  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.
