High-order Lagrangian algorithms for Liouville models of particle-laden flows

Eulerian-Lagrangian models describe fluid flow and particle dynamics in the Eulerian and Lagrangian frameworks, respectively. In chaotic systems the particle dynamics are 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 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 on conformally mapped hypercubes 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 unaffected by the Courant-Friedrichs-Lewy (CFL) stability 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.