Emptying-filling boxes with non-Boussinesq plumes and fountains

In this paper the dynamics of the simultaneously filling and emptying of a box is studied theoretically and experimentally. Both situations, consisting of negatively buoyant (fountains) and positively buoyant (plumes) discharges, are considered. For the sake of generality, no assumption is made about the value of the density deficit (between the release and the ambient fluid), so this study deals with the so-called non-Boussinesq general case. Experiments are carried out with buoyant air-helium mixtures continuously released from the top (fountain) or bottom (plume) into air in a cylindrical tank with an open bottom boundary and a top vent of variable areas. At steady state, for both fountain and plume configurations, a buoyant layer of constant thickness and density forms under the tank ceiling. Based on mass and buoyancy conservation equations applied on the buoyant layer, theoretical models are proposed to estimate its depth and density at steady state. The theoretical models compare favorably with the experimental data. Subsequently, these models allow us to compare the plume and the fountain configurations for identical source conditions, box size, and vent area. Even if, in the majority of situations, the plume configuration allows a better mixing of the buoyant fluid with the environment, it is found that beyond a certain value of the fountain height, the fountain configuration becomes more efficient than the plume configuration for mixing phenomena.

Figure 1

Schematic of the experimental apparatus.

Figure 2

Illustration of (a) plume and (b) fountain filling-emptying box experiments.

Figure 3

Schematic of the plume filling a box which is simultaneously emptied by an opening vent on the ceiling. Note that $A$ corresponds to the vent area.

Figure 4

Dimensionless thickness of the buoyant layer ${\zeta }_a$ as a function of the parameter $\mathrm{\Sigma }$, for three different configurations. The thick solid lines correspond to the numerical solutions of Eqs. (12) and (13), the thin solid lines correspond to the numerical solutions of Eq. (14), the thick dashed lines corresponds to Eq. (15), and the thick dash-dotted lines correspond to Eq. (16). The symbols correspond to the experimental data: $•$, configuration 1; $▴$, configuration 2; and $\blacksquare$, configuration 3. Note that the entrainment coefficient $\alpha$ is set to 0.12.

Figure 5

Schematic of the fountain filling a box which is simultaneously emptied by an opening vent on the ceiling. Note that $A$ corresponds to the vent area.

Figure 6

Plot of the fountain height as a function of the length scale ${\mathcal{L}}_b$. The blue squares correspond to the experimental points, the black solid line corresponds to $z_f\approx 1.8{\mathcal{L}}_b$, the black dotted line corresponds to $z_f\approx 1.84{\mathcal{L}}_b$ given by [6, 34], and the black dashed line corresponds to $z_f\approx 1.92{\mathcal{L}}_b$ given by [20]. The bars correspond to the amplitude of oscillation of the fountain height set to $0.14{\mathcal{L}}_b$ as suggested in [20].

Figure 7

(a) Variation of $z_a$ as a function of ${\mathcal{L}}_b$ for a fixed value of $\mathrm{\Sigma }=14.1$. The experimental data are shaded according to the density deficit ${\eta }_i$; the blue line corresponds to the numerical solution of (23) with ${\eta }_i=0.2$ and the red line corresponds to ${\eta }_i=0.9$. (b) Variation of $z_a$ as a function of ${\mathcal{L}}_b$ for two different values of $\mathrm{\Sigma }$. The colored circles correspond to experimental data and the lines correspond to the theoretical predictions. Red corresponds to $\mathrm{\Sigma }=14.1$ and blue corresponds to $\mathrm{\Sigma }=31.8$.

Figure 8

Variation of ${\zeta }_a$ as a function $\mathrm{\Sigma }$ for two different values of ${\mathrm{\Gamma }}_i$: ${\mathrm{\Gamma }}_i=0.0087$ (blue) and ${\mathrm{\Gamma }}_i=0.0114$ (red).

Figure 9

Comparison between the experimental data and the numerical solution of (23).

Figure 10

Difference $({\zeta }_{a_p}-{\zeta }_{a_f})b_i/H$ as a function of $\mathrm{\Sigma }$ for different values of ${\mathrm{\Gamma }}_i, H/b_i=200$, and (a) ${\eta }_i=0.05$ and (b) ${\eta }_i=0.85$. The thin solid lines correspond to ${\zeta }_{a_p}-{\zeta }_{a_f}=0$.

Figure 11

Difference $({\zeta }_{a_p}-{\zeta }_{a_f})b_i/H$ as a function of $\mathrm{\Sigma }$ for different values of ${\eta }_i, H/b_i=200$, and (a) ${\mathrm{\Gamma }}_i=5\times {10}^{-3}$ and (b) ${\mathrm{\Gamma }}_i=8\times {10}^{-4}$. The thin dashed line in (b) corresponds to ${\zeta }_{a_p}-{\zeta }_{a_f}=0$.

Figure 12

Ratio ${\eta }_p^{*}/{\eta }_f^{*}$ as a function of $\mathrm{\Sigma }$ for different values of ${\mathrm{\Gamma }}_i, H/b_i=200$, and a source density deficit of (a) ${\eta }_i=0.05$ and (b) ${\eta }_i=0.85$. The thin dashed line in (b) corresponds to ${\eta }_p^{*}/{\eta }_f^{*}=1$.

Figure 13

Ratio ${\eta }_p^{*}/{\eta }_f^{*}$ as a function of $\mathrm{\Sigma }$ for different values of ${\eta }_i, H/b_i=200$, and (a) ${\mathrm{\Gamma }}_i=5\times {10}^{-3}$ and (b) ${\mathrm{\Gamma }}_i=8\times {10}^{-4}$. The thin solid line in (b) corresponds to ${\eta }_p^{*}/{\eta }_f^{*}=1$.