# zbMATH — the first resource for mathematics

Direct simulation of flows of solid-liquid mixtures. (English) Zbl 1135.76442
Summary: A finite element technique based on moving unstructured grids is developed to simulate the motion of a large number of solid particles in a flowing liquid. A generalized Galerkin finite element formulation which incorporates both the fluid and particle equations of motion into a single variational equation is developed for Newtonian fluids. The hydrodynamic forces and moments acting on the solid particles are eliminated in the formulation, so need not be computed explicitly. An arbitrary Lagrangian-Eulerian (ALE) technique is adopted to deal with the motion of the particles. In the implementation, the nodes on the particle surface are assumed to move with the particle. The nodes in the interior of the fluid are computed using Laplace’s equation, to guarantee a smoothly varying distribution of the nodes. At each time step, the grid is updated according to the motion of the particles and checked for element degeneration. If unacceptable element distortion is detected, a new finite element grid is generated and the flow fields are projected from the old grid to the new grid. This generalized ALE Galerkin finite element approach gives rise to a set of non-linear algebraic equations which is solved via a quasi-Newton scheme. The corresponding linearized system is solved with an iterative solver using a preconditioned generalized minimal residual algorithm. Initially, the particles are positioned randomly in the fluid, with zero velocity. The particles are then released and the motion of the combined fluid-particle system is simulated using a procedure in which the positions of the particles and of the mesh grids are updated explicitly, while the velocities of the fluid and the solid particles are determined implicitly.
Using the developed numerical procedure, we study the Poiseuille flow of solid-liquid mixtures in a vertical channel. The computation is performed within a unit cell which is periodic in the direction along the channel. The gravity is directed along the channel walls, and a pressure gradient is applied against the gravity and drives the flow. The solid particles are slightly heavier than the liquid. The effects of the applied pressure gradient, the particle Reynolds number and the fraction of the solid loading on the flow pattern of the solid-liquid mixture are studied. It was found that when the applied pressure gradient is large enough to overcome the gravity, the particles migrate away from the channel walls and there is a clear liquid layer next to the wall which lubricates the flow. As the particle Reynolds number is increased, particles interact more strongly and large clusters of particles are formed in the flow.

##### MSC:
 76Txx Multiphase and multicomponent flows
Full Text: