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We propose finite-time measures to compute the divergence, the curl and thevelocity gradient tensor of the point particle velocity for two- andthree-dimensional moving particle clouds. For this purpose, a tessellation ofthe particle positions is performed to assign a volume to each particle. Weintroduce a modified Voronoi tessellation which overcomes some drawbacks of theclassical construction. Instead of the circumcenter we use the center ofgravity of the Delaunay cell for defining the vertices. Considering then twosubsequent time instants, the dynamics of the volume can be assessed.Determining the volume change of tessellation cells yields the divergence ofthe particle velocity. Reorganizing the various velocity coefficients allowscomputing the curl and even the velocity gradient tensor. The helicity ofparticle velocity can be likewise computed and swirling motion of particleclouds can be quantified. First we assess the numerical accuracy for randomlydistributed particles. We find a strong Pearson correlation between thedivergence computed with the the modified tessellation, and the exact value.Moreover, we show that the proposed method converges with first order in spaceand time in two and three dimensions. Then we consider particles advected withrandom velocity fields with imposed power-law energy spectra. We study thenumber of particles necessary to guarantee a given precision. Finally,applications to fluid particles advected in three-dimensional fully developedisotropic turbulence show the utility of the approach for real worldapplications to quantify self-organization in particle clouds and theirvortical or even swirling motion.

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