Boundary Lyer Theory

504 XVII. Origin of k~rbulencc 11
positmion of t,l~c point of tmnsit,ion is shown in addition for aerofoil R 2626. It is
seen that transition occurs sliortly after the pressnre minimum in complete agreement
with t,Iw t,heorctical results in Fig. 17.10. Figure 17.16 shows, furt.her, plot,^ of drag
cocfficicnt.s in terms of thc lift coefficient for three aorofoils of equal tl~ickness but
varying caml)er. It shonld be noted that hy increasing tho camber it is possible to
canse a sllift in the region of vcry small drag in the direction of higher values of
lift,, Intt rvrri so, t,lic rngion of rctlucctl drag still extcntls over a definite witlth only.
Needless to say, in the case of laminar aerofoils t.he int,crart,ion between t.hc estr~.nal
stream antl t.hr bountlary layer is very import.ant; mct.liotls for the c:i~Ic~tlation of
sncli effects have been tleveloped by R. Eppler [BO]. At this point, it, is nrccssary t,(~
remark t.11ut cert,ain rircumstances c:ause consitlcrablc difficulties in t.he pract,ical
application of laminar arrofoils. Principally thcse are dnc to t.he great. drmn~~ds 011
t,he smoot~l~ncss of the surfaces in order t,o exclnde prwnat,ere transit.iot1 owing to
roughness. In this conncxion we wish t,o draw the reader's at.tent.io11 t,o a paper by
I,. Speidel [212] on lnniinar aerofoils placed in a Iiarn~onically dist,urbed free streatn.
Fig. 17.15. I'rr*nw~rn dist,rih~tI,iot~ lor l:~.tninnr
arrofoils at zero incitlmcc (c, 5 0). i\erofoilsOOI2,
65, -012, 66, -012 from rrl. [I];
wrofoil It 2525, nftrr IIort.~ch (Dl]
'I' = posilinlt or point or trxnsilion for R -- :1.5 x 10'
Fig. 13 Ili Corfficirnts of profile drng. c,,,,
plotted ngninst lift coefficient, c,,. for three
Inniinnr nrrofoils with vnrying rwnhrr,
R - 9 x 10" from ref. [7]. The rrgion of
smnll drag mows townrds higher lift roef-
ficients, c,,. as rnmhcr increases
b. Dcterrninntion of the position of the point of instnbility for prrsrrihrtl horly nhnlw 505
'L'hc discussion in this section may bc suminarizctl as follows:
1. The tJicory of std~ilit,y sliows that, tlic prcssure gmdioll; cxcrln an ovcrwl~c~ltnir~g
influence on the stability of the Imninar bountlary hycr; a tlrcrc-as(: in prcssurc
in the downstream directlion has a stal~ilizing cKcct,, wltcrcns increasing prrssurc:
leads tjo instnbility.
2. Jn consequence, tlic position of the point of maximum vclocil,y of t.lic pof.rnLi:~l
velocity distrit~ut~ion function (= point of minimurn pressure) inllucr~c:cs tlccisivcly
the position of the point of inshI)ilit,y antl of t,hc point of t,ransitio~l. It ran I)c
assumctl, as a rough guiding rulc, 1Ji:~t at nwlium 1tc:ynoltls nrirnl)crs (R =--: {Of;
to lo7) the point of inslability coincitlcs with the poinL of minimurn pressure
and that the point of transition follows shortly afterwards.
3. As Urc angle of incitlcnce of an acrof'oil is incrcasctl aL a constanl ltrynoltls
number, the points of instability and transition move forwards on the suction
side and rearwards on the pressure side.
4. As t,he R.cynoltls number is increasctl at const.nnt incitlcnce the points of inst,al,ilil.y
and t.ransit,ion move forwards.
6. At very high Reynolds numbers antl with a flat prrssure minimum, t,l~e polnt
of ins1.nl)ilit.y may, nntler ccrt,ain circum~t~ancrs, sliglit,ly precede the poitit of
niinin~nm prrssurc.
6. Even at low Iteynolcls nurnbcrs (R = 10Vt.o 10" t11c points of inst~:~t~ilil.y :I.IK~
hnsition precede the point of laminar separation; nndcr cerhin circunistnnres
the hminar boundary layer may become soparabed and may re-at,t.ach as a
Flexible wall: Anothcr effective rnethod of stabilizing n larninnr bo~mdnry lnycr is to rnnke
the wetted wnll flexible. In connexion with the obsorvetl antonishing swimming performance of
porpoises [go], it hns been suggested that these nnimnls have n very small skin-friction coefficient
bernuae the boundary lnyer on them remains laminnr even nt very Inrge Rcynolrle numbers
owing to the flcxibility 01 thcir skin. Jri ordrr t,o put t.hk hypot,hesis to the tcst,, M. 0. Krnnter
[110] performed ~ncnuuremenLq of drng on olwt.ic cirrulnr rylin(lcrs plnced in a stmnn~ pnrallel to
their axes. Indeed, reductions of the order of 50% in drng, compared with rigid cylindrrs. have
been observed in the range of Reynolds numbers R = 3 x 10" to 2 x 10'.
Furthermore,T.B. Benjnmin (41 and M.T. 1,nndnhl [I201 instit,uted comprehett~ive thcoretical
analyses on the stnbility of boundnry layers on flexible plntes with the aid ol the method
rxplninetl in See. XVIc. Thcse revealed t,hnt, in nddition to tho Tolltnien-Sclrlirhtitig wnves which
occur in n form ~notlilied by tho flexibility of t.1~ wall, there appcnr tnodifietl c1nst.i~ wavcs in tho
wnll itmlf. Such elnstic waves are creatod owing Lo the prwence of tho flow outttide the wnll.
F~rt~l~ermorc, there appear waves of the Kelvin-Helml~olCz type, rnther like those observed on
free shear layers. The first effect - the n~otlification of the Tollmien-Scblichting wnves by the
flexibi1it.y of the wall - may, taken by iteelf, explain the drnatic displacement of t.hc point, of
neutd utnbility in the upstream direction. However, tho three effectn which depend on t,lic int,ernnl
friction in the wall counteract each other t,o a certain extent. For this rcnuon, we would expect
only a small overall effect. Thus, M.O. Kramer's experiment.al results appear to be confirmed by
the ~1.nhilit:y throry only qrlnlitntively hut not qunntitntivoly. 'l'hc suppo~ition that M.O. Krnrnor's
rwt~lltt wuld lwrhnp 1)o oxplniti~d hy t,ltn inll~~onw 01 wnll llcixihiIil,y on thr~ 111Il.y rlovc~lr~lj~~cl
t.urbulcnt boundary layer induccd U. Zimrnormnnn [25!)) to rlnclcrtakc o thoorctionl invcntigntion
into thin problem. He came to the conch~sion that the flexibility of the wall could lead to a roduction
of the shearing stress on the wall of the order of 10 per cent,, nt lcnut in the presrnce of a fluid
of high density such as water. In the nbsenc~ 01 n co~nplete theory of turbulc~~ce, it. is impossible
to view these rwulta nu more than est,imaks. 'l'he pnpcr. [259], contains references to additional
contributions which concern themselveu with the effect of wall flexibility on the stability and
turbulence of boundary-lnyer flown.