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Flow past disperse solid particles or bubbles induces fluctuations in carrier fluid velocity, which correlate with temperature fluctuations in non-isothermal flows resulting in the pseudo-turbulent heat flux (PTHF). In the Eulerian-Eulerian (EE) two-fluid (TF) model, the transport of PTHF is shown to be an important contributor to the overall energy budget, and is modeled using a pseudo-turbulent thermal diffusivity (PTTD). The PTHF and PTTD were originally quantified using particle-resolved direct numerical simulation (PR-DNS) data, and correlations were developed over a range of solid volume fraction (0.1 ≤ 𝜀𝑠 ≤ 0.5) and mean slip Reynolds number (1 ≤ 𝑅𝑒𝑚 ≤ 100) for a Prandtl number of 0.7. However, the original PTTD correlation diverges to infinity as the solid volume fraction goes to zero, which is physically unrealistic. This singular behavior is problematic for EE TF simulations at particle material fronts where solid volume fraction values can fall below the lower limit of existing data (𝜀𝑠 =0.1) to zero in the pure carrier phase. In this work, additional PR-DNS data are reported for 𝜀𝑠 < 0.1, and improved correlations are developed for the PTHF and PTTD. The new PTTD correlation is non- singular, and both the PTHF and PTTD decay exponentially to zero as the solid volume fraction approaches zero, which is physically reasonable. This improves prediction of PTHF transport in dilute flow using EE TF heat transfer simulations. 

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.