Stochastic Lagrangian model for hydrodynamic acceleration of inertial particles in gas–solid suspensions

Abstract

The acceleration of an inertial particle in a gas–solid flow arises from the particle’s interaction with the gas and from interparticle interactions such as collisions. Analytical treatments to derive a particle acceleration model are difficult outside the Stokes flow regime, but for moderate Reynolds numbers (based on the mean slip velocity between gas and particles) particle-resolved direct numerical simulation (PR-DNS) is a viable tool for model development. In this study, PR-DNS of freely-evolving gas–solid suspensions are performed using the particle-resolved uncontaminated-fluid reconcilable immersed-boundary method (PUReIBM) that has been extensively validated in previous studies. Analysis of the particle velocity variance (granular temperature) equation in statistically homogeneous gas–solid flow shows that a straightforward extension of a class of mean particle acceleration models (drag laws) to their corresponding instantaneous versions, by replacing the mean particle velocity with the instantaneous particle velocity, predicts a granular temperature that decays to zero, which is at variance with the steady particle granular temperature that is obtained from PR-DNS. Fluctuations in particle velocity and particle acceleration (and their correlation) are important because the particle acceleration–velocity covariance governs the evolution of the particle velocity variance (characterized by the particle granular temperature), which plays an important role in the prediction of the core annular structure in riser flows. The acceleration–velocity covariance arising from hydrodynamic forces can be decomposed into source and dissipation terms that appear in the granular temperature evolution equation, and these have already been quantified in the Stokes flow regime using a combination of kinetic theory closure and multipole expansion simulations. From PR-DNS data we show that the fluctuations in the particle acceleration that are aligned with fluctuations in the particle velocity give rise to a source term in the granular temperature evolution equation. This approach is used to quantify the hydrodynamic source and dissipation terms of granular temperature from PR-DNS results for freely-evolving gas–solid suspensions that are performed over a wide range of solid volume fraction ($0.1\leqslant {\it\phi}\leqslant 0.4$), Reynolds number based on the slip velocity between the solid and the fluid phase ($10\leqslant \mathit{Re}_{m}\leqslant 100$) and solid-to-fluid density ratio ($100\leqslant {\it\rho}_{p}/{\it\rho}_{f}\leqslant 2000$). The straightforward extension of drag law models does not give rise to any source in the granular temperature due to hydrodynamic effects. This motivates the development of better Lagrangian particle acceleration models that can be used in Lagrangian–Eulerian formulations of gas–solid flow. It is found that a Langevin equation for the increment in the particle velocity reproduces PR-DNS results for the stationary particle velocity autocorrelation in freely-evolving suspensions. Based on the data obtained from the simulations, the functional dependence of the Langevin model coefficients on solid volume fraction, Reynolds number and solid-to-fluid density ratio is obtained. This new Lagrangian particle acceleration model reproduces the correct steady granular temperature and can also be adapted to gas–solid flow computations using Eulerian moment equations.