Abstract
Particle loading affects the dynamics of buoyant plumes, since the difference between particle and fluid densities is much greater than that in the fluid alone. In stratified environments, plume rise is density limited; after initial overshoot, the plume reaches a terminal level and spreads radially. Particles dropping from this horizontal intrusion may be re-entrained. This recycling of dense matter reduces plume buoyancy and intrusion height and, for sufficient load, can lead to plume collapse. Entrainment-based formulae yield a steady-state plume rise. We identify a new conserved quantity for such plumes. Integrating paths of particles dropping from the intrusion yields the fraction re-entrained. A simple mathematical model predicts from buoyancy ratio at source ($P=$ negative particle buoyancy divided by positive fluid buoyancy) whether a particle-laden plume will collapse. Under this model, for small settling velocity, a particle-laden plume will not collapse if $P<0.368$. Above this, collapse depends also on the amount of particle-free ambient fluid entrained in the overshoot region. For pure plumes, experimental evidence suggests that this is small. For forced plumes, more substantial overshoot and entrainment is shown to increase the critical ratio. An extension, based on successive recycling, estimates time to collapse. To investigate further we develop a simple computational model, coupling a ‘top-hat’ plume model, an analytical formula for radially decaying concentrations in the intrusion and an axisymmetric finite-volume solution for time-dependent settling and entrainment. The model can predict the impact of particle load on final rise, as well as the occurrence and time scales of plume collapse.
Highlights
Turbulent buoyant plumes transporting relatively dense particles can be found in natural and engineered environments, including volcanoes, black smokers rising from deep-sea vents, dredging operations, effluent from marine outfalls and particles from combustion in fires or engines
The paper starts by recapitulating a simple top-hat model for particle-free plume rise in a stratified environment
Using an expression for the fraction re-entrained into the plume during settling, conditions for plume collapse are deduced, based on the particle-to-fluid source buoyancy ratio, P, and the plume buoyancy at the height of the intrusion layer
Summary
Turbulent buoyant plumes transporting relatively dense particles can be found in natural and engineered environments, including volcanoes, black smokers rising from deep-sea vents, dredging operations, effluent from marine outfalls and particles from combustion in fires or engines. For circumstances where plume rise is limited by the ambient density profile (e.g. in the atmosphere), rather than a free-surface rigid lid (as in most laboratory tank-based studies), there is initial overshoot of the final intrusion height, because the plume reaches the neutral-density level with non-zero momentum. This allows us to calculate both the time development of the particle distribution and the effect of re-entraining particles on plume buoyancy and intrusion height (including time to collapse), as well as particles falling out during the rise phase.
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