Computational methods for tracking inertial particles in discrete incompressible flows
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Calculating trajectories of small particles in numerical simulations of incompressible fluids is of great importance for natural and industrial applications, yet it is a difficult and computationally expensive challenge. The problem involves interpolating the fluid velocity field and its derivatives onto the location of the particle, calculating forces and torques, then integrating a set of rigid-body equations, hence this amounts to an interesting challenge from a numerical point of view. In this paper we investigate some computational methods for addressing this problem, including using regularised Stokeslet solutions to the steady Stokes equations to approximate the local fluid field around the particle. We show a simple equivalence between regularised Stokeslets and matrix-valued radial basis function (MRBF) interpolation, which is a well posed interpolation method. The resulting rigid body ODEs can be integrated using a splitting method that utilises the exact rigid body motion of the particle. We show numerically, for a variety of Stokes regimes, that the proposed interpolation and integration algorithm reduces error for trajectories of small inertial ellipsoidal particles compared to conventional methods in discrete Taylor-Green vortices, as an example. We also conduct experiments with 10,000 particles and measure statistical quantities of the particle system as a discrete probability distribution. We show that when compared to a polynomial interpolation scheme, a cheaper MRBF scheme can lead to more accurate distributions, despite having more error, when measured in a more traditional sense. This is numerical evidence to support the claim that interpolation errors are "averaged out" in simulations of many anisotropic particles when MRBF interpolation methods are used.
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