Concentration and segregation of particles and bubbles by turbulence

Understanding the spatial distribution of finite-size massive particles, such as heavy impurities, dust, droplets, neutrally buoyant particles or bubbles suspended in incompressible, turbulent flows is a relevant issue in industrial engineering and atmospheric physics. In a turbulent flow vortices act as cen-trifuges ejecting particles heavier than the fluid and entrapping lighter ones [1, 2]. This phenomenon produces on one side clusterization (also dubbed preferential concentration) on the other segregation (de-mixing) of particle species differing in size and densities. Stated in a rather simplified form, i.e., assuming spherical, not-deformable particles smaller than the smallest scale of turbulence and gravity negligible, the equation of motion for a particle is [3]: x ̈ = β (∂tu+ (u · ∂)u) − (ẋ − u) /τ. (1) Here u = u(x(t), t) is the fluid velocity field described by the incompressible Navier-Stokes (NS) equation, while the parameters β and τ account for the physical properties of the particle. Specifically, β is a dimensionless number connected to the ratio between the particle density (ρp) and the fluid one (ρf), defined as β ≡ 3ρf/(ρf +2ρp). τ instead is the typical particle response-time, that is τ ≡ a2/(3βν), with, a, the particle radius and, ν, the fluid kinematic viscosity. Equation (1) coupled to NS can be considered an accurate physical model as long as the particle suspension is dilute, namely it is almost collision-less and it does not exert feedback on the fluid, that is to say, it is passively advected by the flow