Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-20T01:33:59.677Z Has data issue: false hasContentIssue false
  • English
  • Français

Drag Force on a Sphere Moving in Low-Reynolds-Number Pipe Flows

Published online by Cambridge University Press:  05 May 2011

S.-H. Lee  and
Tzuyin Wu
S.-H. Lee*
Affiliation:
Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
Tzuyin Wu*
Affiliation:
Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
*
*Ph.D. student
**Professor
Get access

Abstract

In this paper, the drag force on a sphere moving constantly along the centerline of a circular pipe filled with viscous fluid (the falling-sphere problem) under low Reynolds number condition is investigated via numerical calculation. The incompressible Navier-Stokes equations are formulated in a pseudocompressibility form. The numerical scheme makes use of finite-volume method and the numerical flux terms are evaluated using the Total-Variation Diminishing (TVD) strategy commonly applied to the compressible flow. Steady solution is obtained by marching (iterating) in time until the artificial time derivative of pressure term in the continuity equation drops to zero.

In the calculation, six different Reynolds number (Re) ranging from 0.1 to 1 and seven different pipe-to-sphere diameter ratios (D/d) ranging from 5 to 40 are selected to study the pipe-wall effect. In each case, the drag force on the sphere is evaluated and the results are compared with the existing approximate theoretical values derived from correcting the Stokes' formula. Both results agree in trend, but with noticeable deviation in values, particularly for cases with large pipe-to-sphere diameter ratios. The deviation is due to the fact that theoretical values were based on the solution to the linearized Navier-Stokes equations (Stokes' creeping-flow equations), while the fully nonlinear form of the Navier-Stokes equations are adopted in the present calculations. Finally, a least-square regression technique is applied to collapse the calculated results into a single expression exhibiting the functional relationship between the drag force, Reynolds number (Re), and the pipe-to-sphere diameter ratio (D/d).

Keywords

Falling-sphere problemCreeping flowDragPipe-wall effect.
Type
Articles
Information
Journal of Mechanics , Volume 23 , Issue 4 , December 2007 , pp. 423 - 432
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Happel, J. and Brenner, H., Low Reynolds Number Hydrodynamics, Kluwer Academic Publishers, pp. 286–357 (1983).CrossRefGoogle Scholar
2.Schlichting, H., Boundary Layer Theory, 7th ed., McGraw Hill, pp. 115–116(1979).Google Scholar
3.Proudman, I. and Pearson, J. R. A., “Expansions at Small Reynolds Numbers for the Flow Past a Sphere and a Circular Cylinder,” J. Fluid Mech., 2, pp. 237–262 (1957).CrossRefGoogle Scholar
4.Wham, R. M., Basaran, O. A. and Byers, C. H., “Wall Effects on Flow Past Solid Spheres at Finite Reynolds Number,” Ind. Eng. Chem. Res., 35, pp. 864–874 (1996).CrossRefGoogle Scholar
5.Francis, A. W., “Wall Effect in Falling Ball Method for Viscosity,” Physics, 4, pp. 403–406 (1933).CrossRefGoogle Scholar
6.Haberman, W. L. and Sayre, R. M., David Taylor Model Basin Report No.1143. Washington, D.C.: U.S. Navy Dept. (1958).Google Scholar
7.Bohlin, T., “On the Drag on a Rigid Sphere Moving in a Viscous Liquid inside a Cylindrical Tube,” Trans. R. Inst. Tech., 155, pp. 1–63 (1960).Google Scholar
8.Fayon, A. M. and Happel, J., “Effect of a Cylindrical Boundary on a Fixed Rigid Sphere in a Moving Viscous Fluid,” AIChEJ., 6, pp. 55–58 (1960).CrossRefGoogle Scholar
9.Chorin, A. J., “A Numerical Method for Solving Incompressible Viscous Flow Problems,” J. Comput. Phys., 2, pp. 12–26 (1967).CrossRefGoogle Scholar
10.Chen, Y. N., Yang, S. C. and Yang, J. Y., “Implicit Weighted Essentially Non-oscillatory Schemes for the Incompressible Navier-Stokes Equations,” Int. J. Numer. Meth. Fluids, 31, pp. 747–765 (1999).3.0.CO;2-F>CrossRefGoogle Scholar
11.Harten, A., “High Resolution Schemes for Hyperbolic Conservation Laws,” J. Comput. Phys., 49, pp. 357–393 (1983).CrossRefGoogle Scholar
12.Laney, C. B., Computational Gasdynamics, Cambridge University Press, Cambridge, UK (1998).CrossRefGoogle Scholar
13.Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag, Berlin (1999).CrossRefGoogle Scholar
14.LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, UK (2002).CrossRefGoogle Scholar
15.Hundsdorfer, W. and Verwer, J. G., Numerical Solution of Time Dependent Advection Diffusion Reaction Equations, Springer Ser. Comput. Math. 33, Springer-Verlag, Berlin (2003).Google Scholar
16.Yoon, S. and Jameson, A., “A LU-SSOR Scheme for the Euler and Navier-Stokes Equations,” AIAA-87–0600 (1987).CrossRefGoogle Scholar
17.Yoon, S. and Jameson, A., “Lower-Upper Symmetric-Gauss-Seidel Method for the Euler and Navier-Stokes Equations,” AIAA J., 26, pp. 1025–1026 (1988).CrossRefGoogle Scholar
18.Van Leer, B., “Towards the Ultimate Conservative Difference Scheme, II, Monotonicity and Conservation Combined in a Second Order Scheme,” J. Comput. Phys., 32, pp. 234–245 (1979).CrossRefGoogle Scholar
19.Fidleris, V. and Whitmore, R. L., “Experimental Determination of the Wall Effect for Spheres Falling Axially in Cylinder Vessel,” Brit. J. Appl. Phys., 12, pp. 490–494 (1960).CrossRefGoogle Scholar