Boundary Lyer Theory

536
XVlJ. Origin of t,urhr~lcnce 11
for which tlic tlrt,nils of the laminar laycr arc known from See. V b. A pliotograph
iI11rst~rat~ing the prnt~css of transition on a rot-,ntirtg tlislr ant1 t8nkcn by N. Gregory,
.J. '1'. Sttiart. rind W. S. Walker [77] is rcprotlucctl in Vig. 17.40. 7'11~ photograph shows
t.hnt. in an nrtnnlar rcgion ttherc nppyr st,at.ioriary vort,iccs which assumc thc shape
of log:trit,litnic spirals. 'l'ht: inricr ratllns of this region marks t,l~c? onset, of inst.nOilit,y
:tntl f.rrlt~sit~ion oc:c:urs :el. I h oltt,t:r mtlills. 'l'lic: intwr r:~(Iius (:orr~spon(ls 1.0 :L ItcynnItIs
nt~titl~rr of R, -1 ltt2 (L)/v = I.!) x 105at~tl ;et tlic outer radius we havc R, -- RO2 (01, =
=. 2.8 x 10". ,J. 1'. Stt~art complcn~ontctl t,he cxpcrimcntal work with an analytic
stmtly of thc stahilit,y of such a motion. In it,, IIC assumctl t.11~ existence of threetlimrnsiorial,
pcriotlic dist~lrbanccs whose forms incl~ldcd 0.s special cases the progrrssing
'I'olltnirn-Sclilicfitirlg waves as well as thc ~tat~ionary, t,l~rec-dirnensiol~d
'I'aglor-Gorrtlcr vor1irc.s 'I'hc rcsults of his calculations sliowcd qrtalitntivc agreement
with Ihc cxpcritnc~ntal results of Fig. 17 40.
Anot,lier case of t,l~is kind occurs on a yawed flat, plate in supersonic flow when
the :~ssorin.t,cti lantinnr lw~~ntlnry Inycr t)rcoinrs unst.at~le. As ~liown cxpcrirncntdly
\,y .J. ,I. (:inoux 1841, t,hc \ IOIIIKI~I.~ hyrr tirvvtops ~ongit.w~ina~ vort.ic:rs which product:
trnnsit ion.
g. The i~~fluer~ce of rougl~rless on transition
I. Introtl~~ctory remark. The prolhrn which we arc about to examine in this
sec.t.ion, namcly the questhi of how t.hc process of transit,ion tlepentls on tllie roughness
of t,hc solitl walls, is one of consitlcra1,lo practical importance; so-far, however, it
has not. I)cen possiblc to annlgzc it t.ltcwrct,icnlly, Tllc prolA:m untlcr cot~sidcmtion
has g:i.inctl in irn~mrt.anc:e in the rcccnt ~~ast,, ~~artic~~larly sincc thc adventl of laminar
arrofds in acronaubic;d applications. 7'hc vcry cxtcnsive cxpcrimcntal material
nolh:ot,cd np 1.0 d:'.t,c incl~drs information on thc efrcct of cylindrical (two-dimensional
roughness elcmc~it~s), point-like (t,hrec-dirncnsional, single roughness clcn~cnts) and
clistril~ut~ccl rorlghncss clcmcnt~s. Many of the investigations include addit,ionnl data
on t,ltc: inflt~crtcc of pressure gratlicnts, t,~~rbulcncc intensity or Mach number.
Generally spearking, the prescncc of rougllncss favours transition in tl~c. sense
t,liat utltlcr oblicrwisc itlcnt,ical cordit.ions transition occurs at a lower lteynolds
nt~nibcr on a rough wall Llian on n smooth wall. That t,liis should bc so follows clearly
from t,lic I,llcory of stability: ti~~_e~i?tencc of roughncss elements gives rise to arldit,ional
tlistur\)anccs in tho laminar stream which have to bc atltlcd to those gcncratG4
by t~~trI~~~Ien(~~
ant1 alrcady present in the boundary layer. If the disturbances creatctl
by roughncss arc bigger tiinn those due t.o turbu~cncc, we ijiiistckji+ !,h