Euler-Lagrange framework

The use of CFD models for gas-liquid bubble columns has also raised considerable interest only Euler-Euler and Euler-Lagrange frameworks have been employed for the description of the gas and liquid phase states [3.38-3.42]. Bubble trays, considered as particular kinds of bubble columns, have lately presented enormous interest for the flow description by CFD. The flow patterns on a sieve tray have been analyzed in the liquid phase, solving the time-averaged equations of continuity and momentum [3.43]. 

Lau, Y, Bai, W., Deen, N., and Kuipers, J. (2014) Numerical study of bubble break-up in bubbly flows using a deterministic Euler-Lagrange framework. Chem. Eng. ScL, 108, 9-22. 

Within the Euler—Lagrange framework (see, e.g., Patankar and Joseph, 2001 Sommerfeld and Decker, 2004), first the flow field of the carrier or continuous phase is calculated in an Eulerian framework then, the flow behavior of the second, dispersed phase is mimicked by tracking the path of individual (usually point) particles in a Lagrangian way. In this procedure, a correct description of the fluid-particle interaction is crucial. This interaction can be treated one way (the particles feel the flow of the carrier phase, while the latter does not feel the particles) or two way and iteratively (rendering the simulation computationally more demanding). 

In the framework of the Euler-Lagrange formalism, we write the equation of motion for the displacements of the atoms as ... 

The term macroscale will be used to denote multiphase models that employ a hydro-dynamic description of the disperse phase. Such models are also called multi-fluid models (because the disperse phase is treated as an effective fluid), or Euler-Euler models. The name of the latter comes from the numerical treatment of the disperse phase (i.e. discretization on a fixed grid), as opposed to Euler-Lagrange models wherein the disperse phase is tracked in a Lagrangian framework as discrete entities. We should note that, in the... 

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