Boundary Lyer Theory

462
XVI. Origin of tnrb~rlmcc 1
numl)er obscrvctl :~t the point of l,ransition. If athnt.ion is fixed on t,hc Ilnw in the
honntlary hyrr along a wall, then t,llc tllc:orctical critical Itcynoltls nnml)cr indicates
the point. on t,111. wall at. wllirlm :~niplific:ntion of somc individual tlisturbances begins
ant1 prorreds rlownstroam of it. The transformatkm of sucli ;~mplificd disturbances
i11t.o t.nrlnllrnw t,altrs I I somr ~ timc, and tho nnstal~lc tlist,ltrl)nncc: has had a chancc:
to t.r:tvel somc tlistn.nrc in the tlownstrcsm direction. It must, therefore, bc cxpcct-cd
that, t.hc o1)scrvctl posit,ion of thc point of transil,ion will be tlownstream of the
calculat~ctl, thcorctica.l limit of stability, or, in othr words, that the experimental
critical Reynolds number cxcccds itFl thcorctical value. This remark, cvidcntly,
applies to Rcynolds n~~mbcrs 1)asctl on tlmc curmnt lcngth as well ns to those bsscd
on the bourdary-layer thickness. In order to distinguish bctwcen these two values
it is usual to call the thcorctical critical Reynolds number (limit of stability) the
pint o/ l:nstabilit?y whcrcas the experimental critical Reynolds number is called
the point o/ trnn~itiont.
Thc st,nbiiitg problem, briefly described in t,hc preceding paragraphs, leads to
cxtremcly difficult mat,llcmntical consitlcmtions. Owing to tllcse, succcss in the
calculation of thc critical Jtcynolrls nnmbcr eluJccl the workers in this field for
several cleoatlcs, in spite of the greatest efforts clircctcd towards this goal. Consequrntly,
in what follows we shall he unable tx, provide a complete presentation of
tl~e stdility t.hrory nnd will be forccd ta restrict ourselves to giving an account of
the most important rcsul th? only.
5. Genernl properties of the Orr-Sommerfeld equntion. Sincc from experimental
cviclcnce thc limit of stability c, =O is cxpectcd to occur for large
valucs of tlmc Itcynolcls number, it is n:~tural to simplify thc eqnation by omitting
the viscous bmms on t.he right-hand side of it, as comparctl with the incrtia tcrms,
beca~~se of t,lmc smallness of the coefficient 1/R. The result.ing differential equation
is known as t.110 /rictionlr.~s .~lnhilily ~q7mlion, or Ru?~lri~~A's equntinn:
(IJ ---c) (4" - a24) -- I/"$ = 0 . (16.16)
It, is imporhrmt. 1.0 note Ilcw that of t h four bountlary contli(.ions (16.15) of t h
cornplctn equation it is now possiblc to satisfy only two, bccausc the fricl ionlcss
stability cqu:~t.ion is of tltc sccon(l ortlcr. 'Umc rcmai~mirmg boundary condition to bc
sat.isficd is t,l~c vanishing of t.hc normal componrnt,s of vclocity near thc wall of
a CII~~IIIICI, or, in l)o~~r~(I:~.r.v-l:~y(:r flow, tlwir vanisl~ing al, I,lmc wall nn(l at infinity,
,.
I bus, in the I:~l.l.cr case, wc 11avc
y=O: +=O; y=m: +=O. (16.17)
,I
Lllc onmission of the viscous tcrms constitutes a tlra,st,ic simplifirat.ion, hccmse the
ortlcr of t,lmo cqnn.t,ion is roclurctl from four to Lwo, and t,llis may result in a loss
of imporl,:mO prol~c!rl.irs of tho gnnom.1 soluI.ion of thc complctc cqnat.ion, as comparctl
wil,ll its simplilictl v(:rsion. Ilnrc we n1n.y rc~pcn.l.'tlmo rom:~.rks nol.c:cl provio~~sly in
C11n.p. IV in conncxion with the transition froi? the Navicr-St,okcs equations of
a viscous IIrritl Lo those for a frictionlcss fluitl.
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t nlrrruly cxplnirtrrl in Src. XVIn, rocnnt, expcritnrntnl rcsulL~ (11. \\I. ICrnrnons 1251, and
hdl~~l)n~tor RINI J