Direct simulation of fluid particle motions

Abstract

Continuum models of two-phase flows of solids and liquids use constitutive assumptions to close the equations. A more fundamental approach is a “molecular dynamic” simulation of flowing “big” particles based on reliable macroscopic equations for both solid and liquid. We developed a package that simulates the unsteady two-dimensional solid-liquid two-phase flows using the Navier-Stokes equations for the liquid and Newton's equations of motion for the solid particles. The Navier-Stokes equations are solved using a finite-element formulation and Newton's equations of motion are solved using an explicit-implicit scheme. We show that the simplest fully explicit scheme to update the particle motion using Newton's equations is unstable. To correct this instability we propose and implement and Explicit-Implicit Scheme in which, at each time step, the positions of the particles are updated explicitly, the computational domain is remeshed, the solution at the previous time is mapped onto the new mesh, and finally the nonlinear Navier-Stokes equation and the implicitly discretized Newton's equations for particle velocities are solved on the new mesh iteratively. The numerical simulation reveals the effect of vortex shedding on the motion of the cylinders and reproduces the drafting, kissing, and tumbling scenario which is the dominant rearrangement mechanism in two-phase flow of solids and liquids in beds of spheres which are constrained to move in only two dimensions.