Boundary Lyer Theory

712 XXI 11. 'I'~~rl~~~lemt bo~~ntlnry lnycrs in comprwsiblc flow b. Rclat,ion bctween velocity and ternpcrnture distrilmlion 7 1 3
Tnhlc 23. I. l'lw conntnntn n nrd 11 for bhc cnlrulntion or the coclficior~t nf hent transfer from cqn.
(23.20) nnd of tho rcrovrry htor l'ron~ rrp. (2:1.27), nfler H. Ibichnrdt [73] nrtd J. C. Rottn [El].
'l'lic tcmpcmturc tlistribution in turbulent boundary layers on flat platcs in the
prescncc of an arbitrarily vn.rying, turbulent T'mndtl number, P,, was studied by
R. R. von Ilricst 1283 :rnd J. C. Itotta 1811. In thc I;~tt.cr rcfcrcncc it is showti t,l~at
only thc vsl~rcs whidi Ll~c turhulcnt I'r:dtl numbcr, P,, assumes c1o.s~ to t,he wall
detcrrnine the rate of hcat transfer antl the tempcraturc distribution; conscqucntly,
tlic details of the variation of P, away from the wall are less important. The variation
of P, with tlist,:incc: from tfl~o w:Jl is hrougl~t to Iw:w only through tho int.crrnctliary
or the cp~ntily A wlion in t.hc rcrnnintlcr thc val~~c of P, at the wall is substit~ited.
A suitable value for this scems to be Pt = 0.9. J. R. Taylor [98] performed such calculations
for boundary layers with variable pressure and temperature along the wall.
2. The transfer of heat from rough mrfaces. tt tins bccn clcmonstratctl in
Secs. XXf and XXIc that rough surfaces develop considerably larger values of
skin friction in turhulcnt flow than do smooth ones. The samc is true of thc coefficient
of heat tmnsfcr. Normally, however, the percentage increase in the rate of
hcat tmn.sfcr is smallcr t,l~arl that in skin friction. This is understandable, because
a part of the turhulcnt sharing stresscs can bc transmitted to the wall through
pressure forcw cxert~tl on protubcranccs; but thcrc exists no analogue for this
mcchanism in Iicat, transfer. Expcrimenhl invcstig,ztions on the transfer of heat to
a rough ttrbc wcrc carried out, among others, by W. Nunncr [66] and V. F. Dipprey
and R. IT. Sabcrsky 1261. 'rh~ latter autliors made mcasurcmenta at different values
of trhc I'rancltl nunibor. l'l~coretical conaidcmtions clue to 1). F. Ilipprcy and R. TI.
Sal)crsky [26] :13 wwrl 113 to P. It. Owen antl W. It. Thomson [67] arc bnscd on thc
hypothesis that the elfcct of roughness on the rncchanism of exchange is confined
to the regions locatd in thc proximity of thc wall. Starting with this hypothesis,
it is possible to derive an equation which has the same structure as cqn. (23.20),
and differs onlg in that the term (P - P,) must be rcplaccd by a quantity, P, which
is a function of the Prandtl numlm, P, and of the roughncss. In the particular case
when P, = I, we othin
I c ' R P
N z - --- -- .--;:r _-%._ _ . .. (r)' ipprcy, Sabersky, Owen, 'I?~omsorr ; P, = I). (23.22)
I -4- 4 c,' /I (a, k/v; P)
Pig. 2:1.4. Tho roughnc~n funct,ion (p 4- 8.5) P-0.44 rts n ft~nr.t,ion of lJ+ k./v for sand FOII~IIIICRR
nt vsrioun Prnnclll nu~~~bcrci, from 1110 lncn~r~ro~~~o~~l~
by I). IT lI)ipprc*y 111rt1 It. II. S1\11t!wI 70. '1'110 grnpl~ ol' fho
function p togctlicr with tlic expcrimcntd rcsr~lt,~ ovcr Lhc wholc rarlgc of roughncss
Reynolds numbcr v, ks/v is shown in Pig. 23.4. Owcn ant1 'l'homson corrclntcd
experimental results from various sourccs, including thosc from refs. [25] and [66],
and coricludcd that
(23.24)
Proccdurcs for thc calculatio~~ of heat-transfer ratcs in turbulcnt flows with nonisothermal
surfaces havc been worked out by D. U. Spdding [88], and J. Kcstin antl
coworkers [36, 46, 461. Extcnsivc mcasuremcnts under such conditions awe performed
by W. C. Itcynolcls, W. M. Kays, and S. J. Klinc [77].
3. Temperature distribution in comprceaiblc flow. In ortlw Lo ut~tlc~rat~ancl the
laws which govern the tcmpcraturc distribution in compressible flows, tho rcadcr
may wish first to rofcr to tho rolovant consirlcrntions for lnminnr I)o1111tlary lnyors
wliicll wcrc advanccd in See. XlItb. Wlicn tlic pressurc remains (:oIIHI.:LI~I~ I I I ~
P = P, = 1, the tcmpcraturc distribution satisfies cqns. (13.12), and cqn. (13.13)
in tho general case with hcat transfer, both owing to the evolution of frictional I~cat,.
When P + P, $r 1, it is possible to cvaluatc thc recovery t~mpcrat~~irc on an (atliabatic)
wall by tlic usc of cqn. (13.19), i. c. by
The rccovory factor, r, is somowhat larger in turbulcrlt flow than it was in inrnir~ar