Large eddy simulation of turbulent gas-solid ¯ows in a vertical channel and evaluation of second-order models Qunzhen Wang a , Kyle D. Squires b, * , Olivier Simonin c,d a Analytical Services & Materials, 244 East Avenue K-4, Lancaster, CA 93534, USA b MAE Department, Arizona State University, Box 876 106, Tempe, AZ 85287-6106, USA c Laboratoire National d'Hydraulique / EDF, 6 Quai Watier, 78400 Chatou, France d Institut de M ecanique des Fluides / INPT, All ee Camille Soula, 31400 Toulouse, France Abstract Large eddy simulation (LES) has been used for prediction of the particle-laden turbulent ¯ow in a vertical channel. Calculations were performed at a Reynolds number based on friction velocity and channel half-width of 180. Subgrid-scale stresses in the ¯uid were closed using the Lagrangian dynamic eddy viscosity model. Particle motion was governed by drag. Particle±particle collisions were neglected and the ¯uid was not modi®ed by the presence of the particles. Results for a particle density ratio of 2118 are pre- sented in this paper, statistics of the dispersed phase were obtained from the trajectories of 250,000 particles. The simulation results were used to perform an a priori evaluation of closure model assumptions in the two-¯uid model of Simonin (1991). In general, there is good agreement between LES results and closure assumptions used for the unknown terms in the particle kinetic stress and ¯uid± particle covariance transport equations. Turbulent momentum transfer from the ¯uid in the particle kinetic stress equation is accu- rately predicted. In the ¯uid-particle covariance equation the greatest discrepancies in closure of the momentum transfer term occur in the near-wall region, indicating the model used for the ¯uid turbulent time scale must be improved. Closure models for triple correlation transport of the kinetic stress and ¯uid±particle covariance are also reasonable. Ó 1998 Elsevier Science Inc. All rights reserved. 1. Introduction and background In two-¯uid modeling of turbulent two-phase ¯ows sepa- rate mean transport equations are derived for both the contin- uous and dispersed phases. As is well known, this procedure leads to closure problems due to the presence of turbulent cor- relations and interphase transfer terms. Of particular impor- tance to prediction of gas±solid ¯ows are the closure approxi- mations used to model the particle kinetic stress tensor (the second-order moments of the particle velocity ¯uctuations) and ¯uid-particle velocity correlations. The assumptions of gradient transport and local equilibrium are often applied to model these correlations, generally assuming the particle kinet- ic shear stresses are proportional to the local mean shear of the dispersed phase and normal stresses are dominated by ¯uid drag (e.g., see Elghobashi and Abou-Arab, 1983). One of the main problems with models based on gradient transport for the shear stress and equilibrium for the normal stresses is that analyses and simulation results show that the Boussinesq approximation is not satisfactory. Reeks (1993) and Liljegren (1993) have shown that the particle shear stresses depend explicitly upon the shearing of both phases. Reeks (1993) and Simonin et al. (1995) and Wang and Squires (1996) showed that the particle stress tensor anisotropy increases with the particle relaxation time and is signi®cantly greater than the ¯uid one. These complex features are con- tained in the approach proposed by Simonin (1991a), which is based on separate transport equations for the components of the kinetic stress tensor and for the ¯uid-particle velocity co- variance. This approach accounts for particle-turbulence inter- actions, the production by mean velocity shear, the mean and turbulent transport by the particle velocity, and the in¯uence of inter-particle collisions. While the particle kinetic stress transport model has been applied with success to many exper- imental gas±solid ¯ow con®gurations (e.g, jets, mixing layers, vertical channel, swirling ¯ows, etc.), one of the primary di- culties with evaluating the closure approximations in such an approach is that there is a lack of detailed experimental mea- surements. In a previous study using the results from large eddy simulation, Simonin et al. (1995) validated the model der- ivation and closure assumptions in homogeneous turbulent shear ¯ows. While very useful for both better understanding particle interactions with turbulence as well as evaluating closure models, measurements or simulation results from non-homogeneous ¯ows are required to validate the model ful- ly. The objective of this work is to present results from large eddy simulation of the simplest non-homogeneous particle- laden turbulent shear ¯ow ± fully developed channel ¯ow ± and to use large eddy simulation (LES) results to evaluate clo- sure approximations in the two-¯uid model of Simonin (1991a). International Journal of Heat and Fluid Flow 19 (1998) 505±511 * Corresponding author. E-mail: squires@asu.edu. 0142-727X/98/$ ± see front matter Ó 1998 Elsevier Science Inc. All rights reserved. PII: S 0 1 4 2 - 7 2 7 X ( 9 8 ) 1 0 0 3 0 - 9