Boundary Lyer Theory

470 XVI. Origin of h~rlwlcnce T
I
R,,,, '5.0
Fig. IB.11
Fig. 16.10. Chrvrs of nc~ltrnl skd)ility for t h tiint.rtrbnncc freqllency /?, and thc wnve velocity c,
n*i n fnnclion of llnynoltls ~nrtnbcr for t.hc hnundnry layer on n flnt plnb nt zero incidence (Blmius
prolilr). 'J'11ror.y ncxwding tn W. 'l'olltnicn (991: nrrlncricnl cnlc~llntions by R. Jordinson [47]; see
nlm 'l'nl~lr 16.1
Fig. 16.1 I. Cnrveu of nerrtrnl nt.nhilit,y for t.lw tlist.~lrhnncc wnvelrngtll n 61 nn n f~rnction of Rayndds
ni1nt1wr for 1.11~ 1~ound:1ry Inyor on n 1hl. plntc at zero inciticncc (Biwills prolilc). 'l'llcory nccording
to W. 'l'olllnicm [W]; nnn~rric:nI mIccllnt,ionn by 1%. Jordin~on 1471; see also Tnblc 16.1. The
nrnplitm~lc? tlist.ribrition for dist.nrlmnc:cn 1 nnrl I I in given in Fig. 16.20
r 7
I Itis is t11c ~wint. (11- inst:~l)ility (iw t .1~ 1~or111tl:i.ry I:~.~cr on n Hat, plxtr.. 11, is wtnnrkal~lc
t.hn.1. only n c~orrly~:l.r:~l~ivc:ly n:Irrow mngo or'wnvclcngtl~s n.ntl Sreq~tencirs is "dnngcrow"
li~r t.110 I:~.rninnr I)o~rncl:wy layer. 011 t,hc one Ilnntl, t.lwrc is n lower limit. for
the Itcytol(ls numl)cr, on t,ho otllcr, thcre is an uppw limit for t h cl~nrnct.erist~ic
mn.gnilwles of t,llc: tlist~urhnccs. Once the Iat,tc:r arc cxcceclcd no inst,nbiLit.y is cnusctl.
The nuinrricnl v:~Iucs :we:
A tlctxilcd c:ompn.rison bot,wccn the precctling thcorctid results and experiment
will be given in the next section. Hcre we shall only remark tlmt the position where
the boundary layer bccon~cs first unstd~le according to theory (point of instabilit,y)
must. always be oxpect~~l tm lit: ~~pst~rcnm of the experimentally observed point of
tmnsition I)ocn.~~sc nctunl tmh~~lencc is created along the path from the point of
itlst~hiiit~y to the poink of ltran~il,ion owing t,o thr nn~plifirntion of thr rrnslable
disturbanrcs. 'rhis condition is satisfied in 1.11~ cnsc untlcr consitlcmtion. \\'(a 11:~ve
rr:l
= 960 (point. of tr:wsil.io~~) ,
. I , he clist,nncc bctrwccn tilo point of ins1,:~l)ilit:y nntl lhn point. of 1m11sil.io11 tlclw~~tl.r
on t h d~grec! of n.mpli/irnlio~, and IJIC kid of tlist8url)nr~ccs 11rcscnt in th(: cxternnl
stream in tens it.^ of turbulence), lmt. t11c R ~IIRI n~ccl~nnism of :1.11111lilicnt,io11 ~:III IN:
ol)t,ninctl from t,he sLdy of the ~nngnit~tlcs of tho ~~nr:trnctr!rs in the inlr:rior of l.l~r:
ctrrvc of ncutml std)ility, P, > 0. (~nlc~~lnt,ions or this Itin(l wc:rc: first, ~)e:rli~rrrlc:tl
hy 11. Scl1Iicl11,ing 1761 l'or 1,110 fhl, 1)111l,c; I,ltcy IIILV~: I)c(:II rq~v~~.I,(;cl l),y S. I*'. SIICII
185 1.
In order to g~in n clenrcr insight intn the mcc:Itnnics or Lltc oscillnLit~g rnot,ion,
JI. Scl~licl~tirlg [77] dctcrmincd the cigcnfi~nct,ions $(?I) for scvcrnl ncutml tlistturhances.
This enabled him to draw the pn.t,tcrn of strc:rmlincs of the tlis1,url)ctl motiotl
for neut1ral oscillnt.ions. An exnmplc of such n pnt.t.crn can he foun(1 in Fig. I(i.14.
7 7
I he tlin.grntn in Fig. 10.12 illr~sI.t.:~l.cs tho nn~j)lilic:nl.ic~n of IIIINI.ILI)I(~
in the bounclary layer on n flat, plah. 'l'h tlingrnm, based on n recent, cnlculnt.ion
performed by 11. G.Onibmwski ct nl. [GR], extcnds ovcr n wide rnngc of R.cynoltls
nurnbrrs. It turns out t.hnt the rnaximrttn nrnplificntion rntc does not pl:~co it.sc4l' at
very higl~ Reynolds n~~mbcr (R -+ m) 11nt is locntntl in the motlorntc rango of R =- 10"
to 10". ?'Itis is due to the fnot, t.llnt. t,lw rrtrvc of nc~lt,rn.l sl.nl~i1it.y fcrr n !In( pln.tr is of
Fig. 16.12. Curves of cot~st~nnt, temporal
amplification for the bonndnry lnyer on a
fiat plnte nt zero incidcncc ovcr n wide
"I 0.25 -
(list.l~t.lt:il~(:(~~