Quantifying divergence and rotation of the inertial particle velocity in high Reynolds number turbulence using Voronoi and Delaunay tessellation

We propose finite-time measures to quantify the divergence and the curl of the velocity advecting point particle clouds in space and time. To this end we respectively determine the volume change rate and the rotation of cells using two subsequent time steps at the particle positions. We consider either Voronoi or Delaunay tessellation and assess the reliability of the two methods. We show a first order convergence in time and in space for divergence and curl for randomly distributed particles using Delaunay triangulation and a good agreement with the exact values. For the Voronoi tessellation we observe some off set in the case of randomly distributed particles due to geometrical effects. We apply these tools to three-dimensional direct numerical simulation data of particle-laden isotropic turbulence computed at high Reynolds number. We discuss and compare the results obtained with the two different techniques. For inertial particles the probability distribution functions (PDFs) of the divergence and of the curl deviate from that for fluid particles and we observe a similar Stokes number dependency for both tessellations. In the Delaunay case the extreme values of divergence and curl of the velocity are reduced and the corresponding PDFs are narrower. We also find different results in analyzing the mean divergence as a function of volume, which leads to different interpretations of the behavior of the particles as a function of scale.