Boundary Lyer Theory

CHAPTER XXV
Determin~ation of profile drag
a. General remarks
The tdal tlrag on a Ijotly placctl in a stream of fluid consists of ski~r./;iction
(equal to thc intcgml of all sllrnril~g strcsscs takcn over the surface o ~~~lle botly)
ant1 of Jorm or prewurc, drag (integral of normal forces). Tho sum of th6 two is called
told or pro/ils tlmg. The skin friction can bc c:LIcIII~~~ with some accuracy by the
rise of the rncthotls of the prccccling chaptrrs. The form drag, dhich docs not exist
in frictionless subsonic flow, is due to the fact that the presence of the boundary layer
modifics the pressurc distribl~t~ion on t,he body as compared with ideal flow, but its
comput~ation is very difficult. Consequently, reliable data on total drag must, in
general, bc obtained by measurement. In more modern times methods of estimating
the amount of profile drag have, nevertheless, been established. We shall discuss them
bricfly in See. d of the present chapter.
1.1, many cascs the tlcbrminatio~~ of total drag by weighing lacks in accuracy
bccarisc, when mensurc~rlrnts arc performed, for example, in a wind tunr~el, the drag
on the suspension wires is too largc compared with the force to be measured. In some
cases even, such as in frec flight cxperimcnts, its direct determination becomes impossible.
In such cases the mctliotl of tlctermining profile dmg from the vclocit,y
tlis~rih~t,ion in the wdzc (I'itot travrrsc method), which has already bccn clescribcd
in Chap. IX, 1)ccotnrs vcr.y ~uscful. Morcover, it is often the only practicable way of
pcrforrning this kind of mcasr~rcment. In priuciple it can ody be used in two-tlimensional
and axially symn~ctricn.l cnscs, but we shall restrict ourselves to the consitleration
of tho two-dimcnsiontd casc.
Thc formula in cqn. (9.2'7) which was tlcdacec~ in Chap. IX and whidi serves to
dctcrn~inc the magriitutle of drag from thc v~locit~y distribution in the wake is valid
only for com~)arat,ivcly large ctisbnces from the body. According to it the total drag
on a botlyt is givcn 11y tlic cxprcssion:
+m
1) ==~Q/U(U,--u)dy. (25.1)
y= -OD
Tlcrc h tlrriot,rs the Irngt,ll of the cylindrical body it, the direction of thc axis of the
cylinder, I/, is the frcc-stream velocity, and u(y) dcnotcs thc velocity distribution
t In Cllnp. TX tho totnl drag or1 a ldy ~.la
111 this chnptcr Llic sy~llbol U is used for iL.
tlrnokd hy 2 I) (for tho two ~idtxi of the plate);
b. The expcrimentol method due to nets 759
in the wake. The integral must be taken at such a large tlistancc from the body that
the static pressure at the measuring section becomes equal to that in tlic untlisturbctl
&ream. In practical cases, whet,hcr in a wind tunnel or in frec flight incas~~remcnt,s,
it is necessary to come much closer to the body. Consequently it becorncs nrcessnry
to take irho account the c~nt~ribution from trhe pressure brtn and eqn. (26.1) nlctst
be modified. 'l'his correction term has an appreciable vhluc whcn mcasummct~t,s arc
performed close to the body (e. g. at distances lcss tlh onc clmd in tho case of
aerofoils) and it is, therefore, important to have a comp~rativcly accuratc exprcswion
for it.. The correction term was first calculatccl by A. I3ctz 141 and later by 1%. Rf.
tJorrcs [2G]. At prcsct~t~ most mcasuremcnb arc bcing cvalu:~tccl wilh t,l~o :&I of t.116
formula clue to Jones because of its compamt,ive simplicit,y. Ncvcrihrlcss, we propose
t.o cliscuss Bet,z's formula as well becausc it* clerivatkm cxhibits scvcm.l very
iritIcrcstting features.
b. The cxperirnentnl method due to Betz
Wit.11 rcfcmnet: to I'ig. 25. I we s~lcct a control surface around tho 1)otly as sliow~l.
In thc rnbry cross-section 1 in front of the botly the flow is loaslcss, its total pressure
being g,. The total prcwurc in cross-scct.iori I1 I)chintl tho hotly is !j2 ,: (I,.,. '1'110
remaining cross-srctions of the control surf;lcc arc imaginctl placed far cnoirgl~ from
the body for the flow in t81iem to be untlistorbecl. In order to satisfy the condit.ion
of continuity, tl~e velocity u2 in cross-sect.ion It niust in some places cxccccl t,he
11ndist,ur1)~tI velocity 11,. Applying tlic moinentwn tlrcorcrn to the cont~rol surf:~c:c
gives t.11~ following expression for the drag on a cylinder of length h:
In order to atlapt this rq~li~tion to the cvnluation of cxpcri~ncn~~nl rcsn1t.s it, is nct~eswary
1.0 t,mnsfornt the above int,cgmls so that they necd hrdy be cvn1unt.c.d owr Illat.
sect,ion of the velocit,y curve which includcs the dcpression of plarlc: I1 in t,hc profile.
The total pressures satisfy the conditions:
I
;it. inlinit.y:
900 = ~ r+ n e um.z
The first integml already hns the tlcsirrcl form, 11cm11sc the total prcssrlrtx is rt111aI t,o
g,, rvrrywlierc outsidr thc tleprcssion In order to transform the scco~~tl inirgml in