Particle-resolved Simulations Research Articles

Three-body correlations and conditional forces in suspensions of active hard disks.

Self-propelled Brownian particles show rich out-of-equilibrium physics, for instance, the motility-induced phase separation (MIPS). While decades of studying the structure of liquids have established a deep understanding of passive systems, not much is known about correlations in active suspensions. In this work we derive an approximate analytic theory for three-body correlations and forces in systems of active Brownian disks starting from the many-body Smoluchowski equation. We use our theory to predict the conditional forces that act on a tagged particle and their dependence on the propulsion speed of self-propelled disks. We identify preferred directions of these forces in relation to the direction of propulsion and the positions of the surrounding particles. We further relate our theory to the effective swimming speed of the active disks, which is relevant for the physics of MIPS. To test and validate our theory, we additionally run particle-resolved computer simulations, for which we explicitly calculate the three-body forces. In this context, we discuss the modeling of active Brownian swimmers with nearly hard interaction potentials. We find very good agreement between our simulations and numerical solutions of our theory, especially for the nonequilibrium pair-distribution function. For our analytical results, we carefully discuss their range of validity in the context of the different levels of approximation we applied. This discussion allows us to study the individual contribution of particles to three-body forces and to the emerging structure. Thus, our work sheds light on the collective behavior, provides the basis for further studies of correlations in active suspensions, and makes a step towards an emerging liquid state theory.

Machine Learning to Predict the Global Distribution of Aerosol Mixing State Metrics

Atmospheric aerosols are evolving mixtures of chemical species. In global climate models (GCMs), this “aerosol mixing state” is represented in a highly simplified manner. This can introduce errors in the estimates of climate-relevant aerosol properties, such as the concentration of cloud condensation nuclei. The goal for this study is to determine a global spatial distribution of aerosol mixing state with respect to hygroscopicity, as quantified by the mixing state metric χ . In this way, areas can be identified where the external or internal mixture assumption is more appropriate. We used the output of a large ensemble of particle-resolved box model simulations in conjunction with machine learning techniques to train a model of the mixing state metric χ . This lower-order model for χ uses as inputs only variables known to GCMs, enabling us to create a global map of χ based on GCM data. We found that χ varied between 20% and nearly 100%, and we quantified how this depended on particle diameter, location, and time of the year. This framework demonstrates how machine learning can be applied to bridge the gap between detailed process modeling and a large-scale climate model.

Direct simulation of liquid–gas–solid flow with a free surface lattice Boltzmann method

ABSTRACTDirect numerical simulation of liquid–gas–solid flows is uncommon due to the considerable computational cost. As the grid spacing is determined by the smallest involved length scale, large grid sizes become necessary – in particular, if the bubble–particle aspect ratio is on the order of 10 or larger. Hence, it arises the question of both feasibility and reasonability. In this paper, we present a fully parallel, scalable method for direct numerical simulation of bubble–particle interaction at a size ratio of 1–2 orders of magnitude that makes simulations feasible on currently available super-computing resources. With the presented approach, simulations of bubbles in suspension columns consisting of more than 100,000 fully resolved particles become possible. Furthermore, we demonstrate the significance of particle-resolved simulations by comparison to previous unresolved solutions. The results indicate that fully resolved direct numerical simulation is indeed necessary to predict the flow structure of bubble–particle interaction problems correctly.

Nonequilibrium dynamics of mixtures of active and passive colloidal particles

We develop a mesoscopic field theory for the collective nonequilibrium dynamics of multicomponent mixtures of interacting active (i.e., motile) and passive (i.e., nonmotile) colloidal particles with isometric shape in two spatial dimensions. By a stability analysis of the field theory, we obtain equations for the spinodal that describes the onset of a motility-induced instability leading to cluster formation in such mixtures. The prediction for the spinodal is found to be in good agreement with particle-resolved computer simulations. Furthermore, we show that in active-passive mixtures the spinodal instability can be of two different types. One type is associated with a stationary bifurcation and occurs also in one-component active systems, whereas the other type is associated with a Hopf bifurcation and can occur only in active-passive mixtures. Remarkably, the Hopf bifurcation leads to moving clusters. This explains recent results from simulations of active-passive particle mixtures, where moving clusters and interfaces that are not seen in the corresponding one-component systems have been observed.

Heat transfer in an assembly of ellipsoidal particles at low to moderate Reynolds numbers

Fixed and dynamic assemblies of non-spherical particles participate in many heat and mass transfer industrial processes. However, most convective heat transfer coefficients are adapted from correlations available for spherical particles by using the concept of equal volume spheres or in some cases the sphericity, which is a measure of deviation from a spherical shape. In this study, Particle-Resolved Simulations (PRS) are performed to investigate the heat transfer in flow through a fixed random assembly of spherical and ellipsoidal particles of aspect ratio 2.5:1. The assembly of particles is simulated for solid fractions between 0.1 to 0.35 using 191 to 669 particles, respectively, at low to moderate Reynolds numbers (10⩽Re⩽200). The random particle assembly is generated using a physics simulation engine SDK-PhysX. The particle-fluid boundaries are resolved using the Immersed Boundary Method (IBM) with a constant heat flux boundary condition. Simulation results in the ellipsoidal assembly show that the local velocity and temperature field are significantly affected by the particle orientation. Mean Nusselt numbers for the ellipsoidal assembly are consistently larger than spherical particles when Reynolds number is more than 10, and the difference increases as the Reynolds number increases. A Nusselt number correlation is proposed based on the simulation data. The proposed correlation is valid in the range 10⩽Re⩽50 for solid fraction of 0<ϕ⩽0.35, and 50<Re⩽200 for solid fraction of 0.1⩽ϕ⩽0.35.

Metrics to quantify the importance of mixing state for CCN activity

Abstract. It is commonly assumed that models are more prone to errors in predicted cloud condensation nuclei (CCN) concentrations when the aerosol populations are externally mixed. In this work we investigate this assumption by using the mixing state index (χ) proposed by Riemer and West (2013) to quantify the degree of external and internal mixing of aerosol populations. We combine this metric with particle-resolved model simulations to quantify error in CCN predictions when mixing state information is neglected, exploring a range of scenarios that cover different conditions of aerosol aging. We show that mixing state information does indeed become unimportant for more internally mixed populations, more precisely for populations with χ larger than 75 %. For more externally mixed populations (χ below 20 %) the relationship of χ and the error in CCN predictions is not unique and ranges from lower than −40 % to about 150 %, depending on the underlying aerosol population and the environmental supersaturation. We explain the reasons for this behavior with detailed process analyses.

Evaluation of drag correlations using particle resolved simulations of spheres and ellipsoids in assembly

Particle-resolved simulations are performed to study the momentum transfer in flow though fixed random assembly of non-spherical particles. Ellipsoidal particles with sphericity (ψ=0.887) are investigated in a periodic cubic domain to simulate an infinite assembly. The incompressible Navier-Stokes equations are solved using the Immersed Boundary Method (IBM). Pressure and viscous force on each particle are calculated based on the resolved flow field. Flow through an assembly of spherical particles is tested, and predicted drag forces are compared with previous particle resolved simulation results to validate the current framework. The assembly of ellipsoidal particles is simulated for solid fraction between 0.1 and 0.35 using 191 to 669 particles, respectively, at low to moderate Reynolds numbers (10≤Re≤200). The simulation results show that the drag force of ellipsoidal particles is 15% to 35% larger than equal volume spherical particles. Widely used drag force correlations are evaluated based on the current simulation results. The comparisons show that for ellipsoidal particles over the range of parameters investigated in the present study, the combination of Tenneti et al.'s correlation with Holzer's single non-spherical particle drag model has the best performance with an average difference of 7.15%.

Particle resolved simulations of liquid/solid and gas/solid fluidized beds

The present work studies particle resolved simulations of liquid/solid and gas/solid fluidization in a cuboid domain with periodic lateral boundary conditions. The focus is on investigating particles’ dynamics, while a particular care is devoted to the spatial grid resolution and statistical time convergence of the results. A statistical analysis of particles’ motion and fluid fluctuations asserts the intrinsic differences in the flow characteristics and mixing properties of these two configurations. Results reveal anisotropic mechanisms driving particles’ motion and highlight the dominance of diffusive and convective mechanisms in liquid/solid and gas/solid regimes, respectively. Following a framework similar to that of Nicolai et al. [“Particle velocity fluctuations and hydrodynamic self-diffusion of sedimenting non-Brownian spheres,” Phys. Fluids 7(1), 12–23 (1995)], we estimate the correlation time and the fluctuation length of particles’ motion. A force budget analysis is discussed to gain more insight into the role of collision in isotropization of the system. Owing to the wide range of employed grid resolutions and accurate error analysis, the present dataset is also deemed to be useful in calibrating the grid resolution for a desired accuracy of the solution in a fluidization configuration.

Modeling pore processes for particle-resolved CFD simulations of catalytic fixed-bed reactors

Abstract The most rigorous description of catalytic fixed-bed reactors is particle-resolved CFD simulations. Pore processes can have a large effect toward reactor performance, since they limit internal mass transport. In this study, three pore models of different complexity are incorporated into the CFD software STAR-CCM+. Instantaneous diffusion, effectiveness-factor approach, and three-dimensional reaction–diffusion model are validated firstly with experimental data of CO oxidation in a stagnation-flow reactor from Karadeniz et al. (2013) . The 3D approach predicts the experiments with high accuracy over the entire temperature range followed by the computationally cheaper effectiveness-factor. In a second study, the effect of the three pore models is evaluated on a catalytic single sphere with Reynolds numbers varying between 10 and 2000. The local interplay between kinetics and transport phenomena are quantified. For all investigated cases internal mass transfer limitations are larger than external ones. This study can be applied for entire packed-bed simulations.

Numerical models for fluid-grains interactions: opportunities and limitations

In the framework of a multi-scale approach, we develop numerical models for suspension flows. At the micro scale level, we perform particle-resolved numerical simulations using a Distributed Lagrange Multiplier/Fictitious Domain approach. At the meso scale level, we use a two-way Euler/Lagrange approach with a Gaussian filtering kernel to model fluid-solid momentum transfer. At both the micro and meso scale levels, particles are individually tracked in a Lagrangian way and all inter-particle collisions are computed by a Discrete Element/Soft-sphere method. The previous numerical models have been extended to handle particles of arbitrary shape (non-spherical, angular and even non-convex) as well as to treat heat and mass transfer. All simulation tools are fully-MPI parallel with standard domain decomposition and run on supercomputers with a satisfactory scalability on up to a few thousands of cores. The main asset of multi scale analysis is the ability to extend our comprehension of the dynamics of suspension flows based on the knowledge acquired from the high-fidelity micro scale simulations and to use that knowledge to improve the meso scale model. We illustrate how we can benefit from this strategy for a fluidized bed, where we introduce a stochastic drag force model derived from micro-scale simulations to recover the proper level of particle fluctuations. Conversely, we discuss the limitations of such modelling tools such as their limited ability to capture lubrication forces and boundary layers in highly inertial flows. We suggest ways to overcome these limitations in order to enhance further the capabilities of the numerical models.

Smectic monolayer confined on a sphere: topology at the particle scale.

The impact of topology on the structure of a smectic monolayer confined to a sphere is explored by particle-resolved computer simulations of hard rods. The orientations of the particles are tangential to the sphere and either free or restricted to a prescribed director field with a latitude or longitude orderings. Depending on the imprinted topology, a wealth of different states are found including equatorial smectic with isotropic poles, equatorial smectic with empty poles, a broken egg-shell like modulated smectic, a capped nematic with equatorial bald patches, equatorial nematic with empty poles, and a situation with 4 or 8 half-strength topological defects. Potentially these states could be verified in experiments with Pickering emulsions of droplets with colloidal rods. The unique nature of dipolar structures consisting of positive and negative half-strength disclinations is revealed. These structures, classified by their density and interaction with other defects in the system, relieve the strain of the poles by separating closely positioned half-strength defects. The proximity of these structures to the half-strength defects might enhance the structural diffusion of the defects across the system.

Contact Modifications for CFD Simulations of Fixed-Bed Reactors: Cylindrical Particles

One of the major issues for particle-resolved CFD simulations of fixed beds are particle–particle and particle–wall contacts. A fixed bed of cylinders is studied with different local contact modifications (i.e., caps and bridges method for line and area contacts, and caps and united method for overlaps resulting from composite DEM particles). Effects are analyzed regarding pressure drop, local velocity, and local heat transfer for 191 < Rep < 763. Modeling particle–wall contacts is becoming more important with increasing Rep. Contact modifications inside the bed are becoming less important because convective heat transfer is dominant. The stable caps method shows promising results in comparison with heat-transfer predictions. However, it overestimates convective heat transfer in the contact regions for high flow rates. The bridges method shows good results for pressure drop and heat-transfer predictions. A difficulty remains the choice of the thermal conductivity of these bridges.

Flow of colloidal solids and fluids through constrictions: dynamical density functional theory versus simulation

Using both dynamical density functional theory and particle-resolved Brownian dynamics simulations, we explore the flow of two-dimensional colloidal solids and fluids driven through a linear channel with a constriction. The flow is generated by a constant external force acting on all colloids. The initial configuration is equilibrated in the absence of flow and then the external force is switched on instantaneously. Upon starting the flow, we observe four different scenarios: a complete blockade, a monotonic decay to a constant particle flux (typical for a fluid), a damped oscillatory behaviour in the particle flux, and a long-lived stop-and-go behaviour in the flow (typical for a solid). The dynamical density functional theory describes all four situations but predicts infinitely long undamped oscillations in the flow which are always damped in the simulations. We attribute the mechanisms of the underlying stop-and-go flow to symmetry conditions on the flowing solid. Our predictions are verifiable in real-space experiments on magnetic colloidal monolayers which are driven through structured microchannels and can be exploited to steer the flow throughput in microfluidics.

Study of Local Turbulence Profiles Relative to the Particle Surface in Particle-Laden Turbulent Flows

As particle-resolved simulations (PRSs) of turbulent flows laden with finite-size solid particles become feasible, methods are needed to analyze the simulated flows in order to convert the simulation data to a form useful for model development. In this paper, the focus is on turbulence statistics at the moving fluid–solid interfaces. An averaged governing equation is developed to quantify the radial transport of turbulent kinetic energy when viewed in a frame moving with a solid particle. Using an interface-resolved flow field solved by the lattice Boltzmann method (LBM), we computed each term in the transport equation for a forced, particle-laden, homogeneous isotropic turbulence. The results illustrate the distributions and relative importance of volumetric source and sink terms, as well as pressure work, viscous stress work, and turbulence transport. In a decaying particle-laden flow, the dissipation rate and kinetic energy profiles are found to be self-similar.

Lattice Boltzmann simulation of particle-laden turbulent channel flow

Modulation of the carrier phase turbulence by finite-size solid particles is relevant to many industrial and environmental applications. Here we report particle-resolved simulations of a turbulent channel flow laden with finite-size solid particles. We discuss how the mesoscopic lattice Boltzmann method (LBM) can be applied to treat both the turbulent carrier flow and moving fluid-particle interfaces. To validate the LBM approach, we first simulate the single-phase turbulent channel flow at a frictional Reynolds number of 180. A non-uniform force field is designed to excite turbulent fluctuations. The resulting mean flow profiles and turbulence statistics were found to be in excellent agreement with the published data based on the Chebychev-spectral method. We also found that the statistics of the fully-developed turbulent channel flow are independent of the setting of some of the relaxation parameters in the LBM approach. We then consider a particle-laden turbulent channel flow under the same body force. The particles have the same density as the fluid. The particle diameter is 5% of the channel width and the average volume fraction is 7.09%. We found that the presence of the particles reduces the mean flow speed by 4.6%, implying that the fluid-particle system is more dissipative than the single-phase flow. The maximum local reduction of the mean flow speed is about 7.5%. The effects of the solid particles on the fluid rms velocity fluctuations are mixed: both reduction and augmentation are observed depending on the direction and spatial location relative to the channel walls. Overall, particles enhance the relative turbulence intensity in the near wall region and suppress the turbulence intensity in the center region. The particle concentration distribution across the channel is also complicated. We find that there is a dynamic equilibrium location resembling the Segŕe–Silberberg effect known for a laminar wall-bounded flows. Our LBM results were found to be in good agreement with results based on a finite-difference method with direct forcing to handle the moving solid particles. Additionally, phase-partitioned statistics are obtained and compared.

Effective Cahn-Hilliard Equation for the Phase Separation of Active Brownian Particles

The kinetic separation of repulsive active Brownian particles into a dense and a dilute phase is analyzed using a systematic coarse-graining strategy. We derive an effective Cahn-Hilliard equation on large length and time scales, which implies that the separation process can be mapped onto that of passive particles. A lower density threshold for clustering is found, and using our approach we demonstrate that clustering first proceeds via a hysteretic nucleation scenario and above a higher threshold changes into a spinodal-like instability. Our results are in agreement with particle-resolved computer simulations and can be verified in experiments of artificial or biological microswimmers.

Traveling and Resting Crystals in Active Systems

A microscopic field theory for crystallization in active systems is proposed which unifies the phase-field-crystal model of freezing with the Toner-Tu theory for self-propelled particles. A wealth of different active crystalline states are predicted and characterized. In particular, for increasing strength of self-propulsion, a transition from a resting crystal to a traveling crystalline state is found where the particles migrate collectively while keeping their crystalline order. Our predictions, which are verifiable in experiments and in particle-resolved computer simulations, provide a starting point for the design of new active materials.

Director field in plastic crystals

The director field in a plastic crystal is calculated by particle-resolved Monte Carlo computer simulations of two-dimensional, slightly anisometric hard spherocylinders exposed to an external periodic substrate potential. We investigate the structure of the director field in the Wigner-Seitz cell and find a topological defect structure that can be controlled with the substrate potential. At zero potential we find a charge −1/2 defect at the corners besides the expected defect in the centre with unit topological charge. When switching the substrate potential on, the corner defects are surrounded by three satellite defects which bear charge −1/2, too. Additionally, we then find two charge +1/2 defects on each edge of the unit cell. Finally, within a simplified model, we obtain a qualitative explanation for this defect structure. Our predictions can in principle be verified by using particle-resolved experiments of colloidal plastic crystals.

An accurate and efficient method for treating aerodynamic interactions of cloud droplets

Motivated by a need to improve the representation of short-range interaction forces in hybrid direct numerical simulation of interacting cloud droplets, an efficient method for treating the aerodynamic interaction of two spherical particles settling under gravity is developed. An effort is made to ensure the accuracy of our method for any inter-particle separation by considering three separation ranges. The first is the long-range interaction where a multipole method is applied. After a decomposition into six simple configurations, explicit formulae for drag forces and torques are derived from an approximate Force–Torque–Stresslet (FTS) formulation. The FTS formulation is found to be accurate when the separation distance normalized by the average radius is larger than 5. The second range concerns the short-range interaction where the interaction force could be very large. Leading-order lubrication expansions are employed for this range and are found to be accurate when the normalized separation is less than about 0.01. Finally, for the intermediate range where no simple method is available, a third-order polynomial fitting is proposed to bridge the treatments for long-range and short-range interactions. After optimizing the precise form of polynomial fitting and matching locations, the force representation is found to be highly accurate when compared with the exact solution for Stokes flows. Using this method, collision efficiencies of cloud droplets sedimenting under gravity have been calculated. It is shown that the results of collision efficiency are in excellent agreement with results based on the exact Stokes flow solution. Collision efficiency results are also compared to previous results to further illustrate the accuracy of our calculations. The effects of particle rotation and the attractive van der Waals force on the collision efficiency are also studied. The efficient force representation developed here is more general than the usual lubrication expansion and thus can serve as a better approach to correct unresolved short-range interactions in particle-resolved simulations.

A particle-based model for predicting the effective conductivities of composite electrodes

This paper develops particle-resolved simulations to predict conductivity within porous composite electrodes. Hundreds of spherical particles (order fractions of a micron) are packed randomly into a cubical region (order of a few microns), using two alternative packing algorithms. The composite structures include both ion-conducting and electron-conducting particles. The particle network is discretized using tetrahedral meshes that fully resolve the interiors of the particles and their intersections. Charge-conservation equations are solved to predict current through the network. These simulations are used to derive the effective conductivities that are required for macroscale simulations at length scales much larger than the particle scale. Because the microstructures are synthesized via random particle packing, multiple realizations are needed to deliver statistically invariant results. The results show that a few hundred particles with a few hundred realizations is sufficient. Predicted coordination numbers, percolation probabilities, and three-phase-boundary lengths are consistent with percolation theory, but the predicted effective conductivities are significantly smaller than those predicted with conventional percolation theory. By adjusting the Bruggeman factor from the conventional value of 1.5–3.5 brings the percolation-theory prediction for effective conductivity in line with the fully resolved results.