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of the 37th National 4th International Conference on Fluid Mechanics FluidPower Power Proceedings ofProceedings the 37th International & 4th&National Conference on Fluid Mechanics andand Fluid December 16-18, 2010, IIT Madras, Chennai, India.

FMFP2010 FMFP10 - AM - 01

December 16-18, 2010, IIT Madras, Chennai, India

FMFP10 - AM - 01

SPHERE ROLLING DOWN AN INCLINE SUBMERGED IN A LIQUID ! ! Pravin K. Verekar Department of Mechanical Engineering Indian Institute of Science Bangalore, Karnataka, India

Jaywant H. Arakeri Department of Mechanical Engineering Indian Institute of Science Bangalore, Karnataka, India

[email protected]!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!jaywant@mecheng.iisc.ernet.in! ! !

ABSTRACT

INTRODUCTION

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A sphere rolling down an inclined plane submerged in water is studied experimentally in order to understand the different forces acting on it. This forms a class of solid-fluid interaction problems that include sediment transport, movement of gravel on ocean floor and river bed due to water currents. The flow development around the rolling sphere is elucidated in order to highlight its implications on the nature of hydrodynamic forces that act on the sphere. Equation of motion for the sphere is solved numerically and the experimental data is fitted on these solutions; the best fit gives the values of the force coefficients. Keywords: rolling sphere, inclined plane, flow development, hydrodynamic forces, force coefficients.

Experiments are done with a sphere rolling down an inclined plane submerged in quiescent water. These experiments are conducted to determine the hydrodynamic force coefficients. The experimental setup consists of a glass tank 15 cm wide by 14 cm deep by 61 cm long. At one end, the glass tank is fixed at the base with two levelling screws on either side, and at the other end, the tank rests on a spherical pivot mounted centrally. The levelling screws allow the tank to be tilted to the desired angle. The motivation for this study comes from the need to improve understanding of the solidfluid interaction problems such as sediment transport, movement of gravel on ocean floor and river bed, etc.

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EXPERIMENTS

FLOW DEVELOPMENT AROUND THE SPHERE

A solid rigid smooth sphere is released from rest on the inclined bottom glass plane of the tank which is filled with water. The sphere rolls down the plane under the influence of gravity. Initial motion of the sphere is with decreasing acceleration until it attains terminal velocity. The spheres used for the experiments along with the inclined plane angles kept during these experiments are given in Table 1. The spheres are so chosen because they have good material homogeneity and are perfectly spherical. The motion of the sphere is photographed in front view using Photron FASTCAM PCI R2 model 500 digital camera. Photography for some experimental runs is done in close view which captures the initial acceleration portion of the sphere motion while for other runs it is done in far view which captures the full motion of the sphere till it attains terminal velocity. The displacement of the sphere in pixels is obtained from its translation in the digital images and the time increment is known from the framing rate. The conversion scale for distance is based on the sphere size in the digital image given by pixels per mm of the sphere diameter. The paired data of the displacement and time are used later for the kinematic analyses of the sphere. Table 1: Experimental settings Sphere description 1

Acrylic sphere No. 1

2

Acrylic sphere No. 2

3

Pool ball

Diameter (cm)

Specific gravity, !

Inclined plane angles, "

Ratio of diameter, D to width of tank, W

2.54

1.18

1.8#, 5.7#

0.17

5.08

1.19

1.7#, 2.8#

0.34

5.24

1.70

2.9#

0.35

The sphere is released using a pair of tongs made from aluminium wire of diameter 1.5 mm. In some experiments a pair of tongs made from steel wire of diameter 1.2 mm is used. To find the angle of the inclined plane, the difference in height of the surface of water from the bottom of the tank is measured at a separation of 50 cm and 30 cm along the incline of the bottom of the tank; and then the average is taken of the two values obtained from the inverse of sine of the ratio of the difference in height to separation distance. !

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The flow development around the sphere is explained here. As the sphere begins to roll, the Reynolds Re based on instantaneous velocity (hereafter called instantaneous Reynolds number) are in the creeping viscous flow regime. For creeping flow or Stokes flow regime, Re is less than 1; this translates for the experiments carried out as: (i) for acrylic sphere No. 1, the instantaneous velocity V of the sphere less than 0.004 cm/s, and (ii) for acrylic sphere No. 2 and pool ball as the velocity less than 0.002 cm/s. What is noticed is that, the Stokes flow is momentary and appears to have no consequence on the later flow development. Creeping viscous flows are explained in Langlois, 1964. The starting flow over the rolling sphere can be treated as irrotational; here the fluid particles glide over the surface of the sphere and the velocity field has a potential which is completely defined by knowing the instantaneous normal velocity of the surface of the sphere; this means that the irrotational motion is entirely without memory (Lighthill, 1986). Cox & Cooker (2000) have found that for an irrotational flow past a sphere touching a tangent plane, the flow velocity is singular at the point of contact and the flow speeds around the point of contact are large. Also it is noted that the influence of the tangent plane is small one radius away and the flow there is very similar to that for an isolated sphere. The starting potential flow over the sphere is brief as the no-slip condition takes hold at the solid-fluid interface. The fluid particles next to the surface adhere to it and the succeeding adjacent fluid layers are sheared until the outer edge of the boundary layer. The boundary layer is a highly strained flow and is the region where all the vorticity is confined. This thin viscous layer is pressed upon by the outer potential flow, and its growth is governed by the momentum diffusivity of the fluid and !

the convective velocity of the outer flow. The fluid particles in the boundary layer in overcoming the viscous friction lose kinetic energy; hence this retarded layer of flow cannot follow the curvature of the sphere at the rear side as the fluid particles with smaller kinetic energy cannot overcome the positive pressure gradient present in this region and the boundary layer separates; initially forming a separation bubble at lower Re, and later in time shedding lumps of vorticity at higher instantaneous Re (see Schlitchting, 1968). Boundary conditions no longer govern the separated flow and as Lighthill (1986) puts it “all of the memory in a fluid flow lies in its vorticity; which, once generated is subject to convection and diffusion.” Stuart (1963) points out that “at large times, the boundary layer flow becomes quasi-steady, in the sense that it behaves like a steady flow of boundary-layer theory with instantaneous outer flow speed.” Since the flow velocities are higher around the bottom hemisphere near the point of contact than those around the top hemisphere, there is a differential shedding of the vortex sheet (boundary layer) that rolls up at the rear of the sphere. The circulating eddies detaching from the lower region are convected with higher velocity; they are pushed upwards by the outer potential flow and are pressed sideways by the slower rotating eddies above them; rotating fluid elements acquire an upward and transverse velocity components. This streamwise spiral vorticity cannot be captured properly in 2-D flow visualisation. Adding complexity to the flow features are two more phenomena peculiar to the rolling sphere: (i) a primary flow in the boundary layer which is opposing the potential flow in the top hemisphere and which is along the potential flow in the lower hemisphere; and (ii) a secondary flow along the surface, spiralling away from the poles towards the equator (Howarth, 1951). Three dimensional boundary layers are discussed in Moore (1956). Visualizations of the wake formation behind a rolling sphere in steady uniform flow have been done by Stewart et al. (2008) using fluorescein dye. The Reynolds numbers are varied from 75 to 350. They have defined a parameter “the rotation rate of !

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the sphere, $” which is the ratio of the tangential velocity on the surface of the sphere with respect to the centre of the moving sphere to the translation velocity of the sphere in ground reference frame. The case $=1 represents sphere rolling down an inclined plane. Their visualizations show a steady wake mode for low Re with transition to unsteady wake mode occurring at around Re=100. In the steady wake mode, they observe that the opposing motion of the top surface of the sphere to the outer flow creates a zone of recirculating fluid behind the sphere and the dye escapes this recirculation zone via a single tail along the centreline of the body. In the unsteady wake mode, they observe that the shedding from the top of the sphere takes the form of hairpin vortices similar to that for an isolated sphere. The flow features that evolve in time around the sphere determine the nature of the hydrodynamic forces that act on it. FORCES ACTING ON THE SPHERE The sphere rolls under the influence of the component of the resultant of the weight force minus the buoyancy force along the plane. This driving force can be expressed as % !s ' ! f & Vol g sin " ! where !s ! is the density

of the sphere, ! f ! is the density of the fluid, Vol ! is the volume of the sphere,! g is the acceleration due to gravity and " is the angle of the incline. When the driving force accelerates the sphere in still water, it needs to also accelerate the surrounding mass of water. This rate of change of momentum of the surrounding mass of fluid appears as a resisting force; this is accounted by considering an increase in the mass of the sphere by adding a separate mass, what is called added-mass (also called virtual mass), to the mass of the sphere. The addedmass is taken as a factor Ca times the mass of the displaced fluid. The added-mass force is !

dV where Ca ! is the added-mass dt coefficient and V is the velocity of the sphere. The other hydrodynamic resisting force to the motion of the sphere, the drag force, originates from the tangential skin friction (skin drag) and the difference in the quasi-steady normal pressure at fore and aft of the sphere (form drag). At the initial times when the boundary layer has not separated, the relative value of the skin drag to the total drag is high; while on separation of the boundary layer, it is the relative value of the form drag to the total drag that is higher; the higher form drag corresponds to the low pressures at the rear of the sphere when the separation occurs. The drag force is given by 1 Cd ! f A V V where Cd is the drag coefficient 2 and A is the projected area of the sphere perpendicular to the main flow. Also opposing the motion is the force of friction at the point of contact. When the sphere is in I dV pure rolling, this force is given by 2 where I R dt is the mass moment of inertia of the sphere about the diameter and R is the radius of the sphere.

given by Ca ! f Vol

Similar inclined plane experiments have been done previously by Carty (1957), Garde & Sethuram (1969), and Jan & Chen (1997). The values of Cd measured by Garde & Sethuram are higher than Carty’s values, while those measured by Jan & Chen are intermediate. The first two studies are not concerned with finding Ca, while Jan & Chen determine Ca as 2.0 using insufficient data. Chhabra & Ferreira (1999) have solved the Eq. (1) analytically using Jan & Chen’s drag and added-mass coefficient data. They use a single expression for Cd–Re relation, which gives the best fit to the Cd–Re equations of Jan & Chen (1997). The Cd–Re relation used by them is given below. 321.906 (3) ) 0.861 Cd ( 0.1 * Re < 10 5 Re

Equations for the potential flow past a sphere touching a tangent plane have been solved numerically by Cox & Cooker (2000). They find the added-mass coefficient Ca as 0.621. EXPERIMENTAL OBSERVATIONS AND ANALYSES

THEORY

The equation of motion of the sphere is given below. dV dV 1 I dV (1) ! Vol ( % ! ' ! & gVol sin " ' Ca ! Vol ' Cd ! V A ' 2

s

dt

s

f

f

dt

2

f

R 2 dt

The above equation is solved for the solid sphere where the ratio

!s is written as ! . The following !f

equation is arrived at. %1 .4 !

) Ca&

dV 3V2 ( % ! ' 1 & g s in " ' C d dt 4 D

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During the early part of the motion of the sphere, when the flow over the sphere is potential, the dominant hydrodynamic resistance comes from the inertia of the added-mass. During the later part of the motion, when the boundary layer has developed and then separated, the hydrodynamic drag is the main opposing force. !

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Collateral experiments on unidirectional, uniform, unsteady flow past the unconstrained acrylic sphere No. 2 resting on a horizontal plane in the unsteady water tunnel facility in the laboratory have shown that the sphere rolls on the glass surface without slipping. A sphere rolls without sliding when the rolling friction is less than the sliding friction. As explained in Starzhinskii (1982), pure rolling occurs when the ratio of the coefficient fr of rolling friction to the radius R of the sphere is less than coefficient fs of sliding friction; this statement is independent of the magnitude of the resultant driving force when this force passes through the centroid of the homogeneous sphere. Practically the coefficient fr depends solely on the materials of the rolling body and of the inclined plane; and if the materials of the !

rolling body and of the inclined plane are sufficiently hard, the force of the rolling friction is very small (Strelkov, 1978). Usually the ratio fr/R is considerably smaller than the coefficient fs of sliding friction (Starzhinskii, 1982). Verification that the sphere is in pure rolling is important for the validity of Eq. (1). Equation (2) is solved for the velocity V of the sphere using MATLAB differential equation solver ode45 (refer Shampine et al., 2003). Cd is taken as given in Eq. (3) and Ca is treated as a parameter; the values of Ca are chosen as 0.5, 0.621, 1.0, 1.5, and 2.0. The value 0.5 comes from the potential flow solution for an isolated sphere; the value 0.621 is taken from the potential flow solution for a sphere touching a tangent plane; the value 2.0 is as given by Jan & Chen (1997) for a rolling sphere on the incline. The values of ! ,! " ! and D come from the experimental settings, which are given in Table 1. The MATLAB code plots following graphs: distance X travelled by the sphere versus time t, velocity V of the sphere vs time t, and velocity V of the sphere vs dimensionless distance X/D. Experimental data points are superposed on these graphs. Velocity for the experimental data is obtained by fitting cubic spline for distance-time points and then differentiating the spline. Typical graphs for an experimental run for acrylic sphere No. 1 (diameter D=2.54cm) and inclination " = 5.7# are shown in Fig. 1 to Fig. 3, and for acrylic sphere No. 2 (diameter D=5.08cm) and inclination " = 2.8# are shown in Fig. 6 to Fig. 8. Figure 1 gives the distance travelled by the acrylic sphere No. 1 plotted against time for the entire motion of the sphere till it reaches terminal velocity. The coloured curves are obtained from the numerical solutions to the differential equation for different values of Ca; and it is found that the experimental points fall close to the green curve (Ca =0.621) for the entire motion of the sphere. This is also seen in Fig. 2 and Fig. 3. The sphere attains terminal velocity at about time t=2.5s and distance X/D=6. Taking the green curve as the best fit for the experimental points, the various forces, namely, the added-mass force, the drag force and the rolling friction, are calculated as the sphere rolls down the !

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Fig. 1. Distance vs time diagram

Fig. 2. Velocity vs time diagram

Fig. 3. Velocity vs X/D diagram

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incline; and the same are normalised by the driving force. The normalised forces are shown in Fig. 4 and Fig. 5. It is seen from Fig. 5 that the dominant influence of the added mass force at starting times is for a distance X/D up to about 0.3.

Fig. 6. Distance vs time diagram

Fig. 4. Normalised force vs time diagram

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Fig. 7. Velocity vs time diagram

Fig. 5. Normalised force vs X/D diagram

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For acrylic sphere No. 2, Fig. 6 gives plot of distance travelled vs time, Fig. 7 gives plot of velocity vs time, and Fig. 8 gives plot of velocity vs X/D. It is seen from these plots that the experimental points at initial times fall close to the green curve (Ca =0.621) but later when nearing terminal velocity (at time t=3.5s and X/D=3) they deviate and fall below the green line. Calculations show that the Cd at the terminal velocity for the experimental points is !

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Fig. 8. Velocity vs X/D diagram

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1.2 times higher than that given by the green line. Taking Ca as 0.621 and Cd as 1.2 times that given by Eq.(3), the differential equation of the motion of the sphere is solved. This solution and the experimental points are shown in Fig. 9. The solution fits the experimental points adequately, except at the region where the acceleration ends and the terminal velocity starts. Taking this solution as the fit, the different normalised forces are calculated and shown in Fig. 10 and Fig. 11. From Fig. 11, it is seen that the dominant influence of the added mass force at starting times is for a distance X/D up to about 0.3. Terminal velocity Vt is found from the experimental data. Reynolds number Ret is calculated at terminal velocity. Coefficient of drag Cdexpt is calculated by equating drag force to the driving force, which it balances when the sphere has attained terminal velocity. Cdlit. is the drag coefficient obtained from Eq.(3), which is given by Chhabra & Ferreira (1999). Results are presented in Table 2.

Fig. 9. Velocity vs time diagram

Table 2: Experimental results

Fig. 10. Normalised force vs time diagram

Sphere description Acrylic sph. 1 Acrylic sph. 1 Acrylic sph. 2 Acrylic sph. 2 Pool ball

D cm 2.54 2.54 5.08 5.08 5.24

D/W

"

0.17 0.17 0.34 0.34 0.35

1.8# 5.7# 1.7# 2.8# 2.9#

Vt cm/s 3.9 7.5 5.6 7.4 14.9

Ret

Cdexpt

Cdlit.

990 1900 2800 3800 7800

1.23 1.06 1.19 1.13 1.09

1.19 1.03 0.98 0.95 0.95

Cd exp t Cd lit .

1.0 1.0 1.2 1.2 1.2

DISCUSSION AND CONCLUSIONS

Fig. 11. Normalised force vs X/D diagram

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It is confirmed experimentally that the Ca for a sphere rolling down an inclined plane is close to 0.621 and it holds good at D/W ratio of 0.17 and 0.35. The equation (Eq. (3)) for Cd for the rolling sphere given by Chhabra and Ferreira (1999) is valid for D/W ratio 0.17 but not for 0.35. At D/W=0.35, the Cd is higher by 1.2 times that given by the Eq. (3). It may be mentioned that the Cd-Re relation given by Eq.(3) is for a rolling sphere down an incline and cannot be used for a flow past a stationary !

sphere touching a tangent plane since the wake profile in the two cases is likely to be different. It also cannot be used when the rolling of a sphere on a horizontal plane is induced by a flow of fluid taking place around it. So the Cd-Re relation for a rolling sphere is problem specific. No such restriction applies for Ca except that the sphere is touching a tangent plane. From Fig. 5 and Fig. 11, it is seen that the dominant influence of the added mass force at starting times is for a distance X/D up to about 0.3.

A Ca Cd Cdexpt Cdlit. D fr fs g I R Re Ret t V Vt Vol W X

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Chhabra, R. P., Ferreira, J. M., 1999. An analytical study of the motion of a sphere rolling down a smooth inclined plane in an incompressible Newtonian fluid, Powder Technology, 104 pp. 130–138. Cox, S. J., Cooker, M. J., 2000. Potential flow past a sphere touching a tangent plane, Journal of Engineering Mathematics, 38 pp. 335–370. Jan, C., Chen, J., 1997. Movements of a sphere rolling down an inclined plane, Journal of Hydraulic Research, 35 pp. 689–706.

NOMENCLATURE

" ! !f !s

REFERENCES

angle of incline specific gravity density of fluid density of sphere projected area of sphere added-mass coefficient drag coefficient drag coefficient from experiment calculated at terminal velocity drag coefficient from eq. given by Chhabra & Ferreira calculated at terminal velocity diameter of sphere coefficient of rolling friction coefficient of sliding friction acceleration due to gravity mass moment of inertia of sphere about its diameter radius of sphere instantaneous Reynolds number Reynolds number at terminal velocity time instantaneous velocity of sphere terminal velocity of sphere volume of sphere width of tank instantaneous distance travelled by sphere

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Langlois, W. E., 1964. Slow Viscous Flow. Macmillan, New York. Lighthill, J., 1986. An Informal Introduction to Theoretical Fluid Mechanics. Clarendon Press, Oxford. Moore, F. K., 1956. Three-dimensional Boundary Layer Theory, Advances in Applied Mechanics, 4 (2) 159-228. Schlichting, H., 1968. Boundary Layer Theory, sixth ed. McGraw-Hill, New York, pp. 24–43. Shampine, L. F., Gladwell, I., Thompson, S., 2003. Solving ODEs with MATLAB, Cambridge University Press, New York, pp. 1– 131. Starzhinskii, V. M., 1982. An Advanced Course of Theoretical Mechanics, Mir Publishers, Moscow, pp. 88–89. Stewart, B. E., Leweke, T., Hourigan, K., Thompson, M. C., 2008. Wake formation behind a rolling sphere, Physics of Fluids, 20, 071704, 1–4. Strelkov, S. P., 1978. Mechanics, Publishers, Moscow, pp. 259–269.

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Stuart, J. T., 1963. Unsteady Boundary Layers. In: Rosenhead, L., (Ed.), Laminar Boundary Layers. Clarendon Press, Oxford, pp. 349–408.

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