 |
Interfacial flows of practical interest, of which liquid sprays are typical examples, generally involve length- and time-scales that span over several orders of magnitude. Their modelling using interface-capturing or interface-tracking methods therefore presents a prohibitively large computational cost. Over the past 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 as Lagrangian particles. The dynamics of the large interfacial structures are 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 the one hand, interfacial structures modelled with the Eulerian approach need to be resolved by at least a few mesh cells; and on the other hand, the accurate account of the effect of the flow on the Lagrangian particles and vice-versa 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.
In this contribution, we propose a hybrid Eulerian-Lagrangian approach that is free of such inconsistencies. It relies upon the filtering of the instantaneous flow equations with a particle marker function; a process in which a length-scale equal to a few particle diameters is chosen. This decouples the size of the Lagrangian particle with the specific length of the discrete Eulerian mesh cell in calculating the volume fraction. This filtering is also applied to the fluid-particle momentum coupling terms, which provides a solid framework for spreading these source terms. Throughout the application of the proposed method to a number of test-cases, we show that the filtering strategy allows for the consistent tracking of Lagrangian particles regardless of their size relative to the Eulerian mesh. Finally, the full hybrid framework is tested on a realistic liquid atomisation case, and the impact of the proposed filtering on the dynamics and the statistics of the spray is investigated.
