In an attempt to develop a reliable numerical method that can deal economically with a large number of rigid particles moving in an incompressible Newtonian fluid at a reasonable cost, we consider two fictitious-domain methods: a Constant-density Explicit Volumetric forcing method (CEV) and a Variable-density Implicit Volumetric forcing method (VIV). In both methods, the mutual interaction between the solid and the fluid phase is taken into account by an additional body force term to the Navier-Stokes equations, but the physical meaning of the forcing is different for the two methods. In the CEV method, which is built on a constant-density Navier-Stokes solver, the net forcing added to the fluid is generally not zero, and must be cancelled by applying Newton’s first law to a rigid particle template which has the same shape as the rigid particle and carries the “excess mass” of the rigid particle, i.e. the excess over the mass of the displaced fluid. The “target velocity” to which one forces the velocity within the particle is evaluated through the equation of motion for the rigid particle template. In the VIV method, built on a variable-density incompressible flow solver, the rigid particle (angular) velocity is determined by averaging the (angular) momentum, within the particle domain, of the fractional-step velocity field, and the net forcing is zero. By design, this method does not require any rigid particle template equations, so it can be applied for both neutral and non-neutral density ratios without any difficulties. We consider two test problems with single freely moving circular disks: a disk falling in quiescent fluid, and a disk in Poiseuille channel flow. At near-neutral density ratios, the CEV method is found to perform better, while the VIV method yields more accurate results at higher relative density ratios.

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