Boundary Lyer Theory

130 VII. Boundnry-layor cquntionzl for two-dimensional flow: bormtlnry lnyer on a plnte
in a diroct,ion normal to t,he boundnry layer is pra.ct,ically con~t~ant.; it mo.y he assumed
equal t,o Lhat at the ontm tdge of the 1)oundary layer whore its value is determined
by Lire frict,ionlcss flow. 'Vho pressure is anid t,o be "impressed" on the boundary layer
by the out.er flow. It, may, therefore, be regarded as a lrnown futlction as far as
houndary-layer flow is concerned, and it depends only on the coordinate z, and on
time t.
At the outm ctlge of the bountlary layer the parallel component 7r becomes
ccl~~a.l lo that in t,lre outer flow, U(x,t). Sinco there is no large velocity gradient
hnrc, the viscous t.rrms in eqn. (7.2) vanish for largo vn.luca of R, and ronscrlnent,ly,
for the o~rt~rr flow we obtain
whore again tho symbols denote dimensional quant,ities.
In the case of stently flow the equation is simplifictl still f~~rlhcr it1 thnl the
pressure dopentis only on s. We shall rmphasize this cirrumstnnrc by writing ttir
tlrrivntivc ns dp/tl~, so that
This rnny also he writ,t.cn in tl~c usllnl form of TZrrnor~lli's cyw~tion
1, -t 4 p 1J2 = eonst, . (7.6)
r 3
I IIC hoi~ttcl:u.y oontlil.ions for t.11~ cxt.rrnnl flow :Ire nrnrly tho snnic :w. for frict.ionlcss
flow. 'l'ho I~onntlnry-lnycr t.hicIzncss is vcry stnnll :tntl t.hr trr~nsvrrsc velocif,y component
v is very smnll at, the edge of t,hc boirntlary hyer (a/ I' - cT/L). 'i'1111s potcnthl
non-viscous flow ahont Lhc hdy nndcr considerntion in wl~ich the prrpcntlicular
vrlocit,y component, is vanishingly smnll nrnr the wall offers n very good npproxin~nt~ion
1.0 tho nct,nnl cxtcrnnl flow. The pressure gmdicnt, in t.he 2-tlireet,ion in the boundnry
k~ycr can I)c oht.ninod by simply npplying t,ho Bcrnor~lli erjl~at.iorr (7.5%) to the st,rcnniline
at t,l~c wall in t.hc known po(.ent,inl flow.
witslt tho I~onntlnry condit,ion.s
1). The ucparaLiotl of n hor~ndnry lnyer 131
!/-=0: u--0, a - - . O ; ?I:-m: 71:-(J(:r). (7.121
11, is necessary t,o prescribe, in ntltiit.ion, n vclocit.y prolilo nt the init.ial sc,cl.iot~,
1 : J,, say, by intlicnting t,he fi~nct~ion ?r(a,,,y). Tho problcrrl is t,I~lts scbrtl t,o ~.t~tll~c-r
it,sclf tto the cn1t:ulstion of tho furtller change of a given vc1ocil.y profile wit.11 n ~ ivc,~~
pot~cntial motion.
:
. 1 , Ile mathemnt,ical simplificntion acllievcd on the prccctling i)n,Aos is col~sitl~:v:~l,lc
it, is (,rue t,lint,, as distinct from the rase of orcoping inot,ior~, tho tro11-lit1car c:l~nmrtrr
of t,ho Nnvirr-Stolrcs oqu~tion 11n.s been prrscwetl, hut of thc t.11rre origir~nl c~tl~rnt~io~ls
for 11, I,, nntl p of t.11~ lwo-tlirnensioniLI Ilow proble~rr, ono, I h cqurlt,iot~ of rnot,iott
normal Lo tho wall, has been clroppcd con~plct,ely. Thus the number of 11rllinc)wrrs
has 1)rrn rrd~~cctl Ijy one. Tl~crr rcrnnins n s?jst,cni of t.wo si~rt~tll~:~.~~col~s t.cll~:~.l.io~~s
Sor tjl~(* LWO II~I~IIOWIIS I& a,n(l 11. '1'11~ pr(:ssllro cc:~sctl lo IN- :III IIII~