[Show abstract][Hide abstract] ABSTRACT: 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 agitation.
Physics of Fluids 11/2013; 25(12). DOI:10.1063/1.4848856 · 2.03 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The effect of particles on turbulence is a key phenomenon in many practical industrial flows (slurries, fluidized beds, …). Depending on the size ratio between the particles and the smallest scale of turbulence, the particle feedback on the flow can either enhance or suppress turbulence. In this communication, we will focus on two generic flow configurations: homogeneous isotropic turbulence seeded with particles and a transitional plane channel flow.
Fully-coupled numerical simulations are used to investigate the interactions between neutrally-buoyant finite-size particles and the fluid flow. The numerical method chosen for this work is the Force-Coupling Method (FCM) (Climent and Maxey, 2009). It is fully-resolved in the sense that the fluid equations are solved at a length-scale smaller than the particle radius. The FCM is based on a low-order, finite multipole expansion of the velocity disturbance induced by the presence of the particles. The presence of the dispersed phase in the fluid is then represented by a body force distribution written as a multipole expansion where the first term is the monopole representing the force that the particle applies on the fluid (due to an external forcing or particle-to-particle contact forces). The second term is the dipole tensor. Its anti-symmetric part is related to external torques applied on the particle. The symmetric part is set through an iterative procedure to ensure that the strain-rate within the fluid volume occupied by the dispersed phase is zero (enforcing solid body response).
Particles in turbulence are experiencing shear flows at different scales. The method has been validated on low Reynolds number shear flows (Abbas et al., 2006) and extension to finite Reynolds will be commented in this communication. Second, the numerical method has been validated in the case of inertia-induced particle migration in a plane Poiseuille flow. In laminar channel flows, neutrally buoyant particles are submitted to a lift force induced by the interaction between the finite-size particle and the parabolic velocity profile when the particle Reynolds number is finite.
Homogeneous isotropic turbulence seeded with finite size neutrally buoyant particles (typically few Kolmogorov length scales) is investigated (Yeo et al., 2010). Results are given on the modulation of the turbulence at volumetric concentrations of 6%. We analyzed the Lagrangian statistics for the velocity and acceleration of the dispersed phase. The turbulent fluctuations are damped at mid-range wavenumbers by the particles while the small scale kinetic energy is significantly enhanced. The pivoting wavenumber characterizing the transition from damped to enhanced energy content is shown to vary with the size of particles.
For neutrally buoyant particles seeded in transitional pipe flows, 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 size-ratio. 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 communication, 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. Particles are randomly seeded into a fluctuating channel flow at a solid volume fraction of 5%, the size ratio of particle diameter to channel height being 1/16. After a transient regime, particles are homogeneously distributed in most of the channel flow. A larger concentration occurs in the near-wall region mainly due to inertial particle transversal migration (Sgr-Silberberg effect). The wall-normal and spanwise flow velocity fluctuations are significantly changed compared to the single-phase flow case. They are larger in the wall vicinity due to larger particle concentration in this region, and they are slightly larger in the channel core due to energy redistribution from the streamwise towards the transverse directions. Hence in the suspension flow, the particles are responsible for the significant agitation growth (in transverse directions) compared to the single-phase case. In our simulations of suspension flows, values of the wall friction coefficient are larger than in the single-phase case. The fluctuating flow exhibits a nearly constant friction coefficient in the transition regime whereas the friction coefficient increases when the Reynolds number is decreased below 1200. Substituting the Reynolds number by a mixture Reynolds number based on the effective suspension viscosity, the evolution of the friction coefficient almost collapses with the laminar law.
These results tend to show that compared to single-phase flows, even at low concentration, finite Reynolds particulate flows relaminarize at lower Reynolds number, in accordance with previous experimental observations of Matas et al. (2003).
References
Matas, J.-P., Morris, J. F., Guazzelli, E., 2003. Transition to turbulence in particulate pipe flow. Phys. Rev. Lett. 90, 014501.
Abbas, M., Climent, E., Simonin, O., Maxey, M. R., 2006. Dynamics of bidisperse suspensions under Stokes flows: Linear shear flow and sedimentation. Phys. Fluids 18.
The Force Coupling Method: A flexible approach for the simulation of particulate flows, E. Climent & M.R. Maxey, (2009) inserted in “Theoretical Methods for Micro Scale Viscous Flows”, Ressign Press, Eds F. Feuillebois and A. Sellier (ISBN: 978-81-7895-400-4).
Modulation of homogeneous turbulence seeded with finite size bubbles or particles (2010) K. Yeo, S. Dong, E. Climent, M.R. Maxey, Int. J. of Multiphase flows, 36, 221–233.
[Show abstract][Hide abstract] ABSTRACT: We present some results issued from the fully resolved direct numerical simulations of a 3-D liquid-solid fluidized bed, experimentally investigated by Aguilar Corona (2008). In these simulations, the flow is solved by a one-fluid formulation of the incompressible Navier-Stokes equations, where the pressure-velocity coupling is provided by an algebraic augmented Lagrangian method and particles presence is modeled by an implicit penalty fictitious domain method, a sub-grid scale lubrication force and soft-sphere collision model. The simulated fluidized bed is a 8 cm diameter, 64 cm height cylindrical column containing 2133 6 mm glass particles fluidized by a low viscosity liquid (3.8x10-3 Pa.s). Particle Reynolds and Stokes number based on terminal velocity are 530 and 7.7 respectively. Simulation results show that homogeneous fluidization regime is obtained for all fluidization velocities investigated, and exhibit large-scale coherent structures in the particle motion. Main features of the Lagrangian velocity signal of the particles are well reproduced by the simulations. Predicted fluidization law nicely fits the experimental curve. Despite the limited time of simulation runs, particle velocity variance is also well predicted as well as the value of the anisotropy coefficient, which is found to be independent of bed concentration. The fluid velocity variance is larger than that of the particles at all bed concentration investigated, following the same trend as Aguilar Corona’s data (2008).
8th International Conference on Multiphase Flow, International Conference on Multiphase Flow (ICMF): Jeju, South Korea. 2013.; 06/2013
[Show abstract][Hide abstract] ABSTRACT: We investigate in this article the macroscopic behavior of sheared suspensions of spherical particles. The effects of the fluid inertia, the Brownian diffusion, and the gravity are neglected. We highlight the influence of the solid-phase inertia on the macroscopic behavior of the suspension, considering moderate to high Stokes numbers. Typically, this study is concerned with solid particles O (100 microns) suspended in a gas with a concentration varying from 5% to 30%. A hard-sphere collision model (with elastic or inelasic rebounds) coupled with the particle Lagrangian tracking is used to simulate the suspension dynamics in an unbounded periodic domain. We first consider the behavior of the suspension with perfect elastic collisions. The suspension properties reveal a strong dependence on the particle inertia and concentration. Increasing the Stokes number from 1 to 10 induces an enhancement of the particle agitation by three orders of magnitude and an evolution of the probability density function of the fluctuating velocity from a highly peaked (close to the Dirac function) to a Maxwellian shape. This sharp transition in the velocity distribution function is related to the time scale which controls the overall dynamics of the suspension flow. The particle relaxation (resp. collision) time scale dominates the particulate phase behavior in the weakly (resp. highly) agitated suspensions. The numerical results are compared with the prediction of two statistical models based on the kinetic theory for granular flows adapted to moderately inertial regimes. The suspensions have a Newtonian behavior when they are highly agitated similarly to rapid granular flows. However, the stress tensors are highly anisotropic in weakly agitated suspensions as a difference of normal stresses arises. Finally, we discuss the effect of energy dissipation due to inelastic collisions on the statistical quantities. We also tested the influence of a simple modeling of local hydrodynamic interactions during the collision by using a restitution coefficient which depends on the local impact velocities.
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: Particle shear-induced self-diffusion is investigated at low Reynolds and variable Stokes (St) numbers. We simulated the suspension hydrodynamics for St<<1 by using the Force Coupling Method. For suspensions with finite particle inertia (finite St), we proposed a new Eulerian prediction based on the kinetic theory for granular flows which have been validated by discrete particle simulations assuming Stokes drag and binary collisions (for low to moderate solid concentration). On the microscopic level, the particle velocity fluctuations have a Gaussian distribution shape for both high and vanishing St, whereas they show a highly peaked distribution for suspensions characterized by St~O(1) and low solid volume fractions. On the macroscopic level, the self-diffusion tensor is strongly anisotropic and the diffusive behavior becomes more prominent when the particle inertia increases. The self-diffusion coefficients decrease with concentration at high St. The results will be analyzed in terms of analogies and differences between the two regimes investigated (hydrodynamic interactions or collisional effects).
[Show abstract][Hide abstract] ABSTRACT: In this work we investigate numerically the dynamics of sheared suspensions in the limit of vanishingly small fluid and particle inertia. The numerical model we used is able to handle the multi-body hydrodynamic interactions between thousands of particles embedded in a linear shear flow. The presence of the particles is modeled by momentum source terms spread out on a spherical envelop forcing the Stokes equations of the creeping flow. Therefore all the velocity perturbations induced by the moving particles are simultaneously accounted for. The statistical properties of the sheared suspensions are related to the velocity fluctuation of the particles. We formed averages for the resulting velocity fluctuation and rotation rate tensors. We found that the latter are highly anisotropic and that all the velocity fluctuation terms grow linearly with particle volume fraction. Only one off-diagonal term is found to be non zero (clearly related to trajectory symmetry breaking induced by the non-hydrodynamic repulsion force). We also found a strong correlation of positive/negative velocities in the shear plane, on a time scale controlled by the shear rate (direct interaction of two particles). The time scale required to restore uncorrelated velocity fluctuations decreases continuously as the concentration increases. We calculated the shear induced self-diffusion coefficients using two different methods and the resulting diffusion tensor appears to be anisotropic too. The microstructure of the suspension is found to be drastically modified by particle interactions. First the probability density function of velocity fluctuations showed a transition from exponential to Gaussian behavior as particle concentration varies. Second the probability of finding close pairs while the particles move under shear flow is strongly enhanced by hydrodynamic interactions when the concentration increases.
Chemical Engineering Research and Design 01/2007; 8(6):778-791. DOI:10.1205/cherd06114 · 2.28 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
Physics of Fluids 12/2006; 18. DOI:10.1063/1.2396916 · 2.03 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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).