Correlative multi-scale CFD

The "correlative" multi-scale CFD, here, refers to CFD with meso-scale models derived from DNS, which is the way that we normally follow when modeling turbulent single-phase flows. That is, to start from the Navier-Stokes equations and perform DNS to provide the closure relations of eddy viscosity for LES, and thereon, to obtain the larger scale stress for RANS simulations (Pope, 2000). There are a lot of reports about this correlative multi-scale CFD for single-phase turbulent flows. Normally, clear scale separation should first be distinguished for the correlative approach, since the finer scale simulation need clear specification of its boundary. In this regard, the correlative multi-scale CFD may be viewed as a "multilevel" approach, in the sense that each span of modeled scales is at comparatively independent level and the finer level output is interlinked with the coarser level input in succession. 

Following the above approach, the correlative multi-scale CFD reduces the computation cost by transforming information over a range of scales into meso-scale models. With these works, it seems feasible to establish a numerical experiment facility to consider thoroughly the... 

Besides the time-averaged and the dynamic characterizations, the correlations between different scales add more complexities to the meso-scale structure. For example, the intensive exchange between meso-scale clusters and dispersed particles will reduce the cross-correlation coefficient and then make it hard to discern a cluster from broth. Such complexity requires more efforts in exploring the dominant mechanisms underlying the correlations, which will be discussed and exemplified by multi-scale CFD in later sections. 

According to the choice of meso-scale models, we may divide the multi-scale CFD into two branches the "correlative" and the "variational," as named in Li and Kwauk (2003) and will be discussed in the following sections. 

Limited to computing capability, the following analysis confines the DNS to the fine-grid TFM simulation, which offers meso-scale closures for the correlative, coarse-grid TFM simulations. For comparison, the variational type of multi-scale CFD takes the EMMS-based models to close TFM simulations. 

Table 1 Comparison between the correlative and the variational types of multi-scale CFD...

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. 

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