Fountain mixing in a filling box at low Reynolds numbers

Pouring a liquid into a container filled with a denser miscible liquid creates a downward jet. The penetration of this jet into the tank is opposed by buoyancy, thus a return flow toward the upper surface occurs and forms a fountain. In such a condition, due to entrainment, a mixed layer of the two liquids is created in the upper part of the tank, while the lower section is only filled with the original denser liquid. In this experimental study, we characterize the behavior of fountains in a bounded container, i.e., a filling box, based on the Reynolds $\mathrm{Re}$ and Froude $\mathrm{Fr}$ numbers of the initial state. Unlike a conventional turbulent fountain, we focus on the regime at lower Reynolds numbers, $\mathrm{Re}<500$, where $\mathrm{Re}$ is defined based on the jet radius, average speed, and the kinematic viscosity of the environment. Furthermore, we measure the fountain volume entrainment coefficient $B$ by modeling the mixing in this configuration as a filling-box problem. We then show that $B$ is a function of both $\mathrm{Re}$ and $\mathrm{Fr}$, in contrast to the turbulent regime where $B$ remains approximately constant. Similarly, the fountain volume entrainment flux ratio (the ratio of the fountain volume entrainment flux $Q_E$ to the injected volume flux $Q_0)$ is also a function of $\mathrm{Re}$ and $\mathrm{Fr}$ for low Reynolds numbers. For a given $\mathrm{Fr}$, the fountain volume entrainment flux ratio at the initial stage of the injection reaches a local peak at an intermediate Reynolds number $\mathrm{Re}\approx 200$. We reason that this local peak of entrainment occurs due to the enhanced penetration of the downward jet and the moderate fountain volume entrainment coefficient at intermediate Reynolds numbers. Consequently, the total volume of the mixture, which is controlled by the injection depth of the fountain and the fountain volume entrainment flux ratio, reaches a maximum value at a similar intermediate Reynolds number, $\mathrm{Re}\approx 200$. Results of this study provide guidelines to achieve effective jet-driven mixing of two miscible liquids in a container.

Figure 1

(a) Schematic of the model experiments. Dyed water (density ${\rho }_0$ and viscosity ${\mu }_0)$ is injected into a tank (cross-sectional area $A)$ of salt solution (density ${\rho }_s$ and viscosity ${\mu }_s)$ with mean injection velocity $U$ through a needle of radius $R$. The injection forms a fountain with mean penetration depth $z_{\mathrm{avg}}$. As the dyed water is injected with a volume flux $Q_0=\pi R^{2}U$, the fountain entrains the salt solution with a fountain volume entrainment flux $Q_E$, and forms a mixed layer with thickness $z_m$ at the top of the tank. (b) Sample parabolic velocity profile at the needle exit measured by particle image velocimetry (PIV). $r$ denotes the radial distance from the centerline of the needle and is normalized by the radius of the needle $R$. $u$ is the magnitude of the vertical component of the jet velocity at the needle exit and is normalized by the maximum velocity on the centerline of the needle $u_{\max }$. The dashed line shows a parabolic curve fitted to the experimental data. Measurements correspond to $\mathrm{Re}=500$, which is the upper bound of Reynolds number in our experiments.

Figure 2

Effect of injection velocity on the penetration depth of the fountain. (a) Sample images of the fountain and the resulting mixed layer at different mean injection velocities $U$. Images correspond to $t^{*}=t{Q}_0/\left(AR\right)=5$ implying a constant injected volume of fluid $t{Q}_0$ in these experiments. The concentration of the salt in the tank is kept unchanged at 20 wt%. (b) The normalized mean fountain penetration depth $z_{\mathrm{avg}}/R$ as a function of the mean injection velocity $U$. The dashed line represents the calculated mean penetration depth, $z_{\mathrm{turb}}=2.46R\mathrm{Fr}$, for a fully developed turbulent fountain [2], as a function of the injection velocity with a constant density difference. Here $\mathrm{Fr}$ varies and is calculated based on the mean injection velocity used in the experiments. The ranges of the Reynolds and Froude numbers are $\mathrm{Re}=50$–500 and $\mathrm{Fr}=3$–30.

Figure 3

Jet-driven mixing of miscible liquids in a container. (a) Images of fountains and mixed layers formed as a result of injecting dyed water into a container filled with salt water solution for different controlled Froude numbers $\left(10\le \mathrm{Fr}\le 35\right)$ at $\mathrm{Re}=100$. Images correspond to $t^{*}=t{Q}_0/\left(AR\right)=5$ implying a constant injected volume of fluid in the experiments. The thickness of the mixed layer and the fountain depth increase approximately linearly with Froude number. (b) Images of fountains and mixed layers at $t^{*}=5$ for different Reynolds numbers $\left(50\le \mathrm{Re}\le 400\right)$ at $\mathrm{Fr}=25$. Enhanced mixing of the salt solution (mixed layer in blue) is achieved at intermediate Reynolds number $\mathrm{Re}\approx 200$. (c) The ratio of the mean fountain penetration depth $z_{\mathrm{avg}}$ to that of the turbulent regime penetration depth, $z_{\mathrm{turb}}/R=2.46\mathrm{Fr}$ [2], as a function of Reynolds number. $z_{\mathrm{avg}}/z_{\mathrm{turb}}$ first increases with Reynolds number for $\mathrm{Re}\lesssim 200$, while further increase in $\mathrm{Re}$ reduces $z_{\mathrm{avg}}/z_{\mathrm{turb}}$. The ratio is a strong function of Reynolds number, while the effect of Froude number is not significant.

Figure 4

Flow separation in the downward jet flow. (a) A sample image $(\mathrm{Re}=375$ and $\mathrm{Fr}=25)$ and measurement parameters for the initial separation depth of the jet $z_{i,\mathrm{sep}}$. We measure the distance from the injection point to the location, where the diameter of the jet core significantly increases to $2eR$ and refer to it as the initial separation depth of the jet $z_{i,\mathrm{sep}}$. Measurements are performed at $t^{*}\approx 0$, at which the penetration depth reaches maximum $z_i$. (b) The separation depth of the jet $z_{i,\mathrm{sep}}$, normalized by the initial penetration depth of the fountain $z_i$, versus Reynolds number $\mathrm{Re}$. $z_{i,\mathrm{sep}}/z_i$ decreases at higher Reynolds numbers thus enhances the radial momentum transfer from the jet core to the surrounding environment.

Figure 5

Estimate of the fountain volume entrainment coefficient $B$ and volume flux. (a) Sample images of the fountain and the growth of the mixed layer at various characteristic times $t^{*}$. $t^{*}=0$ denotes the initial fountain injection and $t^{*}=10$ denotes the termination of the injection process. Images correspond to the measurements at $\mathrm{Re}=375$ and $\mathrm{Fr}=25$. (b) The normalized (over 255) gray scale intensity $\left|\mathrm{\Delta }Y\right|$ within the mixed layers presented in the images in (a) as a function of depth $z$. Reported values are the result of averaging the gray scale intensity horizontally (in the area that is not occupied by the jet). (c) The development of the fountain depth and the mixed layer thickness in (a) as a function of the characteristic time $t^{*}$. The blue curve follows the penetration depth of the fountain. The black and the red dashed lines denote, respectively, the thickness of the mixed layer measured in our experiments and predicted by the model (for $2\le t^{*}\le 5)$. The fit corresponds to $B\approx 0.23$. All results are normalized by the mean penetration depth of the fountain $z_{\mathrm{avg}}$. The inset shows a zoomed-in view of $\log (1-z_m^{*})$ for $2\le t^{*}\le 5$. (d) The fountain volume entrainment coefficient $B$ as a function of $\mathrm{Re}$ for different $\mathrm{Fr}$. The dashed line denotes the fountain volume entrainment coefficient in a fully developed forced turbulent fountain, $B_{\mathrm{turb}}\approx 0.31$ [12].

Figure 6

(a) Volume entrainment flux ratio of laminar and transitional fountains in a container at $t^{*}=0$, i.e., the beginning of the experiment. The ratio of the lower Reynolds number fountain volume entrainment flux ratio $(Q_E/Q_0)$ to that estimated in the turbulent regime ${(Q_E/Q_0)}_{\mathrm{turb}}$ [12] as a function of $\mathrm{Re}$, with controlled Froude numbers. This ratio $(Q_E/Q_0)/{(Q_E/Q_0)}_{\mathrm{turb}}$ increases with Froude number for $5\lesssim \mathrm{Fr}\lesssim 20$ and remains approximately unchanged for $20\lesssim \mathrm{Fr}\lesssim 30$. With a fixed $\mathrm{Fr}$, as $\mathrm{Re}$ increases, this ratio first increases $\left(\mathrm{Re}\lesssim 200\right)$ and then decreases $\left(\mathrm{Re}\gtrsim 300\right)$. For $\mathrm{Re}>100$, this ratio is mostly larger than unity (reaches a maximum of 2.5), indicating that for an identical Froude number, the fountain entrainment flux ratio of a lower Reynolds number fountain is larger than that of the turbulent fountain. (b) The ratio of the total volume of the mixture $V_{\mathrm{mix}}$ to that of the injected fountain volume $V_{\mathrm{in}}$ versus $\mathrm{Re}$ for different $\mathrm{Fr}$. Measurements correspond to $t^{*}=5$, i.e., $V_{\mathrm{in}}=t^{*}AR\approx 26.6$ ml. Similar to the trends observed for the fixed estimated fountain volume entrainment flux, the normalized volume of the mixture initially increases with $\mathrm{Re}$ and reaches a maximum at $\mathrm{Re}\approx 250$, before decreasing at higher Reynolds numbers $\left(\mathrm{Re}\gtrsim 300\right)$.

Figure 7

Practical implications. (a) The dimensionless mean penetration depth of the fountain $z_{\mathrm{avg}}/R$ and (b) the fountain volume entrainment flux ratio $Q_E/Q_0$ (at $z_m=0)$ as a function of Reynolds number $\mathrm{Re}$, for fixed values of Re/Fr. $\mathrm{Re}/\mathrm{Fr}=\sqrt{gR\rho \mathrm{\Delta }\rho /{\mu }^2}$ describes the geometrical and fluid properties of the fountain including the viscosity and density of the fluid and the size of the injection opening, while the Reynolds number $\mathrm{Re}$ reflects the effect of the mean injection velocity that can be tuned in practical applications.

Figure 8

Effect of confinement of the tank on the fountain volume entrainment flux ratio of laminar and transitional fountains. The ratio of the low Reynolds number fountain volume entrainment flux ratio $(Q_E/Q_0$, at $z_m=0)$ to that estimated in the turbulent regime ${(Q_E/Q_0)}_{\mathrm{turb}}$ [12] as a function of $\mathrm{Re}$, with fixed Froude numbers $\mathrm{Fr}=25$ is displayed. Green diamond points denote the experiments with the tank used in all of the previous experiments in this article (cross-sectional area is 9 cm $\times 9$ cm). The purple circle points denote experiments in a larger tank with cross-sectional area of 12.5 cm $\times 12.5$ cm.