Subgrid stresses

M. Moreau, B. Bedat, and O. Simonin. A priori testing of subgrid stress models for euler-euler two-phase LES from euler-lagrange simulations of gas-particle turbulent flow. In 18th Ann. Conf. on Liquid Atomization and Spray Systems. ILASS Americas, 2005. 

The terms that do require modeling are the subgrid stresses, C and R. In fact, most SGS models account for the entire SGS stress given in (19.36). Consequently, LES modeling is formally similar to the modeling of the Reynolds stress tensor in the RANS approach and, therefore, analogous methods are used for LES... 

F. Nicoud and F. Ducros. Subgrid-scale stress modelling based on the square of the velocity gradient. Flow, Turh. and Combustion, 62(3) 183-200, 1999. 

The above equations describe the large-scale motion. Lij represents the interactions among the large scales. The effect of small scales appears through the residual stress tensors (i.e., Cij and Rij). In particular, Cij represents the interactions between the large and small scales, and Rij reflects the interactions between subgrid scales. The tensors Lij, Cij and Rij are known as the Leonard stress, cross-term stress and the residual Reynolds stress, respectively. 

Here, q represents species j or w in gas or solid phase. In aU, we reduce our model to a set of TFM-hke equations. The structure-dependent terms for difRision, stress, drag force, and mass sources distinguish our model from the TFM. The normally used mass transfer and reaction equations of the TFM model can be viewed as a special case of this new model without subgrid structure. 

Schneiderbauer S, Pirker S Filtered and heterogeneity-based subgrid modifications for g3S-sofrd drag and sofrd stresses in bubbling fluidized beds, AICHE J 60 839—854, 2014. 

The weighting faetor m, is ineluded in Eq. (154) to account for the degeneracy in directions for the projeetions of the FCHC lattice that were discussed above. Eggels and Somers also inelude a subgrid-scale stress in their formulation that ean be used for Large Eddy Simulation (LES). However, for sim-plieity, this stress will not be ineluded in the present discussion. 

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