Boundary Lyer Theory

332 XI I I. I,:trni~~:w l)o111111r1ry liiywx in co~nprrssiblr flow c. Thc flnt plntc? nt zero incidcnrc :$:%:I
of tho heat flow is rlrt,erminctl by the gradient (dT/tlu,), at tho wall. In fact, wo ran
tlednce from eqn. (13.13) that
so that, for (tIZ'/tl~,), < 0 t.hrrc is a flow of hcat from tho wall to tho lluitl, :tnd (,on-
versely, for (dT/tlu), > 0 hat flows from tho fluid to tho wall. In this ~nannor
Ilwr 7', (a) (ICIIOI,I,S 1.11~ t~crnprm1.11rc :I.(* 1,llc: mtlw rtlgr of the I)onn(l:~.ry Inycr, :III(I
1,hc- sol~~l.iol~ l)cw)111rs urnz
T," - T, 3 - or
2 C,
ll\v - l', ,Y-1
-
7'-
Irlt.rotl~~c*i~~g 1.11~ Rlacl~ n~lrul)cr M = U/c, whcrc cI2 = (y -- I) cp !ltl we ciu~ rcwrit.r
r'lm (1 3.12%) in the form
Heat flux wall ;'_ fluid, valid for P = 1
Fig. 13.3. Rclationship bct,wcen vclocity and
temperature clistribution for the compressible
laminar boundary layer on a Rat plate including
frictional heat,, from eqn. (13.13)
Pmndt.l nt~mhrr P = I. TI,, = wall tcmpernlllrr;
!Ir, = lrw.sLr?a~n t,rn~p~nIurr, lpnr
$ (Y-1) Ms > (7',,>- Tm)/Tm
wc h%vc (i?7'/i?!,),,,0,1 > 0, nltd l#wL Is trntt4errcd to the
rnll owing Lo the inrpo qu:tntity or lwrt eenrrshvl hy
c. The flat plntc nt zero incitlcl~ce
2
Mm2 :
The boundary layer on a flat plate at zero incitlcnce has been studiccl cxl,er~sively
in numerous publications, and we propose to begin with n more tlctailctl cliscussiotl
of this case. First we shn.ll deduce the ralatio~l bctwccl~ tho vclocil,y and tclnpcral~~~ro
tlist,ribution on a flat plate from the prccctling grnrrnl proposit.ion.
Tn t,l~c case or an rcrlirhlic wtll (flat,-plat,c tl~ernlornclcr) wo s~tl)stitmt,c! -: '/I,.,
nncl [J == (I*, i~tt,o cqn. (l3.12), SO t,h:tt t,ho t,crn~wr;~t,~~rc (lislxil)~~t,iot~ it1 I,IIO IIOIIIII~:I~.V
layer on a llat pla,l,c bcconlcs
and the ntlialmtio mall trmpcrat,nrc, rqrls. (13.128, I)), is
wl~irh follows with M, = U,/c,, ant1 c,2 = (y - 1) I-,, l', . [t is worth noting
that the t~mprrnt~urc of a wall in comprcssiblc flow give11 by eqn. (13.17) is itlcrlticnl