The evolution of the water-entry cavity affects the impact load and the motion of the body. This paper adopts the Eulerian finite element method for multiphase flow for simulations of the high-speed water-entry process. The accuracy and convergence of the numerical method are verified by comparing it with the experimental data and the results of the transient cavity dynamics theory. Based on the results, the representative characteristics of the cavity are discussed from the perspective of the cavity cross-section. It is found that the asymmetry of the cavity expansion and contraction durations is related to the motion of the free surface and the closure of the cavity. The uplift of the free surface suppresses cavity expansion, while the jet generated from free surface closure accelerates cavity contraction. The duration of the contraction of the cavity near the free surface is shorter than the expansion duration due to the change in the velocity distribution caused by the free surface motion. The necking phenomenon during deep closure leads to an increase in the internal pressure of the cavity, prolonging cavity contraction near the deep closure area. This work provides new insights into the cavity dynamics in high-speed water entry.

Emptying–filling boxes have been studied in a wide range of configurations for decades, but the flow created in the box by two plumes rising from sources of arbitrary strength and elevation was previously unsolved. Guided by experiments and simplified analytical modelling, we reveal a rich array of two- and three-layer stratifications across seven possible flow regimes. The governing equations for these regimes show how the prevailing regime and stratification properties vary with three key parameters: the relative strength of the plumes, the height difference between their sources and a parameter characterising the resistance of the box to emptying. We observe and explain new behaviours not described in previous studies that are crucial to understanding emptying–filling boxes with multiple plumes. In particular, we demonstrate that the oft-assumed premise that $n$ plumes leads to a stratification with $n+1$ layers is not necessarily true, even in the absence of mixing. Two emptying–filling box models are developed: an analytical model addressing all combinations of the governing parameters and an extended model for three-layer stratifications that incorporates two mixing processes observed in the experiments. The predictions of these two models are in generally excellent agreement with measurements from the experimental campaign covering 69 combinations of the governing parameters. This study improves our understanding of emptying–filling boxes and could facilitate improvements to natural ventilation building design, as demonstrated by an example scenario in which occupants feel cooler upon the addition of a second source of heat.

A hybrid lattice Boltzmann and finite difference method is applied to study the influence of soluble surfactants on droplet deformation and breakup in simple shear flow. First, the influence of bulk surfactant parameters on droplet deformation in two-dimensional shear flow is investigated, and the surfactant solubility is found to influence droplet deformation by changing average interface surfactant concentration and non-uniform effects induced by non-uniform interfacial tension and Marangoni forces. In addition, the droplet deformation first increases and then decreases with Biot number, increases significantly with adsorption number $k$ and decreases with Péclet number or adsorption depth; and among the parameters, $k$ is the most influential one. Then, we consider three-dimensional shear flow and investigate the roles of surfactants on droplet deformation and breakup for different capillary numbers and viscosity ratios. Results show that in the soluble case with $k=0.429$, the droplet exhibits nearly the same deformation as in the insoluble case due to the balance between surfactant adsorption and desorption; upon increasing $k$ from 0.429 to 1, the average interface surfactant concentration is greatly enhanced, leading to significant increase in droplet deformation. The critical capillary number of droplet breakup $C{{a}_{cr}}$ is identified for varying viscosity ratios in clean, insoluble and soluble ($k=0.429$ and 1) systems. As the viscosity ratio increases, $C{{a}_{cr}}$ first decreases and then increases rapidly in all systems. The addition of surfactants always favours droplet breakup, and increasing solubility or $k$ could further reduce $C{{a}_{cr}}$ by increasing average interface surfactant concentration and local surfactant concentration near the neck during the necking stage.