The dynamics of discrete particles in turbulent flows: open issues and current challenges in statistical modeling

This article is an invitation. It is, first, an invitation to consider as a subject worthy of attention the wide range of situations where small discrete elements, either bubbles, droplets or solid particles, are embedded in turbulent flows. Occurring often at a human scale and in our daily environments, these turbulent dispersed two-phase flows display complex behavior due to the interplay of two fundamental interactions, the fluid-particle and particle-particle interactions, compounded by the turbulence of the carrier flow. This is not a domain where the basic laws are unknown but where the huge number of degrees of freedom involved call for reduced, or coarse-grained, statistical descriptions to be developed. Since we are considering transport and collision phenomena or relaxation processes, it would seem that they can be handled by kinetic theory. In the general case of non-fully resolved turbulent flows, we are however dealing with particles influenced by random media with non-zero time and space correlations. The second invitation is therefore to recognize the limitations of kinetic-based descriptions and to address the challenges driving us to extend the classical framework, for fluid-particle as well as particle-particle interactions. Taking the standpoint provided by the modern formulation of stochastic processes and focusing on the description of the particle phase, this review proposes a step-by-step pedagogical presentation of current models while pointing out new directions and remaining uncharted territories. This is done to provide answers to the question `why?' as much as `how?' and to try to kindle interest into these open and fascinating issues.