A methodology to devise consistent probability density function models for particle dynamics in turbulent dispersed two-phase flows

The purpose of this article is to propose a generic methodology to build consistent Lagrangian models for polydisperse turbulent two-phase flows where the main issue is to devise a stochastic model for the velocity of the fluid seen by discrete particles. By consistent, it is meant that such models should meet the requirements set forth by Minier et al. (Phys. Fluids, 26, 113303, 2014) and, in the limit of vanishing particle inertia, retrieve the state-of-the-art stochastic models referred to as generalized Langevin models (GLMs) used for the simulation of turbulent single-phase flows. The methodology is generic in the sense that the resulting stochastic models for polydisperse two-phase flows are not limited to one particular fluid model but allows extending any GLM formulation to the two-phase flow situation. This is obtained by introducing a specific operator, which represents how statistical characteristics of fluid particles are transformed, or mapped, to the ones pertaining to the velocity of the fluid seen. In practice, this operator can be worked out separately from first principles or by resorting to some physical inputs. Once it is expressed, the present methodology shows how to extend a GLM for fluid particles to obtain a two-phase GLM formulation in a consistent manner. This is helpful to decouple physics-based developments used to obtain such an operator from the construction of practical stochastic models while ensuring that they remain consistent with fluid descriptions.