Boundary Lyer Theory

,,
Ilie t.c!rnprmtt~rc tlist,rib~~tion is also shown in Fig. 13.6, and it is seen that
tlic fric:l.io~~nl ir~crcasc in thc tcmpcrature in tho boundary layer assi~mes large
valurs for Iargc Mach nwnlms. 'Clic pnpcr by W. I~antzsc:l~e and JI. Wcnclt [44], quotccl
cnrlicr, contnins calc~ilat,io~~s for P = 0.7 (air) for the case of a Iicat-conducting plat,e.
It is SII~~VII t.l~:~t, the velocity tlist,ril)~~lion u/rJ, plot,tcd in tcrms of y 1/ U,/Z i,"
drviatcs c:onsitlc:r:~l~ly Srom thnt for P -= 1 when 1.11~ Mnclr nurnlm- :i.ss~~nics largcr
v:rlrlrs. The vchc.it.y . lwofilrs shown in Fig. 13.G can bc mstlc ncarlv t.o coincitlc
A
. . . . . .
when thc dist,a~~cc from t h wall, y, is matlo tlinicnsionlcss with rrfcrcncc to l/a,,, z/cI,,
Fig. 13.7, wl~c:rc v,, tltnotcs the Irinomat~ic viscosity of the air at the wall. This
circn~nstancc dcnot,cs pl~psicnlly that t,llc incrcasc ill I~oundary-laycr tl~iclrncss with
Rlach number (at constant Rcynoltls number) is mainly due to the increase in
volnmc which is nssociated with thc incrcasc in tlrc temperature of the air ncar the
wall. This fact was first noticcd by A. N. Tifford [98].
Jiig. 13.6. Vvl~wit,,~ lind t~cn~prr:~t,~irc (Iistx-
1r11tiori in (;o11111rc,%qil1lc:, 1a111iri:ir l)o~~ncl:~ry
layrr on adinlrnlic flat plate, nfkr Crocco
P1.1
l'rx~idil I I I I I ~ P ~ - ~ I , m = i, y = 1.4. 1)istanr~
rrwn wnll rrfcrrvd lo I-, ~111 I'
In this method of plotling. lhc curvrs fcsr
dillcrcnt Marh nunlbcrn hnvc lwcn rnrcle
nearly lo coincide. It is possible to conclude
from ll~in that tl~c Iargc incrcnsc in llw bovndary-layer
tl~lcknclis will1 nlnch n~~wbcr is
mainly duo to tl~c incrcnac in volomc shlrl~ is
associnted with tho increase in tcmpcrat~~rc of
tllc air ncar tllc wall
7
Jqig. 13.7. Vc-looit,y lint^ il~~~l~ion~ iu t l ~o
lr~ininnr I~ounclnry layor on an acliabatic
Rat plate at zero incidence; data
identical with those in Fig. 13.6. The
dist~nce from tho wall is referred to
I/v,,, Z/U;. For w = 1 , we have
1/ II~,,/V~ = T,/Tm
Jiig. 13.8. (hllicient oldtin frictio~i on dia-
Oolic flat plate with rornprcnniblc, Ian~innr
i)ot~ncl;~ry
ll:~nt~.sd~c :111ql \f1c!n~1ta [44]
a. 'Lh: flat plnto nt nrro inriclci~cc: 337
Fig. IX!). Corfficic~~t, olsliin lric,t,io~i for ediabrrlie
11:1(. plaLc at zero inoiclrnrt: with coin-
layer. P =. I, ), = 1.4 (air), nfl,cr prrssil~lr, laminar borlndnry 1:1yc.r, : ~ h r
IHY]
It111wsi11 1inc1 .1111111son
Adinbntic coefficient of skin friction: 'I'llc rocSfic.ic:i~l, of skill f~.ic:t.ioi~ Sor :II~
adiabatic wall, as cnlculnt.ct1 by W. Ilnntzschc nntl 11. \Vr~~tlt, 11as 1)crn plot.t,ctl in
tcrms of tho Mach nnmbcr in Pig. 13.8. For co -.. 1 t,hc protluct c, R is intlcpr~ltlcnt
of thc Mach number, but Sor tlifircnL val~~cs of rr) t-l~c c:ocKicicwt of slii~~ S~.ic.l.io~~
decrcascs with increasing Mach nurnbcr, the ratc of ~Iccrtasc I~cing largcr for srn:dlcr
va111es of o. Figurc 13.9 contains a comparison bctwccn Lhc valucs of tllc cocl'ficicnt
of skin friction for an adiabatic flat plat,^ obtai~~ctl by scvcm.l aut,l~ors. i. e. for
different valucs of t,hc Pmntltl numbcr, P, ant1 of the cxponcnt in t.1~ viscosity
Fig. 13.10. Bfeasl~ren~mtrr of t,he
velocity diatrihution in nnadiabulic,
Inminnr I~oi~n~lriry
Inyrr in nl1pc.r-
sonic Llow, al'lcr 11. M. O'l)ont~olI
[28]. Mach number Mm = 2.4.
Theory from ref. [I31