Boundary Lyer Theory

158
VIII. General ppropcrtics of the boundary-layer eqr~nLions
l'lcluation (8.27) is a tliffrrent,ial equation for tho totd prrssuro g(x, vi), and its
I)outrtlary rontlit.ions arc
g = p(x) for rl, = 0 and g = .p (2) -1-
Q U2 -- const for )I) = GO .
2
JSq~mtion (8.27) is relat,ed to t,hc hcat,-conduction equation. Tile differcnt~id
rqnn.t.ion for t,he one-dimct~sional case, e. g. for a bar, is given by
whrrc 7' tlcnot,cs the t.cmpernl.t~re, t tlcnoLcs 1.11~ t,in~c, n.nd rc is t,he t,l~rrmal tliKusivily,
scc Chap. XII. Jlowevcr, the transformed 1)oundary-layer cqnation, unlike eqn. (8.28),
is non-linear, ~CCSIIS~ tho thermal tliffusivity is rrplaced by v .u, which tlopentls on
the indepentlent variable x, as well as on the tlepcndet~t~
variable g.
At the wall, VJ = 0, 14 = 0, q -- I), eqn. (8.27) exhibits an unpleasant singularit.y.
Thr Irft.-hn.ntl side becomes ag/ax = dp/dx + 0. On thc right,-hand side we have
16 = 0, and, therefore, @g/avi2 = oo. This circumst,xnce is dist.tlrbing whrn numerical
methods are used, and is intimately conncct,ctl with the singular belraviour of the
velocity profilc near the wall. A detailed tliseussiorr of eqn. (8.27) was given by I,.
I'mndtJ [I I], who had dctlnccd the tmnsfornration a long time before t,he paper by
It. von Misen appcnmd, wit.hout,, however, publishing it?, cI. [I, 12, 161.
11. ,J. 1,11oltcrt [8] applied eqn. (8.27) to tlre example of t.lw boundary laycr
on a flat plat>e in order to test its pm~ticnbilit~y. 1,. Rosenhead and H. Simpson [I31
ga.vc a. rrit.icnl cliscnssion of the preceding pul)lirntion.
e. Tl~c niomcnttlm and energy-integral eqrrntions for the boundary layer
A complete calculation ol the houndary layer for a given body with the aid
of the differ~rit~ial equations is, in many cases, as will 60 seen in more detail in the
next chapter, so cumhersome and time-consuming that it can only be carried out
with t.he n.id of an elcct,ronic computer (sec also See. 1X i). It is, tlicreforc, desirable
1.0 possess nt Im,st approxi~natc methotls of solution, to be applied in cases when an
exact so111I.ion of t,hc bo~lndnry-hycr cqr~at.ions cannot be obtained with a rcasormble
an~oltnt, of work, cvctl if thoir :iccumcy is only limited. Such approximate ~nethotls
can he tlevisctl if we do not insist on satisfying t.he tlifferential equations for every
fluid part.icle. Irtst~catl, t.1~ boundary-layer eqr~ation is ~at~isfietl in a st,ratnm near the
wall nntl nmr t h region of transitior~ t.0 the external flow by satisfying the boundary
rordit.ions, togct.l~er with cert,ain compat.ibilit,y ~ontlit~ions. In t.ho remaining rqion
of flrtitl in the boundary layer only a mean over the tliffcrcrrlial ~quat~iotr is satisfietl,
tlie wcnn heing hken over the whole tlliclrncss of the boundary layer. Such :I mean
vnl~re is oht.ai~red from t,he momentum equation wltich is, in tmrn, tlerivetl from t,he
rr111:ition of niol.iorr I)jr it~t~cgmt~ion over tlrc bor~ndary-1:~ycr t.hicknc:ss. Sinw 1.lris
c-tl~t:~,tiott will Ipc oll,c.rr 11wt1 itr t.110 ~~~)proxitrr~il.(: ~trt~~.l~o~I~, to I)(> ~I~H(:IINS(~~ litt~~r. \v(.
slr~II ~IC~UCC it now, writing it down in it,s motlcrtr Irm. Thc oqrt:ition is know~l :LS t,Ir(:
nlo~ttentunt-integwl equation of boundary-laycr theory, or as von Kiirm;in's irrtcyr:il
cqnntion (7 J
\\'c sltnll rcsl,rict ourselves 1.0 t,lrc cnsc of slcwly, t.\vo-tlitnct~sit~tti~l, :wtl irlcv)tn-
~)ressiblc flow, i. c., we shall refer to cqns. (7.10) tso (7.12). Upon intcgr:ttit~g t,lle
rqu:it.ion of motion (7.10) with rcspert to y, from y = 0 (wall) t,o ?I =- 11, wl~crc
the layer ?/ 1- IL is c?verywhLrc out,sitle t.lrc bo~~ntlnry Iayrr, we obtain:
h
'rhr shenring stress at tho wall, T,, 118s l~rcn substituted for p(au/ay),, so tht
rqrr (8 21)) is sern to br valid both for laminar and turb~~lent flows, on condition
that, in the latter case u and 7~ deuotr the time averages of the respechive velocity
romponents. The normal velocity ron~ponrnt,, v, can be rcplacrd by v -. - J (iIu/r?z)d y,
as sren from the equnt.ion of continuity, and, conscqncttt.ly, we have
1nt.rgrxting hy part,s, we obt,ain for the second t,erm
so that
j~W-wY
0 0
h
1- (Ir' & J(u -U)CIY -= zn e .
(8 mi)
Sincr in both int,rgmls the irrtegmnd vanishes outsitle 1,hc boundary Inyrr, it is
prrmissiblc to put h + oo .
We now introduce the displacement thicknrss, a, and the momcrrtr~nl tlrirl~tr~ss,
d,, which have nlrcady bren liwd in Chap. VIJ. They arc dc4ncd I)y
Y