Boundary Lyer Theory

434 XV. Non-stcxcly boundary layers
From what has I~ecn snit1 hefore it is clear t,I~at the position of the point of
laminar scparat.io~ is aKcct,ctl by the cxt,ernal osciliat.iona nnd t,hnt the point of
separation must osrillat,~ it,sclf. Finally, C. (1. ],in's mcClrotl lends to t,he valr~able
conrlt~sion t,li;~t t,lw f~rncl;uncnI,nI oscill;tt.io~~ iirtlrlcrs 11iglrr.r harmonics in thc boundarylayrr
o~rillnt~ion.
3. Extrrnnl llnw with etnnll, l~nrlnonic prrturLntins. The c ~ when c t.hc extcrnnl flow
perfor~nn small, hnrn~onic oucillnt~ions hnn bcrn tronbed in n nutnher of publications. The method
employcd was lhnt of a scrirs rxpnnsiori in t,ho pcrturbntion pnran~etcr described in Sco. XVn 3.
We nnswne that the extcrnnl flow is of t,l~e form
IJ (x, 1) = Tf (XI -I- rr, (x) ~'"l , (15.70)
and note t.l~nt,, for it, nod, investigntions rest,rict t,l~en~selves to the cnlculnt.ion of t,hc first
approxin~nlion, that. i ~, of thc fnnctions n,,ol, and l', from eqn. (15.30). M. J. I,ighthill [27]
for~nnlntrd an npproxin~nt.c n~etllotl for tlrc solution of cqn. (15.32) for arbitrary forms of tho
fttnc:t.ion o(s) nnd iJ,(s). The particular cnse when both functions cnn be represented in tho
form of power series ltnn been ronsirlcrcd by 1';. Hori [24], whereas N. Ibott and M. I,. Itoscnxwrig
[3!)1 o~nn~inocl t.110 CXRIII~IO W I I the ~ t.wo Ftn~ct.ions #(x) nnrl #,(z) nro ui111p1e powers
or r. 'l'llc c?xrltll~)~o of ~t.;ignat,ion flow st.lldiccl by kt. I$. (:lnllcrt 1131 nlld N. /Lot.t [:!)I nR wc?II
as tlrc Ilow along n Il:lt, plate nt 7,cro inciclcrlrc dincnssccl by A. Gosh (171 nncl S. (:il~bc:l:rLo [I 1, 121
oons1it.ut.c n111)-c:nsw of t.11~ 1nt.tc.r. Finally, A. (:tmll 1171 nncl 1'. (:. Hill ant1 A. 11. Strnning 12.71
pcrforn~cd exporitncnt~nl ~nc:murcmclltR on non-sleady ho~~t~clnry layers.
If the oxtcrnnl flow is of the form
U (r,t) - csm (I + E einl) = if (I + E ci1I1)
(15.71)
thrn rqns. (15.31) Itwl 1x1 Iho familiar d1(rrrrntia1 rqt~ntions for uiniilnr sohltions, C(1119. (9.8) and
(!).Rn), nntnc.ly,
with
Asnutning in cqns. (1G.32) that
u, = E ei"' V @, (6, 11) ,
wo arc? Ircl 1.0 t.hc ftlllowing tlilTcrenl.inl cqnntic~nn for t.lw rwxilinry functions @(E, 7) and O (E, 7):
nt + 1 ln -1- 1
-- -
fDq.t'l
/ (D,," - (€ t 2 m/')@, -1- 1'' @ - ( I - in) 1' E @,,E 4-
1 2
2
+ (1 - m)/"€@t + [ .I- 2m = 0 , (15.76)
with t,hc bonndnry conditions
r. Periodic boundary-Inyrr flows 435
The precwling clilTcrmt.ixl cqnntions arc, normnlly, ROIV~~ in t.hc form of srrics expnn.siorls. first.
for smnll vnlucs of F and lhcn for Iargc vnlucu of F. Assuming thnt
for small vnlucs of t, we arc lerl to ordinnry dilTcrcntinl eqnntions for tho fnnctions dik(l,) nntl
Ok(7). The derivnlivc~
loenl Nnssclt nnmtmr. In this mnnnor we: rnn clwivo ll~nt,
and that
at q = 0 mrvc to cnlculnb tho slrenring ulrcnn nt t.ho wnll IIR wt!II IIH OIC
I
Arrortling Lo 1'. K. Moore 1.711 (am nlno A. ( h l [I71 ~ nntl S. (>il,l)c~lntn 112]), 1 1 r:rsr ~ of t.lw
flat plntc at zero incidence is reprcacntcd by thc cxpreasion:
nntl
SubstiLuting n = 0, wo rewvcr tho uasi steady solution, which signifies thnt nt every inst,ant
the solution behnvw like the shady Jutio; for tho instantnncous cxternal vcIooity The a penrnnco
of an imaginary term nt n =!= 0 moans that the boundary layer aull'ers n phase shift wit( respect
to the cxternal flow, the shift being diflercut for velocity nnd blnpcrntr~rc. Wl~ereas the rnxxirna
in shearing st,rcsa lend thc tnnxima in the cxternnl Ilow (in the limit n x/IJm -+ CT 1.11~ pllnse
ar~gle tmda to 459, the mnximn in lnmpcrnturo Ing hchintl t.l~rrn (in the limit, ?l,:r/!~,., -t m~
lhc pltmc nnglo tendn to '30"). 111 ntldition, iL turns ont t.l~trt nt lnrgt! VIIIIIOR of n ~/ll~., t . 1 nn~pli-
~
tudc of thc ahenring-strrxw oncillntion incrcnaca withont bound, wl~crens t,l~nL or 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 hnd 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
and n ~upplemnntnry, abndy pnrl which in inclopontlt\nl, or 1.itno. 'l'l~c~ Inl.lr.r tnotlific~~ I.ltc\ 1111sic:
flow and cnn I)o intcrprcbtl na a secondnry flow in cot~~plnhr 11111tlogy wi1.11 1.1111t, t.~t(~t~I(~r(~cI
ill
the solutions of tho pwccding seelion.
For shgnalion flow, wc hnvc Ul(a) = consl, anel it in fo~td
t.lmt t.hnrl u,, o, and all
higher-order term vanish, as demonstrated by M. R. Glnuert [13]. Conneqnently, the basic