Abstract
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. 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 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 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 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.
Highlights
Based originally on the Freeman Scholar Lecture 2010 on ’the PDF approach for modelling dispersed flows and subsequently extended to the present time.Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING
Fevrier et al [8] observed that the velocity of suspended par- 2200 ticles subject to Stokes drag in a direct numerical simulation (DNS) of homogeneous isotropic 2201 turbulence consisted of two components: a passive scalar compo- 2202 nent resulting from transport in a smoothly continuously varying 2203 carrier flow velocity field that accounts for all particle–particle 2204 and fluid–particle two point spatial correlations which they 2205 referred to as the mesoscopic Eulerian particle velocity fields (MEPVF); and a spatially uncorrelated component 2206 which is referred to as random uncorrelated motion (RUM) whose contribution to the particle kinetic energy 2208 increases as the particle inertia increases
3360 In these concluding remarks, it is useful to reflect on some of 3361 the important issues that have been raised in this review, to elabo3362 rate on the insights and understanding this kinetic theory has pro3363 vided, and to highlight some of the long standing problems in the 3364 modeling of dispersed flows this approach has resolved
Summary
Based originally on the Freeman Scholar Lecture 2010 on ’the PDF approach for modelling dispersed flows and subsequently extended to the present time. Whilst this separation does occur in an FP formulation, the form of the dispersion term due to the random translational velocity is incorrect More precisely this failure to preserve RGT invariance means failure to reproduce the correct equation of state for the dispersed phase (except in the case of very inert particles for St ) 1Þ Following the approach given in Reeks [23], we generalize the method used to obtain a closed expression for hf Wi by consider543 ing f ðsÞ as continuous random process random referring to ðv; x; tjsÞ, i.e., the aerodynamic force measured at time s along a particle trajectory which passes through the point (v; x) at time t. These results have been derived independently using the Clausius virial theorem of classical kinetic theory [23]
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