Simulation of impulsively started flow past a sphere and a disc using iterative brinkman penalization

We present an iterative Brinkman penalization scheme to enforce the no-slip condition onsolid boundaries in three-dimensional ow simulations. We use a high-order particle-meshvortex method, where the velocity field is obtained from the vorticity field by solving a Poisson equation on a Cartesian mesh as a convolution of the vorticity field with a regularized Green's function [2].

By doing this we can enforce free-space boundary conditions allowing us to consider a minimal computational domain. The Brinkman penalization method [1] is an immersed body method that allows the treatment of solids having complex geometries on a Cartesian mesh. Thereby we avoid the use of unstructured meshes that conventional ow solvers rely on.

In the presented iterative scheme the penalization term is only active in the solid region and in its immediate neighborhood thus the computational costs required for the solution of the penalization problem is kept at a minimum.We apply our method for the simulation of the impulsively started flow past a sphere atRe = 1000 and normal to a circular disc at Re = 500, respectively.

Our results for theunsteady sphere flow are found to be in qualitative good agreement with results obtainedby Ploumhans et. al [3] using a boundary element method. The flow is illustrated by a volume rendering of the vorticity field cf. Fig. 1. The figure shows that the ow is highly unsteady and a challenging problem for accurate analysis.

Furthermore we show that the iterative scheme allows a significantly larger time step than the iterative scheme (more than a factor of 10) in the simulation of the impulsively started flow normal to a circular disc.