Boundary Lyer Theory

70-t XX I1 I. 'Ihrbulcnt bonrwlnry lnycrs in comprcwiblc flow
mass in the thx directions: z, y, z. On averaging, the equation of continuity for a
c~omprcssible strewn, cqn. (3.30), leads to
1bcgn.rcling tho clensity flnctuat.ions, it is possible to say at first that p'/@ is hanlly
likely 10 exccctl u'/S. Sinco, now, u'/d < 1, it appears possible to neglect the last
tcrtn in caoll of cqns. (23.6) with respect to the first. Further simplifications rrsult
when aI.l~(:nt,iot~ is confined to bountlary layers in which d & 12. .I. C. Rottn [SO]
clc:~nor~str:~l.ccI l.l~:rt, in such cnscs if, is possiblo altogct,llor to eliminate thc tlcnsity
Illlct~~ations from t,hc cc(nnt,io'ns for boundary layers if, as is customary, tho nornial
st,rosses tllnmsclvrs arc ncglcctctl. First we notice that 6 < ~i in the eqlmt,ion
for t',, in (23.5), so that only two terms need be retained. Purthcrrnore, since
r?p/(7r < ae7/a!/, thc contitluit,y equation (23.G), written for a t)onntlnry layer
which is two-tlimcnsior~d on t,hc average, aeqr~ircs the form
'l'l~c bountlary-layer equation i.9 clrrivecl from eqn. (12.50b) in that cqn. (12.50n),
mult,iplictl wit,l~ u, is adtlcd, with eqns. (18.1) nntl (23.3) s111~sLiLutccI ; thc ro~ult is
thcn averaged in nccortlanco with cqn. (18.4). When the above-mcntionecl t,ertns arc
neglnct.ctl, tho following, final form for the boundary-lnycr equation is ol)t,ninetl:
It is nol.cd that in tyns. (23.Ga) and (23.7) the tlensity fluctnat.iot~ appears only in
the form of fl adtlctl to @ 6. It is, therefore, convenicr~t to re-introduce the original
- -
expression fo; t.ho mass IIIIX p v -;. 17 -1- p' v' in the y-direction, ancl to tlcfine the
tfurl~l~lnnt,, appnrcnt. strc:ss as
In any cam, the exact value of tho mean velocity component at right angles to the
wall, 17, remains untlet,crnmincd, boing of little interest anyway. The energy equat.ion
(12.19) can ljc: treatctl in like manner. Introducing the turbulent heat flux
we obtain tht: following set, of equations which describe the processes in compressible,
t~rrbolcnt, bour!tlary layers:
Ilere, (.he t,erm ~rcpresents the mean valuo of the (lissip:~t ion, ancl for it,, I hc following
npproximat,iori may bc employed:
, @ = ( p
ail
-- -1- rt )
The set must. t)c nugmcntctl by t.hc a.pproxirnato form of the rqn:rt,ion of st,nl,c for
mean values :
- - -
P"PR?'. (ZI.!))
Tho prccccling syst,crn of cquntior~s for co~npressil)lo, l~rrrl)l~lcnt l)o~lntl;~ry I:~.yc!m rcplnccs
crpnt.ions (1 2.6Oa) to (1 2.ROtl) liw corrcspontlitlg I:~n~in:~r flow. 'l'l~c I)OIII~~~:II~
contlitions rcmn.in 11nc11:rngctl (cf. Chap. XII).
In order 1.0 explore the tlchils of t.rnl~nlcnt motion in comprt?ssiblc mctlin, it, is
necessary to untlert,akc tncasuremcnts with hot, wires. 'l'his is matle cliffic~rlt. 0~1 thr
ncctl to unro~lplc 1.11~ c:fTcrt of Lcmpc:rat,urc ant1 vcloc:it,y Ill~ctnn.l~ions \vit.l~in ;I si~~glr
signal. 'L'hc problems which :wise in this way form f.hc subject of tho pl11)lic.n.t.ions
[49, 651 by I,. S. G. Kovns~nny and M. V. Morkovin, rcspc:cl,ivcly. 1,cavirrg n1)art the
appe~rnncc of density :rnd temperature flr1ct,r1:~t,ions, it is found thatf 1.11~ flow rrmnir~s,
in its gcnar:d oublinc, tho same :IS in :LII inc:oml)rc~ssil)le Il~~itl. Ilowc~vc:r, :IS I,IIv hl:rc.h
nnml)or is incm:~sctl, the volooity Iluc:tu:~Liot~s loso ill iill,t:tlsit.y, :I.$ ~lc!~r~o~~sl,r:tl,~:(I I)y
Lhn oxpc:rim~~n(,n.I rrs11ll.s tlrtc! 1.0 A. I,. IZistlrr 1471 : LII~~ SIIOWII ill Icig. 23. I . '1'11t- t*lli~4~
of tlcnsitv fluctnat,ions which go bcyoncl those inc:lwlrcl in ccltls. (23.8:~) to (23.8~)
Fig. 23.1. Distribution of turbulent vclocity
flnct,nntionn in tire houndnry Inycr
on n flat pink placed st, zero incidence in
a sulvxsonic strcnrn. Monsorc~nc~~b clnr tx,
A. I,. Itiatlcr [47J nntl F. S. Kle11nnoR [48]
In ortler to render the system of cqnntio~~s (23.8;1.) to (23.8tl) more :~mctl:~l)lr
to practical calcnlnt.ions, it is possible, as was (lone in Ch:rp. X [X, to iritrothco i1ll.o
it empirical ass~irnptiorls for momentturn and heat. t,mnsport. lCcluat,iorl (19. I) for thr
app~rent shearing stress t, = t',, is usnnlly taken over ul~changcd. As far ns the Lnrbulent
1ica.t flux is conccrnetl, it is custon~ary to givc it a f rm rc~ninisccr~t of Fourier's
law of ther~nal conduction, cqtl. (12.2), according to which we have
aT
--k.- ?I
Q I -
(Iamiriar) ,