Numerical Simulation of a Small Bubble Impinging onto an Inclined Wall

This paper presents a numerical modeling of the collision between a small bubble –of a few hundred microns, initially moving at terminal velocity, and an inclined wall, with relevance to drag reduction schemes. The theoretical model uses the lubrication theory to describe the film drainage as the bubble approaches the wall, and compute the force exerted by the wall as the integral of the excess pressure due to the bubble deformation. The model is solved using finite differences. The trajectory of the bubble is then determined using equations of classical mechanics. This study is an extension of previous work by Moraga, Cancelos and Lahey, [Multiphase Science and Technology, 18 ,(2),2006] where the simulation and comparison with experiments was carried out for a horizontal wall. In the present study where the wall is inclined, the bubble trajectory is no longer onedimensional and axisymmetry around 1 the vertical axis is lost, allowing for more complex behavior. The influence of various parameters (Reynolds number, Weber number) is examined. Numerical results are compared with the experimental data from Tsao and Koch [Physics Fluids 9, 44, 1997] 1-INTRODUCTION Drag reduction by changing the nature of the fluid and the turbulence associated with it is an idea that has received its share of attention over the years. Back in 1985, Lumley and Kubo [1] observed that injecting polymers seemed to break down the generation of turbulence in the wall layer. More recently, experiments have shown evidence of some drag reduction –typically a few % on fullscale flat plate turbulence [2] such as would be found along a ship’s hull. Injecting micro-bubbles at the base of a Taylor-Couette cell has led to drag reduction of up to 20%[3]. Yet understanding the nature of the Copyright © 2006 by ASME interaction between the bubbles and the wall remains a challenge, particularly as it involves a wide variety of scales. Antal et al. [4] have derived a popular model for two-phase flows, however theirs is only valid for vertical walls. Moraga et al [5] have developed a model for inclined walls and deformable bubbles, but the model fails to account explicitly for the parameters of the flow such as the Reynolds number or the Weber number. Following an approach similar to Klaseboer et al. [6], Moraga et al. [7] have used lubrication theory to build an approximation for the force exerted by the wall on the bubble over a range of Reynolds numbers. The idea is to represent the effect of the wall via a pressure force corresponding to the drainage of the film between the bubble and the wall. The equations of motion in the viscous film describe the evolution of the film height (i.e., the interface fluid/bubble) and are coupled with the trajectory of the bubble centroid through the pressure-dependent wall force and the kinematic condition on the boundary of the interface. By integration of the equations the time evolution of the wall force can be determined. Moraga et al [7] then derived a simple, parameterbased model for the wall force which could in turn be integrated into more complex simulations for bubbly flows. In its present formulation, the model has only been derived in the axisymmetrical case (one dimension). This is not adequate to describe inclined walls – such as those along a ship’s hull – as axisymmetry no longer holds. The purpose of the paper is to evaluate to which extent a two dimensional extension of this approach can reproduce the dynamics of a bubble interacting with an inclined wall. This validation is ! ! a first step toward better modeling of bubbly flows near walls. The paper is organized as follows: we describe the equations in section 2. Practical and theoretical aspects of the implementation are detailed in section 3. Results for a horizontal wall and an inclined wall are presented in sections 4 and 5. A conclusion is given in section 6. 2-THE EQUATIONS OF MOTION The problem is to describe the motion of a bubble of density ρB, of volume V and equivalent radius R, rising initially at terminal velocity Vt a quiescent fluid towards a wall inclined at angle θ (see figure 1). Fig. 1: Physical configuration Let ρl and μl be respectively the fluid density and viscosity. One can define a Reynolds number associated with the flow