Introduction

Garde and Sethuraman [1] have experimentally developeda drag coefficient function for a sphere rolling down aninclined boundary of closely packed spheres, the wholebeing subl11erged in a quiescent fluid. In these experiments,the fluid was stationary relative to the boundary, whereasthe sphere was moving relative to both boundary andfluid. The sphere was moving through a fluid that hadneither ambient turbulence nor a velocity gradient normalto the boundary. Circulation was induced by the rollingof the sphere. Under these conditions, drag coefficientswere substantially higher, for a given Reynolds number,than those for a sphere moving in an infinite fluid.

This paper describes the experimental deterl11inationof the drag coefficient function for a stationary sphereon a horizontal bed of closely packed identical spheres.The experiments were done in a water tunnel. In theseexperiments the fluid was not quiescent but was movingrelative to both the sphere and the boundary. The flowdisplayed both ambient turbulence and a velocity gradientnormal to the boundary. The sphere was stationaryrelative to the boundary, and not rolling. Renee, thecirculation effects present resulted only from the velocitygradient in the approach flow, and not from rotationof the sphere.

C) A contribution from the USDA Sedimentation Laboratory,Southern Branch, Soil and Water Conservation Research Division,Agricultural Research Service, U. S. Dept. of Agriculture, Oxford,Mississippi in cooperation \Vith the Mississippi Agricultural Expe­riment Station and the University of Mississippi.

The purpose of this paper is to illustrate the differencebetween the drag coefficient functions for a sphere restingon a rough boundary and for a sphere rolling on a roughboundary. The drag coefficient data obtained by theauthor have been cited previously [2], in much less detailand in a different context, in a paper on the incipientmotion of alluvial particles.

Definitions

Figure 1 shows the cases of a sphere settling in an infinitefluid, rolling down an inclined rough boundary, andresting on a rough horizontal boundary. The drag, lift,and submerged weight forces are FI), F L , and W, res­pectively, and u is the velocity of the fluid relative to thesphere. Both F D and fi are defined as acting throughthe center of gravity of the particle. This definition isslightly in error for a sphere on a boundary, because inthiscase the skin friction forces which make up part ofthe total drag are not syml11etrically distributed over thesphere.

For a sphere at rest on a horizontal boundary, the so­caHed lift force component has been found by both Coleman[2] and Rao [3] to be acting downward at particle Reynoldsnumbers lower than about 100, and acting upward athigher Reynolds numbers. In the experiments describedhere, the drag force component was measured directly,and the lift cOl11ponent did not affect the determinationof the drag coefficient.

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Article published by SHF and available at http://www.shf-lhb.org or http://dx.doi.org/10.1051/lhb/1972001

Neil L. COLEMAN

The particle Reynolds number is defined as:

uDR=-­

'J

and the drag coefficient is:

(1)

where D is the sphere diameter, A is the projected spherearea, and p and v are, respectively, the density and kine­matic viscosity of the surrounding fluid. In the datagiven by Garde and Sethuraman [1] for the sphere rollingdown a rough boundary, Cn was corrected by the authorsto remove the effect of rolling resistance.

2 FuC -_.~

D- pAu" (2)

Apparatus

As shown in Figure 3, the upper baIl on the transclucerwas attached to a brass rod extending through the trans­ducer casing. This baIl was raised slightly above itsnormal support points on the unclerlying balls so that therod could deflect when a drag force was applied. Twosemiconductor strain gauges, glued to the rod and wiredin a haIf-bridge circuit, measured the drag force. Thetransducer was connected to a commercial strain indicatorfor reading out the drag force. The system was calibratedusing a specially constructed balance to apply knownforces to the transclucer.

The water tunnel used for these experiments is shownin Figure 2. The test section of this tunnel was a rec­tangular plexiglass conduit 10 inches wide by 4-3/4 incheshigh. A bed of closely packed llZ-inch plastic sphereswas glued to the floor of the conduit. This bed was3 feet, 8 inches long. The sphere on this bed, upon whichdrag force measurements were made, was a part of thedrag force transducer shown in Figure 3. This trans­ducer was located 8 inches from the downstream end ofthe fixecl bed. Figure 4 is a view of the inside of thewater tunnel with the transducer installecl.

(cl

~I~+ w

(b)

Fa

1

~w

1u

{al

1/ The forces on a sphcre. with rolling frictionneglcctcd in case (b).

Les efforts sollicitallt IIl1e sphère. POlir le cas (b),nous avons négligé les phénonlènes de jrottenIelllail rolliement.

2/ The watcr tunnel.

Le tlllllle!hydrodynamique.

18

LA HOUILLE BLANCHE / N° 1-1972

3/ The drag forcelransducer.

Sonde de nzesllredes e!.forts de traînée.

4/ The drag forcetransducer installedin the water tunnel.

Implantationde la sonde de mesuredes e!.forts de traînéedans le tunnelhydrodynamique.

19

Neil L. COLEMAN

@ Garde and Sethuraman; rallinll sphere

o Sphere at rest

roHing down incli ned boundary

Sphere in

10

51 Comparison of drag coefficients for rolling sphere,stationary sphere and falling sphere.

COlnparaÎson des coefficients de traînée corres­pondant à llne sphère en roulenlent.. â une sphèrefixe, et à une sphère en chute libre.

20

The velocity 11 was measured with a small total headtube and a piezometer tap located in a section of the tunneljust upstream from the ball on the transducer. The totalhead tube moved through a packing gland in the top of theconduit. It could be lowered to the elevation of the ballcenter for measuring the velocity 11, and then retractedso as not to interfere with the fiow during drag forcemeasurements. The tube and the piezometer tap wereconnected to a differential pressure transducer and strainindicator.

The force and velocity readout devices were connectedto a two-channel milliameter recorder.

The water tunnel was operated with either water orsolutions of sodium carboxymethy1cellulose, so that kine­matic viscosities from 1.0 X ]0--0 to 40 X ]0-0 ft2/s wereobtained. The apparatus provided a working range ofpartic1e Reynolds numbers from about 50 to 9,400, thelower limit being set by the sensitivity of the measuringequipment, and the upper limit by the capacity of thewater tunnel.

Experimental procedure

The first step in performing experiments was to mixa solution of sodium carboxymethyl-cellulose. The tem­perature-viscosity relation for the fiuid was found usinga falling ball viscosimeter in a controlled temperature waterbath. Fluid densities were mea'mred with a Westphalbalance.

Before each series of experiments with a given workingfiuid, the force transducer was calibrated and installed inthe water tunnel. At the beginning of each experiment,the force and velocity indicators were balanced, and therecorder was staTted to record initial no-fiow zero readings.The f10w was then started and increased gently to somedesired discharge, where it was held constant while a one­minute record of drag force was made. The total headtube was then introduced, and a one-minute velocityrecord was made. The temperature of the experimentalfiuid was also observed at this time. At the end of thevelocity recording period the fiow was stopped, and therecorder was allowed to record final no-f1ow zero valuesof force and velocity.

The above process was repeated three to five times foreach sc1ected discharge, beginning at the lowest fiow forwhich force and velocity measurements could be obtained,and proceeding by increments up to the highest fiowattainable. At the end of each experimental series, theworking fiuid was discarded, and a new fiuid prepared.

Analysis and results

The observed data consisted of the 1l0w velocity, àragforce, and fiuid temperature during each experiment, andthe temperature-viscosity relation for each fiuid.

The velocity and force records were analysed for thetime-average ve10city and force by measuring the areasunder the respective chart traces with a planimeter, and

LA HOUILLE BLANCHE / N° 1-1972

dividing the areas by the record length. The Reynoldsnumber R and drag coefficient CI) were then ca1culatedfrom equations (l) and (2).

Figure 5 contains the c1assical drag coefficient functionas given by Rouse [4] for a sphere settling in a quiescentinfinite f1uid, the Garde-Sethuraman [1] data for a sphererolling down a rough boundary in a quiescent fiuid, andthe water tunnel data for a stationary sphere on a roughboundary in a moving fiuid. The drag coefficients fora rolling sphere follow a function higher than that for asphere in free fan. In particular, the difference isgreatest for Reynolds numbers from about 70 to 10,000.In contrast, the drag coefficients for a stationary sphere,as obtained from the water tunnel experiments, clusterabout the free fall function, and this result occurs in thesame Reynolds number range where the data for a rollingsphere shows the greatest deviation.

In the absence of detailed studies of the local fiowpatterns around spheres on rough boundaries, it is onlypossible to speculate that the marked difference betweendrag coefficient functions for a rolling sphere and a sta­tionary sphere is a result of additional circulation inducedaround the rolling sphere. In view of the inherent asym­metIy of the fiow around a stationary sphere on a boun­dary, the close correspondence between the drag coef­ficient function for this case and for the sphere in freefall is remarkable.

Conclusions

The drag coefikient function for a sphere rolling ona rough boundary has been defined by other experimenters[1]. It predicts drag coefficients that are higher thanthose predicted by the c1assical function for a spherein free fall.

In contrast to the above findings, the water tunnelexperiments reported here show that the drag coefficientfunction for a stationary sphere on a rough boundarytends to correspond with the function for a sphere in freefall.

References

[11 GARDE (RJ.), and SETHURAMAN (S.). - Variation of the DragCoefficient of a Sphere Rolling Along a BOllndary. La HouilleBlanche, n" 7 (1969), p. 727-732.

[21 COLEMAN (N.L.). - A Theoretical and Experimental Stlldy ofDrag and Lift Forces Acting on a Sphere Resting on aHypothetical Streambed. Proceedings of the Twelfth Congressof the International Association for Hydraulic Research, FortCollins, Cola., (scptember 1967), vol. 3, p. 185-192.

[31 RAO (M.V.P.). - EfTects of Effluent and Influent Scepage on theHydrodynamic Forces Acting on an Idealized NoncohesiveSediment Particle. Ph. D. Dissertation, Utah State University,Logan, Utah, (1969), p. 133.

[41 ROUSE (H.). - Elementary Mechanics of Flllids New York:John Wiley and Sons, Inc., (1946), p. 376.

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