Particle clouds in homogeneous and stratified environments

Abstract

We examine the settling of monodisperse heavy particles released into a fluid when the resulting motion is sufficiently vigorous that the particle cloud initially assumes the form of a turbulent thermal. A laboratory study is complemented by numerical simulations of particle cloud dynamics in both homogeneous and stratified ambients. In the homogeneous ambient, the cloud generated by a total buoyancy excess $Q= g'N_pV_p$, where $g'$ is the reduced gravity of the $N_p$ spherical particles of volume $V_p\,{=}\,{4\upi a^3}/3$, evolves in a manner consistent with a classical fluid thermal. The cloud grows through turbulent entrainment and decelerates until its speed is exceeded by that of the individual particles $w_s$, at which point the particles rain out as individuals. For particle Reynolds numbers $\hbox{\it Re}_p\,{=}\,w_s a/\nu$ in the range of 0.1–300, the fallout height $Z_f$ is found to be $Z_f/a\,{=}\,(11 \pm 2) (Q^{1/2}/(w_sa))^{0.83}$. For high $\hbox{\it Re}_p$ particles, the fallout height assumes the simple form: $Z_f/a\,{=}\,(9 \pm 2) N_p^{1/2}$. Following fallout, the particles sink at their individual settling speeds in the form of a bowl-shaped swarm. In a stratified environment characterized by a constant Brunt–Väisälä frequency $N$, the mode of fallout depends explicitly on the stratified cloud number, $N_s\,{=}\,w_s Q^{-1/4} N^{-1/2}$. For $N_s\,{<}\,1$, the cloud overshoots, rebounds past, then intrudes at the neutral height, $Z_N$, of the equivalent fluid thermal. The particles fall out between the depth of maximum penetration and the spreading neutral cloud, and may be distributed over a relatively broad area. For $N_s\,{>}\,1$, the particles fall out in the form of a bowl-shaped swarm at a height $Z_f\,{<}\,Z_N$, thus giving rise to a relatively localized deposit. For $N_s\,{>}\,4$, the fallout height is largely uninfluenced by the stratification and is adequately described by the homogeneous result. Regardless of $N_s$, following particle fallout in a stratified ambient, the fluid entrained by the thermal ascends and intrudes at a rebound height given to leading order by ${3} Z_f/4$. Criteria for three distinct modes of particle deposition in a stratified ambient are developed.