Shear-induced self-diffusion of inertial particles in a viscous fluid

We propose a theoretical prediction of the self-diffusion tensor of inertial particles embedded in a viscous fluid. The derivation of the model is based on the kinetic theory for granular media including the effects of finite particle inertia and drag. The self-diffusion coefficients are expressed in terms of the components of the kinetic stress tensor in a general formulation. The model is valid from dilute to dense suspensions and its accuracy is verified in a pure shear flow. The theoretical prediction is compared to simulations of discrete particle trajectories assuming Stokes drag and binary collisions. We show that the prediction of the self-diffusion tensor is accurate provided that the kinetic stress components are correctly predicted

Fully coupled simulations of monodisperse and bidisperse suspensions in a linear shear flow

The dynamics of macroscopically homogenous sheared suspensions of neutrally buoyant, non-Brownian spheres is investigated in the limit of very small Reynolds and Stokes numbers using the Force Coupling Model (Lomholt & Maxey1). In this numerical approach, the velocity disturbance is obtained by a low order multipole expansion (particle forcing on the flow is represented by monopole and dipole terms spread on a finite volume envelop related to particle radius)

Conditional stability of particle alignment in finite-Reynolds-number channel flow

Finite-size neutrally buoyant particles in a channel flow are known to accumulate at specific equilibrium positions or spots in the channel cross-section if the flow inertia is finite at the particle scale. Experiments in different conduit geometries have shown that while reaching equilibrium locations, particles tend also to align regularly in the streamwise direction. In this paper, the Force Coupling Method was used to numerically investigate the inertia-induced particle alignment, using square channel geometry. The method was first shown to be suitable to capture the quasi-steady lift force that leads to particle cross-streamline migration in channel flow. Then the particle alignment in the flow direction was investigated by calculating the particle relative trajectories as a function of flow inertia and of the ratio between the particle size and channel hydraulic diameter. The flow streamlines were examined around the freely rotating particles at equilibrium, revealing stable small-scale vortices between aligned particles. The streamwise inter-particle spacing between aligned particles at equilibrium was calculated and compared to available experimental data in square channel flow (Gao {\it et al.} Microfluidics and Nanofluidics {\bf 21}, 154 (2017)). The new result highlighted by our numerical simulations is that the inter-particle spacing is unconditionally stable only for a limited number of aligned particles in a single train, the threshold number being dependent on the confinement (particle-to-channel size ratio) and on the Reynolds number. For instance, when the particle Reynolds number is $\approx1$ and the particle-to-channel height size ratio is $\approx0.1$, the maximum number of stable aligned particles per train is equal to 3. This agrees with statistics realized on the experiments of (Gao {\it et al.} Microfluidics and Nanofluidics {\bf 21}, 154 (2017)).Comment: 13 pages, 13 figure

Dynamics of laminar pressure-driven channel flows laden with neutrally buoyant finite-size particles.

Since the pioneering work of Reynolds (1883), much effort has been allocated on the topic of laminar-turbulent transition regime in a single-phase flow, with special focusing on the unstable and intermittent natures of this regime (Mullin, 2011). The transition regime of dispersed flows carried less attention even though dispersed flows are used in many industrial processes. As for suspensions of neutrally buoyant particles, Matas et al. (2003) observed changes in the values of the critical Reynolds numbers depending on both the solid volume fraction and the particle-to-pipe sizeratio. Typically, the transition occurs at lower Reynolds numbers when the flow carries macro-sized particles at dilute to moderate concentrations (up to 25%). On the contrary, the critical Reynolds numbers of the onset of transition is shifted towards greater values when particles are micro-sized and their concentration is higher. In this work, we aim at understanding the mechanisms lying behind the shift of the laminar-turbulent transition regime down to lower critical Reynolds numbers in suspension flows of macro-sized particles. Fully-coupled numerical simulations are used to investigate the interactions between neutrally-buoyant finite-size particles and a transitional channel flow. To our knowledge, other than the simulations of Shao et al. (2012) and Garcia-Villalba et al. (2012) performed in turbulent channel flows, there are no direct numerical simulations performed on fluctuating suspension flows in channels or pipes with finite-size particles. The numerical method chosen for this work is the Force-Coupling Method (FCM) (Maxey and Patel, 2001, Lomholt and Maxey 2003). It is fully-resolved in the sense that the fluid equations are solved at a length-scale smaller than the particle radius. In a first step, the laminarization process of a single-phase flow initially turbulent at Re=6000 is statistically characterized (Re is based on the average flow velocity, the channel height and the kinematic viscosity). In a second step, particles are randomly added to the fluctuating channel flow at a solid volume fraction of 5%, the size ratio of particle diameter to channel height being 1/16. The starting point of the calculation of the suspension flow is a snapshot taken from the single-phase flow case at Re=1625 (the smallest Reynolds number at which the flow does not relaminarize)

Dynamics of bidisperse suspensions under stokes flows: linear shear flow and eedimentation

Sedimenting and sheared bidisperse homogeneous suspensions of non-Brownian particles are investigated by numerical simulations in the limit of vanishing small Reynolds number and negligible inertia of the particles. The numerical approach is based on the solution of the three-dimensional Stokes equations forced by the presence of the dispersed phase. Multi-body hydrodynamic interactions are achieved by a low order multipole expansion of the velocity perturbation. The accuracy of the model is validated on analytic solutions of generic flow configurations involving a pair of particles. The first part of the paper aims at investigating the dynamics of monodisperse and bidisperse suspensions embedded in a linear shear flow. The macroscopic transport properties due to hydrodynamic and non hydrodynamic interactions (short range repulsion force) show good agreement with previous theoretical and experimental works on homogeneous monodisperse particles. Increasing the volumetric concentration of the suspension leads to an enhancement of particle fluctuations and self-diffusion. The velocity fluctuation tensor scales linearly up to 15% concentration. Multi-body interactions weaken the correlation of velocity fluctuations and lead to a diffusion like motion of the particles. Probability density functions show a clear transition from Gaussian to exponential tails while the concentration decreases. The behavior of bidisperse suspensions is more complicated, since the respective amount of small and large particles modifies the overall response of the flow. Our simulations show that, for a given concentration of both species, when the size ratio varies from 1 to 2.5, the fluctuation level of the small particles is strongly enhanced. A similar trend is observed on the evolution of the shear induced self-diffusion coefficient. Thus for a fixed and total concentration, increasing the respective volume fraction of large particles can double the velocity fluctuation of small particles. In the second part of the paper, the sedimentation of a single test particle embedded in a suspension of monodisperse particles allows the determination of basic hydrodynamic interactions involved in a bidisperse suspension. Good agreement is achieved when comparing the mean settling velocity and fluctuations levels of the test sphere with experiments. Two distinct behaviors are observed depending on the physical properties of the particle. The Lagrangian velocity autocorrelation function has a negative region when the test particle has a settling velocity twice as large as the reference velocity of the surrounding suspension. The test particle settles with a zig-zag vertical trajectory while a strong reduction of horizontal dispersion occurs. Then, several configurations of bidisperse settling suspensions are investigated. Mean velocity depends on concentration of both species, density ratio and size ratio. Results are compared with theoretical predictions at low concentration and empirical correlations when the assumption of a dilute regime is no longer valid. For particular configurations, a segregation instability sets in. Columnar patterns tend to collect particles of the same species and eventually a complete separation of the suspension is observed. The instability threshold is compared with experiments in the case of suspensions of buoyant and heavy spheres. The basic features are well reproduced by the simulation model

Discrete element study of liquid-solid slurry flows through constricted channels

Discrete element model is used to simulate the flow of liquid-granule mixtures in an inclined channel containing a linear contraction. All the relevant particle/particle and particle/fluid interactions are included in the numerical model. The presence of the contraction induces different steady morphologies of the solid phase or the mixture depending on whether closed or open channels are used. These flows behave quite differently depending on the upstream Froude number and the contraction size ratio. The model is first validated by comparing with the existing results for dry granular (glass particles) chute flows (Vreman et al., 2007). Then simulations of a chute of glass particles in water flowing in a closed channel are compared to the dry granular case. With the same solid flux at the inlet, the hydrodynamic forces in the liquid-solid mixture induce higher particle solid volume fractions in the part of the flow containing the solid phase. The streamwise particle velocity (resp. depth of the solid phase) has the same evolution along the channel with smaller (larger) values than in the dry granular flow case

Modulation of large-scale structures by neutrally buoyant and inertial finite-size particles in turbulent Couette flow

Direct numerical simulations of particle laden flow are carried out with the Force-Coupling Method (FCM) to study the effect of finite-size particles on turbulent plane Couette flow (pCf). The Reynolds numbers considered were close to the laminar-turbulent transition, such that large scale rotational structures were well identified and relatively steady. Thereby, interaction of particles with coherent structures could be evidenced using particle-resolved numerical simulations with two Couette gap-to-particle size ratios (10 and 20), and with particle-to-fluid density ratio ranging from 0 to 5. Regarding the distribution of particles in the mixture flow, the concentration profiles (averaged in the homogeneous streamwise and spanwise directions) suggested a relatively homogeneous distribution of the particles across the Couette gap, resulting from the balance between hydrodynamic repulsive force from the walls, turbulent mixing and shear-induced diffusion. In the case of neutrally-buoyant particles, 2D snapshots of particle positions revealed higher (resp. lower) presence of particles in the sweep (resp. ejection) regions where they are trapped (resp. expelled) for a while. As for buoyant particles, the light ones (ρ_p /ρ_f 1), were rather subject to an outward motion towards the walls, leading to small localized peaks in the concentration profile in that region. Time averaged profiles, in the wall-normal direction, of the mean flow and Reynolds stress components did not reveal significant difference between single phase and mixture flows at equivalent effective Reynolds number, except that the wall shear stress is higher in the two-phase flow. However temporal and modal analysis of flow fluctuations, suggested that particles had an impact on the regeneration cycle of turbulence. While the energy of large scale vortices (LSV) was unchanged by particles (only the rotation rate inside the vortex core was slightly reduced), the level of kinetic energy was increased over the range of intermediate wavenumbers for all considered particle sizes and densities. This is mainly due to flow perturbations induced by the non-deformability of the dispersed phase (finite size effect)

Auto-diffusion de particules dans un ecoulement cisaille : des interactions hydrodynamiques aux effets collisionnels

Ce travail aborde, à l'aide de simulations Lagrangiennes, la description du comportement rhéophysique d'une suspension de particules solides sphériques en écoulement cisaillé. Les effets de l’inertie du fluide, de l'agitation Brownienne et de la gravité sont négligés. Les suspensions étudiées sont classées en deux grandes familles, en fonction de l'inertie de la phase dispersée caractérisée par le nombre de Stokes. A très petit nombre de Stokes, les suspensions sont de type liquide-solide où le fluide est très visqueux. Le modèle de simulation "Force Coupling Method" est utilisée pour simuler les interactions hydrodynamiques qui contrôlent la dynamique de ces suspensions. Cette méthode se base sur un développement multipolaire de la perturbation de vitesse induite par la présence des particules dans le fluide porteur. L'évolution de quantités macroscopiques en fonction de la fraction volumique du solide [φ=1-20%] est analysée dans des suspensions monodisperses. Les résultats (fluctuations de vitesse, auto-diffusion, auto-corrélation des vitesses et distribution spatiale de paires de particules…) confortent les tendances observées dans plusieurs études de la littérature. Nous montrons que l’agitation des particules induit un comportement diffusif dont l’intensité est une fonction croissante de la concentration. Le niveau d’agitation mais aussi le temps de diffusion augmentent lorsque les interactions multi-corps contrôlent la dynamique de la suspension. Les effets de lubrification associés à des particules proches du contact sont résolus précisément. Ceci permet d'utiliser la FCM pour simuler des suspensions de concentration plus élevée (allant jusqu'à 35%), et de quantifier leur viscosité effective. Le modèle de simulation est étendu aux cas de suspensions bidisperses. L'impact de la variation du rapport de taille ou de concentration sur les statistiques (des deux espèces) est examiné pour une fraction volumique constante de la phase dispersée. Pour un rapport de concentration fixe, nous avons trouvé qu'un rapport de taille croissant entraîne une augmentation (resp. diminution) du niveau de fluctuation des petites (resp. grosses) particules. Quand le rapport de taille et la concentration totale sont fixes, l'augmentation du nombre de grosses particules entraîne l'augmentation du taux de fluctuation et de la diffusion des deux espèces. Les suspensions caractérisées par un nombre de Stokes modéré ou grand sont en général de type gaz-solide. Un modèle de simulation basé sur l’intégration des trajectoires de particules assimilées à des sphères dures est utilisé pour simuler la dynamique de la suspension. Le mouvement des particules est uniquement contrôlé par les collisions et par la force de traînée sur une particule isolée. Les simulations montrent que les propriétés de la suspension dépendent fortement de l'inertie des particules et de la concentration. La variation du nombre de Stokes de 1 à 10 induit une augmentation de l'agitation des particules de trois ordres de grandeur, et une évolution de la distribution de vitesse d'une forme très piquée (proche d’un Dirac) à une forme Maxwellienne. Les résultats numériques sont confrontés aux prédictions de deux modèles issus de la théorie cinétique des milieux granulaires adaptés aux nombres de Stokes modérés: la fonction Dirac (resp. Maxwellienne déviée) est utilisée pour décrire les suspensions faiblement (resp. fortement) agitées. Une nouvelle théorie pour déterminer les coefficients du tenseur d'auto-diffusion Lagrangienne est développée et validée avec les résultats des simulations. Les coefficients de diffusion et la viscosité de la phase solide sont également confrontés aux modèles théoriques utilisés pour la prédiction d’écoulements complexes. L'effet de l'inélasticité sur les quantités statistiques est également discuté. La conclusion de ce document fait la synthèse de tous ces résultats en proposant une approche unifiée de l’évolution de la viscosité effective du mélange fluide/particules. Une modification de la méthode FCM est proposée pour modéliser simultanément l'inertie des particules et les interactions hydrodynamiques

Inertia-driven particle migration and mixing in a wall-bounded laminar suspension flow

Laminar pressure-driven suspensionflows are studied in the situation of neutrally buoyant particles at finite Reynolds number. The numerical method is validated for homogeneous particle distribution (no lateral migration across the channel): the increase of particle slip velocities and particle stress with inertia and concentration is in agreement with former works in the literature. In the case of a two-phase channel flow with freely moving particles, migration towards the channel walls due to the Segré-Silberberg effect is observed, leading to the development of a non-uniform concentration profile in the wall-normal direction (the concentration peaks in the wall region and tends towards zero in the channel core). The particle accumulation in the region of highest shear favors the shear-induced particle interactions and agitation, the profile of which appears to be correlated to the concentration profile. A 1D model predicting particle agitation, based on the kinetic theory of granular flows in the quenched state regime when Stokes number St = O(1) and from numerical simulations when St < 1, fails to reproduce the agitation profile in the wall normal direction. Instead, the existence of secondary flows is clearly evidenced by long time simulations. These are composed of a succession of contra-rotating structures, correlated with the development of concentration waves in the transverse direction. The mechanism proposed to explain the onset of this transverse instability is based on the development of a lift force induced by spanwise gradient of the axial velocity fluctuations. The establishment of the concentration profile in the wall-normal direction therefore results from the combination of the mean flow Segré-Silberberg induced migration, which tends to stratify the suspension and secondary flows which tend to mix the particles over the channel cross section

The effect of neutrally buoyant finite-size particles on channel flows in the laminar-turbulent transition regime

The presence of finite-size particles in a channel flow close to the laminar-turbulent transition is simulated with the Force Coupling Method which allows two-way coupling with the flow dynamics. Spherical particles with channel height-to-particle diameter ratio of 16 are initially randomly seeded in a fluctuating flow above the critical Reynolds number corresponding to single phase flow relaminarization. When steady-state is reached, the particle volume fraction is homogeneously distributed in the channel cross-section (φ=5%) except in the near-wall region where it is larger due to inertia-driven migration. Turbulence statistics (intensity of velocity fluctuations, small-scale vortical structures, wall shear stress) calculated in the fully coupled two-phase flow simulations are compared to single-phase flow data in the transition regime. It is observed that particles increase the transverse r.m.s. flow velocity fluctuations and they break down the flow coherent structures into smaller, more numerous and sustained eddies, preventing the flow to relaminarize at the single-phase critical Reynolds number. When the Reynolds number is further decreased and the suspension flow becomes laminar, the wall friction coefficient recovers the evolution of the laminar single-phase law provided that the suspension viscosity is used in the Reynolds number definition. The residual velocity fluctuations in the suspension correspond to a regime of particulate shear-induced agitatio