The settling and dispersion of small dense particles by spherical vortices

Abstract

The gravitational settling of small dense particles (with fall speed $v_T$, response time $\tau_p$) past an isolated spherical vortex (radius $a$, speed $U$) or a random distribution of spherical vortices translating vertical upwards, is examined. As particles sediment past a vortex, they are permanently displaced vertically and laterally a distance $X$ and $Y$, respectively. The bulk settling properties of the particles are expressed in terms of the weighted moments of displacement, denoted by $D_p$, $M_{\hbox{\scriptsize{\it xx}}}$, $M_{\hbox{\scriptsize{\it yy}}}$ and corresponding to the integral of $X$, $X^2/2$, $Y^2/2$, respectively over the particle sheet. When the particle Stokes number ${\it St}$$=U\tau_p/a \rightarrow 0$, the particles are inertialess. Particles starting outside the vortex are excluded from a spherical shadow region when $v_T/U\,{>}\,3/2$. When $v_T/U\,{<}\,3/2$, particles passing close to the particle stagnation points (in the frame moving with the vortex) are held up for a long time relative to particles far from the vortex, but are not displaced laterally. In an unbounded flow, the particle drift volume, $D_p$, is calculated using a geometrical argument, $M_{\hbox{\scriptsize{\it xx}}}=25U^2\upi a^4/8(v_T+U)^2$, and $M_{\hbox{\scriptsize{\it yy}}}=0$. As $v_T/U\,{\rightarrow}\; 0$, the results of Darwin (1953) are recovered. Results for finite values of ${\it St}$ are calculated numerically. The effect of inertia is shown to substantially increase the particle residence time near the vortex because particles overshoot the particle stagnation point, and there is a shadow region within and behind the vortex. $D_p$, $M_{\hbox{\scriptsize{\it xx}}}$, and $M_{\hbox{\scriptsize{\it yy}}}$ all substantially increase with the particle Stokes number. These results are applied to calculate the bulk settling velocity and the dispersivity of particles sedimenting through a random distribution of vortices translating vertically in a bounded flow. This is done by combining information of the particle displacements with a statistical model of their encounter with a vortex. Inertialess particles (${\it St}$$=0$) do not experience the upwards flow within the vortex and the fractional increase in fall speed is proportional to the volume of the shadow region. As ${\it St}$ increases, particles overshoot the particle stagnation point, increasing their residence time and so decreasing the bulk settling fall speed. Particle inertia significantly increases the vertical dispersivity of dense particles compared to fluid particles, but for high $v_T$, particles disperse vertically more slowly than fluid particles.