1、TECHNICAL MEMORANDUM 1292 C3 (S3 c7 l-i 24 b 4 NATIONAL ADVISORY COMMITTEE 3 FOR AERONAUTICS LAWS OF FLOW IN ROUGH PIPES By J. Nikuradse Translation of gStromungsgesetze in rauhen Rohren.“ VDI-Forschungsheft 361. Beilage zu “Forschung auf dem Gebiete des Ingenieurwesens“ Ausgabe B Band 4, uly/August
2、 19 33. I I I ! i I i $ i I i t f i j 1 1 i Washington November 1950 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONAL ADVISORY COMMITTEE FOR iZERONAUTICS TECHNICAL I+EMORANDUM 1292 By J. Nikuradse INTRODUCTION Numerous recent investigations (
3、references 1, 2, 3, 4, and 5) have greatly increased our knowledge of turbulent flow in smooth tubes, channels, and along plates so that there are now available satisfactory data on velocity distribution, on the laws controlling resistance, on impact, and on mixing length. The data cover the turbule
4、nt behavior of these flow problems. The logical development would now indicate a study of the laws governing turbulent flow of fluids in rough tubes, channels, and along rough plane surfaces. A study of these problems, because of their frequent occurrence in practice, is more importamt than the stud
5、y of flow along smooth surfaces and is also of great interest as an extension of our physical knowledge of turbulent flow. Turbulent flow of water in rough tubes has been studied during the last century by many investigators of whom the most outstanding will be briefly mentioned here. H. Darcy (refe
6、rence 6) made comprehensive and very careful tests on 21 pipes of cast iron, lead, wrought iron, asphalt-covered cast iron, and glass. With the exception of the glass all pipes were 100 meters long and 1.2 to 30 centimeters in diameter. He noted that the discharge was dependent upon the type of surf
7、ace as well as upon the diameter of the pipe and the slope. If his results are expressed in the present notation and the resistance factor X is considered dependent upon the Reynolds number Re, then it is found that k according to his measurements A, for a given relative roughness - r varies only sl
8、ightly with the Reynolds number (k is the average depth ,d of roughness and r is the radius of the pipe; Reynolds number Re = u- v in which ti is the average velocity, d is the pipe diameter, and v is the kinematic viscosity). The friction factor decreases with an increasing Reynolds number and the
9、rate of decrease becomes slower for greater relative roughness. For certain roughnesses his data indicate that the friction factor h is independent of the Reynolds number. *“tr$munsesetze in rauhen Rohren. “ VDI-Forschungsheft 361. Beilage zu “Forschung auf dem Gebiete des Ingenieurwesens“ Ausgabe B
10、 Band 4, ul/ust 1933. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-For a constant Reynolds number, h increases markedly for an increasing relative roughness. H. Bazin (reference 7), a follower of Darcy, car- ried on the work and derived from his o
11、wn and Darcys test data an empirical formula in which the discharge is dependent upon the slope and diameter of the pipe. This formula was used in practice until recent times. R. v. Mises (reference 8) in 1914 did a very valuable piece of work, treating all of the then-known test results from the vi
12、ewpoint of similarity. He obtained, chiefly from the observations of Darcy and Bazin with circular pipes, the following formula for the friction fac- tor h in terms of the Reynolds number and the relative roughness: This formula for values of Reynolds numbers near the critical, that is, for small va
13、lues, assumes the following form: k The term “relative roughness“ for the ratio - in which k is the r absolute roughness was first used by v. Mises. Proof of similarity for flow through rough pipes was furnished in 1911 by T. E. Stanton (reference 9). He studied pipes of two diameters into whose inn
14、er sur- faces two intersecting threads had been cut. In order to obtain geometrically similar depths of roughness he varied the pitch and depth of the threads in direct proportion to the diameter of the pipe. He compared for the same pipe the largest and smallest Reynolds number obtainable with his
15、apparatus and then the velocity distributions for various pipe diameters. Perfect agreement in the dimensionless velocity profiles was found for the first case, but a small discrepancy appeared in the immediate vicinity of the walls for the second case. Stanton thereby proved the similarity of flow
16、through rough tubes. More recently L. Schiller (reference 10) made further observations regarding the variation of the friction factor X with the Reynolds number and with the type of surface. His tests were made with drawn brass pipes. He obtained rough surfaces in the same manner as Stanton by usin
17、g threads of various depths and inclinations on the inside of the test pipes. The pipe diameters ranged from 8 to 21 millimeters. His observations indicate that the critical Reynolds number is independent of the type of wall surface. He further determined that for greatly roughened surfaces the quad
18、ratic law of friction is effective as soon Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TM 1292 3 as turbulence sets in. In the case of less severely roughened surfaces he observed a slow increase of the friction factor with the Reynolds numb
19、er. Schiller was not able to determine whether this increase goes over into the quadratic law of friction for high Reynolds numbers, since the 8ttingen test apparatus at that time was limited to about Re = 103. His results also indicate that for a fixed value of Reynolds number the friction factor k
20、 increases with an increasing roughness. L. Hopf (reference 11) made some tests at about the same time as Schiller to determine the function X = f Re - . He performed system- ( :) atic experiments on rectangular channels of various depths with differ- ent roughnesses (wire mesh, zinc plates having s
21、aw-toothed type surfaces, and two types of corrugated plate). A rectangular section was selected in order to determine the effect of the hydraulic radius (hydra.ulic radius r = area of section divided by wetted perimeter) on the varia- tion in depth of section for a constant type of wall surface. At
22、 Hopffs suggestion these tests were extended by K. From (reference 12). On the basis of his own and Fromms tests and of the other available test data, Hopf concluded that there are two fundamenta.1 types of roughness involved in turbulent flow in rough pipes. These two types, which he terms surface
23、roughness and surface corrugation, follow different laws of similarity. A surface roughness, according to Hopf, is characterized by the fact that the loss of head is independent of the Reynolds number and dependent only upon the type of wa.11 surface in accordance with the quadratic law of friction.
24、 He considers surface corrugation to exist when the friction factor as well as the Reynolds number depends upon the type of wall surface in such a manner that, if plotted logarithmically, the curves for X as a function of the Reynolds number for various wall surfaces lie parallel to a smooth curve.
25、If a is the average depth of roughness and b is the average distance between two projections from a the surface, then surface corrugation exists for small values of - b a and surface roughness exists for large values of - b A summary of the tests of Hopf, From, Darcy, Bazin and others is given in fi
26、gures 1 and 2, the first illustrating surface roughness and the second surface corrugation. Hopf derived for the friction factor k within the range of surface roughness the following empirical formula: in which r is the hydraulic radius of the channel 2F (r = F; F = area of cross-section; U = wetted
27、 perimeter). This formula applies to iron pipes, cement, checkered plates and wire mesh. In the case of surface Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-corrugation he gives the formula in which Lo is the friction factor for a smooth surface a
28、nd 6 is a proportionality factor which has a value between 1.5 and 2 for wooden pipes and between 1.2 and 1.5 for asphalted iron pipes. The variation of the velocity distribution with the type of wall surface is also important, as well as the law of resistance. Observa- tions on this problem were ma
29、de by Darcy, Bazin, and Stanton (reference 9). The necessary data, however, on temperature of the fluid, type of wall surface, and loss of head are lacking. In more recent times such obser- vations have been made by Fritsch (reference 13) at the suggestion of Von kt the resistance factor X increases
30、 with an increasing Reynolds number. This transition range is particularly characterized by the fact that the resistance factor depends upon the Reynolds number as well as upon the relative roughness. Within the third range the resistance factor is independent of the Reynolds number and the curves X
31、 = f(Re) become parallel to the hori- zontal axis. This is the range within which the quadratic law of resistance obtains. The three ranges of the curves X = f(e) may be physically inter- preted as follows. In the first range the thickness 6 of the laminar boundary layer, which is known to decrease
32、with an increasing Reynolds number, is still larger than the average projection (6 k). Therefore energy losses due to roughness are no greater than those for the smooth plpe . In the second range the thickness of the boundary layer is of the same magnitude as the average projection (6 Z k). Individu
33、al projections extend through the boundary layer and cause vortices which produce an additional loss of energy. As the Reynolds number increases, an increasing number of projections pass through the laminar boundary layer because of the reduction in its thickness. The additional energy loss than bec
34、omes greater as the Reynolds number increases. This is expressed by the rise of the curves h = f(e) within this range. Finally, in the third range the thickness of the boundary layer has become so small that all projections extend through it. The energy loss due to the vortices has now attained a co
35、nstant value and an increase in the Reynolds number no longer increases the resistance. The relationships within the third range are very simple. Here the resistance factor is independent of the Reynolds number and depends only upon the relative roughness. This dependency may be expressed by the for
36、mula Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-In order to check this formula experimentally the value - was plotted fi in figure 10 against log and it was found that through these points k there could be passed a line The entire field of Reyno
37、lds numbers investigated was covered by plot- 1 r v*k ting the term - - 2 log - against log 7. This term is particularly 6 k suitable dimensionally since it has characteristic values for conditions along the surface. The more convenient value log Re 6 - log might k be used instead of log - v*k as ma
38、y be seen from the following considera- v tion. From the formula for the resistance factor the relationship between the shearing stress T and the friction factor X may be obtained. In accordance with the requirements of equilibriun: for a fluid cylinder of length dx and radius r, or from equation (1
39、) ro in which v, =E is the friction velocity. There results and Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-log(Re fi) - log T k = 10(.66 y) V*k r log = const + log(e G) - log i; From equation (5) there is obtained: 1 r - - 2 log - = 1.74 f-x k I
40、t is evident then that the magnitude of constant wi-thin the region of the quadratic law of resistance but within the other regions is variable depending on the Reynolds number. r The preceding explains why the value log(Re fl) - log - was used as k the abscissa instead of log(e fi) as was done for
41、the smooth pipe. Equation (58) may now be written in the form 1 - - 2 log r = f log - fi k ( v:k) There occurs here, as the determining factor, the dimensionless term which is to be expected from the viewpoint of dimensional analysis. The relationship - - 2 log r = f log “ 6 k ( as determined experi
42、mentally is shown in figure 11 (see tables 2 to 7) for five degrees of relative roughness. The sixth degree of relative roughness was not included because in that the assumption of geometrical similarity probably did not exist. It is evident that a smooth curve may be passed through all the plotted
43、points. The range I in which the resistance is unaffected by the roughness and in which all pipes have a behavior similar to that of a smooth pipe is expressed in this diagram (fig. 11) by the equation Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
44、1 - - r 6 2 log - k = 0.8 + 2 log (9) in which the value of a fhnction f is determined by equation 8. The fact that the test points lie below this range is due to the influence of viscosity which is still present for these small Reynolds numbers. This indicates that the law expressed in equation 3 i
45、s not exactly ful- filled. The transition range, range 11, is represented in figure 11 by a curve which at first rises, then has a constant value, and finally drops. The curves to be used in later computations will be approximated by three straight lines not shown (references 19 and 20) in figure 11
46、. The range covered by the quadratic law of resistance, range 111, in v*k this diagram lies above log = 1.83 and corresponds to equation (5a). These lines may be expressed by equations of the form 1 - - r vk JX 2 log j; = a + b log v vk in which the constants a and b vary with % in the following man
47、ner : vk 1.83 These expressions describe with sufficient accuracy the laws of velocity distribution and of resistance for pipes with walls roughened in the manner here considered. Finally, it will be shown briefly that the Von K here within the range of our experhents the exponent for an increasing
48、relative roughness increases from 1/7 to 1/4. Equation (25) may be written in another form if the velocity and the distance from the wall are made dimensionless by using the friction velocity v,: in which, according to equation (25), n = 117. Then log 9 = log C + n log q Provided by IHSNot for Resal
49、eNo reproduction or networking permitted without license from IHS-,-,-If log 9 is plotted as a functidn of log 7 there results a straight line with slope n. This relationship is shown in figure 18 for various degrees of relative roughness and also for a velocity distribution in a smooth pipe. All of the velocity distributions for rou
