Boundary Lyer Theory

1 64 IX. Exact solutions of the shady-state boundary-layer equations a. Flow past n wedge 165
a. Flow paat a wedge
Thc: 'sirnilar' solutions discussed in Chap. VlIl consLit.utc a particularly sirnplo
class of solutions u(x, y) which have the property that the velocity profiles at different
distnnccs, x, can be made congruent with suitablc scale factors for u and y. The systcr.~
of p;mt,ial differential equations (9.1) and (9.2) is now rednced to onc ordinary
rliffcrcntial cquation. It was proved in Chap. VlIl that such similar solutions exist
when the velocity of t,hc potential flow is proportional to a power of tho length
coordinate, 2, rneasurcd from the stagnetiot~ point,, i. e. for
Jrrorn cqn. (8.24) it rollows that thc transformat.ion of thc int1ol)endcnt. v;l.riablc ?I,
which lends to an ordinary tlifl'crcnt,inl equation, is:
'J'hr r~uation of ~ontinuit~y is intrgratcd by the introduction of a stream function,
as S(~PII from cqns. (8.1 I) ant1 (8.23). 'l'hus the vrlocity romponel~ts become
u = u1 2" /'(r]) = u / '(r]), 1
1 nt.rotluc:ing t,l~osc vnluos into tt~c
nnrl put,t,ing, as in cqn. (8.21),
we ol)Inin the following differential equallion for /(I))
/"' -t / /" -1- p (1 - 1'2) = 0
equation of motion (9.l), dividing by ni. IL~
zZ"'--I,
It. will IN: roc::tllrtl t.li:tt it, was ;drcady given as eilrl. (8.15), antl that its I)OIIII(IR~Y
f
contlil ions a.re
Y] 7 0 : / = 0 , 1' --= 0 ; /l=1.
the velocity profiles have no point of inflexion, whereas in tho case of decolcrat.rti
flow (m < 0, p < 0) they exhibit a point of inflexion. Sepxrat,ion occurs for
= - 0.199, i. e. for nt = - 0.091. This result sl~ows tht the laminar horl~\tlnry
layer is able t.o support only a very small dccelcration witl~o~lt separat,iorl occurir~n.
by IJxrtmr. The additional solution leads to a velocity profile with baclz-flow (cl.
Chap. Xf).
Tl~c potential flow given by U(Z) = 1 ~ xm , exists in thc ricigllbourllootl of the
stagnation poil~t on a wedge, Fig. 8.1, whosc included anglc 8, is given by eqn.
(0.7). Two-dimensional stagnation flow, as well as thc boundary layer on n llat.
plate at zero incidencc, constitut,~ particular cases of the present solutions, the former
for p = 1 and in = 1, the latter for = 0 antl m = 0.
Fig. 9. I. Velocit.y distri-
bution in the 1:~tninar
boundary layer in tile
flow past a wedge given
by U (x) = a, zm. Tllc
exponent m and the
wedge anglc P (Fig. 8.1)
arc connecbd tlirongll
cqn. (9.7)
,, lhc o:~.sc fl :- &, m 2- .j is worL11y of att.c~nI.io~r. 111 I,llis cnac I.llo tlillrc!t~l,i:~l
equation for /(q) hccomcs: /"' .I- / /" 1- 4 (1 = 0; it, t,rnnsforn~s irlt,o ~.II(*
tlifi:rcnt.inl c!qunt.ion ofroL:~(.iordIy symrnct.ric~~l flow with slngn:~l.io~t poit~l,, ocltl. (5.47),
i. e., 4"' -1- 2 4 4" + 1 - 4" = 0 for $(C), if we put r] = 5 1/2 and d//dsj = d+/d