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. For this purpose, a tessellation of the particle positions is performed to assign a volume to each particle. We introduce a modified Voronoi tessellation which overcomes some drawbacks of the classical construction. Instead of the circumcenter we use the center of gravity of the Delaunay cell for defining the vertices. 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. Reorganizing the various velocity coefficients allows computing the curl and even the velocity gradient tensor. The helicity of particle velocity can be likewise computed and swirling motion of particle clouds can be quantified. First we assess the numerical accuracy for randomly distributed particles. We find a strong Pearson correlation between the divergence computed with the the modified tessellation, and the exact value. Moreover, we show that the proposed method converges with first order in space and time in two and three dimensions. Then we consider particles advected with random velocity fields with imposed power-law energy spectra. We study 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.