Analysis of Cluster Size Distribution and Scaling in 2-D Poiseuille Flow
Date Issued
2012
Date
2012
Author(s)
Shih, Che-Ming
Abstract
Solving industrial and engineering problems via the direct numerical simulation (DNS) has been as the third method of scientific investigation besides the theoretical analysis and laboratory experiments for the past several decades. Especially in the past decade the DNS in Computational Fluid Dynamics becomes a major research tool to analyze two-phase flow problems. The major focus of this thesis is to investigate the cluster size distributions and the scaling properties of particles interacting in two-dimensional Poiseuille flow via the DNS. There are five chapters in this thesis. In Chapter One we first recall some important references and introduce the physical phenomena closely related to the particle motion in fluid flow discussed in these references. Based on these fundamental results and theories, we go further in our investigation on the particle clustering and the scaling properties.
The mathematical models for the two-phase flow problems are the Navier-Stokes equations for the incompressible viscous fluid flow and the Newton-Euler equations for the particle motion, which are presented in Chapter Two. Through the integration formulas of the hydrodynamic forces and torque, these two sets of governing equations are coupled in a variational formulation. Then via a distributed Lagrange multiplier term, a fictitious domain formulation has been derived. In this thesis, we have used the distributed Lagrange multiplier/fictitious domain method (DLM/FD) as our numerical method for the DNS.
In Chapter Three, we introduce the space and time discretization first. Then we describe the computational problems encountered in the simulation and the algorithms used to solve those problems. The aforementioned Chapters are the foundation for the numerical results discussed in Chapter 4. We have considered two kinds of particle shapes, namely circular cylinder and elliptical cylinder, with the variation concentration between 5% and 40%. The computational domain is a 4 by 1 rectangle. The detail values of parameters are given in Chapter Four. Following the analyzing procedures given in Ding and Aidun (2006), we calculate both the averaged value of cluster sizes at each time step and averaged number n(s) of each cluster size s after the cluster distribution reaches it equilibrium. For the cases of higher concentration, the averaged value of cluster sizes increases linearly in time during the initial transient period. The effects of the Reynolds number and the diameter of particles have been studied. We found that the effect of the Reynolds number is similar to the results in Ding and Aidun (2006). Finally, the critical concentration as defined in Ding and Aidun (2006) has been identified for each case and used in the normalization factor to scale all data points of (n(s),s) where s varies through all obtained cluster sizes. Moreover, we have obtained a universal curve by using curve fitting to the normalized data points.
Based on the aforementioned results, we conclude that at lower Re (i.e., ) a universal scaling relation for the cluster size distribution is still held for particle interacting in two-dimensional Poiseuille flow for both particle shapes. In aspect of multiple particles moving in two-dimensional Poiseuille flow, we have observed that (1) the particle trajectories are self-regulated at low Reynolds number, which are totally different from those at higher Reynolds number and (2) the particles aggregate in middle of channel and form the long chains at higher concentration due to the higher flow velocity and the drafting between neighboring particles.
Subjects
Low Reynolds number
Cluster distribution
The size of the cluster
Critical concentration
Poiseuille Flow
Type
thesis
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