Experiments on drag of revolving disks, cylinders and streamline rods at high speeds

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Material Information

Title:
Experiments on drag of revolving disks, cylinders and streamline rods at high speeds
Series Title:
NACA WR
Alternate Title:
NACA wartime reports
Physical Description:
37 p., 16 leaves : ill. ; 28 cm.
Language:
English
Creator:
Theodorsen, Theodore
Regier, Arthur A
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

Subjects

Subjects / Keywords:
Propellers, Aerial   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: An experimental investigation concerned primarily with the extension of test data on the drag of revolving disks, cylinders, and streamline rods to high Mach numbers and Reynolds numbers is presented. A Mach number of 2.7 was reached for revolving rods with Freon 113 as the medium. The tests on disks extended to a Reynolds number of 7,000,000. Parts of the study are devoted to a reexamination of the von Kármán-Prandtl logarithmic resistance law and the Ackeret-Taylor supersonic drag formula and conditions for their validity. The tests confirm, in general, earlier theories and add certain new results. A finding of first importance is that the skin friction does not depend on the Mach number. Of interest, also, are experimental results on revolving rods at very high Mach numbers, which show drag curves of the type familiar from ballistics. A new result which may have general applicability is that the effect of surface roughness involves two distinct parameters, particle size and particle unit density. The particle size uniquely determines the Reynolds number at which the effect of the roughness first appears, whereas the particle unit density determines the behavior of the drag coefficient at higher Reynolds numbers. Beyond the critical Reynolds number at which the roughness effect appears, the drag coefficient is found to be a function of unit density. In the limiting case of particle "saturation," or a maximum density of particles, the drag coefficient remains constant as the Reynolds number is increased.
Bibliography:
Includes bibliographic references (p. 36).
Statement of Responsibility:
by Theodore Theodorsen and Arthur Regier.
General Note:
"Originally issued June 1944 as Advance Confidential Report L4F16."
General Note:
"NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003805117
oclc - 123911891
System ID:
AA00009427:00001

Full Text


ACR No. L4F16


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS





WAiRTI'l'ME RI REPORT
ORIGINALLY ISSUED
June 1944 as
Advance Confidential Report L4F16

EXPERIMETS C( DRAG OF REVOLVING DISKS, CYLTDERS
AND STREAMLITE RODS AT HIGH SPEEDS
By Theodore Theodorsen and Arthur Regier

Langley Memorial Aeronautical Laboratory
Langley Field, Va.


,i j-jERSITy OF FLbRiL'R A
.-..- ir.-'.T EPA, BAE, T
SCIENCE LIB3R;P

.. "-FL 32611-7011 .F


NACA-.-.,



WASHINGTON

NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of
advance research results to an authorized group requiring them for the war effort. They were pre-
viously held under a security status but are now unclassified. Some of these reports were not tech-
nically edited. All have been reproduced without change in order to expedite general distribution.


L 226







































Digilized by the Internel Archive
in 2011 wiIh Lunding from
University ol Florida, George A. Smallhers Libraries wilh support roin LYRASIS and the Sloan Foundallonll


hilp: www.archiive.org details e.perimenlsondra00lang







NACA .0 71' o. Ti 16


rT.,TIONAL AD'ilTn.RY CC"':..IITmEE IOR -FiOi,.UTICS


AD"VA'T C.'7. T., ?T


2y^-rI ., -,TS ON D"...0 OF .,"VELV IT,- DISKS, CYLTID'E:..

;" : LI ;7 C.3 .T r -" -d ,, 3

By T.-eodore 7'eoeorsen -ird Arthur Regier

T-": '.,. Y %-


An experimental investi-ition concerned primarily
wit t the extension of test -.ta cn the ."'i-. of revolving
disks, c-. .1:-rs, and stree-li. ro-.s to 1h : ach\
numbers and ':-rnolds r. .Thers is prsent:.. A Mach num-
ber of 2.7 was reached for revolving rods with Freen 115
as the median. .T'* tests on z. I-s exte.--.1, to a
Reynolds n",:.-r of 7, 00,0,C'. Parts of the studc are
cdvoted to a reexaminat on of the von Karman-'r:.;dtl
lo.-rithmic resistance law and the Aekeret-Taylor super-
sonic :;v- formula :,..3 cr '. 11-tiors for the-ir validiL-".
Tne tests c?- 'rim, in general, earlier theories and add
ce-rta.in new results. A "'" .1-" of .:rst importance is
that the sk7in friction does not ..e',nd on the UTch num-
ber. Of interest, also, are e:-.e'vrimental results on
rev,-.lvn- r > '-t .-n 1 'gh 'Mach nia bers, which show
dr_'Rt curves o. ti-. '',r? familiar f''o.' ballistics. A
new r silt w:. r- a nave g?'a:ral :-plicability is that
th.? Ff'st o- su'r-f.ce roughness involves two distinct
e .r meters, r.:i tic. size and particle unit density.
The par.'cle :_._z **. iquel"- determines the Renmolds num-
b.r .at '..'i..c' '-, en2ect of t:'- ro:: Sn'ss first appears,
Wi.-.'-re as ti-3e r.rt ri 1i: unit density determiness the '-,ehavior
of ,-h d.1 o.-c. fc cent at 'i-.-er eynolds n-umbers.
?e-'rin the cr't '1 --*'n..1ds number at which the rour-ness
effect a-reir2, t.'- .rj ; coefficient is found to be a
f-,nction of 1]ift .a nr:i In the limiti:.7 case of
ardric1l "tss iratir.n," or a maximum density o" particles,
thlv dra' coefCie:* remains constant as the Reynolds
n',mber 's inrcre ss ,.









2 C CI1TDE rFT:i L ".C,.: ACR :':o. LT."'16


T 'T T."'rTC*, ..' .CI R' .r' .

\Von ''r .n n-yFrnJtl CL ry ,r 'c r Fipes


a'."esn 'err n nt o t,' e *r-Ine :f the- s:!n fricti on
b- en .a i *- a s:ll- .? t...n titte one of tle .n. ns
S- r ti.'.r' r- cT'-e na'tire of' t2'v.i e'it i lov. V-. o t ofr the
p'onoer *anal-t"ic.al .vork in this fi l1 is founj in the
rap.r-pS i *"on L::'r".n (re ler,. 1 ...:d 2) ind Pran-itl
(re.er .':-.e 3 ). 'i- trest-,er.e t u'e' in th. ir st part of
this sz-c icn f'o lc.s the' work Pr. .nltl whichh in turn,
is 2 osel- 7 rel:er t. the Con I: LI-.:rIl U ,ape.s, The theory,
which concerns ;. Ic.ow in 3;i'. is given jn con-
siJeratle rtq ]. aIs it for.t.s .* b..s s fo.- the su.c zeding
,3 .- i n J ie ..'l.at .i' tes l~'; i t r3, an:1 r sks. The
the'.r- t icaL1. w i ': !n this 'e,.r.iou, o-s 1t.itutes ma nly 1n
atto.'.i'.t to an.il;,ze .:J nrganize e:rli r n r':. found in
r.an sc tLtered 1arti]'-l-s. :.I:s t.ab.e .,. 'rl: along su-ih
lines has alrA.d" oe-n Dn: b '.1 2st in, wl-o is
resr, ers i Ie for --n -,:res 3on ''or thc cra.. on revolving


T.- vcn w.ma n- r.L .-t i th :-r': 7r flow.' in the
tirb lent leyor ,. on t -'I cl ng 'lwO assm-.intic, ns:

(1) Th7 E ril'o ,t tR.h ":.lj;it.; ef'iclcncy to ths
fr cto .n ;.in .c is :n. ion nf om.r rtt.' narpameters


(2] V. ''"c- t, e ,:"'- l ..- be'-rn' the laminar
sitl.- -er, r '- q. rpe o" t'"e :ur," re 'erent-:Ing the's ratio
is ln" rs..l- c:'o orrti'n A to th- is tar.ce from- the wall.
Tl"-e cornstant -f r.recor tiDnal1it is a .Iniv.ersal c.nstant.

The friction velocit- -i 'e in-.i as

IT
UT

-and tiE correFpond in; friction 1enrfi is lcfinod as

L=
(T
(All s'nmb,,ls .s:d in this >.mer are d,-fined in appendix A.)


C C FIDLTI.-.L










:T?.-CA ACT No. I''1.6 CI ',TT l. TIAT, 5


A reference tirs as- be -'v2:1n as

L __
T UTT o
LT T

-' eoetric cc' ~ tions for a rine are given by one
o~rameter, tle rnO~us a. A revolv'- .*-finder of
i.. ite le: 'th represents another s'" ,je-~rjpneter
case, in which "r.Q refe enoe rra.: eter is thr ra. ius
of the r'. 1" .- r.

e dilation of motion can be 'i rttfen in the form.

S- f (
= fl" T'


;:'n, ''- :o:-t z sitably defined !:ean values with
rei-;uct to '', at a -ven cif'e


UT 2TL' L)

"'-.:rce orth u will -:signate su-,. mean velocity. ="
neasur : the velocity with res: -t to a velocity Uo
in a : _- trC- 1 on = ko,


rr







-T


'"1 is q-it? 'e- s't : rc 1 t.cnship, "h'1 has been
P ne.-.1" 1' ,'c:* ;.:e- -,-. ; .'. radse, t.'",..< arf, and
rn''-e:..-- ir-: .r :-:e. ), implies a similarity in
the t "-1- ..,t-f_.e! d ", tr *i',y 'rc.. the walls at all
e-,mnoi: rn-:::te. 7T c-sic re-,. rn for this similarity
r ."::, 3 u ^ ",.', c.'C i.


COI i Pl' T IT- I.









2 CONr ID:..r'T.:rJ. .-.C ; CS .:. i. 1.


't follcv :rc.. a :..'r.:;tion (2) thW t ne,.r the wall


U-- = -
1 y
-- 1,-- + Ccnsc.nt


wer-e ] /K. is tlh constant f p'ropcrtlon&llt :. natural l
c0 .:'"t ti ba1 breer. u9e" th- '.urbo-' t x~- e-t 'v',erte otherwise
in ice ..) -ne u = at -" = 8, this relation


U T" 1
------ 10 -
ITT K

hi's lo ar itY- ii relac'uors'r Y,'i,6 to, a certain 'alue c
f the fini fic.,nt Parameter a (see fig. 1), where
c = 'a wit.n ] a ccst&nt. The (sl oe of 1 1f is
onlr sirA--11 fraction, so ti.-t t.:e pr int c will be
re lat-.'elj; clcse tr the wall. The v.lecit': in the center
of the pipe is t.ere f re l:'.'en as th.r 'iij, of three x slcnS, that is,


+ .c -- + '1


's"*r t,.: lrTni-ar ":ub.1 ',er
TT K





T L

- .nd the equation ay be revwritt n ss


1, a + 1 + log
T'7 K a
1 a
= 71 '- log T + ,2

were
I-
C = a .og a



C O:TF'TDET'T TTAL









:I."A Ar:, To,. IF4 16 C iT-D '..TTAL 5
5---5


2 + 1 c. a


constantt Cl is equal to the '...n::1- .-3ional velocity
meas-r d on the logarithmic velocity pro-"!'e when this
curve is .x.'rolated to ;" = L, .:" the constant C2
is the excess velocity in .. center of the pipe as com-
rar .f-th that of the 17..rithn1ic line extended to
S (- .". 1.) '-,..r those constants are combined,
the foll.: -r.- general relation is c.tained:
"Tmax 1 a
SC + .- -
U K -L

.-: application of this ther'- to cases other than
circular -ir-sj is restrict to _ometric configurations
given r-. a single oarameter. it is interesting to
observe that '.- th C nO. I1 /' are universal constants
reslt!.." ,'r:- the second as :.'t.tion n vmely, that the
flow near a wall is a function of the distance fr"--1 the
waLl only. The secri.i constant C2 which -ives the
excess velocit- as cc' ~pd with the Ic _-,-Lithrnic dis-
tribution t a reference point, the location of which
deo*-.-s on tne ,sometric -: 1si.ons involved, is not a
universal constant but is ,ri,'o.C:ent on t'..? confiu.':ition
n.i] the choice of reference le.(tLh.
I- 3e tn' at i iV.
: 13ar ': r-..r. r t -. i ; --. :.es ar l 'it r 1 3is
1 j.2. t- n. i c c l-t: ln r, t'. 1 '.i : is -b1'. 'isl'- no
effect :t all. Th1s v 1 ,': .f E "L i fcr: oteL.1ri-
S-.e '- :" 1 -. ^ ; ... For > z ...$. ,ay "T l .o3 o n


so-c l --d rs n "' t .o:. .]'- 7".- ... -D e e f i- d





1 E

or

",T, ..m ,- 1 -
1- + -lo -
T
T


CO!T'IDEITTAL










!T,'.CA ACR No. IlJl16


T- :l' : 'tv st ~ t'. ton is 1x9ct!y 1 s if there
"'.-e la.Cin i s .-t nre.ent cf a sicknesss 5 r C
.-.r :. t: --' e -gth were -1 -i ..hen L < e C,
S.3 5.3
rt'.e loc..t I t ':. ti,.n o w lon7'.- r changes **i th an
i.-lre r- In F"ro... n e'rter F. It' seems, therefore,
ti,-.t t .e i-str. ce "r ,m t-e '.vall if tha 'nner,-,ost
.'_ .. c..c ,-r t -. mean '.a vie c 'f i:.r= t ckn ss fof the
]s .v ', "f t' c. -orJe i' .-hr'e- t: four tires
S '-r. ,h t of th: .' r r.".il cities or th rrain size C.
"'- f' ct i r.:t ir.,:, Ist.,: wit' tre ph;:,.leal
i te 'e t tt l" .


T..; .- /[ T 13 Sirw-" tr.
. ..S:', A '. T 1 S


-'.ther,


elaal \1/ .
e 113


T T
Li-
rr
7


a- IT
- a T


a
L v 2


iwherlC P


'.. r3 "e r-re
- X, ,aU '


4-r
t9 Ri iX'J1IU.'~
C


T
- T" .

T


velocit7 and is


+ -: l --
K T


"3-'" -t'- .; be r' t- n


LI
V :~


-K
K


- + 1 -
+ 4


1
F,


C OiFIDFETITT nL


lo-r i,,\/-
102


SR \i' L,


CO TTIDE 'i: .AL ,


q dr p T-I r r re ,








iTACA AC(, ITo. Tl+?16 CO7FID.:-: I. IT.


where
C 1 c ,
C /c \2
C

-' >silartv r.nothesis, 4.e mean velocity in a
DiPe ~if'ars "r'c. the wax'"mmn value '-" a constant, or


UT '

where U isn the mean value of the velocity. Priantl
gives -'f for the volue of K2. (ee reference 5,
r. -1 .) :ote furt.--r that the 7ro'uct RM remains
the same whec-er and C.- re fe, to the mean or the
.alr.,1 v-:lI- of t..e velocity; therefore,

C ...C7 + 1.R
U T K2

and, fiKnll', ith S nd C-, r2'errinK to the mean
velocity i,

1 1 1.i




C -.7 lo 2.



M Y. C r.id 0. 6 ,

-L = .1
T' is val e is nct jacjritely establ'sk d, as the various
auhLors eem .- jif-'er.

rsg n: Flti Flates

In crdar to obtain the Jrag formula for flat plates,
a calculation similar to the "on K'arman-Frandtl trr.trment







3 C DNFIDET-TIAL 7..CA ACR -:o. L'FI6

'?r pipes mlay be rerforr.ed. Th- velocity deficiency Au
is ive by '.he relation


U-Tm
where TTTrm Is a mean value. between 0 and x, the
stance along the rate. The mission ; iuo.antur, may be
written as
.- u \ f.Au
f = -' dy7

or

iT- dy T i )? dy
pu u 0 \0

where U is the stream aelocit;y cad 61 is a significant
length giving the thickness of the boundary layer.
Revrituen, Lhis enubation becomes

T L5 i- ,\L \ 7 A. \ /
5, .5U)- T-" T T


or, by virtue of the L3-ilar-'ty -La,

Si:C. 105 1C6
pl- U 1 (?Tu

S'.nc the -nomentur i-s siven directiv as

!I = ipU2C. x

the following identity is obtained:

1 %m D


or

21 5 = 2Cz x /


COiNFIDENTIAL









II.'CA AfR TNo. T.Fl6 CriT-FDFT.TAT. 9


2:h :- v e


L"1 \ -- "
:: Cc Cc ,
1 -


Urhr e '"ie& Jrritml.iic del i.e.Cncy rt lat 'uor r!ves for' Cc
E.te vhc e -, or .5, ard for LC/C the v r lu5 -, or 5;
7 I LU.S







37 -.2 v .- Tt,-r-Pr '.wt t ,-"pant, the stream
v .,- ...ty is otta_.ed in c a rti-.', t: came form as




7T K
f',o~r ". -, .n 2l i a .1 ~t.,e.~ :,- t,,refore,







t i n -- ..a.ne d
IT l D
= V + Ic
rU c. 4 U
T r]


t/on l ot ai r *rd







Lrv.el VaDue? cf D--..- Co0ffi-.ert for Flat Plates

it :ray be noted trl.t :- rcl,.t.cn for the local drag
Cef' rLenL, or a :' -, c,: ..: .ay ,- ound in a fa-hion
si il.- to ti-. t u:ed -L t:er icr i dcisk. Con.:ider a plate
of u.'i: vidth; for ,rin: lull leron ,

D = ,1 -r -\

L0 CDx (U2)'x



C ONFIDE TT I AL








NACA ACR :I o. L.716


".t:" the subscripts m and x referring to mean and
local values, resnecLively, for the length x,

A- = CDm
2oUu


dCDm
X 4-
dx


= x
10


-Dx dx
-Dx


CDm = CDx


Ru
U


"Dm,
dR
Ir


-Dm = CDx


0 dliog C) ,
r, _Id(lcg R) )


= CDx


There fore


CDx = cDr(n + 1)


n. 0d(log CDm)
d(lo R)


Boundary Relation for Revolving Disks

The ruoment coefficient is defined as

G.. 2-- -

1 p" 2Ia5





COIN T DE NTIAL


where


COFID:LTIALL









:JACA .ACT, No. ITF16


- onent ma- also be written


M = 2, (2-a)UrUta dy



2 5 "ET1 61
= 2p 5 a5 -f -2 Ut
Oa 2 C7 UT, UT 6/





where up Is the variable radial velocity and ut Is
the t :r_..-tlal velocity, fro.i which
1 UT_ 1

or

Con3tant
8

The r^rag formula then roads




A si".il'.u' result was obtained by G-.1dstolr. in reference i.

TE3P2 !.D hSULT'j S'.,-


Te"ts cn ris'(s, c:'linders, an'd strearr.line rods
were conc.iectcd to dc ter-iine dr2c or *.or.nt coefficients.
Fn2 the -lindlr t'-e t-7 ce!' c i.Ats are eq.i.valent; for
the disi: anri the rod it is ;rore c :,n'eni:nt to empl'.,c the
m:mcnt cefficint, v.hich cai le ..iecsureid directly. In
or'er' to ext-i tT rang~ cf ."ch nurtber, se-eral tests
were condvct-?' with I 'r3on 12 cr F'recr. 11) as the :riedium.
The test res.:]ts obtained( ar- of tcehni.c-l interest
bec.puse sc*.e nf t'.: dat-, Farticulrly fr.- the hich .ach
nun .er r2.nge, were obtained for the lirst t1.ie. It
may be ncirnted out that nany of thie earlier tests on


CO.I 0 DZ r TIAL


CC i,'TDT"IT TAL









12 COI-'TDE',TTAT. 'ACA /CR 'To. LlP16


re'v.olvirng -'.sks *an-, in -articulir, on revolving cylinders
were cr:ducted on a rather sn:.'ll scales and in a limited
range of Re-nold3 nunber. It may be noted that a con-
siderable range cf Re:mnrlds number is generally needed
in or-'er to conPfirm with sufficient reliability a par-
ticular theoretical formula. For)- nstnce, It may be
imoossitle to cbt-ain a measurable *difference between
lngarithmic or cower formulas if ;a short range of
?Re'rn.ol's number is available. Th a matter of iist'n-
'uiish; between the various tv.ez of :orr-ulas is of
th'eore tical interns t.


xpnerir.ents on %Rvclvinx Disks

The moment coeffc'ient Is defined as





This -efiniticn :orresoonds to tf-h cne frr 1arrinar flow
on a rev..crving disk given by vcn .risn inr. reference 1
S1



where
La2


The constant aI use- b- von "an ~rn was 1.3 for one
9s'e or 5.68 for both sides; this value was later
ads'.sted by Cochran (see reference 9, vol. T, p. 112)
to aI = 5.*j7. If this corrected value of al is
inserted, the formula for laminar flow reads
1
.,, r 5.'7R~2
The turbulent-flow formula as given by von Karman for
revolving disks is
1
C=- = 0.146R 5

In figure 2 are shown the experimental results for
tests .,of a series of revolving disks. The Reynolds


COiTFIDE 'ITIAL








ThACA CR T'o. I0T 16 CO:7ID:'TIAL 15


n,,t.r r..-. r,-' about 16.0 to more than 1,000,000.
Note that the test points lie along the theoretical
curves .given by Lhe von T.[rmrnn for.nulas. The transi-
tion from laminar flow is seen to occur at R = 510,")00.
This was the largest value reached with the most highly
polish-d disk.

The thickness of the la-.inar boundary layer is,
according to von '*arman,

6 = 2.58j1

or, which is equivalent,
1
6 = 2

T'sing R6 = leads to


-- = 2.'I.
R a

For the transition ReynolJs number, 310,'00,

R6 = 2.5.3J

= 1^0

w ih iS of te sa.-e crcer' as he r,ini''mn critical
v\ lu1- rttainejd "or ir e.

Several tests were 'on'iuctc-d fo" the n;roose of
"r'e -igati g i the Pac 'Lrs Ifec t 1 n the trar, si tion
Re'nol'ns .nmter. q.-e I'irst c:rservation was that the
transLtion `e';nolds nuri.:ber coull not be increased beyond
te value 510, :010 no -atter c.v iignlv the surface was
Dor.li-si,-' ,-r wratever r-.hr .reca. tions were ta'-en. Like-
w'se, t .'jas3 1.:-':D.ectr :l i ffi i t to de-rcasB the
transition ,ev'n,.ilcs n.jtLr. ThI c:p -lice.tinr. of coarse
s-ani (60 .ne h) 6-, .;o the surf-.ce of s d!sk (1-ft
radi.n3) onl-" re. I1ce.-' the tr-.nsion Reynolds number to
aboC t 2 0, 'C, ( 'i 2). Th- r Tluc tion in ch.e te-insition
ney.noldrs carterr b in7 tial ri'rb-iLence ..'as also studied.
A s-:. l hil.-- ez.-ure air 'et azr'ie-_ r.ear the center
of the dis pro.'acee". re -eatest observed redaction
(fig. 2) a-. brought te trr.nsi-ion to a roint near the


0 C,'ID'TIAL








1i CO1ITDE.TL.T.. fA.CA ACR :To. Li,'16


intersecti.: o t-'r:e lines represer'ting the drag formulas
cr lor.inr ani 'Lurb:;lernt fliw, which "s tie absolute
mlni(.rj...1. Note t*'.at the drai-r in rte .urbulent region
is quite aptn.recLably ircreA-e.d jy surf-'ace rogi'hness.

Tbe vzlue.3 of t'-e mo..ient c.oefficlent given n in
fiJ.e 2 re-resent obviously an 1"tec-r.teld i2rs over
t- .'iskk. ;.n 3exress..on may be obtaslncs for the local
drag ,crefficient CDX as i. fux--titcn of lo.-l eynolds
number ag follows:

2 = C(pa.5)


= 2 a CDx p( r r 2)r2) d





2


i 'r i\a







+ -= b,.-( )
r_ do2 + e,. iL/rt,7
-- C=.




r dF



B'- substituting

r d P


2T d7 4 = CDx


CC .FIDC TT TAL








NACA |--t 0'o. IlP16 CrUP'T.''rT, 15


or

-- ,-. .- + i

and 5
S+2
2



d (I o ,;
d(lc R)

If



then




B, use of 4-1.- evm.ression for lon CGr, some of the fata
of fi' r- 2 -.re nlotLc--, in fi'Lz? 5. Alth.' gh the
-eEneral picture does not c":ar.-? much, the abrupt nature
of t. transition becomes sa,.aLaent.

An illustration of the bou,, ry-la:-r profiles
for various radii or 'ynolds nmaibers is given in f"L.-
.re L, in which czirves cf :..-i al velocity ut/wr are
-1, 10ct"c. t cte .: t t:.., thi'ness of the boundary
aye2' in the 1 '.r'-r .>;:ii is &s:entially constant.
The transition .' i rf a, 510,'030, is shown rrc.:'-
r. te." b:., W he l'.-,e ia'rl ed .p':,'-';.. transition" in fil-
ur 'e. The n: "Ir. i i .'" bo,.ndry-layer thickness
consistent': a.- rs -c sc ie.h'Lt in excess of that
r. Lv '' .. r : r :' 1 'Th.re r rr s to
Ss_3.".e iz reD ,.r'-; .. th-:.-retic~ul velocity
'" stri;,uti,:,. ',v,_..:. '._ 3 -."'- ,.. ..... [ :-: e. 1 .,i : totundary
l -a'.. Pcs ..,tain ..' fr:m .-" ., .,.chran. See ref er-
ere 7?i. I, r-. L_. .' T, is ;e c: .- ze I that the
ey:.cerirent, e r'or r I.i t sI s .-se i, o" cci3 '"erable
r-.nit. e. T'-? '. -r. '-t n ..r 1 --r slows almost
Er' t .r -e ,it t:.r 'c' ... c-ur'-e, which
*"s -1:l;tt- c. .- r, 1, i ic r, fL .L r .


C C I T E:- f AL








;L-CA -.CR :,. Il06


S:n-: tr ? eer., :r:c. h.e'c tiac a series of hot-wire
-,e-st 3 v'eP- run L st,' fl.-- t iat.:crin irn the bc.'oundary
Iav.",.-u wlth 'bh' following r,.sultz.

(1) Ic, isturba ices were noted in the l-Tminar
re ;ior.n

(2) \ -ure tor-, of a crequenc ,' of abO'ut 2D'3 -7:'l-S
per second was ntse.'ved in' tl'. tra-s t ion
re .

f() \ r:.ndm Mzi turtbnce '.nvol vi;.- mu:-h higher
I r'q'ienc Les vAL o"bserted in >I turbiulent
re <,n

fi -.r e .,ncer r.n .-n f ,c1he Re-Tnolds nuLnber
.a. on'-n c -r' ] tr,- lP xt-l,'e* The i -ghes c eyn, Ids
n r.ea3c!ed i'. 7,' 0,O '. rhe -p'r..er law holes

f'.'.re".'r v- n :'I "- ..'.ervel r -n -:, w !- h o':.ever, is
tt.c ? t .. r-' t t 'ncti C b l:we--n -. .-ow er
i.*aw an, t,..-: l1 ,-- ,;v'.* lan' o r t; velocit-; d'i t "t,.-
'r-,, i m nr. rir, '-r .r- -- c '-"h tests, .- "' c ] 1
,. '.h :', :re, .. 1 "re 5, as tc i,-.est ir te 3t.he
'f:: ,-f -= '' n, e Ti, '"irs, run -,'-Jrn with
i r- :s tr .:. [ c .\, "'r. 'l s "nu .ber cC '
at '..-it -, C .l ,c r '"c2h nu'",:.- c f ,..:.2. B:. using
l-ecn 12_ t-" .:, th.-- rin." of e'-nold. nu'.iber
v:as e.tL- 1 to '. .'00 2-.t ; 1 i .'.'eL t :?re 3ss3.je,
theo, ; '. "- 'l. IL c f c 3- .. cb .1 ':e:-1: each;-d was 1.6').
,.1 1 a i ita -. 'r n 32 I cw ':v lil tl;' h.ghr.er arag
tt an that e : i- t'. vorn .r';r:..n Ic m* la ap r.-ar n ntly
becc-.iFe of -r,.'- :-s"s -> ...tic rr'r.L'. T-,e si,:In flc'-nt
resrult- of this 1'.nes -tti 13 lisat tle d"ag coeffi-
c ient ,s a'sc lute1 in ec :n' :-nt of the !Fach numb''er.
1, senarate extension o. t:he exp1":imsnc to a i.'ach
number of s3 '-htt. more tha-n two further coni Irmed this
indener.dencc o.f 'the ..c. n'.iber.


lFoor inent3z n evol--r.ng Cylinders

ie exrerti.i nt l results for re vol vin cylinders
a!Re 3" (wn in fl: 'r'e 2 s a ri l p c.t 1 *lo aair3t
ln1 h, vbere ^ m'he dr-c f-rr-ul-a "or laminar
So a ol. e i
fl'w c.,.r a .evolv'2 ,r*inle-. is ,.rt. ineC from L-mb


CO:" ID2. ITIAL


CO T7 ID !L-' I,








NACA ACMR No. T164' nR N FIDF TIAL 17


(r-c.eerece '-C, r. K'e) as




''-
c = qIs



q3

Tn this fmorvula S is the surf'-ce a-ea and a the
radius. In this case it -3 c-n',venient to use CD
instead of C.-, '::hilh was -,:se' fc.r the revolving
disk, because no intra'cti'sn is involved. The laminar
curve :s s1]Cn in fI.Rre '. Tno dra,; relation given
by

"1


for v! tiubul :t fle:., is :1eo slown in figure 6.

7, eyxer:1-ental. res'ilts -.i'e rep? ctter' In fig-
ure 7, where -- ls shown -as a function of

lot',l Eri/5-. Th'e relation for t'-. tArtilnt flow

-- -o.6 .0-- 1 b or ,


anne-rs in figure 7 as a strai-ht li;-e. The cceffi-
ciarLt .3 in this nr.i-ml.r a corresro-nus to a valUe
cf 0.4 for von .c'r,n fii's "i'\xersal cunsKt-nt The
relqition 'or t.ie la:u;..iar regiQn = s ooears as
R
.a-cU.-vad-'l ne near the orf;ii.

It is noted that the drag noefflcient for rough
cylirnders is JependPl!t on the relati-'e grcin size '/a,
wh3re c is the sire of th .end and a is tl.e radius
o" the z'linJer (se-- fiC. C), -n1d 'chht :'rv each .grain
size the Crag ccefficienG re-nuins constant and


C C'IDEITIAL








NACA r..CR :Jo. L4' 16


irndenerndent of '.-e ReT-nelds number beyond a certain
m'n'L-.iL-. or critical value. which lies on the line for
tiroulent flow. In regard to the -.agnituhde of the
drat coefficietle as a fun ijon of relative grain size
for particle "aat.traticn" of the surface, it may be
remarked that the value of 6 is a measure of the
thickness of th., suolayer or, jhat amounts to the
same thin;, a measure of the minimrir.u grain size of
the :urbulence. It is therefore to be expected that
the surface rouhn-s.s will beco.:e effective at the
Reynolds number icr whichh Ccr, the critical value
of C, becomes less than the grain size Inversely,
it may be seen that, if thce ceyi:olds number becomes
smaller than this critical value, the grain size of
the turbulence .s tor. large to be affected by the
surface rouFhn ss. '"ith C greater than v, hi.h
Su e ,hni- i .cr.
is 5..,L, the following relation is aprrox stately true
fcr l-he dra-rg coeffic'ent beyond the critical Reynolds
number f-.r surface rourhness of saturation density:

-, -0.6 + 4.07 loRi0 5 -a

= 2.12 + o.07 1 f1

n f'.,.re 9 t?', pxer'enta-l '.oi:ts are shown to
sstilv: t':is theoretical relation with sufficient
acc-r .i cy.

Tests were made to dete r.-ine the effect of the
dee-isi--- of spaci:g :1' grains of a given n size, and the
results ar-e presented in -irure 10. Such tests were
made '."i.th a certtain mnit raijn size but with tihll sur-
fac- density in grains per squarim inch varied between
cO innd 2200. T.e grain size used corresponds to the
size = 0.03, -.lso usei for the precedin'- exper1-
mental results shown irh fi-ure B. It is verified that
the crItical Revynolds number depenr:Cs on the grain
size rnlv, and t ic f''_r-ther svr.'n that the lone of
the dr-ag curve beyond, the criic:..l 1e',ncld.s number is
a function of the densit-. A saturation condition
evidently always exists, in which the dra.3 coefficient
rer'ains aprrcximately const*it and equal to the
critical value.


C CONFIDENTTAL


COriDEriT .L








.IACA 70 :o. 147 16 CC .'.. .:T'IAL 19


.:neriiments on Streamline Rods

In fi."le 11 results are ,--'ven for certain more
or less streamline bodies, each tested in two or more
different mediums. T-.e tests were obtained 1; us.i3
actual propellers of 12-inch diameter, which are
desl .-.ted rollers B 's id C. Propeller 3 had a
section of double '*.;.etr;, with a circular-arc contour
line. Froncller C was obtar.ed by r-.:'.cing the chord
of propeller B b-. removal of astut one-fifth of the
corr near one extremity to attainn a blunt-nose air-
foil. Py r-,rn:nz propeller C backwar's an airfoil
with a blunt trailn.-g ,'d could also be st'ldied. The
dra, coef'.cient used in f'.vres 21, 12, a." 15 is the
standard torque- coefficient used for rvcr:llers

C --


For c. ..-trieal airfoil -, a value of the
^'Tch ni'n.c-r-o2 about one was re-v'-ed in air, the .mng-
was extended to 1.6 in Freon 12, 7nd the characteristic
decrease In the -. coef1'cient was "n aL Ly reached
i onreon 115. A considerable decrease in i':sg coeffi-
cient w-a noted at the la":',st '-.Th number, 2.7, which
to the knowled i: of the authors is the hiTlst :..tch
iiT.ber rea,'- e- except for a few cases of projectiles.

'- e bl'unt-nose a'rfoll section C showed aoo:-oyi-
mately the same low-sr s resistance as the sym'-ietrical
0 sri-ncse se Lt'or. B, b't :.d a ii "' t"."eque c -'ff -
cie -it ve"' r 'iuc- r. ;-::.-ss r f 3,''t Le-' [.'I.n B. The
test e-ten'ed "'' '. to r-:,a L'-3e (;'- ; ~ ttcr'iue curve
witn -r. on 21 .as t< .redi,' 'j ,.:, re...rcin. the dire -
tion f .- : c'in .- r.'o ell :- C t. -bt i t *a blunt .'e-.i,
e .ec t'=d l ._,e injr as 1.1 i &.-a 1C ; ch w'mbeirs
o seerv .i '..l -e .i. 1 i f r '*-*rc' i e :-,ieic ld
r.i.be'r for ;i, r n..c ,.'r 2 's a1. fp',1 1t :;r tie
u fCff 'ence 'n d ,i of-:-rts t r t-:u r'-jn e beinw a
1..ach -,'inmbr? of ;c.it-,'. -c .i -.n r a.ch nunxnocrs, the
dra- cceffic. ent c-0 t-.e -. cr.oq iti the blunt -cur
e. .es bet'"ve ... t.'- dra c- e 1 r i i 'it :. ., ?-- "oubl.
Ftrear.line sfcct.on aina t.i? -'l.Tu-icse type: the stream-
line leading ,lje is a Ir :'. sat ~ ". twIc' as. eff.-: .tiv:
as che st e.mli- Lr; il'1n r sn1 A ;in r n, ral
agre-mcnt wv to e:.rller csc rvasonlr.. t '.hcu-id be
noted., bo .'e,.er, ti.t Lie .. ....west ta i c abt inL-d with
both le& di-g ,ni' Frai'.ino 'Age -tre.,lin ',.

ot t t tcn : .---r : s:. :.n fi.urz 11, 12, and
15 are based on the tip ra3ias.
COFTI DEiTI AL








20 CC.II7 T ',!r.r-.L I.Ct / ,.': Ho. r_'*'1'


The I effect ,f t' e ..r. ol. nu-n-er is also shown
n re 12, "'V ch 'es the r s : 1r of tests to
st .*' ... the Ie l 'f.ct s rer _*. p ose.J on the
a'-ch nuqer effect. It sycull be n-ted igin that
the '?eynolds nur.ber- eI lfec ar,._--?rs onl" cr.r a rlach
number below unity. A wUde av,'sti 2.. In t.he R.ynol.1ds
rnu'')er sov.as no c.-ii ste2't mie s-r-a', effect on the
dr g icr i..'ach number- g-eater .- nit, .. 3ii-lar
.t or a s.ail1 angle of attj.:K,. in:.tead cf zero anCle
of at rack as u-seJ the -recce it -s :ussion, were used
in *-ne case, :for vhich iesLults are iven in figure 15.

The four piopellers referr.- to in fi-urez 11
to 15 are shown in a rhotograph (fi-. 1') an! LChe
dimensions of The rropellers re-- 1--;n in table I.

It is of so.ne irter-e..t .. ini cer'ect a superficial
analysis of the results rressntid -.erein, in view of
Ackeret's formula as 'iven b', 'i.;slor (r '.e rence 10).
For the local section Ac:eret give3 the dras, coeffi-
cient as
1
\a--/
CD = 2$ 1 a + + '2

where the bar inaic ate s the .nean value.
For z ro n .. cof atta'c: _nd a s,.'.e trick section
with 1- = ,;-- this relaci.an b1 c' es
'-- 1




For a circular-ar- section :2 2, where ^ is
m m' max
the risxtmuiTn tn.le. This in.1e is, in turn, approximately
equal to twice the thickness r-,tio t, which is the
total thick'ness -I'-i .ed by the chcrL For circular-arc
sections, therefore,


y -t1ft2


iure 1 s'ho-ws C r-lotted against Mach number
for different values of t. At ;; = 1.0, the curves
tend erroneously to infinit,,. ThLs effect follows


CoIrFIrETTIAL









,;.IA ".C T7 LT '16


'" a sino' r assumption used in the .,,'rivation of
Acke et' s f'ori ula.

-- us_- the general form f(') instead of the
".ach number function 1, the 3,,G coefficient
may be written a

S- 316


The to:w'u coeff".cient is known e::perimentally to
be a f" r;ction of the .'-cr.: number, or 1/Xl, ;hire x,
is the f. action of radius at which the ,acli number is
unity; thus, '. followi:;g integral relation is obtained:


= 1 }t2


'..ere are several ways of '.,,l1-., this relation. The
n ~e .i---r s Lon l Thord c *:n; t-.. J]-'",.-ss t ma'- be
taken to r-oresent a preferred section at a-. roximato1:-
.0 percent -f1 the radius. '.- as ... .n an initial zr_,,a
ccef ficient CD ar,' desired accuracy may be obtained
7v iteration methods.

The function f(") shown in figure 16 has been
ob'-:..:e for proDeller B b7 such a process based on
the :.: erirtental (fata given in '-- .-: 11. ',te that
the 1Jr. coeff-cient ..pr aches the value -ven ":.
the ...i-. t for ..;s : r l'.- val: s of -,, for
whiic f(.) re:'-s (': 1) 'lote :'rt'-.r that
the :.'. 1, val-ue of the drag coefficient occurs
= 1 :it-. ff sn lm.-st =-:actl- ',.. to unity.
' :', > ...: L -; .."3 r.- c c : ..r ,.." ".:..'. thatc the ."'*',c-
t 2- : ;: r'.. : t 2un :-..on is
- -r r' cr.el i .r .."'pose of co.-






E:per'i ,rentl re3r.. Is o'-. he drag of ,revclving
disks h:.ve bet n r?'e snted, hiani st. .s3t.-r.tiate to a


C o:T7-I DE;L' iA L


CC,! `-I-ol? -r T, T,









?.l C Nr '' J:L i" T ..'.L ,'.CJ," r, .. '', 1^

I I
e. r':ble e.1 r.'. ra c--:rr 1,c b-:sed on the von Karman-
n-. 1 th-er- "' 'in c':. 'e rp e cf the
've3 gat ion w.is e:< t-r.:'edJ cc a Cach nrriber of 1.69,
w!' ic, Is )e;,nrt.' .he range :," an.;. e rl : r tect, snd
t-. a I:eyvnorls n-rimber f 7, .: 0, '.0. t was established
that tKe s':in fric t ii is L:'C-: n': .ci3it of the P.1aah
n,:', -: ? .' to this valijc and ar.2.- rs to e f:..nc ton of
The ols nu:`ber aol'".

I'.e .r:.g at supe_-'s n.3 spe -.'s as studio with
rexvol "rc rt .* -rGWi3 -ec 'on3. .'~Cch nuibors
:s3 '. -i a ., re actainu- l in th:e tsts. The drag
at si.er;o.: spoieds s a .ncti-n of --he :".IA-h nutnber
only, "s i' &rrj i-s c., be r ents1l nl;t.ent cf
bot chL- e T .:-'J.i ru ,er .-:e ta. r.'atur' cf the meoiumn.
TIe r';- _. r', ': in tL.e l ..r "u:ve o e r:. v. d for
p:*c'i ..1. ,':.,3 l--. r r thin .= -. 1- re -o-4ies,
this :.: o rs a :.ch n .- l .r' c l 1i ".t:ly o yo ind
un tt-, n : n tt, t s t j '-.Ch 1.-..be.- o f-
about 1.1'. t- .ti es ts ve u -" 'ct. n stream-
1'nz r.oci, ts v. ", r i ti'.'," o'' r'7 i S vr t l',i l n-rt Ic adinL
-'t:1 tr 1 in ,_-d-- s f t e r -" r ': ,f bt ',i'r the
e t'.- *e nU ts f suh fe t.re' It .s. :rj .that
.ie i.ncres-:i 3 in c-e ne:k '.r'ie C? :hc- c r'g I ce fic ient
re- 1 '? ..'-1 a ': t. :.c3-s i about t'.v t e t. t
r coT L,, n 2 -r 2 t f'ra a L'.n -. 'e.n "'oth dr g
S c'ff -. t.. .. "- crre fr '- v -t.'_ :,.' Ar.r t c.oe"i? c ri t
n :. tlion I ':-. ir-.-liv i -- ar d. trc iling
" ':re 3, v ':.. i .3 e i ,:e -, v.-,l .

-nifirL :: t :'es.'lt: were .it.- reg :n revolving
Tree lijnJir.s .'- v -'c". re:- re 1s3 to earlier tests
see" c.'- be c'.Aic' I. Ic we .c'., tiTt, at '.'erv low
r .ol. s nu.trbers, the 'r-a.T az'-":.t,: ica.lly .a-porrc.achcs
t..e l :.ri r .3 r :..- ccl. I c l t"he -v'- whereas,
at hir'er- c:'nclis n-umnbers, .Lh;- W 3 1 o-ii to
ronifom to log i titan, nc f 'r:c.ila n t. vcn Iar;nan
t"pe. There is no distinct t.'-sition from laminar
to tu'.'ulnt ?flo'., -. s 1 f-..:nd in ,pipes end on
rcvclving disks. The flow vs es 3 ti l to.Jrtlulant
dowr tu the snmalle.4t L.-:,oli: nr.an r3s.

The effect rof initial .tu'-.le.ce was ps-rt.icularl7
studie,' in c,.r.nec_ion .: th best r. ,-.] ving dis3,-s.
Tt was -cui.1 t.h-! the transition eIynolds ni ter wcas
very slightl- affecteA. Tie critical Reynolds number
at which the rcughness effect ar'near-s depends en
-article size on].1 and is .ot a f.ction c" r-article


C' T P i r'T IAL









A'A ;.Cx r"o. LT'j16


dnsi" -- Ec-. t-' bs value of 1'-_ Reynolds number,
the -r-.- coe"'cient is constant only when the sir.ace
is "satiurat-," '-. ..t is, when the '.ensity of the
iv,.,'1:.1 .-i ,_ticles attains a maximum value. For
a r-,u-!-ress of less than this rpticle density, the
drug coefficient decreases with .-olds number.

It is interesting f'.rther to note the persistence
of the l.'ithmic relationship. '."heM I/ /D is
plot 1 as a f.intion of InO A47 wheree CD is the
d:r-d. coefficient and is the .-?nolds nitniher), the
lines re''resentin t-u''alent flow are invariably straight.
A rather critical demonstration of the lo _izthmic
velocity' pattem near the 3'i:'"ace is thus shown. The
rtnrige inves tited is of consi,7rable extent.


L.-.nley ; ..or .al Aeronautical Lbhr.'ator
National A v'.cr- Cromittee for .-Aeronautics
L'-. l.l Fie'l-.,, Va.


CO FIDE. iTI L


CC, I FI rT TAL









2- COi.FID 'v'I, L C. L-.... ,'*:R 14. i 5M


_.' 7 '1 .1 T .




TT, friction veloct P --t


T., hear -c-r in-t area t FLurTce

c ss of r oer '-, t v'r)ht1'

nT me .n ct on ve Ic t;t (from'. 0 to )

IT 3treTr.; velac't, for flat ;-.l es

T' rrx '~i, amy t locitv

-, mei:,n "c-1 ccit:. (in c:.in ea).

T7c refer-nce velocity. (at a Tiver. fr-.action of rar.ius
or of orher reference i'im:sion;r.

T7, vc-lcoit" at &

u -:.I C i'.: v ic r'. :'cle "elo it .. r, fluid in t.ondary
1 ayer

u velocity *d'ricer. strea .elo. tV minis loc1l
valocitv ''Ir -.t llate6J

ur r:cial ve oc t !' Pr i .~ ; s:s

ut t .r i t4E'1 .loci t for '.isi:s

W m,-anular velccitv, rjdia,-s

5 thickness of laminar s.;bla-er

51 b, bundar'-1 r thickn,-ss

L frc ti rn 1enrth ( U /'T I)

7. total length of r.late

S r.-fer nce t rie (L/7iT)

t rP' e; also, thlcil:rcss ratio for prorellcr section,
trilcknoe s of ai.r.fi.ll
chord


'n ,r- -T f.-" iT ,'r ,








:U.CA CC,- "o. 7L'716 CTEFIL-"',T 25


u coefficient of cinematic viscosity

S cot. "ficient of viscosity

r variable radius of p.:-, o.i3k, or propeller

a radius of pipe, cylinder, or disk; also, velocity
of si.i in fluid

x distance from Tes. in," edge of flat *-,,te in
direction of flow: also, fi--ution of rnroceller
r
radius (x = where R denotes r-a.Iii.s of
.rc.peller tip)

xI fraction -I r-.r-or, ller ra"' at h-ich iTach number
is unit,-

y distance normal to sur.'c.,=:

K,- ,n.- ...ensional pro ile constant for turbulent
K flow near walls

c f it.ction of reference dimension a = ; also,
a, chord
nonfi L-ional chord of airfoil,
radius

a ar.l1- of n1ititc' cf "VIr l; also, ,:.rofile


CD tot,'t.1-drs.- C.effric".-t (.'iMur., ".utho-:s uss f, #,
C p-q: sizd o C or n2r0 .)










cr cr.in size cf -ticl r e for pticul

ycr raTra ii s i ze of c -.L t ic-1 rc,..`ne-z; for particular
value of d'i- c '' ic i-nt

C. moment co,- 'ficI-,nt 'or r vo'. vin-. di .s

S r.issino -,omrentr.; moment fur dic.!s; or iMch nunber

r nr T1T7,T "i, T 7-PT T.








26 COC,.'IDiFI.L 'AC.. ACI Fc,. L 76


R 3eynold0- number

R Ry-nclis nu-n' r- bacd on thickness of
boundary L-yr

RPL Reynolds number b-i.e-d on distance frcm le- fading
edre of flat plate or on local radius cf disk

Pd 'Remolds numrner b-aecO on Piie diameter

Ra Re:ynolJs rumb'er oa-'eC oin i. pe radius

v velocity (fAcl:-ret ror-.ula)
1 ,:'
q dyi.amic pre ssure (',or cylinder., q = -o'.0,2a )

S a:rea of cylilder

'^. torque coe.fflicent ( "/,-)I' )

torque

N nu:.ibcr- of blades

n rocational sp-ed, w:''.-olu-ions jer second;
also, cze'f.h e.at in power lav

,,3 an 1';ls w. hi,-h ur;,e and l n wer surfaces of
airf. n ma .,h wit'-- cen.-er line

ax m .a:r-:'. angle ,,hi circular-arc section
r..npes with ce't-r line

C1 no:'r i.-ne:aionDnl velocity measured on logarithmic
velIocity c,'ofile "*"h1an this curve is
cxtraol3ted to = L

C2 nondimens'onU1 excess velocity at y : a over
that of lo,'1rit:u-c line extended to ", = a

C = Cl + U

C ', ,constants
S-r-
K1 ,2,z, .constant s

V constant

a, constant -n equation for moment coefficient
of rovclviLc disk~i

CONFiDENrTIAL









PAI.A I.CR i0. lJ.16 CO; TDE: TAL 27


."FT.:IY B

I'.. 'i.TAL V_.T.U?3 OF FO'.'.1R I' TS' F2R

:. D::lG DIS:-:'- ..: E 1 INLr S


A chart is presented (f.i. 17) which gives the
horsepower --ecuired to drive a smooth 4ick in standard
air (760 rm and 15 C, p = 0.0C'25_ slugs/cu ft and
u = 0.3'0159 ft2/sec). Lines of constant horsepower
r....in. in value 'r-cm 0.01 to ICD'O are plot ed with
d,..'," rotational .3..,d (in r-i) as abscissa and disk
diameter (in ft) as ordinate. "I!-- dashed line in
fi-_,.re 17 represents a ..eynold2 number of about 400,000,
which is consider1- the transition Reynols number.

Th.-v follc-.v.'"nT foriul.as were -ised to calculate the
rvmer for disks r, rating in the turbulent region:



1
C = 0.146R 5
1 2 1
1



LI








1


error m '- br b or te hchr T'.'esr r te h t er
snce th' chart ("ig. 17) covers a rare of tR'eynolds
numnbers3 to 0,C', OOC.


C cPIDE'T iAL









NACA ACL No. 1T416


.-, ,-.- ?s -, ..-o resenced (F i' :. 18) wh' ch -' ves
-e h'j"se r-.' r' quir-d to r'c taste a sm. t':,t cylinJer of
rit ,rn t (1 t i n stan '.," st.,. T ie fol) CLvin,_'
'..' l.s n,..e c .r, ,sr in c.alt latin.; the c';rv-s:

!x' C q,3aw

Cr pr a2
r T. a --- ?_0__
2



r, n

I _
I I ,
,.-,-


~"n~ <-'p ~-n r. tn ''ri-~r-,~ I-.


- -- -- ,1 v' 1 -"





----- ". + 4.' 7 T .
rI ,1 .-
























COflFID.ZJIAL


C O' FID TT'IAL









IA2A A'iT No. Ll.16 CC'TID'!:TTAL 29


APtr-L-r-: G

CrLTJCTKD .3!I:T-'ICTIOJ FORF1 UIA"


FLAT PL;7T3 (l," STDE)


Symbols

7'-r. following o',.rols are ic.'. in the formulas for
flat 'plates collected herein:

CD total r1-h coefficient

C..-- local 91.ir. coef';"cient at -Dint x

x distance from le'1r ... .-: of flat plate in
c section of flow

1, l--gth of flat plate in direction of flow

R -ynrol .s n.i:.1-.e rc on 7

Rx e-:.old' number bj:ed on x


L' rn 1na' Flow

The formula for total dr:.- coefficient

1
CD = 1], '.,'


is as" u 'o *:'-.6 si-ir. li r' h:-ir -i]-,1 ic. : <-',uations
d ev l, 1 :. I -:.-aitc 1 in 1' 'j. ('3.- ref.-re -e 2, p. 2.)
T = Co.1E : V',. i -.. -- j.l C 'L iS sLIS in 1908
3s 1. :", ,-, U e i l u.' .. .-f 'c .r 1'i2% as l. U.
(.e r e"',n:e .', i:U. !c Ti'' fi' :. a for" local dr--g
co ffic- n. isP


Cp. = CI,. :-P' -


CO'TFTDLITT.'-L










3 COFTDEI.TAT. A'ACA ACR :Io. Lrr6


".v0 :. *, F '-oce -1ev*, ar~l ^f-h Ya'.c irdl.&cted tbt,
ct ..L *,r:* -- c .- i .. ;. -it. i


C, Conctanc tn


S J I. r a Co-. i'; ert :.s r.!-et: t -



C,, = 1n + 1)Cp


7' :'- ti'-r, i- ,er.,r c i n .. -.: ti "-i -Litle.i .ccal
. J ..' -i. r '.- r inr t

d -.:-,- -, i ] i 1 'a :.-, n .n .


'-I i ~ : ;'j, ,O 1" "' -,.. 3

r. A. a n,_



"- r L r.e. e r o f


.-, :'.







',: rc rc._' c :_c0 .. t 1,- v.'n '. ': 'i 2' 0. (S i:e
-,-i ,' ur ,- ._ : i: .1 i ,i s I n" on


,'a r- c .1 i i : )'.".- -- .L ,r range,
C .. i".".- '". .

,,,,.. -- '",. -' -" ", "- '., _i'.. ",_,.. .... 'r, -; ~ul ":.- of the sa me
'"7' "' ri ]" q.r t .' clL L n '.rer oa


S- ,-L .





S" r Il."L
I. *. .L L. .. L








.'..CA ACM No. I'1.16 CC'FIDF'.TTAL 51


Of more gene'-al vl.Jlity are the so-called iole-
rithnic ,2r,:- formulas : the t,;


1
-1- = 1.15 log1 r;C.,
CD


The form of this relation was '.Ith constants a:ljust d to ,: -.form with data ty Scl-oelnherr
and others. (Se? reference 2, p. 12.) In the present
pa er a diff-rcnt form ".'-s been develou-d, which is in
somewhat stricter .ec:rt-ical conformity '.'ith the physical
relations involved:


= l 7 1 0-
,--*-.- 10 1 5.5!, C


F:"-sr't. ht' develoredri an explicit e:."r.3ssion which
nives essentially the same results as the 1.o.7ari'hiiic
formulas. It is

-2 -
CD = ..,5 o0 )


(See reS'cr'nc. 5, p. 1Y5.) I'*< lo-al .dr':i coefficient
ha- also te'.6n riv"n by von :'Ar:i'sin n a locarithmi- form
wi-t t'he conrita.nts adjLa-ted to fit the ;*.periments of
K' :, :- icn rnclua-u E .s1r.--.c :ts o-. small -.ova'ble pla;b s
isrt'd 'on a Icn l pontoon. This iTorr:lu is


-- 1.7 + )4.15 lo o-RCDx



(Seec refXrFrnco 2, p. 12. )


COT'IDE rTIAL








-. COiiFTD:':TIAL ".ACA "CR B 7o. i~2.1i6


v'. lc t il ,c* ,: Sur'fac:

S1i i. -.t"'h ree --J t,- -... .e 3, p. o,,2 ) ~ '. es the two
' :. :i f:.n.u -l 1c- thh tcl l -c t-h.. lo l drac coef-
f 1.*i: 5 0". r C'7 f 1 t. pl tc., 1. r*: '-'r-ec: ive l y

62 0 -25
CDr, = (1.0) + 1i.- 1tic 7


C = .2.37 + 1.,.' loj:.0 -


7"! r.-:' r_ hj efe.:-c 2. p. -') .es for the lo c2.. drag
co-. f er: cr ',.. a .:.f.-,: -..r.il f. .he lo a-
1 '-..-- t;-,

1 -e, >: -
= .* : 1- -- L'D.
1 CD--x






S:, -: -, I s


'"" r" ", u':j i- this :tor rPi'cr t':, the
'"-r-n? r n. Ier oC- t--:- :i.: .-'. :n -r a1no thz r:.3an
" i .'.:. c it-,- and te is 1rt I. % rc2*1s Lo t to -e e-,n ld
u i' ., .- ,i pr. u... "." 'f .r -i.-L .-_ U or r
il l : ."- cf C- t1.1.--.'_ 1er-. .'. 7 ct -.-r- ."- C*se h';r- ere

i I -. E:



',.P I"u;.n r..?'- f'l c .' ,i '-. r t'". o' a'. la i ',T r r'r" ;:. coor -




.-


1.-- f .rnuil is attribut-d to PF ireuile and Xierlenman.
(.-S e r.t-cre;-g:1 5, p. J,., .- r -e l p. 293.)


C r T7-.'-n T f
I -i- L-1








.'.CA ACR Yo. I'-+-6 CC:-'iD: : '?IAL 35


i'r'b.lent Flow .ooth u'-,_face

The for.mula for .*:.r coefficient for turbulent flow
in smooth -oi-Is is
1
CD = .o '.- .i

.is formula is b-,s- on the experimental work of Blasius
(see reference 5, p. 156), for which the Ren-.-ilds number
-.:' e was rather limited. Later work by ;i,.-.u..ise (ref-
eronce 5) extnr:3, t'e rL.;: of Ren-clcds number to a
minch "'. -;,. value. The foil.o.1 formula of the t yre
d:-velo. .e t:., von :.,r.man fits the data better:

1 +-
= -0..,0 + ..0 loec1r-'CD'



(3?'" reference 8, p. 7.'3.) In the ,-,sent aperr a
formula of this L:-.c ;:'th different constants is developed:

--- O..X + k.O7 ilORav/






a
Turb E-' nrt 'l"'y: o 1 .-.6.)ce



'- = ,.,6 + L. 0 lc,I0 .



T,- c::lrrz- .c'rnt 1l : ,'rk:' :in C .':. r <..is for tuila was done'
b-r ':i.:u:a.cse. (,' r 2 f- : :zr ..*, .. 'u. and' reference 6.)


C O: 'F DEI'T IA








L U.CA ACR ;1o. 4ir716


'" OT,'T T' D I I'' L




o' l. ncn,- n ? -r u:e7. in th f,-rmul. s for
.-<- ",c 1 < r.:- _] i i l.- :


.z n,. coef zi nt


CT-.. R-c-.l dragj cccff1nient at ra.i-u xT
-a-- ., A
8'... r'."--nold : u.i l-er at rc .' .:E A 2-


,',, l.a-'. r-r I'] o":


1
-~ -7
****


a 'i,1


Z "-" .- -
J n


TrIs fr ; .u'.la foa local orac coefficient is derived from


5 + 2vi
CD: 2_. CT
LT- 1'.


",r t'.: de-:'lop.n.nt i' thi-.s rclatjcr, and for references,
',. t:r- rc ,- n e r it lu 1 ..,- 'r .c-* nts n D, Ra fo]ving Disck "



Tur.- le-Lt Flow

raor, turbulent f o1.'7

1
n -
-,, : c.?.L..o L


CO irL -.T iAL


CC lPD7i'PTIAL









:L'.f ..CR ,. r IT' -6


a1n I'


-105


T;,. formula for the local -]"' coefficient CDx is
d riv.' from the .mvction for the moment coefficient CM
in the sare way as for the case of l.,in.:-r flow. The
local .1-, coefficient in lo)-:.richmic form may be given
as


= -2.0O + 4. .CT lolORxV D



The cost -n..-- -2.05 has been d-.-justed to fit the data
of figures 3.


F.LVOLVING C0iLi.r'}..5S


For laminar flow


R

,o- rurbu~ent f'lo' en 1:,.0.:t7h ,ylin.2rs


1- -0.6 + L.07 1O0TnR\7'-






= 2.3 + a.0 I.-.-
,DE


The d ?,'elop:'cent of thesc forulJs anr thLe references are
given in the action ,ntt'::. "-;:.:er.i'un-ts on Revolving
C;;] in'KrL ."


C O, lTr TDE : T TAL


Ci', T -I L1_ T AL








TDACt 1 .CR .' o. LC4l6


1. -.n :-trman, Th. : o r 1-a'nj.r.iar, ,und turbulente
Telbu.n *.f .a.M\..,., Ed. 1, H-lft 6L, Aug. 1921,
..n. 2 4.-252.

2. "c-n ar:r-i. Th.: Turbul3nce and Skin Friction.
J)T .. .&'ao. Sci., vol. 1, no. 1, Jan. i.95iU,
)'_ .. 1-20.
2. Irsi.l L.: T M -,chani. of Viscous Fluids.
V.l.1 IrI cif Aer'c. .. A ic Th'-ory, d iv. 0,
'i F. D r'ran e d., Juli*s S3r .ir.e-1 (Beri'n),
2?55, P',. 4--Zos.

L.. Cclisein. S.: Cr. th nesi.i-ta-cc to the Rotation
of :. s.-f r.',tei C In Fluid. Proc. Cambridge
PF": c v.l. .*"XT, pU. II, Apr-il 19 -5,


5. Il:.":.cc 3, J.: t.mt -ss i .eitn der turbulentern
3trUI.2 in al tn F ov rn. For.-chungshe ft 356,
.'r, c'u..' a,' dee' r Grbi- tt d.es Tn,. n. ie rwesens.
.-. .. -. t, S' t -0 1952.

6. I!.;l ;',, :'s J. : :.t r.r"lt-S2,s.,'. t 3 "t .In -,en rohren.
1(.r. -, LLes-e'I 3c?., Lse Ir,-e z, Fo'schun ea.uf dem
i'r iet 7 oe.3 T -gsnn efli'rw'-srcns, ..'s -. B, B.J. .,
J"i -.-' 1,;. 1 ?t*.

7. L'atb -r-.2 r. F. L.: A Stud-." cf the E2 i'F ct of
r rv\)t.;re o.-n F.: ll" C, veloped 'Turtl,, ent low.
-oc. rr.. .3ic. (Lordlon), 30 A, vol. 8,
;0o. ,.S', Ftn. 1'?4,, po. n6 -'

C. ?lu'ld r.tcn P-,i.el o': h? .-.eron-iutical Re-earch
C l qit ter: C-:. C. it .- s : 'cd. ron r' v',l.cpm-ents in
la' d D-r-..ni;.,, 'r.ls. T and NI S. 3nldsteln,
-d., 0-I"cr ''.. t t.. Ci r -I'don Presj, 1953.

Lsmb, I:race: IT'Jor.d-Tan"cs. Sixth cd., Cam.br'dge
Cfiv. "P'- s:, 7 I

10. Tl71:r, G. r.: Anr. ILest.ions. to Aeronautics of
4L.cer? c's Thio ry o0' Ae-ofoils iioving at Speeds
Greater Than That of Sound. R. 9 ?.1. No. 1467,
Bditish '.R.2., 1'3Z.


COF-ID" NhTIAL


C C ?:FI D-'Tf i,L







.!.CA ACR No. L?16


TABLE I

DT'"'"S3IONS 0F '?PERLLERS C .,L 'rLVI!G R0 ?S

i-,?. '-"?"S AT '.:-ITC ACH .'i -' S

f.ll prooollers have a straight ta-er in chord and thick-
ness. 'ie tips are rounj.d'i as shown in fig. 14.j

T At 50 percent At 92 -ercent
Propeller e rot-. radius
rde c11r 3i rfall P-tcia ----
nation section (deg) co- T:.t1~ness Chord Thickness
(in.) (in.) (in.) (in.)

B iarc 0 1.75 0.31 1.07 0.14

C Blunt nose 0 1.30 .35 .82 .15

D C[li r Ia25 1.1 .18 1.05 .11
Crce Ia I

Circular 0 .83 .13 .52 .07

aFroDc.ller D was twisted so that a pr::lmately the outer
half of the bl3,e had an anle of attack.


NATIONAL ADVTSQ0R
CCr'.'ITTLE FOR AEROIJAUTICS


Cci GTI D-. IAL


CC-'.T D'ITIAL







NACA ACR No. L4F16 Fig. 1












UT

Universal velocity i ci
16 deficiency function,' -
-Umx u f, .-- -

uT ,


c = ka \ Logarithmic curve,

a = 11.5c __ I i lo( )
12 / Turbulent flow


L 3.5
10/


0- / Laminar flow














NATIONAL ADVISORY
Velocity profile COMMITTEE FOR AERONAUTICS.
2,{I shown by heavy line

a-
I


Figure 1.- Parameters and functions of the velocity profile by the von Karman-Prandtl theory.







NACA ACR No. L4F16


-r



4147



* I --








-A jjj~

-*1


. I k


;. I I I


0
3-a



o 4 0 C

5 a a 0
o S 0
C Ca .- -
-.4.-. 7


- ----4 -f-- -- I I-.


-i j *1



I d 4

tyty TI


I I








'I II
-I---- --.--

-I-
,- [ 11 I I I Li


I I
1~~ -~-


jii


S- -






- 1~


I..







74
-J


i -J
- -


0
>




I --


T -v
ii


SI


-'f- -I:.- t t: : t Tim


-I-- A1 -.
rx q-


I.



S









^i
1o









0
a
'4



I-




*a
S


II.
S


0
U1
6


I I .


IIDI


r -'-W ftfI I i i 7r f"-
:^^-^^^tI I-.J2%T

4-EEE ^S!li!5^
. .-- .- :_ v -L :4_0-s :'40u- i: -
n ^ \--. .**- L^i ii;*'f T P -r .^ ;- t*T fT; .; *a
"^""^ r"^.17 r,7^ ^


--i.--i. ..i.. f\ f j-.^A--|--4--.a.--t--^--*--4- 4--


T- I I I ..II 4. I i


-m-


4 F --


) 1i

i -


Fig. 2








NACA ACR No. L4F16


I-
- I 1/



-


F z
I 2 -

-- Zu

4K- i


.^ {-_-- -! I- a




*( --',^_ _
.I- --
r-A



_ ^ _-_ _~ ~-_A L^ _.- .




1 _ ] ? ..._ ._


4 4 + 4 B- f F 4 4 F I + F 4 + 4 4 -4


I-


'-3-





r~~7


Fi.. 3


I -*








I -1- :
N_
__ _

A


It


--I










7


r
o

a

U


13
&.
r
0






















u


U
g


i
I4
43
14

*








0



0
a
*
J3
I..
I-
Sj
af


&i



ft.


i ii \_u I 11 4 fI L


t


c






NACA ACR No. L4F16


A l*[' .4. 1...!
.. ... T^t -^ tTr Tfft i-9-~--" -- -: -i -E =,:;, -*-* 1 w -'Cy1~- --

7 1; 14 1:i..!,.: .-t 'l. t4 T .:l :.
1D Tt-' 771'"^^iSf~l
.-,' '* -- Ir J ^ ^ | .:* -;iI ? '' '3 -TT !" **' .
T-7; "'t u .0 0 0 t0*

t-.-
*i-u :^ ,,s o ^ -^ ^ ^ ^_ _.!

4r 1
... ... 4:

4Ti
-ZL-;4,H 'i T ,-, :;4 i

.. .. ..
.4e-d- ''r- r --
'j ,' W"'' r

,-* o i" t ) 1 : ."






711
Tl- II : l ; -




at r -__ ,^ U
T, i ,
*~~~~ -t--- rf--- -- ^ n ^ ^ l ^ 'tal



0 t .-..




;': |p:-.' ^ ......g |||

.3
-I '11L, -r 1- :t ri -{ ---1P 1 '- l I r r 1 .17--








-.J : ,
iV -
^ Li ^ i 4;LL: T4. _- _j_1p ^r {L^ *




a 1 E^ ^E^ ^1^ j IE:4

0 T1^r ~i' ^^^ ---* ^ TTTS l ||r -^-
i gliiliiislS~i ^A
4- 1, ,i^ ai il ir ^ ^ ^ 'n ; -
I s^it~le~iap~gl?!^^^^^t+
i rR fi ilflSn~rE iS
i I T^b-1 RR~T3~n- ||
I'SF-1 L. EA.: a 1- I


Fig. 5






NACA ACR No. L4F16


0. c 'Da G C6C
6l-.-
so r0 r- o o Go C Z 3 -
0.-



a t~ ,~ < o ,.,,,, ^B..,,,,, ..., rr' ,,
Is *' U5 ^ ^T -!


i g gl-, -l, ,-,1- --,', l,,- ,,-



I L
a O 0 -, _- -X


3 U U
-0

-1 S. 5.
N -U P %. N.-_
;3 g s a a a3 ^ -*:7 ^ ^r5 -


.0~ .0 g0 b, 0 A0 r.
J .41













7- 1-4
l ---.C -- tK 0 *S S O E C>A -*-- ^-i--.-u'i ^ --


L-.1.'t-'
- -r --
- .l 1 _
-- -r -4p



~t -I- '
- -- -- 4


HHtMfitiKH


S* 7 J

Jt\-

/t ,


- -r


_--
<1'


p


g
-T 1 -,
.1
-i~i ii


* Ii

~..111z

-2-
-II


I..






S- i--

I i -

* I
-.-
l--t --,- -
- -


L44-4-.


(> c
t 0



z -

0'T
-T o

4i


Fil. 6






NACA ACH No. L4F16


FiE. 7


NATIONAL ADVISORY ---H
COMMITTEE FOR AERONAUTICS -r-


S1 : : ::: -- -







--.- 4[b :l lu--td : "*











Diam...4.Leg
TTho 1 6 : tan
.t dt










| a gl 1 12 Liquid


ax 6 18 Air

-acorrected for ddsk drag
-4 4 -of ends.
S -t .. 2 .
4101t0
LT ^ ^"^1^ 1^ ^^f 'tfi~l^ -





















J, ^_Ei_ ^ ,^ E;; ~i^ : _- _
,2^^ _- ^-2; 4 0ll I 40A^ J ^
L0910 W05D


Figure 7--


Drag parameter .- for smooth cylinders as

function of loglo0 tDD.







NACA ACR No. L4F16


I I


>1
o
>*
a
0 .c


Z -A
_o-

z z
.4
* .0
4'.-


ii--

[1


-i-i T -p -t-r i .'i Tj- i~ii^t< C ?s "r' ". *i. .i' "''"r i -''; ij t


-j nljrT^ ""? j'^ "r iT "' 'i"'

T~ -^g -t a ^y ~ .-t-r 1 4-- -}- I


I I J ..


I 0 0000
I --c -- o a .0
-, C =. i. a.f
a-.7



- -1


-H--


r' -





11--
Irj


K t.I .. -.
I-
I __

AL,.
K '. ..


--1

--I


S-- r-i--# -------+- -+- ---- T '-4- F I- -


S.. _+4-


* ]




* 1 L 10

- I -j

-tt L ^'

t ~'N"

-' 4 "i< 1


I


.1 [-i- --4



K'
-ji I


ritt'n ,.


S_ _I 1 ^ _. l i 1
-i ....^1^ -


4


"4 I

01
'-4


zl
-i



44
--I- -
~~1


A


. .. -r' .


F' 1 8. 8


i


9Pr i








NACA ACR No. L4F16


Fig. 9


-- NATIONAL ADVISORY
_4' 4 -COMMITTEE FOR AERONAUTICS
















.-.-.. ..
-n --+,- -- ,"r















|-74





as function of 1 70 *"







NACA ACR No. L4FF16


ITWI


H--++-HF4


Th- 7- ~f4 '' 7 t -------I-


-9-


-:- -
CIEZ


- .? ..-.
-- l ---- -^ ?
--^Fi- ^^'


4. ^~ -S -- ^3 fr 2 |1 I|-



fI, 771
S-. -Ri l -
-^ ^ _- -^-; L __ .jp t Ip S

^l F I ~ll nl i H tl~l!:ll


=2; ;::-AiL7:_


__ .- .i L -
_
A



Z 5
I-
Li
0 V
,)-

u4ili


--Ft-,


I
n

-s i


5 0+
o oooo

o 4 bOO


'4
*I.
0

0

a
C



U
a
Pb
o
C*
U4
I
o
Pb
i

U
0


rC.


I


Fig. 10







NACA ACP No. L4F16


Fi 11


B -r---
+ -_ I_ _I _
----'-+- .--


-A -.L a- NATIONAL ADVISORY
COMMITTEE FOR AERONAUTICS

S ------









I ~ -
7 .1



.Ju LI
-t fllt- -- r-,--i --^ -H-+-- --I -* :=s-I-, : ... ..

_- -- -t-'-- --1- __ -p 1 -- -


















Oaa 1rope11er
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NACA ACR No. L4F16


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NACA ACR No. L4F16


Figs. 15, 16


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Figure 1 .- Theoretical cures of drag unc'on M as fruction oD a number
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NACA ACR No. L4F16


Rotation .ppeed. rpm NATIONAL ADVISORY
COMMITTEEE FOR AERONAUTICS
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NACA ACR No. L4F16 Fig. 18






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UNIVERSITY OF FLORIDA

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