Theoretical analyses of the sub-grid quantities' effect on filtered Eulerian drag force

An expression of the filtered Eulerian drag force is proposed based on the second order Taylor polynomial approximation of the microscopic Eulerian drag coefficient. Theoretical computations of the expression are performed at low Reynolds numbers based on an available microscopic drag model. It shows that four sub-grid quantities, i.e., the gas drift velocity, the solid drift velocity, the scalar variance of solid volume fraction and a third-order moment, defined as the covariance of squared solid volume fraction and the slip velocity, are significant for an accurate estimation of the filtered drag force at low Reynolds numbers. The gas drift velocity is nothing but the drift velocity defined by Parmentier et al. (AIChE Journal, 2012, 58 (4): 1084-1098), and in analogy to the gas drift velocity, we defines the solid drift velocity in the present work. The great relevance of the gas drift velocity and the scalar variance of solid volume fraction to the filtered drag force have been demonstrated through numerous correlative analyses of fully resolved simulations. Present theoretical analysis shows that the gas drift velocity term is primarily responsible for the drag reduction, whereas the solid drift velocity term is prone to attenuate this reduction, especially at moderate and high solid volume fractions. The scalar variance of solid volume fraction term is found to increase the filtered drag force in the full range of solid volume fractions. The third-order term has exactly the same coefficient with that of the variance term, and it mostly attains negative values with the trend to decrease the filtered drag force.