Identification of a particle collision as a finite-time blowup in turbulence

We propose an Eulerian approach to investigate the motion of particles in turbulence under the assumption that the motion of particles remains smooth in space and time until a collision between particles occurs. When the first collision happens, particle velocity loses C¹ continuity, resulting in a finite-time blowup. The corresponding singularities in particle velocity gradient, particle number density, and particle vorticity for various Stokes numbers and gravity factors are numerically investigated for the first time in a simple two-dimensional Taylor-Green vortex flow, two-dimensional decaying turbulence, and three-dimensional isotropic turbulence. In addition to the critical Stokes number above which a collision begins to occur, the flow condition leading to collision is revealed; particles tend to collide in very thin shear layer constructed by two parallel same-signed vortical structures when Stokes number is above the critical one.