  In this work, model closures of the multiphase Reynolds-Average Navier-Stokes
(RANS) equations are developed for homogeneous, fully-developed gas--particle
flows. To date, the majority of RANS closures are based on extensions of
single-phase turbulence models, which fail to capture complex two-phase flow
dynamics across dilute and dense regimes, especially when two-way coupling
between the phases is important. In the present study, particles settle under
gravity in an unbounded viscous fluid. At sufficient mass loadings, interphase
momentum exchange between the phases results in the spontaneous generation of
particle clusters that sustain velocity fluctuations in the fluid. Data
generated from Eulerian--Lagrangian simulations are used in a sparse regression
method for model closure that ensures form invariance. Particular attention is
paid to modelling the unclosed terms unique to the multiphase RANS equations
(drag production, drag exchange, pressure strain and viscous dissipation). A
minimal set of tensors is presented that serve as the basis for modelling. It
is found that sparse regression identifies compact, algebraic models that are
accurate across flow conditions and robust to sparse training data.
