Numerical simulations of random suspensions at finite Reynolds numbers

doi:10.1016/S0301-9322(03)00016-8

International Journal of Multiphase Flow

Volume 29, Issue 4, April 2003, Pages 579-601

https://doi.org/10.1016/S0301-9322(03)00016-8 Get rights and content

Abstract

The sedimentation of solid spherical particles in an initially quiescent fluid is investigated through numerical simulations. Under conditions of Stokes flow, the results provide good agreement with experiments for the mean settling velocity while the velocity fluctuation levels grow with the size of the system in accordance with theoretical predictions for homogeneous suspensions. New results on finite Reynolds number suspensions are presented that illustrate the role of wake-induced interactions between particles. A significant reduction in the average settling velocity is explained by an enhancement of a wake-induced scattering process. Anisotropy in the fluid flow is evident and the evolution of velocity correlations is investigated for various particulate Reynolds numbers. Increases in Reynolds number tend to decrease the integral length scale of the fluctuating flow field and reduce significantly the Lagrangian time scale of the velocity fluctuations. Analysis of the evolution of fluctuation levels when the domain width grows indicates that inertial screening dramatically reduces the divergence.

Introduction

The description of hydrodynamic interactions in dispersed two-phase flows is of great interest for the design of many industrial processes. As an example, mixing of powders in a reactive solvent is omnipresent in chemical engineering. Such configurations are commonly encountered too in environmental sciences: settling of micro-organisms (plankton) and migration of micron-scale particles is widely studied in oceanography, pollutant transport in underground water resource and dusty gas ejection from industrial chimneys are strictly controlled to define safety areas.

Most of the problems involved in the prediction of such complex hydrodynamic interactions arise from the extremely large scatter of length scales present in the flow. Due to an inverse cascade of energy transport, energy supplied at the smallest scales can induce large-scale motions in the fluid. Even at low volume concentration (few percent), the motion of monodisperse spherical solid particles in a suspension cannot be predicted by homogeneous theory due to non-random, multi-body interactions. From the wake-induced flow between two nearby particles to the creation of clusters of particles whose dynamics are mainly controlled by large-scale collective effects, the succession of interactions provides momentum transfer at all scales. Numerical simulations of the motion of both phases (liquid and particles) provide a valuable tool to investigate the evolution of settling characteristics.

Beyond the applications to engineering, the mean settling velocity in a suspension and the relative velocity fluctuations are of particular interest from a fundamental point of view. For low Reynolds number flows (Stokes approximation), numerous theoretical, experimental and numerical results are available (see Davis, 1996 for a detailed review). The evolution of the mean settling velocity for a monodisperse suspension is not a subject of controversy. On the other hand, Caflish and Luke (1985) predicted a divergence of the velocity fluctuations of the suspension with increasing size of the container. Most experiments do not exhibit such a divergence and fluctuation levels saturate for larger containers (see Nicolai and Guazzelli, 1995; Segrè et al., 1997). In recent investigations into the dynamics of Stokes suspensions, arguments involving non-homogeneity due to vertical concentration gradients (Mucha et al., submitted to J. Fluid Mech., 2002; Dance, 2002) or long-range screening related to sidewalls (Brenner, 1999; Ladd, 2002; Bernard-Michel et al., 2002) have been proposed. In the case of finite Reynolds numbers, most of the questions are still open. What is the impact of the loss of fore-and-aft wake symmetry on the average settling velocity and the fluctuations? The purpose of the present paper is to provide new results on random homogeneous suspensions at finite Reynolds number using two-way coupling simulations. The particles are considered non-Brownian and experience only hydrodynamic interactions when settling under gravity. We consider only periodic boxes for the simulations in order to avoid any additional factors from non-homogeneity.

The present paper is organized as follows. Particular features of the force-coupling method (FCM) used to simulate the two-phase flow dynamics are briefly outlined. Results on settling in Stokes suspensions are presented and compared to theoretical and experimental evidence in the literature. New results on finite Reynolds suspensions are then given and the associated flow structure is investigated.

Section snippets

Two-way coupling approach

Direct numerical simulation of dispersed two-phase flows is clearly difficult with present computing resources, even with the notable successes of Hu (1996) and Johnson and Tezduyar (1996). A major difficulty lies in imposing the no-slip condition on the freely moving particles. Such simulations generally require the use of a time-dependent mesh that evolves, following individual particles. Different approaches have been taken by Glowinski et al. (1999), Patankar et al. (2000) and by Esmaeeli

Stokes flow assumption

Most of the studies on sedimenting solid particles have been performed for low Reynolds number flows. This particular situation corresponds to a physical configuration of small particles (10–100 μm) settling in liquids (water, glycerin or silicon oils). The density of a particle should be close to the fluid density to be consistent with the neglect of the excess inertia of a particle. Experimentally, particles could be polypropylene (density 0.91) or polyamide (density 1.12). As an example, a

Finite Reynolds number sedimentation

It is well known that the settling of two particles in Stokes flow conditions is characterized by an increase in the fall velocity while the particle separation remains unchanged. This particular feature is related to the symmetry properties of Stokes flow. Absence of inertia in the motion of the continuous phase suppresses non-linear wake interactions and gives rise to the long-range velocity perturbations that decrease slowly, inversely proportional to the separation distance between the

Conclusion

The purpose of the present paper was to investigate the behavior of non-Brownian suspensions under low, but finite particle Reynolds number. Spherical solid particles are sedimenting in an initially quiescent fluid. Our force-coupling model of hydrodynamic interactions was first tested for Stokes flow conditions. Good agreement with the extensive experimental data and empirical correlations was obtained when both fluid and particle inertia was neglected. The level of the velocity fluctuations

Acknowledgements

We are grateful to the Scientific Computing Facility at Boston University and the French Computing Centers (IDRIS–CINES) for making available computing resources. We wish to thank Dr. Sarah Dance and Dr. Suchuan Dong for valuable discussions and for their assistance with the tests of dipole terms and the parallelization of the code. Partial support by the Defense Advanced Research Projects Agency, Advanced Technology Office, under the Friction drag Program (ARPA order K042/07/39) contract

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^☆: This paper was presented in the 4th International Conference on Multiphase Flow (ICMF-2001). The ICMF-2001 took place in New Orleans, USA during the week of May 27 to June 1, 2001 and was attended by 630 delegates representing 46 countries. Professor E.E. Michaelides was the chair of the conference.

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Numerical simulations of random suspensions at finite Reynolds numbers☆

Abstract

Introduction

Section snippets

Two-way coupling approach

Stokes flow assumption

Finite Reynolds number sedimentation

Conclusion

Acknowledgements

Int. J. Multiphase Flow

Int. J. Multiphase Flow

Computat. Meth. Appl. Mech. Eng.

J. Computat. Phys.

Int. J. Multiphase Flow

Int. J. Multiphase Flow

Fluid Dyn. Res.

Int. J. Multiphase Flow

Sedimentation in a dilute dispersion of spheres

J. Fluid Mech.

Particle velocity fluctuations and correlation lengths in dilute sedimenting suspensions

Phys. Fluids

Stokesian dynamics

Ann. Rev. Fluid Mech.

Screening mechanisms in sedimentation

Phys. Fluids

Variance in the sedimentation speed of a suspension

Phys. Fluids

Bubbles-induced agitation and microstructure in uniform bubbly flows at small to moderate particle Reynolds numbers

Phys. Fluids

Hydrodynamic diffusion of suspended particles: a symposium

J. Fluid Mech.

On the two-way interaction between homogeneous turbulence and dispersed solid particles

Phys. Fluids A

Direct numerical simulations of bubbly flows: Part I

J. Fluid Mech.