Large Droplet Impact on Water Layers

The impact of large droplets onto an otherwise undisturbed layer of water is considered. The work, which is motivated primarily with regard to aircraft icing, is to try and help understand the role of splashing on the formation of ice on a wing, in particular for large droplets where splash appears to have a significant effect. Analytical and numerical approaches are used to investigate a single droplet impact onto a water layer. The flow for small times after impact is determined analytically, for both direct and oblique impacts. The impact is also examined numerically using the volume of fluid (VOF) method. At small times there are promising comparisons between the numerical results, the analytical solution and experimental work capturing the ejector sheet. At larger times there is qualitative agreement with experiments and related simulations. Various cases are considered, varying the droplet size to layer depth ratio, including surface roughness, droplet distortion and air effects. The amount of fluid splashed by such an impact is examined and is found to increase with droplet size and to be significantly influenced by surface roughness. The makeup of the splash is also considered, tracking the incoming fluid, and the splash is found to consist mostly of fluid originating in the layer. Introduction The high-speed impact of a single water droplet onto a previously undisturbed layer of water has a range of applications for example in the chocolate, spray-coating and aeronautics industries but in particular with regard to aircraft icing. For larger droplets splashing is thought to have a considerable influence on the amount of water collected on an aerofoil and therefore a great effect on the shape and quantity of ice produced. Despite this physical importance there has been relatively little previous work on droplet impact at high Reynolds number. Early interest in splashing and impact problems appeared in Worthington whose book ∗Postdoctoral Research Fellow †Goldsmid Professor of Applied Mathematics includes many images of splashing after either a droplet or a solid sphere impacts upon a fluid layer. To date, there has been relatively little previous direct theoretical input and suitable physical modeling on droplet impact, in particular concerning mass and consequent heat transfer and the relationship between input and rebound droplets. However, much work has been done on related aspects both analytically, for example Korobkin and Pukhnachov, Howison et al. who consider solutions at small times after impact (mostly for solidwater impacts), and numerically such asWeiss and Yarin who use a Lagrangian-type approach to examine an inviscid droplet impact numerically. Much work has been done by Josserand and Zaleski and references therein who have developed powerful three-dimensional techniques for capturing droplet impact but tend to examine impacts with Reynolds numbers one or two orders of magnitude less than is typical in an icing context. The motivation of the current work is to help to enhance the understanding of the influence of splashing on the formation of ice on a wing, in particular for super-cooled large droplets where splash is believed to have a significant effect. Given that experiments isolating single droplet impacts and measuring overall splashed volume are difficult to perform, it seems desirable to develop a mathematical model which can describe the process and help guide predictions of mass loss due to splashing. Our approach is to start with a simple model, initially neglecting viscosity (the typical Reynolds number is large in the current practical regime, Re ∼ O(10)), neglecting surface tension (high Weber number, We ∼ O(10)), and neglecting the influence of air and pre-existing flow in order to concentrate on trends in the splashed water as the droplet size to layer depth ratio changes. The idea is to then include other relevant physical effects subsequently, such as those mentioned above as well as ice-surface roughness, aircushioning, compressibility and oblique impacts. Analytical and computational approaches are used to investigate a single droplet impact onto a water layer. The flow for small times after impact is determined analytically, for direct and oblique 1 American Institute of Aeronautics and Astronautics 42nd AIAA Aerospace Sciences Meeting and Exhibit 5 8 January 2004, Reno, Nevada AIAA 2004-414 Copyright © 2004 by R. Purvis and F. T. Smith. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. impact and for very shallow layer depths. The impact is also examined numerically using the VOF (volume of fluid) method, initially treating the fluid as inviscid and incompressible. At small times there are promising comparisons with the analytical solution and with experimental work capturing the ejecta sheet. At larger times there is qualitative agreement with experiments and with related simulations. The method is used to tackle various cases, such as altering the layer depth to droplet diameter ratio, the influence of surface tension and oblique impact, focusing on their effects on the form of splash produced, geometry of the crown formed and make-up of the ejected droplets. In particular the amount of rebounded fluid is examined. Although thermal effects are not included in the current work emphasis is also placed upon the exchange of fluid, tracking the pre-existing (in practice warmer) layer fluid and the (colder) incoming droplet fluid and considering the proportions of each in the splash. This exchange can have a substantial influence on the overall temperature of the water layer. Again the presence of an ice shape beneath the water layer is modelled and its effects on the rebound and the constituents of the ejected fluid are explored. Other aspects include the influence of an air layer, in particular pre-impact air cushioning and pre-existing airflow, and the influence of viscosity and compressibility. In the first instance, then, we consider a single droplet impacting directly upon an otherwise undisturbed layer of water. We treat the behaviour as inviscid and water-only as a first step. The basic set-up is shown in figure 1. The Cartesian coordinates x,y, corresponding velocity components u, v and pressure p used here are non-dimensionalized with respect to a typical layer depth HD (or typical droplet diameter DD) and incoming droplet velocity vD. The Reynolds number Re ≡ vDHD/νD is large, where νD is the kinematic viscosity of the fluid. Values for a typical icing situation are of the order vD ∼ 100m/s, HD ∼ 30μm and droplet diameters ranging from DD ∼ 40μm − 400μm for the large droplets of interest here, although conditions can vary dramatically through different stages and types of icing. The governing equations are the non-dimensionalized, two-dimensional, unsteady Navier-Stokes equations, namely