Extension of EMMS modeling to gas-liquid flow

Albeit originally proposed for gas-solid fluidization, the concepts of structure resolution and compromise between dominant mechanisms embodied in the EMMS model can be generalized into the so-called variational multi-scale methodology (Li and Kwauk, 2003) and extended to other complex systems (Ge et al., 2007). One typical example out of these extensions is the Dual-Bubble-Size (DBS) model for gas-liquid two-phase flow in bubble columns (Yang et al., 2007, 2010). 

From the multi-scale point of view, the total energy dissipation Nj can be grouped into three portions, namely, Nsurf, Nturb, and Nbreak- The last portion is generated from bubble breakage and finally dissipated in the process of bubble coalescence. While Nsurf and Nln rb are considered to be directly dissipated on micro-scale, Nbreak is counted as a kind of meso-scale energy dissipation. Therefore, the stability condition can be either expressed with the minimization of micro-scale energy dissipation Nsurf + Nturb 7 min or conveyed as the maximization of meso-scale energy dissipation Nbreak max. 

With such an understanding on system complexity in mind, the DBS model is composed of two simple force balance equations, respectively, for small or large bubble classes, and one mass conservation equation as well as the stability condition serving as a variational criterion and a closure for conservative equations. For a given operating condition of the global system, six structure parameters for small and large bubble classes (their respective diameters dg, dL, volume fraction 

In fact, extremum tendencies expressing the dominant mechanisms in systems like turbulent pipe flow (Li et al, 1999), gas-liquid-solid flow (Liu et al, 2001), granular flow, emulsions, foam drainages, and multiphase micro-/nanoflows also follow similar scenarios of compromising as in gas-solid and gas-liquid systems (Ge et al., 2007), and therefore, stability conditions established on this basis also lead to reasonable descriptions of the meso-scale structures in these systems. We believe that such an EMMS-based methodology accords with the structure of the problems being solved, and hence realize the similarity of the structures between the physical model and the problems. That is the fundamental reason why the EMMS-based multi-scale CFD improves the 

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