Boundary Lyer Theory

718 XXIII. Turl~ulcnt boundnry layem in comprossiblo flow
von 1Z:irm;in's i~icomprtssiblc resistance formula, eqrl. (21.17). Fig. 23.6 gives a plot
of eqn. (23.31) ant1 a comparison with experimental results. The measure of agreement
bctwccn thcory and experiment is not satisfactory in all cases, but ill this comexion
it, must bc pointd out that mcasurcmcnts at high Mach numbcrs are somewhat
uncertain. R.E. Wilson 11021 cnrricd out similar ealculationq, but based them on
von JZ6rm6n's similarity hypot~hcsis, cqn. (19.39). Limiting himself to the case of an
:ulial~at,ic: w:dl, IIC tlcrivccl a resultf which is quite similar t,o cqn. (23.31). I'urtllcr
cxpcrimontnl msulk arc contninctl in Pig. 23.7 which shows a plot of the ratio of
tltc skin-friction c:ocfficients in compressible and incomprcssiblc flow in terms of t.11~
M:wh n~lmbcr, c:ovcrirlg a rangc which includes very high Mach numbers. The graph
cont.ains two t,heoreticnl curves; the first one due to R. E. Wilson [102] presupposes
an adiabatic wall, and the second one, +rived by E. R. van Driest [27], includes
tl~c cffcct of hcat transfer. The mcasurementa were performed 11y several workers
[7, 14, 38, 63, 871 and show good agreement with thcory. Atlditionnl information
concerning the inllt~cticc of hcat transfer on skin friction is contained in I'ig. 23.8
wllich was also based on van Driest's calculations 1271. The diagram shows that the
skin friction on an adiabatic wall is sornewhnt smaller than is the case when hcat
flows from the fluid to the wall.
Pig. 23.8. Skill fridiot~ codfit~irnt for a
ht pl:rls at zcro il~citlcr~cc in turbulcl~t
flow with lieat transfer as a function of
Itcynolds numhcr for different valrrcs of
t.lic tetnporakrrc ratio !7',/T,, after 1':. It.
van Driest [27]
Coordinate trnnsformntion: 'l'hc coordinate transformn.t.ion dcscrihcd in See. XIIId
and valitl for 1amin:w flow can also be cnrricd through formally whcn applictl to the
cliffcrrnlial cquaf.ions for comprcssil~lc tarl~ulcnt bountlary hycrs. The lteynoltls
stress t',, is trnnsformcd t,o
and with this substit.ntion, the momcnt~~m equation (23.8h) acquires the form:
ati a - ac
6 -1. "j .-'C. = s1 ---' (1 -1- a8a 1
a~ ag a S) 4- v, --I- + -
% eo ag .
(23.33)
(23.34)
The symbols uscd here arc identical with those dcfincd for eqns. (13.24) to (13.41).
Wit.11 the mnthematicnl possil)ilit,y of t.ransforrning thc equations for coniprcssil~le
flow ir11.o a form it1cntic:d with that for iricomprcssihln flow, ninny nut,liors (r. g.
B. A. Magcr [57j, D. Colos [15], L. Crocco [16], I). A. Spencc [$I, 921) t:ouploti a
physical Iiypothcsis, accortling to which the vclocitry prolilcs in the t,r~~rislormc:tl pl:cnc
rct:~in t.hc samc form as tlir~l, valitl for iriconiprcssil~lc Ilow. Conscclucnt.ly, tl~c law
of friot.ion as wcll as othor relations rcrn:~in viditl wl~cn the Imtisforn~otl
arc sub~tit~~t,cd into them. This conclusion, whic:li is ccrtainly valid For Iamir~:w flows,
tlocs riot ncccss:~rily carry over to I.url~ult:nl. Ilows bct::~~lso t,hc l.rnt~sli)rlti:~t.io~~ <
coortlinatcs cannot be applictl to the eqnations which tloscribc the IlrrctunLing motion.
This lcatls to contratlictiorls with rcspcot to all thcorics which arc 1)ascd onI3o11ssillesq's
assumption cmbotlicd in cqn. (19.1). Thcsc inclutle thcorics whicl~ utilize Pr:~litll.l'.s
mixing-length hypothesis or von IGrm:i.n's similarity hypothcsis. If wc accept t h
physically plausible assumption that thc eddy kinematic viscosity E, dcfinrtl in cqn.
(19.2) is intlcpcntlcnt of clcnsity, we arc f:~ccd with blic fncl 1,liaL a t~.nrlsli~r~li:tl,io~i LO
t,hc incomprcssil~lc: form ccasrs to be possil~lc. Ilowcvdr, :L t.mtisforrn:ktion to
~:L~:uII(.~~('~s
can still be cnrric:tl out. In thi~ case (,lit: ricw rtltly Itinc*tnr~l.ic: vis(:o~iI.~. I.,, is II-IIII~.~I
to the original quantity, 8, through the equation
Now, it is known that thc dcnsity mtio e/pl varies consitlcrably with the distance,
y, from the wall whcn the Mach number is large. Conscqucntly, one of two conclusions
forccs itself upon us. If wc, assume that the velocity profilcs remain unchangctl oom-
1)arcd with the incomprcssiblc case, we find that, the clisl,rii)~~l,ion of F has cli:~ngctl.
If, I~owcvcr, wc admit, that c rcmnins unalt.crccl, we cntl up wit.11 rnoclifictl velocity
profilcs. The statements concerning tlic cffcct of Mach numl~or on thc vclot:ity profiles
in tlic original coordinates which can be m:ulc on the I~asis of the two prect:tling
schemes turn out to be exactly opposite. This observation throws a gootl tlcnl of
light on the whole complcx of problems which arisc whcn tho laws ol~lainctl in ihc
incompressible case are tmnslatcd to apply to t.lie comprcssiblo case.
Further dctailer The effect of Mach number on thc velocity profile is hrought to
bear through the increase in temperature in the direction of the wall. Since it is
possible to soppose that the pressure, p, is indepcndcnt of y, it is found that thc
density distribution in the boundary layer is described by
As tho Mach number incrcascs ont.sitlc an atlial~n.t,it: wall, it is seen that the density
nlust dccrease very strongly at small values of y ant1 Chis must cause the l~o~lntla.ry-
Iaycr tl~ickness to incrcasc considerably. On tlic othcr hand, an incrcasc in thc Mach
~lnrnher effects an increase in viscosity and a decreasc in the skin-friction coefficient.
,
I
,
his, in hum, causes I,hc laminar suh-layer lo incrcnsc strongly. An cxa~nl)lc of the