Boundary Lyer Theory

42
11. 011tli11e of Iw~~ntlnry-Inyrr theory
of suc.11 I~otlics csan I x overcome by tl~c bor~ntla.ry layer witho~~t separat.ion. AS we
sl~all also scc Int,cr in grrat,er tlrtail, t,he pressure di~tribut~ion in thc ext,ernal flow
t~xrrt,s a clet~isivc influt:nce on t,hc positmion of t.11~ transition point. Thc bountlnry
Ia.yrr is laminar in the region of prcssurc deereast, i. e. rol~ghly from t.l~e leading
ntlgc? to t.hr pint of minimum pressure, ant1 becomes t~~rhulent, in most cases,
from t.l~:~t point onward througl~o~~t, t.l~r region of prcsslrrc inrrcn.sc. In this corrnexion
it is iml~ort,ant to statc tht, scpamt,ion can only bc nvoitletl in rcgiorrs of incrensing
prcssnrc n h the ~ flow in thc bountlnry layer is turlrulcnt. A laminar 1)ountlary layer,
as wc shall see Int.er, can support, only n very smnll pressure rise so t,hat. scparat,iorr
would occur even wit.l~ very slcndcr botlics. In prt.icular, this remark also applies to
the flow past nn aerofoil wit,li n pressure dist,rit)ut,iorl similnr to that in Fig. 1.14. In
t.llis cnse scpamt~ion is most liltcly t,o ocrur on t.he sncI,ion side. A smoot,l~ flow pattern
nround n.n ncrohil, contlucivc t.o ~ I I C creation of lift, is possihlr only wit.11 a t,~~rhnlent
bountla.ry Ia.ycr. Summing up it, ma.)i be st.at,rtl that, t.hc small drag of slencler bodies
as wrll &s t.11~ lift, of acrofoils are ma.& possible 1,111~ough thc cxist,enec of n t,url)ulent,
t)ountla,ry Inyer.
Bounclnry-lnyer thickness: (~cr~erally spealc~r~g, the thicknesq of a tnrbulcnt
Imr~ntlary hycr is larger than that of n laminar boundary layer owing to grratcr
energy losses in the former. Nenr a smooth flat plate at zero incidence the boundary
layer incrcascs downstream in proportion to xoR (x = distance from leading edge)
It will he ~llown Inter in Chap. XXI that the boundary-layer tl~ieknrss variation
in (nrt)nlrnt flow is given by the rqnntion
f
d lJm,l -'I5 *
= 0.37 ( ) = 0.37 (Rl)-'1' (2.9)
1
whic-ll c:orrcspontls 1.0 rqn. (2.2) for laminar flow. I'ahlr 2.1 givns vnlnes for thn
I~o~~~~~l:r.ry-I:tyc:l. t11i(~Ii11ns~ o:~l~:uIal.r~I from eqn. (Z.!)) for several typical casos of air
:~1d watl~r flows.
c. Twhulent flow in n pipe nr~d in a hourldnry lnycr 43
Tnhle 2.1. Thickness of bormdary Inyer, 6, at t.rniling edge oF flnt plate nt zero inridencc in
parallel t.nrlwlent flow
U, = rrcr ntrenlll vrloclty: I = lrnqth or p1al.e: r = kinrn>nl.le risrasily
Air
v = 150 x 10-e ftZ/~~v:
100
200
2 0
5 0
750
Methods for the prevention of separation: Sopnrnt,ion is mostly nn r~ntlcsir:~.I~lt!
pl~rnomcnon bccnusr, it clltr~il~ lnrgo onorgy losncs. I'nr thin rcnson rnctllo~ls I~r~vo 1,cm
tleviscd for the artificial prcvcntion of separation. Thc simplest met.hotl, from t,l~c
physical point of view, is to move the wall with the stream in order to rcdr~ce hhc
velocity difference between them, and hence to remove the cause of boundary-layer
formation, but this is very difficult to nchicvc in engineering practice. Ilowcvcr,
I'rnndtl t has shown on n rolaling circdar cyli?zP.r tllat this method is very rfrcct.ivn.
On the side where the wall and stream move in thc same direction separnt.ion is oornpletely
prevented. Moreover, on the side where the wall and strenn~ move in oppositc
tlircct,ions, separation is slight so that on the whole it is possible to obtain a gootl
experimental approximation to perfcct flow with circulation ant1 a large lift..
Another very effective method for tlic prcvcnt,ion of separation is hm~d