277739 A Posteriori Study of Filtered Euler-Euler Two-Phase Model Using a High Resolution Simulation of a 3-D Periodic Circulating Fluidized Bed
Gas-particle flows in vertical risers are involved in many industrial scale fluidized bed applications such as catalytic cracking, fossil or biomass combustion. Risers flows are often simulated by two-fluid model equations coupled with closures developed in the frame of the kinetic theory of granular media [1,2,3,4]. However, two-fluid model discretized over coarse mesh with respect to particle clustering size are performed for large units because of limited computational resources [5,6]. Now, it is well established that meso-scales cancelled out by coarse mesh simulations have dramatic effect on overall behaviour of flows [7,8,6,9].
Several attempts of the extension of two-fluid model by accounting for these unresolved structures have been done for coarse mesh simulations through different ways . In the framework of the filtered two-fluid model, Sundaresan and co-workers  performed highly resolved simulations of kinetic theory based two-fluid model equations for gas-particle flow in 2-D and 3-D periodic domain and it was stated that the existence of meso-scale structures causes overestimation of the drag force and underestimation of the particle random kinetic energy production and dissipation. Following this study,  proposed an ad hoc sub-grid models for effective drag force and particle stresses which accounts for the effects of unresolved structures on the resolved flows. [8,11] presented a filtering approach methodology to construct closures for the effective drag force and the effective particle stresses.  proposed an effective drag model dependent on the filter size and the solid volume fraction for 2-D bubbling fluidized bed.
Such high resolutions simulations using standard two-fluid model [1,2] were performed by  for a 3-D periodic circulating fluidized bed (PCFB) where typical FCC particles (A-type by Geldart classification) were interacting with the ambient gas. The mean gas-solid flow were periodically driven along the opposite direction of gravity and concerning the transfers between the phases with non-reactive isothermal flow, the drag and buoyancy (Archimedes) forces were accounted for the momentum transfer. The effect of the fluctuations of the gas velocity at small scales was neglected.  refined computational grids to get mesh-independent result in which statistical quantities do not change with any further mesh refinement. This result were then filtered by volume averaging and used to perform a priori analyses on the filtered phase balance equations. Consistent results with previous studies were obtained. The physical identification of over-prediction of drag force was determined and these results show that filtered momentum equation could be computed on coarse grid simulation but must take into account the particle to fluid drift velocity (sub-grid drift velocity) due to the sub-grid correlation between the local fluid velocity and the local particle volume fraction [13,9,12]. Closure relation is needed for the sub-grid drift velocity and several models were proposed, herein, we discuss only the Functional model (see  for other models). Additionally,  proposed to use the Yoshizawa model  for the trace of particle sub-grid stress tensor and the standard compressible Smagorinsky model  as in the single phase flows for the anisotropic parts of stresses.
The objective of the present study is to verify these models in a posteriori studies. For posteriori studies, mesh independent of a 3-D PCFB was used as a reference for making comparisons.
G. Balzer, A. Böelle, and O. Simonin, Eulerian gas-solid flow modelling of dense fluidized bed. In FLUIDIZATION VIII, Proc. International Symposium of the Engineering Foundation, pages 409-418, 1995.
 A. Gobin, H. Neau, O. Simonin, J.-R. Llinas, V. Reiling, and J.-L. Selo, Fluid dynamic numerical simulation of a gas phase polymerization reactor. International Journal for Numerical Methods in Fluids, 43(10-11):1199-1220, 2003.
 D. Gidaspow, J. Jung, and R. K. Singh, Hydrodynamics of fluidization using kinetic theory: an emerging paradigm: 2002 Flour-Daniel lecture. Powder Technology, 148(2-3):123-141, 2004.
 M.A. van der Hoef, M. van Sint Annaland, N.G. Deen, and J.A.M. Kuipers, Numerical simulation of dense gas-solid fluidized beds: a multiscale modeling strategy. Annual Review of Fluid Mechanics, 40(1):47-70, 2008.
 S. Sundaresan, Modeling the hydrodynamics of multiphase flow reactors: Current status and challenges. AIChE Journal, 46(6):1102-1105, 2000.
 J. Wang, A review of eulerian simulation of geldart A particles in gas-fluidized beds. Industrial Engineering Chemistry Research, 48(12):5567-5577, 2009.
 K. Agrawal, P.N. Loezos, M. Syamlal, and S. Sundaresan, The role of meso-scale structures in rapid gas-solid flows. Journal of Fluid Mechanics, 445:151-185, 2001.
 Y. Igci, A. T. Andrews, S. Sundaresan, S. Pannala, and T. O'Brien, Filtered two-fluid models for fluidized gas-particle suspensions. AIChE Journal, 54(6):1431-1448, 2008.
 J-F. Parmentier, O. Simonin, and O. Delsart, A functional subgrid drift velocity model for filtered drag prediction in dense fluidized bed. AIChE J. 58 (2012) 1084-1098.
 A.T. IV Andrews, P. N. Loezos, and S. Sundaresan, Coarse-grid simulation of gas-particle flows in vertical risers. Industrial Engineering Chemistry Research, 44(16):6022-6037, 2005.
 Y. Igci and S. Sundaresan. Constitutive models for filtered Two-Fluid models of fluidized Gas-Particle flows, Industrial & Engineering Chemistry Research, 0(0), 10.1021/ie200190q.
 A. Ozel. Development of Large Eddy Simulation Approach for Simulation of Circulating Fluidized Beds, PhD thesis, Université de Toulouse, Toulouse, France, 2011.
 A. Ozel, J. F. Parmentier, O. Simonin, and P. Fede, A priori test of effective drag modeling for filtered two-fluid model simulation of circulating and dense gas-solid fluidized beds. In 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings. International Conference on Multiphase Flow (ICMF), 2010.
 A. Yoshizawa, Statistical theory for compressible turbulent shear flows, with the application to subgrid modeling. Physics of Fluids, 29(7):2152, 1986.
 J. Smagorinsky, General circulation experiments with the primitive equations. Monthly Weather Review, 91(3):99-164, March 1963.
 M. Germano, U. Piomelli, P. Moin, and W.H. Cabot, A dynamic subgrid-scale eddy viscosity model. Physics of Fluids A: Fluid Dynamics, 3(7):1760, 1991.