Boundary Lyer Theory

664 XXI. Turbulent boundary layers at zero pressure gradient
We shall now calculate the valuc of k,,,, for a wing of length 1 = 2 m (about 6.5 ft) in
air (v = 14 x m2/scc) at a velocity U, =83m/sec =3OOkm/hr (about 185 mph).
We have R, = U, l/v w lo7. Consider a point on the wing at x = 0.1 1, i. e. at
R, = U, x/v m lo6. The boundary layer can remain laminar as far as this point
owing to the existence of a negative pressure gradient. The shearing stress at thc wall
--
for a laminar boundary layer is given by eqn. (7.32) and is to/~ =0.332 UW2 1/1'/~~X =
= 0.332 x 6000 x lo-? m2/scc2 = 2.20 m2/sec2. IIcnce v, = itole = 1.52 mlsec.
Inserting into cqn. (21.40) we havc
v 15
kc,,, = 15 - = ~ 6 X 3 0.14 x lo-' m = 0.14 mm (about 0.0056 in) .
u 8
This shows that the critical size of a protuhcrancc which causes transition is about
ten times largcr t,linn thc valuc of about 0.02 rnm (0.0008 in) in the turbulent boundary
layer, as calculatctl in Tablc 21.3, for the case in hand (small aeroplane).
Thc laminar bo~~ndary laycr "can stand" much largcr roughness than the turbulent
boundary layer. I(. Schcrbarth [481 carrictl out experiments on the bchnviour
of laminar bountlnry layers on walls provided with single obstacles (rivet heads).
It was mccrtained that behind the obstaclc t,herc forms a wedge-like turbulent
distjnrbcd rcgion whose angle of sprcad is about 14O to 18'.
r 1
I hc very rxlmsivc mmsuremenl~s carried out bv E. G. Feindt 1171 h:~vn I d to
k J - -..
a refincnicnt~ of the criterion for t,l~c critlical height given in eqn. (21.46) as mentioned
in See. XVIIg.
Kg. 21.18. hag on circular cylinders
at varying ronghness, aftm Fage and
Warsap [I41
Thc inflnrnct of rouglmcss on form tlrag can be surnrnarized a$ follows: bodirs
with sharp rtlgrs, such as c. g. a flat plate at right angles to the st,rearn,'nre quite
insensitive to surface roughness, because the poitjt of transition is determined by
thc edges. 011 the othcr hand, thc drag of bluff bodies, such as circular cylinders, is
very sensitive to roughness. Thc valne of the critical Rcyrloltls number for which
thc drag shows a sudtlcn drop (Pig. 1.4) tlrpcnds to a rnarltetl degree on thc roughness
of Ihc surfacc. According to mcasurrmcnts, [I, 141 as shown in Fig. 21.18, the critical
Reynolds nurnher decreases with increasiug relative roughncss k/R (d = 2 R = dia-
meter of cylinder). Tllc boundary laycr appcars to I)c tlisturbctl by rougl~ncss Lo
such a tlcgrec that transition occurs at considerably lower Iteynoltls nurn1)rrs t.lr:~n
is the case with smooth cylinclers. Ronghncss has, Olwrcforc, thc samo clTc:ct as
Prancltl's tripping wire (Fig. 2.25), namely, it does reclncc tlrag in a ccrl.nin rnngc
of Reynolds numbers. Jn any case the drag in thc supcrcritical r:tnge of Itcynoltls
llu~nbcrs is always Iargcr for the rough than for t.11~ smool~l~ cylintlrrs; scc I ~tm IC,OJ.
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