Boundary Lyer Theory

Fig. 7.2. Sepnration of tlin 11ot111dnry layer.
;I) I'low past :i body wit11 sopamtion (S = point
of soparation). b) Shape of st.rrnnllines near
r 7
I l~r. ~inint of sc.lt:~r:~.tio~~ i:; tlofit~ctl :IS 1,I1r limit I)et,wccn forwar(l and rcvcrsc llow in
thc. I:I yvr in tht. immctli:lt.o ncigh1)ourhootl of t,he wall, or
1
In order t.o nnswcr the question of whether and where scpnrat,ion _~~ru_rg,~ it
is nc~ccssn.iy, in gc~lcr;d, first to intcgratc tho boundary-lnycr eqrtntions: Chcmlly
spc:tlting, the I)orlntl;~ry-lxyrr equations are only valid as far as t11c point of scpnrnt.ion.
A short (list,:~ncc tlownst,renrn from thc poinl, of scpnmtion lhc bountlnry-l:lyrr bec:otnvs
so t,l~ic:lc th:lt, I,hc nss~~rnpt,io~~s whic:lt wrrc tn:~tlc in t.I~t! clt:riv:l.l,ion of t,h 1)01111tlary-ln.ynr
rqu:lt,ions no longer apply. 111 tJ~c c?sc of Lotlies with blunt. stcrns_t,ho
sopnr;~.(.etl I)or~ntl:~ry Ia.yor displaces t,hc potcr~t~ial How from t h body by an npprccin1)lr
tlist,;~ric:o xntl t h prrssnro disteribnt,ion in~pressctl on thc bountlary lnycr nus st
I)c tlrt,crmi~~ctl I)y cxpcrimcnt,, I~eoarrsc the cxt,crnn.l flow tlcprntls on tho phcnomcn:~
cwttncc:t.c:tl with scp:lrntion.
'I'hc fact t,l~nt, sepnmt,ion in stcntly flow occnrs orly in tlccclcratcd flow (tlp/tl.~: > 0)
can I)(? rnsily inft:rrctl from a. consitlcmthn of the relation t~et~wecn the prcssurc
gr:ltlicmt, tlp/tl.r and lfhc vclocil,y tlisl.ril)nl,ion II.(?J) with tllc aid of the boundary-layer
t 'I'll~ velonit,y profilc n.t, the point, of urpnrnt,ion is seen to I~ave a perpendicrllnr t.nngent at the
wall. 'Cho ve1ocit.y profiles clownutrennl from tho point of ~eparation will sl~ow regions of reversotl
llow near tho wall, Vig. 7.2~.
c. A remark on the inBgrat.ion of the boundary-layer equntions 183
rquations From eqn. (7 11) wit.11 the bounclnry contlitiorls 71 T v =- 0 wc 11;lvr at
?/ = 0
In the irumediat,e ncighbourhood of the wall the cnrvatnre of t,he volocit,y profile
depends only on the pressure gradient, and t.he curvature of the ve1ocit.y profile
at the wnll changes its sign with the prcssure gradient. Por flow with dccrcnsing
prcssure (accelerated flow, dpldx < 0) we have from eqn. (7.15) that (a2u/ay2),,,, T: 0
and, therefore, a2u/ay2 < 0 over the whole width of the 1)oundary layer, Fig. 7.3.
Jn the region of pressure incroase (dccelcrntctl flow, dp/dx > 0) we fi nd (a2u/ay2) 1 0.
Since, however, in any case a2u./ay2 < 0 at, a largc distance from the wall, Lherc
must exist a point for which a2~~/ay2 = 0. This is a point of inflexiont of the v~locit~y
profile in the bollntlnry Inycr, Fig. 7.4.
Fig. 7.0. Velocity distribut,ion in a borrndnry Fig. 7.4. Vclocil,y dislribution in a borlt~dar~
layer \vit,h pressure derrease layer with pressurc increase; 1'1 - point, of
inflexion
c. A remark on the integration of the bnundnry-lnyer rquntinns
In order to integr:lto t.11~ boundary-layer eq~mtions, whethr in thc non-st.oady case, cqns.
(7.7) and (7.8), or in the shady case, cqna. (7.10) and (7.11), it is ofkn convenirnl to int,rotl~lcr
n stream function yt(x, y, 1) defined by
U= av .=-a~
ay ' ax '
(7.17)
t Tho exisknce of a poi111 of inllcxion in tho vclocil,y profiln in tllr boundiiry Ixyer i4 inlport.ant
Tor its stability (trnrleitiot~
from laminar to turhtllent. flow), ueo Chnp. XVI.