Boundary Lyer Theory

XV. Non-utcncly boouclary lnyers
whrrr tltc bar ovcr tltc symhls tlcnotrs the rcspect,ive conjugate complex quantit.ies.
'I'hc nortnnl and t,sngcnt.ial componrnt,s of the prriotlic cor~t.ril)ut.ion must vanish
at. t,lto w:~ll, whercns :~t :L large tlistnncc! from it only the tangential component
vnnislws. l~utting 11' -- r/l/2 we ot)t.:~in
ltrgardirtg t>he steady-state cont.riht~tion it is found that only the bounclary
rontlikions at the wall can bc satisfied, and that at a large distance from it is possible
t,o n~:tkc the tnngcntial component, finit.(: but not zero. Thus
3 1
Slb' = - ;t + -- exp ( - 2 77') 4- 2 sin 71' cxp (- 17') -k
4
'I'hr srrontl approximatht is seen to rontnin a strady-st,ab tvrm wliich does not
vattislt nb a Iargc tlisLntice from thr body, i, c. oukide tho boundary layer. Its magnit.utlr
is given by
3 dU,
u2 (2, 00) = --- (I5 68)
4n '0 dz .
,.
I itc precetling nrgunlcnt has thus Ircl its t,o the remarkable result that a potentrial
flow whioh is periodic with respect to time induces a steady, secondary ('strenmirtg')
mot,iori at a Inrgc distance from the wnll as a result of viscous forces. Tt,s magnitude,
givrn hy rqn. (I5.63), is inrlrpcntlcnt of the viscosit,y. The steady-state c~mponcnt~
of t.11~ vc+wit,y is ,such t,lml tl~tiil pnrt.ic1r.a arc seen to flow in the direction of decreaqing
nniplit.rttlc of that component of the potential velocity which is parallel to t h wall.
An c~xntnplt: of s~tch a niot,ion, viz. the pntkrrn of streamlines of t.l~c stcatly
flow al)out a riroular cylinder which oscillntes in a Lluitl at rest, is shown in Fig. 15.7.
I'igt~rc- 15.8 cont.:~it,s a of the flow pnttcrn al~out a cylinder which performs
an osci1lnt.or.y mot,ion in a tank filled with watrr. 'I'hc camera with which the photograph
was t.nltrn nlovctl with t . 1 cylintlcr ~ nnd the surfmx of t,he water was covcrcd
with fine ~nct.n.llic pnrt,iclrs which rnntlcrctl the m~tiorl visible. Thc particlcs show
up as witlc I)nntis in the pict,ltre owing to the Ivng exposure time antl to their
rcciprocat.ing mot.ion. 'l'l~e fluid partic:lcs flow t,ouwds bhe cylinder from above and
from hrlow, and move away in both tlircrt.ions pnrallcl tm the reciprocat.ing motion
of t.lic cylintlrr. 'l'l~is is in good agrccn~rnt wit11 t,hc Ihcorct,ical pattmn of streamlines
show'ti in Fig. 15.7. Siniil:~r phdtogrnpI~s wrrc also I,lll,lisl~cd hy 15. N. Antlradc [I],
who it\rlrirc.tl st.:~n(lit~g so1111t1 W:LVCS~~OI~~. ;I circulat nylintlcr and rcndcrctl Lhercsulting
secondary flow visible I)y tlic injection of smokc.
Fig. 15.7. l'n1,tmn of utmnrt~linru
of the fitcnily uccontlnry tnolio~l in
the n~ighl~onrhood of no osrillnting
rirculnr t~ylind~r
e. I'eriodic ho~~ntlnry-lnyer flown 43 1
l'ig. 15.3. Scconclary flow in tho n~.i~lil,owllootl
of n.11 oac:illnI.ing circ~~ler c:yli~~clcr. l'hc
cnmorn niovcs wiL11 tl~c cylilidor. 'l'hr ~rict.n.llic
pnrticltxn which wrvc to rcl~tlrr f,l~o flow viuihlc
sl~ow 111, n.s wide bands owitla tn t,llc
long exposure tilllo nncl to thir rccipror.;~ting
motion, nftcr Scl~licl~l.itlg (441
Tt is importmt to notticc hare that tllo first npproximat,iot~, 11, itt ccp. (16.62),
shows t,lti~t, t,ltc tliffcrcnt layers in the fluid oscillab with clilTcrcnt phase shifts
compnretl with tltc forcing oscillations, and that their amplitudes dccrmsc ont.warcls
from the wnll. 'I'hc sn.tnc f~at~~~rcs were nxhil)it.cd hy tit(: solutions tlisi:~~rtsivl in
Cltnp. V. 'rhc first approximntion, u,, as wcll as the so111t.ions in Chap. V wim 01)t,ninctl
from c1ilh:rcntial eqttntions which (lid not, contain the convcctivc tnrms
It. mn, t,hcrcforc, I)c stated tltnt y-dcpcntlcnt. phase s11ilt.s :rnd amplitirtles tlrc.aging
with distance from the wall are caused cxclrtsivcly by the nct,ion of viscosit,y. On
t,hc other hand, in tho second npproxirnatioll, ul, there appears a tmm which is not,
periodic antl whiclt rcprcser~t.~ stcntlg stmamiug supcrirnpos~l on the oscilln.t~ory
motion. Ilcncc, it can also be st&cd t,l~at sccontfary flow has its origin in t,l~e convcctive
tmms and is due to tlte intcraction bct,ween inertia and viscosit.y. TL slto~llrl
be borne in mind that simplifications in wlticlt the convcctivc tcrms 'tavc I)ccn on~it.t.c~l
le;d 1.0 solutions which arc frw from streaming and may, thcrcforc, give a rnislcailir~g
representation of the flow. Streaming docs, in general, appear only wlicn the? solut.ion
is carried to at least the second-order approximation.
'I'hc phenomena under consideration offcr a sirnplc explanat.ion of I