LIMITING BEHAVIOR OF THE TRANSVERSE TURBULENT VELOCITY FLUCTUATIONS CLOSE TO A WALL KAMALESH K. SIRKAR AND THOMAS J. HANRATTY Department of Chemistry and Chemical Engineering, University of Illinois, Urbana, Ill. 61801
In previous studies of turbulent flow in a pipe the component of the fluctuating velocity gradient a t the wall in the direction of mean flow has been measured by determining the limiting current to a small rectangular polarized electrode mounted flush with the wall and oriented perpendicular to the flow. By measuring the limiting current a t an electrode a t 12$' to 20' as well as a t 90' the root-mean-squared value of the component in the transverse direction has been estimated to be about 0.087 of the time-averaged velocity gradient a t the wall. This is in agreement with recent photographic measurements of solid particles close to a wall made by Fowles. These results suggest that the limiting behavior of the eddy diffusion coefficient close to a wall is such that E -+ y3 as y --+ 0,unless fluctuations in the mass transfer rate are uncorrelated with fluctuations in the gradient of the transverse velocity component.
IT HAS been recognized for some time that turbulent flow
ations in a region closer t o the surface than yf = 0.5. I n extensive studies carried out in this laboratory the electrodes past a solid surface causes the fluid close to the surface t o have been either circular or rectangular with the long side be in a highly agitated state. In recent years a number of oriented perpendicular to the direction of mean flow. Electechniques have been used to investigate the turbulent flow trodes with this shape and orientation are sensitive to the pattern jn the immediate vicinity of a solid surface. These component of the fluctuating velocity gradient in the direction include photographing solid particles (Fowles, 1966; Wenzel of mean flow, s., If a rectangular electrode is oriented a t an and Mathews, 1968), colored plumes (Hummel and Popovich, angle to the direction of mean flow less than go', it should be 1967), or gas bubbles (Kline et aE., 1967) in the fluid, the sensitive to the component of the wall velocity gradient in cooling of heated wires or of heated films (Laufer, 1954), and the transverse direction, sz, as well as in the direction of mass transfer to polarized electrodes (Mitchell and Hanratty, mean flow. Mitchell and Hanratty (1966) did some experi1966). The limiting value of the ratio of the root-meanments with rectangular electrodes oriented a t 45' to the squared fluctuating velocity component in the direction of direction of mean flow. They found a small difference behas been mean flow, (k2)1'2,to the local mean velocity, t, found to be between 0.30 and 0.36. Hot wire anemometer tween these results and those obtained with electrodes at 90'. They concluded that s, is smaller than sz in the immediate studies by Laufer (Hinze, 1959) have indicated that this limiting relation holds out to y+ = 7, where y+ is the distance vicinity of the wall but gave no estimate as to how much smaller. Some calculations presented in this paper, perfrom the wall normalized with respect to wall parameters. If Laufer's measurements of the root-mean-squared values formed to determine the effect of orientation on the response of the electrode, indicated that if w is of the order of t u , a of the fluctuating transverse velocity components, (7)1'2, are normalized with respect to the local average velocity and single electrode oriented a t about 15' would be needed t o extrapolated to the wall, a limiting value of zero is obtained get good measurements of s,. Therefore a series of experi(Hinze, 1959). These results indicate either that close t o ments was performed with rectangular electrodes a t angles the wall w is varying with the square of the distance from the of 123', E', and 20' mounted flush t o the wall of a circular wall or that one must make measurements much closer t o pipe with a diameter of 7.615 inches. Previous work in this laboratory was done in a pipe having the wall to observe the limiting behavior for w than is necesa diameter of 1 inch. These studies revealed that the scale sary for u. Photographs of the motion of solid particles of the turbulence in the circumferential direction is small, obtained by Fowles (1966) have supported the latter explaand the interpretation of measurements with slanted elecnation. Fowles found that the limiting behavior for w cannot trodes was clouded by the uncertainty as to whether averbe observed until yf < 1 and that the limiting value of (G)1/2/o is approximately equal to 0.09. These results of aging is occurring over the electrode surface. Therefore the Fowles are highly significant, and it has been desirable to larger diameter pipe used in the present studies was needed have an additional check on their accuracy, especially since to obtain unambiguous results. The chief inaccuracy in the results reported in this paper arises from slight errors in the a recent study using the same technique as Fowles yielded very different results (Wenzel and Mathews, 1968). alignment of the electrodes. As a result, the calculated Therefore it was decided to explore the possibility of using values of the transverse component of the velocity gradient polarized electrodes to study the limiting behavior of w are regarded as having possible errors as high as 5%. Neverclose to a wall. This technique involves carrying out an theless, they are sufficiently accurate to test the accuracy of electrochemical reaction on the surface of an electrode the conclusions that are drawn from Fowles' study. mounted flush with the wall. At high enough voltages the Aside from their interest with respect to determining a current flowing in the electrochemical circuit is controlled by structure for turbulent shear flow, these measurements are the rate of mass transfer to the test electrode. If the test of use in understanding mass transfer a t high Schmidt numelectrode is small enough, the concentration boundary layer bers, for which the concentration boundary layer can be is so thin that fluctuations in the current reflect flow fluctuthinner than the viscous sublayer. VOL
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Analysis of Performance of Slant Electrodes
The concentration of the reacting species is zero a t the surface of a polarized electrode. A mass transfer coefficient, K , can be defined in terms of the current flowing in the electrochemical circuit, I , and the bulk concentration, Cg, T
where A is the electrode area and F is Faraday's constant. The concentration boundary layer over the electrode surface is so thin that the velocity field can be represented as S y with x, y, and z components of U = S,y, W = S,y, and V = 0. To relate the measured instantaneous value of Z to S, and S, one has to take account of averaging over the electrode surface and the time response of the concentration boundary layer. By making the size of the electrode small compared to the scale characterizing the variation of S one can minimize or eliminate averaging. The velocity gradients, S, and S,, can then be related to K by a solution of the differential equation representing the mass balance over the electrode surface. The mass balance can be simplified by using a pseudo-steady-state approximation (aC/at = 0) if the dimensionless form of the frequency of fluid oscillation, ii = 2mL2/3/D1/882/3, is less than 0.40 (Mitchell and Hanratty, 1966). D in the above definition refers to the diffusion coefficient and L t o the electrode length. Consider a rectangular electrode with its long side perpendicular to the direction of mean flow, the z-axis. Its width, W, is large enough compared t o its length that the effect of flow fluctuations in the z-direction can be ignored, yet small enough that averaging does not occur. Mitchell and Hanratty (1966) have shown that if the pseudo-steady-state approximation can be made
K =U(S~/L)'/~
As can be seen from Figure 1, the value of q will be less near the edges of the electrode than in the central regions. Because of the thinness of the concentration boundary layer molecular diffusion can be neglected in directions parallel to the electrode surface. If the pseudo-steady-state approximation is valid, one can apply Equation 2 to each of these strips K = u (h'/q)1/3 (4 ) where = [ (8, s2)2 s13p (5 1 As long as cp - 0 2 IC, where IC, = tan-'(L/TY), the average mass transfer coefficient over the whole electrode surface a t an instant is S sin
K=u[
tan8 =
--
8,
+
9.
(31
where s, and s, !re the fluctuating components of the velocity gradient, and S, is the time-averaged value of the velocity gradient. The electrode surface can be divided into a number of strips of length q parallel t o the instantaneous flow direction.
L
[ +L
5w cot ($0 - e)]
1
I 8, I cot cp I 8, + 82 I < 1
(6)
(7 1
Let cot cp < 5 (cp > 11°18'), ($)1/2/8zZ 0.10, (?)1/2/8zS 0.32 and W / L = 10 t o 30. Then terms in the series expansion of Equation 6 of second-order and higher can be neglected, so that
+ s,L cot cp s,L cot2cp 158,W + 15[8, &]?Y
+
Ls, cot2 cp
- 5w[&
+
S,]
The time-averaged mass transfer coefficient is then given as
R
=
(T)l!3(1 8, sin cp
+ -)
(9 1
The fluctuating component of the mass transfer coefficient, IC, can be obtained by subtracting Equation 9 from Equation 8. -
8,
- e)]',
('p
The term containing L/W appears because q is varying near the edges of the electrode. As L/W-+O this edge effect can be neglected. Equation 6 has been expanded in a series under the restriction that
(21
where u = 302/3/2r(4/3)91/3. Xow let us consider an electrode a t an angle cp to the direction of mean flow, as shown in Figure 1. The flow is assumed uniform over the electrode surface. The angle 8 between the instantaneous direction of flow and the direction of mean flow is defined by the equation
+ +
s
I?
5
1 92 =--2+-F2 98,
1 2 (cot $0 - [ (2L cot cp)/5w])* 9s, (1 [(Lcotcp)/5W])
+
iz
2- 9
8z2
+
(cot cp - [ (2Lcot cp)/5w]) (10) (1 C(L cot ( ~ ) / 5 ~ 1 1
+
Equation 10 relates the measured to three statistical _ _ properties of the flow field s,Z, si2, and &. Because of symmetry = 0. Therefore 2 can be determined from measurements with the electrode perpendicular to the direction of mean flow and 2 is calculated from measurements with a slant electrode. Inequality 7 implies that if cot cp = 5, Equation 8 is valid only so long as s,/ (8, s), < 0.2. For larger values of cp or smaller values of cot cp the upper limit on sa is higher.
+
Design of Experiments
Figure 1. 190
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Electrode configuration
FUNDAMENTALS
The design of the experiments was influenced by the desire to have electrodes which are sensitive to s, fluctuations, t o have a uniform flow over the electrode surface, and to have a good enough frequency response that the pseudo-steadystate approximation is valid.
From Equation 10 it can be seen if (G)1/2/8, = 0.32 and = 0.10, the value of (@)'l2/R measured with a if (G)'/2/s, 45' electrode with TI'/L = 10 would be only about 4.5% higher than the value measured with a 90' electrode. For the same velocity field the mass transfer intensity is estimated to be about 36% higher for a 123' electrode than for a 90" electrode. The use of la;", l5', and 20' electrodes was a compromise between having the electrodes reasonably sensitive to s,-fluctuations and keeping edge effects small. The measurements of the fluctuations of s, in a 1-inch pipe by RIitchell and Hanratty (1966) indicated that the turbulence scale in the direction of mean flow is over an order of magnitude larger than the scale in the circumferential direction and that the Circumferential scale, A,, is about 0.060 inch for a 7.615-inch 1)ipe a t a Reynolds number, Re, of 30,000. Electrodes having a length, L , of 0.003 inch and a width, It., of 0.03 inch mere oriented a t 90" t o measure the s, fluctuations. At Re = 30,000 the turbulence scale is about two times greater than the width of the electrodes, and the correction factor for the measured ( ~ 2 )due ~ 'to~nonuniform flow is only about 1.03. Values of TV/L and therefore of for the slant electrodes were chosen t o satisfy the restrictions given above. The values of cp and Ti' were such that TI' sin cp, the lateral spread of a slant electrode, was about O.17Az for 15' and 20' and 0.27 A, for 123" electrodes a t Re = 30,000. Therefore, if the scale of the sz fluctuations is of the same order as the scale of the s, fluctuations, no averaging occurred over the slant electrodes Experiments in a 1-inch pipe (Nitchell and Hanratty, 1966) over the Reynolds number range of 8700 to 50,200 indicated that the frequency of the s, fluctuations scaled rrith the diameter of the pipe d and the bulk velocity L'B so that the median frequency corresponding to these fluctuations is given as (nd/UB) = 0.18. ;it a fixed value of Re the increase of the pipe diameter from 1 inch t o 7.615 inches can be expected to decrease the median frequency of the fluctuations by a factor of (7.615)2= 58. The value of 6 corresponding to this median frequency at a Reynolds number of 30,000 for a L of 0.003 inch is 0.12. It is small enough that one can expect the pseudo-steady-state approximation to apply to frequency components which are making significant contribution to the fluctuating velocity gradient at the wall.
+
Description of Equipment
The test electrodes were mounted in an 8-inch o.d., 7.615inch i.d. cast acrylic resin pipe having a length of 40 inches. The loop in which the test section is located has been designed to provide a symmetric fully developed turbulent flow. A &foot vertical run of Van-Cor UPVC pipe having an inside diameter of 7.625 inches precedes the test section, so that the flow has an entrance length of 679 pipe diameters. -4 +-inch long trip ring consisting of a series of $-inch equilateral triangles around the circumference was located a t the bottom of the entry section. To provide a symmetric flow without any large scale disturbances, the entry section was connected to a square 22- by 22-inch stainless steel duct by a 22-inch long stainless steel nozzle. This square duct contained a 9-inch long settling chamber, a 20-mesh wire screen, another 9-inch settling chamber, and an 8-inch honeycomb consisting of 1;-inch square cells. The square duct was preceded by a %foot long vertical stainless steel diffuser with an upstream diameter of 6.06 inches. A 6-inch standard 90" elbow with turning vanes connected this diffuser to the piping coming from the pump. The electrolytic solution circulated through the system was 0.01N potassium ferrocyanide, 0.01N potassium ferrocyanide, and 1.7N sodium hydroxide. The ferricyanide is
1.5 VOLTS EACH
Figure 2.
Electronic circuitry
converted to ferrocyanide a t the cathode, which consists of the surface of a test electrode. The anode consists of several large sheets of nickel located in the piping downstream from the test section. The experiment was conducted with all of the necessary precautions outlined by Mitchell and Hanratty (1966). The platinum test electrodes were mounted in an acrylic resin plug which was glued into the wall of the test section. 0.005-inch-wide slot was cut along the length of a &inch plug which was 1 inch long. A 13inch long X inch wide X 0.003 inch thick platinum sheet was inserted in this slot and glued in place with epoxy cement. One end of the plug was machined to produce an electrode of desired L and TI'. The machined groove a t the end of the plug was filled with epoxy. After drying, the machined end was polished t o produce a platinum electrode flush with the plug surface. A &inch hole was drilled in the wall of the test section. The plug containing the electrode was inserted into the $inch hole and the slit was oriented to the center of a line drawn on the test section. This was a 0.005-inch groove drawn on the test section parallel to its axis and a t the proper distance from the center line of the $-inch hole t o produce the desired value of cp. The plug was cemented into the hole with the electrode end flush with the wall of the test section. After the cement had dried, the surface of the plug was sanded smooth with progressively finer grades of emery paper. Because of the low frequency of the electric signal from the electrode the electronic equipment and circuitry were slightly different frcm those used by 11itchell and Hanratty (1966). The circuit shown in Figure 2 converts the current fluctuations in the electrochemical circuit to voltage fluctuations. The operational amplifier connected to the electrode holds the cathode voltage constant. The value of the resistance R, varied depending on the Reynolds number and the electrode from 0.4 t o 2.8 megaohms. The d x . level in the signal coming from this first operational amplifier is eliminated by supplying a bucking voltage to the positive input of a Nexus SQl0.i operational amplifier in open-loop mode. The value of the bucking voltage was decided by integrating the signal from the first operational amplifier for 150 seconds with a Boonshaft and Fuchs, Inc., hIodel711 CL analyzer. Because of the predominance of low frequencies, an averaging time of about 45 minutes was used to determine the root-meansquared value of the fluctuating signal. The analyzer would not be stable over an integration time larger than 10 minutes, so the average of large number of 150-second analyses was taken. Results
Mass transfer intensities were measured over the Reynolds number range of 26,600 t o 84,200. Four electrodes, each VOL.
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Table 1. Electrode No.
(p;
W/L
200; 10 20"; 10 20"; 10 15"; 13.33 15'; 13.33 123"; 25 121"; 25 90": 10 900: 10 90"; 10 90"; 10 121'; 16
1 1A
1B 2 2A 4 4B 7 8
9
10
3
Measured Mass Transfer Intensities
(@)'/*/Kfrom Different Electrodes
Re
Re
26,600
0.1415 0.132 0.128 0.150 0.148 0.162 0.1585 0.1185 0,1176 0.1195 0.115 0.1558
Re
Re
35,900
46,900
66,600
Re 84,200
0.146 0.143 0.143 0.141 0.139 0.149 0.133 0.144 0.155 0.153 0.149 0.1475 0.151 0.140 0.1595 0.1635 0.155 0.152 0.159 0.168 0.1565 0.1545 0.1194 0.117 0.1146 0.1115 0,1209 0.1183 0.1141 0.1110 0.1218 0.1176 0.120 0.1139 0.1199 0.1170 0.120 0.1112 0.1577 0.1595 0.1574 All electrodes have length L = 0.003inch.
with W / L = 10, were oriented a t 90" to the mean flow direction. Three electrodes, each with W / L = 10, were oriented a t 20". For orientations of 15" and 123", two electrodes with W / L = 13.33 and W / L = 25 mere used. The intensity of the signal for each electrode was constant up t o Re = 46,900 and decreased slightly a t higher Reynolds numbers. This decrease could result from averaging over the electrode surface so only the results for Re 5 46,900 were used to estimate the intensity (see Table I). The average of all the intensity measurements on 90' electrodes at Re 5 46,900 is 0.1186. From Equation 10 a value of = 0.356 is calculated. Using this value of (G)l/*/& values of (G)l/*/& = 0.094, 0.086, and 0.082 are calculated from the average mass transfer intensities measured for angles of 20°, 15", and 123" for Re 5 46,900. The average of these three results, 0.087, is in good agreement with the value estimated by Fowles, and it is therefore concluded that close t o the wall w is increasing linearly with distance from the wall.
(c)1/2/s,
Av. Intensity for Re 5 46,900
(g)l/2/&
(G)l/Z/%
0.139
0.0938
0.149
0.0861
0.162
0.0822
0.1186 0.1586
0.356 0.083
model of turbulent mass transfer with a wall which involves transverse velocity fluctuations dictates that the fluctuations in the mass transfer rate are correlated with the fluctuations in as,/&. Therefore one expects the second term in parentheses t o be significant unless ds,/& is negligible. If the extrapolation of Laufer's measurements is correct then sz = 0 and therefore as,/& = 0. However, from Fowles, measurements and the results reported in this paper we find sZ # 0. Since the turbulence scale in the z-direction is small (Mitchell and Hanratty, 1966), we conclude that as,/& is significant and that E varies as y3 close t o the wall. From the conin the region y+ < 3 shown siderable variation in by Fowles' measurements one could also expect that this limiting behavior should not be seen unless the concentration boundary layer has a thickness of 6,f < 0.5 or that the Schmidt number is greater than 5000. Previous studies of turbulent mass transfer to the wall of a pipe (Son and Hanratty, 1967) carried out in this laboratory at a Schmidt number of 2400 might not have been made a t high enough Schmidt numbers to observe this limiting behavior.
(wz)l/z/c
Limiting Relation for Eddy Diffurivity
The conclusion that close t o the wall w varies linearly with distance from the wall has implications with respect t o the interpretatioii of studies of turbulent mass transfer between a fluid and the walls of a pipe. Mass transfer results at, high Schmidt numbers have been interpreted by defining a n eddy diffusivity
Acknowledgment
The authors are honored t o have the opportunity to contribute to this memorial issue. One of the authors has fond memories of his graduate studies with Richard H. Wilhelm and looks upon the activities during this period as a cornerstone of his own career. literature Cited
where u is the fluctuating velocity component in a direction perpendicular t o the wall, C is the time-averaged concentration, and c is the fluctuating concentration. There has been considerable discussion as t o what is the limiting relation for E as y-+ 0. If t% is expanded around y = 0 one obtains
From Equation 2 it can be seen that E will vary as y4 or higher if the two terms in parentheses are zero. The measurements of Mitchell and Hanratty (1966) indicate a very large scale for s, in the x-direction, and one would expect the first term in parentheses to be close t o zero. Any reasonable
192
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FUNDAMENTALS
Fowles, P. E., Sc.D. thesis, Massachusetts Institute of Technology, 1966. Hinae, J. O., "Turbulence," McGraw-Hill, New York, 1959. Hummel, R. L., Popovich, A. T., Chem. Eng.Sci. 22, 21 (1967). Kline, S. J., Reynolds, W. C., Schraub, F. A., Runstadler, P. W., J . Fluid Mech. 30, part 4, 741 (1967). Laufer, J., Natl. Advisory Comm. Aeronautics, NACA Rept. 1174 (1954). Mitchell, J. E., Hanrattg, - . T. J., J . Fluid Mech. 26, part 1, 199 (1966)'. Son, J. S., Hanratty, T. J., A.I.Ch.E. J. 13, 689 (1967): Wenael, H. G., Mat1iews,' M. J., University of Illinols Water Resources Center Research, Rept. 13 (1968) RECEIVED for review January 10, 1969 ACCEPTED February 28, 1969 Work supported by the National Science Foundation under Grant NSF GK-2813.
