung dcs W5rtnciibergnngrs :in In~ninnrrrrnstriin~~cn Iernt.rlr.Ilias. I5ra11nncl1weigI!l5I ; ZAhlM 53, 89-109 (1953); see nlso J A S 18, 04 - 0 5 (1951). [I21 I)o~longhe.1'. L., nnd 1,ivingoocl. J. N. 13.: 1t:xnc.tsolutions of Iiin~innrbortntlnry liryrr rclrra. t.io~rswith constm~tpro1)rrt.y vnlnes for porous wall with variiihlr Itwporett~re.N,\(:A Ibrp. -122!) - I I!)RR\. ..,[12*J Driest, E. R. van: Convective heat transfrr in gases. 1'rincet.on Univcr.sit,y Serics, High Speed Aerodynnmies and J e t Propulsion, Vol. V , 339-427 (1959). 1131 Ifrkert., E. It. G., and Urnke, R.M.:Hent nnd mass t.mnsTcr. hlcGrnw-Hill, Now York, 1!)5!1. 1141 Eckert, E.: 1':infiihrlmg in den Wiirn~c-untl Stofftitiutnunrl~.:lrd c:tl., Ilorlin. I!)(;(;. I151 Eckcrt, E.,nnti Drewitz, 0.: Uer Wiir~neiihcrgnngnri nitw tnit groBer (:c:sc.l~\r'il~rligkcit, lii.l~psn~~pest.rB~~ito Plnt.lc. Forncl~e.111e.-Wcs. 11. 116 - 124 (1!)10). " ~ c r 1 . :l ~ ~ r iI I r l sr n n ~~ I n ' ~ I . I I I 317 (I !)40). n d W.: Die 'l'ernperntnr unbehciztcr Kiirper in r i n r n ~(:asst.ron~ holler Eckert, E..~ ~ Weisc, Geschaindigknit. Forschg. 1ng.-Wes. 12, 40--60 (1941). Eckert, E., and Urewitz, 0 . : Die Bercchnung dca 'I'e~i~perntnrfeI~It~s in dor la~ninclrenGrenzschicht, nchncll nngcstriirnt,er unl~el~ciztcr 1Ciirpcr. I,n(.t.f~r.l~rl IO~H~:IIIIIIR 19, Is!) l!K (1042). i n tlrr I;uni~~rlrc:n ( : r r n z ~ ~ ~ l nn~nt ~ i ~ ~v ili~~ nt .lCrlterl., 15. : I)ie Ilcrcchnrn~gtlcu Wiir~~~c.iil)orgnr~gt*s t.er Kiirpcr. V J ) l - J ~ o r s c l ~ ~ ~ n p 416 n l ~ e(1942). ft ICckcrt, K., nntl Wcisn, W.: M C R R I Ider I I ~ '~r.(:rt~~~ornl,t~r~~t~rtciI~~~~g nitf rlrr Obcrfllirlrt~n d ~ n r l l angefit,riimIsr unbclieiztrr Kiirper. Porechg. 111g.-Wen.13, 24% -254 (1942). 13rkort. E.R. f:., and Sorhngen, I%.:l>ist,ribut.ion of I1c:~t.t.rnnsfer m ~ f f i e i e n karound rirrulnr cylinders in cross-llow at. Ibeynold~n r ~ n ~ h e rfrom s 20 t.o 500. Triins. ASME 74, 343- -347 (1!)52). \
Fig. 12.29. Intcrferogra~nof a thermal boundnry layer or1 n vcrt,ical I~cntedflat platc, nftcr R. 1%.C. Iklzrrl nnd E. Soehngen [10]
I l~orizorlt.:~l IIC:LL(Y~ O t l ~ e rttlmpes: 'I'IIC mot,ion (111ct o nat,rlral chnvcction s r o r ~ n ( a c i r c u l a r o g l i ~ ~ t l cwr a s t,rcn.t,cd i n a n n.na.logous w a y by R.I l c r m a n n 1551. t i c F ~ I I I for I ~ P := 0 - 7 a I n e m 11cat t m n s f c r coefficient N,, = 0.372 G ' I ~ , w h e r e G is I ~ a s n do n t , l ~ oclinmcter. b l c a s c ~ r c m c n t si n air p e r f o r m e d by I., anel Jnckson. T.W.: ~\nnlysisof tnrbalent frec convection boundnry lnycr on n flnt. plate. XI\(:;\ I (7',,>- T m ) / T m
wc h%vc (i?7'/i?!,),,,0,1 > 0, n l t d l # w L Is trntt4errcd t o the r n l l owing Lo the inrpo qu:tntity or l w r t eenrrshvl h y
c. The flat plntc nt zero incitlcl~ce The boundary layer on a flat plate a t zero incitlcnce has been studiccl cxl,er~sively i n numerous publications, a n d we propose t o begin with n more tlctailctl cliscussiotl of this case. First we shn.ll deduce the ralatio~lbctwccl~tho vclocil,y and t c l n p c r a l ~ ~ ~ r o tlist,ribution o n a flat plate from t h e prccctling grnrrnl proposit.ion. Tn t , l ~ case c or a n rcrlirhlic w t l l (flat,-plat,c tl~ernlornclcr)wo s~tl)stitmt,c! -:'/I,., nncl [ J == (I*, i~tt,ocqn. (l3.12), SO t,h:tt t,ho t , c r n ~ w r ; ~ t (lislxil)~~t,iot~ ,~~rc i t 1 I,IIO I I O I I I I I ~ : I ~ . V layer on a llat pla,l,c bcconlcs
and the ntlialmtio mall trmpcrat,nrc, rqrls. (13.128, I)), is
, = U,/c,, ant1 c,2 = (y - 1) I-,, l', . [t is worth noting w l ~ i r hfollows with M t h a t t h e t ~ m p r r n t ~ u of r c a wall in comprcssiblc flow give11 by eqn. (13.17) is itlcrlticnl
334
XIIT. Lnrninnr 1)oundarylaycra in nompressihlc flow
with t h a t for a n irlromprcssiblc fluitl from eqn. (12.80) provicled t h a t i n thc former rase P = I . IT. W. I'Gnmons and J. G. B r a i n e d [34) have shown t,hat, it1 t h e rasc of T'mntltl 11ntn11crswhich differ from unity the deviations i n wall tempcraturr caused by comprrssibility effects, a s compared with the incon~pressihlccqt~ation (12 SO), arc- only very slight2.TIIIISIhc atIinbat,ic-\vnll trmpcmtnre cqnatiorl
:wi
c. Tllc flnt plntc at. zrro incitlrnct!
The recovery /actor, r , then represcths t l ~ cratio of the frictional lnmpcraturr inc.rcnsr of tllc plntc, (T, - T,), t o t h a t due to adiabat,ic con~prrssion,
urnz
AT, = - -2 c,,- >
from cqn. (12.14). 0 1 1 cornprrirtg rqns. (13.lH) ant1 (13.19) il. is scwl l . I ~ : , t , t,l~o mcovc:r,y factor has the val~rc remains vnlitl for rompressil~loflows with a vrry gootl tlcgrrc of npproxirn:lfio~l F o r nir, with y -- 1.4 : m i P -- 0.7 1 , wt. ol)t,nill
r = Ilcncc: for air r
dF-
-- d0.71
= 0.84
(Inminnr) ,
(lXl!):~)
(Inniinnr) .
(l:!.I!)l))
Thv r r s ~ ~ l l i ntlrprntlrnce g of t h r ntlinbatic-wall trmpcraturc on thc Mar11 n u n ~ b e rhas Iwrn r r l ~ r r s r r ~ t r tgmpl~icwlly i by tho plot in Pig. 13.4. For rxamplc, at, a Mnrli nl~nil)rr M,, -= I t h r wall 1)ccornrs Ilc~atrtlby 4.5O C (or 80° F) in roirnrl fig~rrrs. A[. M,, - 3, t llc t r ~ n p r r a t l ~ rinorc:rso r brc~orncans l~iglla s 400' C (or 720° P), ; i ~ l t l n l M,, = 5, i t is a s rnlrrll a s 1200° C (or 2200° 1').
-
Fig. 13.5. Mrnunrrrl rrcovrry
10 I Z br? TO 33 019 to 25 %I lo 18
t'Z
factors, r, for laminar boundary layen on conra nt s q w sonic veloriticq lor difkrcnt Mnrh n11111brmnnd Ilrynoltln numbcra, al1r.r C. 11. I ~ rJ2]; Y ronipnrison \\ it11 theorel icnl vnlncs lrotn rqn. ( I 3.19%)
v
%I0
A
60'
0
Boo
r
'.rhc diagrams in Fig. 13.5 rcprcscnt the rcsults of ~nnn.suronirnlson t,llc rccovcry factor in t h e cast of laminar boundary layers on conos in supcrsonie strca.rns, poris s w n t o be ~ o n f i ~ ~ n ~ t : ~ I formed by G . R . Eber !32J. The numcrirnl valnc r = lly t,llcse men.surernents. Similar results follow from ~nc:~surcnlcr~l,s p(di)rtllt:~lon various cones and a paraboloid pcrformcd by B. dcs Clcrs ant1 J. St,ert~l~crg 1271 :d It. Scl~crrcr[89]. Velneity nncl tctnpernturc distributions in thc nbnetlce n l lwnt trmslcr: 'I'wo ppt:rs by W. Ilant.zscht? ant1 11. Wcntlt. [44, 461 ant1 :L p:l.prr by 1,. (Irorxo 121 1 coll(,nill cxplici(, formll]ac for the c:rlc~rln.l.ionor Ll~cvvle~cil~y :1,11tl I,c:ln~)c~l~:li.llt~t* tlish'i~)llt,iorl ill Illlmbcr of spceific cnscs. J'ignrc 13.0 conI.:iilw 11lot,sof lSllt: vtdor.il,y rlisl.l.ihlt.ion in t,lle l ~ o l ~ n t l a r1;~ycr y for sovcral M ; ~ numlmrs. I It, reprrst!~lt,sCl-oc:c:o's c:~l(:i~Intions for a boulldary h y c r o n a n arlirrhcttic /kt plnlc o n tho :~,ssnn~l)l ion of ;I. vise-11sil.y law = 1 and for P = 1. T h e distance, y, from t h e wall has 11cc11rnntle (litncnsiolllcss \vit,ll rcfcrcnco 1.0 where 11, tlrnotcs 1,lle Itinrrn:rl,ic visc80sit,y in t,llo oxt,crrln,lflow. I t is seen l,Il:~t for incrcasirlg I\l:rcl~tl~tml)rwl,llc:ro is :I. c : r ~ l ~ s i t l ~ : ~ . : ~ . l ) l e : t~lickcrlillg of tllo Ilollntln.ry laycr and tll;~I,for very ~ : L I . ~ I :hl:11.11 n11t11111.t.s (.)I(: v~.lor,i(y C clist,rihllt,ion is approximately lincar over i b W ~ I O ~thicloless.
~GTu,
a. ' L h :
flat plnto nt nrro inriclci~cc:
337
,, Ilie t.c!rnprmtt~rctlist,rib~~tion is also shown in Fig. 13.6, and i t is seen t h a t tlic fric:l.io~~nl ir~crcascin thc tcmpcrature in tho boundary layer assi~meslarge valurs for Iargc Mach n w n l m s . 'Clic pnpcr by W. I~antzsc:l~e and JI. Wcnclt [44], quotccl cnrlicr, contnins c a l c ~ i l a t , i ofor ~ ~ sP = 0.7 (air) for t h e case of a Iicat-conducting plat,e. I the velocity tlist,ril)~~lion u/rJ, plot,tcd in tcrms of y U , / Z i," I t is S I I ~ ~ V It.l~:~t, drviatcs c:onsitlc:r:~l~lySrom thnt for P -= 1 when 1.11~Mnclr nurnlm- :i.ss~~nics largcr v:rlrlrs. The vchc.it.y . lwofilrs shown in Fig. 13.G can bc mstlc ncarlv t.o ..coincitlc . . . .. . when t h c d i s t , a ~ ~from c c t h wall, y, is matlo tlinicnsionlcss with rrfcrcncc t o l/a,,, z/cI,, Fig. 13.7, wl~c:rc v,, tltnotcs t h e Irinomat~icviscosity of the air a t the wall. This circn~nstanccdcnot,cs pl~psicnllyt h a t t,llc incrcasc ill I~oundary-laycrtl~iclrncsswith Rlach number ( a t constant Rcynoltls number) is mainly due t o the increase in volnmc which is nssociated with t h c incrcasc in tlrc temperature of t h e air ncar the wall. This fact was first noticcd by A. N. Tifford [98].
1/
A
Jiig. 13.8. (hllicient oldtin frictio~ion d i a Oolic flat plate with rornprcnniblc, Ian~innr i)ot~ncl;~ry layer. P =. I, ), = 1.4 (air), nfl,cr l l : ~ n t ~ . s d:111ql ~ c \f1c!n~1ta[44]
I n this method of plotling. lhc curvrs fcsr dillcrcnt Marh nunlbcrn h n v c lwcn rnrcle nearly lo coincide. I t i s possible to conclude from l l ~ i nthat t l ~ cIargc incrcnsc i n llw bovndary-layer tl~lcknclis will1 nlnch n ~ ~ w b cisr mainly duo to t l ~ cincrcnac in volomc s h l r l ~i s associnted with t h o increase in t c m p c r a t ~ ~ of rc tllc air ncar tllc wall
7
Jiig. 13.6. V v l ~ w i t , lind , ~ t~cn~prr:~t,~irc (Iistx1r11tiori in (;o11111rc,%qil1lc:, 1a111iri:irl)o~~ncl:~ry layrr on adinlrnlic flat plate, nfkr Crocco P1.1 l'rx~idil
I I I I I ~P ~ ~
rrwn wnll rrfcrrvd lo
I, m
I'
I-,
= i , y = 1 . 4 . 1)istanr~ ~111
Jqig. 13.7. Vc-looit,y lint^ i l ~ ~ ~ li ~ u tilo~no ~ lr~ininnrI~ounclnrylayor on an acliabatic Rat plate a t zero incidence; data identical with those in Fig. 13.6. The dist~ncefrom tho wall is referred to I/v,,, Z / U ; . For w = 1 , we have 1/ I I ~ , , /=V T,/Tm ~
Fig. IX!). Corfficic~~t, olsliin lric,t,io~i for ediabrrlie 11:1(. plaLc a t zero inoiclrnrt: w i t h coinprrssil~lr, laminar borlndnry 1:1yc.r, : ~ h r It111wsi11 1inc1 .1111111son IHY]
Sor : I I ~ Adinbntic coefficient of skin friction: 'I'llc rocSfic.ic:i~l,of skill f~.ic:t.ioi~ adiabatic wall, a s cnlculnt.ct1 by W. Ilnntzschc nntl 11. \ V r ~ ~ t l11as t , 1)crn plot.t,ctl in tcrms of tho Mach nnmbcr in Pig. 13.8. F o r co -.. 1 t,hc protluct c, R is intlcpr~ltlcnt of thc Mach number, b u t Sor tlifircnL v a l ~ ~ cofs rr) t - l ~ cc:ocKicicwt of s l i i ~S~.ic.l.io~~ ~ decrcascs with increasing Mach nurnbcr, t h e ratc of ~IccrtascI~cinglargcr for srn:dlcr va111esof o.Figurc 13.9 contains a comparison bctwccn Lhc valucs of tllc cocl'ficicnt t l scvcm.l aut,l~ors.i. e. for of skin friction for a n adiabatic flat plat,^ o b t a i ~ ~ c by different valucs of t,hc Pmntltl numbcr, P, ant1 of the cxponcnt in t . 1 ~viscosity
Fig. 13.10. Bfeasl~ren~mtrr of t,he velocity diatrihution in nnadiabulic, Inminnr I~oi~n~lriry Inyrr i n nl1pc.rsonic Llow, al'lcr 11. M. O'l)ont~olI [28]. Mach number Mm = 2.4. Theory from ref. [I31
338
X I I I . Im~linnrI>orirtclt~ry Inytm i n con~prcnnil)lrflow
funrtiot~ 'I'hr plot sltows t11:11 the I'rantlfl numbrr exert*^ a much smaller influcncr on the rorffic.ienlf of skin friction than the cxponcnt to.
Vrlority n i d trniprrnlr~rctlistril~~~lianq in l h r prcsrnrr of lwnt 1rn1tnft.r: In ~ I I V grnrml ca:lqr, 1111thhrrrl trtr?i+r p ~ r s r n l ,t h r rrhtiort I)ctwcen tho vclocity and t,oml)cra t t ~ r rtlist,ril)ution ( m i I)r tlctlucrtl from rqn. (13 13a). Wl~eri P = 1, it can b r wril t t.11
w h t w 'I1,, is givc.11 I)y rqn. (13.17). '1'11r prcccding rtluation can bc cxt.c~rltlctl1,o I'r:~ntltl 1111n1l)t~rs tlifYrring froni 1111it~y I)y the int~rt~tlucl~ioti of trhc rcrovrry factor, whctl wc: ol)tn.in
In 1.lliscq~rntion,t h r acli:~.l)n.t,ic wn.11 t c n i p r m t , ~ ~ rl',,, e , shoultl be ~ % l c u h t efrom d eqn. (13.18). 1)11t, il, n i ~ ~ sIIC t , rr;clizrtl 1.l1n.t~ this is only a n approximation. Thn direction in which 11(-:1.1, is tr:lnsft:rrcxl c:i.n I)c tlctlncotl from eqn. (13.21) n.nd written Fig. 13.11. Vcloci1.y and tc?mpcroturc
clistrihrtt.ion in oorn~)maqihloIiitninnr bountlnry lnyer on flat plate nL zcro incidence with hrnt tranrler, aftor IIantzaclre anti Wendt [44]
-
-
Wall tempernlllrf! free r t r r r m letllpernt~ltre. T,. T,: P 0.7. = 1 ; y = 1-4
Since for tt) :1 1,111: corfficicnt of slzin fricfion is intlcpcntlcrlt o f the Mach tlumlw.r (IG% 1:3.8), thc r:cl.c :el, which 1 1 t d is tr:~nsk~rrt:tlbrcornrs equal t,o t h a t in nil inc:omprcssil~lt: strmrn. cvlri. (12.81). A survry of lirnt.-l.r:lt~sfc-rcocffirirtlt,s ant1 rct.ovc:ry Ij~t.tt~rs liw I : I I I I ~ I I : I , ~:1.11(1 t ~ u r 1 ~ 1 1IIow ~ 1 1:I.I.~ I ~ i g lMi1t41 ~ I I I I I ~ I ~ ) ( : I . S t:wt I)e (o1111(1 i l l :I ~ m p I)y r . I . IC:tyo 1551. In t,l~isrpr~nc!xion rrf. (1051 may also I)(% mcrit,ionetl.
Calculations conccrnjng compressible boundary layers on flat plntcs which are based on the momentturn-inhgml eqnat,ion (Chap. X) have been pcrfornictl by Th.. vorl ICBrmrin and 11. S. T s i e ~ l[ S ] ; see also Pig. 13.9. Approxiniate solutions for t h e flat plate were also published by F. R o ~ ~ n i oand l 15. A. Ric!l~cll)rrtlricr(71, 1). Colrs [I 71, 1,. Crocco 1221 ant1 11. .I. Monngltrcn 1751. S ~ l ~ ~ t i ofor n s1,110 tyu~lt.iow of lnmir~nrI)ountlary lrrycr~lwith v n r i a l h pro1)crtiw W C ~ Cg i v t : ~l)y ~ I,. I,. Moort: ( 7 7 1 and G. B. W. Young and E. Janssen [1081.
340
XI 11. Iml~innr1)oundnry Inycm in co~nprcwil>lo flow
(1. noundxry layrr with non-zero prrRunrr grndirnt
d. Ilor~ritlnrylayer with n o ~ ~ - z e rpressure o grnclient
The surceeding derivation aims a t expressing the hountlary-layer equations (13.5) antl (13.6) in terms of' t h e new coordinatcs 5 and i.'rhc continrrit~yrqitation (13.5) is satisfied iclrntically by the introduction of t h r slrc:m function y)(x,y) tlclil~crl through its derivatives
1. Exnrt solutions. T h e ralculaiions conccrning boundary layers with non-zrro pressure gratlicnt s are more difficult I Imn t hose concerning flat plates, owing t o t h r Iargr nrlmbrr of intlrprntlrnt variables 1,. Crorco [21] tlisrovered quite early a t~mnsformat.io~t which simldifics t h e task of int,egra.ting t h e equations for t l ~ e cases when r i t h r r (1) P = 1, ant1 t.hc viscosil,y function / A ( ? "is) arbit,rary, or (2) whcn thn 1'mntll.l n u m l m has a n arltitrary value but, p/T' = const (i.e. when w 1). 1 1 1 1 . 1 1 ~spcri:rl cases of ;MI atlinl~nt~ic wnll wi1.h P -- 1 and to = 1, I,. Ilowarth [481, C. It. Illingwort,h 1701 a . ~ ~It~~l'orc~, ittlxi~tsit~:rIl~y vt*r.y i t r t ~ ~ o r h ~ l'vrI~:ri)s rl.. cvcn ~ n o r citnport;lntIy, solutions of this Itintl art: cntployctl a s t.011c41st.ont~s :~g:r.inst, whir11 t.hc n.ronmay of n.pproximal.c prorrtl~trasran 11a j~itlgotl. For l.l~rsc:rrnsons. we 11ow proposo roughly to s l t e l d ~t h e line of rcnsoning which It::uls l,o s i n ~ i h solut.iot~s r starl.ing wit,l~the lllingwortl~-SLc~vart~so~~ t,mnsforn~nt,im.We sh:tll ror~c:lutls t,l~i.s topic with n number of r~~lrncricnl results. Wc shall postulate t l ~ cvalitlilsy of t.l~e I n.ro implic:tl. In t,l~ot::~scof viscosit,y Inw from eqn. (13.4n) so t h a t ro = I ant1 P bountlary layers wilh herrt lrn?is/er, a n n r l ~ i l r : ~ r y1,111, , c:onst,:lnt, w:~ll I ~ : I I I ~ ) ( ~ ~ : I . ~ . I I I . ~ : , II',,, will be assnrncd, so t h a t A', will 11ccornc a oonst:~nf,.In prol)lwns i~tvolving;HI nrlirrbnlic wall, t,ltc stagnation ent,l~nlpyis given by cqn. (13.12):
-
I t is easy t o sco t h a t these 11011ntl:wycor~tlit.ionst.mnsforrn as follows:
ant1 rentains cot~st,nntover t.hc 1)ountlnry-lnycr t.l~ic~ltt~c~ss, itnplying S : 0 (c/. also end of preceding section). I n this cnsc, the sirnila.ril,y of tho st,agl~:~(,iort-cr~tl~:~l~ty prof les assumes a trivial form. 7
-
I/irnilirtq crrsrs: If P = : 1 Iltrn S 0 is a spccid solut,ion of the cr1crg.y cquaI.ion (13.47). 'I'ogcl~l~rr wil,l~rqn. (13.:%0),it. Ict~tlst o t.lta rcl:~~tion between t,cmperat,l~roand volorit,y for nn ntli:~li:~.l.ic \vall tlisrovcratl r:~rlirr as cqn. (13.12). I n this case, cqn. ( I 3.11 ) assumrs 1,hc "incotnprrssil)lc" form of r q n . (9.I ) rnnctly.
Employing the stream funct.ion form :
111,
we r r w r i k cclns. (13.41) i l t t t l (14.-t'i) in f,ltr
T h r similarity vnriablr is int,rotluccd wit11 f l ~ caid of I I I C following assttrnl~t,io~ts. 1.2. Srlf-similnr S O I ~ I ~ ~ O 'I'lttI I U . Illi~~g~vtt~~tl~-Sl~t~~v:~rl~so~t t . r : ~ t ~ s f ( > r ~ ~ ~11a.s n t , ilo~r r~( ~ n usrtl t.o rlvrivc cs:wt, sol~tf.ionsnntl l,o formulntc n I:~.rgcnntnl)cr of a l ~ p r o x i n ~ a l , c proac~tlnrt~s. Srlf-sirniln.r solrttions piny a n important, part, wit,l~int>heclass of exact, solnt~ions.111 tlrc sonLrxL of incomprrssil)lc Ilows, we consitlerctl t h a t n solntiot~1)nlongvtl to this group if I.lm vchcit.y 1)rolilrs 11 ( R : , y) atf two clill'crcr~tst,at,ions n: cor~ltlIN: n~:~.tlc t~ongrncnl.by I.hr npplicat,ion of :t singlc scale l;l.t:t,orritc:I~for IL and y (Scc. VI 1 I 1)). It was t.11c.n sht)wn t . l ~ n surh t sin~ilnrsolut,ions existrsd in t , l prosrnco ~~ of a dcfinit.e gronp of t~slnrnnlIlows II,(:I:). In cnscs of t,l~isI t i r ~ t l .I,ltr pnrli:~ltliffrrant.inl orlr~ntion for t.Iw st rmtn frlttc~l~ion rrtl~trctlt.o nn ortlinary ililrrrrnt.i:ll aqrtnt.ion wlrich is conI sitlrr:~l~ly r:~sirrl o solve I,II:III111s I'orn~rr. Rl:~liillgIISC of :L 111ln11wr of sl.t~(lit~s, l'nr (:x:~tnl)lt:148, 40, 50J, 'I.'. Y . I i nnct 11. 'l'. N ~ ~ : I I I I : I ([(XI. . S I I611 ( I ~ ~ I I I O I I S ~ . ~ ; L ~ in ~ ~ : a( I n ~ t l n l ~ cofr pr:liscwort,l~yinvrst,igntions t.l~at. S I I ~ Isin~il:rrsol~tl.ionsrxist i t 1 1 . 1 1 ~c-nsr of comprossiitlc bo~tntl:rry layrrs ns wcll. l ~ c r ct.oo, ~irnilarit~y rsl.cntls t,o As liw a s 1I1c vrlorit.y Ito~tr~tl;~ry h y r r is cor~ccrr~rcl, 1 . 1 1 ~longil.~ttlirt:~l vrlovily rotnponrnt~,76: wi1.h rcsl~octt o the t,l~armalla.ycr, similarity
/(?I) is a n I I ~ I ~ ~ I I O I V I I where A , 11, r, s , t pl:~ythe part,s of rtntlcl.rrrninotl COIIS~.:III~.S, stream funct,ion, ant1 S ( q ) is the tcrnpcr:tt,~trcS~lnct,iontlclinctl in oqn. (13.35), n o w cor~ccivcdt,o be n functi& of 77 alone. 1Squat.ions (13.50) and (13.51) arc now bmnsfomlcd t o tlto coortlin:~tcs2 an.ntl 17, and i n t h e result.ing cxprcssiorls i t is clcmantlctl t h a t t,ltc terms in 3 m u s t tlisapprar. I n this manner we obtain ordinary tliffcrcnt.inl cqrinl,ions for t.110 fnncl.ions / ( q ) :r.nti S(17). Snch c:~lcnl:~t,ions 11:~vc11ecn pcrforn~nclby 'I'. Y. I,i :wcl 11. 'I'. N:~~irrnir.t.su JOOJ who found t h a t there exist,ctl four clnsscs of solut,ions for 7i., ( Z ) . I~ollowingthis work, C. U. Colten [I61 dcmonst~rht,cdt h a t t,hree of t.11csc classr,~can be rcd~lcctlt o thc
XI 11. T,ntninar I~or~nrjary hycrn in cornpressihlo now
346 (Ic' nntl
VL
arc c:onst,:~rlls).'1'111, fourth case ?it
7--
is
I 0) il, is possible t o find sufficient,ly small valrtc~sof p -- P,,,,,, for w11ic:h I)ot,l~vxlncs of I," arc ncgnt,ivc, t h a t is for wl~iohthe flow 1 1 : ~rcvorsc~tlit,s dirccLion. 1 n the c:~soof :I coolctl wall (8, < O ) , 11oIh valrles of I,," ran Im positive, 1.11al. is 1)0(,11can r ~ p r c s c n tnon-separated ~ flow patrlmns. It is s w n , fin:~.lly,Ifhat, snp:~m.t.ior~ (I,,," = O ) rnovcs i n the tlirect,ion of smnllcr pressure risw :IS t11c l.cmprr:~.l,~~rn of 1,Iw wall is incrrn.sctl. 1.0
I I I ortlvr t,o t,r:~nsformfro1111,11ovn.ri:~l)lc11 f,o the p l ~ y d o at1isLnnc:c l y, it is ncccssnry vqns. (I :1.H), ( 1 3. IO), (1:1.24), (I R.25) nntl (13.62). I t is then found t h t
111 ili.xv
Y'ho f:rct,or nl~oatlof t,hc ir~togmlis comrrrlt,cd from cqn. (13.53), and t h e func1,ionnl rcl:tI.ion b(:t,woon z nntl 2 I I I I I S ~ ,bc tdic11 from cqn. (13.5G). According t o eqns. (13.46) : L I I ( ~ (l3,(;2), I,IIc int,~gr:bnclis
It~ P = I ; f~~rl,llc.r, ot~t.sho~tltl rncvll ion l , I ~ t : proc:c:tl~lrc~s tlt!vc~lol)ctl1)y IS. (:lwsrllwitz [ U ] ,113'.J.:\. %ant 11 101 n t l t l I. l~lnc*ggc.-l,otz :LIKI A . I?. .J(IIIIISOII (:!(i]. :dl w l i ( 1 for arl)it.rnryv:~lucsof the 1'rnndt.l n ~ ~ m b cIn r . the spcc;i:~l(YLSPI v I I w ~ I ~ / ( I , I - - 0 , the last-tncnl.ionct1 nlclhod can I)c motlilictl t o inclwlo the tr:u~sfc.r of l~a;~l;. All proccdrlrc~sarc bnsctl on thc assrlnlpt.ions t h a t o) ::: 1. In l.hc course of the last y a m , \vorlr proccctlcti m:~inlyon l.hc solut,ion of liroI)lrms wit.l~hccct trrr~sfcr.I'ron~ :~tnong1.11(: p r o r ( ~ I ~ twIli(~11 r ( ~ nro rcst.ri(-I.t:,Il o P 1, i t is ncccwlrg t,o ~ n c n t ~ i t.l~osc o r ~ d r ~ t.o c M . h l o r c l ~ ~ r l ~17!)1.(:. ow I%.COII~:II:I.II(I I(. I~VSIIOI,ICO [lGn]. I t . ,J. Monng11n.n['i(i] nncl G. I'oots [851. 1111 of' 111rtnf ~ ~ r O ~ ( ~ ~ . l1.0i t(.II(* ~ ~ :iI Sl S. IrI I~I Il~ ) tion t h a t m -- 1. The second : ~ n dthirtl tncllloti on t,his list serve to tlctcrminc the momrnt~urnt,llickncss, sl go. The pressure rise which lratls to scpzmtion is independent of the deflexion angle and has a value of ahont p / p , = 2. The incidence of transition and srparation in the nciglhourhood of an impinging shock wave are governed principally by the Reynolds number of t h e boundary layrr and by the Mach numbcr of the extcrnal stream. When t h e shock is weak and the lteynoltls nnmbcr is very small, thc boundary layer remains laminar thronghout. Tncrrasing the Reynolds numbcr a t a fixed,.'small Mach number, causes transition towccur at tbc point of impingement,. When thc shock is strong (largo Mach number) and thc Itc.ynolds number is small, tho laminar boundary layer will scparatc ahratl of t h r shock front owing to prcssnrc diffusion; i t may also undergo transition ahrad of the shock front.. When the lteynolds numbor is large enough, transition in thc t)onr~tlarylnyrr occurs ahcad of the shock, w!~cther tho boundary layer has
separated or not. According t o observations made by A. Fage and R. Sargcnt [RBI, turbulent boundary layers d o not separate when the pressure ratio p J p l is smaller than 1-8, which corresponds to a Mach number M, < 1.3 for a normal shock wave. T h t l ~ c cxperimcnt.al r rcw~ltson t h it~l.c:racIio~l I~c!l.wcr?nsl~oc:lcwirves itr~tlI~oilnclrl.r.y lagcrs car1 bc fonnd in the pu1,licntions by W. A. Mair [G9], N. I l . .Johanncscn 1521, 0. Itartisley and W.A . Mair [R], and .J. Lulcasicwicz and J. I
p ~ w w l i ~ irrl:l.l.iotts g
:ire
'-
dy
(14.12)
( -&j j. I'
tlu -. (Irn /L
-
valicl for
:LII
: ~ r l ) i l . r : ~ rV y : I . ~ Iof' I I ~I.Iw
I'I~:IIIIII,I 11111111~1..
I I,(: sl~c*:tringsl.r~,ss:,I. I.IIc w:ill is now
7 ,
T,,
=- 0 dcnotm s ~ ~ c t ~ iI(0) on, clcwh~nI~lowing,and
:.
ill
0
1 : L I I ~ I I,IIc W:III is : ~ c l i : ~ l ~ ; t ti if ,c ,is l ~ o s s i l ~ l v (14.16). '1'11~ rg:t~~g nuatnnsch hri Vrrtlnnol.nng oincn l~liin~igkeitrnfiltns i i l w cinrr p:irrrllcl n ~ ~ g r n t . r i i t ~ I'1:iLt.c ~lr~~ untnr I3oriic:I~~i~~l1l.i~11tig v~rii~~tlt-rIi(:l~cbr Sl.olThri\v(:rt~~. lnl.. . I . I I W I L MILHH' I ' ~ : I I I H ~ ~11, Y 1537 . 1550 .. 110711. ,[22] Emmons, H. W., and h i g h , D.C.: Tabrilnt,iori of Illnsinn funat,ion with blowit~gund ~uct.ion. ARC CP 157 (1954). r231 Borcchnnng e inn~irinrcrrind t , ~ ~ r b ~ ~AII~I~II~(~-(:~(~II~,R~~II lcnl~~~ - - E ~ p l n r ,R..: r r n . k t . ~ ~ c h ' I & . - A ~ ~ I$2, I . 221 2 4 6 (I!wB). 1241 lCjil)lrr, It.: ~ ~ c m c i n n ~ t m ( ~or c ~ ~ z s c l ~ i ~ ~ l ~ t fiir ~ n ll ~l oo~n~~I~~g: ~t ~~~~S~IIII,! tg. r i (S* l~~~ I I I I ~ ~.lh IIIII~. WGI, 140---149 (1962). -3
404
References
XIV. 13oundnry-laycr control
1251 . . Ihnldrrs, C.R.: A notc on 1;irninar Inyrr skin friction under the influence of foreign gnn injnction. .JASS 28, I(i6 - 167 (I!WI). 1261 Ihvrc, A,: ( : o t ~ t , r i l ~ ~ ~ht ,I'6tude i o ~ ~ expi:ritnent;rle des mouvetnentn I ~ ~ d r o d ~ n a r n i ~ u e s b d c u x elitncnsie~ns.'lXhesis Univcrsit,~of Paris 1!1:38, I-- 192. [27J I~lnt.1,.I.: 'I'hn I~isLoryof boundary Iqycr contxol rcsearch in the United S t a h of America. In: Honntl:iry I:ivcr and flow control ( G . V. I,achm:um, cd.), 1, 122-143, London, 1961. 1281 l?liiecl. (:.: l~r~e:l~nissn BUR dem S l , r i ~ ~ n l ~ l ~ g ~ i t dcr l s t i'I'eeltni~~l~nn L~~t 1 ~ 0 c h s c h nT)anzig. l~ .Jb. ' ~ch~ff1;arrtcc:h;;.(:csellsr:ll:ift 31, 87 -- I l:f (l!)RO). [2!)] Fox, H., and I,ihby, P.A.: Ilcliu~ninjection into the boundary layer a t an a x i ~ ~ m t n e t r i c stagnation point.. JASS 29, 921 (1962). 12!ta] (:C~RI.CII, Ic:bt.o 1,Itc Imly ant1 assuming t.I~att h r ll~titl ~ assl~mplionth:~t.I,he vrlocit,y niovrs willt rcy)c~c.I, t,o 1 . 1 1 ~I)orly a t mst-, \vv W I I I I I : I ~1.l1r. is c.ontlws~~(I of' 1.1~01.rrtns
ITttclw lsl~rscc*ontlit.it>ns1.hr first, n.~)~)rositrt:lt.io~~, it,,, snt.islics t.lte 1incn.r tlifTorcnt.inl (v111:tti011
whore
nntl
General rcmarka on tho calcolntion of non-st.cady boutlclary layrrs
41 1
412
XV. Non strarly Imnndnry hycrs
'J'ltc c~sontial.simplification of t h e theory consists in retaining only the throc underlinetl tnrrns in cqn. (16.22), which is thereby linearized and reduces t o
Ry rstimnting ordcls of magnitntlc it can be shown t h a t t h e preceding approximation is a valitl ont? if the r:~tioof the so-callcd "ac" boundary-layer thickness,
formrtl with the frequency n of t h e oscillation, is small compared with t h e steadys t a t e 1)oundary-layer thickness 0 which would exist if IJ(x, 1 ) were equal t o T J ( s ) . JIcncc, for the approximation t o bc valid we must have
wltiol~,in ~r;lc!l.icc,restricts the t,hcory to vcry high frcqucncics. I t will be recalled that, t h e quantity a,, cqn. (15.24), occnrcd in t h e solution t o t h e problem of a n oscillating plate which has h e n considered in Scc. V a 7. Equation (15.23) which is linear a n d related t o the so-called heat-contluctio~i oqu:~t.ion(6.17) describes t h e oscillating component ul of t h e boundary-layer profile and can he solvccl in terms of the given oscillating component U 1 of t h e potential flow alone, b c o a ~ ~ st,lic c process of linearization has made i t independent of t h c mcan mot.ion. T h c normal componcnk of t h e flow can be calculated from the equation of cont.inuit.y (15.1) ivliich can be split into a n average part
a. General remarks on the calculation of non-steady boundary laye
413
from t h e outaet. The difference is clearly brought into evidence by tlte appcaranep of t h e function F ( x , y) ;i t has its origin in the non-linearity of the differcnCial equation.
It will be stated later in Chaps. X V I I I a n d X I X t h a t the essential charaoteristic of a steady turbulent stream consists in t h e faet t h a t on the mean velocity of flow there is superimposed a random, three-dimensional, quasi-periodic oscillat,ion. Cbnsequently, problems involving turbulent frce slrcnms c x l ~ i b i ttho same featnrcs a s those being discussed now; they involve changes in d i r e ~ t ~ i oans well a s in t h e magnitude of t h e free-stream velocity I J . Tn most cases i t is cust,ornsry t,o tlcglcct, t h e free-stream oscillation and to calculate a s if t h e flow wore stcady and :LR if the potential velocity were given b y 0 (x) instead of lJ (x, t). 'l'his is cquivalcnt t o omitting t h e additional term F ( x , y) in eqn. (15.20) a n d necessarily leads t o a n average velocity profile which isdiffercnt from ti ( X J ) Tho preceding remarks show clearly t.11:~t~ tltc order in which t h e two operations, averaging and solving the c:cjn:~l.ions,arc pt:rSortr~cd is not immaterial and aKccts t h e final msult. 4. Expan~ioninlo a series when a steady stream is per~urbedrligldy. Very oftr:n, pr.c~l~lcn~s in non-steady bonndary l~iycrsinvolve nn c:snenl.ially nfencly flow c r l l \vlric:h 1,lrt.n: is ~~~lwrirt~l,c~srtl lhal 1110prL1rr1mLi011 is ~rn:dlt:nn11):irc(l wiL11 a small non-stcndy pcrlurbntion. If i l in 1~~sumct1 the steady basic flow, it is porisiblc In split the eqnalions into a non-linGwbounclnry-layerequation for the steady pcrturbetion. A well-known exatnple is that for wlrich t.bc cxternnl st,rr:un 11n.s tllo form U(z,t) = d ( z ) -1 s U,(z,t) + . . . , (15.28)
whcrc E d e n o h a very small nntnbcr. T l ~ crnosl itnporhnt ~poc:i;dcilst: ~ I I C I I the estrt.n:d pt:rtarbation is purely harn~onicwx.s studied e ~ l t a ~ ~ ~ byl M. i ~.I.~ I~igl~t.l~ill ly 127).'I'll(: S:HII(\ t , y ~of) ~ linrarizalion can be c111ployecIwhen the i~n~pcrnlure e t Lhe wall is rr~~rosc:r~lntl 1 ) ~ .l . 1 c:xprcssio~r ~ lTw (z,t)
=
pw (z) +
E
TTw, (z,t)
(15.29)
or when the wall ikelf performu smell, norr-steady. pcrt.t~rl>ing rnot.ions (oscill:~ting1)oclics). for 1 . l (Iyn:~n~it. ~ :LR \~~cll as In such cases we start with the assnmplion t.l~;rl t l ~ csol~~l.ion.q for the thermal boundary layer nrc of the following forn~s:
Ilaving solvotl for the oscillation ~ ~ (y,zl ) , n,(x, y, 1) we can rctnrn t o eqn. (15.21) nncl c:~lculate tlic function F ( a , y) which appears in eqn. (15.20). Tho lattcr now dcscrihcs tho mean motion d ( z , ?I). I t should he notctl t h a t t h cquation for t h e mean flow, cqn. (15.20). has a form wl~icliis identical with the steatly-state version of t h e boundary-layer equation. I'hc only tliiroronco consist.^ in the a.ppearance of t h e acl(litiona1 term F ( x , y); t,his now plays tJic same part, a s tht: term If . tlV/dz which originates in the pressure graclicrit,. Both tt?rm.s mprcscnt, known f~lnct,ionsin the diffemntial equation. The only tIifirrnc:c consists in the fact tliat, t ~ i cmean pfcssure gratiient 17. dIf/dx is "irnpressed" on tho hor~nclarylayer and is intfepentlcnt of t h e trsnsvcrse coordinate 11, whcrcas the :~tldit.ional term F ( z , y) dcpcnds on it. Owing t,o the existence of oscillatory compont:nt&, the average flow is tliffcrcnt, from l.hnt. which wonlrl be ol,t,aincd if Llie potential velocity I l ( z , 1 ) were averageti
'J'lie postulated forms from eqns. (15.30) arc introdnced into eqns. (16.1) to (15.3) and Lho losnlting terms are ordcrcd with respect to the powers of E. From the rcquirctncnt that, the tlini?rcntial expreaaions which mult,iply enell power of s muut vanish singly, we obtain a msc:rtlo nf cliffcn.ntinl equations. We list them for t.he cmc when Q = const, wlicn t h external llow is of 1.11(. (i)rn~of eqn. (15.28), and when the wall temperature is given by eqn. (15.29): Equations for zeroth order (steady bmic How):
414
XV.
h. llorlntlnry Inyrr for~nnt,ionnltrr impr~lsivcutnrt of nlotion
Non-st.cncly hour~tlnryInycru
with tho I)o1111(1nry contlit,itms ,J- 0 : 16, -. a, -- 0 ; 7'" = T-'," ),( , U" = (7 (z) ; T', = T ', . 1, m:
-
ISq~~ntionn of fir& ortlor (purely non-stcndy):
au,
au, 1
au" =
415
5. Sinlilnr mid ~emi-similnrsolutions. When we stuclictl t,he throry of' st.rntly, two-dimensionel boundary layers (see Sec. VIIIb), wc clcscri\)otl a s similar that, class of solutions for which t h e depcntlenco on t#hc two vnriablcs 3: : ~ n dy c-oi~ltlI,(: rcdl~cctlt o t h a t on n singlc variable 71 hy t h e npplicat,ion of a st~iL:~l)lo simil:~rity tran~format~ion. In s.11analogous manner, we say t h a t a solution of a non-steady two-dimensional problem 1)elongs t o the class of similar solutions whnn thc three independent variables x , y, 1 can be reduced t o n single variable TI. 11. S c l i ~ ~1461 l t and Th. Geis 1101have intlicatxd all such solutions for which a rctluct.ion t o n single v ~ r i n h l c is possiMc, t.liat; is, s i ~ c has arc of tho form
a~
For example, cxtmnal flows of the form IJ (z, 1 ) = mx/L and the cascs when IJ ( x ,1 ) .- (=tn mentioned in See. X V c belong to this class. T h e similar solutions for ctn cxtfrrnnl stream of tho form 1J ( 2 , I ) -- x / ( n - 1 Id), whcrc a and b a r e c o n s t a ~ ~ twcrc s, : i ~ i : ~ l y ~ ~ l by K. T. Yang 1711. I f a transformation can he found whiol~reduces Lhc IJlrcc indopcndcnt, vn.rial)lcs x , y, 1 t o t.uro, we say t h a t the resulting solution is scmi-similar [21]. I n particnl:~r, when t h e vnriablcs are r c t l ~ ~ c etdo y and x / t , t h e solutions arc also called pseutlostmcly ( r / . 171). A soll~tionof t h i ~t,ypc was tliscovorctl by 1. Tani I.561 for the (::tsc wltcn the cxtcrnal flow is given by U ( 2 , 1 ) = ( l o- x/('Z' - - I ) , with 11" and rl' tlonoting constar1t.s. A wider class of semi-similar solutions W:LS consitlcretl by I f . A. Ilnssan [In]; scc also rcf. 1211.
,
.
I Iw c y ~ ~ : ~ t i r of ) ~ ~11igl1c.r rr ttrtlrrs I~nvc:corrrapo~~~linp nl.r~lrt~l~rrn. 'rho prccoding nyst.c~nnofrqr~nt~ionn I)(* rolvrd one. :I l'lcr l . 1 1 ~ol,lwr, it, I I C ~ I 11ol.w1 I~ LI~rrttrll, oxccpt tl~oncof zc.rcrt 11 ortlnr. nro litwrr. I l oq~~:llions (15.1) t o (15.3) wrrc t,o psscss rsnvt solutionn of t.he postul.zktl lorn1 (l5.:10) I I Lo ~ orilrr P , t.I~rn,g(!ncr:tIIy slwdting, tho solutiolm nrrivctl nt by t,llc prccrtling scl~c~nn worlld tlill'er l'ro~nt.lw C*SILI% solut,ion I I ler1118 ~ of or(1cr 1 4 1 I I . IYIII
XV. Non-steady boundary layers
b. Boundary-layer formation after irnp~tlsivcntnrt or nrotior~
where TJ(x) tfcrrotcs t h e potential flow about t h e body i n t h e steady state. I n this particular casc we have a U p l = 0, and equation (15.12) of t h e first approximation hccornc.~simply
with t h e boundary conditions 5, = 5,' = 0 a t 17 = 0, anrl to' = 1 a t ?I =- oo. Equation (15.42) is identical with eqn. (5.21) and the solution for C,' is intliratctl in eqn. (15.39). The function 5,' is shown plotted in Fig. 15.1.
416
at
- va2L1,
av1
=(,
(15.37)
with 11,~ 0 h r - ~ =0, ;I.IUI M,) = ( J ( x ) for ?J -= m. This cqu;~tionis itIcntic:~l wit,l~ t.llat, for one-din~ot~siorrd ltcat contluctior~.I t was solved irr Sec. V 4 for the casc of a plnt,v st.;irt,ctl in~p~rlsivcly in its own plirnc, while tho fluid was a t rest a t a large tlist.;~ncc:frotn it. I t was tl~crrpossiblc to introtlucc a new tlimcnsionlcss varin.blc (sl:in.iltr~il~y Imn,s/orn~wlion.) : !I
'l=21/;i.
417
Combining eqn. (15.13) with (15.40) we obtain t h e differential equation for the second approximat,ion C1 ( 9 1 ) in t h e form :
C,"' -1- 2 q
el" - 4 el' = 4(5.,'2
- ~ O ~ O-" 1) ,
wit11 the bounclary contliLions C1 = 5,' = 0 a t 11 -= 0 ant1 solrrt,ion tlcrivctl hy XI. 13l;~siusis:
[,' = 0
a t 11 --
m.
'I'ho
(15.38)
In t.lris m;rnnor w c ol)t,ain the solution in t h c form IL,(Z, y, 1 ) = U (x)
x Cljt(q)= U (z) erf q .
(15.39)
'I'his is l,hc first, ;~pproxirnntionbotjh for the two-dimensional and for t h e axi-sym~nct.rirnlcase. I'nrt.lrcr, if the pot,cnt~ialvolocit.y is inclopenclcnt of z , i . c. if TJ = : (1, :- const (II;rt plate ;I.!, zero incitloltc:c:), oqn. (15.39) constitr~tcs the exact solution of cqn. (16.2). since thc: c:onvot:tivc t r r m s in eqn. (15.13) vanish together with the prrssurc torm so t h a t T I , E 0. I Iowcvcr, t h e solution arrived a t in this way does not c:onst.it.t~t.c t,he complctc solubion t o t h e prol)lc~nand applies only sufficiently far downst,rcmn whcrc thc influonce of the ctlge is negligible and where the flow behaves a s if th(: plate wcrc infinitely long. Strictly spraking, t h e complete solution must also satisfy the condibion t h a t ~ ( 0?J,, 1) = 0 for all values of I/ and 1. T h e complct,e solution is givcll in ref. 1541. 111 the gonrrn,l casc, wl~c:n Llrc external flow U(x, 1) tlcpends on t h e space coortlinatn, i t is nwcssary t o make a d i s t i n c t i ~between ~~ t h e two-dimensional and the
I. Two-climenuionnl cnue. Wc shall begin by considrring the two-climrnsiot~d c.:tsc. 19,r this r m o wc assume a power series in t,imc for t h e stream function ~ t i p u l a t i n g th:rt, it, has tl~t,form
Fig. 15.1. 'l'hc fw~ctionst;l nut1 = and t i b For t . 1 velocity ~ distribution in tho 11011st,cndy horlr~dnryInycr, cqns. (15.41) ;u~tl(15.50).for itnpulsive tno(.ion The function (1' is shown plotted (as funct,ion (I,') in Fig. 15.1. 'I'hc initial sl01)cs of the two functions, required for the calculation, of sci)nratior~ arc givcn 11.y
An exact expression for t h e next term of t h e expansion of tlrc stream f u n o h u it1 t,nrms of time was obtxinctl by S. (:oltlst.cin ant1 1,. 1Losonhr:cd 1141. Iil,l)c~lntn112]), 1 1 r:rsr ~ of t.lw flat plntc a t zero incidence is reprcacntcd by thc cxpreasion:
nntl with
Asnutning in cqns. (1G.32) that
u, =
E
ei"'
V @ , ( 6 , 11) ,
wo arc?Ircl 1.0 t.hc ftlllowing tlilTcrenl.inl cqnntic~nnfor t.lw rwxilinry functions @(E, 7) and O (E, 7): ln -1- 1 nt 1 1'' @ - ( I - in) 1' E @,,E 4-- - / (D,," - ( € t 2 m/')@, -1fDq.t'l 1 2 2
+
+ (1 - m ) / " € @ t + [ .I-2 m = 0 ,
(15.76)
SubstiLuting n = 0, wo rewvcr tho uasi steady solution, which signifies thnt n t every inst,ant the solution behnvw like the s h a d y J u t i o ; for tho instantnncous cxternal vcIooity The a penrnnco of an imaginary term n t n ==! 0 moans t h a t the boundary layer aull'ers n phase shift wit( respect to the cxternal flow, the shift being diflercut for velocity nnd blnpcrntr~rc.Wl~ereasthe rnxxirna in shearing st,rcsa lend thc tnnxima in the cxternnl Ilow (in the limit n x/IJm -+ CT 1.11~pllnse ar~gletmda to 459, the mnximn in lnmpcrnturo Ing hchintl t.l~rrn(in the limit, ?l,:r/!~,., - t m~ lhc pltmc nnglo tendn to '30"). 111ntldition, iL turns ont t.l~trtn t lnrgt! VIIIIIOR of n ~ / l l ~t .. ,1 nn~pli~ tudc of thc ahenring-strrxw oncillntion incrcnaca withont bound, wl~crenst,l~nLor I.l~c? I~nnt,llnx slowly decays tm zero na n %/Urn is mndc Lo incrcnac. When thc solution of the system of eqttntions (15.33) is corrictl to second ordor, it is h n d thnt tho functions u,(z.y,t), v,(z,y,l), and l',(z,!/,l) cont,nin n Irnrmonic pnrt of donldt: f'rotp~cncy Inl.lr.r tnotlific~~ I.ltc\ 1111sic: and n ~upplemnntnry,a b n d y pnrl which in inclopontlt\nl, or 1.itno. 'l'l~c~ 11111tlogy wi1.11 1.1111t,t . ~ t ( ~ t ~ I ( i~l l r ( ~ c I flow and cnn I)o intcrprcbtl na a secondnry flow in cot~~plnhr the solutions of tho pwccding seelion. For shgnalion flow, wc hnvc Ul(a) = consl, anel it in f o ~ t dt.lmt t.hnrl u,,o, and all higher-order t e r m vanish, a s demonstrated by M. R. Glnuert [13]. Conneqnently, the basic
436
e. Periodic boundary-lnycr floe.s
XV. Non.stmdy borlndnry lnycra
flo\\- n t ~ g ~ ~ r n I)y t w lt.l~c: t.rr111stl1 :1.11(1 v I rol~slit~utrs nn exart ROIIIL~OII, one, ~nor~ovrr, ~ l ~ iisr l ~ :~lsoc - s : t r , l for lhc; v o ~ ~ ~ p ~ Nlivirr-Stokrs (~Lv cqllntions ('/. also rvf. [67]). I$y :t ~llit~I110 1XR114f i ~ r t t i : i I i o t of ~ v:iri:~l)lrs, t h 11rtwdi11g r:isc c:i~t bv I I I : L ~ C tn yield l,Ito solt1l~ion8for st~agt~aliot~
~ ~ first. Rivcl; in rrfs. 113. 67, 21. A solution for tho caso of nn ililinite IIo\r O I I M I o s c . i l l : l t i ~ \wll ll:it pI;itv w i t h suc~t.iol~ tui 0 )
antl for cxpnnsion waves (1I,/1JS
< 0)
\Z'hrn P -r I, wr have r ( 0 ) - I, antl t h c adiabatic wall t.cmperatVnreIwcomcs identical with the stagn:~tiont c n ~ p r r n t u r rIc/. rqn ( 1 3 17)] Wlicn the Prandtl number of t h r gas difrrrs little from unity, it is possible, according t o H Mirels 1291, t o rmploy t h r npproxirnnt ion t h a t r ( 0 ) = P" , with
I --
a-039a:
0.50
0'02
(U,./Vs) 0.13
--
I -j
for for
urn :-0 (comprcsaion (1.9 Urn
u, < 0
waves)
(15.104)
( c x p n s i o n waves)
,. I hns, finally, t h r t,cmprmt,~lrcdistril)ution becomes,
exccccSs tho so-nnllctl Itnyloigli v a l ~ i cwl~cnthe The I ~ o ~ i n d a r y - h y thickness cr wave is cornprcssivc; this en11sc.41 h s h ~ ~ r iahrrus, t t ~ the ~ k i t ~ . r r i ( : t (:ot:l'fi(:i(-t~t, io~~ 11ttcI tho Nt1ssc11, n ~ t ~ ~ t l1.0w rII~:(YXIIII XIIIILIII:~ ~ott~l)~~re!cI wit.l~I.11oir ICnyIt:iKI~v ~ ~ l t t t'I'l~t, *~. opposita is trlio for c x p n s i o n W:LVCR. In t11c spnci:~l c;rsc when P - - I , t.ttt: Ilt::rt.transfer formulae rct111cc to Chc simplc Itcynolds a l d o g y
known t o the render a s oqn. (12.55). r .
For the skin-frict.ion rorfficirnt,
1110 precrtling problom which cliscussctl the hounc1:rry layer brl~irrd n shock wave of constxmt velocity c o ~ ~ s t i t , ~at b n sidcnlizccl special c:rsc in thaL iL call bv mduccd t o a stoatly problcm 1)y the fr1iciI.011~ c h o i ~ cof :L coordinate syst,rm in wllic11 the shock wave is at, rest. More gcnrrxl sol~rt.ionsof t h same problem have boon trmtctl in the works of R. 13ccltcr 13, 4, 0, 71 ant1 11. Mircls and .J. Ilammnn [301. 2. Flnt plnte nt zero incicler~ccwith vnriable free-strenrn vrlocity nrd mrfncc Inycr temperature. In our scmmtl osaniplc wc col~xitlcvtho cotnpressil)lc I)or~nrl:~r,y on a flat platc whcn t h r frcc-sbrcnm vcloc:ity, 11,(1), a s wcll a s tho tcmpcmt,urc a t the surfacr, T,(t), vary in t2hccorlrsc ol't,imc. 'l'hc strc:brn rt~nctiony) rrom cqn. ( 16.90). and the t~rml)crnt.urcclistribution
Onrr ngnin. ncw~~eling to 11. Mirrls [29l, whrn t11r Prnntltl numl)cr is near to unity, it, is possiblr t o rrsort to Lhc following approximations: in which the pmss~lrc-grntlicnttcrrn lixs I~ocndclotctl. 'rho variablc has I)rcn ele-firwd ~~~~ in cqn. (15.9l), :tnd (I,, anti 7', tlvnolo the tlcriv:rtivos of f r ~ o - s t r c : :vclo,'il,.y R I I ~swface I t,rmpcri~t.urcy i t h rcspcct. t o kinrc, rc~spo~t,ive;ly. 111 orclcr t,o :rrrivcx :if. solutions, thc following series cxpn~tsionsarc postulal.ctl :
444
XV. Non-st,mdy I~oundnryInyora T h e t h e o r y of l a m i n a r , n o n - s t e a d y b o u n d a r y l a y e r s has been dcveloprtl c o n s ~ t l -e ~ ably i n t h e l a s t years I n f o r m a t i o n o n t h i s phnac c a n 1~ fortntl i n t h r r c v o l ~ t t n r sof c o n f c r c n r c pro~w-tlings. '1'11~ fitst, rtlilrtl Ity 15 A 15icl1rll)rrnttrr, rc%l)orls(111 1 1 1 ~
IU'I'AM S y m p o s i u m " l t c r e n t R e s e a r c h o n U n s t c a d y I3ountlary Layers", Q r w l ~ r c1072 [74]. T h e s e c o n d , e d i t e d by R R. K i n n ~ y[76], concerns a symposirtm o n "IJnstcndy Aerodynamics" held i n I 9 7 5 a t t h e U n i v e r s i t y of Arizona,. l'hc t h i r d is tlcvotcd l o n n r] =
--
22
AGARTI m e c t i n g lrcltl i n 1077 [7BJ. A rcvicw p n p c r b y N l t i l r y m a y also tnct it c o m -
Urn2
parison [37a].
voo
tit,finrs a I I ~ \ Vt l, i r n t ~ n s i o t ~ l r scoordinntc, s ant1 t h e following a l ) b r c v i a t f i o r ~hs a v e b e e n mi p l o y r d :
r
7
I h r lwct:c(ling fortns a r c s n l ~ s t i t ~ ~ ti nn t lo t h e d i f k r c n t i a l cqrtations f o r t h e b o u n t l a r y Inycr ; ~ n t lit, is l i ~ t t r dt h a t , t h c futlctiotis F ( q ) , /O(tj) , . . . s a t i s f y o r d i n a r y diITcrentia1 ctlttnt.ions. Solrrtions f o r t h c m wlron P .-. 0.72 11avc been g i v e n i n refs. [35, 491. 'I'lre functiorts F ( , , ) , O,,(T]) ant1 A'(71) a r c i d r n t i c a l w i t h t h e s o l u t i o r ~ sf o r t h e s t e a d y s ( q u a s i - s t e a d y flow). prol~lt-1x1witlr CI,., in(.rrprc:t,cxl as t h e i n s l . a r ~ t a n o o r ~vrlocil,y ,I h c rcrn:iining t.c:rt~~s tlcst:rilm t.hc? c l o p : ~ r t ~ t r ofsr o m t h o q ~ ~ : ~ s i - s t ~ tsw ~ Jl y ~ttion.
.
( : o r r c ~ s l ~ ) ~ ~ t l i r t gIJIV I y , m l i o of hcv~h f i r ~ x r s :it, tltc wall for P = 0.72 ( c / . drscrilwtl Ily I
+# ,
--- .- , , - - k . . . ,
I ,
--
I",
1501) is
[I] Andratlc, E.N.: On tho airculnt.ion musod by tho vibr1lt.ion of :rir in 11I.IIIII..I'rov. 1111y. Stw. A 134, 447-470 (1931). 12) Arduini, C.: Strnto limite incomprcnail~ilcInrninnro ncll'int.orno do1 pnnt,o tli rist.ngt~ot l i 1111 ciliuclro intlofinito oac:illanlo. I,'Aorolcc:nii:l~ .I/, 34 1 34lL (l!)lil). [:)I I)ct*kor, I?:.: 1)m A I I W : L C ~ dcr I U CC~OII~.H(:II~(:II~~ ~ in I I I I ~ I11iut.t:r (:illor I ~ ~ x ~ I I L I I H ~ ~ I I I H ~ v IIIU.. ~~II~~. Arch. 25, 155.- 103 (1957). [4] Reckor, E. : lnahtioniirc Crcnzscl~icl~tcn l~intorVarrlicl~tr~~~gsst.iinn(!~~ I I ~ I ~I I' : x l ~ n ~ ~ s i o ~ ~ ~ \ v ( ~ l ZPW 7, 61-73 (1959). [5J Bccker, E.: Dic lnminnre inkomprcasible Grcr~zscl~irl~t nn rinrr tlural~I:rufrntlc \Vrllcu deformierten ebenen Wnnd. ZFW 8, 308-310 (1960). [fi] Bccker, E.: Instationnre Grenzschicl~tenhintrr Vcrclict~t~~ngsst,iiase~~ unrl E x ~ ~ n n s i o ~ ~ s a ~ c ~ I I e Progress in Aero. Sci. I (A. Ferry, D. Kiichc~nnnn,nnd L. Iulctitint,ervals;at positions closer to thc pipc wall, conditions arc revcrscd. Sincc during thc cxpcrimcnts care was taken t o maintain a constnnt rate of flow over long intervals of timc, it is concluded t h a t i n t h e region of intcrmittcnt, flow tho velocity distribution alternates between n corresponding dcvclopcd laminar distribntion, nntl a corrcsponding fully developed turbulent distribution, a s shown in Fig. 16.1 a and 16. I b respectively. The physical nature of this flow can be aptly clescribcd with the aid of the intermittency factor y, which is defined as t h a t fraction of time during which t h e flow a t a given position rcmains turbulcnt. Hence y = 1 corresponds t o r~s flow. The intercontinuous turbulcnt flow, nnd y = 0 dcnotcs c o n t i n ~ ~ o laminar mittcncy factor is shown plottctl in Fig. 16.3 for vnriol~sReynol(1s nnnihcrs in terms of tho axial distnncc z. A t n consf,nnt Itcynoltls nirrnl)c:r, the ir~l.crr~~it,l.t:t~c:y factor increases continuously with t h c distance. Thc Rcynolds numbcrs cover tho
a. Some exprrinicnlnl results on transition from laminar 1.0 L~~rl~ulrnt flow
453
important ones being the prcssure distribution in tllr rxternal flow, tthr rl:lfurc- or tlir wall (ronghr~rss)and the nature of t h r disturbanrrs in t,hc frro flow ( i r ~ i r t i s i I j ~ of turbulcncc). Blimt Ilodies: A pnrticularly rcvn:wkal)lo phenomrnorl c:onrlrnt,rtl wit,l~i,rt~nsilit111 in thc hour~tlarylaycr occurs with blunt bodies, for c x n l ~ ~ p spl~cros lc or (:irc:~tl:ir cyIintl(:rs. I t is seen frorn Figs. 1.4 : ~ n d1.5 t h a t t h c r1r:i.g c:ocflicic~rlto r a sj)l~c:rr:or cylindcr tlccrcascs : ~ l ) r ~ ~ p ia, tI yItcyr~oltlsrn~mbcrs R :-:I r I ) / v of al)orlt :1 x lo5. ,.I his a l m p t drop in t h c t l r : ~corfficicnt, ~ noticctl l i d 11y 0. 12ifli.l [231 in rcsl;rt ion I
ITig. 10.2. Variation of flow vclority in a pipc i n thc tranuition range a t dihrent distances r from pilw axis, as 111cas11rrt1 by .J. I n d ~ kn u r n 0 ~ rA = ii.d/v = 2 5 5 0 ; axial t l i s t n n ~ ez / d = 322; E x 4.27 m/scc ( P 14.0 ltlsec); vclocitien given in n r l s r r . .rhr.rr vclorily plols, o b l n i n n l with t h o m i d o l n Iwt-wire nncmomatrr, rle~nonstratethe i ~ ~ t e r m i t l o naluro nt vf l l m I l o w i n f l t x l pcrindr n l Inrninnr n n d l u r h u ~ l r n lflow aurcerrl crch othrr In time
Fig. 16.3. Intermittct~cyfactor for pipe flow in the transition range in ternls of the axial distance z for different Itrylrolds nurnbers R, as rneasr~rpdby J. Rotta [75]
y
I l r r c y = i dcnotcs r n n l i n l l n l l ~ l ytllrIHIIPIIL, I I I I ~ y = 0 1-onti1iunil5~y larnlrlnr Bnw
d
rango frorn R 2300 t,o 2600 ovcr which transitpion is completctl. A t Rcynolds rtllrnhtw rlcar l , l ~ lowcr c limit, the process of tmnsit,ion t o f ~ l l ydcvclopcd turbulent Hr)li(:J1 1,l1(, (lo,,r pS~,t:tl(lS ~ V ( Tvery I ~ g tIist,anccs c mcasurctl in thousa~idsof tlialnctcrs. ol. i,llis kind have been reccntly amplified by J. Meseth [GO]. Mc~asrlrc:tnt~nt.s 1
I : ~ ~ '1'r:l.tlsil.iotl t.;tttsi*x t o sphc:rc?s, is a conscc~~~cnc:c of Lr:lrlsiUor~it1 the I ~ I I I I ~ 1:iyc:r. tltc jminl. o f scp:lr:lt.ion t o move clowt~st.rcar~~ wllicll consitlrr:~l)ly tlt:crc~:~sosi,l~(: width of the walrc. 'l'hc truth of this cxpl:in:~tionwas tlc1nonsi.r:il,nt1i ) y I,. I'r:r11111.1 1411 w l ~ onlor~ntctln thin wire hoop sotncwl~:~t .-llrntl of tht: cquator of a si~llorc:.'l'liis causes artificinlly t h e b0undar.y layer t o 1)ccomc turbrllcnt a t a lowcr Ilcynol~ls numhrr antl protluccs the same drop in drag a s occurs w11t:n I,lic Itoyt~oltlsI I I I I I I I I W is nmtla to incrrasc. 'l'hc stnolrc photogrii~)l~s in l'ig. 2.2.1- nntl 2.26 S I I ~ )+~: ~~ r~I j ~ t h r cxtcnt of t,hc waltc on a sphcrc: it1 thc sub-critic:~lIlow rcgirnc t,llc \v:~ltc is uitlt. arid t h e drag is large, antl in t-bcsupcr-crit,icnl regime i t is narrow nntl t h c clrag is stnall. T h e l a t t r r flow rcgimc was here crcatctl witll-the a.itl of L'rantltl's 't,ripping wire'. These experiments show coriclusively t h a t the jump in the drag curve of a sphrrc is due t o n boundary-layer cKect and is caused by t m r ~ s i t i o ~ ~ . Flat plate: Thc procrss of transition on a flat plate a t zrro incitloncc is sonrrwhat. simplcr t o understand t h a n t h a t o n a blunt hotly. T h c prorcss of t.r:rnsit.ion in t.11~ bountlary layer on a flat plate was first stntlictl by ,J. nl. nrwgrrs 161. 13. (:. van der IIcggc Zijncn 1411 antl Iat.er by M. ITanscn antl, it1 grca1.c.r clrt,ail. I1.y 1 1 . I,. 1)ryclrn 116: 17, 181. According t,o Cl~:lp.VlT, t,llc bo~~ntlnt~y-layer t.l~ivlztlrsson :i flat plntc incrcases in proportion t o whcrc s tlcrmtlcs t l ~ ctlistancc from thc Ic:atling edge. Near t h e lending edge the bountlary Iaycr is always I a m i t ~ a r t ,l~cnorningi,urbulent further downstream. On a p l ; ~ t cwith a sharp lratling edge ant1 in a. tlormal ~ ~:I air stream (i. c. of int,crisit,y of turbulcncc T = 0.6 %) t,mnsit.ior~t . : i l ~ sp l ; at, distance z from it, a s detcrminecl hy
j/z,
On a Axt plaLe, in t h c same way a s in a p i p , the c:rit,ic:il Itcyt~oltlsn11~11)c.r ran 1)r incrcasctl by provitlitig for a dist,urbancc-frcc cxt.orna1 flow (vc.rp low ir~t.c*~lsil,y or tllrl)ulcncr), c / . Scc. XI1 tl 2. i l l (11t'I'r:~rtsit,ionis casirst t o pcrrrivo Oy a s111tlyof t.lw vrloc~i1.ytlisl.t~il~i~iiot~ 1)oun~Iaryla,ycr. As SCCII from Fig. 2.23, t~r:insit,io~~ is shown ~ ) r n t ~ t i ~ ~1j.y ( - ~:I, ~s Ii t, ~l ~l tyl t : ~ ~ incrrase In t h e boundary-layer tliick~lcss.111a lalninar l)ou~iclarv . . .. Iavor .. tlrc t l i ~ n c q ~ i o n -. . less I~ountlary-1;~ycr trhickricss, d / i v :,;/Urn, rcrnnins rorlst,:~ntant1 rtl11;11, :I l)pro,yimatcly, t o 5. 'I'he dimcnsiortlcss boutidary-layrr t01icknrss is s c c t ~plot,t.c~tl: ~ z : ~ i r ~ s t . :I(, thc length Rcynolds numbcr R, =-: U , z/v i n l'ig. 2.2:) nlrmtly mc!~lt~iot~rtl: R, >. 3.2 x 105 n sutltlcn incrcaso i l l 1.llo I ) o ~ ~ t ~ t l : ~ ~I.l~i(.l 0 t l ~ n o t c sil~stabilit~y. Thc limiling case c, - 0 corrrsponds 1.0 nrnLml (indiKcrmt) dist~rrl~ancrs.
I
11nm1)cris t h e critical Reytrolds nuinher or limit of I a n ~ i n a rflow untlcr cot~sitlcrxLion.
Fig. 16.8. Curves of neutral stabi1it.y for two-di~nensionnlborlntlary layer wi(.tr two-dimensional disltlrbrmccs (a) "non-visrolls" inslal,ilily; i n t h c rnsc nC vclocity ~ ~ r n i l l or r s 1 y p :t wilh p i n 1 *di ~ ~ f l r x i o n
:,
P I , t h r rllrrc or 1w111rnl s l n b i l i t y is or t y p c Instahllity; in I11e raw or vriorit,) proIllrr nl type 8 uilhotrt poi111of lnflrxion, l l w r u r v e o l erutml s h b i l i l y is or l y p c b T l ~ nnryrnglotm Tor l l ~ nrrlrvc or rwulral stnbility a al. R - - + r, arc ohlxincd rroln l l w "rrirlionlrar" slal~ilit?.rrlllalio~i(16.16)
(1,)"visrow"
r ,
I he r x p r r i t n r n t d rvi(Irncr eonwrnit~gt,ransit,iot~fro111 l:~n~in:t,r 1-0 I , I I ~ I ~ ~ I I I ~ I I ~ , flow rcfrrrrtl 1.0 ~)roviorlslyI(.nds 11s t,o c s ~ ~ r ct hl ,a t , ;LI, SIII:LIII b ( ~ , y ~ ~ oI II ~I Il sI I I I ~ for ~~ wl1ic.11I:~minarllow is ol~servrtl.all \v:rvrl~ngt.l~s wo~lltll)rorluc*vo t ~ l gsl.:rl)lr tlist.llrl):~l~c ~ s ,wl~rrrn.s:lip I:~r.gt-r ltc!g~~oltls rtlln~l)rrs,l i ~ rwllic~l~I . I I ~ I I I I I ~ : I I I IIow , is 011svr\~1~1, . ul~st,:~l)lc tlist~lirbarlccso~tght,t,o corrcs~~ontl to at, I(ywt, sonto \ v : L v ~ I ~ I I ~ ~ . I I s IIo\r.(:v(~, it is nwrssnry t,o r c n ~ n r ka t tallispoitlt t.l~:rt~ t.11~ vrit,ic:n.l I f,l 5 4 , ,_,
s
5
4 s
500
XVTT. Origin of turbulence 11
1). I)etcrn~inaliot~ of t l ~ posibion r of tlw poinl, of il~stat)ilityTor prrsrrilwtl I,otly slt:qm 601
of minimttm prrssttrr irlstal)ilil.,v and, ronsequently, transition sets in almost a t onrp r v m a t low lteynolds numbers. I'igurc 17.11 SIIOWS, furt.l~cr,the pmit,ion of (,hepoint of instnbilit,~,a s clet.ermined cxpcrimrnl:tlly for a. N A V A n r r o f i l , which possrssrd :in almost, idrnt,ical pressurr disl~rib~tl.ion with l,l~:il, ol' t l ~ cZlt~tliovsltii:ic:rofoil ~ttttlrrronsitlt.r:~f~ion.11, is sccn t , l ~ : ~ tl,Iir, , lwit~t,oC tr:tnsiI.io~~ 1it.s I w l ~ i t ~ tl I l t ~l)oit~t, ol' it~sl~:~l)ilil~y 1)11t,i t 1 fr0111,of l , l t t * p o i t t L of l:~.l~lill:~r ~r~):il.ilI.ioll (i)r :III \':1111t'~ of Ib~ytlol(ls11111nbcr ant1 Lift rocffioict~l, :IS rsprcl,rtl f r o n ~t l ~ r o r c ~ l i ct.onsitlcr:it~iot~s. ~~l Sorontlly, the shift, of thc point of l,ratisibion wiI.11 a v;iryitlg Ibcynoltls n~tnll~c:r:rntl lift, cosfficirnt, follows l,h:~t,or 1 . 1 1 ~ poinl, of insl:iI~ilil~y. I ~ ~ ~ s I I of I I . sysl,ctn:il,iv ~ ~-:~I~ml;il~ions on t,hc position of ~ , I I ( ? lwi111, of t r a ~ ~ s i l iI'or ~ n:~(*roli)ils of varying t , I ~ i c l i t ~a< ~: ~cambrr d~ ca.n be f o ~ t ~ w inl a rcbl)ort l)y Iilit.yand of p i n t of t.mnsibion as a ffnnctionof l i f t cncffiricnt nnrl Iloynoltln I I I I I I I ~)~ ~ .t.ltroroLicnl point, of inritnl)ility:J 0015; - - - - ~neasurcdpoi111of tr:~tisil,io~~: NA(-!A 0018
A s a, roltgh guide in approximate calcttlations it is possible t o cletluce t,Im rule thnL lhc poir~i~ of t,ra.nsit.ion nlmost, coincitlcs with t h e point of minimum pressure of 1,llc pot,ont,inl llow in 1 . l ~range of Rcynoltls numbers from 106 t o 10'. At very I:wgc 1to.ynoltls nl~rnl)msthe point of tr:insition may lie a short distnncc in front of t,lt:~l, i)osit,ion nntl it, may move a consitlcr:~.ljlcdist,ance bchind i t a t small Itcy~toltlstt~ttt~l)c~rs, parl.icttl:irIy whrn f.he prcssttrc: grndicnt;, w l ~ c t l ~positive cr or nca;itivc, is slna.ll. 0 1 1 t , l ~ cothcr I l a ~ ~ tit, l , will I)c not,rd t h a t tho point o f instability always lirs in fro111of (.It(: point of I:imin:ir scp:trnt.iot~irrrspcct,ivc of the v:iluc o f ' t h h:ynoltls I I I I I I I ~ ) ( T . 'I'hns wc ciin rst.nl)lish t,hn r t ~ l rI.l~atI,llc p o i t ~ tof inst.nbility lips bcltintl t,hr p o i ~ of ~ t m i t ~ i n ~ ~prrss~ire ttn 1)11t,in f~0n1,of l.l~c'pointof laminar scpnrnt,ion, a t :III c ~ x r ~ lvvry ) l . Inrgc Iptw-is(: r l ish n c ~ :I)c~t,~v~;cv~ tlvprn(ls on 1.l1r: r:~.l,co f : ~ . t ~ ~ ~ ) l i l i r of : ~ .~,II(: l , i ottnst.:il)l~ ~~ tlist,nrha.nccs and on tlw infjcnsit.y ~ I ' I ~ I I I . I ) I I Ii Il l- I.IIP I I ( -frw ( ~ S ~ . ~ ( * : I . I I I111 . h ~ r n(.II(: , r:il.r of nlnplificatior~is st.rongly i n f l ~ m ~ c c t l
(17.7)
This discovery was confirmed, a t about t h r same tirnr, by , J . L van Ingsn [!14]. S r r also a paper by R. Michel 11661. Jn nod ern times this discovery was confirmed I)y tnnl1y rr~ras~rrctnc.nts ( 1041 which intlicn.te an amplification factor of about cxp 1 0 -- 22,026.
Fig. 17.12. I)cl.cr~ninationof tlrcsn~plification rntc rxp (/fit dl) for trnshlh di~tltrbanccs extendctl over the path from the theoretical limit of stabilit,yto the cxpcriment,al point of transition, after A. M. 0. Smith [211]
Tlw tlisl.nr~ccIwt,wccn t h e point, of iwt.n.l~ilit,ya n d t.11~point. of t,mrlsit,iolt can bo rcprescnlccl in t,llc form o f (.he tli1li:rrrlc:c: Irct,wce~tt h e Ttnynoltls numl)orn forrnctl wit11 the aid of t h e rnonwnt.~ltnt.lli~lzncssi ~ t.llcsc t tewo points, a s wn.s a.lre.zdy clotlo ) ~ ,(116,/v),. Fig. 17.14 sllows a 111ot.of thin q u a n t i t y in Fig. 10.21, t h a t is, a s ( l l r ? , / ~ ~ -J? : ~ n t lis hn.srcl on t,hc v n l ~ l c sfonntl in tmlns of t h o lncan I ' o h l l ~ : l l ~ s o1)~r:1n1rtcr ~~ by 1'. S. Gm.nvillc 1751. llorc we 11:~vc:
i,rr~ninnrn r r d t d ~ : 'i'hr st.:llrilil,y c~:~lc.lll:~liolls s~~nlln:rrizrtlill I?igs. 17.9 nrltl 17. 10 clvrnnnst.rat~c.very c:onvinrit~gl,vt,llnt.I It(, ~)~~cssu~.c~gt.:~tlit.nt. llaxnclvc~isi\~c: i~r(lrtrl~c.c* O I I sl:~l~ilit,,v ant1 tmnxition i l l V O I I I ~ I I ~ I :1.xrf~c.tn(-111, V wit,l~~ I I ( * : I S I I ~ I'-l ~' h~ I(lnsig,~ ~*II~~~.
504
XVII. Origin of k~rbulencc11
positmionof t , l ~ cpoint of tmnsit,ion is shown in addition for aerofoil R 2626. It is seen t h a t transition occurs sliortly a f t e r t h e pressnre m i n i m u m i n complete agreement with t,Iw t,heorctical results in Fig. 17.10. Figure 17.16 shows, furt.her, plot,^ of d r a g cocfficicnt.s i n t e r m s of t h c lift coefficient for three aorofoils of equal t l ~ i c k n e s sb u t varying caml)er. I t shonld be noted t h a t h y increasing t h o camber i t is possible t o canse a sllift in t h e region of vcry small d r a g in t h e direction of higher values of lift,, Intt rvrri so, t,lic rngion of rctlucctl d r a g still extcntls over a definite witlth only. Needless t o s a y , in t h e case of laminar aerofoils t.he int,crart,ion between t.hc e s t r ~ . n a l s t r e a m antl t.hr bountlary layer is very import.ant; mct.liotls for t h e c:i~Ic~tlationof sncli effects have been tleveloped b y R. Eppler [BO]. A t this point, it, is nrccssary t , ( ~ remark t.11ut cert,ain rircumstances c:ause consitlcrablc difficulties in t.he pract,ical 011 application of laminar arrofoils. Principally thcse a r e d n c t o t.he great. d r m n ~ ~ d s t,he s m o o t ~ l ~ n c sofs t h e surfaces in order t,o exclnde prwnat,ere transit.iot1 owing t o roughness. I n this conncxion we wish t,o d r a w t h e reader's at.tent.io11 t,o a p a p e r b y I,. Speidel [212] on lnniinar aerofoils placed in a Iiarn~onicallydist,urbed free streatn.
b. Dcterrninntion of the position of the point of instnbility for prrsrrihrtl horly nhnlw 505
'L'hc discussion in this section m a y bc suminarizctl a s follows: 1. T h e tJicory of s t d ~ i l i t , ysliows that, tlic prcssure gmdioll; cxcrln a n ovcrwl~c~ltnir~g influence on t h e stability o f t h e Imninar bountlary h y c r ; a tlrcrc-as(: in prcssurc in t h e downstream directlion h a s a stal~ilizingcKcct,, wltcrcns increasing prrssurc: leads tjo instnbility. 2. J n consequence, tlic position of t h e point of maximum vclocil,y of t.lic pof.rnLi:~l velocity d i s t r i t ~ u t ~ i ofunction n ( = point of minimurn pressure) inllucr~c:cstlccisivcly t h e position of t h e point of inshI)ilit,y antl of t,hc point of t,ransitio~l. I t r a n I)c assumctl, a s a rough guiding rulc, 1Ji:~ta t n w l i u m 1tc:ynoltls nrirnl)crs ( R =--: {Of; to lo7) t h e point of inslability coincitlcs with t h e poinL of minimurn pressure a n d t h a t t h e point of transition follows shortly afterwards. 3. As Urc angle of incitlcnce of a n acrof'oil is incrcasctl aL a c o n s t a n l ltrynoltls number, t h e points of instability a n d transition m o v e forwards o n t h e suction side and rearwards o n t h e pressure side. 4. As t,he R.cynoltls n u m b e r is increasctl a t const.nnt incitlcnce t h e points of inst,al,ilil.y a n d t.ransit,ion m o v e forwards. 6 . A t very high Reynolds numbers antl with a flat prrssure minimum, t , l ~ epolnt o f ins1.nl)ilit.y m a y , nntler ccrt,ain c i r c u m ~ t ~ a n c r sliglit,ly s, precede t h e poitit of n i i n i n ~ n mprrssurc. :I.IK~ 6. E v e n a t low Iteynolcls nurnbcrs ( R = 1 0 V t . o 10" t11c points of inst~:~t~ilil.y h n s i t i o n precede t h e p o i n t of laminar separation; nndcr c e r h i n circunistnnres t h e h m i n a r boundary layer m a y become soparabed a n d m a y re-at,t.ach a s a
Flexible wall: Anothcr effective rnethod of stabilizing n larninnr bo~mdnrylnycr is to rnnke the wetted wnll flexible. In connexion with the obsorvetl antonishing swimming performance of porpoises [go], it hns been suggested that these nnimnls have n very small skin-friction coefficient bernuae the boundary lnyer on them remains laminnr even nt very Inrge Rcynolrle numbers owing to the flcxibility 01 thcir skin. Jri ordrr t,o put t.hk hypot,hesis to the tcst,, M. 0. Krnnter [110] performed ~ncnuuremenLqof drng on olwt.ic cirrulnr rylin(lcrs plnced in a stmnn~pnrallel to their axes. Indeed, reductions of the order of 50% in drng, compared with rigid cylindrrs. have been observed in the range of Reynolds numbers R = 3 x 10" to 2 x 10'.
Fig. 17.15. I'rr*nw~rndist,rih~tI,iot~ lor l:~.tninnr arrofoils at zero incitlmcc ( c , 0). i\erofoilsOOI2, 65, -012, 66, -012 from rrl. [I]; wrofoil It 2525, nftrr IIort.~ch (Dl] 5
'I' = posilinlt or point or trxnsilion for R
--
:1.5 x 10'
Fig. 13 Ili Corfficirnts of profile drng. c,,,, plotted ngninst lift coefficient, c,,. for three Inniinnr nrrofoils with vnrying rwnhrr, R 9 x 10" from ref. [7]. The rrgion of smnll drag mows townrds higher lift roefficients, c,,. as rnmhcr increases
-
Furthermore,T.B. Benjnmin (41 and M.T. 1,nndnhl [I201 instit,uted comprehett~ivethcoretical analyses on the stnbility of boundnry layers on flexible plntes with the aid o l the method rxplninetl in See. XVIc. Thcse revealed t,hnt,in nddition to tho Tolltnien-Sclrlirhtitig wnves which occur in n form ~notliliedby tho flexibility of t . 1 ~ wall, there appcnr tnodifietl c1nst.i~wavcs in tho wnll itmlf. Such elnstic waves are creatod owing Lo the prwence of tho flow outttide the wnll. F ~ r t ~ l ~ e r m othere r c , appear waves of the Kelvin-Helml~olCztype, rnther like those observed on free shear layers. The first effect - the n~otlificationof the Tollmien-Scblichting wnves by the flexibi1it.y of the wall - may, taken by iteelf, explain the drnatic displacement of t.hc point, of n e u t d utnbility in the upstream direction. However, tho three effectn which depend on t,licint,ernnl friction in the wall counteract each other t,o a certain extent. For this rcnuon, we would expect only a small overall effect. Thus, M.O. Kramer's experiment.al results appear to be confirmed by the ~1.nhilit:y throry only qrlnlitntively hut not qunntitntivoly. 'l'hc suppo~itionthat M.O. Krnrnor's rwt~llttwuld lwrhnp 1)o oxplniti~dhy t,ltn inll~~onw 01 wnll llcixihiIil,y on thr~111Il.yrlovc~lr~lj~~cl t.urbulcnt boundary layer induccd U. Zimrnormnnn [25!)) to rlnclcrtakc o thoorctionl invcntigntion into thin problem. He came to the conch~sionthat the flexibility of the wall could lead to a roduction of the shearing stress on the wall of the order of 10 per cent,, nt lcnut in the presrnce of a fluid of high density such as water. In the nbsenc~01 n co~npletetheory of turbulc~~ce, it. is impossible to view these rwulta nu more than est,imaks. 'l'he pnpcr. [259], contains references to additional contributions which concern themselveu with the effect of wall flexibility on the stability and turbulence of boundary-lnyer flown.
c. ElTect of suction on tmnuition in n bounclnry lnycr
c. E f i c t of ~ u r t i n t lon trnt~nitionin
n
hn~rntlnry lnyer
It, has alrrntly Iwrn poitrI.ctl 0111, in CII:I~I.SIV t.hnt Ihc application of suction I.o a 1:rminnr ho11rtt1:~ryh y r r is an rKcc1,ivc m m n s of rrtlucing drag. The clrect of s u r t i o ~is~t.o st.nOilizr I hc h ~ n t l a r ylaycr i n n way sirnilnr to the cffcct of t h e prcssrlrc gratlirnt tliscussctl in t l ~ oprccntling srctior~,antl the rrduction in drag is nchicvctl 1)y 1)rt:vc11lingtmnsilion fro111I:~tnit):~r t,o I I I I . I ) I I I C I I ~ . flow. A marc (Ichilc(1 :~n:~lysis rrvcnls t,l\nt.t.lrc i t ~ f l u c ~ ~ of~sc~c ~ v l i oisn ( 1 1 1 ~t o t.wo c f i c t s . First, s ~ ~ c t i oret111ccs n thc l)or~~l~l:~ry-l:~.yrr t~l1ic4tnrssn.ntl a t,l~it~nc.r I)ol~ntl:~ry h y r r is loss p r o m t,o I)ocome l.t~rbrllrnl.. St~:ontlly,s~~t:l.ion crr:~l.rs:I.I:~n~it):~r. vc~loc~il.y 1)rofilc which ~~ossc:ssc~h i g l ~ r r t i vcIot:it..y profilc wit11 no s~~(:l,ion. limit. of stal1ilil,y (c:rit.ic::d I 1 wit,ll phnnc vc.loc:il,y c , c:" --- (I,, m, a n d n = 0. Ncn1,ral supersonic tlist,urb:mccs wc: p o s ~ i l ) l ci ~ l l wrt.:litt fIows, I)ut 110 general conditions for their existence h a v c bccn given. Figurc 17.24 shows 1,11c:clirncnsionlrss phnsr vclocitics cJUm a n d co/U, of t.ho n c ~ ~ t . rsul)sonir nl and sonic: tlist.~~rl~:incv as fullct.ions of Mm for n fnrnily of ntli:rb:~t.ic flat.-pln.t,c I~onntlnt~y I;ryws. 1 1 1 c ~ 111(~:111 bountlary laycr profilcs which were usrtl in t.hc cnlculat.ion of c,*, ant1 will h: usc:tl throughout this Section, a r e aceurat.c numerical soluLions of I,hc coml~rcssiblc:In.tninn.r boundary layer equations for air with both t,he viscosit.y cocflicient, nnd I'rnntltl n u m l ) r r fr~nct~ions of t.emperature, a n d with a frcc-st.rcatn stngnation t,cmpcrnl,~lroof 3 1 11< up t o M, = 5.1 where T m = 50 K. A t higher Mach numbers, T m r e m a j n sa t 50 I co > 0 in Fig. 17.24, all of the b o u n d k ~ ylayers of t,his family sat.isfy t h e conditions of t h e extended theorem a n d a r e unst.al~lctto frictioulcss ({isturbnnces. T h e movement of t h e generalized inflexion poinL 1.0 larger y/S wit.ll i l l creasing M, is similar t o t.he movement of t,hc inflexion point \vil.ll increasing nt1~r.t.s~. pressure gradient. in incompressible flow. l'igure 17.24 also givrs t,hc tlimct~sior~lt.ss displaccmcnt thicltncss dl vl/,/x v, ns a f n n o t h of M, for Lltr fnmily of ntli:~I,;,.t,ic: boundary layers. I,. 1,ees and C.C. Lin were gblc t o prove th:~.tl.llc wnvo n ~ r r n l ) cO~~~'.I . I I V neut,rrtl subsonic disturbance is unique a s in inco~nprcssil)lrflow, provitlccl t,llnt. t,Ilc mean flow relative t o t h e phase velocity is everywheresubsonic, i. e. h2 < I t,h~.or~gho u t t.he boundary layer, where M = ( I J - c,)/a is t h e locnl rrlntivc~M:tch I I I I I I I ~ ) ~ ~ . Although t-heir proof t,ltat cqn. (17.17) is a s~ifficionl,contlil.ion for 1,l1c:inst.:ll)lit,y h:stl t,he same restrict,ion, i t appcnrs from rxlensive nnmrricnl c : r k ~ ~ h t . i o nl.l~:rl, s rtltt. ( 1 7.17) is a true snff cient condition even when M2> 1. On t h e contrary, L.M. Mack [ 1521show-
--
..
518
XVI I. Origin of turhr~lcncc11
c. Effects duc to heat transfer and cornprcsnihilil.y
510
rtl 11y n~lrncrirnlrnlrulnt,ions thnt with n region in tho boundary layer whcrc lk > 1 tI1rr.c-nrc- :In inlinitc nrlmbcr of nculml wnvc numbers, or modes, with the sarnc plrasc vrlocity c,. 'l'hr mult,iplc modes arc n result. of the change in the govcrning tlilPcrc:nt.inl rquation for, say, thc: ~ m w u r coscillntictn from rlliptic whon MZ < 1 t o hyperbolic \vhrn M" I. 'l'l~efirst motlo is t.lx snmn a s in incomprrssiblc flow, nntl was first r o t n l ~ ~ t rI'IW t l c~otnpr(~ssil~lc flow 1)y I r . IATS tu10 1':. Itc:sllot.ko 11421. Thc ntltlit.ionn1, or I~ighrr.rnotlw hnvo n o incornprc~ssil~lc ronnt,crpxrts. c, = c,, MZ, first, reaches 11nit.y a t M, 2.2, nntl the uppcr I~ountlaryof t,hc rcgion of supersonic relative flow is nt ?I/(> -- 0.16, 0.43, 0.50, Lor &f =- 3, 5, 10, rcsprctivcly.
..l h c multiple 11tmt.rn1d i ~ t ~ u r l ~ n nwit,lr r c s phnsc vdocit,y c8 are not tho only ones
possiblo wllcw ME, > 1. Thcrc nrr also rnult.iplc ncut,rnl dist,urbances wit.h U , < c, I l , - t a,. Tllrse dist,nrbnnccs (lo not. tlcpond on the boundary layer having a gc.nrrnlizcd inllrxion point,. I~urt.llcrmore,there nre always adjacent amplified tlist~rlrlmnrrsof t IIC .w~rret?/pr i d h , plrass 11e1ocitie.sc, < I/,. Co~tacquentl?/,the co~rr.prossil)le Ooi~?tdar?/ /fl?/rri s isnslable lo frictioltkss dislicrba~~ces rrgardloss of any other f~atureno/ thc idoci/?/ rr~rtlhrg~cr(r/rtrrprofilrs (1,s lonq n.p th,rre i n a rqlion idtrre M2 > 1.
I , three-tli~ncnsionnl disLurbanees arc t8hc most unstable. I n the sccond region, from M, = 2.5 t o 5.0, the increasing frictionless instability shown in Pig. 17.25 begins t o make its influence felt a t lower Reynolds trumbers. As n rcsult of c, - co increasing wit,ll {I t.ho ~naxirrlr~tn nmpliBcnt,ion fnet,or of t 400 :
laminar Coucttc flow, laminar llow with 'I'nylor vorl,ic:os, lurl~tllor~t Ilow.
Agrccmcnt bctwccn theory and c x ~ w r i m r n tis cxecllrnt in thc first I.wo rangost.
An extension of Taylor's thoory can bc found in a study hy I i . Iiirchgarssner [IOG]. A detailed experimental investigation of Couettc Row, particularly in transition, was carried o u t in 1965 by 1) Colcs [291 E k c t of an axinl velocity: The preceding stability calculations have been extended by 13. Ludwieg [I 32, 1331 t o includc the case when the two ryl~ntlcts arc also axially displaced with respect t o each other. Let u ( r ) denote the tangential velocity, a n d let w ( r )denotc t h e axial velocity. If we now introduce the dimensionless velocity gradients r dzl r dw u=-a n d GI=--, u dr u dr
-
wc can writc t h e stability criterion for n non-viscous fl~tidin the form
Kg. 17.34. I'low hot\rsc:cnt.wo conoentxic rylintlrr.q: tor+io cocflicicnt for inner cylinder in t,rrms of t.hr 'I'nglor nnml)cr, T,,.
t
l'lic cxperirner~tnlmsulkq displnycd in Fig. 17.34 dernonstrntc furtl~orthat an increase in the Taylor number, that is, that an increase in tho lteynolrls number a t a constant value of d/.R,, cansen n trflnnition from cellular to tnrhulcnt flow. Whcn thc flnw is tc~rI)nlcnt(1,> 400), wo have CM Td-0.2. and I~cncc,nt constnnt d / R t a l ~ o CM w ( ( I , d l v ) - o . Z R (1 2. 'l'lm sarno ~(:RIIIL WIW discov(:red IIY { I . It~it:J~nrtlI, ((201 i n Cl~np.XI X ) WIICII 110 ~t.ndi(:~II h I:ILH(: o r I ~ I I I : : ~ ~ Couette flow between flat parallol walls. I t is remarkable that l l ~ csame dependence of the torqm coefficient on Iteynolds number exists for a disk rotating in a llnid at rcst, eqn. (21 3 0 ) .
-
-
630
XVII. Origin ot t.~~rhr~lrncc Jl
This ineqnnlity contnins Rayleigh's criterion from eqn. (17.19) a s a special case ant1 r c s ~ ~ l twherl .s 7o = 0 is n s ~ n m c dhere; we then find t h a t 1 5 > 0. The stability calcnlntion which led t o eqn. (17.23) took into account disturbances which were not ~icccssnrilyaxially symmetric; the 1n.th.x turned out, t o be the "moat dangerous" o m s ant1 detcrminctl ( h e litnit of st.:~l)ilil.yimplied by t h c ineqrlnlity (17.23). l'ignro 17.36 shows a n example of a n unstable flow which contains vortices in the shape of spirals. I f . Ludwirg's tllcory has bccn compared withexpcrirnent.al resulk [134] in Fig. 17.36. Every bnse flow invcstigated experimentally is represented by a point in t,Ilc I;, 271 plane. The opcn ant1 full circles characterize stable and unstable flow, respectivciy, i t being riotcd t h a t vortkcs were observed for t h e latter. It is seen t.hn.t, I T . I,utlwicg's st.:~l)ilil.y crit,crion from rqn. (17.23) is fully confirmed hy cx-
+
Fig. 17.37. R.nngen of Inwinnr nnd l,rrrl)~tlet~t, flow i n n~~tii~luu I ~ C ~ I Y C ~ two I I concctltric. cyli11tlc.r~; innrr cylinricr rotntrs. outm cylitldrr nt. r n ~ Ii,n prmRltro of nxi:d flow: plot. i n t.crr~luo f ' l ' r ~ ~ l oI I rI I ~ I I . I,t.rT,, 1111rl IIIIIIII#('). R.,; t l l l ~ l l ~ 1 1 ~ l ~ l l l l ' l 1))' 1 ( H # I . I < I I ~ I $I I I I ~1':. ~ 151gt1rI I I!)J 14' = axial velocity (I.
T l ~ cflow t,l~rought h e annulus between two concentric cylinders, rnitsl1thc inner cylinder rotating and the outer cylinder a t rest, on which a n axial velocity component is superimposcd is of great, practicnl import,nnce. Such flow patterns occur in the hydrodynamic lubrication of journal I~caringns well ns in t.hr air rooling of c k t r i c gcnernl.ors. A Itnowlctlgc of Ihc Ilow p~ttd~c~m is rcclr~irvtlfor (.llcc~rlrl~lrtl.io~loI'l lit. 1 . 0 1 ~ 1 u v of the bearing a s well ns of thc cooling cfTcctivencss of t,hc gcncmbor. 'l'hc cling~.ntn in Fig. 17.37, drawn on the basis of measurements pcrfortnccl by J. Kayc and E.C. Elgar [I 191, nllows us t o d ~ t ~ e r m i t,hc n r prevailing flow rcgimc - laminar or t.urbrilcnt - for a n nnnultrs with n givcn nxinl strcam. 'l'llis is tletrrrninctl by two cl~nr:~.t:tcristic I. nntl by n Ilcyni~ltlsti11111Irt~r nurnlws, l . 1 1 ~l'itylor nurnl)vc. T, tirfitivtl in W ~ I (17.20) formctl with t,hc nxinl vc1ocit.y If' nrd tlic witlth, (1, of l.llc nnnulus, viz.
Experinwnt points to thc cxistcnrc of four zones:
< T, < 300 will1 low t o tnotlr~.ntclic~yr~oltls laminar flow with n system of Taylor vortices;
(I)) Tnylor nunll)cr in the rnngr 41.3 numbers R;,
XVII. Origin of t~~rbr~lenco 11
632
f.
d/N1 < 0.2. 'l'hc. out,c,r s p l ~ c r cwas a t rest, whereas the inner sphere rottat.etl. 'l'he charact.t-r o f the Ilow in ~ u c ha spltcri~alannulus is also tlct.crn~incdby tho Taylor nurnlwr from rqn. ( 1 7.20) and t,hc Iteynoltls number formed with the annulus width, d , and I.hc poriphcral velocity, (II, t h a t is by
I n t.11~ range of validit,y of linrar theory, t h a t is bcforc t,hc appearanoc of Taylor vortices of t h e kind shown in Wig. 17.32, the torque noting on t,he inner sphere is
Stability of n boundary layor in the prcacnco of thrcc-din~rnsionnltlisturbnncc.~
533
vortices) which arc known a s Taylor-Gocrtlcr vortiacs. 'rhay arc of the sarno Itind a s t h e Taylor vortices fiom Fig. 17.320. The crtlculation of the amplificat,ion of thcsc thrrc-dimensional vort.iccs with time bawd on t h e method of small disturlmnccs leads t,o a n cigcnvalue prol)lctn in a manner similar t o t h a t discussed in connexion wit.h two-tlimensionnl tlistnrbances (Chap. XVI). The influence of viscosity was tdren into account in the invcsligation under discussion. The first, approximate, solution of this vcry tlifficnlt cigcr~vnlue prohlern was publishcil in 1940 by 11. Gocrt.lcr (721. I,nt.cr, in 1973, 1'. Svhn1tzGrunow [204a] formrllntccl a more nocnrntc thcory in t.hnt, he troolt in1.o :~.c:c:ct~tnt. :dl Lcrrns of first ortlcr of s ~ n n ~ l n c s'l'ltc s . tlingrrim in Fig. 17.38 t~onl.:sinxl ~ i r r I I I I I I I : \ I . ~ ~ ~ rcsults. It is seen that, the minimum of the limit, of stnl1ili1.y octwrs I~:t.wt*t.nR, U o fi/v = 4 to 6 whcn thc rclr~tivccurvatrtrc in ()/It! := 0.02 1.0 0.10.
-
with
rlenoting t.11~torqnc, and R r the inner radius.
Whereas in t.hc ~ ~ r c r e t l i nmsc g with rot,at>ingconccnt,ric cylinders thc entire flow fidtl is either laminar or t . ~ ~ r l ) ~ t l cdcprntling nl,, on t.hc valucs of t,he Taylor and Rcync~ltlsnr~mI)ers,t,l~c:case of l.l~csphere is more complcx, bccause different flow regimes cat1 occur s i r r l ~ ~ l l , a t ~ r osidc ~ ~ s by l y sidc. AS t.hc Reynolds number is increasctl, Taylor vorticrs. and hence also t coortliwll,(? of (.II(* poirtIs or l,r:itisiliot~: l . t ~ t l ~ , I I P Ih:yt~ol(lsI I I I I ~ I I ) (1, ~ X-,7/~ forruc,ct wil,lt Lhc s:it~tl gr:titl si7.c, P,, for clilli:rc.nt. I I ~ ( - S S I I ~ Cg r a < l i ( ~ nI ItI~~ : : L R I I ~ C ~ Ov IC. (:. Vcit~tll,.'l'hr vnluvs for n. snlool.l~wall rnrtgcd from Ill r , J v -- 2 x IOVf.o \V(W
11. Axinlly ~ y ~ n ~ ~ ~ c tflows ricnl r
I h c most itnporlanl, c:l.sc of :in :~.si:~lly s y m n ~ c ~ t , r i r flow n l is t.Itnt, exist,itlg in n st,r:l.igltf ~ ) i p r i., c. w l w t ~ the vc-loc:it,y ~)rolilcis pn,ral)olic. This case \vns invcst.igntrtl vc-ry early by '1%. Snxl 120.51 \vho \\,:IS nrln.l)lc t.o discovrr a n y inst.nl)ilit.y; ho wns c q ~ t n l l yr t ~ ~ : ~ l )howcvcr, ln, t.o provo f,hc c x i s t c ~ ~ cofc s h l ) i l i t y for ,711 R c y t ~ o k l s n ~ ~ n l l w t Sotnc: s. t i m r I:rlcr, ,I. I'rc-l,sc:h 1 177J st~c~cc.c?t\c~l in proving (.hat t h r nttnlysis of IJIC sl.:~,l)ilil,y o f t.l~cw:~):~r:ilmlic: vvloril.,y 1)rofilos can IN? rntlrrcccl t o that, of p l n r ~ c ( ' o t t ~ ~ flow l ~ l ~(i. ~ ~r , p r c sltvnr llo\v). Sine(: I,IIV lnl,l.rr is st~:~l)le at d l l ~ ~ c y n o l~~il~s ~ n i l > r r s , l . 1 1 ~S:IIII(* is svrtt 1.0 be t . r ~ ~ :I,IK)II~, c 1110 l):~r:~l)olic vcIo(~ityprofilrs it1 a, pipe. 'I%r s n , t ~ ~ c conclrrsion w:ls rc~nel~otl by (:. R.1. C'otw)s nntl .I. 12. ScIInra ( I HI, by C. I,. I'c*l.: Sl,ut,tgnrt.cr Profilk:~t.alog.Inst. Acrodynamil~of SI,tlttprt Univ. (19721. 1:)) ARC Ilk1 24!)!): 'l'riinsi t.ion and t h g rnrasnrct11c.nt8on the 1lonll.ot1I'xrtl sample of Inminnr flow wing c.on.rt,ruc:t.ion.Pnrt I: by . J . H . I'rcston nnd N. Gregory; J'art 11: by K.W. Iopartnrwfrom isotropy l~coomonwc11 of isotropic L11r1111lcncc larger in pipe Ilow, bonndary laycm, ctc. Nevcrtlmlrm, the ~~ot.ion acquires witlcr npplicnhilit.y if i t is rcstrictcd to tlislrib~~tion fnnctions of velocity dilrcrcnccs inate:~dof to those formcd with rrspect to vclocitics the~nselvrs.Following A. N. Kolmogorovt, wc ronsider correlation fnnctions of the form
(r/. rqn. (18.13) and Fig. 18.7), and dcsignatc the t ~ ~ r b ~ ~ l :cLVn "locally cc isolropic" whcn the correlation function rrmnins invariant with r c s p c t to rotatinnu and rcflcxions of the coordinnLo ayskm in n rmf.rir:bd domain, that ia in a rrstrictctl range of distnnccn r hctwcrn poinlrq 1 and 2. I t is rollnil that ultch lorn1 isotropy r.xisln in uny L11rbuht flow in a snficicnb~y~111:lllinhrvnt r G I,, whcrc 1, has Ijecn defined in cqn. (18.14) on condition t.l~ntthe I L r y ~ r ~ number ld~
p r r s s w c I l ~ ~ c t ~ r i n t i oi t 1n st,hcsc znncs move, nt, n n inst.n.ntmumr~sconvect,ivc velocity of i l l t.hr tlircct,ion of t01c m r n n s t w n m . T l l c wnvo f o r m l~ c l ~ a n g c sslowly ~ i t . 1 1titnc. S e e also t h e p a p e r s by W. I 70 l , l ~ c1amin:rr con1 riI)ut,ion is twgligil~lc rotnparwl wil,l~I 1tr1)11Itwl, frit:l,ioii. '1'1111s:
.
t 11. lkic:har~ll.[fir,] intlirntcd
a refined cxpremiort for thp vclocity distribution. I t covcrs the wholc rangc of distances, froni the wnll of the pipe a t ye= 0 to the centre-line a t y = R, i. c., it is also true for thc la~ninnrsub-layer, to which eqti. (20.13) does not apply. I t is also valid in the ncighbortrl~oodof bhe ccntrc-line, wherc ~ricasurcdvelocity-distxibution curves show systctnatic:clrviations fron~cqn. (20.13). In particrtlur, the transition region shown aa curve (2) in velocity-distriblltionhw wan dcducccl Fig. 20.4 iu wcll rrproduccd by thc forrnltln. Thin ~~nivcrsnl with thc aid of bl~rorcticnlrabin~ntinnsarid vrry r:rrrf~driicns~trc*~ncnlrt of t l ~ cturb~llrntmixing coefficient '4 rlrlinrcl 11y cqtt (I!).I). (hmparc also a pnpcr by \V. Sznbicwski (741.
'* > 70
:
p ~ i r ~ lt y~ i r l n ~ l c nfriction t. .
I
XX. 'hrl)r~lcntnow t,hro~tgllp i p s
604
We now propose t,o compare the cxpcrimcnt.n.l rrsnlts on vrlocity-rlistril)utior~ mcasr~rerncntsin pipc flow with t.hc altcrnntivc ~ ~ n i v e r s aequation, l which was clcdl~rc~l in Ch:~p. X I X in the form ( I / - - u ) / v * .-/ ( l / / R ) . It will be rccalletl t h a t it followcd hot11 from von I 30. I n t h e t i ~ r b u l c n case t I t o 1271 hns sl~ownthcoreticnlly t h a t the ratio of 1.11~ ~.osintance coefficients, I/,Io, may bc exprcs3ed in terms of t.hc ditncnsionlcs-q vnriahle R ( R / r ) z .The experiment.al results of J t o [28Jcan hc represent,ed with sufncirnt i ~ c c ~ l racy by the equations mentioned in thc footnote. I n flow through a bend or elbow t.herc is not only some loss of enorgy within t,he bend itself, bnt n part of the Ions p r o t l ~ ~ c cby d the bent1 h k c s place in the stmight. pipe following it. 15xtensive measurements of the loss cocfficicnts for smooth pipe bends and n correlation of results wore given by 11. Ito 1291. Thcorcticd r r s ~ ~ l tnrc .s reported by W. M. Collins e t ai. [Sb]. In flow through a radially rotating st,raight pipe, n. secondary flow sirnilnr t,o t h a t found in a curvcd p i p is sct, up by the action of n (hriolis forc.n; it. fiirrs risv 1.0 n Inrgc incrcc~scin resisl,r~ncc.I3xt,c11sivc:~ncnsurrrncn~a :1.11cl1.11corct.ic:~l c:t~lcl~l;~l.ions on this subject were carried out by 11. Tto and l i . Nanbu 1301.
t
H. I t o [27] givcs:
and Fig. 20.27. Flow in Prnntltl [52]
n
c.nrvrtl pipe, aflrr These differ somewhat from, hut are in general nprecmcnt with, C. M. IVl~itc'~ cqnnt,iot~ RI)OVC.
628
XX. T~lrbnlmtflow through pipes
15xttcnsivc mcasurcmcnt~s and tlicorcLica1 calculations on frictional losses in turbulcr~tflow hnve also bcen carried out. hy R. W. Dctra [ll] who includcd curved p i p s of nonrirculer cross-scclion in his investigations. I t is found t,haS the resistranee offered by a n cllipt,ic pipe is grcater whcn the major axis of the ellipse lies in the plnnc of C I I ~ V A ~ J Ithan ~C when it is pcrpcndiculnr t o it. 11:. l 0 x 10" Tho numerical factor in the cquntion for ((he1)ound:~ry-Iayrrt h i & n ~ s s which was left unclctcrrninctl bccomcs
for a disk wetted on both sidcs, is given by cqn. (5.56), and is equal to
C, = 3.87 R-"'
G49
(2 I .3 I )
and the volume of flow in the axial direction is given by
where R = R2rn)/v is thc Roynolds numhcr, Fig. 5.14.
It is now proposcd to make the same estimation for the turbulent casc basing i t on the same resistance formula for turbulent flow as was used in the case of the flat plate, i. e., in the simplest case, on the +-th-power law for the velocity distribution. A fluid particle which rotates in tho boundary layer a t a distance r from the axis is acted on by a centrifngal forco per unit volume of magnitude e r w2. The centrifugal force on a volume of area d r x ds and height S becomes e r w2 d r x ds. The shearing stress t oforms a n anglc 0 with the tangential direction and ita radial component must balance the centrifugal force. IIcnce we have t osin 0 d r x d s = e r w V ~ J rx d s or
On the other hand, the tangential component of shearing stress ran be expressed with the aid of eqn. (21.5) which was used in the case of a flat plate, replacing U, by the Lmgential velocity r o.Thus t ocos
Equating
T,
-e -
0
(or)"' (v/S)ll'
.
in thcsc two cxprcssions, we find that
8
ral' (v/Up6 .
It is sccn that in the turhnlcnt casc thc bonntlary-lnyer thickness increases outwards in proportion to r 3 / b n d docs not rcmain constant as in the laminar case. Further, t oR3 N e R W ~ ( V / C O RRI6 ) ~ B3 / ~ so t h a t the torque becomes M
-
as comparcd with cqn. (5.57) for laminar flow. An approximate calculation based on the logarithmic vclocity-dist~ribr~l,iot~ Inw u/v* = A, In(?/v,/v) -1- Dl was performed by S. Coldsteiri [21], who found i.hn following formula for the torque : 1
7"M
= 1.97 log (R
1/c) + 0.03
It is n ~ t e w o r t ~ hthnt y this equation has tho same form as the ~~niversnl pipe-rraistancc forrn~rln,cqn. (20.00). Tho nnmcricnl fac1,or~have bccn 'atlj~rstctllo obLl~i11I,II(- IwsI~ possible agreement with experimental rcsults. This equation is sccn plotted as cnrvc (3) in Pig. 5.14. On this topic see also P. S. Granville [22]. 2. The disk in a housing. The dislc in turbines or rotwy compressors 111os1Iy revolve in very tight housings in which the width of thc gap, a, is small compared with the radius, R, oC the disk, Pig. 21.7. Consequcntly, i t was found necessary to investigate the case of a disk rotating in a housing. Laminar flow. The relations become particularly simple when thc flow is laminar, R < lo5, and when t h e gap is very small. If the gap, s, is smaller than the boundarylayer thickness the variation of the tangential velocity across the gap becomes linear in thc m m n c r of Co~~ctto-flow. lFcncc, tho shcaring strcss a t a distancc r from the axis is equal to T = r(up/s and thc torquc of the viscous Corccs on onc sitlc of a disk is given by n
J
Th. von ICiirrnhn [30] investigated the tnrbulcnt boundary layer on a rotating disk with the aid of a n approximate method based on the momentum equation and similar to the one applied in the preceding section ijb the study of the flat plate. The variation of the tangential velocity component through the boundary layer was assurnctl t40obcy the 4-th-power law. The viscous torque for a disk wetted on both sidcs'wa.. shown t o be equal t o
(turl)ulcnt)
Consequently for both sides we have 2 M = n w R4,+
,
and the torque coefficient from eqn. (21.27) becomes
t
Soc
refe. [10] and [31] in Chap. V.
650
b. The rotating disk
XXI. T~lrhulentboundary lnyers at zero preaaure ~rndicnt
66 I
flow both for thc laminar antl for the turbulcnt case. T l ~ ccxprrssion for t,ho t m q w is of the same form as for thc free disk in eqn. (5.56). only thc numerical factor has a tlifforent valnc. 'Che frictional momcnt of a disk in laminar flow and wrl,tctl on both sitlrs I~cromcs2 M = 1.334 ,A I 3 x 10' tJw flow arountl a clislc rot :tLing in a housing becomcs turbulent as usual. This crwc was also solvctl by 1" S~r1111ltzGronow who usrtl an approxitnalc mclhod bnsctl on t.hc sc~hcmcof Fig. 21.7. 7 ' 1 1 ~ tangential vclociLy was assnmctl to obey the 4-th-powcr law and i t was sl~ownthat t,hc eorc rcvolvrs with nl~out,half t h angnlnr vc1oc:it.y in t,his rasc t,oo. The monirnt cocficicnt wc~sshown bo bc aqua1 to
C, R
Fig. 21.7. Rxplnriatiori of symbola for the problom of n clink
- RL
Viacottn drag of dink rotating in o houning C u r v r ( I ) , rrom cqn. (21.341, Imnllnnr: rurvo (2), rrorn cqn. (21.35). laminar; crlrvc (3). rrom eqn. (21.36), tarbulcnt. Theory with no Lousi n g (Tree disk) sce Pig. 5.14
9
R
(laminar) .
J his equation is secn pk)thd as curve (1) in Fig. 21.8 for a value of .v/R = 0.02. It shows very good agreement with thc cxperimentnl values due t o 0. Zumbusrh (sw rrf. [54j). C!. Schmir~lcn[49] invrstigat.rtl I,l~oinflucncc of 1.11~witlt,ll n of t,lm lateral spacing of R clislc in a cylindrical housing, Pig. 21.7, on the assumption of vcr.y small Rcynolcls numl~ers(creeping n~otion).Thc Navicr-Stoltcs equations can bc simplified because of t,hc vcry low ltcynoltls nurnbcrs (scc Soc. IVtl) and the solution for the moment cocffirionC appears in tho form C,, = Ii/R, in analogy with cqn. (21.34). The const.ant I\' tlrl)cnds ou t,he two tlimcnsionlcss ratios n/R ant1 a/R.' I n t.ho case of vcry s m d l vnllws of o / R ( < 0.1) tho valurs of C,,arc! n~arltctlly1:trgcr I.l~n.nthoso in cqn. (21.34), wl~c~rrns liw hrgo vnlncs of cr/II' cqn. (21.34) retains ib valiclit,y ( K = 2 n Ills). . . 'I'lw flow pal,tnrn in t.11~case of larger gaps diffors considerably from the above sin~plcscl~rnic.'l'llis latrl.crcase was invrst.iptct1 t.~~rorcticaily and cxpcrimcntally by 1'. Srll~~lt~z-(:r~lllo~v [54]. I f 1 . h ~gap is a mnlt,ipl# of t,he boundary-layer thickness, thcn an ntlditional boundary layer will bc formed on thc liousing, Fig. 21.7. The fluid in t h r bount1:ary Ia.ycr on tho r o t d i n g disk is centrifuged outwards, and this is co~nprnsatctlby a llow inwartls in t.hc I~onnclarylayer o n thc housing a t rrat. Tllcre is I I O apprrcinldn radial component. in tllo int,t:rrnctlial.c Iayer of fluid which rot,ntcs wi1.11: ~ l ~ oIrn n tIf t,llo angular velocity of 1.l1cclislc. P. Sc1111lt.z-Grunowinvcstigatcrl this
(R)-"'
(turbulent) .
(21.36)
This equation has bccn plottccl in Fig. 21.8 ns curvc (3). Comprotl wilfh Inwwre!ment i t laacls to vnluea which arc too small hy ahout 17 par cont., n r d this m w t I)o atLribut,rtl to the arntlc n.q.sl~tnl)t.ionumntlo in t l ~ ccr~loulnl,ion. It is particularly noteworthy that, apart from thc casc of vcry small gaps, eqn. (21.34), the momcnt of viscous forecs is complctrly inclcprnttcnt of I(hc witltlr of the gap, as secn from cqns. (21.35) nntl (21.36). Comparing tho frictionsl rnon~culon a "free" disk and on one rotating in a housing, cqns. (21.35) and (21.36) as against e q n s (21.28) arid (21.30), i t is seen that tho n ~ o m c n ton a frcc tlisk is grcatm than t h a t on a disk in a housing, Fig. 21.8. This fact can bc explained by the existence of the core which moves a t half the angular velocity. This dccrrascs t.hc transverse gradient of the tangential velocity to approximately onc half of what it woultl hr on a free disk and, consequently, the drag is also smaller than on a "free" disk.
7-
Fig. 21.8.
C,, = 2n -
= 0.0622
t 3
The flow process depict,ctl in Fig. 21.7 in which thc boundary layer on thc rotating disk flows outwards and that on the casing flows inwards was lat.er invcstigntcd e~periment~ally by J. Dailey and R. Nece [Bb]; their measurcrncnts covcred the wiclc range of gap widths s / R = 0.01 to 0.20, and a range of lteynolds numbcrs R = R2w/v = 103 to 107 and included bot.11 laminar and turbulent, flows. Thc rcsults shown in Fig. 21.8 concerning the torque have bccn largely confirnicd.
I
coolcr casing at, rest. is irnport,anb in the design of gas turbines. T l ~ ctcmpcrnt,urc ficld which develops in t,he gal) bctwcen t.he disk ant1 the casing is strongly infinrnrcd by the complcx flow pat.t,nrn which prcvails in it; in t,urn, this has a large i ~ ~ l l u r n con c the flux of heat from tlisk to ho~lsing.The simpler case of a rot.nt,ing "frcc" disk wxs invest,igated some t.imc ago by K. Millsaps nntl I
which is shown dotted in Hip. 21.14 and 21.15. llcrc T, ant1 r, clcnotc Lhc shcaring strcsscs on the rough hnd srnoot11 wall, rcupectively, both for fully clcvelopccl How, z is Lhc disbncc d o n g the wall measured from the border line bctwccn Lhc two portions of thc plate, y is tho distance irom the wall, and h denotca the height of the channcl. For tho rcversc order of transition (rough smooth) the same formula may bc uscd, exccpt that 7, and rr munt bc inhrchnnged. The influence of a pressure gradicnt on the transition fro111a smooth to o r o ~ ~ gsurfacc h ha8 been investigated by W. 11. Schoficld [Fill and It. A. Antonin [Is]. Scvcre locnl prcssnrc fluct,ua, tions havc been observed by P. J . Mulhenrn 136~1downstrcntn from RIICII nn ~ h r u p t (:h:~tlgc.
-.
d. Admissible roughness
1 Fig. 21.13. Curves of conatant velocity in tho flow field1bchind a row of spheres (full lines), as n~easurcdby 11. Schlichting 1451, and accompanying i t the secondary flow (broken lines) in the bounr1:iry Inyrr bchilld sphere (I), ,zs calrulnted by F. Schultz-Grunow [65a]. I n the neighbourhood of the wall, tho vcloeity behind the nphorcs is larger than that in the gaps. The spheres produce a "ncgntivc wake cff&t7' which irr rxplnincd by the existence of secondary flow 1)ismrtrr of n1rIirrt.q d 4 mm
-
The amount of roughness which is considcrctl "admissible" in engincoring applications is that maximum heighl of individual roughness elements whicl~causes no increase in drag compared with a smooth wall. Tho practical importance of determining the amount of admissible roughness for a given set of circunistances is very great, because i t determines the amount of labour which it is work11 spcntling in manufacturing a given surface. Thc answer to this question is essent,inlly different depending on whethcr the flow under consitlcralion is laminar or tur1)ulenL.
658
XXI. Torhulcnt boundary layera at zero preuaure gr~liorit InJ!le case of lur.bu~?nt.~bo~m+ry h y r s roughness has no effect, and the wall is hy&aulic$llfjmooth if all_ potuberanc& iife co_n_tai+ -within the laminar subG e T A s mcntionctl befor&, the thicltnrss of the lnttcr in only a small fr,zvtion of the boundary-laycr t.hickness. I n conncxion with p i p flow i t waq founcl that, the condition for a wall t o be l~ydraulicallysmooth is given I)g rqn (20 37) which s t a t r d t h a t the dimensionless roughness lteynolds numbert -v* - k-
.
.
2
J
-
>
~
. -
.
a
-
a. Some experimontnl rusulta
CHAPTER XXII
The incompreesible turbulent boundary layer with preseure gradient JI n tho present chaptcr we sliall discnss the bchaviour of a turbulcnt boundary layer in the prwrnce of a positive or nrgativc prcssr~rcgradient along thc wall, thus providing an extension of thc sobjcrt matter of the preceding chapter in which the boundary layer on a flat plate with no pressure grhdicnt was considered. The present case is pzrticnlarly important for thc calculation of the drag of a n aeroplane wing or a tutbinc blade as well as for thc untlcrstanding of the processes which takc plarc in a tliffuscr. Apart from skin friction we arc intcrcstctl in knowing whether the boundary layer will scparal.c under given rircumstanccs and if SO, wc shall wish to detcrmine t l ~ cpoint of separation. The existmcc of a ncgstive and, in particular, of a positive prcssnrc gratlicnt exerts a strong influcncc on the formation of the laycr just as was thc case with laminar layers. A t the present timc these very complicated phcnomcna arc far from being understood complctcly but there are in cxistcncc several scmi-empirical mctbods of calculation which lead to comparatively satisfactory results.
i n coni~ergcnl Fig. 22.1. Vclocit,ydiutri1~11Lion and. divergent cl~nnnols with flat wall^, as n~cmrircclby J. Nikurnduo [71] - ImIr Included nnglr; It - wicllll of ctlnnrlcl
I n the year 1962, J.C. Rotta [86] prepared a comprehensive and careful review of this vast ficld of knowledge. I n order t o develop methods of calculating incompressible, turbulent boundary layers with pressure gradients it is necessary to derive from experiment relations which go beyond thosc employed for pipes and flat plates at zero incidence. For this reason we shall begin by giving a short account of some experimental results. a. Some cx~mrirncntalresults
ILrly systc~naticcxpcrimcnts on two-dimensional flow^ with pressure drop and prcssuro rise in convcrgcnt, and divcrgcnt clianncls with flat walls have been carried out by F. Doench [28], J. Nikuradse [71], II. Hochschild [45], R . Kroener 1571 and J. Polzin [76]. Measurements on circular diffusers, and particularly on the efficiency of the process of energy transformation, are described in papers by F. A. L. Winternitz and W. J.Ramsay [123]. These experiments demonstrate that the shape of the velocity profile dcpcnds very strongly on the pressure gradient. Figure 22.1 shows the with velocity profiles which were mcasurcd by J. Nikuradse during his g~~erirnent.3
t
Tho new veruion of tliiu chnplcr wo.8 propnrcd by Profemor E. Truckenbrodt whose nssistance I I~ercl~y grnbhlly ac:knowletlgo.
-1.0
-0.6
-0.2 0 0.2
0.6
L6
LO
Fig. 22.2. Velocity distribulion in a divergo11 chnnnel of l d f includotl angle n = 6" and a = Go, as measured by .J. Nilruradse [71]. The lnck of Qmmetry i n the velocity distribution signifies incipient separation
Fig. 22.3. Volocity distribution in n diaergent cl\aiincl of hnll inclrtdect analc n =: X", rnctwr~redby .I. Nikr~ratlnc[71]. Itcvcrsc flow is coniplebly dcvclopcd. Tlicr flow oncillntcs nt h g t ? r iiikrvah het.wm\ pnthrll~(a) twd (b)
670
XXII. Tho incomprcaniblo t~urbnlontbound~rylaycr
slightly convergent or divergent channels. T l ~ chalf included angle of the channels ranged over the valrles a = -ao, -4", -2", 0°, lo, 2", 3", 4". The bo~~ndary-layer thickness in a convergent channel is much smaller than that a t zero pressure gradient, whercas in a divergent channel it becomes very thick and extends as far as the centxeline of thc chn.nne1. For semi-angles up to 4' in a divergent channel the velocity profile is fully symmetrical over the width of the channel and shows no features associated with sepamtion. On increasing the semi-angle beyond 4" the shape of the velocity profile untlcrgoes a fnndament.al change. The velocity profiles for channels with . 5 O , Go and 8" of divcrgence, respectivcly, shown in Figs. 22.2 and 22.3,cease t,o bc symmctricnl. With a 5" nnglc o f divcrgcncc, Wig. 22.2, no barlz-flow can yet bc disccrncd, but separation is about to brpin on one of t,he channel walls. In addition the flow bcnomcs unstable so t,l~nt,depending on fort-uitous disturbances, the stream adheres alternately to the one or the other wall of t,he channel. Such a n instability is characteristic of incipient, separation. J. Nikuradse observed the first occurrence of separation a t an nnglo bet.ween a = 4.8" and 6-1". At a n angle of a = Go, Fig. 22.2, the lack of symmetry in the ve1ocit.y profile is even more pronounced, and the reversal of t.he flow intlicnt~csthe start, of separakion. At n = 8" the witlt,h of the region of Pressure distribulion
reversed flow is considerably larger than for n = 6",and frequent oscillation of the stream from one side to the other is observcd, the phenomenon being absent a t a = 6" and Go. However, the duration of one particular flow configuration is sufficiently long for a full sct of readings to be obtained. As tho nnglc of divcrgence is incrcnscd, the region of reverse flow becomes wider, and the beats are more frequent. The diagram in Fig. 22.4 shows an example of a turbulent boundary laycr formed on an ~erofoiland measured by J. Stueper [lo61 in free flight. I n the case represented here, the boundary layer on the pressure side is turbulent from the leading edge onwards, because here the pressure rises over the whole width of the wing. On tho suction side, t,hc point of t,ransition plnccs itsclf a short distancc behind t h r pressure minimum in agreement with the description given in Scc. XVII b. The fact that the boundary layer has become turbulent is inferred from the sudden iorrrasc in its thickness. Very thorough expcrimcntal in~est~igations into t,ho bchnviour of l.url)ulcnt, boundary layers with pressure gradients have been later perfnrmctl by G . 1%. Schubauer and P.S. Klebanoff [97], by J. Laufer [68], and by F.H. Clauser [21]. The first two of the abovc papers contain, in particular, rrmlts of mcnsurcments on tmrbnlent, fluctuations and on thc correlation cocfficicntw which wcrc clolincd in on shnaring Chap. XVTII. Thc last paper contains cxtcnsive rcsu1t.s of mca~urcmrnt~s sLrcssns. 'l'hc c:nlcul:~l,ionscloscril)otl in tho following oonl.ic,~t~ c::ru c:vitlonI.ly nlq~lgo11l.y t o [flows which adhere comptctcly t o the walls, tirat is, to cnscs which are sitnililr to the one shown in Figs. 22.1 and 22.4. b. The cnlculntion of two-dimc~~siorlnl turbulent lto~~nclnry lnyers
Try-
Fig. 22.4. llol~~lrlnry lnyrrotl wing nrrofnil. nn rncrmltrcd by Stilrprr [IOR]; mensuremmtrr in flight; R - 4 x loo; chord 1 = 1800 nlm. 't'l~eboundary 1a.s-er lift rooffic:iollt r,, =- 0.4; IZoynolrln nt11111)cr in turhulrnt all nlonp tho prrn.wrc sirfr owing l o xtlvotnr, prmnttro grnrlintlt; on tlle a e r t i o ~ sridr it. iu l n n ~ i n n ruprtrcntn of prcfxwrr t n i t ~ i ~ ~nnd i ~ ~trlrl~rtlc~~t tn downut,rcnl~lfrom it
1. General remarks. T o this day, all methods for the calculation of turbulent boundary layers rely on semi-empirical procedures, because the apparent nornml and tangential stmss componcnta crent.cd by the turbulent fluclmations as well as the thus released energy losses cannot be ~alculat~cd by purely theorct.ical means. Furthcrmorc, it. is still necessary to int,rotlucc hcrc empirical relations of the t,ypc of Prandtl's famous mixing-length formula invented in 1925, because the statistical t,hrory of t,urbulcnce has yet t.o produce a replacement. for it. [ t is nst,onishing thatf 1'rnndt,l1s hypot,hesis, half a cent,ury after it8 discovery, still plays a vcry important role in the lit,erat,urc on the calculat~ionof tu~.bulcntboundary Iaycrs. Mosl contcmporary met,hods are approximate; t h y make use of t,he momentum and energy equat.ions of t,he velocit,y layer (as distinct from t.he t,hcrmal layer which will not be discussrcl in this scction) and of certain relations t.hat follow from them. Thc corrcs l m d i n g rclitl,ions for Inmi11n.r I)ouneln.ry In.ycrs wrro tic-tivctl in Clr:~ps.S :me1 XI. The procedures for the calculatio~~ of turbulent boundary laycrs available today can bc tlividcd into two classes: methods based on in,legral fornts of t,hc principal equations and methods based on diffarcnlid equations. The former can be traced t,o work t.hat was done by Th. von IGirrn.in in 1!)21 1.: + ! : i s procedure, thc partial tlifferer~t,ialequations are reduced to a system of ordinary dillercnbia~cquat,ions in that. an ana1yt.i~int,cgrat,ion in the t,ransversc dircot,ion is first performed, cf. C11n.p~.V l l I arc int.cgratctl and XITI. It1 the ot,l~erclass of cnscs, t h r pnrt.inl tliITrrential ~clnat~ions dircct.ly I)y the n.pplicnt,ion of nnrnoriral rnrt,hotls, suc:h as the mrtltod of fi11it.ct l i f i rcnces outlinctl in Scc. IXi, or by finitc clcmcnt.s. It, is c:vitlrnl, thn,t t.hc nniount, of work
672
X X l l . 'l'llc incon~prcssiblcturbulent bol~ndarylnycr
1). 7'110 cnlci~lnLionof two-rlimcnsional lrtrbulont boundary lnynrs
673
involved when differential equation methods are used is substantially larger than in tho case of integral methods. The former require the use of a very large digital computer equipped with a large memory, whereas the latter can be done on a small calcrllator or, even, with the aid of a slide rule. I n the following paragraphs we shall confine ourselves to the des~ript~ion of methods which rcsult merely in the calculation of time-averaged values of such variables of the turbulent flow as t,he velocity, the local shearing stress and the region of separation, because we subscribe to the view that only such mean values are of real interost to the engineer. Thus we rcfrain from calculating all those quantities that result from fluctuations, for example the correlation coefficients, the intensity of turbulence and i t s scale. Readers interested in these aspects are referred t o more specialized publications, e. g. [lo, 813. Rcsearch into turbulent boundary layers was considerably advanced by the Stanford Univcrsity Conference organized by S. J . Kline in 1968. The results achieved a t the time have been published in two large volumes edited by S. J . Kline, M.V. Morkovin, G. Sovran, D. J. Cockrell, D. E. Coles and E.A. Hirst [64]. I n the appended [79] "morphology" prepared by W. C. Reynolds, the reader will find a de~cription of 20 integral and 8 differential methods and characterized according to their respective physical basis (status = of 1967). They differ, principally, in the empirical closure functions which are introduced in ordcr to malre the system of equations solvable. I n addition, the conference had a t its disposal 33 sets of experimental data which served as testing material for the computational algorithms. About ten years later, W.C. Reynolds [81] provided once again a summary review of the very large number of computational schemes; this appeared in his contribution to the Annual Reviews of Fluid Mechanics of 1976 (cf. the same author's 1974 contribution in Chemical Engineering [80]). I n 1974 there appeared the book b y F.M.White [I191 which describes 20 integral and 11 differential procedures. It is difficult, and we shall not attempt, to select a "best method" from among the very large number proposed so far. A summary of many of these methods, principally integral ones, was prepared earlier by A. Walz [116] and J.C. Rotta [86, 871. A review of differedial methods is conlaincd in P. Bmdshaw's contributions [9, 12, 13, 141. Further, the book by T . Cebrci and A . M . O . Srnit,l~[20] and two earlier papers by the same authors [18, 191, contain good reviews of many calculational procedures. The two earlier reviews by L. S. C. Ih t,hc aid of boundary-layer theory in the following way: in t,he potential-flow cnlcr~lnt~ion the 1ocal.ion of the point of sepnrdion is treated as a free paraniet,cr. TIIS determinat.ion of this parameter is achieved by combining the cnlc~ilnt.ionof the pressure disl,ribution of thc potential flow with separation wit.11 the calc~~lation of t.he laminar or turbulent boundar layer generated by this pressure distribution. An "atlecluatc flow" demands t,hat t le point of separat,ion of the boundary laycr must coincicle wit,h the point of' separation of the potential flow with a tlmd-n.ir region; the rcquirctl rcsult is achieved by iteration. In this way the point of scpnmt.ion cart be located. The calenln.l.ion bring.9 to hear t.hc influence of the Reynoltls numlwr, h ~ c a u s ethe lomlinn of the point of sepnrat,ion of n tnrbulent bound-
I
c. Turl~~ilcnt boitndnry l n y n r ~on ncrolniln: mnxinintn lilt
S
691
- sepnrntlon;
T = trnnsltlon
Fig. 22.13. Prwure distribution on an wrofoil in aepnrated flow, after K. Jacob [47], nt two different Reynolds numbers R = Vllv
NACA ZL12
R
Fig. 22.14. Lift coefficient e L against angle of incidence a for an nerofoil with a slat. Thsory by I 1000.
1'. S. Granville [39] formulated a multiparamcter procedure for tho cnlc~~lntion of t.urbulcnt boundary layers on rotationnlly symmctric bodies placccl in an a s i d l y tlir~ct~ctl st,rcam. 'l'hc nlcthod hinges o n trlic cnlculat~io~? ol' momcntmm t~l~irkncss and of a shapc factor and can bc used for the aft portion of the body where t.hr I)onntlnry layer thickness is of the same order of magniturlc as thc local radius of t . 1 1 ~I~otly.
r
The numerical constants b and v' should be taken from eqn. (22.22b) and the constant of integration is
I n the more recent formnlation [114], the equation for the modificd shape /aclor in thc axially symmetric case contains the function describing thc variation of the body radius.This is in contrast with the earlier formulation [l 111 according to which the modified shape factor was the same for bodics of rcvolut,ion and t,wo-tli~t~c~t~sior~al bodies. The generalized form of eqn. (22.25) is now
whore the influence functions for the radius and cxternal velocity distril)ut.io~~s nrc Z
I
~ ( 1= ) ~ ( x +~ ) Ill+b U2(l+b)d z ;
Z
N(x) = N ( q ) 1 c
2,
/ IIl-'h lJ2("h)I'GC-' 21
The constants of integration are IJrre R(T) tlrnolcs thc radius of the local cross-section of thc hody of revolution.
G(x1) = v' [ l € ( r l ) {IE(x1)}'+"U(x1)}'+26 N ( x i ) = [U(xi) G(.zi)/I~(xi)lC. 'I'hr nr~mcricalronstants follow from rqn. (22.2Gb).
{R3(rl)}1+b];
tlz.
Tho diagrams in Fig. 22.10 show a comparison bctwccn tlmwy and mrnsurcrnrnt in a flow past a n axially symmetric body; the diagrams plot the Itrynoltls number formed with thc cncrgy tl~icloicssand the modificrl shape factor. Tn order to take into account correct,ions due t o thrcc-di~~rc~~sio~ralit~y r:111sr(1by the possible convergence or divergence of streamlines, J . C . Itotta [8F] proposcs t o base the calculntion on a n clTccLive radius R ( z ) . Numcrioal valuos for R ( n ) nrr snmrnnrizctl in [86] for all ~ n c a s r ~ r c m o lcatalog~~rcl ts in 1541; comI)nrc Iirrt: t,lic I I I ~ ~ : ~ , s I I I . ( ~ mrrils 1)y Mi.\\'. \Villmart,h c t xl. 1122~1nntl A . M . O . Stnit,lr [ I O h ] .
2. Bn~tt~clnry lnycrs on rntnting hndiea. 'J'l~t: calt:ul:~t.io~~ ol' I:intin:~t. 1~)111itl:i1.v layers oti rotating bodies placcd in an axiril s t r c n ~ nwas clisoussotl in Scr. S 1 c. The m c t l ~ o d of calculation which maltcs usc of n t o m r n t t ~ nint.rgl;ll ~ t~~n;ltions, formulated for the meridional ant1 t:irc~~mfcrent.ialtlirrc:tions rcsprc~tivc4y, h s been cxtc~~cletl by R. Trucltenbrotlt 11121 to inclutlo t.hc ~ u r b u l r n t c.:~sr. , IT(: wn.s, ion ol' moreover, fortunato t o succcctl in giving convcnicnL intcgr:ils l i ~ rthe rn.lt~~tI;tt the parameters of thc boundary layer. JCxperimcntnl and furtltcr t11cwrct.ic::l.li ~ ~ v r s ( . i gntions into the boundary layer on rotnting strcamlinc botlics wcrc c:nrrirtl o t ~ I)y t 0.Parr 1741. Jn this casc, tho bountlary laycr grows rajjitlly wit.11 t,lro rot.;~l.iotr paramctcr 2 = (11 RIU,,,; hcrc (1) tlcnotcs tho nng111wvelocity, It t , l ~ t : I:~,t.grst,r;uIius of thc body, nntl ( 1 , is tho axial rofrrcnc:~vcloc:il,y. 'I'lrc t,~~rln~lt!r~l. I~ott~i(l:~ry 1:iyt:r on a rotating body of revolution placctl i l l an axial st.re:ltn can IN: (::dt:~tl:~.lt:~l witl~ t h e aid of thc system of equations (11.45) t o (11.48), in whicl~thc shr:wit~gstrcss must bc ass~tmcdt o vary with the rotat.ion pnr:mct,rr. 'I'hc tlin.gr11.m i l l I'ig. 22.16 compares Lhc onlculatctl nntl n~cas~trt:d vnlt~cso l tho rnomcntr~nrI.l~ic:lit~c:ssrs A,, :111t1 ,a, as rcportcd by 0. l'nrr [74] for :I cylintlrical botly provitlctl wit.11n. sp11vric.nIIlosr. 'L'hc ngrccrncnt is good. 'L'lrc rcgion of tmnsition l'ron~ laminar t o Lnrl)ulcnt flow moves forward a s thc rotation paramctcr incrcascs; its position coincitlos with t h e point a t n~hiclrthc momentum tlrickncsscs incrcnsc a l ) r ~ ~ j ) ( JScc y . also Scc. X1112. A mctliocl for thc calculntion o l tll~rrr-tli~~~c~tisionnl bo11ntl;~ry I:~yc:rs or1 st.;~t.io. rtnry botlics a s \vdl a s on rotating orics, suc:lr a s pro[)cllors or 1)l:rtlcs of rol,;lr.y cornpressors and turbinw, was inclicatcd by A. Magcr [Fl]; comparat,ivc mcasurcm e n b are contained in ref. [621. H. Himmclskamp [44] carried out mcasurcmcnts in thc boundary laycr on a rotating airscrcw ant1 tlctcrminctl iooal l i l L cocffioic~rb of t,hc blade from mcasurcmcnts of prcssurc tlisLribuLions. Somc of his rosulbs arc? sccn reproduced in Fig. 22.17; they arc given in the form of plots of lthc local lift cocfficicnt., c,, a t various radial seclions, in term^ of thc nnglc of irlciclcnce, or. Corresponding mcasurc~nentson n stntionary bladc placcd in a wind tnnnnl are also shown for comparison. Figure 22.17 shows t h a t ~narlrr?tlly itrt:rcasctI lift c o e f l i r i ~ ~ ~ f , ~ arc obt.aincd near the hub, and the d c c t can LC trnoctl Lo soparation 1)cing clc1:~yccl t o larger angles of incidence. F o r cxamplc, the scc:t,ion closcst to the hub has I , maximum lift coefficient of 3.2 compnrcd with 1.4 on thc stat,ionary blatlc. The tlisplacemcnt of s ~ p i r a t ~ i otowards n larger anglcs of incidcncc is cxplai~rctlby the appearance of a n additional acceleration which acts in thc flow direction ant1 which is crcatcd by Coriolis forces; i t has t l ~ samo c cffcct as n favonrablc prnssurtr gmtliont. 111addition, b u t t o a lesser extent, the ccntrifugd forces acting in the bound;rry layer carried with the blade exert a bcncficial influcncc with rcspcot Lo soparntion. Ii'luict p,zr~iclesin the hountlarg layer are actctl upon by a centrifugal forcc which ~
J'ig. 22.17. 1,ovnl lift. cwefhienb. c,, nt. vnriot~~ rntlinl sections on 1wrfor111cc1by 11. lli~ntnolskntl~p [44] tn IIII('ARIIRIIICIIL~
n rot,llt.ingpropeller
nccordirlg
is prol)nt~t,iot~nl t o t h e rnclius. Consctl~lcl~tly, loss fluid is t r a n s p r t c t l t o each blntfc from t h contm tAan a w r y from i t a n d outwarclu, ant1 t l ~ cbor1ntlnry layer is thinner t.llnn woultl be I ( I I ~casc in t~wo-t~itiicnsiotl:rI flow a b o u t t h e s a m e slrape. A. Betz 161 gave some t(t~eorct.icnl argumcnta o n l.his point. F. G u t s c l ~ e[42J m a d e tile flow o n g former with a tlyc. Ccl~t~riftlgal forces also a propellor I~lntlcvisible: I)y ~ ~ a i n t i nChc n sllowctl ill r x c r l :r h r ~ itl~lut:ncc c o n the J)rocc:c%s of 1.mnsiI.ion. I[. M ~ l c s n l a l ~[(j8] his t.hc.sis t,l1:itp,o t l ~ c rt h i t ~ g sbeing cqnal, trarlsition occurs or1 a rotatirlg propeller I)la.tlc at. :t. considcrat)ly lowcr Ltcynolds number t h a n o n one wl~icllis stationsr.y.
Fig. 22.18. ConvcrgrnL :tntl clivcrgent hounclnry Inycrs; ~yrrtmno f coordinnten; a -1 z > 0 ; a) divcrgc~~t~, 11) oonvcrgcnl., a I- :r -: 0
La- '
a n d t h e momentum i t ~ t ~ c g r acquat.iot~ l m u s t b e supplcrncntatl wilh t,l~o:~tl~lil~ion:rl term 0 , 1J2/(z -1-a). Conscqucnt.ly, l.l~c: m o m c n l r ~ minI,rgr:1.1 c q ~ ~ a I , i ofor n (.II(: pl:~.r~c, .of s y m m e t r y which replaces eqn. (22.711) is now
"n*,
3. (:rcnvcrgc~lt ~ I I C clivcrgeni I b o ~ r r d n r ylayers. 'rhc methods for tllc calculat,ion of t , r ~ r \ ) t ~ l c1)ortntlary nt l a y r r s wlrith were tlcscribed in Scc. XXlTb h a v e been ext,cntlntl 11y A. lIcier l6Oj - - - Illwry ns 111 eqn. (23.378) wlllt (XI+'= 0-9; A/AI 1.3 I ) l n ~ e n a l o ~ ~heat l e r ~tranafer roefflrlent hc'lw1cp T W Q W 1 1.ocnI nk111-Trivtlnnrorfflrlnnt r j - r , , / p , IJ' ; -
;
rJ
A number of authors, including G. W. Englcrt [31], IC. Rcsltotlto and M. Tucker [76], N.B. Cohen [12] and D.A. Spence [92], applied the Illingworth-Stewartson transformation with respcct to thc momcntunt-intcgml cquation (23.39) and thus reduced i t to its incompressible form. A. Walz [loo] rcduced the two cqnnLior~s (23.39)nnd (23.40) to a rclativcly convcnicnt form from tho point of vicw of n~rrncricnl computation and oncornpassed the required universal f ~ ~ n c t i o nins a s r t of tnblcs of numerical values. J.C. Rotta [84] described a similar procedure for two-dimensional and nxisymmetric Rows as well as for the calculation of a body of revolution in subsonic and supersonic flow [105]. The agreement between calculations and mensurrmcnt is , = satisfactory u p to a Mach number of M, = 2. The deviations which occur a t M 2.4 and 2.8 are cxplnined, partially, by the fact that the curvature of the strcnmlincs
724
X X l l l . 'rur1)ulcnt boundnry Inyera in coinpressible f l o ~
i n c o n j ~ r n c t i o nwith t . 1 vnrintions ~ of tlenait,y exert^ n n unexpcetetlly large inflrlcnec o n t h e tlevclol~mcnt.o f t.ltc b o u n d a r y Iayc-r - a n effect n o t a c e o r ~ n t r dfor ill t,lle c a l c n l a t ~ i o ~T~h. e rcnsona for t h i s effect, of s t r e a m l i n e cnrvatrrre were invest.igat,etl b y J.C. R o t t a 1821; n cont.rihulion t,o t,llis p r r ~ l ~ l c rwna s also rnatlc b y 1'. Rrarlsltaw (4). Methotls of finite tlilTerr.nrrs h n v c also becn atlnpt.ct1 t o d e a l wit.11 l , u r l ) u l e ~bountlal.y ~t layers in comprrssilde st~rc:ltns.7'.Ccbcci a n d A . M . O . Smit.11 [I)] dcveloprtl n mtttllotl Imsrtl o n ~ n i s i n gt>lwory(srv Scc. X I X r ) wllosc wlitlil.y lins 1tcc.n rxtcntl(,tl t o inc:l~ltl(* t . h r o e - t l i r ~ ~ c n s i o ~1)ountlnry ~al Inycrs I101. 'l'he ~ n e t . l ~ oddu e t,o 1.' 13rntlsltnx\. (sc.e Sec. X I X f ) that mnlres u s e of t.he equat.ion for kinetic e n e r g y h a s n.lso becn e x t e n d e d t o a.pply t,o cornprcssihlc flows (61. P. I~mtls11,zw[5]rcnchctl t,hc conclusion t.hat t,hc volumet,ric tlilxtation e x e r t s a d e e p influence o n t h e st,ructure of t h e t u r b u l e n c e in t h e b o u n d a r y laycr. A g r e e m e n t between rnensnrement, a n d calculnt.ion could b e considera b l y i m p r o v e d b y t h e introduction of an a d d i t i o n a l t e r m in e q n . (19.42). A m e t h o d of int,egrat.ion for three-dimensional cornprcssible b o u n d a r y layers w a s (leveloped b y P T ) . S m i t , l ~(941; a ~ ) r o p o s n li n t h i s rnaI.(.cr w a s tnntlc by J . Cor~st,cix[ g a l : colnparcalso I). Arnnl ct, al'. [ I n l n.nd J. Consteix c,t al. [Db]. References [I] Anon.: Con~preasihlcturbulent houndnry Inyers. A symposium held a t Langley Itesearcll Center. Hnmpton, Virginin. Ilecenibcr 10-11, 1968; NASA S P 210 (1969). nvrc pwli(mt do [In] Arnnl, I ) . , Courrl.c:ix, .I., and Mid~cl,I 0.4. An example of this effect is shown in Fig. 25.11b. The plot represents the loss coefficient for a cascade producing a small angle of turn in a subsonic flow. The Mach number Mz is the independent variable and t,he thrce curves refer to three different Reynolds numbers. The pressure distribution for M = 0.7, Fig. 25.118, shows that a t Rz = 4 x 105 the loss coefficient increases sharply as the Mach number is increased. The sharp increase occurs as a rcsult of shock formation in region8 where the local value of the velocity of sound, c p , crit, has bcen exceeded in the flow. For the two lower Reynolds numbers, Rz = 1.0 x 105 and Rz = 2-0 x 105, the pressure distribution points to a separated flow. The results displayed in Figs. 25.10 and 25.11 demonstrate t h a t the Mach number exerts a deep influence on the flow through cnacades in the range of Reynolds numbers from R = 104 to 105, in addition to the large effect of the Reynolds number itself. The preccding measurements were performed in the high-specd cascade wind tunnel in Brunswick [54] in which the Reynolds number and the Mach number can be varied independently. The diagram of Fig. 25.12 illust.mt,es the effect of the Mach number on the loss coefficient of a cnscade that produces a large angle of turn in the flow. 'rhc cnscadc was designed for incompressible flow. The loss coefficient remains nearly constant a t the value Ct2 M 0.03 up to M2 = 0.7; it increases sharply as the Mach number is further increased. The reason for this behaviour is clear from Fig. 25.13 in which it is possible t o discern the existence of shock waves on the suction side of the blade. These cause separation of the boundary layer. The effect of the Mach number and of the turbulence intensity on the loss coefficient of cascades has been studied in two theses presentcd to the Engineering University at, Rraunschweig by J. Bahr [2] and 1%.ITcbbeI 1211, respectively. Rcference [50] may also be consulted on this point.
XXV. 1)ctormination of profile drag
776
References
Fig. 25.12. Loss coefficient of a turbine cascade, t t z from eqn. (25.34) in brnis of the Mach nnmber Mp aftm 0. Lawaczeck [34]
blnclc nnule: Ps A,
-
snllclily rnt,lo: 111 -0.81 = 00'; I l e y ~ ~ o i dnrtmbcr: n
= 56';
n~iulrnl inlet: /1, 0 x 10'
Fig. 25.13. Transonic Row through n tmrbinn cancndc. Phot,ogrnph obtainnd wiI.11 the aid of Schlieren tnetlrod by 0. Lawaczeck and H.J. Neinernann [32]. Exposure 20 x 10-9 sec. The strong shock waves on the suction side of the nerofoil cause separation and hence large losses, see also Fig. 25.12
I n modern times, the development of steam turbines of increased powcr density has caused the outer bladc sections of the low-pressure stages to operate in the transonic vclocitpyrregimc. This made it neccssnry t o undertalte systematic investigations into l,hc behnvio~trof t,ransonic turbine blades. Here tho Mach nrimber of the a p pronching stream is lower than unity (MI < l), ,whereas that a t exit exceeds i t (Mz I ) ; cf. 1.11 1. Rcfcrcncrs 133, 341 contain a n ~ a c c o r ~ of n t t,ransonic flow across cn.scadc with n Inrgc angle of turn.
,
Abhott, J.H., v o t ~Doenhoff, A.E., and Stivers, L.S.: Summary of airfoil data. NA('A IIrp. 824 (1945). Uahr, J . : Untersuchungen ribw den Einflnss der Profildirke nuf die komp~rusil)leebcnc Stromung durch Verdichtergittcr. D i m Braunsch\\eig l!)62 Forschg. 1ng.-\Vcq. 30, 14- 25 ( 1 !I641, Jhmmcrt,, I
