1、REPORT No. 793 EXPERIMENTS ON DRAG OF REVOLVING DISKS, CYLINDERS, AND STREAMLINE RODS AT HIGH SPEEDS By TODOEE TFLBODORBN and AE- RDQIER SUMMABY An experimentid investigation concerned pimarily with the e therefore u- u, -=fa U, (z) 367 Provided by IHSNot for ResaleNo reproduction or networking perm
2、itted without license from IHS-,-,-. I . .- 368 REPORT NO. 793-NATIONAL ADVISORY COMhLT?TEE FOR AERONAUTICS This quite remkable relationship, which has been generally confirmed by Nikuradse, Wnttendorf, and others (references 5 to 7), implies a similarity in the turbulenbfield pattern nmay from the
3、molls at all Reynolds numbers. The basic reason for this similarity remains unknown. It follows ram ssumption (2) that near the wall =I K log $+Constant where l/ is the constant of proportionality. (Natural logarithm haa been used throughout except where othenvise indicated.) Since u= Ua at y=6, thi
4、s relation reduces to This logarithmic relationship holds to a certain value c of the signi6cnnt pameter a (see fig. l), where c=ka vith k a constant. The value of 1-k is only a small fraction, so that the point c will be relatively close to the mall. The velocity in the center of the pipe is theref
5、ore given as the sum of three exprssions, that is, For the lnminar sublayer ua 6 u. E=a - and the equntion mny be rewritten as la =a+; log E+“* whero and 1 C,=a- log a K c;= f I! +-log; c (a): : The const.ant C1 is equal to the nondimensiond velocity measured on the logarithmic velocity profile when
6、 this curve is extrapolated to y=L, and the constant Cr is the exw velocity in the center of the pipe 8s compared with that of the logarithmic line extended to y=a. (See fig. 1 .I When these constants axe combined, the following general relation is obtained: qy =“+; la log E The application of this
7、theory to caw other than circular pipes is restricted to geometric configurations given by a single parameter. It is interesting to observe that both Cl and 1/ are universal constants resulting from tho second assumption-namely, that the flow near a wall is a function of the distance from the wd onl
8、y The second constant C; which gives the excess velocity na compsred wit11 tho logarithmic distribution at a reference point, the locntion of yhich depends on the geometric dimensions involved, is not a universal constant but is dependent on the configurntion nnd the choice of reference length. The
9、effect of surface roughness may be treated in a similnr manner. If the roughness parameter e/L js less than a cerhin ma-dtude, there is obviously no effect at d. This vnlue of a/L is found experimentally to be 3.3. 3.3, U UT - or The velocity distribution is .exactly ns if there were n laminar layer
10、 present of a thickness 6c3.5 or ns if the length 1 1 L were - E- When L the velocity dishibution no 3.3 longer changes with an increase in Reynolds number R. It seems, therefore, that the distance from the wall of the innermost disturbance, or the mean value of the thickness Q.“ Provided by IHSNot
11、for ResaleNo reproduction or networking permitted without license from IHS-,-,-DRAG OF REVOLVING DISKS, CYLINDERS, AND S!CREAMLIAW RODS AT HIGH SPEEDS 369 of the laminnr layer, is of the order of three to four times the height of the irrcgularitim or the grain size E. This fact is not inconsistent w
12、ith the physical interpretntion. The quantity U therefore, (See reference 3, p. 142.) and, finally, with R and CD referring to the mean velocity, where With 0=5.5 and =0.4, P, 0.4 This value is not accurately established, as the various authors seem to differ. DRAG OF FLAT PLATES In order to obtnin
13、the drag formula for flat plates, n calculation similar to the von KBrmh-Prandtl treatment for pipes may be performed. The velocity deficiency Auk given by the relation where UTm is a mean value between 0 and 2, the distance along the plate. The missing momentum may be mitten as or where U is the st
14、renm velocity and 61 is a significant length giving the thickness of the boundary layer. Rewritten, this equation becomes or, by virtue of the similarity law, Since the momentum is given directly as the following identity is obtained: or Using the logarithmic deficiency relation gives for C5 the val
15、ue I/., or 2.5, and for c6/c5 the value 2/, or 5; thus Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-370 REPORT NO. 703-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS By use of the von KhBn-Prandtl treatment, the stream velocity is obtained in essenti
16、ally the same form as for pipes. With small adjustments, therefore, By use of the expression for for the full length I, With the subscripts m and z referring to mean and local values, respectively, for the length x, or Therefore where BOUNDARY RELATION FOR REVOLVING DISK8 The moment coef6cient is de
17、fined as The moment may also be mitten M=2p (24urup. dy where u, is the variable radial velocity and ut is the tnngen- tial velocity, from which or !=constant a The drag formula then reads A similar result was obtained by Goldstein in referonce 4. . TESTS AND RESULTS Tests on disks, cylinders, and s
18、treamhe rods wore con- ducted to determine drag or moment coefficionts. For tho cylinder the two coefficients are equivalent; for tho disk and the rod it is more convenient to employ tho momont coo5- cient, which can be msasured directly. In order to oxtond the range of Mach number, several tests WO
19、TC conducted with Freon 12 or Freon 113 as the medium. Tho test results obtained are of technical interest bccause mmo of the data, particularly for the high Mach numbor rango, were obtained for the first time. It may be pointed out that many of the earlier tests on revolving disks and, in particula
20、r, on revolving cylinders more conductod on n rather amah scale and in a limited range of Roynolds num- ber. It may be notrd that a considerable mngo of Roynolds number is generally needed in order to confirm with su5- cient reliability a particular theoretical formula. For instance, it may be impos
21、sible to obtain a meosurable difference between logarithmic or power formulas if a short range of Reynolds number is available. This mattor of distinguishing between the various types of formulas is of theoretical interest. EXPBRIMENTB ON REVOLVING DISKS The moment coefficient is defined as M Cx=, 7
22、Pwa6 This definition corresponds to the one for laminar flow on a revolving disk given by von K6rmfin in referenco 1 as: Cdd = alR-In where CLULz E=- 0 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-DRAG OF REVOLVING DISKS, CYLINDEEtS, AND STREAMLIN
23、E RODS AT HIGH SPEEDS -f. 0 %2, 34 371 won Kdrms lamnor-ffow formula, o 24-1k-diam. this value was later adjusted by Cochran (see reference 8, vol. I, p. 112) to ul=3.87. If this corrected value OI al is inserted, the formula for laminar flow rends Ck= 3.87R-“ The turbulenbflow formula as given by v
24、on K:=0.03, also used for the preceding os- perimenhl results shown in figure 8. It is verified that tho critical Reynolds number depends on the grnin size only, and it is further shorn that the slope of the drag curvo beyond the critical Reynolds number is a function of the density. A saturation co
25、ndition evidently always exists, in which the drag co the streamline leading edge is approximately tmice as effective as the streamline trailing edge, a result in general agreement with earlier observations. It should be noted, however, that the lowest drag is obtained with both lading and trailing
26、edges streamlined. The effect of the Reynolds number is tbe D m twisted m that appmdmatdy the ontar half of the blade had an angle It is of some interest to interject a supeficial analysis of the results prssented herein, in view of Ackerets formula as given by Taylor (reference 10). For the local s
27、ection Ackeret gives the drag coefEcient as where the bar indicates the mean value. For zero angle of attack and a symmetric section mith g=x this relation becomes 1 3 For a circular-arc section, p=- prn.2, whom pmo+ IS tlic maximum angle. This angle is, in turn, approrcimntoly equal to twice the th
28、ichess ratio t, which is tho total thick- ness divided by the chord. For circular-arc scciions, therefore, Figure 15 shows CD plotted against h4acli numbor for different values of t. At M=1.0, the curves tond erronoously to idnifiy. This effect follows from a simplifying assumption used in tho deriv
29、ation of Ackerets formuln. . -. - - -.- _ piSare l4.-prOler8 B, 0, D, and E. I FIQE K-Theoretical mes of the drag dolaat CD Ma thus, the fol- lowing integral relation is obtained: There are several ways of handling this relation. The non- dimemional chord c and the thickness t may be taken to roprca
30、ent a preferred section at approximately 80 percent Mach mber.M Flame lB.-Velue of ding function I(M) an innotion of Mach number from anal the function is given here for propeller B for the purpose of comparing the data with the Ackeret theory. CONCLUDING REMARKS Experimental results on the drag of
31、revolving disks have been presented, which substantiate to a remarkable degree drag formulas baed on the von Ktkmh-Prandtl theory of skin friction. The range of the investigation wm extended to a Mach number of 1.69, which is beyond the range of any sarlier test, and to a Reynolds number of 7,000,00
32、0. It was established that the akin friction is independent of the Mach number up to this value and appears to be a function of the Reynolds number only. The drag at supmonic speeds was studied with revolving rods or propeller sections. Mach numbers as high as 2.7 were attained in the teats. The drs
33、g at supersonic speeds is a function of the Mach number only, as it appears to be essen- tially independent of both the Reynolds number and the nature of the medium. The characteristic peak in the drag curve observed for projectiles was obtnined. For thin trsamline bodies, this peak appears at Mach
34、numbers only slightly beyond unity; in fact, it appears at 8 Mach number of about 1.2. Systematic testa were conducted on strenm- line bodies with combinations of sharp and blunt leading and trailing edges for the purpose of obtaining the relative merits of such features. It was found that the incre
35、ase in the peak value of the drag coacient resulting from a blunt nose is about twice that resulting from a blunt trailing edge, when both drag coefficients are compared with the drag coefficient of a section with streamline leading and trailing edges, which has the lowest value. Sicant results mere
36、 obtained on revolving free cylin- ders for which references to earlier tests seem to be lacking. It was found that, at very low Reynolds numbers, the hm asymptotically approaches the laminaz drag of the classid theory whereas, at higher Reynolds numbers, the drag is found to conform to a logarithmi
37、c formula of the von Kh8S type. There is no distinct transition from laminar to turbulent flow, as is found in pipes and on revolving disks. The flow is essentially turbulent down to the smallest Reynolds numbers. The effect of initial turbulence was particularly studied in connection with tests of
38、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 on particle size only and is not a function of particle density. Beyond this value of the Reynolds num- ber, the drag coeflicient i
39、s constant only when the surface is “saturated,” that is, when the density of the individual particles attains a mmimum value. For a roughness of less than this particle density, the drag coefficient decreases with Reynolds number. It is interesting further to note the persistence of the logarithmic
40、 relationship. When l/G is plotted as a func- tion of log BG (where CD is the drag co also, velocity of sound in fluid distance from leading edge of flat plate in direction of flow; also, fraction of pro- peller radius X=B where R denotes radius of propeller tip fraction of propeller radius at which
41、 Mach number is unity distance normal to surface nondimensional prof also, profilo tot,al-d.rag coefficient (Many authors us0 f, mean drag coefficient (from 0 to 2) local drag coefficient drag; also, propeller diameter drag of plate (from 0 to 2) grain size of roughness grain size of critical rougll
42、noss for particu- moment coefficient for revolving disks missing momentum; momont for disks; or iMach number Reynolds number Reynolds number based on thickness of boundary layer Reynolds number baaed on distnnco from leading edge of flat plate or on local radius of disk Reynolds number based on pipe
43、 diamotor Reynolds number based on pipe radius velocity (Ackeret formula) dynamic pressure for cylinders, q=spw2a2) area of cylinder torque coefficient (Q/pn2Ds) torque number of blades rotational speed, revolutions per second ; also, coefficient in power law angles which upper and lower surfaces of
44、 airfoil make with center line maximum angle which circular-arc soction makes with center line nondimensional velocity measured on log- arithmic velocity profle when this curve is extrapolated to y=L nondimemiond excess velocity at y=a over thnt of logarithmic line extonded to y=a constant (6/L) 7,
45、or X/4 instead of CD for pipes.) lar value of drng coefficient 1 ( constants constants constant constant in equation for moment coeffi- cient of revolving disks 380 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-APPENDIX B NUMERICAL VALUES OF POWER
46、REQUIREMENTS FOR REVOLVING DISKS AND CYLINDERS A chnrt is presented (fig. 17) which gives the horsepower required to drive a smooth disk in standard air (760 mm and 16 C, p=0.00238 slugs/cu ft and u=0.000159 fi?/sec). Lines of constant horsepower ranging in value from 0.01 to 1000 are plotted with d
47、isk rotational speed (in rpm) as abscissa and disk diameter (in ft) as ordinate. The dashed line in figure 17 represents a Reynolds number of about 400,000, which is considered the transition Reynolds number. The following formulas were used to calculate the power for disks operating in the turbulen
48、t region: U-llSa-4/6p-1/5 =0.146 /p/s Horsepower=m MU 0.146 ,- os 16 e8 0.2 650x2 a Innsmuch na the formula for Cu is based on the 1/7 power for velocitg distribution, the calculated values of C, are too low for high Reynolds numbers. This error may become appreciable for the lllghest power, since t
49、he chart (fig. 17) covers a range of Remolds numbers to 60,000,000. A chart is also presented (e. 18) which gives the horse- power required to rotate a smooth cylinder of unit length (1 ft) in standard air. The following formulas have been used in calculating the curves: Mo=C 4% Rofofiwml speed rpm prom Ig-Powa reqalrement for mth oyllndars (14 length). 381 Provided by IHSNot for ResaleNo reproduction or networking per
