How to generate valid local quantities of particle–fluid flows for establishing constitutive relations

There are continuum and discrete approaches to describe granular flows. A continuum approach relies on local average quantities which can be derived through an averaging method based on a discrete approach. However, the selection of averaging domain and the validity of local quantities for constitutive relations are not well established, particularly for transient particle–fluid flows. Here, it is demonstrated that converged local quantities can be achieved on an averaging domain with proper spatial and temporal sizes. Furthermore, the relation between solid pressure and solid volume fraction is established, agreeing qualitatively to all the existing monotonic ones in the literature. However, it is quantitatively different, showing a bifurcation at a high solid volume fraction, which is essentially linked to the variation of short and enduring contacts among particles with flow state and solid volume fraction. This bifurcation must be properly recognized in developing constitutive relations for granular materials.