A second-order method for three-dimensional particle simulation

This paper describes a numerical method for the direct numerical simulation of Navier-Stokes flows with one or more solid spheres. The particles may be fixed or mobile, and they may have different radii. The basic idea of the method stems from the observation that, due to the no-slip condition, in the reference frame of each particle, the velocity near the particle boundary is very small so that the Stokes equations constitute an excellent approximation to the full Navier-Stokes problem. The general analytic solution of the Stokes equations can then be used to “transfer” the no-slip condition from the particle surface to the adjacent grid nodes. In this way the geometric complexity arising from the irregular relation between the particle boundary and the underlying mesh is avoided and fast solvers can be used. The method is validated by a detailed comparison with spectral solutions for the flow past a sphere at Reynolds numbers of 50 and 100. The existence in these situations of a Stokes region near the particle is explicitly demonstrated. Other numerical experiments to show the performance of the code are also described. To illustrate the power and efficiency of the method, a simulation of decaying homogeneous turbulence in a cell containing 100 moveable spheres is described. As implemented here, the method can only be applied to simple body shapes such as spheres and cylinders. Extensions to more general situations are mentioned in the last section.