A formula for the wall-amplified added mass coefficient for a solid sphere in normal approach to a wall and its application for such motion at low Reynolds number

作者：Yang F L^*

来源：Physics of Fluids, 2010, 22(12): 123303.

DOI：10.1063/1.3518764

This work re-examines the potential flow theory for a sphere in normal approach to a wall, based on the classical results derived by Lamb [Hydrodynamics (Dover, New York, 1932)] and Milne-Thomson [Theoretical Hydrodynamics, 5th ed. (Dover, New York, 1968)]. These authors generated an expression in which the kinetic energy for a sphere in an unbounded fluid is augmented by a wall correction function in terms of an infinite series that depends on the scaled center-to-wall distance, h* = h/a, with a denoting the sphere radius. By truncating the series at the order of h*(-3), the resulting one-term correction function, 3/8h*(-3), is widely employed to approximate the wall-amplified added mass coefficient, C(AM)(h*), in multiphase flow research. Nonetheless, this work shows that this one-term correction deviates greatly from corrections including higher order terms when the interstitial gap drops below the half sphere radius. Thus, an explicit formula is developed, for all h*, using a near-wall Pade approximation, an intermediate bridging function, and a far-field approximation. This proposed formula provides an efficient and reasonable approximation to the infinite series and thus may serve as an improved wall correction function as compared to the one-term formula. The developed formula is applied to compute the unsteady approach of a nonrotating sphere toward a wall in a viscous fluid at low Reynolds number condition. In addition to Brenner's wall correction on the quasisteady viscous force [H. Brenner, "The slow motion of a sphere through a viscous fluid towards a plane surface," Chem. Eng. Sci. 16, 242 (1961)], the current formula is employed to modify both the added mass coefficient, from 1/2, and the history force. This latter force is modified by integrating the wall-modified potential flow theory with the boundary layer theory. If the one-term correction is used in the equation of motion, underestimation of the sphere motion and the force magnitudes are observed. Lastly, the limiting value of the infinite series derived by Lamb and Milne-Thomson as h -> a is analytically evaluated, leading to a result in terms of the generalized zeta function, zeta(3, 1). When this limiting value is compared to the one-term correction at h=a, a 38% deficiency of the wall-augmented kinetic energy is revealed, resulting in an underestimated C(AM)(h*) that plays a crucial role in presenting the near-wall normal approach.

• 出版日期2010-12