Boundary Lyer Theory

644 XXI. Turbulent boundary laycrs at zero pressure gradient a. The smooth flat plate 645
K. Wiegl~artlt [M] ntlvanccd an cxplanntion for the difference between tlic velocity profile
in a pipo and tlrnt on n phk, pointing or11 tllnt t,he influence of t1tr1)ulrnce at the outer edge
of t h boundary lilyor tliNcm in the two cnRca. In t,ho cam of n platc a low degree of turbr~lcnce
in the cxtmtd ~t~re~i~ii gives risc to vclorily fl~~ct~~ntions which arc practically mro at the oubr
edge of Ll~o honnrl~~ry Iiycr, wllcrcas in tlrc crntro of the pipo thcy would hnve an apprcciablc
mngnit.r~do bccarrsc of t,lw inllitcncc of 1l10 other side. To the srnallcr intensity of turbulence
on a plnlo there corrcsporirls a slacpcr incrcnsc in velocity and l~criro a thinner total houndary
Inycr. lle wns idso able t.o ~llow t,hat thr vclocil.y profile on a platc bccon~ca vcry clono Lo tlrnt
in pip flow if tlrr cxtcrnal llow in niatlc Iiiglily I.tlrbtrlcnt.
J. Nilz~~rndsc [38] alxo condnctcd a vcry comprcl~enxivc series of cxpcrimonh on flat plnlcs.
Ire found that in the range of l:qe Rnynolcls twtnbcrs of R, - 1.7 x 10' to 18 x 106 the volocity
profilcs arc similar, if ?i/U is plot,lcd against y/dl, whcrc 6, clcnotes the displaccmcnt thickness.
'J'lic univcrnnl vclor:ity-clist.ribrltiol~ law w/ll = /(y/~!~) turns out to be indepcndcnt of
lhc Iteynolcls number. 'l'l~r loca! nncl total cocfficicnb of skm friction have heen calculated from
tho nicns~~red ~rlooit~y proliles wit11 thc aid of tho ~noment,um tl~corem.
Tho following intcrpolnlion forn~ulac wcrc ol)t,ninctl for tho velocity distribution, dis
lhic4~nc~, and roefficicnt~ of &in friction, rcspcct,ively:
plarcrncnl thickness, IIIOI~CII~~I~III
.- ~ ~ =
urn
0.1315
0737 ( ) ,
'.?.!L = 0.01738 ~zo'86L ,
v
In conncxion with t,lie calculation of skin friction on a platc, the papor by V. M. Falkner [15]
may also be consulted. In a paper hy D. Colerr [Ea] the velocity profiles are reprcselntrd by n
lincnr con~hinat,ion of two universal functions, one of which is c:allcd the law of the wake, the
other being the law of the wall ns already mentioned.
Mc,muretnent,s pcrforrnrtl hy 11. hlotzfcld [3GI concerncd tlre~nnclves with the turbulent
boundary lmyer on a wavy wall. II. Schlichting [46] gave some eutimatcs concerning trrrbulcnt
boundiiry layers with suction and blowing. When homogeneous (that is, continuously and
unifornily dist.ribnted) suct.ion is applied, thc asytnptotio boundary-layer thicltncss remains
constant in lho name manner an for a laminar boundary laycr. However, in the turl~nlcnt case
thc borrntl:~ry laycr is ~nr~clr inorc scnsitivc to clrnnges in the snction HOW-rate than in the
laminar. Vcry cxknsivc tncasurcmcnb pcrforn~ccl in tr~rbtllcnt boundary layers on porous flat
walls by A. l'avre, R. D~~mna and E. Vorollet [lo] show that the npplication of suction exerts
a st.rong inll~~cncc on t.he 1.11rb11lcnt motion.
4. Errect 01 finite dimensions; boundary Inycrs in corners. Wl~cn a flat platc of finite span
is pllrccd in a ulrrn~n which llow~ in t h tlircction of ik Icngl,l~, il is I;)ontI t.l~nl, nonr tJre docdgc
1110 honnclnry layer is no longcr two-din~rnsionnl, ns it is along tho centre-linc of t,he platc.
~xperinicntri pcrfi~rtnctl by .I. W. 1Tlder [13] dcnionslrated that near the edges there arise
sccolldary flows wl~icli arc similar to'lhosc ol)scrvcd in pips of iron-circulnr cross-~cct,ion (cf.
See. XXc). 'l'l~ix causes n large incrc;mr, in the locd skin-friction cocfficicnt along the edges.
I\ccording tn 1':ldcr'~ mcasuron~cnL~, and rcmnrltnbly cnongh, thin additionnl dmg, always
avcmgcd over t,lic span, hrns out Lo ho intlcpcndcnt o[ tlrc lcngl,l~ Ilcynolds number, Rl, or
the width of thc plate. Irowcvcr, t.hc rcgion sit,uatd vcry close to tlic lcacling edge of thc plate
forms an cxcrption, tlic lorn1 skin-friction coefficient varying irregularly in the flow direction
and gt right anglcs to it. Still weording to ISltler's mcasurcmants, the incrcasc in drag is given
hv
The second term in this equntion accounta for the rapidly deoaying erect of the lending edge
(on this detail the reader nmy also refer to A. A. Townsond [04j).
A similar effect arim when two platea ali ned with the flow are made to form a concave
corner. The interaction between the two bounfary layers for the cam of a rectan lar corner
waa studicd by K. Ceraten [20] who indicates the existence of an additional drag oKnsgnitude
where, according to K. Geratcn, the interaction contribrrbion is
and
8.76
Ac = - -- in laminar flow,
I Rl
Ac = - -- @;:
in turbulent Row .
f
The supplementary drag hna turned out to be negative, which mcnna that the drag of two platcs
which are wettcd only on the inner side of the corner and which arc joined at riglrb angles, is
smaller than the drag of a flat plate of equal total area.
E. Eichelhrenner [12] examined the case of a corner of arbitrary angle.
5. Boundary layers with suction and blowi~~g. Mensurenrent: In this section wc nhnll intaroduce
brief remarks concerning turbulent boundary layers on a flat plate with suction and blowing
which may serve as an cxtcnoion of tho conaidcrntiona of Chap. XIV on Inminnr hounclnry layora
with suction. Thc first tllcorcticcil study of this Inpic wan ~nntlo
(21.21 a)
nfl erirly IUI 11M2 11y II. S~~l~lir~l~l.i~~g
146, 471. In modern t.imcs experimental as well as tl~eoretical studics have been perfortncd by
J.C. Rotta [44].
Some of Itotta'a expcrimentol results are shown grapliically in Fig. 21.4. This is a dingriim
showing the variation of the momentum thickness dl(%) along a porous flat plate with I~onrogcncoua
suction and blowing at various values of the auction velocity, v,,,, at the wall. The external
velocity was Urn = 20 to 30 m/sec and the normal wall velocity ranged from o,, -= -0.10 n~lsec
(auction) to 0.13 m sec (blowing). The volume coefficient varied from ca = v,,,/fJ, = --0.005 to
+0.005 and was, t I IIIR, vcry smnllt. Thcse rncmurements confirmod thc well-kno\vn fwt. that tlio
rate of hountlary-layer thickness growth in the downstream direction increases MI the blo~ring
Fig. 21.4. Turhulent boundary
lnyer on a flat plntc with mi-
form suction or injection: nio-
mentum thickness 62, according
to eqn. (7.38), along the plate;
mea~urements by J. C. Rotta
[441
t Suction and blowing startd at a short distanrc from the lending edge rathcr th-11 nt t,hc leadi~lg
edge itself.