Dynamic Taylor-based gradient model for subgrid heat flux in turbulent fluidization: An a priori analysis

In this study, we propose closure models for the subgrid turbulent heat flux and drift
temperature components. These terms arise as unresolved contributions in the filtered two-fluid model (TFM) for heat transport equations of turbulent gas-particle flows. In the context of moderately dense turbulent fluidization, the drift temperature appears as a correction to the filtered resolved interphase heat transfer component and expressed by a covariance of gas phase temperature and solid concentration. The derived models are mainly motivated by our findings showing that the most commonly used subgrid heat flux models associated linearly with the gradient resolved temperature are invalid in this kind of multiphase turbulence. They reveal a clear misalignment with the actual gas and solid phase turbulent heat flux vectors. Thus, the proposed models are principally based on Taylor development of the subfilter terms, identified as Clark’s gradient model [Clark et al., J. Fluid Mech. 91, 1 (1979)]. They differ only in the dynamic forma-
tion of model coefficient using the test filters. We find that the conventional dynamic
Smagorinsky-type model leads to large errors in predicting the subfilter turbulent heat flux and solid concentration variance, and our dynamic procedure based on a Taylor series
expansion of test-filtered resolved terms (Leonard term) [Balarac et al., Phys. Fluids 20,
035114 (2008)] substantially improves the predictive accuracy. This last approach was
originally assessed based on the optimal estimator concept, i.e., conditional expectations and statistical minimization of modeling errors; thus we use this terminology in referring to all of our models adopting this dynamic procedure. Two dynamic drift temperature closures are accordingly presented based on optimal estimator dynamic procedure coefficient. They basically approximate the covariance of gas phase temperature and solid concentration by the product of its variances or the gradient Clark’s expansion. All of our models are
examined in a framework of an apriori study, where filtering fine-grid TFM simulation
data of laboratory-scale wall-bounded turbulent fluidization of Geldart B group particles is employed. The presented models provide more accurate and robust tools for heat transfer TFM simulations of industrial-scale fluidized bed reactors.