Lagrangian simulation of fiber orientation dynamics using random walk methods

Abstract

This thesis focuses on developing a two-way coupled framework for the numerical simulation of fiber suspension flows. The influence of fibers on the flow is accounted for by evaluating a non-Newtonian stress term incorporated into the Navier-Stokes equations. The accuracy of the analysis depends on a second-order tensor field used to approximate the orientation distribution of fibers. In this context, the disperse phase can be treated in the Lagrangian or Eulerian manner. We conduct a comprehensive comparison of these frameworks for one-way coupled scenarios in both two- and three-dimensional homogeneous flows. With a special focus on the Lagrangian approach, the algorithm for solving the two-way coupled fiber suspension flow in a segregated manner is proposed by incorporating the fiber-induced stresses in the finite element formulation of the Navier-Stokes equations.

In non-dilute suspensions, fiber-fiber interactions may cause spontaneous changes in the orientation of fibers. Applying the theory of rotary Brownian motion, the effect can be studied using a rotary diffusion term with a Laplace-Beltrami operator. In this work, we develop random walk methodologies to emulate the action of the diffusion term without evolving or reconstructing the so-called orientation distribution function. After deriving simplified forms of Brownian motion generators for rotated reference frames, several practical approaches to generating random walks on the unit sphere are discussed. Among the proposed methods, this research effort presents the projection of Cartesian random walks, as well as polar random walks on the tangential plane. The standard random walks are then projected onto the unit sphere. Moreover, we propose an alternative based on a tabulated approximation of the cumulative distribution function obtained from the exact solution of the spherical heat equation.

In the last part of this work, the random walk approaches are compared through several numerical studies, including the study of the orientation distribution of fibers in a three-dimensional homogeneous flow. Then, the two-way coupled solver is validated in a simple geometry, followed by performing a few three-dimensional numerical simulations to study the rheological behavior of the fiber suspension flow through an axisymmetric contraction. The effect of fiber-fiber interactions is also incorporated using the random walk methodology.