Computing differential operators of the particle velocity in moving particle clouds using tessellations

We propose finite-time measures to compute the divergence, the curl and the velocity gradient tensor of the point particle velocity for two- and three-dimensional moving particle clouds. To this end, tessellation of the particle positions is applied to associate a volume to each particle. Considering then two subsequent time instants, the dynamics of the volume can be assessed. Determining the volume change of tessellation cells yields the divergence of the particle velocity and the rotation of the cells evaluates its curl. Thus the helicity of particle velocity can be likewise computed and swirling motion of particle clouds can be quantified. We propose a modified version of Voronoi tessellation and which overcomes some drawbacks of the classical Voronoi tessellation. First we assess the numerical accuracy for randomly distributed particles. We find strong Pearson correlation between the divergence computed with the the modified version, and the analytic value which confirms the validity of the method. Moreover the modified Voronoi-based method converges with first order in space and time is observed in two and three dimensions for randomly distributed particles, which is not the case for the classical Voronoi tessellation. Furthermore, we consider for advecting particles, random velocity fields with imposed power-law energy spectra, motivated by turbulence. We determine the number of particles necessary to guarantee a given precision. Finally, applications to fluid particles advected in three-dimensional fully developed isotropic turbulence show the utility of the approach for real world applications to quantify self-organization in particle clouds and their vortical or even swirling motion.