The Development and Application of a Kinetic Theory for Modeling Dispersed Particle Flows

Abstract

This Freeman Scholar article reviews the formulation and application of a kinetic theory for modeling the transport and dispersion of small particles in turbulent gas-flows, highlighting the insights and understanding it has provided and some of the long standing problems in the modeling of dispersed flows it has resolved. The theory has been developed and refined by numerous authors and now forms a rational basis for modeling complex particle laden flows. The formalism and methodology of this approach are discussed and the choice of closure of the kinetic equations involved which ensures realizability and well posedness with exact closure for Gaussian carrier flow fields. The historical development is presented and how single particle kinetic equations resolve the problem of closure of the transport equations for particle mass, momentum and kinetic energy /stress (the so called continuum equations) and the treatment of the dispersed phase as a fluid. The mass fluxes associated with the turbulent aerodynamic driving forces and interfacial stresses are shown to be both dispersive and convective in inhomogeneous turbulence with implications for the build up of particles concentration in near wall turbulent boundary layers and particle pair concentration at small separation. It is shown how this approach deals with the natural wall boundary conditions for a flowing particle suspension and examples are given of partially absorbing surfaces with particle scattering, and gravitational settling; how this approach has revealed the existence of contra gradient diffusion in a developing shear flow and the influence of the turbulence on gravitational settling (the Maxey effect). Particular consideration is given to the general problem of particle transport and deposition in turbulent boundary layers and near wall behavior including particle resuspension. Finally the application of a particle pair formulation for both monodisperse and bidisperse particle flows is reviewed where the differences between the two are compared through the influence of collisions on the particle continuum equations and on the particle collision kernel for the clustering of particles and the degree of random uncorrelated motion (RUM) at the small scales of the turbulence. The inclusion of bidisperse particle suspensions implies the application to polydisperse flows and the evolution of particle size distribution.

A model for drift velocity mediated scalar eddy diffusivity in homogeneous turbulent flows

“…A common model form for the turbulent scalar flux is gradient diffusion, written analogously to Fickian diffusion as u i c = −D ij (∂C/∂x j ), with D ij representing eddy diffusivity and C the full scalar field. Foundational work has shown eddy diffusivity decays with increased drift, but few extant models for capturing the flux are algebraic closed-form expressions (Yudine 1959;Moraga et al 2003;Reeks 2021). An exception is Csanady (1963), which derives an expression for particle diffusivity scaling as a function of u d from theoretical arguments about the form and relevant parameters of the velocity autocorrelation, but Squires & Eaton (1991) showed disparities between it and measured experimental and computational turbulent data.…”

Dynamics of particle-laden turbulent Couette flow. Part2: Modified fluctuating force model (M-FFS)

“…Some seminal literature can be found in continuum modelling for the particle phase, where kinetic theory based Phase-space Probability Density Function (PDF) based approach is established. Evolution of these works can be understood from the review articles of Reeks [3] and Reeks [4]. The main goal is to find out a suitable master-equation to account for the statistical nature of the processes e.g.…”