This paper demonstrates the performance of PHYSALIS, a numerical method for the direct numerical simulation of Navier-Stokes flows with many 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, the fluid velocity in the immediate neighborhood of the particle differs little from a rigid-body velocity field so that it is legitimate to neglect the square of this difference. With the aid of a suitable transformation, it is possible to exploit this observation to introduce an auxiliary velocity field that satisfies the Stokes equations in a thin layer of cells adjacent to the particle. The known general solution of the Stokes equations can then be used to bridge the gap between a finite-difference solution on a regular grid and the particle surface. After a brief summary of the method, the paper presents the results of validation studies for the flow past a stationary sphere and the motion of a sphere settling in a container under gravity. The main motivation for the development of the method was however its ability to deal with a large number of particles. To demonstrate this point, some results of the simulation of the gravitational settling of 1,024 spheres are shown. The time evolution of the spatial arrangement of the particles is investigated by means of the two-particle distribution function.

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