  Eulerian-Lagrangian models of particle-laden (multiphase) flows describe
fluid flow and particle dynamics in the Eulerian and Lagrangian frameworks
respectively. Regardless of whether the flow is turbulent or laminar, the
particle dynamics is stochastic because the suspended particles are subjected
to random forces. We use a polynomial chaos expansion (PCE), rather than a
postulated constitutive law, to capture structural and parametric uncertainties
in the particles' forcing. The stochastic particle dynamics is described by a
joint probability density function (PDF) of a particle's position and velocity
and random coefficients in the PCE. We deploy the method of distributions (MoD)
to derive a deterministic (Liouville-type) partial-differential equation for
this PDF. We reformulate this PDF equation in a Lagrangian form, obtaining PDF
flow maps and tracing events and their probability in the phase space. That is
accomplished via a new high-order spectral scheme, which traces, marginalizes
and computes moments of the high-dimensional joint PDF and comports with
high-order carrier-phase solvers. Our approach has lower computational cost
than either high-order Eulerian solvers or Monte Carlo methods, is not
subjected to a CFL condition, does not suffer from Gibbs oscillations and does
not require (order-reducing) filtering and regularization techniques. These
features are demonstrated on several test cases.
