Particle-resolved Direct Numerical Simulation Research Articles

A particle-resolved direct numerical simulation method for the compressible gas flow and arbitrary shape solid moving with a uniform framework

Compressible particle-resolved direct numerical simulations (PR-DNS) are widely used in explosion-driven dispersion of particles simulations, multiphase turbulence modelling, and stage separation for two-stage-to-orbit vehicles. The direct forcing immersed boundary method (IBM) is a promising method and widely applied in low speed flow while there is few research regarding compressible flows. We developed a novel IBM to resolve supersonic and hypersonic gas flows interacting with irregularly shaped multi-body particle. The main innovation is that current method can solve the interaction of particles and high-speed fluids, particle translation and rotation, and collision among complex-shaped particles within a uniform framework. Specially, high conservation and computation consumption are strictly satisfied, which is critical for resolving the high speed compressible flow feature. To avoid the non-physical flow penetration around particle surface, an special iterative algorithm is specially derived to handle the coupling force between the gas and particles. The magnitude of the velocity difference error could be reduced by 6–8 orders compared to that of a previous method. Additionally, aerodynamic force integration was achieved using the momentum equation to ensure momentum conservation for two-phase coupling. A high-efficiency cell-type identification method for each step was proposed, and mapping among LPs and cells was used again to select the immersed cells. As for the collision force calculation, the complex shape of a particle was represented by a cloud of LPs and the mapping of LPs and cells was used to reduce the complexity of the algorithm for contact searching. The repetitive use of the mapping relationship could reduce the internal memory and improve the efficiency of the proposed algorithm. Moreover, various verification cases were conducted to evaluate the simulation performance of the proposed algorithm, including two- and three-dimensional moving and motionless particles with regular and complex shapes interacting with high-speed flow. Specifically, an experiment involving a shock passing through a sphere was designed and conducted to provide high-precision data. The corresponding results of the large-scale numerical simulation agree well with those obtained experimentally. The current method supports flow simulations at a particle-resolved scale in engineering.

Deterministic drag modelling for spherical particles in Stokes regime using data-driven approaches

In this paper, we develop a deterministic drag model for stationary spherical particles in a Stokes flow using a cascade of data-driven approaches. The model accounts for the variation in drag experienced by each particle within fixed random arrangements. The developed model is a symbolic expression that offers explainability, ease of implementation, and computational efficiency. Firstly, we generate particle-resolved direct numerical simulation data of the flow past periodic random arrangements of stationary spherical particles with volume fractions between 0.05 and 0.4 using the method of regularized Stokeslets. Secondly, we train graph neural networks (GNs) on the generated data to learn the pairwise influence of neighbouring particles on a reference particle. The GNs are converted to symbolic expressions using genetic programming (GP), unveiling repeated subexpressions. Finally, these subexpressions constitute the foundation of the proposed algebraic model, further refined via non-linear regression. The proposed model can qualitatively mimic the pairwise influences as predicted by the GN and can capture the drag variations with accuracy from 74% and up to 84.7% when compared to the particle-resolved simulations. Due to the interpretability of the proposed model, we are able to explore how neighbour positions alter the drag of a particle in an assembly. The proposed model is a promising tool for studying the dynamics of particle assemblies in Stokes flow.

Flow past a random array of statistically homogeneously distributed stationary Platonic polyhedrons: Data analysis, Probability maps and Deep Learning models

We perform particle-resolved direct numerical simulations (PR-DNS) of the flow past a random array of stationary Platonic polyhedrons seeded in a tri-periodic box at Reynolds number Re=1, 10, and 100 and solid volume fraction 0.05, 0.1 and 0.2. Using Platonic polyhedrons enables us to examine the effects of particle angularity on the hydrodynamic force and torque exerted on the stationary particles in a controlled manner. We report the average drag, lift, and torque coefficients as well as their respective distribution. We explore the relationship between the particle angularity, particle angular position, flow disturbances created by the neighboring particles, and the hydrodynamic force and torque exerted on each individual particle. We attempt to extend our probability map based model (MPP), our Physics Informed neural network (PINN) model previously developed for spheres to Platonic polyhedrons, and propose an additional NN architecture. We show that ignoring the angular position and actual shape of the reference and neighboring particles constitutes a reasonable first order approximation that significantly simplifies the construction of deterministic models of hydrodynamic forces and torques at low Re=1 to moderate Re=10. At high Re=100, both the MPP model and the PINN model without any additional knowledge of the particle shape and angular position fail to properly predict the force and torque fluctuations. We then design a Convolutional Neural Network (CNN) architecture that is able to maintain a satisfactory performance at Re=100 but requires additional input data in the form of a coarse-grained velocity field around each particle. Overall, predicting the torque fluctuation remains a significant challenge.

Neural-network-based drag force model for polydisperse assemblies of irregular-shaped particles

In this study, a drag force model of irregular-shaped particles in gas-solid flows is investigated using neural network approaches. Pre-developed variational autoencoder, multi-layer perceptron model, and transpose convolutional neural-network model are utilized to obtain intermediate parameters of a pair-wise interaction point-particle (PIEP) model. These parameters are utilized to predict individual drag force coefficients of polydisperse, particulate flows. To apply the PIEP-based model to medium-concentration flows, the average drag force coefficients are obtained through an in-house particle-resolved direct numerical simulation, and a regressed model is developed to predict solid volume fraction factors. With the regression model, the PIEP-based model using the neural-network approaches shows good prediction in particulate systems from low to intermediate concentrations. This study provides computationally efficient models of practical, realistic particulate systems for large scale fluid dynamics simulation, in that it allows predicting the drag forces of polydisperse particles with a small amount of data.

Computational study on interaction between rotating non-spherical particles and shear-thinning fluids

In this paper, the drag coefficient (Cd), lift coefficient (Cl), and torque coefficient (Ct) of rotating non-spherical particles in shear-thinning non-Newtonian fluids are investigated based on particle-resolved direct numerical simulation. The Carreau model is used to describe the rheological behavior of non-Newtonian fluids, and the numerical model is validated against previously published data. Then, the effects of aspect ratio (Ar), spin number (Spa), flow index (n), and Carreau number (Cu) on Cd, Cl, and Ct of rotating non-spherical particles are investigated at different Reynolds numbers (Re). The numerical results show that the closer the particles are to the spherical shape, the smaller the fluctuations of Cd, Cl, and Ct curves. The peaks and valleys of Cd, Cl, and Ct of oblate and prolate ellipsoidal particles are reversely distributed. The fluctuations of Cd and Cl curves increase with increasing Spa. Cd decreases with increasing Spa at low Re, contrary to Newtonian fluids' results. Cd and Ct decrease with increasing shear-thinning properties, Cl increases with increasing shear-thinning properties, and the effect of shear-thinning properties decreases with increasing Re. The variation of viscosity and pressure is the main reason for the variation of Cd, Cl, and Ct under different variables. Predictive correlations of Cd and Ct are established based on Re, Spa, n, Cu, and α. The findings indicate that particle rotation and shear-thinning properties must be considered when evaluating particle-fluid interactions, which provide important guidance for predicting and controlling the orientation and distribution of non-spherical particles in non-Newtonian fluids.

Novel drag and Nusselt number models based on direct numerical simulations of a bidisperse gas–solid system

AbstractFlow and heat transfer in a bidisperse gas–solid system with freely moving spheres are simulated by particle‐resolved direct numerical simulation (PR‐DNS). Gas–solid coupling is enforced by the direct‐forcing immersed boundary method. Compard with the DNS database, it is found that the existing polydisperse drag correction model developed from static systems combined with various monodisperse drag models underestimates the drag force on dynamic arrays of particles. The existing Nusselt number correction model developed from static systems combined with various monodisperse models overestimates the Nusselt number of dynamic arrays of particles. Sensitivity analysis indicates that the effects of the granular temperature on the drag force and Nusselt number are negligible. Novel polydisperse drag and Nusselt number models are derived based on the database. The advantages of the derived polydisperse drag and Nusselt number models are demonstrated and confirmed by comparing the results of the computational fluid dynamics–discrete element method using various drag and Nusselt number models with experimental or additional PR‐DNS data.

Interparticle friction in sheared dense suspensions: Comparison of the viscous and frictional rheology descriptions

In the literature, two different frameworks exist for describing the rheology of solid/liquid suspensions: (1) the “viscous” framework in terms of the relative suspension viscosity, ηr, as a function of the reduced solid volume fraction, ϕ/ϕm, with ϕm the maximum flowable packing fraction, and (2) the “frictional” framework in terms of a macroscopic friction coefficient, μ, as a function of the viscous number, Iv, defined as the ratio of the viscous shear to the wall-normal particle stress. Our goal is to compare the two different frameworks, focusing on the effect of friction between particles. We have conducted a particle-resolved direct numerical simulation study of a dense non-Brownian suspension of neutrally buoyant spheres in slow plane Couette flow. We varied the bulk solid volume fraction from ϕb=0.1 to 0.6 and considered three different Coulomb friction coefficients: μc=0, 0.2, and 0.39. We find that ηr scales well with ϕ/ϕm, with ϕm obtained from fitting the Maron–Pierce correlation. We also find that μ scales well with Iv. Furthermore, we find a monotonic relation between ϕ/ϕm and Iv, which depends only weakly on μc. Since ηr=μ/Iv, we thus find that the two frameworks are largely equivalent and that both account implicitly for Coulomb friction. However, we find that the normal particle stress differences, N1 and N2, when normalized with the total shear stress and plotted against either ϕ/ϕm or Iv, remain explicitly dependent on μc in a manner that is not yet fully understood.

Preferential accumulation of finite-size particles in near-wall streaks

The preference for particles to accumulate at specific regions in the near-wall part is a widely observed phenomenon in wall-bounded turbulence. Unlike small particles more frequently found in low-speed streaks, finite-size particles can accumulate in either low-speed or high-speed streaks. However, mechanisms and influencing factors leading to the different preferential concentration locations still need to be clarified. The present study conducts particle-resolved direct numerical simulations of particle-laden turbulent channel flows to provide a better understanding of this seemingly puzzling behaviour of preferential accumulation. These simulations cover different particle-to-fluid density ratios, particle volume fractions, particle sizes and degrees of sedimentation intensity. We find that the large particle size is the crucial factor that results in particles accumulating in high-speed streaks. Large particles not only are difficult to be conveyed by the quasi-streamwise vortices to low-speed streaks but also can escape from the near-wall region before moving spanwisely out from high-speed streaks. The sedimentation effect allows particles to gather closer to the channel wall and stay longer in the near-wall regions, reinforcing the sweeping mechanism of quasi-streamwise vortices that transport particles from high- to low-speed streaks. As a result, sedimenting particles tend to accumulate in the low-speed streaks.

Study of flow resistance coefficients acting on regular non-spherical particles in simple shear flow at moderate Reynolds numbers

This contribution is devoted to the study of the aerodynamic characteristics of non-spherical particles of definite regular shape, such as prolate and oblate ellipsoids as well as cylinders, immersed in a locally linear shear flow. The flow resistance coefficients of drag, lift and pitching torque are computed by means of the Particle Resolved Direct Numerical Simulation (PR-DNS) technique for such shapes as function of the Reynolds number Re. The parameter space comprises particle aspect ratio, AR, fluid spin ratio, ζ, and particle orientation angle, α. The Reynolds numbers of interest are in the intermediate range 1≤Re≤100, common in industrial and environmental processes. To properly understand and explain the obtained results, the role of the friction and pressure contributions to total flow coefficients is analyzed in detail for the different cases, allowing the differences between the considered shapes to be pointed out. On the other hand, the behavior of the pressure and skin friction coefficients in the particle plane of symmetry parallel to the flow is investigated as a function of the previous parameters (shape, AR, ζ, α), which provides further insights into the features of the shear flow around the non-spherical particles at finite Re. Finally, the influence of the shear flow magnitude and incidence angle on the location of the center of aerodynamic force is devised for the three shapes considered as function of Reynolds number and particle aspect ratio. It is expected that the information generated in this work will be useful for researchers to enhance the modeling of non-spherical particles immersed in non-uniform flows.

Hydrodynamic force interaction of two fixed spheres in a wall-bounded linear shear flow

We examine the modulations of the drag coefficient Cd and lift coefficient Cl on a spherical particle caused by the presence of a neighboring sphere when positioned in close proximity to a wall. Data are generated through the utilization of a fast and accurate Particle-Resolved Direct Numerical Simulation (PR-DNS) method that relies on the Direction-Splitting algorithm and that was extensively validated in our prior study (Goyal and Wachs, 2023a, 2023b; Morente et al., 2023). We further validate the method in the flow configuration of the present study by comparing them with data published in the literature. We consider three Reynolds numbers Re=20, Re=50 and Re=100, which describe the flow conditions in the vicinity of the wall. The primary sphere is positioned at a distance δ from the wall, while the neighboring sphere is positioned at a constant distance from the primary sphere encompassing all possible angular positions θ. We report the modulations of Cd and Cl compared to a situation without any neighboring sphere as a function of δ, θ and Re. The inclusion of a wall intensifies the influence of the neighboring sphere on the primary sphere and significantly modifies the values of Cd and Cl. We attempt to relate the values of the modulations of Cd and Cl to the modulations in the contours of the pressure and of the spanwise vorticity on the surface of the primary sphere. We take advantage of the periodic nature of Cd and Cl as a function of the relative angular position θ of the neighboring sphere and propose a Fourier Predictive Model (FPM) to effectively represent the data using a finite number of modes in the cosine Fourier series. We show that our FPM is able to estimate the modulations of Cd and Cl excellently with a coefficient of determination R2≳0.99 in the parameter space investigated in our study.

Particle-resolved direct numerical simulation of the oscillatory flow and sediment motion over a rippled bed

The flow field and the sediment motion generated over a rippled bed by a uniform and oscillating pressure gradient are evaluated by means of a direct numerical simulation of Navier–Stokes and continuity equations and assuming that small spherical particles, free to move, mimic the sediment grains. The numerical approach determines the flow around each sediment particle, along with the force and torque that the fluid exerts on it. Then, the particle trajectories are computed by means of Newton-Euler’s laws. The parameters of the problem are the same as those of one of the laboratory experiments described by Blondeaux and Vittori (1991a) and the initial ripple profile is similar to that observed during the experiment. The numerical results are qualitatively compared with the experimental observations and support the reliability of the numerical simulation. Moreover, the results provide further valuable information on sediment dynamics under surface waves which propagate over a rippled bed. In particular, the obtained results allow a detailed analysis of the dynamics of the vortices generated by flow separation at the ripple crest and of the mechanism through which the sediments are lifted up from the bottom and carried into suspension.

Particle resolved direct numerical simulations for heat transfer analysis of a spouted bed

In this paper, fully coupled particle resolved direct numerical simulations are used for heat transfer calculations. For modelling fluid velocity and temperature, the immersed boundary method (IBM) is used. Particle-particle heat transfer is modelled using well-established relations from the literature. Two-way coupling between particles and fluid is obtained using a body force that is a function of the solid volume fraction present in the fluid grid cell. Results are compared with those of spouted bed experiments to validate the simulations, and good agreement is observed. At the end of the paper, several interesting characteristics of spouted bed are obtained using simulations. The speedup of the method used for simulations was tested on multi-core CPU clusters. It is observed that the simulations scales almost linearly with the number of CPUs (tests were done up to 500 CPUs), showing the capability of the particle resolved DNS to be used on multi-CPU architectures.

Development of Particle Resolved - Subgrid Corrected Simulations: Hydrodynamic force calculation and flow sub-resolution corrections

Particle Resolved - Direct Numerical Simulation (PR-DNS) of fluid–solid particles is conducted to study hydrodynamic interactions at the interface between phases. An original distinction is proposed between Particle Resolved Simulation (PRS) and Particle Resolved - Direct Numerical Simulation (PR-DNS). In PR-DNS, velocity gradients and pressure are fully resolved, which required several dozens of meshes in the diameter of the particles (40 in Stokes configuration), whereas for PRS, with grid resolution of 10 meshes per diameter, the gradients are only partly resolved, resulting in a sub-resolution of hydrodynamic forces exerted by the fluid to the interface of solid particles. The very high computational cost, associated to PR-DNS, limits its application to academic cases, even though, it was originally developed to study the collective physical effects of a fluid–particle assembly. It explains why numerous simulations with a grid resolution of only a dozen meshes in the particle’s diameter – typically between 10 and 16 – referred to here as PRS, can be found in the literature. In these cases, the velocity gradients and pressure are not accurately computed, inducing an error in the hydrodynamic forces exerted by the fluid to the interface of particles. This paper aims to clarify the hydrodynamic interactions at this scale for a single particle settling in an infinite medium under Stokes configuration. For this purpose, an original method, designed for solid particles, is proposed to compute hydrodynamic forces. A mesh-dependent correlation is then built to numerically correct the partially resolved hydrodynamic forces obtained with PRS, in what we call Particle Resolved - Subgrid Corrected Simulations (PR-SCS). Finally, the correction is assessed for various Reynolds numbers and density ratios under the Stokes assumption.

Modeling pseudo-turbulent heat flux in gas-solid heat transfer

Flow past disperse solid particles or bubbles induces fluctuations in carrier fluid velocity, which correlate with temperature fluctuations in non-isothermal flows resulting in the pseudo-turbulent heat flux (PTHF). In the Eulerian-Eulerian (EE) two-fluid (TF) model, the transport of PTHF is shown to be an important contributor to the overall energy budget, and is modeled using a pseudo-turbulent thermal diffusivity (PTTD). The PTHF and PTTD were originally quantified using particle-resolved direct numerical simulation (PR-DNS) data, and correlations were developed over a range of solid volume fraction (0.1≤εs≤0.5) and mean slip Reynolds number (1≤Rem≤100) for a Prandtl number of 0.7. However, the original PTTD correlation diverges to infinity as the solid volume fraction goes to zero, which is physically unrealistic. This singular behavior is problematic for EE TF simulations at particle material fronts where solid volume fraction values can fall below the lower limit of existing data (εs=0.1) to zero in the pure carrier phase. In this work, additional PR-DNS data are reported for εs<0.1, and improved correlations are developed for the PTHF and PTTD. The new PTTD correlation is non-singular, and both the PTHF and PTTD decay exponentially to zero as the solid volume fraction approaches zero, which is physically reasonable. This improves prediction of PTHF transport in dilute flow using EE TF heat transfer simulations.

Hydrodynamic force and torque fluctuations in a random array of polydisperse stationary spheres

We report data on the fluctuations of hydrodynamic force and torque exerted on polydisperse stationary spheres randomly distributed in a cubic tri-periodic box. We generate our data with an accurate Particle-Resolved/Direct Numerical Simulation (PR-DNS) method that relies on an Immersed Boundary/lattice Boltzmann method implemented on a Cartesian octree grid (Cheng and Wachs, 2022). The local grid refinement capability offered by the octree grid enables us to properly resolve the flow dynamics even around the smallest stationary spheres in the system in the case of a wide sphere diameter distribution. We verify that our data are indeed accurate and reliable through a thorough comparison with results produced by other numerical methods and with data published in the literature. We evaluate the accuracy and limitations of various average drag models on different levels of polydispersity. We consider two Reynolds numbers Re=1 and Re=100 representative of low and moderate inertia. We analyze results in terms of hydrodynamic force and torque fluctuations around the average value and relate these fluctuations to the flow disturbances created by the neighboring particles, i.e., the local microstructure. Finally, we discuss the implications of the presented results on the extension of hydrodynamic force and torque models recently proposed in the literature that take advantage of the local microstructure to predict fluctuations in the case of arrays of monodisperse stationary spheres to the case of arrays of polydisperse stationary spheres.

Particle-resolved direct numerical simulation of particle-laden turbulence modulation with high Stokes number monodisperse spheres

Graphic processing units (GPU)-accelerated direct numerical simulations of particle–fluid systems based on lattice Boltzmann method combined with discrete element method are successfully performed. This method is validated to be accurate and efficient in the four-way coupling simulation of particle–fluid systems. Particle-laden homogeneous isotropic turbulence under particle size dp=30.1η (the Kolmogorov length scale), a wide range of solid volume fraction ϕV=2%−20% and particle Stokes number St=503.87−50 387 are systematically studied. The additional energy dissipation rate εaddi induced by particles is quantitatively calculated. One criterion for turbulence enhancement or attenuation based on energy balance is proposed and successfully used in different particle-laden turbulence. The relative magnitude of two-way coupling rate ψ and particle-induced dissipation εaddi is the direct factor determining the modulation effect on turbulence. Turbulence laden with high Stokes number particles is enhanced. Turbulence laden with high volume fraction particles transited rapidly from enhancement to attenuation, which is attributed to the more rapid decay of ψ relative to εaddi in the freely decaying turbulence.

Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime

In this brief paper, a novel set of correlations for the drag, lift and torque coefficients for non-spherical particles of irregular shape is proposed. Such correlations are developed for a sphericity range between 0.7 and 0.95 and for intermediate Reynolds numbers in the range 1,200, common in industrial and environmental processes. The proposed expressions are derived by fitting the results obtained by means of Particle-Resolved Direct Numerical Simulations (PR-DNS) for a uniform flow around different sets of irregular particles and considering a large number of random orientations. The resulting flow resistance coefficients can be approximated by normal distributions whose first and second order statistical moments (i.e., mean and standard deviation) are described in this work by fitting functions depending of particle Reynolds number and sphericity. The derived correlations are simple, can be easily implemented and allow the tracking of irregular particles in a Lagrangian stochastic framework.

Inertial effects in sedimenting suspensions of solid spheres in a liquid

Particle-resolved Direct Numerical Simulations have been performed on the gravitational settling of mono-disperse solid spheres in a viscous fluid and triply periodic domain. In a comprehensive study, the bulk solid volume concentration was varied from ϕ=0.5 to 30%. To study the effect of inertia, three different Galileo numbers were considered in the inertial regime, Ga=144, 178 and 210, for which a single settling sphere exhibits distinctly different wake and path characteristics. The particle/fluid mass density ratio was fixed at 1.5. We find that for ϕ=2−30% the suspension microstructure and dynamics depend predominantly on the bulk concentration. In qualitative agreement with previous studies in literature, three different sedimentation regimes can be distinguished: (1) the dilute concentration regime for ϕ≲2% with preferential settling of particles in vertical trains, (2) the moderate concentration regime for 2%≲ϕ≲10% with preferential settling of particles in horizontal pairs with an interparticle distance of ∼ 1.5 particle diameters, and (3) the dense concentration regime for ϕ≳10% with a nearly random (“hard-sphere”) distribution of the particles in space. The clustering of particles is dictated by, respectively, trapping of particles in the wake of other particles, a drafting–kissing–tumbling (DKT) instability by which two vertically aligned particles quickly reorient themselves into a horizontally aligned particle pair, and short-range multiparticle interactions through viscous lubrication and to a lesser extent collisions between particles. In all cases, hindered settling at a reduced speed is observed as compared to a single settling sphere. The well-known Richardson–Zaki relation for the mean sedimentation velocity appears valid only for the dense concentration regime. We provide ample evidence that in the dense regime the characteristic velocity and time scales of particle motion are proportional to gDp and Dp/g, respectively, with g the gravitational acceleration and Dp the particle diameter. We also observe an ω−3 scaling of the particle velocity spectra for ωDp/g≳0.4 and we propose a model to explain this scaling behavior, based on the inertial response of the particles to small-scale flow perturbations. Kinematic waves, i.e., vertically propagating plane waves in the local concentration field, are observed in all cases, though unrelated particle motions are responsible for significant loss of the spatio-temporal coherence of the waves. The wave speed was determined from repeated space–time autocorrelations of the local concentration field and appears in reasonable agreement with Kynch sedimentation theory using the Richardson–Zaki relation. The passage of kinematic waves causes perturbations in the particle velocity at a frequency that matches well with peak frequencies in the particle velocity spectra for concentrations up to ϕ≈10%. The time-lagged cross-correlation of the vertical and horizontal particle velocity suggests that kinematic waves may trigger DKT instabilities, while conversely DKT instabilities may be responsible for the onset of kinematic waves. Finally, we suggest that obstruction and perturbation of the particle wake by neighboring particles could offer an explanation for the small influence of the Galileo number on the suspension behavior for ϕ=2−30%.

A numerical study of the settling of non-spherical particles in quiescent water

Settling of non-spherical particles is poorly understood with previous studies having focused mainly on spherical particles. Here, a series of particle-resolved direct numerical simulations are conducted using FLOW-3D (commercial computational fluid dynamics software) for spheres and five regular, non-spherical shapes of sediment particles, i.e., prolate spheroid, oblate spheroid, cylinder, disk, and cube. The Galileo number varies from 0.248 to 360, and the particle Reynolds number Rep ranges from 0.002 77 to 562. The results show that a non-spherical particle may experience larger drag and, consequently, attain a lower terminal velocity than an equivalent sphere. If Rep is sufficiently small, the terminal velocity is less affected by particle shape as characterized by the particle aspect ratio. For relatively large Rep, the shape effect (represented by the Corey shape factor) becomes more significant. Empirical correlations are derived for the dimensionless characteristic time t95∗ and displacement s95∗ of particle settling, which show that t95∗ remains constant in the Stokes regime (Rep &amp;lt; 1) and decreases with increasing Rep in the intermediate regime (1 ≤ Rep &amp;lt; 103), whereas s95∗ increases progressively with increasing Rep over the simulated range. It is also found that in the Stokes regime, particle orientation remains essentially unchanged during settling, and so the terminal velocity is governed by the initial orientation. In the intermediate regime, a particle provisionally settling at an unstable orientation self-readjusts to a stable equilibrium state, such that the effect of initial orientation on the terminal velocity is negligible. Moreover, an unstable initial orientation can enhance the vertical displacement and may promote vortex shedding.

Numerical study on turbulence modulation of finite-size particles in plane-Couette flow

Turbulent plane-Couette flow suspended with finite-size spheroidal particles is studied using fully particle-resolved direct numerical simulations. The effects of particle aspect ratio on turbulent arguments and particle statistics are explored, leading to the same conclusions as the previous experimental findings (Wang et al., J. Fluid Mech., vol. 937, 2022, A15). By performing stress analysis, we find that the presence of particles introduces extra stresses to the system and accounts for the global drag increases. The particle-laden flow cases exhibit spectra that are consistent with the scalings $k^{-5/3}$ and $k^{-3}$ in the large and small scales, respectively. While the $k^{-3}$ scaling observed in the particle-laden flow is reminiscent of bubbly flow, an examination of the particle Reynolds number suggests that the mechanism responsible may not be attributable to the pseudo-turbulence induced by particles as in the case of bubbles. In the view of particle statistics, we observe that spherical and non-spherical particles cluster preferentially in the near-wall and the bulk region, respectively, and that the orientations of non-spherical particles are affected by their aspect ratios, especially in the near-wall region. The present numerical results, combined with previous experimental findings in Wang et al. (J. Fluid Mech., vol. 937, 2022, A15), provide in-depth information on both the fluid and the particle phase, contributing to a better understanding of particle suspension in shear flows.