On the numerical solution of a convection–diffusion equation for particle orientation dynamics on geodesic grids

In this work, numerical methods for computing the dynamics of rigid rodlike particles suspended in a Newtonian carrier fluid are investigated. Such elongated particles or fibers have the potential to reduce drag in turbulent wall flows and are therefore of considerable interest in several application fields. One of the main computational challenges is the approximation of the orientation distribution function for the fibers in the domain. We focus here on this topic by expressing the dynamics of the distribution via a Fokker–Planck equation on the unit sphere, where each point on the sphere represents a particular orientation. The classical approach to solve this problem numerically is a Galerkin projection onto spherical harmonics. However, in the presence of shear the solution is approaching a delta distribution, leading to the problem that a high number of modes is needed to resolve it properly. As alternative, we present a new approach based on a geodesic icosahedral type grid in combination with the Finite Volume Method. We compare the new approach on a quasi-uniform grid to a Monte-Carlo reference solver and the IBOF closure as well as a spherical harmonics method and demonstrate at several test cases that this approach provides high quality solutions.