Moving Grid-Multigrid Fictitious Boundary Method for the Simulation of Viscous Flow Past Many Moving Disks

Viscous flow past many moving disks is frequently found in many applications. Examples are pneumatic conveying, fluidized beds, vertical risers, particle separation in cyclones, foods containing particles, slurry flows, etc. Efficient numerical solution of viscous flow past many moving disks problem is still a challenging task. Different differential equations must be satisfied on each side of the interface between fluid and moving disks and the solutions are coupled through relationships or jump conditions that must hold at the interface. The movement of both the interface and the moving disks is unknown in advance and must be determined as part of the solution. In this paper, we address the numerical simulation of the viscous flow past many moving disks via using multigrid fictitious boundary method coupled with arbitrary Lagrangian-Eulerian (ALE) and moving grid techniques. The flow is computed by a special ALE formulation with multigrid finite element solver. The solid disks are allowed to move freely through the computational mesh which is adaptively aligned by a special mesh deformation method (r-type moving grid method) such that the accuracy for dealing with the interaction between the fluid and the disks is highly improved. The deformed grid, created from an equidistant cartesian mesh in which the topology is preserved and only the grid spacing is changed such that the grid points are concentrated near the surfaces of the solid disks. Only the solution of additional linear Poisson problems in every time step is required for generating the deformation grid, which means that the additional work is significantly less than the main fluid-solid part. The main advantage of this methodology is that they allow the numerical treatment on a fixed structured mesh on a simple shape auxiliary domain containing the actual one, independent of the actual boundary of moving disks, allowing therefore the use of fast solvers. Thus the time-consuming construction of a boundary-fitted mesh for each different position of moving disk can be skipped. A numerical simulation of a benchmark configuration of 2D flow around an self-induced motion of airfoil in a channel is first presented to show accuracy improvement. Then numerical examples of two circular, elliptic or rectangular disks falling down as well as sedimentation of 200 circular disks in a cavity filled with incompressible viscous flow are computed to show that the presented method is potentially powerful to simulate the viscous flow past many moving disks.