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We derive analytical solutions for hydrodynamic sources and sinks to granular temperature in moderately dense suspensions of elastic particles at finite Reynolds numbers. Modelling the neighbour-induced drag disturbances with a Langevin equation allows an exact solution for the joint fluctuating acceleration–velocity distribution function $$P(v^{\prime },a^{\prime };t)$$ . Quadrant-conditioned covariance integrals of $$P(v^{\prime },a^{\prime };t)$$ yield the hydrodynamic source and sink that dictate the evolution of granular temperature that can be used in Eulerian two-fluid models. Analytical predictions agree with benchmark data from particle-resolved direct numerical simulations and show promise as a general theory from gas–solid to bubbly flows. 

This study provides a detailed account of stochastic approaches that may be utilized in Eulerian–Lagrangian simulations to account for neighbour-induced drag force fluctuations. The frameworks examined here correspond to Langevin equations for the particle position (PL), particle velocity (VL) and fluctuating drag force (FL). Rigorous derivations of the particle velocity variance (granular temperature) and dispersion resulting from each method are presented. The solutions derived herein provide a basis for comparison with particle-resolved direct numerical simulation. The FL method allows for the most complex behaviour, enabling control of both the granular temperature and dispersion. A Stokes number $$St_F$$ is defined for the fluctuating force that relates the integral time scale of the force to the Stokes response time. Formal convergence of the FL scheme to the VL scheme is shown for $$St_F \gg 1$$ . In the opposite limit, $$St_F \ll 1$$ , the fluctuating drag forces are highly inertial and the FL scheme departs significantly from the VL scheme.