Direction-Splitting Schemes For Particulate Flows

This thesis introduces a new temporally second-order accurate direction-splitting scheme for implicitly solving parabolic or elliptic partial differential equations in complex-shaped domains. While some other splitting schemes can be unstable in such domains, numerical evidence suggests that the new splitting scheme is unconditionally stable even when using non-commutative spatial operators. The new direction-splitting scheme is combined with other splitting schemes to produce an efficient numerical method for solving the incompressible Navier-Stokes equations. Finite differences using staggered grids and sharp boundary-fitting is used to achieve second-order spatial accuracy. The numerical method is extended to perform direct numerical simulations of particulate flows where each rigid particle is used as Dirichlet boundary conditions for the Navier-Stokes equations, and forces on each particle are computed by performing surface integrals of the fluid stress. The method is validated by reproducing experimental results, reproducing numerical results of other independent authors, and demonstrating second-order convergence on manufactured solutions. Particle collisions are handled using a dry viscoelastic soft-sphere model with sub-time stepping. An additional model based on lubrication theory is proposed and shown to agree with experiments of submerged collisions. The complete numerical method is suitable for parallel computing. Weak scaling results of a 3D fluidized bed simulation containing two million particles suggests that flows containing one billion particles could be computed on today's supercomputers.