Boundary Lyer Theory

28
TI. Ot~(,linc! of bor~ndnry-layer thoory b. Srparation antl vortex fortnn(.ion 20
Dalinition of Imnndnry-layer thickness: Thc clefinition of lhc bountlary-laycr
t.lrickncss is to a ccrtain extent arbitrary l)ccausc transitsion from the velocity in
t,l~c borlndary t,o that o~~t.sitlc it t,:~.ltcs plncc asympt,olically. Tlris is, IIOW~VC~, of
no pract,icn.l import,ancc, I~ccnusc t,hc vclocil~y in thc bor~ntlnry laycr at.t,:iins :I. vnl~lc
whic:h is vrry c:losc t,o fho cxl,crt~n.l vcloriLy drcatly at, a small tlistancc frotn the
wnll. 11, is Ijossil~ln to tlcfino Lhc I)o~lnd;~~.y-l:~yc:r thioltncss :IS l.l~nl rlis1,:~noo from lllc:
wnll wllorc: t,hc vclonity tlilTcrs 11y I pcr ct:111 from the oxt,crnn,l vrlociLy. \Vil.l~ titis
dnfinition the rtrtmcric:d f:~.ct.or in cqn. (2.2) has thc value 5. [nst,ead of t,hc bonntlarylaycr
t.lricknc~s, anotlrcr qunnt.it$y, thc dinplr~cement thickness a, is somct.imcs used,
Fig. 2.3. It, is dcfinetl by thc cqnntion
(2.6)
'Ilc displnccment tl~icltncss indicates l.llc tlistancc by which the external strcamlines
arc shift,cd owing to tire fonnat,ion of t,l~c boundary Iaycr. In the case of a plate
in parallel flow nntl at zcro incidcncc tlrc tlisplaccmrnt thickness is about & of the
bountlary-layer IJ~icltncss 0 givcn in cqn. (2.1 a).
..
b. Srpamlion and vortcx forrnntion
llte bo~~ntln.ry laycr ncnr a fht plate in par:~llcl flow and al, zcro incitlencc is
part,icrllarly sirnplc, Ijccausc the static prcssurc remains conshnt in the whole field
of Ilow. Sincc orlt,sitlc the 1m11ntI:~ry lnyrr tho vclocily rcnmins constant t,hc samc
qjplics to the prcss~~re l~ecausc in the frictiorrlcss flow Bcrl~orrlli's cquation remains
vnlitl. Furthcnnorc, tlrc prcssnrc rcmnins scnsibly constnnt over thc width of t,hc
\)o~~~rrlary layer at 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 at the samc
tlist.ancc, ant1 the same applies lo cnscs of arbit,mry body pl~n.pcs whcn tho prcssnrc
o~rt.sitlc 1.h~ I)o~ln(l:~ry I:~yt:r vnrics along t,lrc wall wit11 t,l~c 1cngl.h of arc. 'l'his fnct
is cxprcsscd by saying 1,h:~L t,lrc cstcrnnl prcssnrr is "i~n~rcssctl" on thc boundary
Inycr. Ilcncc in the cnsc of the motion pst a plate l,hc prcssnrc rcmains constant.
througIrouL t,llr: bountlnry Inycr.
'j'lrr phrnonrrnon of 1murrtl:~ry InycrsrpnraLiot \ ~nrt~tiot~c~tlprc~viously
--. - - isi!rtinral~ly
c~onnrclctl wrtl~ tlrr prcssurc t1istril)ution in ti16 orintlary layrr In the boundary
lnycr on a plate rro srpnmlion takrs phrr as no back-fldw occurs
In ortlcr to r\plnitr t IIV very import nrrt pl~rnornrr~on of bountlary-lnycr s~paration
let us rorrritlrr 1 hr Ilow :~ljouI n Ijlrrnt hotly, r g abont, a rirrnlar rylintlrr, as shown
it1 IClg 2 4 111 ft ic.1 inl~lcw flow, t l ~c flu~tl par1 irlrs nrr :~rc.rlrmlrtl on tlw npstmam
half frorn D to E, and decelerated on the downstream half from E to F. Ifcnce the
pressure decreases frorn D to E antl increases from i' to F. Wltcrl the flow is stmtcd
up the motion in the first inst,arlt is very nearly frict,ionlcss, ant1 rcmains so as Img as
t h bounthry lnycr remains thin. Outsitlo lhc I~onntl:~ry lrtycr lllcro is n tprr~l~s~ornlctl.io~
of pressure into 1tincl.ic energy idong 11 R, 1.110 rcverso hlting pl:~c:o r~lottg IC I(', so
IJtaL IL parlidc nrrivo~ ILL 11' with Llto HILIII~> vclocil,y 11s it, IIIL~ nl, J). A lIrci(l ~~:~rl.iclt:
wltich lrroves in IJIC i~nmctlinlo vioi~til~y of tho wtdl in I,llc bo~lntl:r.ry I:~.yor rc:~n:iit~s
under the influence of the same pressure field as that 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
Fig. 2.4. Doundary-layer scpara-
tion ~ind vortex forrnntion on a
circular cylinder (dingran~n~atir)
S - point nf scl~nrnllo~~
energy on its pat.h from D to E that thc remaintlcr is too slnall to srlrmount t.hc
"pressure hill" from E to 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 pl~otogra~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 to tlte short traces. In Fig. 2.5s, Lakcn shortly aftcr the start of lhc rnot,iorl;
the rcvcrsc motmion has just begun. In Fig. 2.5b the rcvcrsc nrotion lrns pci~-t,r:.tctl
a consitlcrablc distancc forward :~nd l,l~c boundary Iayor lrns tllicltcnctl n.pprcci:~l)ly.
Fig. 2 .5~ shows how this rcvcrsc mot,ion givcs risc to a vortex, whoso sizc is incrc,iscd
still furthx in Fig. 2.6tI. 'l'hc vorLcx bccorncs scp:~mlctl shortly afLcr~:~r~Is n.td rnovc!s
tlow~~strearn in tho fluid. This circnn~stancc changcs complctcly blrc fiolcl of flow
in tho waltc, and Lhc prcssnrc clisLrib~lI,ion suKcrs a rntlical change, as cornparctl
with frictio~rlcss Ilow. 'L'llc find 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
At a larger distance from the body it is possible to discern a rcgul:~r patt,ern
of vorticcs which move alternately clockwise and courrt~crclocltwise, and wllich is
known as a IGirmiin vortex strect [20], Fig. 2.7 (scc also Fig. 1.6). In Fig. 2.6 a vortex
moving in a clockwise direction can be seen to be about to detach it,sclf from the
body before joining the pattern. In a further pzpcr, von Kilrmhn [21] proved
that such vorticcs are gcncrally nrrstablc with rcspcct to small tli~t~urbancrs pnrallcl