Coarse-graining particulate two-phase flow

TL;DR

The paper develops a general, exact coarse-graining framework for particulate two-phase flow that does not rely on scale separation, producing coupled macroscopic two-fluid mass and momentum balances valid for arbitrary particle size relative to the flow scale. By using particle indicator functions $X^p$ and a flexible averaging operator, it ensures macroscopic volume fractions sum to unity and reveals additional stress contributions, including interfacial contact stresses and particle-rotation effects. The framework is fluid-rheology agnostic and applicable to diverse particle shapes, densities, and mild deformability, with practical formulations that can be extracted from DNS-DEM data. The authors tailor the method to Immersed Boundary Method (IBM) implementations, deriving two exact IBM adaptations and validating them against sediment-transport simulations, demonstrating that first-order $R/L$ approximations can fail when scale separation is absent. Overall, this work enables faithful prediction of macroscopic rheology in heterogeneous two-phase flows and provides actionable computational tools for high-fidelity coarse-graining of DNS-DEM data.

Abstract

To acquire the ability to numerically study the rheology of particulate two-phase flows that lack scale separation, we present a general method to average or coarse-grain the equations of motion of a mixture of a continuous fluid of arbitrary rheology and non-Brownian particles, interacting via contacts, of arbitrary shapes and compositions. It universally covers ensemble and typical spatio-temporal averaging procedures and overcomes two shortcomings of existing methods. First, the derived micromechanical expressions for the coarse-grained fields are mathematically exact and formulated in a manner that allows a computationally cheap extraction from Direct Numerical Simulation-Discrete Element Method (DNS-DEM) simulations, avoiding the unlimited-order derivatives appearing in previous exact formulations. Second, the microscopic volume fraction of each particle is its corresponding indicator function, rather than the traditional volume-weighted delta distribution at its center of mass, to ensure that the resulting macroscopic fluid and solid volume fractions add precisely to unity. This leads to an additional contact stress contribution not seen in standard coarse-grained expressions for granular matter, and, for non-spherical particles, to particle-rotational contributions to translational solid phase balance equations. Many implementations of DNS-DEM simulations are based on Immersed Boundary Methods (IBMs), for which modifications of the coarse-graining method are necessary due to certain peculiarities of IBMs, such as the replacement of the particles' interiors by pseudo-fluid. We therefore derive mathematically exact adaptations of the coarse-graining method for two distinct common IBM versions, implement one version to obtain coarse-grained fields from sediment transport simulations based on this version, and validate the implementation.

Figures (3)

• Figure 1: Summary of the derivation of the macroscopic momentum balance equations. The bluish boxes mark the most important derivation steps. They are responsible for the novel elements in the final expressions for the effective fluid phase ($\bm{\sigma^\mathrm{f}}$) and solid phase ($\bm{\sigma^\mathrm{s}}$) stress tensors when compared with expressions from previous studies.
• Figure 2: Snapshot of a DNS-DEM simulation of steady, bed-tangentially homogeneous sediment transport.
• Figure 3: Vertical profiles of the effective stress components (a) $\sigma^\mathrm{f}_{zx}$, (b) $\sigma^\mathrm{f}_{zz}$, (c) $\sigma^\mathrm{s}_{zx}$, and (d) $\sigma^\mathrm{s}_{zz}$. Data correspond to a IBM-based DNS-DEM simulation of sediment transport, calculated in three distinct manners: directly using the derived micromechanical expressions, Eqs. (\ref{['SigmaSolid']}) and (\ref{['SigmaIBM']}), indirectly via integrating the right-hand sides of Eqs. (\ref{['SigmaFluidSteady']}) and (\ref{['SigmaSolidSteady']}), or directly using the stress expressions approximated in the first order of $R/L$ (see Appendix \ref{['Implementation']} for implementation details).