A new approach to modelling the interaction between droplets and the carrier phase is suggested. The new model is applied to the analysis of a spray injected into a chamber of quiescent air, using an Eulerian-Lagrangian approach. The conservative formulation of the equations for mass, momentum and energy transport is used for the analysis of the carrier phase. The dispersed phase is modelled using the Lagrangian approach with droplets represented by individual parcels. The implementation of the Discontinuous Galerkin method (ForestDG), based on a topological representation of the computational mesh by a hierarchical structure consisting of oct-quad-and binary trees, is used in our analysis. Adaptive mesh refinement (h-refinement) enables us to increase the spatial resolution for the computational mesh in the vicinity of the points of interest such as interfaces, geometrical features, or flow discontinuities. The local increase in the expansion order (p-refinement) at areas of high strain rates or vorticity magnitude results in an increase of the order of the accuracy of discretisation of shear layers and vortices. The initial domain consists of a graph of unitarian-trees representing hexahedral, prismatic and tetrahedral elements. The ancestral elements of the mesh can be split into self-similar elements allowing each tree to grow branches to an arbitrary level of refinement. The connectivity of the elements, their genealogy and their partitioning are described by linked lists of pointers. These are attached to the tree data structure which facilitates the on-the-fly splitting, merging and repartitioning of the computational mesh by rearranging the links of each node of the tree. This enables us to refine the computational mesh in the vicinity of the droplet parcels aiming to accurately resolve the coupling between the two phases.
KeywordsDroplets, Sprays, Vortex Rings, Discontinuous Galerkin, Adaptive Mesh Refinement.
IntroductionThe need to accurately model the interaction between droplets and carrier phase (coupling) in various engineering applications is well known [1]. Although various approaches to modelling this coupling have been suggested [1,2,3,4,5], this problem is far from being solved. The main focus of this paper is on the description of the new approach to this coupling and the application of this approach to modelling realistic sprays in gasoline-engine-like conditions. This new approach is based on the application of the adaptive mesh refinement in the vicinity of the droplet parcels. The Discontinuous Galerkin (DG) method [6,7,8,9] is used for solving the equations for the carrier phase. The latter method combines high order accuracy with the ability to handle complex geometries described by hybrid unstructured meshes by incorporating a minimal computational stencil. However, the computational efficiency of this method (alongside the spectral volume [8,10] and spectral difference [8,11,12] methods is generally believed to be inferior to more commonly used methods as the Finite Differenc...