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 Re and Froude Fr numbers of the initial state. Unlike a conventional turbulent fountain, we focus on the regime at lower Reynolds numbers, Re<500, where 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 Re and 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 QE to the injected volume flux Q0) is also a function of Re and Fr for low Reynolds numbers. For a given Fr, the fountain volume entrainment flux ratio at the initial stage of the injection reaches a local peak at an intermediate Reynolds number Re≈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, Re≈200. Results of this study provide guidelines to achieve effective jet-driven mixing of two miscible liquids in a container.