Phase-space formulation of the Full Lagrangian Approach for dispersed multiphase flows

Dispersed multiphase flows occur in a variety of industrial and environmental contexts, with their omnipresence giving rise to the development of a range of modelling approaches for describing their behaviour. Of primary importance in the modelling of such flows is accurate representation of the spatial distribution of particles, which can exhibit wide variation in unsteady or turbulent flows. The full Lagrangian approach addresses this by tracking individual particles through the flow, and by computing the Eulerian-Lagrangian transformation tensor along trajectories as the flow evolves this method is able to provide the evolution of the local particle number density along each trajectory. Despite being an established method, the inherent drawback to this procedure is that singularities in the particle number density occur whenever trajectories cross in physical space, giving rise to unphysical contributions to the concentration field. This phenomenon occurs for inertial particles due to the multi-valued nature and nonzero compressibility of the particle velocity field, and the inherent dependence of trajectories on the particle velocity. To resolve this problem, it is necessary to consider particle motion within the full phase-space of particle
variables which are tracked, specifically position and velocity. The uniqueness of trajectories in this phase-space is then guaranteed for a given set of initial conditions, with the implication that phase-space trajectories no longer intersect. Tracking the evolution of the phase-space probability density function (pdf) along trajectories therefore provides a well-defined methodology for avoiding the singularities which appear in number density where particle trajectories cross in position space, and results in a single-valued field representation for this pdf. In practice, reconstruction of the Eulerian pdf presents an issue in terms of computational tractability due to the high-dimensionality of the phase-space, however this can be mitigated by
using an appropriate interpolation procedure. Specifically, it is demonstrated that using radial basis function interpolation with a judicious choice of kernel enables the Eulerian number density field to be directly computed from the Lagrangian phase space pdf contributions. Results from a synthetic turbulent flow field are contrasted with both the conventional full Lagrangian approach and direct trajectory methods in order to exemplify the accuracy and computational efficiency of the proposed framework.