Boundary Lyer Theory

256 XI. Axially ~yn~motricnl and three-dimcneionnl boundary layers
tho oxperi~nent,nl pnt,tnrr~ in Pic 11.17 b. It is, tllcreforo, not at all eaay to establish a critorion
for scparat,ion in a thrco-clirnc~~siol~nl boundary layer, if proper weight is given to this type of
bel~aviour. At this point, we wish to draw tho reader's ~t~tention to the investigations on yawed
cmwn he to W. J. Itninbird, It. S. Crnbbe and L. S. Jurewicz [91].
It, ap~)rars Lo bo possible to attempt, a theoret,ical analysis of t,hree-dimenaional boundary
layers wilh t.ho aid of ir scheme snggcsted by L. Prandtl [RBI who proposed to introduce a
cwrviiincnr systrrn of coordinatm in which the potential lines and streamlines of the free stream
would play t.lm part of coordit~atos. This progmmme wns cnrrictl out by E. A. Eichelbrenner
and A. Owlnrt [22] whon Lhcy calmllnlad tho laminar cmo ment,ioncd earlier. It hna already
Iwrn mcnt.ioncd l.l~nt good qualit,ativc agreerncnt rrsultetl. as shown in Fig. 11.17c. See also
It. 'I'imtnnu [I 141.
'I'he mcthotl of c:nlculnt,ion proposed by L. I'randtl r86] was recently developed
numerically by W. Gcisslcr 135, 36, 371. Figure 11.18 illnstrntes the result.s referring
t,o tlrc t,hrcc-tlilnensional l~oundary layer on a yawed ellipsoid of revolution. In
acldition t80 t1hc potmtial lines and st.reamlines of Llre external flow, Figure 11.18~
shows the separation line A'; the latter has a course ~imilar to that in Fig. 11.17. Figures
ll. 18 11 and 1 1.18 c represent t,he v~locit~y distribution in the boundary layer nt
various st,alions on a particular potc,nt.ial line.
The lamir~ar I~oundnry layer on a yawad rotating circular cone in a supersonic
stream was earlier invcst,ignted by R. Sedncy 11041, whereas ,J. C. Martin 17.71
irivrstigat.ecl the Mngrrns eKect8 on bodies of revol~tt~ion at, R, small angle of incidence.
Fig. 11 .lR. \'rloc.il.y tli~trilmlion in t.11~ t,hrco-di~r~cnsio~~ai bor~ndary-layer on an ellipsoid of mvo-
Iution ornxis mtio LII) - 4 nt an angle of inci~lcnco ? = is0, after W. Geisslcr 136, 871. a) SysLcm
of potrnt,inl linrfi ~ ~nd st,rrn~nlinm in outer flow; S = sopaintion line. b) Primary flow velocity profilrs.
~r/lJ,, in t,hr tlircvt.ion of the outer flow strcaniliue~. c) Secondary flow velocity proBles,iu/Um,
nt ripht anplr~ to thr dircrtion of t,lw outer flom strcatnlines. 'rhc velocit,y profiles arc given for
pot.rnt.inl line 1 - (13) nt, di(Torrnt st,ntions 111, wit.h a7,i~n~it,h nnplc 4 and st.nt,ion x as pcr table above
(6 - 0'' - wir~dwarrl sytnn~rt,ry)
Another irnport,ant, example of a t,hree-dimensional bountlary layer can 11e fount1
in the corner formed by two mutually pcrprndicular planes in a slrcarn prallrl t,o
their line of intersection. This flow config~rat~ion was invrst,igat,ed t.llrorrt~irnlly 1)y
V~~sant,n 1dn.m 1921. 'I'lin est.rrnnl rcloc:it.y at, fnr t1ist.ntlc.c I1as hen nssunrc~l lo Iw of
Ilnrt~.co's l,ypc, i. c!. givcn by
It, is recalled from See. TXa that this type of external strrnln leads to sirnil:w vclocity
pofilcs in the boundary layer. This feat,ure continues t,o hold in the case of flow in a
corner. Some of [.he results of these studics arc givcn in Fig. 11.19; t,llis shows the
vclooity distril)utions in the corner for three cliflkrcnt vnl~~cs of the prcssnrc parallletcr
nt. A comparison between the distribulions for different values of ns demonstrates
that the boundary layer in s corner thiclrc~~s apprrcia1)ly in t,hc prcsrnce of a
pressure increase in the external flow.
Expcrimcnt,al ol)scrvat.ions [82, 391 suggest that t,he flow in t,lle corncr s(:p.:~t,rs
carlicr than th:~l on llic portions of t01e walls at a larger dislnncc from it, cvv11 in thc
prcscncc of sn~nll ntlversc pressure grntlicnts. 'J'lris pliysic:nlly r~r~tlcrst~r~.~~tl~~~I~I~
111011(:
of I)clinvior is fnlly conlirrnetl by Lllesc thoorctionl r~sult~. 011 a flnt plr~l,c: H(-~):LI.:I(~
occurs at m = -0.091 (see Fig. 9.1), separ:rtion in a right-nnglcd corncr occ~rs as
cnrly ag for m = -0.05. At na = -0.08, Fig. 11.19, the flow in the nciglll)ou~.lrootl
of t,he corner displays a separation region with revrrse flom (IL < 0). By oontrnsl, at
a large distance no reverse flow occurs. M. Za.mir and A. 1). Young [120, 1271 carrid
out extensive experiments on the laminar bomndary layer xlot~g a right,-anglctl corner
a.t zero incidence. See also S. G. Rubin [93a].