NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

ORIGINALLY ISSUED July 1944 as

Advance Confidential Report L4G1.4

A MEmIOD FOR THE RAPID ESTlMATION OF

TURBULENT BOUNDARY -LAYER THICKNESSES

FOR CALCULATING PROFILE DRAG -----By Neal Teterv1n

Langley Mamorial Aeronautical Laboratory Langley Field, Va .

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· nlcally edited. All have been reproduced without change In order to expedite general distribution.

L-16

https://ntrs.nasa.gov/search.jsp?R=19930092716 2018-04-26T07:18:41+00:00Z

~TACA AC~'~ Ho . L4Gl~-

l'TATIONAL ADV ISORY COMMITTEZ FOR AERONAUTICS

-----_ ..

ADVANCE CONF'ID~mlIAL REPORT

A r~THOD FOR "T'RE TIAPID ESTI:;:ATIm- OF

T1.JRBULEI'TT B OUNDATIY - LA ::'l'~R TEl CKr-1I.. SSES

FOR CALCULATIEG I-{OFIV~ DRAG

By Neal Tetel'vin

SUl'':u..YRY

An anaJysis is develo~ed tbat m~~es it possible to integrate von =Carman i s bouncl8.ry-.lcS_ye:> mo,,::sntum ec~ut1tion directly . For this purpose the slin-friction coeffi­cient is expresped as a func.tion of t~e Reynclds number based on the boundary-layer mO:;lentum thickness and the boundary -layer shape j s a.3sllined to bl~ constant:;. The inte.:;rat ed equation permits the bo - .nc.l~iry-J ayer ncyaentum thickness at the airfoil traillnr edce to be rapidly deter~ined for the esti~at'on of ai~:oil profile-drag coe ,"'f4_ cj ant s .

IrITR ODUC1' ION

Pec.ent incre8 ses in size and speed of modern air­craft ~ave emphasized the need for & r~thod of esti­TIl8tiYlg ra.:;idly t~he profile drag of win[s. One of the most time ·-ccnsuming oper" .. ti0ns in a :)rofile-drag estimate has been the co.lculation· of the t1:...ic cness of the turbu­lent bound8ry layer . Squire and Younr (reference 1) calculate t~_e thicknes :o c:: t -::t e turbule~t ~o1..md8.ry layer by a step-b:- step inte gr~tion of von ~ar~anis houndary­layer rromentum equation (reference 2) . A logarlth'7J.ic s1:in-·fri ct i on formula de SiGned to f1 t Sc'h.l:l chtinr, IS np­pl'oximatG for;nula for the sl/ 1n friction (r e ference 3) is used in the integration . Reference 4 gives a m6thod for the determination of the t:;urbulent bOLAndury-1Bycr thick­ness v,,-bich eliminates the 8tep-':)'Y-step c0I19utations of reference 1 but requires tLat suitable ap~roxlmations to the airfoi l pressure d.istribution be made. The method of reference 5 eliminates both the ste,-by- scep cor~utations of re.:t"erence 1 and the ne:::ecsity for making approximations to t he airfoil pressure distribution required in reference 4;

/l

I I I

I '

2 CONFIDENTIAL NACA ACE No . T.,4G14

however , reference 5 is restricted to a particular boundary- layer shf'pe ara does "lot; eXjJlic itly indicate how to use an arbitrary sJdn- friction law with the integrated von l\:armai1 equation .

In t~e present 9sper the ster - by- s tep computation of reference 1 is eliminated along with the necessity for Flakin.=- a:)proximations to the airfoil pressure distribu­tions, which were req lired in reference ~__ A mol" general specification of the boundary-layer shape than that given in r0ference 5 is used and a method for us e of any skin­friction lav with the integrated von Kirman equation is pre sented . The integr'B ted form of the von E.8'rman momentum equ~tion appears applicable either to the laminar o~ the turoulent boundary layer provided that t:he appropriate and constant s~8pe of the bcunct~ry laYBr is assl~ed and the a~pro:oriate fornr: .. lla for the: skin friction is l:sed . Por the tm'bu18nt boundary layer , a Elkin-friction formula based djractly on the data obtainGd from experiments in ~)ip'3s (reference 2 ) ,is presented in a form readily usable ~or boundary-layer calculations .

rme pre sent work provj.de s 0. met~od for the calcula ­tion of the thickness of the t.urbuJent boun<iary layer whj cl1 is mOl'e rapid than that of refe;oence 1 , more flexible than that of reference 4, nd ~ore ge neral than that of reference 5. The flexibil'.t..y of the r.lethod arises from the fdCt that trw effects ':)11 the calculated turbulent boundary- layer thickness ot loc~l chanses i~ the airfoil pres...,UI'e distribution, c:1an;6~ in the b oundary-layer shape, and c1:.ange.:l jn t:!1.e skiE- frletiol1 1' orll1ulas can all be easily evaluated . The method , however , like those of re.ferE)DCe s 1 , )~. , and 5 cannot prodict the onset of turbu­lent separation 2.nd , if the dange r of turbulent separation is thOl'.ght to exist , the met-Lod of reference 6 may be used to check for such a condition . Although the present method may be used in such cases to ootnin moment"UIll thickness up to the point of separatjon, it should not be used in the deteY'rni nat ion of the drag coeffl cient s . The pre sent method can be used to calcula:i:;e momentum thicknessos for flow ov'er rough a8 we 11 as smoo'th "lu,t' face s, if a curve of 8kin­fr'iction coefficient for the roug'h surface is available .

The skin- friction equations presented 8hould provide a good estimate of the turbulent boundary- layer thiCKness for use in profile - drag 8stHo:ates when the airfoil is sn!ooth and there is no danger of turbulent separa tion .

COIJFIDEl'JTIAL

NACA ACR No . rhG14 CO NT-IDE HI' IAL

SY~,rBCLS

c airfoil chord

Cdo s3 ction profile-dr.ug coeffic .. ent

H boundary- layer si1ape parameter: fo ... ' v:hich values.

( cr'~) pre glven in re fer-ence 6 -f k constant In equat ton for sldn-fr lct ion cceffi cient

n conBtant in e~ubtion fo~ &kin-friction coefficient

q d 'ynamic pressure , pcun;i3 pc" square foo"c (~J) nc wine P.eyno lds number b~lsGd on chord and free-

stream velocit-y IUS) Re boundary-layel' Reynolds :lumber ~ U

s

u

x

y

o

P

8

distance along airfoil surface, usuG.lly taken equal t o ~x.

position on surface at start of integrat:i.on

10c8.1 ve loci t:y' inside boundary layer

10('.a1 velocity out jde ooundary layer

free - stream velocity

velocity at so/c

djstance alone airfoil chcrd

perpendi cular eli stane"" from wi ng su.rface

fll.ll boundary··layer t"lickness

d 1. splacement tbJ ckne 38 r (6 (1-tr)dY]

:':::::~1ffi t~ickne ss [t~O(:_ ~)dYl o -

momentum thickness at solc

TO 8 1:in friction per unit area , :)Oun::13 per ~qua.re foot

u ki~emat!c viscosity

'rhe suhscript t i.s used to indicate the condition at the trailing ed.ge .

CONPIDEN?IAL

r

4 CONFIDE (TIAL NACA ACR No . L4GlL~

ANALYSIS

The von Karms<n b01.IDdary- l ayor mor:1entu.;.~ equation , which is used to co l c ul£ te the T'107ilen tUM thic.kn e s s of the boundary layer , is given in r e f erence 2 a:J.d may ~)e writ ­t e n o.s

de + E + 2 (~ dq = To ds 2 q ds 2q

1'11e mo;::en tum eq lation 5.s a diffe r en tial equation that can ue integrated if t~e bo1mdary- layer shape , and t~ere ­fore E , is assumed to be constant and if the local skin-

TO fricti on coefficient is expreasi~le as a power

func tioY'. of R 8. 2q

In order to in t egrate the equation the s k in - friction coefficient is written a s

~here k and n are constants that are to be determined . The mo '(,entuIl1 equation is lJIrri tten as

de + }I + 2 d U 8 = k\)n 8-n ds U ds 'J'~

v'.7hic h is a differentia l equation of t '1e 3ernoulll type . Aftep the equation i s lnt8 (;rat ed and the arbitrary con·· stant arising from the inteGration is evaluated , the result is

(9) 1 [ (1 +n ) k J~s/c (..Q..)(H + 1 ) (n + 1 ) + Id ~ 'C" sic = (~ )H+2 L Rc n solc Uo c

o sic

( )

!'l+l( )( H+2) (n+l )],L 80 Ul L+n + - -

C Uo

CONFIDENTIAL

( 1 )

NACA ACR No . Il~G 14 C ONF IDE fJT I AL 5

The de fi n i te integral occur r i n; in equation (1) may be eva l ua t ed b y e i ther an analyt ica l or a e:;ra-=Jhical l:1et~~od , whi c heve r is more convenien t .

DISCUSS I ON

\Jhen t he momentum e 1.uat ion wes integruted, the as s IDl1p t i on was made t ha t t h e s hape of the boundaJ:y l ayer , and t her efor e ~-{ , wa s c onstant over the chord . Al -though the a ss wnp tion is not t r ue in ceneral , the error in 9 will us ua lly be sma ll be c ause of the manner ill whi ch J: appear s i n t he moment 11IIl equbtion .

Sxper ienc e ha s indic a t ed t :1at , for tl16 c a l culo. t ion o f tho t hickn e s s o f the turbulent boundary layer , the v a l ue of 3 ma y be cho s en c l ose to 1.4 f or v i rfoils t ha t heve sma l l pr ess ure rec overies but , for airfoils tha t h.s.ve high pres sur e r e cover i(;s , #111 ch are t:lick or oper atin~ a t h i gh lift coefficien ts , ~ should bG chosen c l oser t o 1 . 6 . In refe r enc e s 1, 4, and 5, H was chosen as const ant and equal t o 1.1-!- . r.:he t'11ckness of the l aminar b oundary l ayer c a n 'usual ly be obtained \,llth suf ­fici en t a cc 'lracy for yr ofi l e - draG calculations lj' the i ntegr a t e d moment um e quati on is used , together 'with a value o f H of 2 . 592 , which i s the value for the :Slas::' us fl a t - p l a t e prof ile (refe r ence 2 ).

I n or de r · to u s e the pr esent form of t2e inte r rated mome''ltmn eC!.ua t ion , i t is nece s sary t '1.1l t t :16 sk1.l. - fr iG t ion coe f ficient be written a s

T O

2 q :::

n Re

For t he l am nar 'oolLYldary 1a:rer :l.aving the Blasius s~ape , n ::: 1 and k = 0 . 2205 .

A sk i n - f ri ction formu l a for the turbulent ~)oundary l .s.yer w~ich ha s the r equired for m is thdt ot Falkner (r eferenc e 7) who, after analyzing vdl"ious data obtained fr om f l at p l a t es , s U8gested the relation

TO ::: 0 . 00 6535 2q 1/6 Re

CONFIDENT I AL

6 CO~~T?IDENTTAL NACA ACR No . r4c14

A skin - friction for!':ula ments c~n be derived by (22 . 10 ) , and (22 . 17) This forn.ula

based directly on pipe ex~eri -+' ,. (,.,r 21) use o~ equ~~lons ~~ . J

of refer0nc8 2, paCes 14.2 and 143 .

T O

2q = (2 )

i s plot ted in fig1..lY'e 1. T:le c m've of eliua t ion (2) and the relation sugg~sted by palkne? (reference 7) are plot ted in fl :."ure 2 . t .. t value s of He above about 2000 , the C1..TI"7E:;S are fOlmd to be in good c.2,reeInent ut at values ~elow 2000 diverge cffilslderably .

::r":le sldn- friction formula de::,ived in reference 3 indicates skin- friction coefficients· at small values of Re t~at &re in good agree .. e~t with equation (2) . The skin - friction coefficients given "")y Fall-:ner is equation are tl18reforv pr obably too small at small values of Re •

.2::1 .ution (2) 1s based directl:r on t::e pipe experi ­men.ts to which the for;i1ula of Squire and Young (refer ­ence ~ l) , although not derived In a direct manner, also owes its crigin . This equation and a cODvenient plot of the equat10n are (Tiven In refere:1ce 6. C,::ms idert:1.t ion of flat - plate skin- friction data ~nd approxi~ations, wn:ici V18re made in the interest of ::: ir;r::Jl:!.ci ty of ex~res ­SiOll, account .for the difference between the Squire and Yo un.; formula 8.nd equation (2). Trie skin- friction coef­fic iEnts indicated 'uy use of the Squire and Young for:~ula are sliihtly lower than those o~tain6d ~y use of equn ­tio~ (2) at lLrce values of He but are about the same as those obtained by equatlon (2) at s 1;1a l1 values of Re . T~e i .. troduction of equa~ion (2) also serves to indicate how an a::>bltrary s1-:ie - friction fOrl:lula may be used with equo.tion (1 ) .

In order to permit the i'-,tec-:rnt2.on of the r(omentum equation , the skin- friction r elation chosen for use is approximated by a si ~le power fmction of Re o This app:"'oxi~na tion is made by usin3 two values of Re and the skin- friction coefficientu assQciated ~ith the two Re

values to evaluate the two constants , n 8.nd k , in the equation

..

.. .

NACA ACR No . ~-cnh C ONB'1 DE NT ILL 7

The f ormulas to ~c used are

whe r e the subscript 1 identifie s the sl,in - friction coeffi c ient occurring a t ReI ' the s '1aller value of Re , and tho subscript 2 denotes the skin-friction coefficient occurring at RS2 ' the 12rger va l ue of Re ' Experience

has indicated that, if RS? is chosen about ten times as

l arfe as ReI ' the a~~)roxLnatioi1 to e CL'...lation (2) will be

good over t'l!e entire r ange of Re val"Ll.cs of the turbu­l ent i,)Olmaary layer likely to be enC01L'1tered in a spe-cifi c case . The beg inning of t'1e turbulent boundary layer has been fotmd to be the re["2.on in \':hich it is mos t i mpor t ant to have a food 8. y prOJr:imat-i on of tlLc sldn­fricti on curve . Secau~e of this fact , ReI should be

chosen clos e to the in.itial value of Be of the turbu­l ent bOUl1dary layer, ,<"hieh is assu.r.led equal to the final value of' He in the 12mina~ ')ound8..Y':r layer.

After t~1e moment u:.rn thic1r ness of the boundary layer at the t rail ing edge has been obtained, the D~ofile -drag coe fficien t is calculated y tho relation developed by Squire and YO'.mg in Y'E:;f'erenC0 1

The subscript t is used to i dentify t he condition at the tr~iling 6dge .

CONFIDZ1JTIAL

-------~~--~ - - -

\ \

8 CONFIDENTIAL NACA ACR No . L4G14

COHPARISON ." ITH EXPERIr'JENT

Calculations of profile - drag coefficients have been made o:t the method de scrib .Jd herein and the results havo bo e n compared with those obtained by other inv8stigators. The profile - drag coefficient of the 25 - percent - thick airfoil at a lift COefficient of 0 . 25 and ~ Reynolds number of 8.2 x 10 6, for which data are given in refer ­ence 1, was calculated and compared with the experimental value of 0 . 0080 given in reference 1. . lIr!len the equation

T O = 2q

0.006 535 R 1/6 e

was used , the ca l culated pro ile - drag coefficient was 0 . 0076. "':hen the equation

T O = 0 . 00905 2q Re 0 . 2028

was us ed to fit equation (2) for values of Re between 1 , 100 and 11 , 000 , the calculated drag coefficient was 0 . 0077 . Squire and Young in reference 1 obtained a calculated value of 0 . 0079. In all three cases the value of H was taken equal to 1.4.

Calculations of the boundary- layer momentum thi ck ­nesses . along the chord and the profile - drag coefficients of t he NACA 0012 airfoil at zero angle of attack for four Reynolds numbers have been made and compared with the expe rimental results obtaine d from reference 8 . The comparison is shown in the following table :

Rc Experi -mental

(ref -l erence 8 ) -_. __ ._- - ---Gr-'- _.. ..-" ,--

2 . 675 x 10 ! 0 . 0071 3 . 780 . 0070 5 . 3 50 . 00 68 7 . 560 . 0067

!calculated, calculated , lcalculated, 'Squire and Falkner ' s I equation ~YOlli~C (ref-, equation: (2) : erence 8) I f '_. -- .. _- . ~ 'i '- ' - . -- -~. - - . ______ _ 9 _ ' __ ' __ '

0.0074 I 0 . 0067 . 0 . 0069 . 0072 l . 00 69 ' .0070 . 0071 . 0069 . 0070 .0071 .0069 . 0069

... _.- "- '-- - .. -. __ .---- _._-_._ .-

C OlJFIDE l'l'-r IAL

NACA ACR No . I4G14 C ONP IDS rJ1l:1 IAL

The profile - drag coefficients that were calculated by using Falkner ' s fo r mul a i n one calculation and the r elation

TO

2q = 0 . 00934 ReO . 2068

9

a s an approximation to equation (2) between ~8 values of 900 and 9000 in the s econd cnlculation are ~lven in t hi s table, tosether vii t h the experi'llental values from r eference 8 and profile - drag coefficients c&lculated by t he Squire and YOlLYlg met:o.od in roference 8. 'rhe b O Lmc~[lry- l8.yer 1:10mentum tllickn3sses over the surface of the ;·r.LCA 0012 a':'rfoi l ; Wl1ich VJere cdlculated by usinG t he apprOXimation to e~uation (2) and the 10r~ula sug­gested ;yy Falkner , are given in fiL~m"e 3 tOGether r:ith the cxper1L'1ental values o.f n" o_nel1t-w.ll t:~icJ::::1ess obtc:ined fr~~ reference 8 . The differences between the t~o cal ­culated ::rJ.Oiilentum thicknesses are ca'..lsed 1::l:r t!1e use of the two different si-~ in - fri c t ion for_ .u1as, the value of H ~elng taken as 1 . 5 for bot~ calculatio~s.

In order to ind ~ cate the effect of the cholce of H for t:.:.e tm'bulent boundary layer on t:-:e calculated prof11e ~ drag coefficient of a thin airfoil at a small l ift coefficient , the profile - draG coefficient of the NA~A 0012 airfoil at zero anGle of attack and a Reynolds

I'

n um'Jer of 7 . 560 x 10° was also calcl~lated for value s of S of 1 . 4 and 1 . 6 . The profile - drag coefficients we~e 0 . 0068 for a value of H of 1 . 4 and 0.0070 for a value of H of 1 . 6 . The change in H caused a rela ­ti vely s"11all change in the profile - drag coefficient of t he [lTLCA 0012 airfoil at 3ero angle of attack bocause for this airfoi l at zero angle of attack the chan~e in velocity ove r t~1.e airfoil surface i_s s!'1all. !.s can be seen from equation (1) , a change in the vL.lue 01: l{

wi ll have Greater effect on the calculateJ profi1e-draz c oefficient for cases in which larce changes in velocity occur (for exar:1p le , on VIick airfoils or on airfoils at l arge lift coeffIcients) than for cases Ll which t~le chance in velocity is small .

C orJF' IDE NT IAL

\

l

10 CONFIDE1~IAL NA CA A C::1. No . r1:.0 lL,.

SAl'1PL:5.: CALCTJLA'I' IO)T

~ calculation is given for the profile - drag coef ­ficient of the NACA 0012 airfoil at a Reynolds number of 2. 675 x 106 and zero angle of attack . The velocity distribut ion , which was o~tained from reference 8, is eiven in figure 4 and is the experinental velocity dis ­t r L:lUtion , uncorrected for tun.~el - wall effects . At the

Reynolds number of 2 . 675 x 106 the t r ansition point

was at ~ = 0. 48 . c

The equation t hat was used for the calculation of the momentum thickness of the laminar boundary layer is obtained from equat ion (1 ) by placing 90/c = 0 , n = 1 , k = 0.2205 , and H = 2 . 592 . The resulting equation is

1 ~ 12

(8) 0 . 66L~ 1 Ij"s/c IU ) 8 . 184 s-l ;-S/C= V~) l 4.S92 Rc 1/ 2 1_o ( Uo d ~

& Uo s/~ o~ for t _e particular case ,

= O. 66)+ 1

4 C,92 ., (1.144) .. / V2 .675 x

j 'O . !~.8 (U)8 .184 s 'l'he value of - d -o Uo c

(.§.) = 0 . 0002 901

c ~ = 0 . 48 c

is equal to 1. 760 .

The final va l ue of Re in the laminar boundary , which

is ·assu.':ied to De the initial value of Re of the

-------- - "~- -

NACA .AC~ No . r.L:-a14 CONF IDB N'I'IAL 11

turbul ent b oundar y l aye r , is t hen c a lculated :

-R (~JL \ Re c c TJo )

~=O . )+8

= 2 . 675 x 106 x 0 . 0002901 x 1 . 144

= 888

A value of 900 was c ~10 sen for RS I and 90()O for RS2

fc,:, the a-)pl'ox i mu tion of e y'uat i on (2 ) by a power function of Re o The c omput a tions f or n and k &re as fol ­lows : Fr Ol71 f i gur e (1)

The n ,

1"' = 900 H8 1

To 0. 002289 =

2q

Re2 - 9000 -

TO 0 . 001~2 l = 2 q

0 . 002289

== logO . 0011 121 - 0.1~76 n -

l og 10 2 . 302

= 0 . 2068

1;: == 0 . 002289(9 00)J · 20G8

== 0 . 00934-

TO =

2q 0 . 009 3L+­

D G . 2068 "\8

C OnPID'I:2'JT IAL

\

/ 12

S~ll'3 for._.u1a for the turbulent portion of the bouncLfH'-Y layer then becmlles , :I.'or H = 1.5,

The vahw of

I-l. flEul' e 5 are shovm Lu've s of (~;t184 and

(_U \ [1 . • 017

I that apply to the la~inar and turbulent Yo/

pOl'tion3 of the boundapy layer , respectively . Then

(~)~=l - 0 . 002005 for one s=face a nd

r-

CONCU.TSJONS

1 . If the skIn- friction coeffIcient is expressed as a ]Jower function of the Feynolds number based on the b01lJ.1Ci llry--l8.ye:l' r.nomentur,l thiCKness and. the boundary- layer shape lS ass,umed to be cons-G 8.:1t , j.t is )ossib1e to inte ­grClte von Karman I s boundary-layer nOl1 ,entum equation d 11' oS c t 171 .

2. The integrated bounda.ry- layer :':nomentUl'i equation permits t't'e "boundary-layer :nO':lEJntll;'il tll.ic}{ness E~t the airfoil trailing ed£e to be raDidly determined for esti ­mation of airfoil )roflle -drag coe fficients.

Langley Femorie..l ,1I.eronnutlcal La'::loratory Na t 10nal JI"d vi sory Com.mittee for Aeror..aut ics

Langley Field , Va .

COi-,PIDLIITIAL

NACf. ACE No . lhGl}-t- 13

1.

2 .

:5 •

5·

6 .

7 .

8 .

SCl1lire , H . B . , and Youn.: , A. D.~ Tr,e Calculation of the Pr-ofiJ..e Drag of Aerofoils. H. 8- I;{, :To . 1838 , British A . R . C. , 1938 .

Prandtl , L . ~ The Vechanics of Vi seous Pluids. Vol. III , div . G, ~ecs. 17 and 22 of l-i.erodynamic l 'f't.'.eory , 1!' . r . DLJ.rar..d , ed ., Julius Springer ( Berlin) , 1935 .

Schultz - Grunow, F .: for S1:1ooth Plate s .

Kew }~ic~ional Ras!stance Law NACA 'rl: No. 986, 1941.

nitzbers , Gerald E .: A Concise TheoretIcal ~ethod for Profile - Drag Calculation. Nl-l.CA ACR No. 4B05, 1944 .

Holt , .aur:ice : Calculation of ',tiinS Profile Drag. Air craft Engineering > vol . XV, no. 176 Oct. 1943, pp . ·'278-280 .

VOlJ. Doenhoff , Albert E ., a:c.d ':'etervin, Neal: Deter­mination of General Relations for t~e 3ehavior of 'Pl'rbulent 30unda"'y Layer s . rACA. t.CR. No. 3Gl3, 1943 .

Fal\ner, V . ~- .' A ?Iew I,&'v for Calculatins Drag. ~he Pesistance of a ~moot~ rlat Plate with ".'urb 11er..t Boundary Layer . l'.irc:r'aft :::r.glneering, Vol . XV , no . J.69 , I'~arC(l 191~.3, p;> . 65-69.

von Doenhoff , Albert c: . ~ In~Testigatj on of the Boundary Layer about a Sy.ulietric8.1 Airfoil in a Wlnd Tunnel of Lo\" Tur'Julence. NACA ACR, A.ug . 1940 .

CONFIDEl'lTIAL

NACA ACR No. L4G14 CONFID ENTIAL Fig . 1

Figure 1.- Skin-friction coefficient To/2Q as a f une tion of He obtained by equation (2) .

Figure 1.- Concluded. CONFIDENTIAL

CONF i DENTIAL

;- ~-~lfnTI 1_C r~ .,. ti,

F:"=i ~ :~ I j ',

T

-t--"1'-----1 'H

Cf4 f-++i- Ict::i- 'i= "~-=b£i 1--'

. I:-Gf ~, '" I I cf4 I I ,= I , lit, ~' I I~~' " ~k 1+_E:fI ~ I cit l" 'I flil$ L: "1tl}Un l=r~g--- I~I ,

~~ -d l'-mlT 1 ~1 1¥i llWl'WlJr l;;oohi h'1 If2.)-l- Ii 11 . 1 + I~i

I#lW;td;.; I I~ I~ I I. I 'li f1 lH t 1~l=±TI~; _ I -pg "O~~L I'\UP t9°1'

t-- v I E$:IT~I-:-l mltl1f" 11 1 1 l'.3:=ffi-r~ 1 4 I x;.qu&1~p~lTlfOl11~ERJJ~~m,--s- , .-.::r<::T7

t--- ti--I4JJ 1-.1 III 1. IWI~rtt ltr~ I*IOOLl1 I '~ , i 'f I , I -. lifT J l:f 111 • I , .1 Iml 1t! · I' 1'!'IW lrJ I , 1 :~' l ll; +

I ! :-i _:-:;vc. ~: I '. I¥· -'1 il

, "t I I p~~t~~:~"Tr 1IIIII flillt Vt' . II I I ~ .oor-

I I · ... llll~~~~L

r r , 1

iF'" /' F --

..-ooe8 t;'n"A~_"""'" II? \ j

~Jt~ ". I •. ~~ I ~n

11:1 11 I I -i-+ g I fH ... IHf:hidi:l l I ... 11 ~

TI ...

~~- _ ~ I j - -:1:' T ~-Tu~T'1!

~ " I j lP I ~ I ip I il5 1 <P pliP lilltill ill ; +- t--I-~.-~llJlW.-IJJI ,-~ft I I j ~_JwLI I IR~I I , 1 ,1 I I ~)

I I

Figure 2.- Plot of Falkner's equation and equation (2) against Re . CONFIDENTIAL

_._----- _. ------

z » 0 » » 0 ::0

z 0

L' ,p. Q I-' ~

'%J 1-"

OQ

l\:)

a/c

.0024

o Exper1lllental (reference 8)

f--tNATIOINAL JVISOI -- calcula ted from pipe

skin-friction relation -- Calculated from Falkner' s COMMITtEE FOR AEROI kin-fr iction re l ation

.0020

J l

j' .0016

}' VI . 0012

)0

f I tD

l£': .0008

.0004

~ o o

V ~

.2

L r~

~

---~ ~ r---.. r---- 'l'rjna1yon Piint

.4 .6 .8 1.0

a/e

(al R, 2.675 x 106 •

Figure }.- Experimental and calculated momentum thickness on the NACA 0012 airfoil .

,

I

CONFIDENTIAL

y

AUT1C~

a/c

.0024

. 0020

. 0016

.0012

.0008

• oo~

0 Experimental (reference S) -- Calculated from pipe

skin- friction relation -- Calculated from Falkner I II

akin-friction relati on

)

I /~

II )tD

I'l' I- &

/~ e

7 N TIONAl ~VlSOR1 COMM ~EE FOR AERONA TICS .

7 f--r- r--_ 'l'rana1t1on point

---..--

I I I o o .2 .4 -' .8 1.0

./0

CONFIDENTIAL (bl R, 3.780 x 106.

Figure 3.- Continued.

I I

I

z >­(")

>->­(")

::0

Z o

L' P o f-' P

'XJ .......

OQ (J)

(J.l

III

cr

I

CONF IDENT IAL 0 Experimental (reterence 8)

-- calculated trom pi~ skin-triction relation .0024

0 Experimental (rererence 8) -- calculated trolll pipe

akin-triction relation .0024 --Calculated trom Falkner'.

skin-triction relation -- Calculated trOIll Pallmer'.

akin-triction relation

a/c •

0016

1 1 1 1 1 lit 1 If 1 1 I/c .0016

.0012, .0012

.e0081 I f I .0008

I , I

1/ Je

II L

/ 1;

/ (

IA TlO~'Al AOVISOII'

.0020 I

'f

r, / II

1. / /

/ " , /

k:' NATIONAL ADVISOR~

.0020

H . _n °IUneS 1 .,,1 jCOMMITTEE t OR AERONAUTICS .0004 1 I / I I . ()~

/ ft r,OMMITlEE FOR AERON~

V /

/ CO MMITTEE t OR AERON~

,/

o o ---I--I----

.2 .4 .6

s/c

(c) R, 5.~50 x 106 •

Figure ~.- Cont1nued.

'l'rantitioln POijt I

I .8 1.0 0

CONFIDENTIAL

/ I - /. 'l'rajiUOr pOit -...---.2 .4 .6 .8 1.0

./e

(d) R, 7.560 x 106.

Figure ~.- Concluded.

Z :J> (")

:J>

:J> (")

::0

z o

r ,j:::.

o I-' ,j:::.

'xJ 1-"

OQ

CJl

VJ ()

a.

[ I

(~)2

CONFIDENTIAL 1~6

"""" r----.

1.2 / ----I----I----t--. {

......

------~ t--... ~

.8

.4 NAT PNAl AI VISORY

~MMITI foE FOR; RONAUl CS

o o .2 .4 .6 .8 1.0

'SIc CONFIDENTIAL

Figure 4.- Velocity distribution for NAC~ 0012 airfoil at a = 0°. (Taken from reference 8.)

z ~ o ~

~

o ~

z o

L' ,p. o f-' ,p.

'XJ 1-'.

OQ

,p.

NACA ACR No . L4G14

5 .• 0

I' ~

+ 4.0

1/ \ /

I I

1.0

0

CONFI DENTIAL

) I __ (~~t184

~ ~

NA TlONAl AO~ COMMITTEE FOR ~ f.

~U)4.017 ~ V Uo

~r---r---

""" ...........

F i g . 5

ISORY RONAUTICS

0 .2 .4 .6 .8 1.0

sic

(U )(H + 1)(n + 1) + 1 Figure 5.- Curves of - for the Uo

NACA 0012 airfoil a t a = 0°.

CONFIDE NTI AL