Stability condition EMMS model

The "variational type of multi-scale CFD, here, refers to CFD with meso-scale models featuring variational stability conditions. This approach can be exemplified by the coupling of the EMMS model (Li and Kwauk, 1994) and TFM, where the EMMS/matrix model (Wang and Li, 2007) at the subgrid level is applied to calculate a structure-dependent drag force. 

In summary, we may expect that the correlative type of multi-scale CFD can be used for the problems with clear scale separations between the micro-scale and the meso-scale, while the variational type, provided with appropriate stability condition, seems free of such limitation. In what follows we will detail some examples of the variational approach by introducing its basis of the EMMS model. 

These six equations are insufficient to give a closure of the EMMS model that involves eight variables. The closure is provided by the most unique part of the EMMS model, that is, the introduction of stability condition to constraint dynamics equations. It is expressed mathematically as Nst = min, which expresses the compromise between the tendency of the fluid to choose an upward path through the particle suspension with least resistance, characterized by Wst = min, and the tendency of the particle to maintain least gravitational potential, characterized by g = min (Li and Kwauk, 1994). 

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... 

A combination of the EMMS/matrix model for clusters and the EMMS/ bubbling model for bubbles was presented in this work. To close this set of equations, they adopted a simple criterion to quantify the phase inversion phenomenon, which is if/>l-/, then the system has dispersed bubbles in continuous dense phase, else it has continuous gas phase with dispersed clusters. Furthermore, clusters wiU exist only when input solids flux is greater than zero. The traditional stability condition of EMMS was, however, adopted to determine these structures, irrespective of bubble or cluster. 

Despite the complexity of mesoscale structures and mechanisms, we highlight a heuristic mesoscale modeling approach starting from a zero-dimensional conceptual model (EMMS model) and ending at the SCMF CFD model. While the stability condition determines the direction of structure evolution of the system, the stability-constrained CFD model further describes the dynamics of structure evolution. The relationship between the two approaches is more or less like that of thermodynamics and chemical kinetics. 

By resolving the structures and dominant mechanisms, it is possible to establish a stability condition reflecting the compromise between different dominant mechanisms for multiphase reaction systems, and the stability condition supplies a mesoscale constraint in addition to mass and momentum conservation equations. The calculation of EMMS model... 

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