Application of the Modified Force-Coupling Method of Tracing the Trajectories of Spherical Bubbles with Solid-Like and Slip Surfaces

Abstract

The force-coupling method (FCM) developed by Maxey and Patel (2001) was modified and applied to trace the trajectories of spherical bubbles with solid-like and slip surfaces. Careful comparison was made to the experimental results of Takemura et al. (2000, 2002a, 2002b). First, the result obtained by use of the original version of the FCM was compared to the experimental results. It was found that the original FCM was not feasible for tracing spherical bubble trajectories. Then, a correction was made to the FCM calculation of the bubble velocity by renormalization in terms of the bubble Reynolds number, which could very well trace the trajectory of the bubble with a solid-like, no-slip surface, but not that of a bubble with a slip surface. Finally, a substantial correction was made to the monopole term of the FCM, which could trace the trajectory of a bubble with a solid-like or slip surface very well even for the Reynolds number up to 20.

Numerical Study of the Lift Force, Velocities and Pressure Distribution of a Single Air Bubble and Two Interacting Air Bubbles Rising in Quiescent Liquid

“…In the present paper, we proposed the calculation method with a more appropriate definition of the effective ranges. Specifically, the effective range of the force monopole term and that of the force dipole term of OFCM were changed to be applied to air bubbles, which had not been made in the previous papers [23] [24]. As a result, we reproduced more accurately the results of the experimental data of Takemura et al [13] than those in the previous study [23]…”

“…In the present study, we used the similar method as Guan et al [23], and derived the following equation for ( ) n i U according to Mei et al [26]: (8) in order to apply MFCM over a wide range of the bubble Reynolds number. Here, ( ) n Re ∞ is the bubble Reynolds number determined by the terminal velocity of a bubblerising in quiescent liquid, ( ) n U ∞ , given by…”

“…Note that in Guan et al [23], the equation corresponding to Equation (8) (Equation (12) in Guan et al [23]) was developed for the flow including a solid particle. This is a new correction introduced in the present paper for the air bubble case.…”

“…represents a stresslet acting on the fluid and has been already given by the following equation for an air bubble [23].…”

“…Very recently, Guan et al [23] [24] proposed the modification of OFCM in order to simulate the multiphase flow composed of liquid and air bubbles, where they called it the modified FCM (MFCM), and obtained fairly good agreement between the numerical results and the experimental data by Takemura et al [13] [14]. However, inconsistency remained in the definition of the effective ranges for the force terms of MFCM in the previous papers [23] [24], where the values valid for flow containing solid particles were used for that containing air bubbles. In the present paper, we proposed the calculation method with a more appropriate definition of the effective ranges.…”