Boundary Lyer Theory

636
XXI. Td~nlcnt houndnry layers at zoro promure gredicnt a. The smooth flat platc '
R = I/, 1/v (Urn - frcc-strca~n vclocity; 1 - length of plate) are so large t1hat they
cannot bc strbjcctcd to mcasurcment in a laboratory. Moreover, even at motlerate
Itcynol(1s numhcrs it is murh morc difficult to carry out mcasuremcnts in the
bounclnry Iaycr on a plate than in that inside a pipe. It is, therefore, very atlvantageons
that it is possiblc to calculate the skin friction on a plate from the extensive data
availnblc for piprs hy thc nse of a method dne to 1,. Prandtl [40] and Th. von
IGirmi~n [30J. This ralcnlation of t,hc skin-friction drag on a plate can he rarrictl out.
both for smooth nnd for rough walls. A good summary of this work was given by
P. R. Hama [23].
a. The smooth flnt plate
The approximate method to bc applied to this problcm is based on the momentum
integral equation of I)onntlary-laycr theory as givcn in eqn. (8.32) of Chap. VIIf,
the vclocily prolile ovcr the boundary-laycr thickncss bcing approximatccl by a
suitable empirical equation. Thc morncntum equation thcn provitlcs a relation
hrtwccn t,hc chr~mderislic prumelers of thc boundary laycr, i. e. bct,wcen displacement
tdlicltness, morncntum thickncss and shcaring stress at the wall.
Tn thc following argument we shall assume at first that the bountlary layer is
turbulent already at thc leading edge (x=0) antl we shall choose a system of coortlinatcs
as shown in Fig. 21.1, h tlcnoting the width of the plate. The boundarylayer
thickness O(x) increams with x antl on translating the tlatn for n pip into
t.hose for a j)lat.e we not ic:c that the maximum velocity, U, of the former corrrsponds
to thr free-stream vclrx:ity, U,, of thc lnttcr, the rntlius, R, of thc pipc corresponding
to the boundary-laycr thickness, b.
At this stage wc introduce with I,. I'ranrltl the fundamental assumption that the
vr4ocit.y dis~ribution in tlic bonnclary lnycr on a plate is idcnticnl with tht inside
a c.ircular pipc. This assumption cannot, ccrtainly, be exact, because the velocity
rlist,ribnt.ion in a pipc is formed unclcr the influence of a pressure gradient, wllcrcas
on a platr t,hc prcssurc gradient is zero. However, small differences in thc velocity
clistribntion arc unimportant, bcca~~sc the drag is calculated from the intrcgral of
morncntum. I~urtllrrrnorc, thc cxperimcntal results obtained by M. IIansen [23a]
ant1 .J. IW nnrgrrs [6] prove that this assumption is well satisfied at least in the
Y
Fig. 21.1. Tnrbulent, boundary lnycr
on n flnt plab nt zero iricidcnce
mrrgc of rnotlrratcly large Rrynoltls nurnbcrs (IJ, l/v < 10" They hot11 found
that thr vrlority profilc in thc bountlary laycr on a plate can be described fairly
well hy a powrr formnh of thc form of eqn. (20.6), as found for a pipe We shall
revert once more to this problem (p. 643), when we 'shall discuss some systematic
deviations between the velocity profiles in pipes and on plates at larger Rcynolds
numbers.
The skin-friction drag D(x) of a flat plato of lenght x on onesidcsatisfics the follo-
wing relation as seen from eqns. (10.1) and (10.2) in Chap. X:
IEerc to(%) dcnotes Lhe shcaring stress at a distnrlcc x from tl~c Icading rtlgo, :~lld
the secor~d integral is evaluated at x over the boundary-layer thickness. Introd
ducing the momentum thickness d2, dcfincd by A, 1Jm2 = / 7l(r!, - TI) dy ill
eqn. (8.31), we can rewrite eqn. (21 .l) as follows:
From eqns. (21.1) and (21.2) we obtain the local shcaring strcss :
Equation (21.3) is identical with thc monlent,lrn~-int,cgral equation of ho~~~~dary-
layer theory, eqn. (8.32), in thc case of uniform potential flow U(x) .= IJ, -- COIIS~,.
We shall now perform thc calculation of the drag on a flat plate on the assump-
tion of a f-th-power law for tho velocity profile which is trrro for modcrntc Royriolds
numbers, and we shall then confinc onrselves to quoting thc results for thc hgarith-
mic law which is valid for arbitrarily largc Itcynolds numbers, Fig. 20.4, bccarrsc
the complete calculation for this case is fairly tedious.
1. Resistance formula deduced from the 4 -th-power velocity distribution Inw. 111
accordance with the preceding argument and with eqn. (20.6) it is seen that the
+-th-power law of velocity distribution in a pipc leads to the following volonity
distribution in the boundary layer on a flat plate
whcrc d = B(x) tlcnotcs tho I~oun~lnry-layer ~II~CICIICSR wl~ioh is a function of rlist,nnt-r-,
x, and is to bc clclcrminctl in thc course of thc calculation. 'l'hc :msumption in
cqn. (21.4) implies that the velocity profiles along a Rat pletc arc similar, i. e. that
all velocity profiles plot as one curve of n/U, versus y/d
Tllc equation for shcaring stress at the wall is also taken ovrr tlirwtly from
the circular pipe, cqn. (20.12a) :
.From eqns. (8.30) and (8.31), togcthcr wilh cqn. (21.4) we o:~n ca1culat.c t,l~c tlispl:tcemcnt
thickness, dl, and thc momcntnn~ tliickncss, (1,:
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