Characterization of the instability of a Rankine vortex in semi-dilute dusty flows using Linear stability analysis and Eulerian-Lagrangian simulations

This study investigates the effect of inertial particles on the stability of a two-dimensional Rankine vortex in semi-dilute dusty flows. Unlike the particle-free case where the vortex is stable to infinitesimal disturbances, we show that the feedback force from the suspended inertial particles triggers a novel instability specific to two phase flows. For weakly inertial particles with a typical Stokes number smaller than 0.01, we perform a linear stability analysis using continuum conservation equations, namely, the Two-Fluid model. These equations are coupled with the asymptotic particle velocity field for low Stokes particles whose velocity deviates from that of the suspending fluid by an inertial correction. The growth of the instability is characterized in terms of the disturbance azimuthal mode, the particle Stokes number, particle volume fraction, and mass loading. We show that the instability persists even for tracer particles (St = 0), provided that the mass loading exceeds a certain threshold. Comparison with Eulerian-Lagrangian simulations show that the linear stability analysis is able to predict the instability properties for small Stokes number. Additional Eulerian-Lagrangian simulations are presented for moderate and high Stokes number particles (St~1), where the effects of preferential concentration, and discrete nature of the particle phase are shown to control the dynamics of the particle-laden vortex.