Lagrangian–Eulerian methods for multiphase flows

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 sub-models 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.