Role of dispersed particles on the dynamics of an umbrella cloud of a forced plume in a linearly stratified environment

Abstract

The vertical release of a lighter buoyant fluid, commonly referred to as a forced plume, into dense environment is a common occurrence in ocean and atmosphere. Such releases may have heterogeneity in them in form of particles with varied size, shape, and volume fraction. Normally in field conditions, the particle size ranges from a few microns to millimetres, and the volume fraction ranges from $$\phi _{v}=0.1{-}10$$ %. In this study, the effects of low values of $$\phi _{v}$$ (corresponding to a two-way coupled system) on the dynamics and structure of a plume umbrella cloud formed in a linearly stratified ambient were examined. Spherical particles with mean diameter $$d_{p}=100\, \upmu \hbox {m}$$ , density, $$\rho _{p}=2500\, \hbox {kg}\,\hbox {m}^{-3}$$ , and $$\phi _{v}=0{-}0.7\%$$ were injected along with the lighter plume fluid. Due to the phenomena of “particle fall-out” and “particle re-entrainment”, it was observed that a plume trough characterized by radius, $$R_{c}$$ , and depth, $$L_p$$ , forms below the neutral buoyant layer of an umbrella cloud. The plume trough formation is linked to the draw-down of the fluid from the neutral buoyant layer by the sedimenting particles. This trough either sustains or collapses depending on the plume conditions at the source, namely, the diameter $$d_{0}$$ , fluid buoyancy, $$g^{\prime }_{0}$$ , vertical velocity, $$W_{0}$$ , and $$\phi _{v}$$ . In all the experiments, $$g^{\prime }_{0}$$ and $$d_{0}$$ were kept constant while $$\phi _{v}=0{-}0.7\%$$ and $$W_{0}=0.2{-}0.65 \,\hbox {m}\,\hbox {s}^{-1}$$ were varied. The experiments revealed that the sustaining and collapsing trough regimes could be qualitatively demarcated based on a source effective Richardson number, $$Ri^{*}_{0}$$ , that accounts for the combined effect of $$\phi _{v}$$ and $$W_{0}$$ . It was found that when $$Ri^{*}_{0}<0.018$$ , the plume trough collapses, otherwise it is sustained. However, $$Ri^{*}_{0}$$ failed to predict the variations in $$R_{c}$$ and $$L_{p}$$ and hence was unsuitable for quantifying the plume trough. Therefore, it was established that the characteristics of a plume trough (i.e. $$R_{c}$$ and $$L_{p}$$ ) depend independently on the particle volume fraction $$(\phi _{v})$$ and source Richardson number $$(Ri_{0})$$ , while the demarcation of the trough regimes could be done using $$Ri^{*}_{0}$$ . For a constant $$Ri_{0}= \frac{-g^{\prime }_{0}d_{0}}{W_{0}^{2}}$$ , with an increase in $$\phi _{v}$$ the trough radius, $$R_{c}$$ , was found to decrease. In contrast, $$L_p$$ increases for a sustaining trough and decreases for a collapsing trough. For a constant $$\phi _{v}$$ , a decrease in $$Ri_{0}$$ caused both $$R_{c}$$ and $$L_{p}$$ to increase irrespective of a sustaining or collapsing plume trough. Using equations of motion for sedimenting particles, an analytical expression for the trough radius, $$R_{c}$$ , was formulated that accurately explains the experimental results. A relation for trough radius, $$R_{c}$$ , that elucidates the dynamics of particle re-entrainment was also obtained. If a particle is within the value of $$R_{c}$$ it is re-entrained into the plume, otherwise settles at the bottom. These results provide useful information needed for modeling particle-laden forced plumes.