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turbulent component of velocity at the centre of a circular pipe of radius a and that at radius r,

f rRu' =0Jo

where u'= Vu2 and u is the component parallel to the axis at radius r.This integral forms a useful check on the accuracy of the methods used in measuring turbulence. It is verified in measurements made at the National Physical Laboratory by Mr. L. F. G. Simmons.

546 G. I. Taylor

Fluid Friction Between Rotating Cylinders I—Torque Measurements

By G. I. T aylor , F.R.S.

( Received31 August, 1936)

The stability of fluid contained between concentric rotating cylinders has been investigated and it has been shown that, when only the inner cylinder rotates, the flow becomes unstable when a certain Reynolds number of the flow is exceeded. When the outer cylinder only is rotated, the flow is stable so far as disturbances of the type produced in the former case are concerned, but provided the Reynolds number of the flow exceeds a certain value, turbulence sets in. The object of the present experiments was partly to measure the torque reaction between two cylinders in the two cases in order to find the effect of centrifugal force on the turbulence, and partly to find the critical Reynolds numbers for the transition from stream-line to turbulent flow.

The apparatus is shown diagrammatically in fig. 1.*

D escription of A pparatus

The outer cylinder was made of brass turned inside and out. It measured 8T1 cm. inside diameter by 84-4 cm. long. The centre portion was a brass casting which contained two circular brass trapdoors made so that, when in place, they fitted flush with the inside surface

* This sketch merely indicates the general nature of the measurements. More detail is given in fig. 1, Part II.

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Fluid Friction Between Rotating Cylinders 547

without a break. The cylinder was balanced so that it could rotate without vibration and was then fitted centrally on a turn-table which ran between two ball-bearing collars and rested on a ball-bearing thrust

Fig. 1—Sketch illustrating method employed.

block. This table could either be rotated by means of a tight belt or held against friction when the inner cylinder was revolving by a thread which ran over an accurately made ball-bearing pulley and was attached to a scale pan.

The inner cylinder was supported on a single steel ball rotating on a



548 G. I. Taylor

flat, hard, bronze end-plate, being kept in position by a plain bearing which was lubricated by grease forced in from below.

The inner cylinder was made of a pile of cylindrical blocks of ebonite threaded on a turned steel rod. The blocks were kept in place by circular end nuts so shaped that they cleared the bottom and top of the inner cylinder by about 1 mm. As the experiments proceeded, the inner cylinder was turned down so that an increasing series of thicknesses were obtained for the annular space between the cylinders.

The top of the inner cylinder was slightly conical in shape and the cover of the outer cylinder was similarly shaped on its under side. A hole was bored horizontally through the cap to the highest point of the cone, and this hole was sealed with a needle valve as soon as the overflow began, while the apparatus was being filled with fluid. To fill the apparatus, a long funnel was screwed to a cock at the bottom of the outer cylinder. It was filled with the fluid while the cock was closed and all bubbles allowed to rise to the surface. The cock was then opened and the level of the fluid in the funnel was kept up till overflow started at the top. The needle valve was then closed and the cock turned off. The inner cylinder was set rotating to free the bubbles at intervals during the filling process.

The upper bearing between the two cylinders was a greased ball race, set between two narrow disks which only just cleared the spindle.

The upper bearing between the fixed upper support and the inner cylinder was a ball-bearing carried on a brass plate. In setting up the apparatus the inner cylinder was trued up. The brass plate and ball bearing were slipped over the top. The upper’fixed support was then brought into position so that a hole in it was concentric with the top of the inner cylinder. The plate containing the ball-bearing was then fixed to the upper fixed support by means of three bolts which held it firm against three adjustable pins. In this way the whole apparatus was set up true so that the friction between the cylinders when empty was very small.

A driving belt or torque thread could be fitted to the inner cylinder in the same way as to the outer one.

To make a measurement, the inner or outer cylinder was driven by a tight belt and its speed measured by timing a revolution counter attached to it. The torque required to hold the other cylinder in position was measured by putting weights in the scale pans (a) till the cylinder was just able to move in the same direction as the belt driven cylinder, and ( ) till it just moved in the opposite direction. The difference between the two was due to the friction against fixed supports, so that the friction torque between the two cylinders was taken as the mean of and ( ).




The torque between the cylinders is due partly to friction between the bearings and partly to fluid friction. The torque was measured both when the apparatus was full and when it was empty. The difference is due to fluid friction, nearly all of which was due to the tangential stress on the cylindrical walls, though a small fraction must have been due to fluid friction between the top of the inner cylinder and the top of the outer cylinder and between the bottom of the inner cylinder and the bottom of the outer cylinder.

Fluids Used—The following fluids were used: water, pentane, aviation spirit, and various mixtures of glycerine and water. Their viscosities depended on the temperature, and they were determined by means of calibrated capillary tubes for a range of temperatures.

The relevant properties at 15° C. are given in Table I.

T able ILiquid Density Viscosity v

Water ....................................... 1 00 0 0114 0 0114Pentane ................................... 0-634 0-0024 0-00379Glycerine and water ............... 1*120 0 -0524 0-0468

„ ............... 1*171 0-153 0-131„ ............... 1-083 0-0263 0*0243„ ............... 1-046 0-0174 0*0167

Aviation spirit (18° C .) .......... 0-729 0 -00457 0 -00625

With pentane great precautions were necessary to ensure that the apparatus was full and that there were no bubbles.

D ata

The radius of the torque pulley on the outer cylinder was 9*78 cm The radius of the torque pulley on the inner cylinder was 5*01 cm. As an example of the action of the apparatus the results of a set of observations made with water are given below.

Outer cylinder diameter 8*11 cm.Inner cylinder diameter 7*78 cm.W = weight of scale pan = 22*5 gm.The tension in the thread round the torque arm is

W -f load in pan = when the pan is just falling = P2 when the pan is just rising.

The weight which counterbalances the torque due to fluid friction is taken to be

P = 2 ( P i + P 2 )full ~~ ( P i + P2)em pty-



550 G. I. Taylor

G = torque expressed as Pg X (radius of torque pulley in cm.). N = number of revolutions per second.

O bservations

Outer cylinder rotating, water

/N = 16, apparatus empty Px = W + 20

G = 7-38 x 105G/pN2 = 3-03 X 103, N/v = 1 -45 x 103.

.N = 22*4, full Px = W + 350 ^P = 330 gm.

T = 17-5° C. P 2 = W + 360 JG = 16-23 x 105

G/pN2 = 3 - 2 x 103, N/v = 2-17 x 103.

/N = 11-48, empty P- = W -4- 10

G/pN2 = 3-10 x 103, N/v = 1 -04 x 103.

N = 15-0 P = 145, G = 7-12 x 105, G/pN2 = 3-17 x 103N/v = 1-37 X 103

N = 10-78 P = 75, G = 3-69 X 105, G/pN2 = 3-17 X 103N/v = 1-00 x 103

N = 17-65 P = 190, G = 9-33 x 105, G/pN2 = 3-00 x 103N/v = 1-62 x 103

N = 20-55 P = 280, G = 13-7 X 105, G/pN2 = 3-25 X 103N/v = 1-96 X 103

N = 22-8 P = 355, G = 17-5 x 106, G/pN2 = 3-36 X 103N/v = 2-23 X 103

P 2 = W + 10 ]N = 15-6, apparatus full Px = W + 170

P = 150 gm.

P 2 = W + 160

|N = 11-28, full' r 2 = w t 5U

G = 3-94 x 105




Inner cylinder rotating, water

("N = 20, empty P, = O + 30 ,P 2 — O + 0

P = 245 + W - 15 = 253 gm.1 = 21-1, full P, = W + 260

p2 = w + 230

G = 253 x 981 x 9-78 = 24-3 x 105 G/pN2 = 5-45 = 103, N/v = 1-92 x 103.

N = 17-58 P = 185, G = 17-8 x 105, G/pN2 = 5-75 x 103N/v = 1-58 x 103

N = 24-4 P = 323, G = 31 -0 x 103, G/pN2 = 5 -2 1 x 103N/v = 7-22 x 103

N = 29-7 P = 478, G = 45-9 x 103, G/pN2 = 5-21 x 103N/v = 2-7 X 103

\

'/

1Inner cylinilei rotating

Outer cylindt^ ■■■r rotating <=H 1 1 1 I

/ L \©Critical

i for instavalue x bility \o >c ^

\ o\x\ *-V n

' a~eV-ta- o" °

)

i L.

ZCL

o60

3

Log Np/[xFig. 2—Rx = 4 05, R 2 = 3 -9 4 . Upper curve, inner cylinder rotating. Lower

curve, outer cylinder rotating. Full line, calculated critical line for inner cylinder rotating. + v = 0-131; O v = 0-047; x v = 0-024; Q v = 0-011; <C>

v = 0-0038.

C omplete R esults

In representing the results, it must be remembered that the theory of Dynamical Similarity demands that for any given pair of diameters



Log

G/p

N-

Log

G/p

N:

552 G. I. Taylor

\ C ritica U" for inst.

Ny_____1

v a lu eability

\ +

\\ x h p - f n , - Q Q\ *3

\10 1*5 2-0 Z 5 3 0 3*5

Log Np/pt,Fig. 3—Rx = 4-05, R 2 = 3-89. + v = 0-131; © v = 0-047; x v = 0-024;

H v = 0-011; (©) v = 0-0062; <$> v = 0-0038.

\ Critical value for instability

N\

*

\ ^ f i x .

X

\ \s

&'&GK3-q _ —

■Q~'Q-Gr-Q-^_"O'- 'O —

Log Np/ptFig. 4—Rx = 4-05, R2 = 3 -8 3 . + v = 0-131; 0 v = 0-047; x v = 0-024;

□ v = 0-011; <$> v = 0-0038.



Log

G/p

Ns

Log

G/p


0 G

Log Np/pt

Fig. 5—Rx = 4 05; R 2 = 3-74. 0 v = 0 047; x v = 0 024; □ v = 0-011;<•> v = 0 0038.

Log Np/ptFig. 6—Rx = 4 05, R a = 3 -6 8 . © v = 0 047; x v = 0 024; Q v = 0-011;

<3> v = 0*0038.



Log

G/p

N2

Log

G/p

N!

554 G. I. Taylor

Log Np/[xFig. 7—Rx = 4 05, R 2 = 3-59. + v = 0-131; © v = 0-047; x vO-024; Q

v = 0-011; <$> v = 0-0038.

Log Np/p.Fig. 8—Rx = 4-05, R 2 = 3 -4 5 . + v = 0-131; © v = 0-047; x v = 0-024;

H v == 0-011; O v = 0-0038.



G/pN2 must be a function of N/v. In each case, therefore, these two quantities were calculated. In figs. 2—9 all the results are represented on diagrams giving log10(G/pN2) and log10(N/v). A separate curve is given for each diameter of the inner cylinder, namely 7-89, 7*78, 7-66, 7-48, 7*36, 7-18, 6*91, 6*40 cm. The outer one is the same for all, namely, 8*11 cm. The points for both types of measurement, inner


ZQ.060O

x § !

N

X\

~<Sk\ ifix

\ oV •x 1 §L_. A ____

>1----

ver limit o f-n >s rbulence i1 i

i1 »

1 1 ^ '

v Upper .n. i steady

i \

* <S>

imit of flow

Log NpFig. 9—R = 4 05, R 2 = 3 -2 0 . + v = 0 1 3 1 ; © v = 0 047; x v = 0 024;

0 v = 0 0 1 1 ; (©) v = 0 0062; <•> v = 0 0038; A glycerine and water, p = 1 046, v = 0017 .

cylinder rotating and outer cylinder rotating, are given on the same figure. The points corresponding with the former case are always above those corresponding with the latter except for the lowest values of N/v. When there is any possibility of confusion the points are distinguished on the diagram.

In order to see the effect of rotation on the turbulence, the theoretical value of G for steady motion was calculated in each case. Since in that case G is proportional to N, G/pN2 is proportional to 1/N, so that with



556 G. I. Taylor

the logarithmic coordinates of figs. 2-9 this theoretical curve is always a straight line at —45° to the horizontal axis. The calculated value for G is

n 2tt2D12D 22 /(jtNG = (D1 + D 2)(D 1 - D 2) > 0 )

where Dx is the diameter of the outer cylinder, namely 8T1 cm., D 0 is the diameter of the inner cylinder, / is the length, namely 84-2 cm.

(1) may be written in the form

log (G/pN2) - log (N/v) = log (2tt2/D 12)+ log D2 - log (D, + D 2) - log (20 = C, (2)

where t — \ (Dx — D 2) is the thickness of the annular space between the two cylinders and C represents the right-hand side of (2).

Table II gives the values of C for the values of D 2 used in the course of the work

T able IID 2, cm. . . . . 7-89 7-78 7-66 7-48 7-36 7-18 6*91 6-40C* .......... . . . 6-286 6-10 5-956 5-794 5-708 5-598 5-462 5-256t (cm.) .,. . . 0-110 0-165 0-225 0-315 0-375 0-465 0-600 0-855log (t/Rx) .. 2-433 2-609 2-744 2-890 2-966 1-059 1-170 1-324

t/R r ..........Number

. . . 0-0271in

0-0407 0-0555 0-0776 0-0924 0-1146 0-1480 0-210

fig. 10 1 2 3 4 5 6 7 8

* See equation (2).Inner cylinder used for velocity distribution measurement Part II), = 0-226.

Criterion of T urbulence

1— Outer Cylinder Rotating—In all the results given in figs. 2-9 it will be seen that for low values of N/v the observed values of G/pN2 are close to those calculated on the assumption that the flow is steady. At a value of N/v which depends on the ratio of the radii of the cylinders, the observed values of G/pN2 begin to leave the straight line which represents the conditions in stream line motion. The value of N/v at which this takes place is taken to be the critical value at which turbulence sets in. In the neighbourhood of this point the flow can be either steady or turbulent. In general there is a lower critical point below which the flow is in all cases steady, and an upper critical point above which the flow is always turbulent. At intermediate points the flow is usually stable if the rotation of the outer cylinder is steadily increased through the lower




critical point, but a slight disturbance, such as that produced by a slight rotation of the inner cylinder in the opposite direction to that of the outer one, will make the flow permanently turbulent.

As an example of the limits found in this way, experiments with the inner cylinder 6-40 cm. diameter m aybe cited. The fluid used was a mixture of glycerine and water of density 1 -046. The relevant observations are given below:

Apparatus Empty rP = lOv

N = 211 I mean 9 gm .

Apparatus Full

Speed increased gradually fp i — w + 10\ p = 4- 71 — 9 = 21 gmN = 20-8 lp 2 = W + 5 /

Speed increased gradually J p i = W + 10j _N = 21 -0 lp 2 = w + 7 /

After turning inner cylinder backwards to start turbulence, N —. 21*0

Pi W + 25-v IP

p 2 = W + 2036 gm.

The points corresponding with these observations are marked with points A in fig. 9. The value of log Uf/v corresponding with N = 21 -0 rev. per sec. is marked at D in fig. 11. It will be seen that it is in the middle of the range between the points A and B.

The critical range in which turbulent and steady flow were both possible could not always be determined from the torque measurements. A pair of glass windows was therefore fitted to replace the brass windows in the outer cylinder. The inner surface of the glass was ground to the same radius as the inside of the outer cylinder and was fitted flush, so that there was no obstruction to the flow. The steel axle of the inner cylinder had a hole down the centre so that a thin film of coloured fluid could be spread over the inner cylinder.* The upper and lower criteria found in this way with the smallest inner cylinder (6-40 cm. diameter) are shown in fig. 9, and are marked in fig. 11 by the points A and B.

* This method is described in “ Stability of a Viscous Liquid Contained Between Two Rotating Cylinders,” ‘ Phil. Trans.,’ A, vol. 223, p. 289 (1923).



558 G. I Taylor

2 __Inner Cylinder Rotating-The stability of the flow when the innercylinder is rotating was discussed by the present writer* some years ago, and it was shown that when the speed of rotation is gradually increased instability sets in as soon as the calculated criterion of instability isreached.

This criterion is■ - I -------1 - ■ S B

where

p = 0*0571 (1 — 0*652

p s

+ 0*00056(1 - 0*652 -f- V 1 2'

tc4v2 (Rx + R 2) 202/3R22

(3)

(4)

and t is the thickness of annulus.When t\R2 is smah, P - 0*0571 and Rx + R 2 may be taken as 2R, so

that (4) may be written approximately

0-057! (5)nr

log t/Rx + 2 log Uf/v = 4 log 7C — log 0 • 0 5 7 1 ;.3 • 232, (6)where U * O R i ^ 2 7 tNRx (7)

is the velocity of the outer cylinder relative to the inner one and N is the number of revolutions per second. Another expression equivalent to

[log (N/v)]crlt = 0*818 — log Rx I log (8)

and when R2 = 4*055 this becomes

[log N/v]crit H 0 • 514 - f log (9)

In most cases the lowest speeds of rotation were above the critical speed, but in the cases D xB 7*89 and Rx |§7*78 cm. the critical value of N/v falls within the range of observations. Values for [log (N/v)]crit given by (9) are:

Dx

T able III

1 * [log N/v];rit7*89 0 1 1 1*957*78 0*165 1*68

* Loc. cit., p. 318, equation (7.11). In the present paper Rx is the outer cylinder and R2 the inner, whereas in the paper here referred to Rx was the inner cylinder and R2 the outer cylinder.




E ffect of R otation on T urbulent Stresses

between the cylinders can be seen by comparing the values of G/pN* a, the same value of N /v (a) when the inner cylinder is revolving and (b) when the outer cylinder revolves. Up to the critical value in case fat the torque m the two cases is identical for a given speed of rotation (see fig. 2). Above that the torque becomes greater in (a) than in (6). Above the critical speed in case (b) the ratio of the torques in (a) and (b) may decrease, though that in (ci) still remains greater than that in The ratio G (case a) /G (case b) may be taken to represent the effect of rotation on the Reynolds stress in turbulent motion.

To compare the effects of rotations for various thicknesses of the annulus, the results have been plotted in fig. 10 so as to show log x/pU2 as a function of log U t/v for each value of t\Rx. Here t is the tangential stress on the outer cylinder, and U = 27tN R x as before. In fig. 10 the curves are traced directly from figs. 2-9, and in each case the origin shifted by the amount necessary to make the curves represent x/pU2 and XJt/v instead of G /pN2 and N /v. The correspondence between the data given in Table II and the numbers of the curves in fig. 10 is given at the foot of Table II.

The full lines represent observations taken with the inner cylinder rotating. The dotted lines with the outer cylinder rotating. It will be seen that the full lines are close together, whereas the dotted lines show a rapidly decreasing friction as tj Rx increases. It seems, therefore, that rotation does not very greatly affect the Reynolds stresses when the inner cylinder rotates, but has a large effect when the outer cylinder rotates.

The former of these effects might have been expected because it has been shown that when f/Rx = 0-49 and the inner cylinder rotates, the distribution of velocity is such that over 83% of the thickness of the annulus U r is constant. * In that region, therefore, the effect of rotation is likely to be small. The effect of decreasing t/Rx might be expected to reduce the proportion of the whole volume of the annulus in which Ur = constant without affecting very greatly the transition layers close to the surfaces of the cylinders.

* “ Distribution o f Velocity and Temperature Between Concentric Cylinders,” ‘ Proc. Roy. Soc.,’ A , vol. 151, p. 494 (1935).

VOL. CLVII.— A. 2 P



560 G. I. Taylor

Gnd/0 soiFig. 10—Results compared in non-dimensional form

Log

(U

//v)




The distribution of velocity when the outer cylinder rotates will form the subject of the second part of this paper. It seems that the distribution is then of a type which might be expected to be greatly influenced by rotation.

(b) Outer cylinder rotating

Couette

Calculated(a) Inner cylinderrotating

Experimental

-2 0 -1-5Log (*/Rx)

Fig. 11—Critical speeds.

E ffect of R otation on C ritical S peeds

The observed upper and lower critical speeds for the case when the outer cylinder is revolving are shown in fig. 11. The ordinates represent the critical values of log (t\J/ v) or log (tQRJ v). The upper and lower critical values are shown joined by a line to represent the range of states at which the change from steady to turbulent flow can take place. The abscissae are log (tjRj).

The calculated critical speeds (see equation (9)) when the inner cylinder

2 P 2



562 G. I. Taylor

is revolving are also shown, and an experimental point in one case, namely when R 2 = 8T1, Rx = 7-89, t — O i l .

Comparison w ith Couette’s Observations on the C ritical

Speed

The smallest ratio 7/Rx used in the experiments here described was 0-0271. The experiments were not carried to a lower value because it was thought that if t were less than 1 mm. difficulty would be experienced in securing accuracy and uniformity in the thickness of the annulus and at the same time preserving sufficient freedom from non-fluid friction between the cylinders. The apparatus of Couette* had a larger radius and less length than mine. This is permissible when small values of tjRx are being used, and Couette’s value of Rx was only 0-01685. His

experiments were designed primarily to measure viscosity, but he measured the critical speed of the outer cylinder when turbulence set in. He found it to be N ' = 55-69 rev. per minute at a temperature of 16-3° C.

Taking [x = 0-01108 at this temperature, 0-2465 cm. R 2 = 14-639 cm., [log(UY)/v]Crit = 3-28, and lo g ^ /R ^ = — 1-77.

The point corresponding with Couette’s experiments is shown in fig. 11. It will be seen that it fits in well with the present results.

E xtrapolation to C ase of T w o P arallel P lates

The limiting critical value rtJ/v when Rx is very small, must be that corresponding with the case of two parallel planes distant t apart and moving tangentially with relative velocity U. In fact, the critical values for the two cases shown in fig. 11 must ultimately coincide for large negative values of log t/R ±.If we consider the course of the upper curve ( b), fig. 11, it will be seen that it descends rapidly till about log t /Ri = — 1-3. At that point it flattens out, gradually descending. This flat part of the curve (from Couette’s log t /R1 = — to log t\Rx = — 1 -3) indicates that the critical value of log Uf/v for parallel plates must be less than 3-28, i.e., [U?/v]crlt < 2000. On the other hand, it hardly seems likely that the effect of a very minute curvature in the flow will be felt, so that it is unlikely that the critical value of U t/v will fall very far below 2000.

Now consider the lower curve (a), fig. 11. It seems that steady motion will break down into cellular vorticesf as soon as U//v exceeds the values on the line ( a). If, however, this line is produced till it cuts the pro-

* ‘ Ann. chim. (Phys.),’ vol. 21, p. 433 (1890). t Loc. cit., p. 327.



longation of ( b) it seems certain that the line will no longer represent the critical condition for values of Uf/v which are lower still. In fact, turbulence due to another cause, the cause which causes turbulence in case ( b),must now set in before rtJ/v has risen to the value necessary for the cellular vortices. Since the curve ( ) is descending slowly towards the rising line (a) their point of intersection* must fall in the limited range between the points X 1 and X 2 where the line (a) cuts (1) the prolongation of ( b) produced as a straight line with the same slope that it had in the lowest part of the observed range of t/Rx; (2) a horizontal line through the lowest observed point on (b), namely log (Uf/v) = 3 -28. In this way Xx corresponds with t\Rx = 0-001, and X 2 corresponds with tj R, = 0-00045.

It appears, therefore, that in order that curvature may have no effect on the critical Reynolds number of the flow between concentric rotating cylinders, t/R ± must be less than 1/1000.

The corresponding limits to the critical value of UY/v for flat parallel plates are, for Xl5 tJJ/v = 1260; for X 2, rtJ/v = 2000.

C omparison w ith C ouette’s T orque M easurements

In Couette’s apparatus the outer cylinder rotated and the inner cylinder was held in position by a load P. Couette measured the number of revolutions N ' per minute so that N ' = 60N. Some of his observations, for values of N ' above the critical value, are given in columns 1 and 2 of


Table IV.T able IV

N' P/N' log (r/pU2) log (Uf/v)56-08 0-3067 4-916 3-29771-56 0-4704 4-996 3-398

107-9 0-6883 4-993 3-581249-7 1-327 4-903 3-929453-3 2-111 4-846 4-192

Using the dimensions of Couette’s apparatus, 1 find that

log (t/pU2) = log (P/N') - log N ' + 1-1777 )( 10)

log (U//v) = log N' - log V + 1-5773 J

The values of t / pU2 and U t/v calculated from (10) are given in columns 3 and 4 of Table IV and are marked in fig. 10. It will be seen that they compare very well with the present results for larger values of //

* Loc. cit., p. 327.



564 Fluid Friction Between Rotating Cylinders

Stress C alculated from V elocity M easurements

In Part II of this paper the distribution of velocity is given when the outer cylinder rotates and the inner cylinder has a diameter of 6-27 cm., the outer one being 8*11 cm., so that ?/Rx = 0-227. The tangential stress at the outer surface was calculated from the velocity measurements and the results shown in fig. 10. It will be seen that the stress is very low, being rather lower than the lowest stress observed directly with the slightly larger inner cylinder 6*40 cm. diameter. This might have been expected because the direct measurements are necessarily too high owing to the fact that the fluid friction on the ends of the inner cylinder was not allowed for. On the other hand, the friction deduced from velocity distribution measurements taken near the middle of the cylinders is likely to give too low a result. It seems, therefore, that the true values ar/pU2 for the values of t/ less than 0*227 are not likely to be very much in error.

Summary

The torque between concentric rotating cylinders was measured in two cases, (a) inner cylinder rotating, outer cylinder fixed, ( ) inner cylinder fixed, outer cylinder rotating.

It was found, as was to be expected, that the critical speed at which turbulence begins is very much lower in case (a) than in case (b). The difference between the critical speeds in the two cases becomes rapidly greater as t\Rx increases, t being the difference between the radii, and Rx the radius of the outer cylinder. When //R x = 0*38 the ratio of the critical speeds in the two cases exceeds 1000, when Rx =--0*1 it is about 50, when t/R ± = 0-017 it is 6. It appears that Rx must be less than 0-001 before the effect of rotation on the critical speed disappears.

When the flow is turbulent the effect of rotation on the torque is small in case (a) but large in case ( b).



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