Orthotropic closure approximations for flow-induced fiber orientation

A new family of closure approximations, called orthotropic closures, is developed for modeling of flow-induced fiber orientation. These closures approximate the fourth-order moment tensor for fiber orientation in terms of the second-order moment tensor. Key theoretical concepts are that any approximate fourth-order tensor must be orthotropic, that its principal axes must match those of the second-order tensor, and that each principal fourth-order component is a function of just two principal values of the second-order tensor. Examples of orthotropic closures are presented, including a simple form based on linear interpolation and a formula that is fitted to numerical solutions for the probability density function. These closures are tested against distribution function solutions in a variety of flow fields, both steady and unsteady, by integrating the orientation evolution equation. A scalar measure of the difference between the exact and approximate second-order tensors quantifies the errors of various closures. The orthotropic fitted closure is shown to be far more accurate than any earlier closure approximation, and slightly more accurate than Verleye and Dupret’s natural closure. Approaches for further increasing the accuracy of orthotropic closures and ultimate limits to the accuracy of any closure approximation are discussed.