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Research Papers: Multiphase Flows

# Influence of Torque on the Lift and Drag of a Particle in an Oscillatory Flow

[+] Author and Article Information
Paul F. Fischer, Gary K. Leaf

Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439

Juan M. Restrepo

Department of Mathematics, and Physics Department, University of Arizona, Tucson, AZ 85721

J. Fluids Eng 130(10), 101303 (Sep 04, 2008) (9 pages) doi:10.1115/1.2969456 History: Received October 21, 2007; Revised June 05, 2008; Published September 04, 2008

## Abstract

In the work of Fischer (2002, “Forces on Particles in an Oscillatory Boundary Layer  ,” J. Fluid Mech., 468, pp. 327–347, 2005; “Influence of Wall Proximity on the Lift and Drag of a Particle in an Oscillatory Flow  ,” ASME J. Fluids Eng., 127, pp. 583–594) we computed the lift and drag forces on a sphere, subjected to a wall-bounded oscillatory flow. The forces were found as a function of the Reynolds number, the forcing frequency, and the gap between the particle and the ideally smooth rigid bounding wall. Here we investigate how the forces change as a function of the above parameters and its moment of inertia if the particle is allowed to freely rotate. Allowing the particle to rotate does not change appreciably the drag force, as compared to the drag experienced by the particle when it is held fixed. Lift differences between the rotating and nonrotating cases are shown to be primarily dominated in the mean by the pressure component. The lift of the rotating particle varies significantly from the fixed-particle case and depends strongly on the Reynolds number, the forcing frequency, and the gap; much less so on the moment of inertia. Of special significance is that the lift is enhanced for small Reynolds numbers and suppressed for larger ones, with a clear transition point. We also examine how the torque changes when the particle is allowed to rotate as compared to when it is held fixed. As a function of the Reynolds number the torque of the fixed sphere is monotonically decreasing in the range $Re=5$ to $Re=400$. The rotating-sphere counterpart experiences a smaller and more complex torque, synchronized with the lift transition mentioned before. As a function of the gap, the torque is significantly larger in the fixed particle case.

###### FIGURES IN THIS ARTICLE
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Copyright © 2008 by American Society of Mechanical Engineers
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## Figures

Figure 1

Velocity profiles, given by Eq. 3, shown at equal intervals in time over the course of one period of oscillation. Re=100. (a) τ=80 and (b) τ=300. The horirontal lines depict the top and bottom positions of a unit-diameter sphere in (a) for gaps ϵ=0 (dashed) and ϵ=0.5 (solid), and in (b) for gaps ϵ=1 (dashed) and ϵ=0.5 (solid).

Figure 2

Maximum rotation angle θmax (deg) and peak normalized torque CT as a function of R for Re=100, ϵ=0.5, and τ=80

Figure 3

Dependence on Reynolds number. R=0.95, τ=80, and ϵ=0.25. Time series of the drag, lift, torque, and angular deflection for a freely rotating particle: (a) Re=5, (b) Re=100, (c) Re=200, and (d) Re=400. For a nonrotating particle: (e) Re=200 and (f) Re=400.

Figure 4

Dependence on the Reynolds number. R=0.95, τ=80, and ϵ=0.25. (a) Maximum torque of the nonrotating particle and (b) maximum torque of the rotating particle. (c) Maximum drag, with and without rotational effects; (d) the mean lift of the freely rotating and nonrotating cases. The DML is the difference between the two curves in (d).

Figure 5

Symmetry-plane velocity profiles and vorticity contours (unit spacing on [−5,5]) during deceleration: (a) Re=100, fixed; (b) Re=100, rotating; (c) Re=200, fixed; (d) Re=200, rotating; (e) Re=300, fixed; and (f) Re=300, rotating. R=0.95, τ=80, and ϵ=0.25.

Figure 6

Phase portrait for the instantaneous lift and drag. Dependence on Reynolds number. (a) Re=5, (b) Re=100, (c) Re=200, and (d) Re=400. Freely rotating (light); fixed (dark). R=0.95, τ=80, and ϵ=0.25.

Figure 7

(a) Maximum rotational angle (deg) and (b) peak normalized torque for the freely rotating sphere, as a function of ϵ and τ. (c) Peak normalized torque, comparing the rotating and nonrotating cases for ϵ=0.125 for all τ. R=0.95 and Re=100.

Figure 8

For Re=100, Re=0.95, and the DML dependence on τ. From top to bottom: ϵ=0.125, ϵ=0.375, ϵ=0.5, and ϵ=1.0.

Figure 9

For Re=100, DML (solid), its pressure (circles), and viscous (dashed) components as a function of τ; (a) ϵ=0.125, (b) ϵ=0.375, (c) ϵ=0.50, and (d) ϵ=1.00

Figure 10

Phase portraits for τ=40, Re=100, and R=0.95, and (a) ϵ=0.125, (b) ϵ=0.5, and (c) ϵ=1.00. Phase portraits for τ=120, (d) ϵ=0.125, (e) ϵ=0.5, and (f) ϵ=1.00. Phase portraits for τ=220, (g) ϵ=0.125, (h) ϵ=0.5, and (i) ϵ=1.00. (Light) freely rotating; (dark) fixed.

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