Uncertainty quantification in Eulerian-Lagrangian simulations of (point-)particle-laden flows with data-driven and empirical forcing models

An uncertainty quantification framework is developed for Eulerian-Lagrangian models of particle-laden flows, where the fluid is modeled through a system of partial differential equations in the Eulerian frame and inertial particles are traced as points in the Lagrangian frame. The source of uncertainty in such problems is the particle forcing, which is determined empirically or computationally with high-fidelity methods (data-driven). The framework relies on the averaging of the deterministic governing equations with the stochastic forcing and allows for an estimation of the first and second moment of the quantities of interest. Via comparison with Monte Carlo simulations, it is demonstrated that the moment equations accurately predict the uncertainty for problems whose Eulerian dynamics are either governed by the linear advection equation or the compressible Euler equations. In areas of singular particle interfaces and shock singularities significant uncertainty is generated. An investigation into the effect of the numerical methods shows that low-dissipative higher-order methods are necessary to capture numerical singularities (shock discontinuities, singular source terms, particle clustering) with low diffusion in the propagation of uncertainty.