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Migration of a sphere in tube flow

Published online by Cambridge University Press:  27 September 2005

B. H. YANG ,
J. WANG ,
D. D. JOSEPH ,
H. H. HU ,
T.-W. PAN  and
R. GLOWINSKI
B. H. YANG
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
J. WANG
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
D. D. JOSEPH
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
H. H. HU
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA
T.-W. PAN
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA
R. GLOWINSKI
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA
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Abstract

The cross-stream migration of a single neutrally buoyant rigid sphere in tube flow is simulated by two packages, one (ALE) based on a moving and adaptive grid and another (DLM) using distributed Lagrange multipliers on a fixed grid. The two packages give results in good agreement with each other and with experiments. A lift law $L \,{=}\, CU_s (\Omega_s-\Omega_{\hbox{\scriptsize{\it se}}})$ analogous to $L \,{=}\, \rho U\Gamma$ which was proposed and validated in two dimensions is validated in three dimensions here; $C$ is a constant depending on material and geometric parameters, $U_s$ is the slip velocity and it is positive, $\Omega_s$ is the slip angular velocity and $\Omega_{\hbox{\scriptsize{\it se}}}$ is the slip angular velocity when the sphere is in equilibrium at the Segré–Silberberg radius. The slip angular velocity discrepancy $\Omega_s-\Omega_{\hbox{\scriptsize{\it se}}acute;$ is the circulation for the free particle and it changes sign with the lift. A method of constrained simulation is used to generate data which is processed for correlation formulas for the lift force, slip velocity, and equilibrium position. Our formulae predict the change of sign of the lift force which is necessary in the Segré–Silberberg effect. Our correlation formula is compared with analytical lift formulae in the literature and with the results of two-dimensional simulations. Our work establishes a general procedure for obtaining correlation formulae from numerical experiments. This procedure forms a link between numerical simulation and engineering practice.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 540 , 10 October 2005 , pp. 109 - 131
Copyright
© 2005 Cambridge University Press

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