No abstract

No abstract

The Enskog kinetic theory is used as a starting point to model a suspension of solid particles in a viscous gas. Unlike previous efforts for similar suspensions, the gas-phase contribution to the instantaneous particle acceleration appearing in the Enskog equation is modelled using a Langevin equation, which can be applied to a wide parameter space (e.g. high Reynolds number). Attention here is limited to low Reynolds number flow, however, in order to assess the influence of the gas phase on the constitutive relations, which was assumed to be negligible in a previous analytical treatment. The Chapman–Enskog method is used to derive the constitutive relations needed for the conservation of mass, momentum and granular energy. The results indicate that the Langevin model for instantaneous gas–solid force matches the form of the previous analytical treatment, indicating the promise of this method for regions of the parameter space outside of those attainable by analytical methods (e.g. higher Reynolds number). The results also indicate that the effect of the gas phase on the constitutive relations for the solid-phase shear viscosity and Dufour coefficient is non-negligible, particularly in relatively dilute systems. Moreover, unlike their granular (no gas phase) counterparts, the shear viscosity in gas–solid systems is found to be zero in the dilute limit and the Dufour coefficient is found to be non-zero in the elastic limit.

No abstract

This review article aims to provide a comprehensive and understandable account of the theoretical foundation, modeling issues, and numerical implementation of the Lagrangian-Eulerian (LE) approach for multiphase flows. The LE approach is based on a statistical description of the dispersed phase in terms of a stochastic point process that is coupled with a Eulerian statistical representation of the carrier fluid phase. A modeled transport equation for the particle distribution function-also known as Williams' spray equation in the case of sprays-is indirectly solved using a Lagrangian particle method. Interphase transfer of mass, momentum and energy are represented by coupling terms that appear in the Eulerian conservation equations for the fluid phase. This theoretical foundation is then used to develop LE submodels for interphase interactions such as momentum transfer. Every LE model implies a corresponding closure in the Eulerian-Eulerian two-fluid theory, and these moment equations are derived. Approaches to incorporate multiscale interactions between particles (or spray droplets) and turbulent eddies in the carrier gas that result in better predictions of particle (or droplet) dispersion are described. Numerical convergence of LE implementations is shown to be crucial to the success of the LE modeling approach. It is shown how numerical convergence and accuracy of an LE implementation can be established using grid-free estimators and computational particle number density control algorithms. This review of recent advances establishes that LE methods can be used to solve multiphase flow problems of practical interest, provided sub-models are implemented using numerically convergent algorithms. These insights also provide the foundation for further development of Lagrangian methods for multiphase flows. Extensions to the LE method that can account for neighbor particle interactions and preferential concentration of particles in turbulence are outlined.