Invariant-based optimal fitting closure approximation for the numerical prediction of flow-induced fiber orientation

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Abstract
An invariant-based optimal fitting (IBOF) closure approximation is proposed to approximate the fourth order structural orientation tensor in terms of the second order structural orientation tensor and its invariants. IBOF adopts the most general expression of a full symmetric fourth order tensor using a symmetric second order tensor and an identity tensor. The six coefficients that appear in the expression are represented by polynomial expansions in terms of the second and third invariants of the second order orientation tensor, similar to in the natural (NAT) closure approximation. Unknown parameters in the polynomial expansions are determined by following the method introduced by an orthotropic fitted closure approximation, which is a least-square optimization fitting technique of various flow data generated from solutions of the probability distribution function. IBOF is a hybrid of the NAT and the orthotropic fitted approximations, which are types of eigenvalue-based optimal fitting (EBOF) closure approximations. The accuracy of IBOF is as good as EBOF, and IBOF requires less computational time to obtain a solution. Also, IBOF does not suffer from the singularity problems encountered in using the NAT approximations.