Key Sub-Grid Quantities Affecting the Filtered Drag Force
Based on numerous correlative analyses of fully resolved simulations, the filtered drag force is found to correlate well two sub-grid quantities: the drift velocity and the scalar variance of solid volume fraction, with the drift velocity as the primary marker. We refer to the drift velocity as the gas drift velocity since it is defined as the difference between the filtered gas velocity seen by the particle phase and the resolved filtered gas velocity and a drift velocity for the solid phase can be defined similarly. In present study, the dependence of the filtered drag force on sub-grid quantities are analyzed by employing the second order Taylor polynomial approximation to the microscopic drag coefficient. Through theoretical analyses of the coefficients of the new appearing sub-grid quantities in this approximation, it is found that, other than the gas drift velocity and the scalar variance of solid volume fraction, the solid drift velocity and a third-order moment defined as the covariance of squared solid volume fraction and the fluctuation of slip velocity, are significant for an accurate estimation of the filtered drag force at low Reynolds numbers. The data from fully resolved simulations support this finding and also reveal that the contributions of these four quantities vary at different solid volume fractions.