Finite-size Particles Research Articles

Bounds on quantum confinement effects in metal nanoparticles

Quantum size effects on the permittivity of metal nanoparticles are investigated using the quantum box model. Explicit upper and lower bounds are derived for the permittivity and relaxation rates due to quantum confinement effects. These bounds are verified numerically, and the size-dependence and frequency-dependence of the empirical Drude size parameter is extracted from the model. Results suggest that the common practice of empirically modifying the dielectric function can lead to inaccurate predictions for highly uniform distributions of finite-sized particles.

Effects of the finite particle size in turbulent wall-bounded flows of dense suspensions

We use interface-resolved numerical simulations to study finite-size effects in turbulent channel flow of neutrally buoyant spheres. Two cases with particle sizes differing by a factor of two, at the same solid volume fraction of 20 % and bulk Reynolds number are considered. These are complemented with two reference single-phase flows: the unladen case, and the flow of a Newtonian fluid with the effective suspension viscosity of the same mixture in the laminar regime. As recently highlighted in Costa et al. (Phys. Rev. Lett., vol. 117, 2016, 134501), a particle–wall layer is responsible for deviations of the mesoscale-averaged statistics from what is observed in the continuum limit where the suspension is modelled as a Newtonian fluid with (higher) effective viscosity. Here we investigate in detail the fluid and particle dynamics inside this layer and in the bulk. In the particle–wall layer, the near-wall inhomogeneity has an influence on the suspension microstructure over a distance proportional to the particle size. In this layer, particles have a significant (apparent) slip velocity that is reflected in the distribution of wall shear stresses. This is characterized by extreme events (both much higher and much lower than the mean). Based on these observations we provide a scaling for the particle-to-fluid apparent slip velocity as a function of the flow parameters. We also extend the scaling laws in Costa et al. (Phys. Rev. Lett., vol. 117, 2016, 134501) to second-order Eulerian statistics in the homogeneous suspension region away from the wall. The results show that finite-size effects in the bulk of the channel become important for larger particles, while negligible for lower-order statistics and smaller particles. Finally, we study the particle dynamics along the wall-normal direction. Our results suggest that single-point dispersion is dominated by particle–turbulence (and not particle–particle) interactions, while differences in two-point dispersion and collisional dynamics are consistent with a picture of shear-driven interactions.

Lattice Boltzmann simulation of dense rigid spherical particle suspensions using immersed boundary method

We present a lattice Boltzmann model designed for the simulation of dilute and dense finite-sized rigid particle suspensions under applied shear.We use a bottom-up approach and fully resolve the mechanical interaction between fluid and particles. Our model consists in coupling a lattice Boltzmann scheme for Newtonian and incompressible fluid flows with an immersed boundary scheme to simulate two-ways fluid-particles interaction. We introduce a simple yet robust contact model that includes repulsive elastic collision between particles, and neglects lubrication corrections. We apply this model to simple sheared flow with rigid spherical particles and we provide results for the relative apparent viscosity of the particle suspension as a function of the particle volume fraction and strain rate of the flow.We show that, using the proposed approach, there is no need for a lubrication model in the Newtonian regime, provided that an elastic contact model is included. Our algorithm, therefore, can be based only on physically sound and simple rules, a feature that we think to be fundamental for aiming at resolving polydispersed and arbitrarily shaped particle suspensions.Comparing our results with Krieger–Dougherty semi-empirical law, we confirm that the simulations are not sensitive to the particle Reynolds number for Rep ≪ 1 in the Newtonian regime. We show that the proposed model is sufficient to obtain a correct description of the rheology of particle suspension up to volume fraction equal to 0.55 (approaching the critical random packing fraction for monodispersed spheres), which goes beyond the state of the art.

Inertial focusing of finite-size particles in microchannels

At finite Reynolds numbers, $Re$, particles migrate across laminar flow streamlines to their equilibrium positions in microchannels. This migration is attributed to a lift force, and the balance between this lift and gravity determines the location of particles in channels. Here we demonstrate that velocity of finite-size particles located near a channel wall differs significantly from that of an undisturbed flow, and that their equilibrium position depends on this, referred to as slip velocity, difference. We then present theoretical arguments, which allow us to generalize expressions for a lift force, originally suggested for some limiting cases and $Re\ll 1$, to finite-size particles in a channel flow at $Re\leqslant 20$. Our theoretical model, validated by lattice Boltzmann simulations, provides considerable insight into inertial migration of finite-size particles in a microchannel and suggests some novel microfluidic approaches to separate them by size or density at a moderate $Re$.

Experimental study of the flow in the wake of a stationary sphere immersed in a turbulent boundary layer

In many applications, finite-sized particles are immersed in a turbulent boundary layer (TBL) and it is of interest to study wall effects on the instantaneous shedding of turbulence structures and associated mean velocity and Reynolds stress distributions. Here, 3D flow field dynamics in the wake of a prototypical, small sphere (D+=50, 692lReDl959) placed in the TBL's outer, logarithmic, and buffer layer, were measured using time-resolved tomo-PIV. Increasing wall proximity increasingly tilted the mean recirculating wake away from the wall implying a negative lift force. Mean velocity deficit recovery scaled with the mean wake length with minor effects of wall proximity. Farthest from the wall, streamwise Reynolds normal stresses encircled the mean wake as an axisymmetric tubular shell, while transverse and wall-normal stresses extended off its tip as axisymmetric tapered cones. Wall proximity removed axisymmetry and attenuated values near the wall. Reynolds shear stresses were distributed as antisymmetric lobes extending off the mean wake displaying increasing values with reducing sphere-wall gap. Instantaneous snapshots revealed a wake densely populated by archlike vortices with shedding frequencies lower than for a sphere in uniform flow except in the buffer layer. Tilting of the wake away from the wall resulted from self-induced motion of shed hairpinlike vortices whose symmetry plane was increasingly wall-normal oriented with reduced sphere-wall gap.

An immersed lubrication model for the fluid flow in a narrow gap region

A lubrication model is proposed to correct the solution of the Navier-Stokes equations in a poorly-resolved region between two solid surfaces. In the model, the velocity and pressure fields of fluid in a thin film between two solid surfaces are estimated from the Navier-Stokes equations, and those are modified by incorporating the solution of the Reynolds lubrication equation. This immersed lubrication model also modifies the motion of the solid objects in response to the lubrication pressure distribution. A fluid-film bearing case in 2-D is studied to assess the validity of the proposed lubrication model for the fluid flow in the narrow gap with varying the resolution of the domain, and a head-on collision problem of two spherical particles is investigated to discuss the effectiveness of the lubrication model on the motion of finite-sized particles by modifying the fluid flow in the unresolved area of the inter-particle region. The result shows the proposed lubrication model essentially improves the accuracy of the thin-film flow and the motion of the finite-sized objects via the momentum exchange.

Simulation of Finite-Size Particles in Turbulent Flows Using the Lattice Boltzmann Method

Particle laden turbulent flows occur in a variety of industrial applications. While the numerical simulation of such flows has seen significant advances in recent years, it still remains a challenging problem. Many studies investigated the rheology of dense suspensions in laminar flows as well as the dynamics of point-particles in turbulence. Here we will present results on the development of numerical algorithms, based on the lattice Boltzmann method, suitable for the study of suspensions of finite-size particles under turbulent flow conditions. The turbulent flow is modeled by the lattice Boltzmann method, and the interaction between particles and carrier fluid is modeled using the bounce-back rule. Direct contact and lubrication force models for particle-particle interactions and particle-wall interaction are taken into account to allow for a full four-way coupled interaction. The accuracy and robustness of the method is discussed by validating the velocity profile in turbulent pipe flow, the sedimentation velocity of spheres in duct flow and the resistance functions of approaching particles. Preliminary results from the turbulent pipe flow simulations with particles show that the angular and axial velocities of the particles are scattered around values of mean axial velocity and shear rate obtained from the Eulerian velocity field.

Concentration-dependent transport in finite sized composites: Modified effective medium theory

Current models for transport in dispersions, while grounded in well-established effective medium theory (EMT), rely on the assumption of uniformity of the driving force. As consequence, theoretical approaches cannot accommodate driving force inhomogeneities as well as variations over the space occupied by the dispersed phase particles, and existing EMT-based models therefore fail to represent finite particle size effects. Moreover, because transport coefficients are generally considered uniform, such models largely pertain to the Henry's law region. Here, using the context of permeation in mixed-matrix membranes (MMMs), we introduce a self-consistent theory for transport in dispersion-based composites, which captures effects of isotherm nonlinearity and dispersant size without introducing fitting parameters, and accurately predicts concentration-dependent permeabilities. The model is validated against rigorous 3d simulations of transport in filler-polymer composite MMMs, with excellent agreement between theoretical results and those from simulation. Both model and simulations confirm isotherm nonlinearities to have a very significant effect on effective MMM permeability, which is found to be more sensitive to isotherm nonlinearity in the filler phase than in the continuous phase. These effects disappear when the filler phase is much more permeable than the continuous phase, although additional system size effects related to exclusion regions at the ends due to finite particle size lead to decrease in permeability with increase in particle size even for linear isotherms.

Influence of Rayleigh number and solid volume fraction in particle-dispersed natural convection

To study the heat transfer in a natural convection under dense particulate condition with finite-size spherical particles of uniform diameter, a direct numerical simulation is conducted. For calculating the momentum-interaction between the particle and fluid, our original immersed solid method is applied. The heat transfer in the particle-dispersed flow is treated in Eulerian way with the interfacial flux decomposition method. The results shows that, with fixing the thermal conductivity ratio (of the solid to the fluid) to be 100, the temporal- and horizontal-average Nusselt number 〈Nu〉t increases monotonically with solid volume fraction (vf) at Rayleigh number Ra = 104, while 〈Nu〉t at Ra = 105 exhibits a local maximum at around vf = 40%, although 〈Nu〉t at Ra = 105 is always larger than that at Ra = 104. The heat flux in the particulate system is decomposed into the contributions by convection and conduction through the particles, fluid and interface, and the result shows that the conduction through the interface is the dominant factor to the vertical heat flux in the media. Through visualization of the heat flux through the particle surface, the importance of directly resolving the local heat conduction within the individual particle and through the interface is highlighted.

Corrective interface tracking approach to simulate finite-size bubbly flows

This work reports on the development and application of a new approach, corrective interface tracking, to simulate finite size bubbles. Finite-size bubbles, which are by definition bigger than the grid cell size but not well resolved, are not capable of being modeled with any standard two-phase flow approaches, such as interface tracking (IT), Euler-Euler (EE), or Lagrangian particle tracking (LPT). This poses a problem when simulating bubbly flows with different bubble sizes on the same computational grid. The Finite-size Lagrangian particle tracking (FSL) approach (Badreddine et al., 2015), aimed at simulating finite-size bubbly flows by inheriting features of IT and LPT approaches, simulated a single bubble with good accuracy. However, deficiencies with the FSL approach led to a newly developed approach based on interface tracking with the addition of a correcting force. The correcting force, derived from modeling the hydrodynamic forces on a bubble, attempts to correct for errors introduced when a coarse grid is used and the flow and pressure fields around the bubble are under-resolved. Therefore, as a finer grid is used the correcting force decreases. The corrective interface tracking approach is validated against a single bubble rising in stagnant and linear flow, and then results are compared to FSL and to finely resolved IT simulations.

Frequency-dependent hydrodynamic interaction between two solid spheres

Hydrodynamic interactions play an important role in many areas of soft matter science. In simulations with implicit solvent, various techniques such as Brownian or Stokesian dynamics explicitly include hydrodynamic interactions a posteriori by using hydrodynamic diffusion tensors derived from the Stokes equation. However, this equation assumes the interaction to be instantaneous which is an idealized approximation and only valid on long time scales. In the present paper, we go one step further and analyze the time-dependence of hydrodynamic interactions between finite-sized particles in a compressible fluid on the basis of the linearized Navier-Stokes equation. The theoretical results show that at high frequencies, the compressibility of the fluid has a significant impact on the frequency-dependent pair interactions. The predictions of hydrodynamic theory are compared to molecular dynamics simulations of two nanocolloids in a Lennard-Jones fluid. For this system, we reconstruct memory functions by extending the inverse Volterra technique. The simulation data agree very well with the theory, therefore, the theory can be used to implement dynamically consistent hydrodynamic interactions in the increasingly popular field of non-Markovian modeling.

Fully resolved direct numerical simulation of multiphase turbulent thermal boundary layer with finite size particles

In order to investigate the effects of the buoyancy and finite size particles on the heat transfer process in the turbulent thermal boundary layer, we have carried out three simulations in the present paper, a single-phase neutral boundary layer, an unstable boundary layer with buoyancy effect, and a multi-phase boundary layer with thousands of finite size particles. This is the first time that particle-resolved direct numerical simulation (PR-DNS) is used in the study of thermal boundary layer. The DNS results show the turbulent statistics as well as the thermal structures. It turns out that both the buoyancy and the finite size particles will dramatically affect the turbulent statistics of the boundary layer. Detailed comparisons between the three simulations reveal that the buoyancy effect reshapes the coherent structures in the boundary layer, while finite size particles mainly induce additional disturbance all over the computational domain. Specifically, the Reynolds shear stress and the wall normal turbulent heat flux are remarkably enhanced in the log region by the effect of buoyancy. On the other hand, the finite size particles cause remarkable increment of velocity fluctuations all over the boundary layer, while have the effect of stabilizing temperature fluctuation near the wall.

Fully resolved simulations of turbulence modulation by high-inertia particles in an isotropic turbulent flow

We performed direct numerical simulations of decaying isotropic turbulence laden with finite-size particles using an immersed boundary method combined with a soft-sphere collision model. The particle diameter is about 25 times the Kolmogorov length scale, and the particle-to-fluid density ratio ranges from 10 to 1000. The turbulence is enhanced by heavy particles but suppressed by light particles. The radial and spectral distributions of the main quantities are examined. The enstrophy budget is analyzed to reveal how the particles enhance the dissipation rate. The dependence of the two-way coupling term on the particle Reynolds number is loosely predicted. The relation between the particle-induced dissipation rate and particle Reynolds number is also estimated. The results indicate that the work done by particles outweighs the particle-induced dissipation at high particle Reynolds number, thereby enhancing the turbulence. Thus the particle Reynolds number should be a key parameter for turbulence modulation by finite-size and high-inertia particles.

Effects of finite-size neutrally buoyant particles on the turbulent flows in a square duct

Interface-resolved direct numerical simulations of the particle-laden turbulent flows in a square duct are performed with a direct-forcing fictitious domain method. The effects of the finite-size particles on the mean and root-mean-square (RMS) velocities are investigated at the friction Reynolds number of 150 (based on the friction velocity and half duct width) and the particle volume fractions ranging from 0.78% to 7.07%. Our results show that the mean secondary flow is enhanced and its circulation center shifts closer to the center of the duct cross section when the particles are added. The reason for the particle effect on the mean secondary flow is analyzed by examining the terms in the mean streamwise vorticity equation. It is observed that the particles enhance the gradients of the secondary Reynolds normal stress difference and shear stress in the near-wall region near the corners, which we think is mainly responsible for the enhancement in the mean secondary flow. Under a prescribed driving pressure gradient, the presence of particles attenuates the bulk velocity and the turbulent intensity. All particle-induced effects are intensified with increasing particle volume fraction and decreasing particle size, if other parameters are fixed. In addition, the particles accumulate preferentially in the near-corner region. The effects of the type of the collision model (i.e., if friction and damping are included or not) on the results are found significant, but not so significant to bring about qualitatively different results.

Effects of particle-fluid density ratio on the interactions between the turbulent channel flow and finite-size particles.

A parallel direct-forcing fictitious domain method is employed to perform fully resolved numerical simulations of turbulent channel flow laden with finite-size particles. The effects of the particle-fluid density ratio on the turbulence modulation in the channel flow are investigated at the friction Reynolds number of 180, the particle volume fraction of 0.84%, and the particle-fluid density ratio ranging from 1 to 104.2. The results show that the variation of the flow drag with the particle-fluid density ratio is not monotonic, with a larger flow drag for the density ratio of 10.42, compared to those of unity and 104.2. A significant drag reduction by the particles is observed for large particle-fluid density ratios during the transient stage, but not at the statistically stationary stage. The intensity of particle velocity fluctuations generally decreases with increasing particle inertia, except that the particle streamwise root-mean-square velocity and streamwise-transverse velocity correlation in the near-wall region are largest at the density ratio of the order of 10. The averaged momentum equations are derived with the spatial averaging theorem and are used to analyze the mechanisms for the effects of the particles on the flow drag. The results indicate that the drag-reduction effect due to the decrease in the fluid Reynolds shear stress is counteracted by the drag-enhancement effect due to the increase in the total particle stress or the interphase drag force for the large particle-inertia case. The sum of the total Reynolds stress and particle inner stress contributions to the flow drag is largest at the density ratio of the order of 10, which is the reason for the largest flow drag at this density ratio. The interphase drag force obtained from the averaged momentum equation (the balance theory) is significantly smaller than (but agrees qualitatively with) that from the empirical drag formula based on the phase-averaged slip velocity for large density ratios. For the neutrally buoyant case, the balance theory predicts a positive interphase force on the particles arising from the negative gradient of the particle inner stress, which cannot be predicted by the drag formula based on the phase-averaged slip velocity. In addition, our results show that both particle collision and particle-turbulence interaction play roles in the formation of the inhomogeneous distribution of the particles at the density ratio of the order of 10.

Limit cycles for the motion of finite-size particles in axisymmetric thermocapillary flows in liquid bridges

The motion of a small spherical particle of finite size in an axisymmetric thermocapillary liquid bridge is investigated numerically and experimentally. Due to the crowding of streamlines towards the free surface and the recirculating nature of the flow, advected particles visit the free surface repeatedly. The balance between centrifugal inertia and the strong short-range repulsive forces a particle experiences near the free surface leads to an attracting limit cycle for the particle motion. The existence of this limit cycle is established experimentally. It is shown that limit cycles obtained numerically by one-way-coupled simulations based on the Maxey–Riley equation and a particle–surface interaction model compare favorably with the experimental results if the thickness of the lubrication gap between the free surface and the surface of the particle is properly taken into account.

Effect of gravity on the development of homogeneous shear turbulence laden with finite-size particles

ABSTRACTFully resolved simulations of homogeneous shear turbulence (HST) laden with sedimenting spherical particles of finite size have been performed to clarify the effects of gravity on the development of particle-laden turbulent shear flows. We consider turbulence in a horizontal flow subjected to vertical or horizontal shear. Numerical results show that the development of HST laden with finite-size particles are significantly altered by gravity. The effects of gravity lead to a slower increase in the Taylor-microscale Reynolds number, whose value is found to be well correlated with the average particle Reynolds number. The gravity also causes a slower increase in the turbulence kinetic energy (TKE) through the enhancement of energy dissipation. The change in the Reynolds shear stress (RSS) due to particles also significantly contributes to the relative change in TKE. In vertically sheared cases, RSS has high values between counter-rotating trailing vortices behind the particles, which causes a transient relative increase in TKE. In horizontally sheared cases, on the other hand, RSS is reduced in the wakes of particles, which contributes to a significant relative reduction in TKE.

Particle sizes in Saturn's rings from UVIS stellar occultations 1. Variations with ring region

The Cassini spacecraft's Ultraviolet Imaging Spectrograph (UVIS) includes a high speed photometer (HSP) that has observed stellar occultations by Saturn's rings with a radial resolution of ∼10 m. In the absence of intervening ring material, the time series of measurements by the HSP is described by Poisson statistics in which the variance equals the mean. The finite sizes of the ring particles occulting the star lead to a variance that is larger than the mean due to correlations in the blocking of photons due to finite particle size and due to random variations in the number of individual particles in each measurement area. This effect was first exploited by Showalter and Nicholson (1990) with the stellar occultation observed by Voyager 2. At a given optical depth, a larger excess variance corresponds to larger particles or clumps that results in greater variation of the signal from measurement to measurement. Here we present analysis of the excess variance in occultations observed by Cassini UVIS. We observe differences in the best-fitting particle size in different ring regions. The C ring plateaus show a distinctly smaller effective particle size, R, than the background C ring, while the background C ring itself shows a positive correlation between R and optical depth. The innermost 700 km of the B ring has a distribution of excess variance with optical depth that is consistent with the C ring ramp and C ring but not with the remainder of the B1 region. The Cassini Division, while similar to the C ring in spectral and structural properties, has different trends in effective particle size with optical depth. There are discrete jumps in R on either side of the Cassini Division ramp, while the C ring ramp shows a smooth transition in R from the C ring to the B ring. The A ring is dominated by self-gravity wakes whose shadow size depends on the occultation geometry. The spectral “halo” regions around the strongest density waves in the A ring correspond to decreases in R. There is also a pronounced dip in R at the Mimas 5:3 bending wave corresponding to an increase in optical depth there, suggesting that at these waves small particles are liberated from clumps or self-gravity wakes leading to a reduction in effective particle size and an increase in optical depth.

The effect of polydispersity in a turbulent channel flow laden with finite-size particles

We study turbulent channel flows of monodisperse and polydisperse suspensions of finite-size spheres by means of Direct Numerical Simulations using an immersed boundary method to account for the dispersed phase. Suspensions with 3 different Gaussian distributions of particle radii are considered (i.e. 3 different standard deviations). The distributions are centered on the reference particle radius of the monodisperse suspension. In the most extreme case, the radius of the largest particles is 4 times that of the smaller particles. We consider two different solid volume fractions, 2% and 10%. We find that for all polydisperse cases, both fluid and particles statistics are not substantially altered with respect to those of the monodisperse case. Mean streamwise fluid and particle velocity profiles are almost perfectly overlapping. Slightly larger differences are found for particle velocity fluctuations. These increase close to the wall and decrease towards the centerline as the standard deviation of the distribution is increased. Hence, the behavior of the suspension is mostly governed by excluded volume effects regardless of particle size distribution (at least for the radii here studied). Due to turbulent mixing, particles are uniformly distributed across the channel. However, smaller particles can penetrate more into the viscous and buffer layer and velocity fluctuations are therein altered. Non trivial results are presented for particle-pair statistics.

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

Effect of neutrally buoyant large particles on turbulent plane Couette flow is numerically investigated at $R\phantom{\rule{0}{0ex}}e$ near transition. Second moments of the flow are relatively unchanged. However, slightly inertial particles modulate the streak dynamics and flow intermittency.