Robust interpolation for dispersed gas-droplet flows using statistical learning with the fully Lagrangian approach

A novel methodology is presented for reconstructing the Eulerian number density field of dispersed gas-droplet flows modelled using the fully Lagrangian approach (FLA). In this work, the nonparametric framework of kernel regression is used to accumulate the FLA number density contributions of individual droplets in accordance with the spatial structure of the dispersed phase. The high variation which is observed in the droplet number density field for unsteady flows is accounted for by using the Eulerian-Lagrangian transformation tensor, which is central to the FLA, to specify the size and shape of the kernel associated with each droplet. This procedure enables a high level of structural detail to be retained, and it is demonstrated that far fewer droplets have to be tracked in order to reconstruct a faithful Eulerian representation of the dispersed phase. Furthermore, the kernel regression procedure is easily extended to higher dimensions, and inclusion of the droplet radius within the phase space description using the generalised fully Lagrangian approach (gFLA) additionally enables statistics of the droplet size distribution to be determined for polydisperse flows. The developed methodology is applied to a range of one-dimensional and two-dimensional steady-state and transient flows, for both monodisperse and polydisperse droplets, and it is shown that kernel regression performs well across this variety of cases. A comparison is made against conventional direct trajectory methods to determine the saving in computational expense which can be gained, and it is found that $10^3$ times fewer droplet realisations are needed to reconstruct a qualitatively similar representation of the number density field.