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Eulerian-Lagrangian models of particle-laden (multiphase) flows describefluid flow and particle dynamics in the Eulerian and Lagrangian frameworksrespectively. Regardless of whether the flow is turbulent or laminar, theparticle dynamics is stochastic because the suspended particles are subjectedto random forces. We use a polynomial chaos expansion (PCE), rather than apostulated constitutive law, to capture structural and parametric uncertaintiesin the particles' forcing. The stochastic particle dynamics is described by ajoint probability density function (PDF) of a particle's position and velocityand random coefficients in the PCE. We deploy the method of distributions (MoD)to derive a deterministic (Liouville-type) partial-differential equation forthis PDF. We reformulate this PDF equation in a Lagrangian form, obtaining PDFflow maps and tracing events and their probability in the phase space. That isaccomplished via a new high-order spectral scheme, which traces, marginalizesand computes moments of the high-dimensional joint PDF and comports withhigh-order carrier-phase solvers. Our approach has lower computational costthan either high-order Eulerian solvers or Monte Carlo methods, is notsubjected to a CFL condition, does not suffer from Gibbs oscillations and doesnot require (order-reducing) filtering and regularization techniques. Thesefeatures are demonstrated on several test cases.

Lagrangian Liouville models of multiphase flows with randomly forced inertial particles