Boundary Lyer Theory

d(i.4 XVT. Origin of hrbulcnce I
of wavol~:ngblts; ill t.l~c tlircction of (lcwmsing Itoynoltls numbers, this rangc is
scpnmtctl fron~ tl~c stal)lo rnngo by tho anrvo of nc:utral stability.
tn contrnst, with t.11~ l~rccctling C:LSC, IJ~SC~LS inatahilily is associatcd with a curvc
or ncnl,r:~l st:l.i~ilil,y of sl1n.p b, also sl~own in Fig. 10.8, and with 1)ountlary-laycr
prolilrs pnsscssi~~g no point of i~tflcsion. At Itey1101~1s nwnl)crs tending to infinity,
I,hr r:~.t~gc of t~nsl,:~.l)lc w:~.vrlrngl~l~s is rnnl.r:~cl~~~l to a point, anql (1om:tins or ~~nst,:~l~Ic
osc:ill;~l.iot~s :IIV swt~ 1.0 oxisl, otlly for fi~~il,o Ilnyrioltls t~nrnl)ors. (~cncrally spr:~.Iting,
t,ho n.tno~tnt, of :~.rnplificnt.ion is m11c11 largcr in t,hc casc of frictionless in~tabilit~y than
in tho c:~sc: of visc:ous it~sl.:~.l~ilit,y.
, ,
I llc vsislrncSc: of visco~ls insl.:~l)ilily (XII bc tlisc:ovr~~ctl only in c:onncsion wil,ll
a discussion of the fitll Orr-Sommcrli4(1 equatior~; it const.itut,cs, tl~croforc, t,hc
moro tliffirrrlt, nn:~l~t,icnl c:tsc. 'l'hc simplcst case of flow, nan~cly t,hat along a flat
plal,c: with zero Iwrssurc gmclicnt belongs l,o the kind for which only viscous inst,abilit,y
tlocs occur; it, W:IS s~~cc:rssl'r~lly taclilctl only comparatively recently.
TI1 corc. rn I I : Tllc sccontl impor1,:lnt goncr:~l theorem sl,atxs that tsho vrlocit,y
of 1~011nptio11 of nnutrnl tlis1mrl):anccs (c, = 0) in n I~ountlary I:~ycr is srnnllcr tll:~n
the m:urtnlrnl vrloci1.y of 1.11~ mcnn flow, i. o. tht, c, < (I,,,.
'I'his t,l~c:orwn was :rlso first provctl by J,ortl IXnylcigll [70J, albcit itnclcr somc
~.(~sfli(*t.iv(* :~s~r~mpl~ions; if, was ~I.OVC(I ngnin by 14'. 'Yollmien [I001 for more gcneral
conclit,ions. It, :lsscrt.s th:tt in t,hc intcrior of the flow there cxist-R a layer wllcrc
IJ - c = O for nc~ltral tlist.r~rl~anccs. 'I'l~is fact,, too, is of funtlanlental importance
in t,hc t.llcor,y of st.:~.l)ilit,y. 'I'l~o Ia.ycr for which 11 - c = 0 rorrcspontls, namely,
to a singttlar point, of t,lto frictionless st.n.l)ilit,y cqrration (16.16). Att this point #"
I~ccwncs ir~fittit~c! il 11" tlocs not v:~t~isll 1,ltcrn simttltnncor~sly. Tllc (1isl~:tnt:c =
wltorc: 11 -~. c is c::i.ll(~l t,11(: crilicctl. Inyrr or 1.110 mr:an flow. If [I," 4: 0, thxl 4" tends
to infinil,.~ :w
' ' 1 1
. - ..
(I,/,. !/ ---y/<
in 1.11~: ~lc:iglll)o~lt.l~oo(l of 1.11(* c:rit,ic::d 1:tyc.r wllrrc it, is pcrtnissil)lc t,o pnt IJ - c =
--- 1Jlfr(?y - yJi) :~.[~l)roxit~~:~l.cl.y; (:onsrq~~~~nt,ly IJIC x-ootn~mncnt of lhc vclocil~y can
br writ.t,rr~ :r.s
'I'hrts, a.ncortling t,o Ihc frictionloss stability cquation, the component, IL' of the
vrlooi1,y wl~idt is p:~rallcl to t,llc wall l)cc:otnos infinite if the curvature of the velocity
profile at, the critical layer tlocs not vatdsli simnltnnconslg. This mn.thmatical
sing~tI:~.ril,y in t.ho fric:Lionloss sl.a.bilit,y ccl~l:~l~ion poin1.s l,o the f:i.c:t tll:~t t.11~ cff'rcl
of viscosil.y on (.Ire c(111:~I.ion of motion tnttst noL be ncglcct,ctl in Ihc ncighbor~rhootl
of t,hc oriticxl I:~ycr. 'J'llc irlnl~~sion of t,hc effcct of viscosity removes this physica.lly
a11~11rd sing111:1rity of 1.11~ frictionlrss st,:~l)ilif.y ccl~lat,ion. 'l'hc n.na.lysis of the cfial,
of (,his so-c:rllotl viscous corroct.ion on 1.h~ soluticfn of the st,abilit.y cquation plays
a fut~tlnmcnl.n.l part in tltc tliscussion of st.nbility.
'J'l~c two tllnorcms tluc to Lord Raylcigh sl~ow that tho curvature of thc vclocity
profile n.ni.ct,s st~al)ility in nfi~ntlatncnt.:~l w:~y. Sini~~ltar~cously it has hen dcmonstmtcd
t11:lt the c~:rlaul:ttion of vrlocit,y profilcs in laminar bountlary layers must proceed
with vrry high accnraoy for the investigation of stddity to bc possible: it is not
enongh t,o cvsluato U (y) wit11 sllfficicnt dcgrcc of accnrnry Imt. i(,s srconcI (~rrivativc
d2Ultly2 must also bc nccurotely hown.
c. Itcsultn of tl~c theory of stnbility us tltry upply to the bo~r~~tlr~ry Iayr nn XI flut pl;~tc
at zero ir~citlcncc
Velocity U
ffm