ACR No. IAF16 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED June 19hk as Advance Confidential Report lAFl6 EXPERIMENTS CK DRAG OF REVOLVING DISKS, CYLINDERS AND STREAMLINE RODS AT HIGH SPEEDS By Theodore Theodorsen and Arthur Regier n 1 Langley Memorial Aeronautical Laboratory Langley Field, Va. UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT ) MARSTON SCIENCE LIBRARY P.O. BOX 117011 ^ iMQ A GAINESVILLE. FL 32611-7011 USA NACA 1 i 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 Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/experimentsondraOOIang 3SOOI7 TVT / AC A ACR No. IijElb NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ■CRT - 1 REVOLTING DISKS. CYLINDERS E RODS AT HIGH SPEEDS Dre Theodorsen and Arthur Pegier ■ \RY An e; ' ental investigation concerned primarily with the extension of test data en the drag of revolving disks, cyl* lers, ' streamline rods to Lch numbers and Reynoli \ ■ ' ,;bers is presented. A Mach num- ber cf 2.7 was reached for revolving rods with Pre on 115 as the medium. The testa lisks ided to a ynolds number of 7,' . 00. Parts oi the study are devoted to a rec Lon of the von Karman-Prandtl logarithmic resistance law and the Ackeret-Taylor super- sonic drs hitlers f or their valid! b; . The tests cc i , Ln general, earlier theorie d add certain new results. A ' I of first importance "s that the skin friction dees not depend on the Mach num- ber. Of interest, also, are experimental results on revolving rods at very ' h Mad ers, which show drag curves of the familiar from ] "sties. A new result w] Lc general applicability is that the effect of surface roughness involves two distinct parameters, particle size and particle unit density. e particle ::iz^ uniquely determines the Reynolds num- ber at which the effect of the roughness first appears, reas the particle unit density determines the behavior of the drag coefficient at higher Reynolds numbers. Beyond the critical :n Ids number at which the roughness effect appears, the t?rag coefficient is found to be a function of unit 3 snsj -. In the limiting case of oarticle "saturation," or a maximum density of particles, the drag coefficient remains constant as the Reynolds number is increased. CONFIDENTIAL NAOA ACR !fo. Ihg-'lo THTS0R5 T T : AL BACKGROUND Von Karman- Prandtl Theory for pipes Measurements of the value of the skin friction between a fluid en'- 1 a solid constitute one of the means for studying the nature of turbulent flow. Most of the pioneer analytical work in this field is found in the papers by von Karman (references 1 and 2) and Prandtl (reference J). The treatment used in the first part of this section follows the work of Prandtl which, in turn, is closely related to the von Karman papers. The theory, which concerns the flow in pipes, is given in con- siderable detail as it forms the basis for the succeeding discussion on flat plates, cylinders, and disks. The theoretical work in this section constitutes mainly an attempt to analyze and organize earlier work found in many scattered articles. Considerable work along such lines has already been done tj Goldstein, who is responsible for an expression for the drag on revolving disks . The von Karman- Prandtl theory for flow in the turbulent layer is based on the following two assumptions: CI) The racio of the velocity deficiency to the friction velocity is a function of geometric parameters only. (2) : cent to the wall, but beyond the laminar sublayer, the slope o^ the curve representing this ratio Is Inversely proportional to the distance from the wall. The constant of Droportionality is a universal constant. The friction velocity is defined as Vj ' V and the corresponding friction length is defined as (All symbols used in this paper are defined in appendix A.) CONFIDENTIAL "■ ' - '16 CONFIDE A refe rence bj ' ' as u u i" = T Jt V \ • "trie con ' ' ns for a pipe are •' m by one leter, the r ' i a. ' ' blinder 'inite lei 'epresents anot] - irameter case, lr ■ the re "eience parameter is the radius of the cylinder. be wi '. " - e Form and, by adopt] litabl? le fined me an values with (t tc tire, at a given profile u wl" 1 ^ " ite such mean velocity. measuring :it with respect to a velocity U In a fixed gee n c = k , u - a ill- t (1 .\ In id. AtDout 1/;>C von ' ir m shewed that for the t layer bhis uncti n is essentially independent of L and c Lent o. : y n the try as indicated in assumption (1); therefore u - U c (1) Ls quite remarkable relati : . which has been generally " : ' * Kikuradse, i ! at ', and 3 (re' a T - to 7)* implies a similarity in t- '..-Id pattern i 'r the walls at all old£ 3rs . The basic reason for this similarity Lknc wn . COHPIDEH! Ij. CONFIDENTIAL NACA ACR Mo. Lqi'16 It follows fron assumption (2) that near the wall U T V.L/ 1 - I L log ~ + Constant where 1/k is the constant of proportionality. (Natural lor.ir-.tlm has beer, used throughout except where otherwise indicated.) Since u = U= at y = 6, this relation reduces to U - T T g 1 T T T K log This logarithmic relationship holds to a certain value c of the significant parameter a (see fig. 1), where c = ka with k a constant. The value of 1 - k is only a small fraction, so that the point c will be relatively close to the wall. The velocity in the center of the pipe is therefore given as the sum of three expres- sions, that is, i y.-a ax tj q 1 c Rl)l For the laminar sublayer L '5 _ 5 U T " L " a and the equation may be rewritten as T ^H= a -± log a +1 logf + i log^ + [>m] K k L 3 a L W J 1 a °1 + k l0g I + C 2 >a 1 a w> ere 1 C- = a -*— log a CONFIDENTIAL ja acr No. T.4FI6 CONFIDENTIAL r 3 . f UJ 1 , c + 7 lo S a • constant C] is equal to the nondimensional velocit^r me as r ' or the logarithmic velocity pro:" .hen this 1.3 extrapolated to y = L, and the constant G£ is see ss veloc y in the ter of ' pe as com- it of the logarithmic line extended to -: . • Mg. 1.) /Vhen these constants are combined, 5l] ng general relation is obtained: " U_ n + — 1 K cirm-l pes given by t bo : lting flow near a wal 1 onl; . excess veloc it y tr Lbution £ t a depends on I re r sal c t ] e , , :■■ bo eases other than is r 3d to . trie c< ■. 'ations le iter. is interesting to th G] and 1/k are universal constants scond as3 Lon - namely, that the ■ function of the distance from the secon .stunt C w] Lc jives t the logarithmic dis- reference point, the location of which geometra .j involved, is not a but is dependent on the configuration of reference 1". I . irface roug 33 may be treated in a similar mann . If bee r ess parameter e/L is less than a certain magnitude, there is obviously no effect at all. of e /L is found experi- mentally to be 3.3. For — > ; to be c :i s t an t , or ent so-called unsat ' ' t on later. Thus ••3. Uniax/ U T i3 shown of ~°> max ■ j t 'or the • : . ] '. e 'e fined - x = c + iiog3.: ' r K = C + 7; log 3.3 + 1 i a — log — or Umax 1 i a = K]_ + - log - Tr K -"-Of ONFIDENTIAL C CONFIDENTIAL NAG A ACR No. LI4PI6 The velocity distribution is exactly ?s if there were a laminar layer present of a thickness 6 ~ y . c . e 1 1 or as ; f the length L were ?. When L < e, the velocity c is tribi bion no longer changes with an increase in Reynolds number R. It seems, therefore, bhat the distance from the wall of the innermost disturbance, or the mean value of the thickness of th= lai inar layer, is of the order of three to four tines the height of the irregularities or the grain size e This fact 13 not inconsistent with the physical interpretation . quantity U raa x/l T T ^ s shown to equal W77-. Further, L = -2- TJ ax TT T T •-max L t and, therefore, a fD L ' •'• V 2 where R is referred to the maximum velocity and is equal to %ax a A ■ '- 1 - 6 equation '~rib.x 1 a = + - leg 7 TT_ K & 1 lay thus 'be written /! = . W 2 h — log R 1 /— or 1 1 i ;.~ - Cz + 1 log R JCn \/C D ' K ^2 V L C CONFIDENTIAL NACA ACR No. lijpl6 COMFIDEETIAL - re 1 , ~ \i -Ice V2 " 5 - t tl s : ; -.ili?'t- hypothesis, t v e mean velocity In a '-'•-■ Jiffers from the :imum value by a constant, or ' jT max _ U T JL 2 whore T 7 n 13 the mean valus ~" ■ velocity. Prandtl for the value cf ■:>. (See reference 3, p. iLl.) Note further th I - product R\/Cd remains the S; ~- cher R and Cd refer to the mean or the Lue of the velocity.* therefore, 1 pT) — = C - 1...C7 + - log Rj— and, finally, with B and Cq referring to the mean velocity, 1 11 ,— ■ - -j log p./: D V 4 * & -*re CI i^.07 - - log \/'2 K \/2 ' r :, C = 5.5 and k = O.J . This value is not accurately established, as the various authors aeem bo differ. in order to obtain the drag formula for flat plates, a calculation similar to the von Karman- Prandtl treatment CORFIDENTI i 3 CONFIDENTIAL MCA ACR No. li|Pl6 for pipes may be performed. The velocity deficiency Au is given by the relation u Tm where TT Tm is a mean value between and x, the distance along the plate. The missing momentum may be written as M = /ptf (a - *) f 4y or f u l Au ^l/iu\ £ pu J Q J Q where U is the stream velocity and 5^ is a significant length giving the thickness of the boundary layer. Rewritten, this equation becomes V _ u Tm ■ o- — ^itAMW^mfiik) or, by virtue of bhe similarity law, M _ U Tm _ / TJ Tm\ 2 « „ — - = 6 X C5 - ) P1C5 pu 2 u V u J Since the momentum is given directly as M = Ipu^Cj^x the following identity is obtained: 1 2 °Dm x or - / C Ira o „ ^Dm = y~ 5 i c 5 - ~ 6 i c 6 c Dm 5 1 C 5 = o C Dm* ( x + T c ^) CONFIDENTIAL NACA ACR No. li|Pl6 CC FT v/hicfc -:ves »1 ] . / I)m V 2 X % °6 /^ c 5 V (Jsing ^he logarithmic deficiency relation rives for Cc the value — , or 2.5, and for C^/Cc the value — , or 5; .us 61 1 ■ • 5 , _ _^ \i 2 3y u ' von Farma'n-Prar.dtl treatment, the stream u.ty is obtained in esser.tiallj same form as for pipes. With '-.-..all adjustments, .fore, - K* + — lc - TBI By use of the = 3sion for 61/x, the following equa- tion is obtained; D -4= = Kj, + t.07 log 1Q ' A , , Local Values of Draj . efficient for Flat Plates It may te noted that a relation for the local drag coefficient or a flat ' y :>e found in a fashion similar to that used ! for r . disk. Consider a plate of unit width; for the full length I, r ? -L U C Dx (iP u2 )^ COM IDENTIAL 10 CONFIDENTIAL NACA ACR No. I;. r ?l6 Vith the subscripts m and x referring to mean and local values, respectively, Tor the length x, or u x _ :oU^ = r Dx dx 'Dm x + Crm - Cdx dx x Ru T J Therefore where •Dm J^~ + u Dm °Dx d_R R C Dm d(log G Dm ) [ d(log r) = o C Dx = G Dm( n + 1) n = d ( lQ S c Dm) d(log R) Boundary Relation for Revolving Disks The moment coefficient is defined as Cm = M - p a2 a 5 CONFIDENTIAL TJACA AOK No. Ii+Fl6 CONFIDENT IAI 11 - also be written ao- ■re Uj. Is the variable radial velocity and u t is the tangential velocity, from which or f u ™\ 2 £i " V a7 a" C ' 6l — s Constant a The drag formula t'-.er roads T" „ A similar result was obtained by Goldstein in reference h. RESULTS Teits on disks, cylinders, and streamline rods were conducted to determine drag or moment coefficients. For the cylinder the two coefficients are equivalent; for the disk and the rod it is more convenient to employ the moment coefficient, which can te measured directly. In order to extend the range of hach number, several tests were conducted with Pre on 12 or Pre on 113 as the medium. The test results obtained are of technical interest because some of the data, particularly foi the high Mach number range, were obtained for the first time. It mar be pointed out that many of the earlier tests on CONFIDENTIAL 12 C0N7 T DENTIAL blACA ACR Mo. lliFl6 revolving disks and, In particular, on revolving cylinders were conducted on a rather small scale and in a limited range of Reynolds number. It may be noted that a con- siderable range of Re:/nclds number is generally needed in order to confirm with sufficient reliability a par- ticular theoretical formula. For instance, it may be impossible to obtain a measurable difference between logarithmic or power formulas if a short range of Reynolds number is available. This matter of distin- guishing between the various types cf formulas is of theoretical interest. Experiments on Revolving Disks The moment coefficient is defined as M ~ 1 -po> 2 a5 This definition corresponds to the one for laminar flow on a revolving disk given by von Karman in reference 1 as : n M — a 1 n 1 n 2 where R coa 2 The constant ai used by von Karman was 1.8i|. for one side or 3.68 for both sides; this value was later adjusted by Cochran (see reference 8, vol. I, p. 112) to a^ = 3.07. If this corrected value of ai 'is inserted, the formula for laminar flow reads 1 C M = 3.C7R _ 2 The turbulent- flow formula as riven by von Karman for revolving disks is _1 Or = 0.l!i.6R 5 In figure 2 are shown the experimental results for tests of a series of revolving disks. The Reynolds CONFIDENTIAL NACA ACR No. Tl|Pl6 CONFIDENTIAL 15 -■..• 1 ;ed from about 1600 to rrore than 1,000,000. Note that the test points lie along the theoretical curves given by ;;he von Ka^rman formulas. The transi- tion from laminar flow is seen to occur at R = 310,000, T v " s was the largest value reached with the most highly oolished disk. The thickness of the laminar boundary layer is, according to von Karman, or, which is equivalent, 5 - ^ 2 Using R~ = leads to u 1 — = - = 2.50R - R a For the transition Reynolds number, 310,000, Rg = 2.53 J~ = il^o which is of the same order as the minimum critical value obtained '"or oir>es. Several tests were conducted for the purpose of investigating the factors affecting the transition Reynolds number. The first observation was that the transition Reynolds number could not be increased beyond the value 510,000 no matter how ' the surface was polished ^r whatever other precautions were taken. Like- wise, it v;as unexpectedly Jifficult to decrease the transition Reynolds number. T' : 1 plication of coarse i (60 mesh) glued co the surface of a disk (1-ft radius) only reduced the transition Reynolds number to about 220,000 (fig. 2). The reduction in the transition Reynolds number by Initial turbulence was also studied. A small high-pressure air jet applied near the center of the disk produce c ; greatest observed reduction (fig. 2) and brought the transition to a point near the CONFIDENTIAL lij. CONFIDENTIAL MCA AGR No. liji'lo intersection o f the lines representing the drag formulas fcr laminar and turbulent flew, which is the absolute minimum. Note that the drag in the turbulent region is quite appreciably increased by surface rorghness. • The values of the moment coefficient given in figure 2 represent obviously an integrated drag over the disk. An axpre salon may be obtained for the local drag coefficient Cp x as a function of local Reynolds number as follows; M = C r ,/ipto 2 a5j '\2' f a = Z f C DxQp~ 2r2 ) ( 2Trr2 ) dr = Cm -pa-- a 7 nl Jo f °-(ff fl $ r Qj ivi a "7r\ + 5 °M = ^Dx By s ub 3 1 i t u 1 5 n g =*£ l c ^-- 5 2tt dF + ^ °M = C Dx CONFIDE ITIAL T To. li\Fl6 CONFIDENTIAL or : nd ] ': .) . il : " _ «a(iog r) ; + t iTT 3 re If tl en = d ( lQ s QmJ d(los R) n CR _ ! + 2n„ arr n use of the expression for log C rx , some of the data of figure 2 are nlotteo in figure 3. Although the general picture does not chai luch, the abrupt nature of the transition becomes apparent. An illustrati u ary-layer profiles for vario\is radii rs is given in fig- ure [j., in which curves of equal velocity u;-/oor ar - also plotted. Note that t] ■ thicknes i cf the bound- layer in the laminar region is essentially constant. The transition ■ R, $10,000, is shown approxi- - 1 t , tl e Line . asition" in fig- ure l r . The nominal 1 rr.inar boundary-layer thickness consists] anoears ' rhat in excess ol' that given by / : Karraan In reference 1. There appears to be some discrep ;he t i tical velocity stribution whic] Ls 0' 000, 000. The — newer law holds 7 fairly well in the observed range which, however, is toe limited to ^crmit a distinction between the power haw and the logo '.thmio lav; .or the velocity distribu- tion.. T-e main nuroose of the tests, the results of w] Lch fire shown in figure 5 , was to investigate the effect of the Mach number . The first run taken with air as the medium extended bo a Reynolds number of about 2,000,000 are a "'ach number of 0.62. By using Freon 12 a3 the medium, the rang of Reynolds number was extended to 7,00 3.000. at bhe lowest pressure, the higtesi vz 1 le of bhe Mach number reached was I.69. All the data for Preon 12 show a slightly higher drag than that given by the von Karman formula, apparently becaase of some systematic error. The significant result of this investigation is that the drag coeffi- cient 'S absolutely independent of the Mach number. A seoarate extension of the experiments to a Mach number of slightly more than two further confirmed. thi3 independence of the Mach number. Experiments on Revolving Cylinders The experimental results for revolving cylinders are shewn in figure 6 as a plot of log-inCr* against 1 °Sl0 li J where P. = --— . drag formula f or laminar flow on a revolving cvlinder is obtained from Lamb CONFIDENTIAL MCA ACR No. 1I4FI6 CONFIDENTIAL 1? (reference Q , p. 5S8) * s v;here R D = £ qS _ _f_ q3a In this formula 5 is the surface area aid a the radius. In this case it is convenient to use Cn instead of Cjj, which was used for the revolving disk, because no Integration is involved. The laminar curve is shewn in figure 6. le drag relation given bv 7== "°- 6 + : '-°? lo Sio R \/^ V 1 for the turbulent flow is also shown in figure 6. experimental results are replotted in fig- ure 7, where — — is shown as a function of l°SlO^ |/^D* T ' ie re l at i° n f° r the turbulent flow l=r -- -0.6 - U-07 log 10 R\/cJ V"f appears in figure 7 as a straight line. The coeffi- cient Cp in this formula corresponds to a value of O.lj. for von Karma'ii's universal constant k. The relation for the laminar region. Gri = aopears as ..— •-■ R .a -cu^-vad -line near the origin. It is noted that the drag coefficient for rough cylinders is dependent on the relative grain size f/a, 3 re e is the size of the sand and a is the radius 0^ the cylinder (see fig. 8), arid bhat for each grain size the drag coefficient remains constant and CONFIDENTIAL ic CONFIDENTIAL NACA ACR No. li|Fl6 Independent or" the Reynolds nusiber beyond a certain minimum or critical value, which lies on the line for turbulent flow. In regard to the nagnitude of the drag coefficient as a function of relative grain size for particle "saturation" of the surface, it may be remarked that the value of 6 is a measure of the thickness of the sublayer or, what amounts to the same thing, a measure of the minimum grain size of the turbulence. It is therefore to be expected that the surface roughness will become effective at the Reynolds number for which c cr > the critical value of e, becomes less than the grain size e . Inversely, it may be seen that, if the Reynolds number becomes smaller than this critical value, the grain size of the turbulence is too large to be affected by the surface roughness. With £ greater than e ~ r , which is 3.51, the following relation is approximately true for the drag coefficient beyond the critical Reynolds number for surface roughness of saturation density: -£= - -0.6 + 4.. 0? log lc 5.3\/T^ - 2.12 + J+.O? log 10 f In figure 9 the p-xcer?" mental points are shown to satisfy this theoretical relation with sufficient accuracy. Tests were made to determine the effect of the density of spacing of grains of a given size, and the results are presented in figure 10. Such tests were made with a certain unit grain size but with the sur- face density in grains per square inch varied between c -0 and 2200. The grain size uaed corresponds to the size — = 0.03, also used for the preceding experi- a mental results shown in figure 8. It is verified that the critical Reynolds number depends on the grain size only, and it is further shown that the slope of the drag curve beyond the critical Reynolds number is a function of the density. A saturation condition evidently always exists, in which the drag coefficient remai] proximately constant and equal to the critical value . CONFIDENTIAL NACA ACR To. li|TTl6 CONFIDENTIAL 19 xperimsnts on Strer.r-'.lins Rods In figure 11 results are Lven ^or 1 certain more or less streamline bodies, each tested in tv;o or more different mediums. The tests were obtained by using actual propellers of 12- inch diameter, which are designated propellers B and C. Propeller 3 had a jjection of double symmetry with a circular-arc contour line. Propeller C was obtained by reducing the chord of propeller B by removal of about one-fifth of the chord near one extremity to obtain a blunt-nose air- foil. By running propeller C backwards an airfoil with a blunt trail! ;o could also be studied. The drag coefficient used in fi 11, 12, and 13 is the standard torque coefficient used for propellers C~ = 5 pn 2 ■ ' 1 airfoil B, a value of the Mach numt t ne was reached in air, the range was extended to 1.6 in Freon 12, and the characteristic decrease in I 'ficient was finally reached In Freon l n 3- LderabTe decrease '•. Irag coeffi- it was not 3d at the largest Macb number, 2.7, which to the knowledge of the authors is the highest Mach number reached except for a few cases of projectiles. The blunt-nose airfoil ceoti.cn C showed approxi- mately - : resistance as the symmetrical irp-nose section B but had a maximum torque coeffi- cient very much 'r excess of that of section B. The test extended ir the peak of the torque curve wit Dn 12 1 the medium. By reversing the direc- tion of ' r propeller C to obtain a blunt rear, the expected large : . se in drag at low Mach numbers s observed. • ' le ference Ln Reynolds number for air .. ; ! is a; difference in n • coefficients ir the ran^ce below a Mach number of unit; . )r higher Mach numbers, the drag coefficient of the section with the blunt rear lies between the drag coefficients of the doubly streamline section una the blunt-nose type; the stream- line leading edge is approximately twice as effective as the streamline trailing ec >, a result in general agreement with earlier observations. It should be noted, however, that the lowest era:; is obtained with both le?dii md trailing edges streamlin Note that t ich numbers used in fi 11, 12, and 13 are based on the tip radius. CONFIDENTIAL 20 CONFIDENTIAL NACA ACR No. L4FI6 The effect of the Reynolds number is also shown in figure 12, which jives the results of tests to study how the scale effect is superimposed on the Mach number effect. It should be noted again that the Reynolds number effect appears only for a Mach number below unity. A wide variation in the Reynolds number shows no consistent measurable effect on the drag for a Iv'ach number greater than unity. Similar data for a small angle of attack, instead of zero angle of attack as used in the preceding discussion, were used in one case, for which results are given in figure 13. The four propellers referred to in figures 11 to 13 are shown in a photograph (fig. lip) and the dimensions of the propellers are given in table I. It is of some interest to interject a superficial analysis of the results presented herein, in view of Ackeret's formula as given by Taylor (reference 10). For the local section Ackeret gives the drag coeffi- cient as 1 G D = 2^ ; l) 2 ^ 2 + Pi 2 + P 2 2 ) where the bar indicates the mean value. For ze ro angle cf attack and a symmetric section with ]? 2 = p*2 , this relation becomes 'v 2 1 p 72 For a circular-arc section 3^ - —3 2 where 6 is yraax ' -max the maximum angle. This angle is, in turn, approximately equal to twice the thickness ratio t, which is the total thickness divided by the chord. For circular-arc sections, therefore, 1 !d = £{5§ _ iV-2 t< Figure 15 shows Cp. "lotted against Mach number for different values of t. At II = 1.0, the curves tend erroneously to infinity. This effect follows CONFIDENTIAL - lb CONFIDENT!/ 21 from a s" : ] ig as 3nir.pt ion used in the derivation of " ke 'et's formul . using the general form f(M) instead of the . bar function — - 1, the drag coefficient nay he written a 3 f ( '-■■ ) The torque coefficient is known e-perinentally to be a function of the Mach m , or l/x]_, where xj_ is the fraction of radius at which the Mach uu-nber is unity; thus, th ] .ving integral relation is obtained: 1 2 iH*)%) ire are several • andling this relation. The and th s s t -nay be taken to r3ores2nt a nrefsrred section at aeproxi'':;at3ly 30 percent of the radius. By assuming an initial dr coefficient Cq any desj ■ .ccurac- ie cbcainsd ] beration methods . The function f(M) shown in figure Id has been propeller B by such a process hased on the experimental data gSven in figure 11. Note that ■ coefficient approaches the value given b; 3t for la for large values of M, for Lch f(M) 'aches (h~ - 1) IT. rther that i j Lmum " lue of the dr Lent occurs M = 1, " I ( alncst exactly equal to unity. It is. of rs not tc he func- tion f(M) 3 as jeneral \ by? on i3 given hers for propeller 5 for ; se of com- paring the ciata tfith the Acherec ir3o\ . :o: t clul t i rks Experimental results on the drag of revolving disks have been presented, which substantiate to a CONFIDENTIAL 22 C0K5TEI -""■ : " C . . to, ] remarkable degree 3rag formulas based on the von Karman- Frandtl theory i* skin friction. The range of the investigation was extended to a Macs number of 1.69» which is beyond the range of any earlier test, and to a Reynolds number c± 7,000,000. It was established that the skin friction is Independent of the Mach number up to this value and appears to be a function of the Reynolds number only. 3 drag at supersonic speeds was studied with revolving rods or propeller sections. Mach numbers as high as 2." .'/ere attained in the tests. The drag at supersonic speeds is a function of the Mach number only, as it appeal's to be essentially independent of both the Reynolds number and the nature of the medium. The ^cteristic peak in the drag 3urve observed for projectiles was P tained. For thin streamline bodies, this peak appears at Pack numbers only slightly beyond unity; in fact, it apoears at a Mach number of about 1.2. >ys1 smatic tests were conducted on stream- line bodies with combinations of sharp and blunt leading and trailing edges for the purpose of obtai ' ; bhe relative merits of such features. It was found that the increase in bhe peak value of the drag coefficient resulting from a blunt nose Ls about twice that resulting ?vr.r^ a blunt trailing edge, when both drag coefficiei bs are compared % with the drag coefficient of a section with streamline leading and trailing pages, which has the lowest value. Significant results were obtained en revolving free cylinders !"or which references to earlier tests seer to be lacking. It was J '< and that, at very low yno] Is numbers, the drag asymptotically approaches the laminar drag oi the classical theory whereas, at higher Reynolds numbers, bhe drag is found to conform to a logarithmic formula of the von Karman type. There is no distinct transition from laminar to turbulent flow, as is found in pipes end on revolving disks. The flow Is essentially turbulent down to the smallest Reynolds numbers. The effect of initial txirbulence was particularly studied in connection with tests of revolving disks. It was found that the transition Reynolds number was very slightly affected. The critical Reynolds number at which the roughness effect appears depends en particle size only and is not a function of particle CONFIDENTIAL 3A ACR ?T o. LhFl6 CONFIDENTIAL 2J isitTt Beyond tM.a value of th ynolds number, 'icient is constant only when the surface Is "saturated," that Is, when the density of the il particles attains a maximum value. For a roughness of leas than this particle density, t ,g coefficient decreases with Reynolds number. It is interesting further to note the persistence of the logarithmic relationship, '/.'hen l/yCp is plotted as a function of log R\JCj) (where Cj) is the drag coefficient and R is the Reynolds number), the lines representing turbulent flow are invariably straight A rather critical demonstration of the logarithmic velocity pattern near the surface is thus shown. The range invest!,- is of considerable extent. Langley ' morial Aeronautical Laboratory tional Advisory Committee for Aeronautics Leld, Va. CONFIDE i\ITIAL - ': t . i C. 2i| CONFIDENTIAL NACA ACR To. 14FI6 APPENDIX A IJ T friction velocity t shear per .in it area at surface p was s of air oer unit volume XL. m mean friction velocity (from to x ) i m U stream velocity for flat plates T J,.„ aJ . max imum ve 1 o c i t y U m moan velocity (in pipes) U c reference velocity (at a given fraction of radius or of other reference dimension) Ug velocity at 5 u absolute variable velocity of fluid in boundary layer Au velocity deficiency, stream velocity minus local velocity for flat plates u r radial velocity for disks U£ tangential velocity for disks w angular velocity, radians 5 thickness of Laminar sublayer 5]_ boundary- layer thickness L friction length fu/U T ) 7, total lengt'n of nlate T reference time (L/l T T ) t time j also, thickness ratio for propeller section, thickness of airfoil chord 3A ::Cl. No. lifPl6 CO] 2S u coeff" ■ of kinematic viscosity H coefficient of viscosity r 'iable radius of pipe, disk, or propeller a lius of pip 3 , cylinder, or disk; also, velocity of sound in fluid x dis ;ance from leading i of flat plate in direction of flov/: also, fraction of Dropeller r radius (x = — -"here ?. denotes radius of propeller tip) Xn ction of propeller radius at which Mach number is unii j ] vi: ■ :• normal to surface no v nal profile constant l'o^ turbulent K - ow near walls / c \ o fra ■ of reference dimension \— = k): also, a chord nondimensional chora of airfoil, ■ radius a angle of a cf airfoil; also, profile constant ( S/l) C-q total-dr* ; ^efficient (Many authors use f, v, or t- instead of C-> for pipes) Gjvp mean drag coefficient (from to x) Ci) X local drag coefficient D dra -; also, propeller r .r D,. drag of plate (from to x) e grain size cf roughness e cr , grain size of critical roughness for particular value of drag coefficient moment coefficient for revolving disks missing momentum; moment :~or disks; or Mach number nrMtfTPTTrtriOTTA i 26 confidential naca acr eo, l fi6 R Reynoldc number Eo- Reynolds number based on thickness of boundary layer R x Reynolds number based on distance from leading edge of flat plate or on local radius of disk R^ Ee;>molds number ba°ec! on pipe diameter R a Reynolds number based on pipe radius v velocity (Ackeret formula) q dynamic pressure (for cylinders, q = r-o- &■ ) S area of cylinder "3, torque coefficient (Q/pn'D^/ 3, t or que N number of blades n rotational speed, revolutions per second; also, coefficient in power law p-j. j i^2 angles which upper and lower surfaces of airfoil make with center line Pjnax maximum angle which circular-arc section makes with center line Cj nondimensional velocity measured on logarithmic velocity profile when this curve is extrapolated to y = L Cj nondimensional excess velocity at y = a over that of logarithmic line extended to y = a C = C]_ + Cp_ Cz,C],, ... constants K]_,K2*Ez, • • .constants k constant &]_ constant in equation for moment coefficient of revolvi .;• disks CONFIDENTIAL 1TACA LCR No. ll\Fl6 CONFIDENTIAL 2T> x AP] ' B • L VALUES CT POWER ENTS FOR DISKS Al D CYLINDERS A chart is presented (fig. 1?) which gives the 'sepower required to drive a smooth dick in standard Lr (76O mm and 1S° C, p = 0.00233 slugs/cu ft and u = 0.000159 ft^/sec). Lines of constant horsepov/er ranging in value from 0.01 to 1000 are plotted with disk rotational speed (in rpra) as abscissa and disk meter (in ft) as ordinate. The dashed line in figure 17 represents a Reynolds number oT about ij.00,000, Lcb is con?, id ere.} the transition Lds number. The foil ' ■ formulas were used to calculate the power for disl rating in the turbulent region: (-pa%3j '.'<*:• = Cm 1 C M = C-.ll4.6R ' _1 _2 _1 . , x ^a ^c 5 = o.ih.6 - — " v 1 '5 Horsepower = 550 = 0-346 pO.8^.^2. 8^0.2 550 x2' Inasmuch as the formula for Cm is based on the —power for velocity distribution, the calculated values of Cjfl are tco lovi for high Reynolds numbers. This error may become appreciable for the highest power, since the chart (fig. 17) covers a range of Reynolds numbers to 60,000/000. CONFIDENTIAL 20 CONFIDENTIAL MCA ACR No. 1J4PI6 A chart is also presented (fig. 18) which gives bhe horsepower required to rotate a smooth cylinder of unit 1 (1 ft) in standard air. The following formulas have been used in calculating the curves: *o fvlo = Cr^qSaco CT-pa> 2 a 2 = 2rra — ? , aw 2 Horsepower = C D Trp a 4a)3 where, for smooth cylinders, -L = -0.6 + k. 07 log 10 R/^ v -m: i - = 2.12 + i|.. ; log 1f ^ V C D /q- ' ' -^ e COKPIDiSHTIAL 3A ACR No. ll\Fl6 CONFIDENTIAI 29 APPENDIX C i [ jti;d skin-friction formulas S SIDE) Symbols e following r; nbols are used in the formulas for flat elates collected herein; C D >ti (oefflcient local drag coefficient at point x x distance from leading e^i[;e of flat elate in directic ow I length of flat plate in direction of flow R ids number eased on I R x Reynolds number based on x Laminar Flow le formula for total drag coefficient C~ = 1.328R : is based on the simplifie Lrodynsnaic equations developed by Prandtl in 19 C\. (See reference 2, p. 2.) The 3C - Lc 'ate J 1 Lasius in 1908 as I.327, was calcu! • in 1912 as 1.323. (See reference 3* P. 89.) I '•■ • tula for local drag fficient is r - - = . x 2 CONFID 30 CONFIDENTIAL TACA ACR No. li|Fi6 Von an, Scboenherr, and others have indicated that, ir the total drag coefficient is Cj ~ Constant I R n ;he local drag coefficient is given as C D:: = (n + 1) : Jr> T] is r lation is derived in the section entitled "Local Values of Drag Coefficient for Flat Flates" ir this paper. All formulas giver L i b] Lc i p idix for the local drag on flat plate are in confer* ■■' Ltb bhis lei- 3 vat ion. ■anient "" 1 ow formr.] jinooth Surface 0.074a •> i i no. j. Ctw = 0.059^ were first calculated by von ;. : l in 1920. (See references 1 and ?..) sed on results from pipes and on tie --power law -"or veiociby distribution, they are co!:' I Lid Ir ere lover* Reynolds nunber range, R < 10,CC0,00C, So] e writers i ! follow!] fc u] s of bhe same tyre, - > ere ft Irly ^fceurate ! o] . r u j r of 500,000,000; 1 r; Ox ~ ' ° ■ "' •: "' ■ LAL I 3A ACR No. L':-?l6 confidential 31 Of more general validity are the so-called loga' 'ithmic drag formulas of the tj — = I4..I5 log 10 RC D >rm of this relation was determined by vcn Karraan constants ad;j\isted to conform with data by Schoenherr and cth< 1 . (See reference 2, ,?. 3.2.) In the present paper a different form has been developed, which is in at stricter theoretical conforr.it" with the physical relations i:v/clved? :c D J== I4..O7 log lor — X - ,.• . Prandtl has developed an explicit expression v;hich r es essentially the same results as the logarithmic formulas. ft -2.53 C D = 0.!i55(log 10 R) (See reference 3, p. 153«) .ocal drag coefficient has also been given by von Karma'n in a logarithmic form bhe constants adjusted to fit the experiments of , ich incluaea measurements on small movable plates inserted on a long pontoon. This formula is -= =1.7+ li.15 log 10 R x C Dx vGdx (See reference 2, p. Yl.) CONFIDENTIAL 52 CONFIDENTIAL MACA ACR No. lifFl6 Turbulent Flow - Rough Surface Sch.3ich.ting (see reference 3, p. ?o2 ) gives the two following formulas for the total and the lo^al drag coef- ficients for rough flat plates, respectively; -2 , S C D = (I.89 -i- 1.62 log 10 f ) c Dx = (2.07 + lob iog 10 -; Von Karman (reference 2. p. lS) gives for the local drag coefficient for "owg'a surfaces a formula of Che loga- rithmic type — — = 5.8 + U-.15 log 10 -v C Dx vc Dx e PIPES Symbols The symbol R^ used In this section refers to the Reynolds number based on the pij 3 diameter and the mean flow velocity, and the symbol R a refers to the Reynold: number based or pipe radius. Some writers use f or y instead of Cp, vised herein, and others use X where ; ■■.. i vir Plow For laminar flow in piper the formula for dreg coef ficient is C D ~ ~ s formula is attributed to Poise*ille and Wiedeman. (See reference 3, p. 38, and reference 8, p. 298.) CONFIDENT CAL ;a acr Ho. i'. f ?i6 cc- cal 33 nt "lev - Smooth Surface ula for drag coefficient for turbulent flow in smoot] is -> , = 0.079% ^ La is based or. the experimental work of Blasius T , p. loo), Voi' ■:; the Reynold:: number mited. "er work by Nikuradse (ref- f ) extended the range of Reynolds number to a muc her value. The folio-.. La of the type developed by von rma'n fits the ^ctter: -±= = -0.1+0 + I4..OO lociC^dv'^D v c D (See reference '.' , p. 333.) In the present paper a formula of this type iifferent constants is developed: = O.I4-O + U..C7 logioR a -/CD Turbulent Flow - ' -race 'or turbulent flow in rough pipes — = 3.L.6 + L.00 log 10 - The experimental work in deriv -"..is formula was done by Nikuradse. (See reference , . 38Q and reference 6.) co:.Tir" 3I4. CONFIDENTIAL NACA ACR No. LljFl6 REVOLVING DISKS Sy.no els The following symbols are used in the formulas for r e vo Iv ir:r d i. sks ; n : ,r moment coefficient V-- local drag coefficient at radius x> R v Remolds number at radius x [— j A Laminar Flow for laminar flow ana C M = 5.^?R L'TN ... U/ ;x n This formula for local drag coefficient is derived from the relation 5 + 2n L.TT For the development of this relation and for references. see the section entitled " Experiments on Revolving Disks" in this paper. Turbulent PI ow 17 or turbulent flow _1 C M = O.lliiR 5 CONFIDE NTIAL NAOA ACR No. L ! ;Flo CONFIDENTIAL 35 and C D: , = 0,O53R x 5 formula for the local drag coefficient C- is derived fro:n the equation for the moment coefficient Cj v j in the same way as for the case of laminar flow. The local coefficient in logarithmic ^or:. nay be given as — - 1 — = -2.05 + 1^.07 logiolWQD x constant -2.05 has been adjusted to fit the data of figure 3« IG CYLIND For laminar flew D R ~ ! or turbulent flow on smooth cylinders _. = -0.6 + k,07 lon; 10 R-/C^ For turbulent flow or roi b c; ] u lers 2.1 + L.o ior ]0 7 VCd The development of these formulas and the reierences are given in the section entitled "Experiments on Revolving Cylinders . i( CONFIDENT U 36 CONFIDENTIAL NACA ACR No. L4FI6 REFERENCES 1. von Karman, Th.: IToer laninare und turbulente Reibung. Z.f.a.M.M., Bd. 1, Heft k, Aug. 1921, pp. 233-252. 2. vcn Karman, Th. : Turbulence and Skin Friction. ■Tovir. Aero. Sci . , vol. 1, no. 1, Jan. 193^4-j pp. 1-20. 3. Prandtl, L.: The Mechanics of Viscous Fluids. Vol. Ill of Aerodynamic Theory, div. G, l¥. F. Durand, ed., Julius Snringer (Berlin), 1935, PP- 3V-208. k. Goldstein, S.: On the Resistance to the Rotation of a Disc Immersed in a Fluid. proc. Cambridge Phil. Soc, vol. XXXI, pt. II, April 195S pp. 232-2J4.I. 5. Nikuradse, J.: Gesetzmassigkeiten der turbulenten Strbmung in glatten Rohren. Forschungsheft 35^> Forschung auf dem Gebiete des Ingenieurwesens, Ausg. B, Bd. 3, Sept. -Oct. 1952. 6. Nikuradse, J.: S.trb'raungsgesetze in rauhen Rohren. rschungsheft 361, Eeilage zu Forschung auf dem Gatiete des Ingenieurwesens, Ausg. B, Bd. h, July- Aug. 1953. n 1 Wattendorf, P. L. : A Study cf the Effect of Curvature on Fully Developed Turbulent Flow. Proc. Roy. Soc. (London), aer. A, vol. llj.8, no. 865, Feb. 1935, pp. 56^-598. 8. Fluid Motion Panel of the Aeronautical Research Committee and Others: Modern Developments in Fluid Dynamics. Vols. I and II. S. Goldstein, ed., Oxford at the Clarendon Press, 1938. 9. Lamb, Horace: Ilvdro dynamics . Sixth ed., Cambridge Univ. Press, 1932. 10, Taylor, G. I.: Applications to Aeronautics of kckeret's Theory of Aerofoils Moving ab Speeds Greater Than That of Sound. R. & M. No. lij-67, British A.R.C., 1952. G r '. ] . D IMTIAL UACA ACR No. Lip 1 16 CONFIDENTIAI 37 TABLE I DIMENSIO] ? I? p r ' OR REVOLVING RODS FOR 1 RS T/.H propQller3 have a straight taper in chord and thick- tipa are rounded as shown In fig. lJ+.J Prooeller desig- nation 1 Airfoil " bch section i ( 4- Circular arc Blunt nose Circular arc Circular arc At SO percent radius Cho^J (in.) 1.75 1.30 .83 Thickness (in.) 0.31 •35 At 92 percent radius Chord (in.) ,18 1* 1.07 .82 I 1.03 c 1 Thickness (in.) O.1I4. ±1 • 07 a proooller D was twisted so that approximately the outer If f the olade had an angle of attack. iTIONAL ADVISORY COMMITTEE FOR AERONAUTICS CONFIDENTIAL NACA ACR No. L4F16 Fii Figure 1.- Parameters and functions of the velocity profile by the von Karma n-Prandtl theory. NACA ACR No. L4F16 Fig 1 ■ i r . • qz it ■- ■ ■ a Eg ' t : ' ; i": 1 : | «f' :f £_ i J & * ■ ■ ■ 1 £ ; i 8 a {- - J it c a u n a o -* 41 . 4 2 B a ■ • -< E ■ It i 3 I ^ h li O 24 a * o -h ■ h 1 c c c a J± -« ^ ^ hja^ • • 4 4 -. -t o c t» -o c o o . . ° e, B a a c m 3 m a d ~< -* Q I I O + x D O "j K J ^ ! — -- _fi * jS ^ 11 /P Mi - a l thi A 4 4h « :jffj ; ft ! i r ut. r o >i k-' It ' i gr :.{ • -,j : 5 ft ^ : - • * - £ a => ° 3 £ o > a 1 < rf f ' \\ ^ : i 5 ■ - h- 7 1 * -1 £- £ - 1 : ■ : 1 I ' < at - 1 9 3 to H 1 / ! I I Z UJ O £ - (- t- j fc _! / ± u 7 -- i : o ? - - 4 ■ — -4 -6 i < ? 5 * V < > < ■ - •■ 1 v j 1 1 J i 1 i t < 3 - - 1 1 i i 1 NACA ACR No. L4F16 Fife -< u > t- a 2 ° t H| SO 2 o // ■ Jt <1 -> P < "- Z UJ o £ II O f c it I C sO t rH -=J c o vc irl J II •u * ' t \ o S — ; 1 a vc j ? c ; 1 ' ~ / sj -I / O u o jt / / JD 11 i / \ / 8 / / a 1 ' V h / as 1 / 1 / lr i 3 A I / V o 1 i / ■ 1 v : ... / A- vc IT / \ / ^ 1 *> ■^ ~ / uu t 'i li - 1 i — / J i- r j / U" K / 1 i / / •/ b H 1 ■_j ? IV 1- / # If . ^ / // i / y :> / / o .* / . / il- , * 1 ► J / / !fc *i f // CO c .1 // S? J i (i'i 7 \ \, r ^ ^ y // -1 M >// ! s£] // r -3 ' # t // I X / '/ : J // / -J /i / — / / 1 ' / // txi / / ' J / / i / / / ~ c D / C / c i * > H" f "^ \ M 1 P c 3 < * C c i - ' rr i 1 i ■ 1 ■ i (<.*' r "7 1 1 if < - I ►> « NACA ACR No. L4F16 Fig. i M^ nTfT x:.]- : i ■T- — — u A « M 1M vO (\| C\ no cr* _d- so O H rH O O O O *} *} ^ JJ _4 O K\ t— N 4 UN C- o ■ °> W n H o t« • ft, c o O _^ IT. CD rH n S Air Preon 12 Preon 12 Preon 12 r-l O t 0) o + x □ ~t~ — .. _l MIT f— ■ > u n .u m *§ •a x o a « • n 33 b 43 CO a> « u » -4 <~ U HH O a> .o e a o a i o l/N 0) f> 3 ;l a NACA ACR No. L4F16 Fife. 6 U fiu .-•KM rflnj ,~i|(\l <-i|c\l i*\U rH|rvj ,-i|<\j icjoo rtvOvO OJ O + X O O > O + X< 4 i 4 NACA ACR No. L4F16 Fig. 9 Log 10 | Figure 9.- Drag parameter — : for rough-surfaced cylinders as function of log 10 — . NACA ACR No. L4F16 Fig. 10 I — - — | ■ — — 3 f ~- ■1 ~ 1 u- | + o S n I t !M x J Sf f» fa * 8 u X J X) c a si a Q e i o - 9 a ■d « C C (. •-( *4

£c ■ur.. J o o o c ' ?*"« ^ O OS H H\ O B K\ P- (\l ■Vi ^ + v< ■u >* v ..-L j ? i K- ES3 ::.' $ i ll r— i © j i r- u 6- /] CJ h- ° 3 £ o > a 1* -I O < "" Z u; O £ K h o ^n \ \ I - — — * \ J j frt ij t 1 5 B f 3 1 3 O tt a ■ i -< ■ ^ -1 ( * <: 3 ' 6 3 } 1 J ► -" ■ " - •— V NACA ACP No. L4F16 Fife. 11 I i ■ OlSlj i&i A i~ - - - / fti r / i U * : i le kfli / X X i NATIONAL ADVISORY k Tv s v luhpiiiilc. ruK aekviiwiiu uo J *\ < : tLLe^- », 1« M 3 \ _»sa rp ed Sfi Rf I? -r * » X j * rf V I 1 - / ( - ~- «£. &+ 1 i V rt^-Ot jsTT'n, il ' 1 t* "j^ 4- < / s \ ■ / f / c - > o \ »T B r -u ^ ?\1 c f ' J o ' i (>\ ■> k // 1 *k- X Pr"eon 12 ) C • 3har P ed ^ leading C?5^ D Preon 12 "| O Preon 11} > E, symmetrical circular- <- ^ V Air J arc airfoil ? s JL °] TO r* l<* kj O -- rf> a i 9 #F — Jli 3 ftf rf" f* 3 2 'i a id -iV IP — 4 ' fe 1 - I i '"' .t .f= .f 1. Q_ T. 2 1 ^ fc i, £- ju r. ■ n p 4- 2^ ft ? c 1 n Hfc Sli nui -.fa 1 1 Figure 11.- Torque coefficient C Q = * pn 2 D^ as function of Kach number for propellers B and 0. NACA ACR No. L4F16 Figs. 12, 13 1 1 1 : i . .' N > i- O H a = O 2 »t ir~~ a % - - -Jt ••4 v o < * at A r-t - 1 9 - \ Is ^ Z tu O £ 1- t- * o 1 3 ' o z X o y j ■ >_> i 5 * / - r eg a o O X 1 g o a i,| .1 --- -- • r- rr- — c < 3 r i s 5 I * j r < 5 ? <>" 3 3 c . > 5 5 c 1 -- 1 r .... ... • . 1. '. CO CM l~ jj o o O fc, ® c ■" i» b < £ fi li a O o c— A . *r» fc» 9 C "• w +■ X vise ERON/ o <«< 7 - nj - 1 J? i CO < u - Z UJ / % H O £ f J 1- t- < 7 1-1 2 * ' v N 3 O i ■ f-i J •%. rvj ' *» ~> t ,-H C * *-, -** v., o o ^ K * "6 * rn , ? • y P ( O -; 7 5 3 p f V D 1 c g - > r^ c i < ) < J> i *; ' 1 II NACA ACR No. L4F16 Fig. 14 w •d C o OQ o I u NACA ACR No. L4F16 Figs. 15, 16 Figure 15.- Theoretical curves of the drag coefficient C D against Hach number for various thickness ratios for circular-arc airfoils by Ackeret's formula. :; ; A* - ■ ' 1 ;nfei»>»Jtlaa'. r Tl i ■ 1 . 4 i ■ t : : l- : ■ • t -~i M ,M^~- NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS. B f" - i 1 ; 1 ; ■■ ■!■■ \ ' " " j'-" {' ! ! \ : " \ :'.'. tr . 5 h s 1 _£ rv \ . \ J V I f >s _j__ h> 1 ' JBx T"»l lis ftnl «i ' 4 ■r 1 ) 1 •4- 1 / " t .2 4 .6 .: !i. ii. 2 1. h nu i. 6 1. e 2. ? ? ? il ? 6 ?,f i— — — u L-l J --J. -fc- msiL mbe V, K = Figure 16.- Values of drag function f(ld) as function of Mach number from analysis of experimental moment curves for propeller E In figure 11. NACA ACR No. L4F16 Fig. 17 Horse 1000 _ 500 V - \ ioa 50 10 J 1.00 •}o .10 - .0} .01 \^\ \\ ^ y N s s > x V N \\ \ V s. s V . V s, s, V, V v -\ \ N s V \ , s S V V . \ S L V s L \ \ V N "•« s \ \ ^ V s s. "\ "-v X s s s ^ \ X •> • H v ~~. \ k; \ \ '»> V \ s Dluitir, ft ** ^ r " *■ \^ Tr»naltl< to turt n from ulent lftml •low nar * % >^ s \ -s V x * ^ V X, s ■< s X N . s % 5 ■ \ \ N s ^ \^N s ^5 <^ 1,000 10,000 40,000 Rot.tlon.1 .peod, rpm NATIONAL ADVISORY :0MMITTEE FOR AERONAUTICS Figure 17»- Power requirement for smooth disks . NACA ACR No. L4F16 Fig. 18 Horsef 1000 500 - - 100 JO 10 5 1.00 .jo .10 I .05 .01 ower \ 100 . X V \ X \ s N K V X \ Z\ s ^ s N \\ \ S > v X X V V s. v \ "s V s 1.. s "^~ X N « ^ \ s V V X S k «i \ v \ 'X \ \ V V s s r S k X \ \ > \ 4 x s s s \ v S X X \ X X \ k^ X s \ N N x\ Dl«m«t«r, ft S XX N s. \ ■\ \ s. N ' I \ v N V \ ■■. X V x, \ s \ \ X X X \ X, v V^ xZi N, N \ s \ \ X \ 1,000 10,000 1*0,000 NATIONAL ADVISORY Rotational apeed, rpm COMMITTEE FOR AERONAUTICS Figure IB.- power requirement for smooth cylinders (1-ft length)- UNIVERSITY OF FLORIDA ■ mi II Hill Hill '|| 3 1262 08106 540