Concentration waves and flow modification in a particle-laden circular vortex

Abstract

Evolution of a two-dimensional axisymmetrical vortex laden with solid heavy particles is studied analytically and numerically. The particulate phase is assumed to be dilute enough to neglect the effects of particle–particle collisions. Only sufficiently small particle Stokes (St) and Reynolds numbers are considered, for which an approximate solution for the particle velocity can be derived. An analytical solution to a Cauchy problem is obtained for initially uniform concentration of particles in a circular flow describing the accumulation of particles in the form of a kinematic wave and the corresponding modification of the carrier flow. According to this solution, a steep peak of the concentration develops forming the wave crest which propagates out of the vortex. Due to the interaction between the two phases, a fluid velocity component directed towards the vortex center is generated, so that in the vicinity of the crest the vortex acquires a spiral-like shape. At later stages, the growth of the crest is inhibited and its propagation velocity decreases. Analysis of the problem for particles with larger Stokes numbers shows that the accumulation process is most intense when St is close to a critical value St* which generally depends on the vortex structure and, for the flow considered, is of the order unity.