A multi-scale approach to simulate atomisation processes

Abstract

Most interfacial flows of practical interest typically involve length- and time-scales that span over several orders of magnitude. Their accurate modelling, using interface-capturing or interface-tracking methods, therefore presents a prohibitively large computational cost. Over the past few years, hybrid Eulerian-Lagrangian approaches have been proposed with the aim to predict the behaviour of such flows both efficiently and accurately. These approaches rely on the assumption that small detached interfacial structures are spherical due to the dominant influence of surface tension, therefore allowing for their modelling using classical Lagrangian particle tracking techniques. The dynamics of large interfacial structures are thus fully resolved on the Eulerian grid, whereas the smaller interfacial structures, resulting from breakup instances, are transferred to a Lagrangian frame of reference and tracked as point-particles. In doing so, a main issue arises due to the competing requirements of both the Eulerian and Lagrangian approaches: on one hand, interfacial structures modelled with the Eulerian approach should be resolved by at least a few mesh cells; and on the other hand, Lagrangian particle tracking classically requires the tracked particles to be much smaller than the Eulerian mesh cells. Interfacial structures that lie between these two limits – typically shortly before and/or after their transfer from one framework to the other – are therefore poorly represented by any of the two approaches. So-far, a successful strategy to deal with this spatial paradox has not been proposed.To address this issue, a hybrid Eulerian-Lagrangian approach relying on an enhanced Lagrangian particle tracking method is proposed in this paper. Through the derivation of the filtered flow equations for a gas-droplets mixture, it is shown that the filtering of relevant quantities allows for the consistent tracking of Lagrangian particles that are of a similar size as, or larger than the Eulerian mesh cells. Numerical simulations of a particle settling under the influence of gravity are conducted to demonstrate the ability of the framework to accurately and consistently deal with Lagrangian particles that are larger than the Eulerian mesh cells. Finally, the full hybrid framework is tested on a realistic spray atomisation case, and the impact of the proposed filtering on the dynamics and the statistics of the spray is investigated.