  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.
