Low Particle Reynolds Numbers Research Articles

Kinematic waves and collision effects in dense fluid–particle flow during hydraulic conveying

The effects of fluid–particle and particle–particle interactions on the kinematic waves and particle dynamics in a vertical pipe with continuous upward fluid flow are investigated. The unresolved computational fluid dynamics-discrete element method, in which fluid flow field is modeled using large eddy simulation with dynamic Smagorinsky model for the eddy viscosity, is used to simulate the conveying process for particle Reynolds numbers within 8000 and 40000. The results imply that it can show kinematic waves and determine wave velocity based on the wave frequency and wave number relationships, which is also found to closely agree with the theory of kinematic waves based on the Richardson-Zaki relation. One feature is that the kinematic wave velocity in hydraulic conveying manifests itself in relation to the upward fluid velocity. In addition, two different flow regimes are found in the simulation. For relative low particle Reynolds numbers (8000<Rep<16000), the collision and hydrodynamic effects are equally important (regime 1). As the particle Reynolds number increases, the relative significance of collision effects increases. At Rep=16000, the collision effects go beyond the hydrodynamic effects, leading to the flow regime transition (regime 2). The increase of the collision effects also makes the particles uniformly distributed.

Shear-induced motion of a bead on regular substrates at small particle Reynolds numbers

We study experimentally the impact of substrate topology on shear-flow-induced motion of a single bead at low particle Reynolds numbers. The substrates are regular quadratic and triangular arrangements of fixed spherical particles. Their topology is varied by using different spacings between the spheres. Here, we show that it has a strong impact not only on the critical Shields number for incipient bead motion but also on its motion above threshold. We focus on Shields numbers where the bead velocity is smaller than the settling velocity. For the different substrates, the data on the average bead velocity collapse on a master curve, showing the impact of the critical Shields number on the bead motion. To describe the bead motion, we develop a model for creeping flows based on expressions by Goldman, Cox and Brenner for the flow-induced forces and torques on a moving bead near a plane. Our model considers rolling and sliding motion. The bead detaches from the substrate on the downhill side at larger substrate spacing or higher Shields numbers, and flies through the interstices of the substrate until hitting the neighbouring substrate spheres. While sliding has only a minor effect on the average bead velocity, detachment has a strong impact. At large substrate spacings, it leads to a bistability, usually associated with inertial flows, even for adhesionless particles under creeping-flow conditions. The model shows good agreement with the experimental results.

The discrete Green's function paradigm for two-way coupled Euler–Lagrange simulation

We outline a methodology for the simulation of two-way coupled particle-laden flows. The drag force that couples fluid and particle momentum depends on the undisturbed fluid velocity at the particle location, and this latter quantity requires modelling. We demonstrate that the undisturbed fluid velocity, in the low particle Reynolds number limit, can be related exactly to the discrete Green's function of the discrete Stokes equations. In addition to hydrodynamics, the method can be extended to other physics present in particle-laden flows such as heat transfer and electromagnetism. The discrete Green's functions for the Navier–Stokes equations are obtained at low particle Reynolds number in a two-plane channel geometry. We perform verification at different Reynolds numbers for a particle settling under gravity parallel to a plane wall, for different wall-normal separations. Compared with other point-particle schemes, the Stokesian discrete Green's function approach is the most robust at low particle Reynolds number, accurate at all wall-normal separations. To account for degradation in accuracy away from the wall at finite Reynolds number, we extend the present methodology to an Oseen-like discrete Green's function. The extended discrete Green's function method is found to be accurate within $6\,\%$ at all wall-normal separations for particle Reynolds numbers up to 24. The discrete Green's function approach is well suited to dilute systems with significant mass loading and this is highlighted by comparison against other Euler–Lagrange as well as particle-resolved simulations of gas–solid turbulent channel flow. Strong particle–turbulence coupling is observed in the form of turbulence modification and turbophoresis suppression, and these observations are placed in context of the different methods.

Competing flow and collision effects in a monodispersed liquid–solid fluidized bed at a moderate Archimedes number

We study the effects of fluid–particle and particle–particle interactions in a three-dimensional monodispersed reactor with unstable fluidization. Simulations were conducted using the immersed boundary method for particle Reynolds numbers of 20–70 with an Archimedes number of 23 600. Two different flow regimes were identified as a function of the particle Reynolds number. For low particle Reynolds numbers ( $20 &lt; Re_p &lt; 40$ ), the porosity is relatively low and the particle dynamics are dominated by interparticle collisions that produce anisotropic particle velocity fluctuations. The relative importance of hydrodynamic effects increases with increasing particle Reynolds number, leading to a minimized anisotropy in the particle velocity fluctuations at an intermediate particle Reynolds number. For high particle Reynolds numbers ( $Re_p &gt; 40$ ), the particle dynamics are dominated by hydrodynamic effects, leading to decreasing and more anisotropic particle velocity fluctuations. A sharp increase in the anisotropy occurs when the particle Reynolds number increases from 40 to 50, corresponding to a transition from a regime in which collision and hydrodynamic effects are equally important (regime 1) to a hydrodynamic-dominated regime (regime 2). The results imply an optimum particle Reynolds number of roughly 40 for the investigated Archimedes number of 23 600 at which mixing in the reactor is expected to peak, which is consistent with reactor studies showing peak performance at a similar particle Reynolds number and with a similar Archimedes number. Results also show that maximum effective collisions are attained at intermediate particle Reynolds number. Future work is required to relate optimum particle Reynolds number to Archimedes number.

Assessment of behavioral modification techniques through immersed boundary method simulation of binary particle interactions in isotropic turbulence

Behavioral modification effects for particle-laden turbulent flows are developed and assessed through high-fidelity modeling using an implementation of the mirroring ghost-cell based immersed boundary method in conjunction with direct numerical simulation. The continuous phase uses the open-source spectral element method-based solver, Nek5000. A dynamic form of the mirroring immersed boundary method is described that also solves for interparticle attraction and repulsion forces allowing for nontrivial collision outcomes such as agglomeration. The solid-phase solver is validated against empirical drag coefficient data as well as spherical bouncing experiments with excellent agreement obtained at low particle Reynolds numbers. Periodic boxes of homogeneous isotropic turbulence are generated using the linear forcing method at Reλ=29, 51, and 120. Ensembles of structure-resolved binary particle collisions are then studied within these boxes, considering the variation of six key mechanical and chemical parameters. These are the coefficient of restitution, Hamaker constant, surface charge potential, inverse Debye length, temperature, and Reynolds number. It is established that the coefficient of restitution, inverse Debye length, and Reynolds number have the greatest impact on the resulting particle motion and interaction by considering probability density functions of intersurfacial distance and relative particle velocities. Suggestions for real-world procedures that modify these parameters in order to either encourage or discourage particle interaction and potential agglomeration are discussed.

Effects of interphase coupling in one-dimensional particle-laden flow

In Euler–Euler particle-laden flow models, the interphase coupling terms might include, in addition to the drag force, the effects of fluid pressure interaction with the particles and the volumetric displacement of the fluid phase due to the presence of the particles. This paper presents an analytical study on these effects for laminar, one-dimensional, steady, inviscid particle-laden flow. The interphase drag force is represented by a linear Stokes formula that is valid for low particle Reynolds number. It is shown mathematically that a full two-way coupling model, which includes all the aforementioned effects, depends on three parameters: inlet particle volume fraction (concentration), inlet density ratio–between the fluid and particle phases–and inlet velocity ratio. When the inlet particle concentration and density ratio are kept low at 0.01 and 0.001, respectively, both the interphase pressure coupling force and the volumetric displacement of the fluid phase are insignificant. For density ratios of 0.001 and 1 and for the inlet particle concentration values examined, the effect of pressure interaction force is not significant. However, it becomes significant for higher density ratio of 10 at inlet particle concentration of 0.1.

Fluid–particle drag and particle–particle drag in low-Reynolds-number bidisperse gas–solid suspensions

Particle-resolved direct numerical simulations (PR-DNSs) of dynamic bidisperse gas–solid suspensions are performed at low particle Reynolds numbers. Unlike the fixed-bed suspensions, the mobility of particles allows particles of different size types to develop different slip velocities relative to the fluid phase. The scaled slip velocity, defined as the ratio of the slip velocity of one particle type to the mean slip velocity of the mixture, varies profoundly depending on the specific properties of the bidisperse mixture. For large particles, the drag force, scaled by the mean drag force of the mixture, is reasonably predicted by the models obtained from fixed-bed suspensions, while for small particles, these models tend to underestimate the scaled drag force as the scaled slip velocity decreases. By introducing the scaled slip velocity, a new model for the fluid–particle drag on each particle type is proposed and agrees well with the PR-DNS data. For the situation where the monodisperse drag models are employed to predict the mixture mean drag force, a new mean diameter that is variant with the total solid volume fraction is suggested. This diameter increases as the total solid volume fraction decreases and approaches the Sauter mean diameter in the close-packed volume fraction. In dilute suspensions, due to the strong influence of surrounding fluids on the particle phase, the simulated particle–particle drag is significantly smaller than the predictions of models based on kinetic theory of granular flow. Based on the PR-DNS results, new relations for particle–particle drag are also proposed.

Theoretical and numerical analysis of key sub-grid quantities' effect on filtered Eulerian drag force

An expression of the filtered Eulerian drag force is proposed based on the second order Taylor polynomial approximation of a microscopic drag coefficient. Theoretical computations of the expression at low particle Reynolds numbers show that four sub-grid quantities, the gas drift velocity, the solid drift velocity, the scalar variance of solid volume fraction and a third-order moment, are significant for an accurate estimation of the filtered drag force. Various drag models with different combinations of the four sub-grid quantities are proposed and their coefficients are optimized based on fine-grid two fluid model results. The gas drift velocity is demonstrated to be the most important marker in predicting the filtered drag force. The model with the solid drift velocity and the scalar variance of the solid volume fraction is found to be promising in computational fluid dynamic-discrete element model simulations due to the ready availability of the two sub-grid quantities.

Crystal Fractionation by Crystal‐Driven Convection

AbstractFractional crystallization is an essential process proposed to explain worldwide compositional abundances of igneous rocks. It requires crystals to precipitate from the melt and segregate from its residual melt, or experience crystal fractionation. The compositional abundances of volcanic systems show a bell curve distribution suggesting that the process has variable efficiencies. We test crystal fractionation efficiency in convective flow in low to intermediate crystallinity regime. We simulate the physical segregation of crystals from their residual melt at the scale of individual crystals, using a direct numerical method. We find that at low particle Reynolds numbers, crystals sink in clusters. The relatively rapid motion of clusters strips away residual melt. Our results show cluster settling can imprint observational signatures at the crystalline scale. The collective crystal behavior results in a crystal convection that governs the efficiency of crystal fractionation, providing a possible explanation for the bell curve distribution in volcanic systems.

Heterogeneous dispersions as microcontinuum fluids

We present a non-local ‘microcontinuum’ constitutive equation describing particle suspensions undergoing heterogeneous flows in the low Stokes and particle Reynolds number regime. Most prior models rely on mesoscale averages of structural dynamics that smear out non-local effects or otherwise place strong restrictions on flow type. We relieve this restriction here by ensemble averaging the exact equations that govern the suspension at the microscale, thereby obtaining explicit structure-property connections. The result is a pointwise, heterogeneous ensemble average of the suspension stress valid for suspended particles of arbitrary shape, size, composition and concentration, as well as arbitrary sources of heterogeneity, including non-uniform shear fields, heterogeneous force fields or spatially varying volume fractions. This non-local model accounts for spatial and temporal variations in the flow structure over arbitrary length and time scales. We express the microcontinuum constitutive equation as a superposition of gradient operators that automatically accounts for flow heterogeneity over any length scale larger than the particle size. Batchelor’s result for a homogeneous suspension is recovered in the limit of zero gradients; non-zero gradient terms provide non-local corrections to the average stress and account for statistical heterogeneity in the suspension. We utilize energy methods to compute the influence of these gradient operators on the pointwise-averaged stress tensor, revealing a deep connection to the microhydrodynamics formalism. We apply this general framework to a dilute suspension of spherical particles and show that the resultant constitutive equations are consistent with experimental observations.

Performance Simulation for Gas Flow through a Porous Media in Packed bed Columns using CFD

Two dimensions of Computational Fluid Dynamics simulation for gas flow in the column with filling material as bed at low particle Reynolds numbers and high particle Reynolds numbers have been presented. The simulation test has been done with the column length 0.9 m. Porous media model was used to simulate the gas flow inside porous granular media. Model of standard k-e for the flow turbulent range was used; the difference on the mesh size used to study effect it on the performance of gas flow through porous media. The result of flow rate of the gas was an important variable influencing bed and there are good results between the CFD simulation of pressure drop and the predicted values from Ergun model, and Computational Fluid Dynamics gave us simulation with high details for porous flow in packed bed equipment.

Effect of particle size ratio on shear-induced onset of particle motion at low particle Reynolds numbers: From high shielding to roughness

We study incipient motion of single beads on regular substrates made of spherical particles of a different size in steady shear flow at small particle Reynolds numbers. We cover a large range of sizes: from small beads that are highly shielded from the shear flow by the substrate spheres, and hence, are susceptible to the flow through the interstices of the substrate, to beads fully exposed to the flow, where the substrate rather acts like roughness of an otherwise flat surface. Numerical and experimental studies agree within measurement uncertainty. To describe the findings, we extend a recently derived model for particles of equal size which was validated over a wide range of substrates [Agudo et al., “Shear-induced incipient motion of a single sphere on uniform substrates at low particle Reynolds numbers,” J. Fluid Mech. 825, 284–314 (2017)]. The extended model covers the entire spectrum of size ratios, where the critical Shields number varies from about zero to infinity. The model properly describes the numerical and experimental data. For well exposed beads, we find a scaling law between the critical Shields number and the size ratio between mobile bead and substrate spheres with an exponent of −1.

Dynamic particle ordering in oscillatory inertial microfluidics

Particles suspended in conduit flows at small and intermediate Reynolds numbers cluster on specific focal positions while also forming particle pairs and trains due to flow-mediated interactions. The recent introduction of oscillatory inertial microfluidics has enabled the creation of virtually infinite channels, allowing the manipulation of particles at extremely low particle Reynolds numbers (Rep ≪ 1). Here, we investigate experimentally the dynamics of formation, the robustness and the stability of particle pairs, and the precision of the inter-particle distance in an oscillatory flow field, in microchannels with a rectangular cross section. Our results indicate that the cross-sectional arrangement of the particles is fundamental in determining the characteristics of the resulting particle pair.

Coal particle volatile combustion and flame interaction. Part II: Effects of particle Reynolds number and turbulence

Direct numerical simulation (DNS) is performed to characterize volatile combustion of isolated coal particles and closely spaced particle ensembles in laminar and turbulent flow. In part I of the paper the transient evolution of devolatilization and group flame effects were studied, Tufano et al. (2018). This analysis was limited to laminar flows and relatively low particle Reynolds numbers Rep. Here, we investigate the effect of large Rep and considerable levels of turbulence on the devolatilization and burning behavior of the particles. The complex physico-chemical interactions during PCC are characterized for laminar flow conditions first, before arrays of infinite particle layers are subjected to turbulence. Increasing Rep in laminar flow leads to delayed ignition of single particles with local extinction due to high upstream scalar dissipation rates and the formation of wake flame structures downstream of the particle. An attempt is made to recover the single particle results with a standard steady laminar flamelet approach, which is shown to work well at low Rep, but fails at high Reynolds numbers, where multi-dimensional effects occur and must be incorporated into flamelet modeling. It is found that applying standard film theory to model the effect of convection on devolatilization rates can lead to qualitatively wrong trends and up to 66% error in the peak devolatilization rate compared to the DNS results at Rep=8. The analysis of particle arrays in laminar flow shows a strong dependence of flame interactions on the values of Rep due to the different extent of the particle wakes. The occurrence of significant levels of turbulence introduces a wide range of additional chemical states due to the randomness of the turbulent fluctuations that can either act to increase or decrease the local strain, in turn weakening or enhancing particle interactions. For the studied conditions turbulence slightly promotes the mass release from the most upstream particle set, but considerably delays the volatile release from the downstream particles, which is explained by the different extents and degrees of interaction of the up- and downstream volatile flames. The present results are considered useful for the development of LES sub-grid scale combustion models for pulverized coal flames, such as flamelets and others.

Oscillatory inertial focusing in infinite microchannels

Inertial microfluidics (i.e., migration and focusing of particles in finite Reynolds number microchannel flows) is a passive, precise, and high-throughput method for microparticle manipulation and sorting. Therefore, it has been utilized in numerous biomedical applications including phenotypic cell screening, blood fractionation, and rare-cell isolation. Nonetheless, the applications of this technology have been limited to larger bioparticles such as blood cells, circulating tumor cells, and stem cells, because smaller particles require drastically longer channels for inertial focusing, which increases the pressure requirement and the footprint of the device to the extent that the system becomes unfeasible. Inertial manipulation of smaller bioparticles such as fungi, bacteria, viruses, and other pathogens or blood components such as platelets and exosomes is of significant interest. Here, we show that using oscillatory microfluidics, inertial focusing in practically "infinite channels" can be achieved, allowing for focusing of micron-scale (i.e. hundreds of nanometers) particles. This method enables manipulation of particles at extremely low particle Reynolds number (Rep < 0.005) flows that are otherwise unattainable by steady-flow inertial microfluidics (which has been limited to Rep > ∼10-1). Using this technique, we demonstrated that synthetic particles as small as 500 nm and a submicron bacterium, Staphylococcus aureus, can be inertially focused. Furthermore, we characterized the physics of inertial microfluidics in this newly enabled particle size and Rep range using a Peclet-like dimensionless number (α). We experimentally observed that α >> 1 is required to overcome diffusion and be able to inertially manipulate particles.

Direct numerical simulation of magnetic particles suspended in a Newtonian fluid exhibiting finite inertia under SAOS

A direct numerical simulation approach is utilized to understand the oscillatory shear rheology of a confined suspension of magnetic chains formed by paramagnetic circular cylinders under the influence of an external magnetic field. The common assumption of gap-spanning chains made in the literature is relaxed in this work, so that a fully suspended (periodic) array of magnetic chains is formed. In this sense, the effective rheological parameters are only influenced through a layer of fluid adjacent to the walls. All tests are conducted at very low but finite particle Reynolds numbers, and typical inertial effects are discussed. The main aim of the present study is to investigate the apparent viscoelasticity of the system as a function of the external magnetic field and frequency of the input strain. This work concentrates on cases with large blockage ratio in order to have pronounced viscoelastic behaviours.

Effects of granular temperature on inter-phase drag in gas-solid flows

Direct numerical simulations with a second-order immersed boundary-lattice Boltzmann method are used to investigate the effect of particle fluctuation on flows in fixed random arrays of spheres at low and moderate particle Reynolds numbers. Random velocities obeying the isotropic Maxwellian distribution are imposed to all particles in the computational domain to mimic the granular temperature due to particle collisions. The simulation results show that the effect of particle fluctuation on the drag force is significant especially when the solid volume fraction and the particle Reynolds number are small. The drag increment due to the particle fluctuation increases with the increase of the granular temperature-based Reynolds number, however, it decreases with the increase of the solid volume fraction and the particle Reynolds number. On the basis of the simulation results, a new drag force relation based on the stationary drag in random arrays of spheres for arbitrary solid volume fractions, granular temperature-based Reynolds numbers, and particle Reynolds numbers is formulated. The fluctuations of the drag force on individual particles with respect to the mean drag are also analyzed. It is found that the drag on individual particles can differ up to 40% from the mean at small granular temperature-based Reynolds numbers and the difference increases rapidly with the increase of the granular temperature-based Reynolds number, indicating the particle fluctuation can affect the individual drag force significantly. The present results also show the drag force on individual particles well follows Gaussian distribution. The mean relative deviation is found to increase with the increase of the granular temperature-based Reynolds number and the solid volume fraction but is not affected much by the particle Reynolds number.

Shear-induced incipient motion of a single sphere on uniform substrates at low particle Reynolds numbers

We study the incipient motion of single spheres in steady shear flow on regular substrates at low particle Reynolds numbers. The substrate consists of a monolayer of regularly arranged fixed beads, in which the spacing between beads is varied resulting in different angles of repose and exposures of the particle to the main flow. The flow-induced forces and the level of flow penetration into the substrate are determined numerically. Since experiments in this range had shown that the critical Shields number is independent of inertia but strongly dependent on the substrate geometry, the particle Reynolds number was fixed to 0.01 in the numerical study. Numerics indicates that rolling motion is always preferred to sliding and that the flow penetration is linearly dependent on the spacing between the substrate particles. Besides, we propose an analytical model for the incipient motion. The model is an extension of Goldman’s classical result for a single sphere near a plain surface taking into account the angle of repose, flow orientation with respect to substrate topography and shielding of the sphere to the linear shear flow. The effective level of flow penetration is the only external parameter. The model, applied to triangular and quadratic arrangements with different spacings, is able to predict the dependence of the critical Shields number on the geometry and on the orientation of the substrate. The model is in very good agreement with numerical results. For well-exposed particles, we observed that the minimum critical Shields number for a certain angle of repose does not depend sensitively on the considered arrangement. At large angles of repose, as expected in fully armoured beds, the model is consistent with experimental results for erodible beds at saturated conditions.

Improving particle drag predictions in Euler–Lagrange simulations with two-way coupling

Euler–Lagrange methods are popular approaches for simulating particle-laden flows. While such approaches have been rigorously verified in the dilute limit (where particles do not noticeably alter their carrier flow), much less verification has been attempted for cases where the coupling between the two phases leads to non-negligible modifications in the local fluid velocity. We review one of these techniques for coupled fluid–particle flows, the volume-filtered Euler–Lagrange method, and show that it (like many similar methods) provides erroneous predictions for the interphase drag force due to the presence of the particles. We show that these errors are tied to inaccuracies in the numerical implementation of the drag model for systems with two-way coupling. We therefore introduce a simple approach to correct the implementation of this drag model, and show that this corrected implementation provides accurate and grid-independent predictions of particle settling in two-way coupled flows at low particle Reynolds numbers. Finally, we study the effect of the corrected implementation on a more complicated, cluster-induced turbulence flow.

Influence of Inertial Particles on Turbulence Characteristics in Outer and Near Wall Flow as Revealed With High Resolution Particle Image Velocimetry

A fully developed turbulent particle-gas flow in a rectangular horizontal channel 100 × 10 × 4000 mm3 is disclosed with high spatial resolution two-dimensional (2D) particle image velocimetry (PIV). The objective is to increase the knowledge of the mechanisms behind alterations in turbulent characteristics when adding two sets of relatively large solid spherical particles with mean diameters of 525 and 755 μm and particle size distributions of 450–600 and 710–800 μm, respectively. Reynolds numbers are 4000 and 5600 and relatively high volume fraction of 5.4 × 10−4 and 8.0 × 10−4 are tested. Both the near wall turbulent boundary layer flow and outer core flow are considered. Results show that the carrier phase turbulent intensities increase with the volume fraction of the inertial particles. The overall mean flow velocity is affected when adding the particles but only to a minor extent. Near the wall, averaged velocity decreases while fluctuating velocity components increase when particles are added to the flow. Quadrant analysis shows the importance of sweep near the wall and ejection events in the region defined by y+ &gt; 20. In conclusion, high inertia particles can enhance turbulence even at relatively low particle Reynolds number &lt;90. In the near bottom wall region, particles tend to be a source of instability reflected as enhancement in rms values of the normal velocity component.