Boundary Lyer Theory

638 XXT. Tnrbnlent, honndnry Inyeru at zero prcmnre gmdirnt.
l'ront rqns. (21 3 ) oncl (21.0) we have
6 - . a
7
,-g; 6,=;i,jd.
wllicl~ is the- tlill'vrcnl in1 rqltation for (Y(x). lntrgrntion from thc initial vnll~c: fi = 0
at 1 =- 0 gives
S (x) = 0.37 z (y)-" (21.8)
The l)o~~nclary-layer t.hickncw is seen to iricreasc with thc power 21-f the
dist,ancc, wl~creas in I:iminar flow wc had S - ~ ' 1 ~ The . total skin-friction drag on
a flat, plate of length 1 and width b welf,ctl on one side is, by cqn. (21.2), given by
'l'he drag on a plate in turl~ulcnt flow is sccn to hc proportional to iJWR15 and l4I5
compnrctl with 11,"2 and 1'12, rcspcctivcly, for laminar flow, eqn. (7.33). Introducing
tlimensionlrss coefficicnta for the local and thc total skin friction by putting
we obtain from eqns. (21.3) and (21.2) that
1ianc.0, fromeqn. (21.9). wecan writccff =0-0876 (U,~/v)-J1~andc, = 0.072 (U,Z/V)-'~~.
Tllc last equation is in very good agreement with experimental resulh for plates
wlloso hountlnry layers are turbtllent from thc Icading edge onwards, if the
numerical constant 0.072 is changed to 0.074. Thus
'1'111: rrsist,ancc formula (21.11) is seen plott.cd as curve (2) in Fig. 21.2. The range
of vali~lit~y of t.his formt~la is restricted to U, 6/v < 105 in accordance with the
limitation on Blasius's pipe resistance formula. JJy eqn. (21.8) this corresponds to
U, l/v < 6 x IOfi. Since for R, < 5 x 10"he boundary layer on a plate is fully
. .
t In. tho genornl cam of a power law u/U = (y/d)lln we have:
Fig. 21.2. histance formula for amooth flnt platm at zero incidcnccr; cornpnri~on brt.wrcn
theory and rncnsnremcnt
Tl~eorellcnl carvcn: curvc (1) mom eqn. (7.30. Inn1111nr. Illnrrlnrr; cwvc (2) Tm~tt eqn. (21.11). l u r l ~ ~ ~ I'r~1ldt.1; l ~ i ,
curve (3) from eqn. (21.18), lurbnlonl. Prandll-Scl~lirl~llnp; curvo (3n) from rqn. (21.IOn), la~si~~nr-lr,-tt~rh~~lrnl
tranaibion; curve (4) rrom efin.('L1.10~).111rb11im(. Selmlt~-Ctr~~now
laminar, it is possiblc to specify tl~e following mngc of valitlitty for cqn. (21.1 1 ):
6 x lo5 < R, < lo7, using round numbers. Int.roducing tl~c ~iccessary corrections for
the numerical coefficients we obtain the following expression for the local corfficiant
of skin friction
Equation (21.11), as already ment,ioned, is valid on the assuml)t.ion t.hat the 1)ourlciary
layer is t,urbulcnt from tthe lending edge onwards. In rcalit.y, t.he boundary layer
will be laminar t.o Iwgin with, and will cl~at~go to :I t,~~rbulent, onc furtlwr tlowns(,reanl.
The position of tho point of transition will clepentl on tho intlensit,y of t,ltrl)~rlcnce
in the extternal flow and will bc tlcfinecl by t.he value of the oritirnl ltoyrlolds 1111tnI)er
which ranges over (TI, x/v),,,, = R,,,, = 3 x 10Qo 3 x 10" (sro Sco. XV1 a).
The existmce of t,hc I:minnr scc:t.ion cntlscs tho tlrtq t,n tlcc!rcnsc> II.II(~. l'ollowit~g
I,. Prantlt,l, the dccrcasc can be cstimatcd if it is assumed tallat 1)c:hintl lhc point,
of tmnsition tthc turhulont boundary layer hel~avcs as if it* wcrc tnrbulc~tt, from
the leading edge. Thus, from the drag of a wl~olly turl~ulcnt boundary l;~pc?r it is
necessary to subtract tlic turbulent c1ra.g of the lengt,h 111) to thc poiht of t,ransitiott
at xcr,, and to add thc laminar drag for the same IcngtJ~. 'Sltus, the dccrcnsc I)rc:onw,s
AD = - (~12) Urn2 6 xcrl, (eft -el,), whcre c!, and cf, donote tl~c cocfficicnt of
turbulent and laminar skin friction, rcspect~vely, for the total drag at the: scct.ion
where tran~it~ion occurs, i. e. at R,,,,. Hence the correction for cf is