Boundary Lyer Theory

714 X X I I I. 'hrhulcwt. I)oc~ntl:~.ry I:rycrs in conq)rcrrsiblc flow
flow, rxpcrirnr~~ts showing tltat on the avcragc its value placcs itaelf between 0.875
ant1 0 88 (see lj'i'ig 17 31). 'I'hc diagram in l'ig. 23.5 rcproduccs 1,. M. Mack's [56]
comparison of valucs of the rrcovcry fartor, r, measurcd on concs at cliffcrcnt Mach
numhrrs and at cliffcrcnt Itrynoltls numbrrs In order to rstimate the cffcct of
Pr:tntlt,l nurulwr, many rtutllors quote thc formula
whic:l~ yioltls r .-I 0.896 at. P :-: 0.72. It is rtlso possildc t.o obtain this csthatc thcorcti-
oally, in a mallncr analogous to that used I,r thc crtlculntion of tltc cocfficiont of
11eaL transfer. For this I)urpost? it is ncccssary to start with the cncrgy equation
(23.1 1 (:) ant1 10 irlclutle Lhc cffoots of Cllc molecular and of tho turbulcnt transfer
mechanisms in aocortlancr! whicll the hypot.hcsis contraincd in eqn. (23.14). Proceeding
in this way, .J. C. Rotta [81] obtaincd the cquation:
Tltc quantity h is a function of the raLio PIP, and accounts for, like the quantity a
in cqn. (23.20), tho procwscs taking place in the laminat' sublayer. It is given by the
intrgral
Wind lunncl 1 Mm 1 Typo of pone
I
k~,ercirca
(:AI.CIT 5 x G In.
Aatm I x 3 lt No. 1
JI'L I8 x 20 in.
I ' 18 x 20 in.
J I I?( x 20 in.
J I 12 x 1% in.
I , 12 x 12 in.
2.18 10' wood
6.0 20" ccrnrnlc
2.0 PO' l~olluw; slrcl
4.60 5" nl~rcglnss
I.o:l 13' lucitx
4-50 13" llrcitc
1.W 13" lf~cilc
2.54 10" lllcitr
c. Influcncc of Mach nunher; laws of fricbion 715
Numerical values have bcen incluticd in Table 23.1. The factor 13 dcpcnds on P, and
somewhat on dc,'/2 . According to Rotta, we may take
When the turbulrnt I'mndtl number varies over the thicknrss of thc bouutlary
layer, it is necessary to insert into eqn. (23.24) the value assumed by it at the wall.
Whcn the Pmndtl n~lml)er, P, as well as the turbulcnt Prantltl number, P,, differ
from nnity, it is worth noting that, normally, eqn. (13.21) givcn in Chap. XTII for
laminar bountlary layers constitutes a usable approximation for the tcmpcmt~lrr
distribution in a comprcssiblc turbulent boundary layer. 13. SchulCz-Jnncl~ 1!15]
dcvelopcd n procedure for the calculation of temperature distributions in turbuicnt
comprcssible boundary layers.
c. Infl~~enee of Mnch nun~ber; lnws of friction
To date, the calculation of turbulcnt boundary laycrs in inconl pressiblc flow
has not developed to a point where it could be classccl as bcing morc than a semiempirical
theory. It is, therefore, not surprising that the same remark applies to the
cnlc~tlation of comprcssil)lc tur1)ulcnt boundary layers. In tho rnsc of incomprcssil)lr
turbulcnt bounclwy laycm a starting point is proviclctl by tllc hypothcscs whicll wc-rc
tliscusscd in the prcccding chnptcrs, narnrly by l'rnntltl's mixing-lcrlgt.11 Ilyl~otlw~is,
by von Kilrmh's similarity rule or by Prandtl's universal velocity-distribution law.
The authors of numerous contemporary papers have cndcavourcd to create a semiempirical
theory of comprcssiblc turbulcnt boundary layors by transposing thcsc
hypotheses and by adapting them to thc compressible case. This ncccssitatcd thc
introduction of additional ad hoc hypotheses. In the absence of detailed investigations
into the mechanics of compressible turbulent flows, thc transposition of thc semiempirical
theories of turbulent flows from the incompressible to tho cornprcssible
case involves a good deal of arbitrariness.
Prom the practical point of vicw, thc tlimcdtics incrcnsc bccnusc, on tl~c one
hand, there arc two additional pamrnetcrs- thc Mach numbcr, A&,, of thc: froc
stream and the temperature, T,, of tllc solid surface- which influence the flow, and,
on trhe other hand, tile available experimental results are not cntircly frcc of contmdictions.
Tllrcc mcthotls should bc singlcd out from among tl~c numcrous propos:ds
for handling the problem, bccause they havc bcen employcd parlit:ularly freq~lentIy:
(1) Introduction of a reference temperature for the density and viscosity of the gas.
(2) Application of PrantlLl's mixing length hypothesis or of von Kitrmbn's similarity
hypothesis.
(3) Transformation of the coordinates.
Over and above, the litcrature of thc subjcct contains expositions of mc4lotls
which cannot be classified under any onc of the thrcc proceding hcdding~. In an
impressive comparison, D.R. Cllnpmnn and R.11. Kcstcr [I I] brought to thc forc
the large diffcrcnces which result when cliffercnt methods arc usctl to calculat~c skin
friction (cf. [30]). An extensive comparison betwcen twcnty cliffcrcnt ~omput~ational
schemos and cxisting, expcrimental rcsults was carried out by D. R. Spnlding and
S. W. Chi [89].