Deriving fluid-particle correlation closures for Eulerian two-fluid models through use of Langevin equations

The correlation between the fluctuating particle and gas velocity in isotropic turbulence is studied with a set of stochastic differential equations taking into account both particle–particle collisions and the particle feedback on the turbulence. The principal aim of this work is to use the Langevin equations to formulate closures for two-fluid gas–particle flow models. Using Itô calculus we derived solutions for the turbulent kinetic energy of the particle phase and the particle-gas velocity correlations. If particle–particle collisions and particle feedback on the turbulence are neglected the new relations approach the ones derived by Tchen and Hinze but if these effects are included additional terms in the relations appear. In this study we only use a very simple model for the particle–particle collisions. The new relation and the classical relation of Tchen and Hinze for the particle turbulent kinetic energy as well as a relation based on the kinetic theory of granular flows have been implemented in a two-fluid model for turbulent gas–particle flow in a channel in order to make comparison for different particle Stokes numbers. Results show that while the two-fluid model using Hinze’s relations only gives good results for small Stokes numbers, the new relation yields significant improvements for a large range of Stokes numbers.