C
convection heat transfer parameter, between wall and fluid, dimensionless = drag coefficient, dimensionless = specific heat at constant pressure of gas, B.t.u./lb.,-
CP
= specific heat of solid particles, B.t.u./lb.,-OR.
Bw CD
=
O R .
D
DH DR d
ED
f
7k L MP * mb‘ ?izp
n (Shu)DH
diameter of flow passage, ft. hydraulic diameter of flow passage, ft. = radiation parameter, wall to particle, dimensionless = diameter of solid particles, ft. = drag parameter, between solid particles and gas, dimensionless = local friction factor, dimensionless = average friction factor, dimensionless = thermal conductivity of gas, B.t.u./sec.-ft.-OR. = length of flow duct, ft. = loading ratio, mass of solids to mass of air, dimensionless = mass flow rate of gas phase, lb.,/min. = mass flow rate of particulate phase, lb.,/min. = exponent in i.he axial pressure relation, dimensionless = =
=
Nusselt number of flow passage,
(LF), . ,
di-
mensionless = Nusselt number of convection between solid and gas phase,
(y),
dimensionless
= Reynolds number of relative motion between
= = = = =
= = =
solid particles and gas, dimensionless (particle Jug* - U,*/m,d Reynolds number), A!-%* static pressure, lb.f/sq. ft. initial static pressure, lb.f/sq. ft. gas constant of the fluid, ft.-lb.f/lb.,-OR. temperature of gas phase, OR. initial static temperature, O R . temperature of solid particles, OR. wall temperature, OR. characteristic: velocity (2gcTo)’’*, ft./sec.
UQ
=
velocity of gas, ft./sec.
UP
= velocity of particles, ft./sec.
X
=
space coordinate in the direction of motion, ft.
GREEKLETTERS = dynamic gas viscosity, lb.f-sec./sq. ft. I.r = gas phase density, lb.,/cu. ft. PO = particle density, Ib.,/cu. ft. PP e = emissivity of particle, dimensionless = Stefan-Boltzmann constant, B.t.u./sec.-sq. U = wall shear stress, lb.f/sq. ft. rw
~L-OR.~
SUBSCRIPTS g = gas phase 0 = reference condition = particle phase P = condition at wall W SUPERSCRIPT * = dimensionless variables Literature Cited
Hultberg, 4.J., Soo, S. L., Astronautics Acta 11 (3), 207-16 (1965). Lel’chuk, V. L., “Heat Transfer and Hydraulic Flow Resistance for Streams of Hich Velocitv.” Natl. Advisorv Comm. Aeron.. NACA TM-1054,7943. Orr, Clyde, “Particulate Technology,” p. 147, Macmillan, New York, 1966. Schlichting, H., “Boundary Layer Theory,” 4th ed., McGrawHill, New York, 1960. Soo, S. L., “Fluid Dynamics of Multiphase Systems,” Sect. 7.2. 280-93. Blaisdell Publishinv Co.. LtTaltham. Mass. 1967. Sob, S. L.’, A.I.Ch.E. J . 7 (3),-384-91 (1961).’ Trezek, G. J., Soo, S. L., “Gas Dynamics of Accelerating Particulate Flow in Circular Ducts,” Proc. Heat Transfer Fluid Mech. Inst., 1966. RECEIVED for review May 29, 1967 ACCEPTED January 4, 1968 , I
The continued investigation of the accelerating gas-solid suspension was made possible with funds from a faculty research grant while the authors were at the Gas Dynamic Laboratory, Northwestern University, Evanston, Ill.
TURBULEINT FLOW IN ROUGH PIPES J. M. ROBERTSON, J. D. M A R T l N , ’ A N D T. H . BURKHART2 Department of Theoretical and Apfllied Mechanics, University of Illinois, Urbana, Ill.
61801
Surface roughn’ess effects on frictional and temporal-mean velocity distributions in conduit and other flows past rigid boundaries are considered and modes of correlating transitional effects between smooth and rough flow are indicated. From study of air flow in 8-inch “natural” roughness and a 3-inch sand-roughened pipes, frictional pressure drop, near wall velocity profiles and turbulence profiles for pipe Reynolds number range of 1.3 )( lo4 to 28 X l o 4 are presented. These verify transition functions, clarify an uncertainty in regard to pipe factor formulation, and present new information on turbulence structure.
HE nature of turbulent flow near rough and smooth surfaces T i s a long-standing problem of fluid mechanics. Whereas for laminar-viscous flows past solid boundaries surface roughness has no effect (so long as it does not change the general surface contour appreciably), for turbulent flow the nature of the flow is intimately associated with the surface roughness. Knowledge of the nature of turbulent flow along smooth surfaces is extensive and reasonably complete, as t o both the
Present address, Cornsdl Aeronautical Laboratory, Buffalo, N. Y . 2 Present address, McDoiinell-Douglas, St. Louis, Mo.
temporal-mean velocity distribution and the general characteristics of the turbulence. I n many cases the surface over which flows occur are rough and, in spite of extensive studies, there is still much t o be learned from the mean-flow standpoint. As t o the nature of the turbulence appearing near rough surfaces there is very little information. Characteristics of Mean-Flow Occurrences
For conduit and flat-plate boundary-layer flows, surface roughness effects appear in the frictional and velocity profile formulations. I t is conventional to present these two results VOL. 7
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2
MAY 1960
253
For the fully rough regime Nikuradse’s tests yielded the simple relation 1
I
-
___ = 1.14
, I
61
I 7-H
Q7’
I
lo4
dfzh
I20
(4)
and in the smooth regime
D
1 I
I
‘OS
+ 2 log-Dk
R
3omr
= 210g [I
? /-
v Jsmooth
106
Figure 1 . Variation in pipe factor with Reynolds number as indicated by Nikuradse (1 933) for sand-roughened pipes
separately-in terms of the variation with flow Reynolds number of the frictional occurrences, as characterized by the friction factor (or skin friction coefficient) and the velocityprofile effects, as characterized by the pipe factor and/or the shift of the wall-region velocity function-however, these indications are interrelated. As commonly explained in terms of the relative magnitudes of the size of the surface roughness elements and the thickness of the viscous sublayer, rough surfaces do not always act rough. There are three basic regimes of turbulent flow past rough surfaces: flow as though the surface were smooth, transitional smooth-to-rough flow, and fully rough flow in which the frictional and wall-region velocity shift effects are independent of the fluid viscosity. Although the occurrence of these regimes was demonstrated as early as 1913 in the classic paper of Blasius, it was not clearly shown until Nikuradse’s (1933) well known experiments on flow in sand-roughened pipes; Figure 1 indicates this is a chart of his results for the pipe factor, P, as a function of the mean flow Reynolds number, R = UD/V,and relative smoothness ( D / k , where D is the pipe diameter and k is the sand grain roughness size). Here the pipe factor is defined as
(5)
Now, which of these regimes occurs depends upon the relative magnitudes of the roughness size and viscous sublayer thickness. A nominal definition for the sublayer thickness is 6, = 11.6 Y/u,. When a buffer zone is taken t o occur between the regions of predominantly viscous and predominantly turbulent action, an appreciably smaller numerical factor appears. The ratio of k to this thickness is expressed by the roughness Reynolds number
with u, the shear (or friction) velocity defined in terms of the wall shear stress, rtD,as
d:
-
.-
UT
=
=
ud+
(7)
The smooth regime of frictional occurrences for Nikuradse’s sand roughness is found for Rk values less than 3 and the fully rough regime for RI, above 70. A compact mode of coalescing the family off us. R curves for various k/D values into a single curve for a given type of roughness is obtained by subtracting the reciprocal of the square root of the actual friction factor from both sides of Equation 4. Thus we define the function
U
p = -
U1
where U is the cross-sectional average flow velocity and u1 the maximum (center line) velocity is the conduit. This factor also has boundary-layer flow significance, in that
where frough is the friction factor reached by the surface in the fully rough regime. Now the @ function is expected to vary with Rk in a manner dependent only on the type of rough surface. One such relation was propounded by Colebrook (1939), for surfaces of “natural” roughness, in the form
p = 1 - - 6* 6
-= 1
where 6* is the displacement thickness, a factor of considerable import, and 6 is the boundary layer thickness (corresponding t o the factor D/2 of pipe flow). T h e three regimes of turbulent flow past rough surfaces appear clearly in Figure 1. Thus the line marked p = 3.75 represents the first, smooth-pipe regime and the series of horizontal lines indicates the variation of P in the third, fully rough regime. Identical regime occurrences appear in the frictional representations. Before considering the nature of the functional variation in the second, or transitional, regime, several modes of correlating frictional and velocity occurrences will be presented. T h e frictional occurrences for channel flow are correlated by the Darcy-Weisbach friction factor, f, defined by the relation for pressure loss (3) where Ap is the pressure (and Ah the head) loss found with fully developed flow in a conduit of length L and diameter D. 254
I&EC FUNDAMENTALS
47
-2log[-+-I k 3.76
2.51
R ~ T
(9)
or
(
3
@=21og 1 + -
This transitional relation, and that for Nikuradse’s sand roughness are shown in the upper sketch of Figure 2, together with the smooth and fully-rough lines to which these relations are asymptotic. T h e @ function summarization for rough surface effects involves a coalescence on the fully rough occurrences. As an alternate we employ the smooth-regime relation due to H a m a (1954) and obtain the $ function; thus
where the numerical constant is introduced to yield equivalence with the velocity shift factor, Au/uT, which arises in considerations of the near-wall velocity profile representation. For flat-plate boundary layer flows Hama has shown that $ is
The last method of correlating rough-surface occurrences is in terms of a , the ratio ofy, t0.k. This may be evaluated from either the friction factor or the wall-region velocity profile, as clearly demonstrated by Prandtl (1933) with the Nikuradse sand-roughened pipe results. In terms of a , Equation l l then takes the form
Graphical Appearance S =Smooth C=Colebrook
R =Rough NSNikuradre
- --
5.6 log
k/k) -
5.6 l o g a
(1lc)
UT
so that
(12)
(IO)
log a = log ( y / k )
1
--
u
-
(13)
5.6 u,
This function is also obtainable from the friction factor; thus
(14)
loga =
(13)
logs =$(@-3.0316) Figure 2.
-5.6 W
-log Rk-I
Correlations for smooth to rough pipe flow
given by a relation similar to Equation 10 with thef’s replaced by the skin friction coefficients and the multiplicative factor changed to T h e logarithmic relation for the velocity profile in terms of a coordinate distance y (measured from the wall) was derived by von Karman and Pra.ndt1 in the early 1930’s and verified through reference to Nikuradse’s classic tests on smooth and rough pipe. This relation has the form
di.
=
5.6 log (y/ro)
(11)
UT
where y o is the wall distance a t which this equation indicates zero velocity. I n the early 1950’s it was shown that this equation-as the “law of the wall”-is properly applicable only from the outer edge of the viscous sublayer t o y distances of one-tenth to one-seventh D / 2 or 6. Now for smooth pipes it is found thaty, = 0.1 v/u,, that = 5.6
+ 5.6 log Y”_r Y
according t o Ross’ (1953) careful reanalysis of Nikuradse’s smooth pipe data. Near rough surfaces Tillmann (1945) and later Clauser (1954) more formally, following Nikuradse’s (1933) and Moore’s (1951) observations, took the rough surface effect as a downward shift in the wall-law relation. Thus for flow past a rough surface
2
=
5.6
+ 5.6 log YU, - - Au -
UT
v
(1 1b)
UT
where the shift Au/u, is the $ function defined as Au = 5.6 $ =u,
+ 5.6 log YU-‘ v
-
u
-
(12)
UT
This function varies with Rk in a fashion dependent upon the type of rough surface. T h e middle sketch of Figure 2 depicts this occurrence with the (Colebrook “natural” and the Nikuradse sand-roughness transition curves presented as possible modes of variation between smooth and fully rough trend lines.
1 ~-
The variation of a with RI: for the two types of rough surface so far considered is also sketched in Figure 2. For the two limiting flow regimes a is simply prescribed, as a = O.l/Rk in the smooth-surface behavior and a = 1/30 for the fully rough. A general similarity between the variations in 4, 4,and a functions with roughness Reynolds number Rk is apparent in Figure 2. This is hardly surprising, since these functions all convey the same information. In fact they are simply related by the equations loga =
4
-
3.038 2 5.6
- logRk -
1
(15)
Although consideration of more than one is thus redundant, two of these are employed to present the authors’ experimental results-4 for the friction factor and I$ for the wall-region velocity data-in order to check further the identities just noted for other than the sand-roughness case checked by Prandtl. Artificial vs. Natural Roughnesses
The nature of the kind of surface roughness has been noted as influencing the mode of variation for the 9,4,or a functions in the transitional region between smooth-surface and fully rough surface flow. Many factors-such as the relative spacing or wavelength of the roughness elements, their size distribution, their sharpness, and lateral spacing-undoubtedly enter into the definition of these functions in the transitional region. But as yet we know little about which factors are responsible and even less about how they enter. In an attempt to establish which factors, in addition to the effective roughness size, specify the transitional variation as well, a number of investigators have studied the flow past artificial roughnesses. Thus, besides Nikuradse’s (1 933) sand roughness tests, we have the study of a two-parameter sand-roughness by Colebrook and White (1 937), Schlichting’s (1 936) study of two- and three-dimensional roughnesses (as slats or distributed particles), Moore’s (1951) study of slat roughnesses, and numerous others. As an indication of the difficulty of properly evaluating all the factors, we note four transition curves for sand roughness of one nominal size-those of Nikuradse, Colebrook and White, Hama, and the present authors-and in fact only one investigator (Schlichting \vith sand from the same source) has duplicated an effective sandroughness size (corresponding to the Nikuradse standard) in agreement with his actual sand-grain size. Other investiVOL. 7
NO. 2
MAY 1968
255
I ‘w
I m naiona
! o i s D o &evensbn, 1960 #tats: t Jonaaren 1942 Powell, i h 4 rn Moore, 1951
T h r e e Dimenaion H e m i a p h : r F r e e t a n , 1892 Sph.Seg. Conea Schlichting, 1936 Spherrr I 0 Cr. Stone: e Chando, 1958
z}
E
-kh I
oo.ool
0.01
Figure 3. Indications of effective roughness height
1.0
0.1
roughness density
effects
on
gators find the effective size t o be larger by ratios of 1.13 or 1.35 (present study based on the coated or uncoated dimensions) to 1.64. I n correlating the effects of various kinds of roughness, some linear measure of the roughness size is necessary. I t is common practice to employ an effective roughness height-the effective (Nikuradse) sand-roughness size k-as obtained from Equation 4 when f is taken in the fully rough regime. Now Morris (1955) has urged the use of the longitudinal spacing of the roughness elements as the proper dimension and his presentations show that this is indeed appropriate for slat-type roughnesses; however, for natural-type roughness there is no appropriate spacing dimension t o employ. Hinze (1962) has objected to the use of the equivalent sand roughness, since the value obtained sometimes varies with Rk. However, for Rk values above 100 the friction factor-and thus the effective sand-roughness size-is constant for most roughnesses. For slat and corrugated roughness types the fully rough regime appears a t much higher Reynolds numbers; for the corrugated roughness there is even a question as to whether the friction factor ever reaches a constant, as recently pointed out by Robertson (1966). An indication of the manner in which density and shape affect the effective roughness height has been provided by the studies of Schlichting (1936). The density effect is well correlated on the basis of his factor F, here defined as the projected area of the roughness on a plane normal to the flow divided by the unoccupied plate area, whereas Schlichting took the total plate surface area for the denominator rather than the net area. Some of Schlichting’s experimental results are presented in Figure 3 as the ratio of k to the actual roughness height h us. F. A general tendency, found in all such density studies, for the effective roughness height t o peak a t an F value of 0.1 t o 0.2 is clearly evident. A tendency for k/h at any F value to increase with the amount of roughness particle projecting into the flow also seems indicated; thus for threedimensional roughness k/h increases from low values for the hemispheres and spherical segments, through intermediate values for cones (base down) to high values for spheres, for which a peak value of the order of 4 is indicated. Sharper roughness elements would seem t o lead to even larger values of k l h , as suggested by the single point given for a distributed crushed-stone particle roughness tested by Chanda (1 958). A similar peaking effect a t some density appears from the re256
l&EC FUNDAMENTALS
sults of Colebrook and White’s (1937) study of the roughness induced by two sizes of sand grain, when one refers k to the average sand grain size. Thus for their case 111, in which the small grains covered 96% of the surface and the large ones (10 times as large) 4%, the equivalent Nikuradse sand roughness of the surface was 1.96 times the average grain size. Although natural roughnesses are hardly two-dimensional, many experimentors have studied two-dimensional roughnesses because of their simple geometry. Thus the results found by Stevenson (1960) with rods, a simple rounded-type roughness related to the sphere roughness of Schlichting, are also summarized in Figure 3. These seem to yield a peak klh of over 6 at about the same F value for which the spheres peaked. T h e wire screen roughness studied by H a m a (1954) yielded a roughness value k about twice its height and a transition function curve very similar to that of his sand roughness. Numerous investigators, starting with H. Bazin in 1855, have studied the roughness effects of slats, since these are extremely simple items. Here we have a n additional variable as the ratio oft, the slat thickness (in the flow direction), to its height, h. Some differences in the k l h us. F = h/(s - t) curves appear with variation in t/h, over the range of 1 to 4, but there is considerable scatter due to differences between experiments. As a n indication of the results which appear for square slats a trend line, based on the tests of Jonassen (1942), as given by Johnson (1946), Powell (1944), and Moore (1951), is also shown in Figure 3. Although one expects the sharp corners of this type of roughness to produce greater effect than rounded shapes, this slat curve differs little from that of t h e rod roughness and the peak k/h value is nearly 6. Schlichting (1936) studied the limiting case of slats-namely, metal angles for which t l h is approximately zero. Based on the same F density function, these studies seem to indicate a peak k/h of over 12 a t F of about 0.1. This implies a spacing of ten times the roughness height and it is difficult to envision any great difference in the separated flow due to these angles from that of the square slats a t about the same spacing, and yet the effective roughness is twice as great. Even larger effective roughness heights are found when the slats or angles are of finite length, across the flow. Thus Sayre and Albertson (1960) found a k j h value of 16 for one such combination. The above noted effects of roughness density and shape on the effective k are suggestive, although hardly complete. When one extends his concern to the effects on the transition formulation between smooth and fully rough flow, the occurrences are less clearly defined. For simple roughnesses the nature of the transition functions are relatively well established, although the existence of at least four such curves for dense distributions of sand-grain roughnesses has already been noted. The truly sad feature of this situation lies in the fact that we cannot explain why these differ. For sand grains of loose density (case V, of 47, density of coverage) the Colebrook and White (1937) studies indicate a much flatter transition curve tending to lie above the Colebrook natural roughness curve. Conversely Morris (1955) indicates a transition curve for slats (based on spacing s as the pertinent length dimension) which drops much lower and more sharply than that for sand roughness. Some of the various transition curves that have been found are indicated in Figure 4. The occurrences when the roughnesses are not simple is far from clear. Hinze (1962) covers this situation by the succinct statement: “Not yet satisfactorily solved is the case of composite roughness.” About the only systematic studies have been those by Colebrook and White (1937), who varied the percentage of rough to fine sand grains.
the Prandtl mixing length from mean-velocity measurements and found its rate of increase with wall distance in the rough case t o be some 1.3 times that found with a smooth surface. I n addition, the ratio of the greatest mixing length to boundary layer thickness was some 20% larger. This is suggestive, but such evaluations are somewhat uncertain and do not necessarily represent how the turbulent scale may vary. Moore (1951) measured some microscales and obtained values of the order of 0.02 foot, in general agreement with smoothsurface results. However, this signifies little, since microscale values of this order of magnitude are commonly found in most, if not all, shear flow studies in air. The macroscales (integral scale of turbulence) are more indicative of the nature of the production phases of the turbulence where one expects that the roughness might have some effect. I n their study of the boundary-layer flow downstream of a coarse sandpaper boundary roughness, Klebanoff and Diehl (1952) found the ratio of macroscale (15,) t o boundary-layer thickness to be constant a t a value of 0.37 for downstream distances from 1 to 8.5 feet; beyond this distance the apparent roughness effects were negligible.
+2
+I
-Ik I s- o
+
4
-B N I1
* -I
Objective of Present Study
-2 I
Figure 4.
IO
IO0
I o3
I‘0
Smooth to rough transition function relations
Turbulence in Flow Past Rough Surfaces
O u r eventual understanding of the occurrences in the turbulent flow past rough surfaces must involve knowledge of the turbulence structure. This need is implied by the loose terminology sometimes employed in characterizing the regimes of rough conduit flow, whereby the transitional regime is incorrectly termed “pardy turbulent” and the fully rough regime is called “fully turbulent.” Despite the extensive literature on the effects of surface roughness on the tempo.ra1-mean velocity field of turbulent flow, as suggested by the preceding discussions, the available information on the turbulence structure is limited. I n further contrast to the case for frictional and mean-velocity information for which pipe flow data are abundant, almost all the turbulence information has been obtained in boundary-layer flows. Some information on the turbulence intensities in flow past the very rough wall of a square conduit was obtained by Fage (1 933) in water using the fluid-motion microscope technique. Turbulence intensities in the boundary-layer flow of air past rough surfaces were obta.ined by Tillmann (1945) and Baines (1950) for sand-roughencd surfaces, by Chanda (1958) for a roughness of distributed crushed stones, by Moore (1951) for three sizes of square slat roughness of spacing 4 times the height ( F = 0.33), and by Corrsin and Kistler (1954) for a corrugated roughness. When compared with corresponding measurements in the flow past smooth surfaces the turbulence intensities, u ’ (r.m.s. fluctuations), are found t o be significantly larger. Since one major effect on temporal-mean flow occurrences in the flow past rough surfaces is an increase in friction, and thus the friction velocity u,, it is suggested that essential agreement between smooth- and rough-wall turbulence should occur if the intensities are divided by u,. And indeed, with the exception of the Baines data, which appear t o be low even for the smooth boundary, -this seems to be the case. Information on the sca.le of turbulence in flow near rough surfaces is almost nonexistent. Tillmann (1945) did evaluate
The experimental study of the mean-flow and turbulence occurrences in flow past rough surfaces summarized in this paper had as its objectives the clarification of some of the uncertainties noted in the preceding discussions. T h e studies of the temporal-mean velocity and frictional occurrences were planned t o verify simultaneously the Colebrook transition relation for a “natural” roughness in terms of the 6 function, as obtained directly from the friction factor, and the $ function, as found directly from the near-wall velocity profile. The verification of the 6 function from friction loss measurements has been indicated by Colebrook and other investigators, while Robertson (1957) indicated a modicum of verification of the Aulu, expectations, in rather spotty fashion, with data obtained from tests in several different pipe lines. However, these results are scattered and there has been no independent verification of both formulations for 6 and from the same pipeline (except for Nikuradse’s family of similarly sand-roughened pipes). As a secondary objective of the mean-flow studies there was need for verification of the pipe-factor expectations for rough conduit. In fact, as suggested by Robertson (1957), the indications of Figure 1 for sand roughnesses are in some doubt. Since he verified the similarity of velocity profile relations for the away-from-the wall region, concern here is limited t o P and Aulu,. T h e need for information on the turbulence structure in flow past rough surfaces has been established above. The study, therefore, had as its basic objective the obtaining of such information for a pipe of natural roughness, over the range of transitional roughness behavior. O n e additional reason for the mean-flow measurements was to establish and permit proper characterization of the mean-flow field in which the turbulence structure was obtained. Besides the pipe of natural roughness, it was desirable to obtain turbulence information in a pipe with a rather well-known type of roughness and the sand roughness was the logical choice in this case. H a m a (1 954) indicated that the mean-flow occurrences-at least for friction and flow in the near-\\,all region-are the same for boundary-layer and pipe or channel flows. I t is not clear that the turbulence structure is as simply correlated; certainly the away-from-the-wall, or core, region of floiv differs in a fundamental way because of the appearance of intermit-
+
VOL. 7
NO. 2
MAY
1968
257
100
7
J
siek Sizes
i50 el
u
0
a 0
0.02
0.03
0.04
0.05
Figure
6, Roughness in &inch pipe
S i z e , Inch Figure 5. inch pipe
Sand grain roughness used in 3-
Upper. Sand grains bonded lo inside of pipe lower. Dislribution of grain size [before otlashingl
tancy in boundary-layer flows. It has been suggested that in the near-wall region the turbulence level should be similar in boundary-layer flow near rough and smooth surfaces when it is related to the friction velocity. An objective of this study was to verify this via pipe-flow studies. Experimental Apparafus and Procedure
T h e experimental studies involved mean-flow and turbulence measurements in a 3-inch diameter artificially roughened pipe and in an 8-inch diameter “natural roughness” pipe. For reference purposes the 3-inch pipe (nominal inside diameter) was also studied in the smooth condition; it consisted of 30 feet of cast acrylic resin tubing. Air flow to the smaller pipe was supplied with an axial blower at first and later with a variable-speed centrifugal blower, followed in each case with a honeycomb and screens and a contraction nozzle. A centrifugal blower fed the 8-incb pipe through a contraction and vaned turns followed by a honeycomb flow straighfen!r. . At ....
~~~~~
~
~
the atmosphere. T h e artificial roughness was produced by bonding sand grains (white Ottawa sand) to the inside surface with “spar” varnish. I n contrast to the technique used by Nikuradse, this was done by splitting sections of the pioe loneitudinally and later fastening them back together. Statistical analysis of dimensional m e a s w m e n t s of the sand grains before application (Figure 5, bottom) indicated that the average size was 0.035 inch with a standard deviation of 0.006 inch. T h e sand grain distribution and appearance, as applied and coated, are shown in Figure 5 (top). T h e measured surface density was found t o be 670 =t30 grains per square inch. After application, a light cover coat of varnish was applied. T h e mean effective diameter of the pipe, determined by weighing known lengths empty and full of water, was found to be 2.960 inches, with a standard deviation of 0.021. Using 258
16EC FUNDAMENTALS
the average values for the size and density of the grains and assuming they were spherical in shape, the average size of the varnish-coated sand grains was 0.042 inch. T h e 8 i n c h line consisted of some 144 feet of spiral-welded steel pipe which had been used in basic laboratory tests with water for nearly 40 years. T h e natural roughness in the pipe had a very nonuniform circumferential distribution, with the larger roughness nodules formed on the bottom. To give somewhat more uniformity the largest ones were scraped out by dragging a pair of 7.75-inch disks, rigidly spaced some two diameters apart, through the line. This still left considerable roughness which was more nearly uniform circumferentially; the nature of the roughness is shown in Figure 6. T h e diameter of this pipe was established at 8.00 inches (again via volumetrical determination). Rate of flow in the pipelines was controlled by changing the blower speed. I n a few tests the flow rate was partly changed by bleeding downstream of the axial blower for the 3-inch pipe or by throttling at the end of the 8-inch pipe. Measurements were taken over the velocity range of 21 to 98 feet per second for the smooth 3-inch pipe, 5.5 to 82 for the rough 3-inch pipe, and 7.4 to 68 for the 8-inch pipe. T h e static pressure at various paints in both pipelines was measured with small static-pressure probes inserted into the flow through the pipe wall. I n the 3-inch pipe, the probes were L-shaped portions of 0,035-inch diameter hypodermic tubing with a hemispherical nose cap pointing int? theflpw - and .... . r~~~~~~~~~~~
~
~~
~
inch (8 diameters) from the nose. I n the 8-inch pipe, the probes were also L-shaped, were made of 0.134-inch hypodermic tubing with a hemispherical nose, and had four piezometer holes (0.0300-inch diameter) oriented at 45 degrees to the shank and located 0.670 inch from the nose. I n both pipes static pressure measurements were made at locations where fully developed flow was assured. Gross friction loss occurrences in these pipelines were obtained from differential pressure readings between three locations (the exact locations are indicated in the report by Robertson et al., 1965). T h e additional loss introduced by the L-shaped pressure probes was negligible; thus for those in the &inch pipe the loss coefficient was evaluated as 0.002 which, when divided by the number of diameters between locations (50 and more), is negligible compared with the measured f values. I n most cases, the mean velocity was measured with a Pitot tube (constructed from 0.028- or 0,035-inch diameter hypodermic tubing projecting 1.25 inches forward from a ‘/cinch diameter shaft) which could be traversed acmss the flow. T h e
flow rate could then be determined by integrating the velocity profile over the cross-sectional area. At very low speeds in the 8-inch pipe, a mean velocity hot-wire anemometer was also used for independent measurement. I n the 3-inch pipe most of the velocity measurements were made just inside the pipe exit (120 diameters from entrance), while in the 8-inch line, the traverse was taken sosme 13 feet from the end of the pipe (194 diameters from the entrance). T h e static pressure \vas measured a t the same plane as the impact tube with wedgetype and round-nosed probes. .4t the higher Reynolds numbers the differential pressure was measured with inclined draft gages. At small differential pressures a precision micromanometer (accuracy 0.001 inch of water) was generally used. T h e apparatus and techniques used for the turbulence studies during each phase of the experiments were identical, except for some minor differences in probes and probe alignment. Turbulence measurements were made exclusively \vith a Hubbard-Ling constant temperature hot-\vire anemometer described by Hubbard (1957). T h e signal was linearized in the control amplifiers. T\vo \vires could be handled \vith the signals combined, if necessary, in the signal analyzer. This analyzer circuit incorporated a number of convenient networks, including a resistance octagon [for adding and bucking signals) and a differentiating network (for taking the time derivative of the fluctuating signal) ahead of the actual metering device, which was a thermal-type rms ammeter. T h e connections between the control amplifiers and signal analyzer were external; this provided a convenient point to insert a band-pass filter network for obtaining spectral measurements. A General Radio 760-B sound analyzer was used for these; the band width of this analyzer was checked at four frequencies in the range of 25 to 7000 cycles and found to increase slowly from 2.9 to 3.77, in this range. T h e analyzer gain as a function of frequency was calibrated at the conclusion of each test. T h e hot-wire probes usecl in the study \vere fabricated in the laboratory using 0,00014-inch diameter tungsten \\.ire. T h e active length was approximately 0.025 inch, which yielded a wire resistance of about 5 ohms. T h e wire was soldered across the tips of t\vo needles fastened with epoxy to the end of a '/*-inch member located in L shape at the end of a 1/4-inch shaft. T h e smaller member holding the wire pointed into the fluid generally 1.25 inches ahead of the end of the '/'4-inch traversing shaft. Mean-Flow Measurements
Friction losses and the velocity profiles were measured for various flow rates t o obtain information on the pipe factor, the friction factor, and the d.etails of the velocity profiles over a maximum range in flow Reynolds number. Besides f us. R and P us. R results, these measurements yielded indications on the variations in the 6 and functions with Rk. ,4lthough the range of flow rates covereld with each pipe line was not enough t o encompass the full extent of the transition curves between the smooth and fully rough regimes, it covered the significant portion thereof. T h e baric and conventional mode of indicating occurrences in a rough-surface pipe line is via the chart of f us. R, as introduced by Blasius (1913) and Stanton and Pannell (1914). Such a Blasius-Stanton graph for the 3-inch and 8-inch pipes of the present study is presented in Figure 7 . I n verification of the measurmement technique, the 3-inch smoothpipe results are seen to group closely about the established smooth-surface line off 0.;. R-the Blasius relation and Equation 5. Although the 8-inch pipe line had a somewhat unnatural roughness and the measurement of friction and floiv in it involved appreciable variati'on, thef us. R results seem to establish a definite trend line, which has been verified a t the lower Reynolds numbers by wai:er tests with the flow rate evaluated gravimetrically. T h e tests, although not extending t o high enough Reynolds numbers t o reach the fully rough regime, seem to conform t o one of the Colebrook transition lines reasonably well. Thus it is strongly suggested that the effective k / D is 0.0015 and this is taken as defining the effective sand-
+
io4
4x104
io5
4x105
R = UD/Q
Figure 7.
Blosius-Stanton friction factor chart f o r pipes
roughness height of k = 0.012 inch for this pipe line. Some attempts were made to determine the actual roughness heights in this line and surface perturbations of the order of 0.0029 inch in 0.05 inch and 0.013 inch in 0.2 inch were noted; although these need not agree with k , neither are they incompatible with it. The frictional results for the 3-inch pipe with sand roughness contain one anomaly which is clearly evident in Figure 7-that is, experiments carried out a year apart (series A and B) with the same pipe and sand covering yielded appreciably different results. The fact that the second set of data indicate lower friction factors, and thus roughness, suggests that during the disassembly of sections, storage, and reassembly of the pipeline some of the sand grains may have become dislodged or dirt may have collected in the space between the grains. I t is seen from the figure that the studies of this sand-roughened pipe covered most of the transitional range and did reach the fully rough regime, so that the effective Nikuradse-type k may be evaluated unequivocally. Thus k was found to be 0.0474 and 0.0429 inch for series .4 and B, respectively. I n terms of the size of the mounted grains the effective size was 1.13 to 1.03 times the coated size and 1.35 t o 1.23 times the original sand grain size. A k value of 1.42 has been found for the same sand dispersed over the surface of a 90 degree V-notch flume; in addition, values of 1.3 and 1.64 were obtained by Colebrook and Schlichting, respectively, using different sands. Therefore the k results for the 3-inch pipe appear reasonable. Two additional data points from tests of sand-roughened pipe are included in the Blasius-Stanton chart and the consideration of the pipe factor. These were obtained by Black (1961) in 1.25-inch copper pipe with two sizes of sand, which unfortunately was specified only as "fine" and "coarse." Both roughened pipes were tested in the fully rough regime. From Figure 7 it appears that the fine sand in the pipe had the same relative roughness k / D = 0.016 as found for the series A 3-inch pipe, while the coarse sand yielded a k / D of 0.044. VOL. 7
NO. 2
MAY 1968
259
I
0.0145 0.016
r I
I
I
I
0.7I V.
Io4
Io5
166
A = UD/v Figure 8.
Pipe factor variation found for pipes considered
0 1 .
4i
0
I
I
19
""1 r "
h
J.
Io4 Figure 9. number
R =UDT
IO6
Factor /3 as a function of Reynolds
Pipe Factor Variation. As a parallel presentation to the frictional occurrences summarized in the Blasius-Stanton chart we have the gross indication of velocity profile occurrences in the pipe factor chart of P us. R. The P results found for the 3- and 8-inch pipes together with those of Black are presented in Figure 8 and appear to follow the trends suggested by earlier studies [thus refer to Figure 1 and Robertson (1957)l. T h e smooth pipe trend line shown is that established by Robertson based on an extensive review of smooth pipe results. Despite considerable scatter in the 8-inch pipe data, a definite trend of the P variation with R is indicated for this natural roughness. Heretofore the variation of P for natural roughness pipes in the transitional regime has been very poorly defined by the extant experimental data. One factor often employed in attempts to correlate P andf is the /3 factor
I n the early 1930's, the expectation that the Karman-Prandtl logarithmic velocity formulation for turbulent pipe flow should apply universally across the flow from the viscous sublayer to the pipe center line led t o the conclusion that p should be a universal constant, of value 3.75. Uncertainty as to such a universality arises from Prandtl's (1933) evaluation of /3 as 4.07 from Nikuradse's smooth pipe studies and reviews of other 260
I&EC FUNDAMENTALS
available data. The /3 results from the present study of sandroughened and natural roughness pipes are presented in Figure 9, in comparison with the line established by Robertson (1957) for smooth pipe. The scatter in the 8-inch pipe results is considerable; however, many of these data and all of the 3-inch pipe results verify the smooth line trend reasonably well. I n view of the difficulties of obtaining consistent results in the 8-inch pipe, the significance of the considerable divergences from the smooth line evidenced is dubious. Another Nikuradse Shift. Certain anomalies, if not contradictions, in Nikuradse's (1933) presentations of his P and @ results for sand-roughened pipe were pointed out by Robertson (1957). The present study of the flow in the 3-inch sandroughened pipe was partly motivated by a desire to verify Nikuradse's indications that @ had the universal value of 3.75 and that the rough-surface pipe when in the smooth-pipe regime (on the basis of the friction factor) followed the relation predicted by /3 = 3.75 rather than the trend line established for smooth pipes. The f, P, and /3 results obtained from the 3-inch sandroughened pipe studies show no unusual features; however, the test Reynolds numbers did not reach a low enough value for smooth pipe flow to occur, so that the contradiction of P in the smooth pipe regime is neither directly observed nor refuted. Values of P obtained for the 3-inch pipe in the fully rough regime, together with the two results of Black, as a function of k / D , when compared with the corresponding results of Nikuradse indicate a 0.032 difference in P . Similar comparison of the latter's P values for the smooth regime with the established smooth-pipe trend yields a difference in P of 0.026. Thus it is strongly suggested that Nikuradse raised his P values by the order of 0.026 to 0.032, presumably in order to obtain the 3.75 value for p. As a check on this point a number of his special 41 rough-pipe traverses were corrected by 0.026 in P and were found then to have @ values of about 4.5, in reasonable agreement with the smooth pipe trend line of Figure 9. Wall Law Formulation near Rough Surfaces. Roughness effects modify the velocity profile formulation only in the nearwall region, A shift in the logarithmic velocity formulation (given by Equation l l b ) results from a modification in the y, distance due to the roughness effect. The velocity profiles in the wall-law region ( 2 y / D less than one tenth to one seventh) from seven of the traverses in the 3- and 8-inch pipes are presented in Figure 10 for comparison with Equation l l b . At R, = u y , / ~values above 30 this formulation is well verified and the velocity shift function $ = Au/u, well characterizes the occurrences. The upper limit t o the wall-law formulation indicated by these lines is, of course, specified not by R, but rather by the 2yiD restriction. The A u l u , evaluation from plots such as presented in Figure 10 permit direct evaluation of the 4 function for the transition region for the natural and sand-roughened pipes. The results of this evaluation, as $ = function (RJ,are summarized in Figure 11. Included in this figure for comparison purposes are the few evaluations of $ = h : u r for natural roughness which Robertson (1957) was able t o glean from the literature. The $variations with Rk indicated for the two kinds of roughness are seen to follow the expected trends rather well. Transition Function Formulations. Discussion of the "characteristics of mean-flow occurrences" has identified the +, $, and a! functions as useful in correlating the variation in meanflow effects of surface roughness. I t has also indicated the interrelationships between them which permit one t o be determined from the other. Despite this feature, the desirability of independent evaluation of + and # for the same natural-
I.2f
I\
Data Symbols 8
A 3 in. Sand Ser. B} Roughened
09 8 in. Pipe
IO -0.0
1-
Ham0
3
Figure 12. Smooth to rough friction-factor transition function curves obtained b y several investigators
I I I 1000
100
IO
5-Y%/$ Figure 10. region
Profiles of temporal-mean
velocity in wall
Verification of the expected qi function variation with Rk for the two roughnesses is indicated in Figure 12. T h e sandroughened pipe results appear much like Nikuradse’s, except that the dip of the line below the fully rough result is greaterLe., closer t o the results found by Colebrook and White and by Hama. The 4 results from the 8-inch pipe again follow the Colebrook transition formualtion reasonably well, although the scatter in the data is considerably greater than in the $function results. Turbulence Results
8 in. Stoel Pipcj *Stoyko 1938 k/D- 0.00187 0 RBM 1961-62 k/D*0.0015
1
I
100
1000
F3: k%h Figure 1 1 . Variation in velocity-shift factor +b = Au/u, with roughness Reynolds number
roughness pipe was noted and has been another achievement of the study summarized herein. T h e $ us. Rk function variation has been indicated in Figure 11. T h e 8-inch pipe results are distributed evenly about the Colebrook natural-roughness line, which thus may be deemed t o have been directly verified even though the whole range of transitional behavior was, not covered. T h e additional naturalroughness pipe data presented are seen to complement the present studies directly. Of particular interest are the data of Stoyke (1938), since these were obtained using the same 8-inch pipe (although of a different age and roughness). T h e trend of the sand-roughened results is seen to be similar, although not identical, t o Nikuradse’s results.
One of the prime objectives of this study was t o ascertain the nature of turbulence in the flow near rough surfaces and specifically to see how it differed from that past smooth surfaces. Intensity. T h e turbulence intensities, u’ longitudinal, v’ radial, and w’ circumferential, were obtained from hot-wire anemometer measurements in the two pipes as a function of the radial position for a number of flow Reynolds numbers. T h e intensity was found t o be larger in the rough-pipe than in smooth-pipe flow, but when it is compared to the wall shear velocity, ur, the variation across the flow is much the same as found with flow past smooth surfaces. Results for the u’ component found in both the 3-inch and 8-inch pipes are thus shown in Figure 13. T h e u’ data also vary in much the same manner as in the smooth case. T h e u / intensity results from this study have been collated with those obtained by other investigators of shear flows under mildly favorable or zero pressure gradient, in order t o evaluate the variation with Reynolds number. Thus Figure 14, a, summarizes most of the available smooth-conduit flow results found on the center line, as the u t values divided by ul. A similar correlation in flat-plate boundary-layer flow a t a relative wall distance of 0.1 has been indicated by Robertson and Martin (1966). I n both cases the variation with Reynolds and there seems t o be no variation with number is as R-0*084 conduit shape or kind of test fluid. T h e rough conduit results a t the center of the 3-inch and 8-inch pipes are presented as u‘/u~ in Figure 14, b; the effect of roughness in increasing the intensity is clearly evident. As noted in the introduction, the turbulence intensities in smooth and rough flow are expected to agree if they are expressed in terms of ur. That this is the case appears in the presentation of Figure 14, c. A similar conclusion for both conduit and flat-plate boundary-layer flows, in terms of intensity results found a t relative wall distances of 0.1, VOL. 7
NO. 2
M A Y 1968
261
-
Run 84 88 .*-a- R u n 42 -b R u n 85 7.- Run 47 - k ~Run -
2
R, = 2.2 x IO4 RI = 9.85 x IO4 RIi 13.7 X IO4 RI = 14.5 x IO4 RI= 15.6 x IO4
0.06
U'
UI
u'/u,
0.04
o*02t
@
I
I
@E-lnch I
01
0.2
,myth
/
Conduit
0.076 IR,0'084
Pipe I
0.4
I
I
0.6
0.8
I
1.0
2Y/D Series A
R , = 4.1 x I 04 4 R u n 34 R,= I .9 x I 0
R u n 38
I
I
1.01
I 0'
@3-lnch
0'51
Pipe
I
I
I
0.2
0.4
0.6
I
0.8
lo
I
I
I @ Smooth and
1
Rough
I
1.0
2y/D
Figure 1 3. Variation in relative turbulence intensities across rough pipe
has been indicated by Robertson and Martin. The -0.084 exponent on the Reynolds number dependency in Figure 14, a, is based on the constancy in u'/u,. At the conduit center line u ' / u , has the value 0.78, whereas a t a distance 10% away from the wall, this ratio has a universal value of about 1.7 near rough and smooth walls. The results are independent of the flow Reynolds number and the particular type of flow (pipe, channel, or flat plate boundary layer). Of course, because of intermittency occurrences in the outer boundary layer region, there is no correspondence of boundary-layer results a t y = 6 with conduit center line values. The u' and u t results permit an indication of the variation in turbulence anisotropy across the flow. There was a great deal of scatter in these results, as shown in Figure 15. However, an arithmetic mean at each local shows good agreement with results obtained by Laufer (1954) in a smooth-pipe study. No systematic variation with Reynolds number or roughness is evidenced. The results indicate that u'/u' increases nearly linearly from a value of about 0.6 at 2y/D = 0.1 t o about 0.9 a t 1.0. The absence of an approach t o isotropy a t the pipe center may not be significant in view of the large scatter. From the small amount of w' data obtained, the second anisotropy ratio, w'/u', is found to be lower than Laufer's (1954) results; but in terms of the data scatter this difference may not be significant. Scale of Turbulence. Microscale determinations were obtained via the time derivative of the turbulence signal. I n many instances these values were checked by calculating the microscale from the second moment of the spectra using the following equation,
Figure 14. Turbulence intensity at center of smooth and rough conduits Fluid Water Air Air Air Air Air Air Air Water
1932 1938 1951 1954 1956 1955 1955 1962 1966
Fage Reichardt Laufer Laufer Eskinazi Sandborn Mickelsen Gaviglio Jones
Conditions sq.
Rect. Rect. Cir. Rect. Cir. Cir. Cir. Cir.
where u is the temporal mean velocity a t the measuring locale while E,, is the fractional spectral energy (utZ per cycle) at frequency n. Except for some dropoff near the wall, the microscale is found t o be essentially constant across the flow. The microscale values are somewhat lower than those of about 0.01 foot found by Laufer (1951) in a wide rectangular channel; however, the value was essentially the same regardless of Reynolds number of wall roughness. The macroscale was obtained from the spectrum by extrapolating the data into the lower frequency region by using a curve derived from Dryden's isotropic relation
UEU,
4
If this relation is written as
a family of parabolas can be constructed and the most appropriate one fitted to the lower frequency portion of the spectrum data. Then L,, is specified by the factor M or N . 262
I&EC FUNDAMENTALS
3in. Pipe Bin. Pipe Sym Rl S y m R, x ~ 4 . 1 ~ 1 0 0~ 13.6~10~ + 1.9 0 0 15.6 v 2.4 Q b 15.6 A 4.2 De 9.8 0 0 9.8
1 1.2 1.0 0.8
.
V?U‘
0.6 0.4 Smooth
0.2-
L0.4 ’
OO
0.2
1
0.8
0.6
I
I.o
2y/D
Figure 15. flow
I --
Anisotropy ratios obtained in rough pipe
Cross Ctream Variation
‘0.6
2Lx 0.4
Figure 17.
0.2
0
0A
0.6
2Y D
0.8
1.0
0.6
0.4
2Lx
Center line energy spectra for 3-inch pipe
indicate a Reynolds number dependence, as suggested by Figure 166 for the center line values. Similar Reynolds number dependence is shown for the integral scales found by Baldwin and Walsh (1 961) from longitudinal correlation measurements. No explanation for the apparent effects of size and roughness is apparent. Isotropic turbulence theory suggests a simple relation between the length scale of the primary energy containing eddies (approximately L ) and the Taylor microscale, A, namely, that a t high turbulence Reynolds numbers,
-L _- -Rx
D
X
0.2
0 I o4
5x104
R, =$
lo5
Figure 16. Macroscale variation with locale and Reynolds number
T h e cross-stream variation in the macroscales thus found is shown in Figure 16a and agrees fairly well with trends found by other investigators. ‘There is a general constancy in the mid-region with a definite decrease as the wall is approached. T h e magnitudes are be1o.w those found in straight rectangular channels by Laufer (1951) and Eskinazi and Yeh (1956) and
(19 )
15
Comparison of the center line results from the present study with this expectation shows a certain amount of verification a t Rx = XU‘/V values above 70 indicated by Robertson et al. (1965) and Robertson and Martin (1966). In any case there is a clear tendency for L/X t o increase with the turbulence Reynolds number. I n view of the suggested constancy in A, it is not surprising that L increases with Reynolds number. Spectrum. Turbulence energy spectra were found using the equipment and techniques described. I n general, the spectral distributions obtained appear as typical of those found in other turbulent flow studies. A comparison between the spectral energy distributions a t the center line in the 3-inch smooth and rough pipe is presented in Figure 17. The general variation of E,, with frequency is seen t o be about the same for the two kinds of wall. As would be expected, the rough pipe results lie above the corresponding points of the smooth pipe case a t the same flow Reynolds number, since this involves a larger intensity. For a middle range of frequencies, each curve evidences a small regime where the spectral energy decays as 12-5’8, thus suggesting an inertial subrange in eddy sizes, in accord with VOL. 7
NO. 2
MAY 1960
263
Symbol X
3'5
A 0
+ 0
RI R, Rk 14.5 X IO4 304 18.9 10.8 X IO4 152 14.2 6.3 x IO4 88 8.4 3.0 x IO4 42 4. I 2 . 1 9 ~IO4 24 3.0
A
I
0.I
uE(n) d2LX
0.0
i,
152 88
0.oc
I
n, scc-I
I02
lo3
to4
Figure 18. Reynolds number dependence of center line energy spectrum as found in 8-inch pipe
Kolmogoroff's similarity hypothesis. This range seems to be largest for the rough pipe and t o start a t the highest frequency for the highest turbulence Reynolds number RA and to be smallest (or possibly nonexistent) and start a t the lowest frequency for the lowest Reynolds number test (with the smooth pipe). The variation in the spectral energy distribution at the center line of the 8-inch pipeline with a change in the flow Reynolds number is shown in Figure 18. Again the curves are widely spread vertically because of the considerable range in turbulence intensities involved. Also a significant inertial subrange appears a t the highest Rh Reynolds number with a limited to nonexistent range a t the lowest number. Measurements in both the 8-inch and 3-inch pipeline indicate that the n-513 range disappears below a turbulence Reynolds number of 40, whereas the range is clearly evident above values of 100 or so. Since the intensity increases with approach t o the wall, the near-wall energy spectra may likewise be expected to be larger, and this was the result found experimentally. This seems to be the only effect of rough-wall proximity. The difference between the spectral energies a t the various locations across the pipe was found to be essentially independent of frequency. T h e 72-5'8 range vanished very near the wall (at 2y/D = 0.025) despite the high RAthere. Direct comparison of the turbulence spectra obtained under various conditions is complicated by differences in intensity and scale as well as mean flow velocity which specify the level of E , and its rate of change with n. The most widely accepted presentation is through the use of Dryden's 1938 relation, Equation 18. Some of the data from the 8-inch pipe have been collected in this fashion and are presented in Figure 19. I n general, the experimental results 264
l&EC FUNDAMENTALS
)I
Figure 19.
\ * I
0.I
\
"LX U
\ I
\
1.0
Universal plot of spectral data
correspond well with this formulation; however, the data drop below the relation a t the higher frequencies and sooner at the lower turbulence Reynolds numbers.
Conclusions
Despite certain vagaries in the results, some tentative conclusions can be drawn from this study. The mode of pipe factor variation with Reynolds number in the transitional regime for pipes of natural roughness has been established via the results from one relative roughness. The factor p , which correlates the pipe and friction factors, appears to have a value of about 4.5 for both smooth and rough pipe flows. The 3.75 value and its implications, such as for the pipe factor P, are incorrect. The variation with roughness Reynolds number of the correlating functions IJ and 9 have been directly verified for pipe of natural roughness, thus substantiating Colebrook's formulations. The structure of the turbulence does not seem to be altered in rough pipe flow except for an increase in intensity associated with an increase in the frictional occurrences. The turbulence macroscale appears to increase with Reynolds number for a given pipe. I t also seems to depend on pipe size and roughness in a manner in need of clarification. The effect of roughness on the spectrum of turbulence seems to be prescribed by the increased intensity of turbulence and by the macroscale changes. In view of the uncertainties as t o the occurrences causing this latter to vary, the changes in spectra due to roughness are not defined at present.
Acknowledgment
Special acknowledgment is due M. E. Clark and t o undergraduate student assistants C. H. Ahlenius, J. Swanson, T. M. Mulcahy, R . Kowalski, (2.C . Chen, J. Werth, and W. G. Steiner. literature Cited
Baines. \V. D.. “Exuloratorv Investiration of Boundarv-Laver Developmen; on SAooth a i d Rough”Surfaces,” Ph.D. disse4tation, State University of Iowa, 1950. Baldwin, L. V., \\‘alsh, T. S., A.I.Ch.E. J . 7, 53-61 (1961). Black, R . E., “Mechanics of Square Elbow Losses,’’ Ph.D. thesis in civil engineering, Graduate College, University of Illinois, 1 _ I .
IYUI.
Blasius, H., Forschr. Gebiete Ingenieurw. VDI, Heft 131 (1913). Chanda, B., “Turbulent Boundary Layer over Heated and Unheated, Plane, Rough Surfaces,” Dept. Civil Eng., Colorado State University, Rept. 1 (CER 58BC21; AFCRC TN-58-428; ASTIX-.4D152599) (19%). Clauser, F. R., J . Aeron. Sei.21, 91-108 (1954). Colebrook, C. F., J . Inst. Civil Engrs. 11, 133-56 (1939). Colebrook, C. F., \l’hite, C. M., Proc. Roy. Soc. London A161, 367-81 (1937). Corrsin, S., Kistler, A. L., “Free Stream Boundaries of Turbulent Flows,” Natl. Advisory Comm. Aeronaut., T N 3133 (1954); Rept. 1244 (1955). Eskinazi, S., Yeh, H., J . A m n . Sei.23, 23-35 (1956). Fage, A , , “Fluid Flow in Rough Pipes,” British Aeronautic Research Council, R 8; M 1585 (1933); “Studies of BoundaryLayer Flow with a Fluid Motion Microscope,” in 50 Jahre Grenschichtforschung, € 1 . Goertler and \C. Tollmien eds., pp. 132-46, F. Viewig and Sohn, Braunschweig, 1955. Gaviglio, J., “Sur quelques probl2mes de mesures de turbulence effectuees a l’aide de l’antmom6tre a fils chaud paracourus par un courant d’intensitt constant,” Publications Scientifique et Technique du Ministre cle l’Air, No. 385 (1962). Hama, F. R., Trans. SOC..l’azal Arch. Marine Engrs. 62, 333-58 (1954). Hinze, J . O., “Turbulent Pipe Flow,” in “Mechanique de la Turbulence,’’ Colloques Int’mmtionaux du Centre National de la Recherche Scientifique, No. 108, Paris, pp. 129-165, 1962. Hubbard, P. G., “Operating Manual for the IIHR. Hot-\%re and Hot-Film Anemometers,” University of Iowa, Studies in Engineering, Bull. 37 (1957). Johnson, J. \V., Trans. A m Soc. C k i l . Engrs. 111, 556 (1946). Jonassen, F., “Artificial Roughness,” unpublished paper, University of California F1,id Mechanics Laboratory, 1942 (see Johnson). Jones, B. G., “Experimental Study of the Motion of Small Particles in a Turbulent Fluid Field Using Digital Techniques for Statistical Data Processing,” Ph.D. thesis in nuclear engineering, University of Illinois, 1966. Klebanoff, P. S., Diehl, Z. \\‘., “Some Features of Artificially Thickened Fully Developed Turbulent Boundary Layers with Zero Pressure Gradient,” Natl. Advisory Comm. Aeronaut., Rept. 1110 (1952). Laufer, J., “Investigation of Turbulent Flow in a Two-Dimensional Channel,” Natl. Advisory Comm. Aeronaut., Rept. 1053 (1951).
Laufer, J., “Structure of Turbulence in Fully Developed Pipe Flow,” Natl. Advisory Comm. Aeronaut. Rept. 1174 (1954). Mickelsen, W.R., “Experimental Comparison of the Lagrangian and Eulerian Correlation Coefficients in Homogeneous Isotropic Turbulence,” Natl. Advisory Comm. Aeronaut., TN3570 (1955). Moore, W. L. “Experimental Investigation of the Boundary-Layer Development along a Rough Surface,” Ph.D. dissertation, State University of Iowa, August 1951. Morris, H. M., Trans. A m . Soc. Ciuzl Engrs. 120, 373-410 (1955). Nikuradse, J., Forschr. Gebiete Ingenieurw. VDI, Heft 361 (1933). Powell, R. W.,Proc. A m . S o d . Civil Engrs. ( T r a n s . A m . SOC.Ciuil Engrs., 111, 1946), 531-54 (1944). Prandtl, L., “The Mechanics of Viscous Fluids,” Vol. 111, Sec. G, of “Aerodynamic Theory,” \Y. F. Durand, ed., Julius Springer, Berlin, (reprinted by Dover Publ. 1963), pp. 135-45, 1933. Reichardt, H., Z A M , M 18, 358 (1938). Robertson, J. M., discussion of “Factors Influencing Flow in Large Conduits,” Proc. A m . Sac. Ciuil Engrs., J . Hydraulics Dizision, 92 HY4, 168-73 (1966). Robertson, J. M., “Turbulent Velocity Distribution in Rough Pipe,” Proceedings of Fifth Midwest Conference on Fluid Mechanics, pp. 67-84 (1957). Robertson, J. M., Burkhart, T. H., Martin, J. D., “Study of Turbulent Flow in Rough Pines.” Universitv of Illinois. TAM v Rept. 279 (May 1965). Robertson, J. M., Martin, J. D., A m . Inst. Aero. Astron. J . 4, 2242-45 (1966). Ross, D., “New Analysis of Nikuradse’s Experiments on Turbulent Flow in Smooth Pipes.” Proceedincs of Third Midwestern Conference on Fluid rvlechanics, pp. 631-67, 1953. Sandborn. V. A.. “ExDerimental Evaluation of Momentum Terms in Turbulent ’Pipe ‘Flow,” Natl. Advisory Comm. Aeronaut., TN 3266 (1955). Sayre, \V. \C.,Albertson, M. L., Proc. A m . Soc. Civil Engrs. J., Hydraulics Dicision 86 HY3, 131-50 (1960). Schlichtinn. H.. Ineenieur-Archiu. 7. 1-32 (1936). Stanton, ?.’E.,‘Pagnell, J. R., P h i . Trans: Roy.’Soc. London 92, 199224 (1914). Stevenson, M., “Roughness Effect and \Vake Influence for TwoDimensional Wire Roughness,” Inst. for Fluid Dynamics and Applied Math. Univ. of Maryland, Tech. Note BN-219 (1960). Stoyke, L. T., “Experimental Study of the Distribution of Velocity of Water in Various Pipes,” M.S. thesis in TAM, Graduate College, University of Illinois, 1938. Straub, L. G., Bowers, C. E., Pilch, M,, “Resistance to Flow in Two Types of Concrete Pipe,” St. Anthony Falls Hydraulic Laboratory, University of Minnesota, Tech. Paper 22B (1960). Tillmann, \V., “Investigations of Some Particularities of Turbulent Boundary Layers on Plates,” Kaiser-\\’ilhelm Institut fur Stromungforschung, Gottingen, UM 6227 trans. in British 41. A. P. Volkenrode MAP-VG-34-45T, Joint Intelligence Objectives Agency File No. B.1.G.S-19 (1945). A
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RECEIVED for review June 16, 1967 ACCEPTED January 8, 1968 Study sponsored by the Bureau of Ships Fundamental Hydromechanics Research Program S-R009 01 01, administered by the David Taylor Model Basin, under contract Nonr 1834(32). Part of the fluid mechanics research program of the Department of Theoretical and Applied Mechanics through the Engineering Experiment Station, University of Illinois.
VELOCITY PROFILES IN A HEATED ROTATING ANNULUS D . K . P E T R E E , W . L . D U N K L E Y , A.JD J . M . S M I T H
UniriersiQ of California, Davis, Calif. ELOCITY and temperature patterns in a vertical, rotating Vannulus of fluid are affected by both natural convection and rotation speed. This paper is concerned with a n annulus of water contained between a n inner cylinder, heated t o give a uniform energy flux, and a n outer cylinder which rotated. Horizontal end plates, a.ttached t o the outer cylinder, also rotate.
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When the outer cylinder is stationary, heating of the inner cylinder above a critical energy input causes fluid motion in the annulus due t o density gradients. Below a critical Rayleigh number (Dropkin and Globe, 1959), corresponding t o very low heat inputs, natural convection does not occur, because the small density gradients are offset by the viscosity and thermal conductivity of the fluid. This work is conVOL.
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