The Dependence of the Effective Gas-Solid Drag in Filtered Two-Fluid Models on the Sub-Filter Solid Turbulence

We present a new approach for the closure of the effective drag force appearing in the filtered two-fluid model (TFM) for gas-solid flow. In particular, we show that the effective drag can be systematically described by using the filtered volume fraction and the kinetic energy of the meso-scale velocity fluctuations, k_s. Characterizing the effective drag as a function of these two markers appears insensitive to the actual configuration of the fine grid simulation as well as to the particle diameter. In addition, the fractional correction, which is a measure for the sub-filter heterogeneities and thus, determines the effective drag, can be easily collapsed to a single band for a wide range of filtered solid volume fractions. This collapse can be simply modeled by the logarithm of k_s. We will further show that this special form of the fractional correction is an intrinsic property of the microscopic drag employed in this study by theoretical considerations.

However, employing these findings to coarse grid simulations requires closures for k_s. Our analysis demonstrates that in a first step k_s can modeled by Smagorinsky type closure. Nevertheless, the presented constitutive relation for the kinetic energy of the cluster fluctuations requires a modification of its single phase counterpart to include the considerable impact of the sub-filter generation of k_s (due to microscopic right hand side of the microscopic solids momentum equation) as well as the transport of k_s at low macroscopic scalar shear rates. At large macroscopic scalar shear rates the presented model approaches the original form of the single phase Smagorinsky model.